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UMAP: Uniform Manifold

Approximation and Projection for

Dimension Reduction

Leland McInnes

Tue Institute for Mathematics and Computing

leland.mcinnes@gmail.com

John Healy

Tue Institute for Mathematics and Computing

jchealy@gmail.com

James Melville

jlmelville@gmail.com

September 21, 2020

Abstract

UMAP (Uniform Manifold Approximation and Projection) is a novel

manifold learning technique for dimension reduction. UMAP is constructed

from a theoretical framework based in Riemannian geometry and algebraic

topology. e result is a practical scalable algorithm that is applicable to

real world data. e UMAP algorithm is competitive with t-SNE for visu-

alization quality, and arguably preserves more of the global structure with

superior run time performance. Furthermore, UMAP has no computational

restrictions on embedding dimension, making it viable as a general purpose

dimension reduction technique for machine learning.

1 Introduction

Dimension reduction plays an important role in data science, being a funda-

mental technique in both visualisation and as pre-processing for machine

1

arXiv:1802.03426v3 [stat.ML] 18 Sep 2020

learning. Dimension reduction techniques are being applied in a broaden-

ing range of elds and on ever increasing sizes of datasets. It is thus desir-

able to have an algorithm that is both scalable to massive data and able to

cope with the diversity of data available. Dimension reduction algorithms

tend to fall into two categories; those that seek to preserve the pairwise

distance structure amongst all the data samples and those that favor the

preservation of local distances over global distance. Algorithms such as

PCA [27], MDS [30], and Sammon mapping [50] fall into the former cate-

gory while t-SNE [59, 58], Isomap [56], LargeVis [54], Laplacian eigenmaps

[6, 7] and diusion maps [16] all fall into the laer category.

In this paper we introduce a novel manifold learning technique for di-

mension reduction. We provide a sound mathematical theory grounding

the technique and a practical scalable algorithm that applies to real world

data. UMAP (Uniform Manifold Approximation and Projection) builds upon

mathematical foundations related to the work of Belkin and Niyogi on

Laplacian eigenmaps. We seek to address the issue of uniform data distri-

butions on manifolds through a combination of Riemannian geometry and

the work of David Spivak [52] in category theoretic approaches to geomet-

ric realization of fuzzy simplicial sets. t-SNE is the current state-of-the-art

for dimension reduction for visualization. Our algorithm is competitive

with t-SNE for visualization quality and arguably preserves more of the

global structure with superior run time performance. Furthermore the al-

gorithm is able to scale to signicantly larger data set sizes than are feasible

for t-SNE. Finally, UMAP has no computational restrictions on embedding

dimension, making it viable as a general purpose dimension reduction tech-

nique for machine learning.

Based upon preliminary releases of a soware implementation, UMAP

has already found widespread use in the elds of bioinformatics [5, 12, 17,

46, 2, 45, 15], materials science [34, 23], and machine learning [14, 20, 21,

24, 19, 47] among others.

is paper is laid out as follows. In Section 2 we describe the theory un-

derlying the algorithm. Section 2 is necessary to understand both the the-

ory underlying why UMAP works and the motivation for the choices that

where made in developing the algorithm. A reader without a background

(or interest) in topological data analysis, category theory or the theoretical

underpinnings of UMAP should skip over this section and proceed directly

to Section 3.

at being said, we feel that strong theory and mathematically justied

algorithmic decisions are of particular importance in the eld of unsuper-

vised learning. is is, at least partially, due to plethora of proposed objec-

2

tive functions within the area. We aempt to highlight in this paper that

UMAPs design decisions were all grounded in a solid theoretic foundation

and not derived through experimentation with any particular task focused

objective function. ough all neighbourhood based manifold learning al-

gorithms must share certain fundamental components we believe it to be

advantageous for these components to be selected through well grounded

theoretical decisions. One of the primary contributions of this paper is to

reframe the problem of manifold learning and dimension reduction in a dif-

ferent mathematical language allowing pracitioners to apply a new eld of

mathemtaics to the problems.

In Section 3 we provide a more computational description of UMAP.

Section 3 should provide readers less familiar with topological data analysis

with a beer foundation for understanding the theory described in Section

2. Appendix C contrasts UMAP against the more familiar algorithms t-SNE

and LargeVis, describing all these algorithms in similar language. is sec-

tion should assist readers already familiar with those techniques to quickly

gain an understanding of the UMAP algorithm though they will grant lile

insite into its theoretical underpinnings.

In Section 4 we discuss implementation details of the UMAP algorithm.

is includes a more detailed algorithmic description, and discussion of the

hyper-parameters involved and their practical eects.

In Section 5 we provide practical results on real world datasets as well

as scaling experiments to demonstrate the algorithm’s performance in real

world scenarios as compared with other dimension reduction algorithms.

In Section 6 we discuss relative weakenesses of the algorithm, and ap-

plications for which UMAP may not be the best choice.

Finally, in Section 7 we detail a number of potential extensions of UMAP

that are made possible by its construction upon solid mathematical foun-

dations. ese avenues for further development include semi-supervised

learning, metric learning and heterogeneous data embedding.

2 eoretical Foundations for UMAP

e theoretical foundations for UMAP are largely based in manifold theory

and topological data analysis. Much of the theory is most easily explained

in the language of topology and category theory. Readers may consult

[39], [49] and [40] for background. Readers more interested in practical

computational aspects of the algorithm, and not necessarily the theoretical

motivation for the computations involved, may wish to skip this section.

3

Readers more familiar with traditional machine learning may nd the re-

lationships between UMAP, t-SNE and Largeviz located in Appendix C en-

lightening. Unfortunately, this purely computational view fails to shed any

light upon the reasoning that underlies the algorithmic decisions made in

UMAP. Without strong theoretical foundations the only arguments which

can be made about algorithms amount to empirical measures, for which

there are no clear universal choices for unsupervised problems.

At a high level, UMAP uses local manifold approximations and patches

together their local fuzzy simplicial set representations to construct a topo-

logical representation of the high dimensional data. Given some low dimen-

sional representation of the data, a similar process can be used to construct

an equivalent topological representation. UMAP then optimizes the layout

of the data representation in the low dimensional space, to minimize the

cross-entropy between the two topological representations.

e construction of fuzzy topological representations can be broken

down into two problems: approximating a manifold on which the data is

assumed to lie; and constructing a fuzzy simplicial set representation of

the approximated manifold. In explaining the algorithm we will rst dis-

cuss the method of approximating the manifold for the source data. Next

we will discuss how to construct a fuzzy simplicial set structure from the

manifold approximation. Finally, we will discuss the construction of the

fuzzy simplicial set associated to a low dimensional representation (where

the manifold is simply Rd), and how to optimize the representation with

respect to our objective function.

2.1 Uniform distribution of data on a manifold and

geodesic approximation

e rst step of our algorithm is to approximate the manifold we assume

the data (approximately) lies on. e manifold may be known apriori (as

simply Rn) or may need to be inferred from the data. Suppose the manifold

is not known in advance and we wish to approximate geodesic distance on

it. Let the input data be X={X1, . . . , XN}. As in the work of Belkin and

Niyogi on Laplacian eigenmaps [6, 7], for theoretical reasons it is benecial

to assume the data is uniformly distributed on the manifold, and even if that

assumption is not made (e.g [26]) results are only valid in the limit of innite

data. In practice, nite real world data is rarely so nicely behaved. However,

if we assume that the manifold has a Riemannian metric not inherited from

the ambient space, we can nd a metric such that the data is approximately

uniformly distributed with regard to that metric.

4

Formally, let Mbe the manifold we assume the data to lie on, and let g

be the Riemannian metric on M. us, for each point p∈ M we have gp,

an inner product on the tangent space TpM.

Lemma 1. Let (M, g)be a Riemannian manifold in an ambient Rn, and let

p∈Mbe a point. If gis locally constant about pin an open neighbourhood

Usuch that gis a constant diagonal matrix in ambient coordinates, then in a

ball B⊆Ucentered at pwith volume πn/2

Γ(n/2+1) with respect to g, the geodesic

distance from pto any point q∈Bis 1

rdRn(p, q), where ris the radius of the

ball in the ambient space and dRnis the existing metric on the ambient space.

See Appendix A of the supplementary materials for a proof of Lemma

1.

If we assume the data to be uniformly distributed on M(with respect to

g) then, away from any boundaries, any ball of xed volume should contain

approximately the same number of points of Xregardless of where on the

manifold it is centered. Given nite data and small enough local neighbor-

hoods this crude approximation should be accurate enough even for data

samples near manifold boundaries. Now, conversely, a ball centered at Xi

that contains exactly the k-nearest-neighbors of Xishould have approxi-

mately xed volume regardless of the choice of Xi∈X. Under Lemma 1

it follows that we can approximate geodesic distance from Xito its neigh-

bors by normalising distances with respect to the distance to the kth nearest

neighbor of Xi.

In essence, by creating a custom distance for each Xi, we can ensure

the validity of the assumption of uniform distribution on the manifold. e

cost is that we now have an independent notion of distance for each and

every Xi, and these notions of distance may not be compatible. We have

a family of discrete metric spaces (one for each Xi) that we wish to merge

into a consistent global structure. is can be done in a natural way by

converting the metric spaces into fuzzy simplicial sets.

2.2 Fuzzy topological representation

We will use functors between the relevant categories to convert from metric

spaces to fuzzy topological representations. is will provide a means to

merge the incompatible local views of the data. e topological structure

of choice is that of simplicial sets. For more details on simplicial sets we

refer the reader to [25], [40], [48], or [22]. Our approach draws heavily

upon the work of Michael Barr [3] and David Spivak in [52], and many

of the denitions and theorems below are drawn or adapted from those

5

sources. We assume familiarity with the basics of category theory. For an

introduction to category theory readers may consult [39] or [49].

To start we will review the denitions for simplicial sets. Simplicial sets

provide a combinatorial approach to the study of topological spaces. ey

are related to the simpler notion of simplicial complexes – which construct

topological spaces by gluing together simple building blocks called sim-

plices – but are more general. Simplicial sets are most easily dened purely

abstractly in the language of category theory.

Denition 1. e category ∆has as objects the nite order sets [n] =

{1, . . . , n}, with morphims given by (non-strictly) order-preserving maps.

Following standard category theoretic notation, ∆op denotes the cate-

gory with the same objects as ∆and morphisms given by the morphisms

of ∆with the direction (domain and codomain) reversed.

Denition 2. Asimplicial set is a functor from ∆op to Sets, the category of

sets; that is, a contravariant functor from ∆to Sets.

Given a simplicial set X:∆op →Sets,it is common to denote the set

X([n]) as Xnand refer to the elements of the set as the n-simplices of X.

e simplest possible examples of simplicial sets are the standard simplices

∆n, dened as the representable functors hom∆(·,[n]). It follows from the

Yoneda lemma that there is a natural correspondence between n-simplices

of Xand morphisms ∆n→Xin the category of simplicial sets, and it

is oen helpful to think in these terms. us for each x∈Xnwe have

a corresponding morphism x: ∆n→X. By the density theorem and

employing a minor abuse of notation we then have

colim

x∈Xn

∆n∼

=X

ere is a standard covariant functor |·|:∆→Top mapping from

the category ∆to the category of topological spaces that sends [n]to the

standard n-simplex |∆n| ⊂ Rn+1 dened as

|∆n|,((t0, . . . , tn)∈Rn+1 |

n

X

i=0

ti= 1, ti≥0)

with the standard subspace topology. If X:∆op →Sets is a simplicial

set then we can construct the realization of X(denoted |X|) as the colimit

|X|= colim

x∈Xn

|∆n|

6

and thus associate a topological space with a given simplicial set. Con-

versely given a topological space Ywe can construct an associated simpli-

cial set S(Y), called the singular set of Y, by dening

S(Y) : [n]7→ homTop(|∆n|, Y ).

It is a standard result of classical homotopy theory that the realization func-

tor and singular set functors form an adjunction, and provide the standard

means of translating between topological spaces and simplicial sets. Our

goal will be to adapt these powerful classical results to the case of nite

metric spaces.

We draw signicant inspiration from Spivak, specically [52], where

he extends the classical theory of singular sets and topological realization

to fuzzy singular sets and metric realization. To develop this theory here

we will rst outline a categorical presentation of fuzzy sets, due to [3], that

will make extending classical simplicial sets to fuzzy simplicial sets most

natural.

Classically a fuzzy set [65] is dened in terms of a carrier set Aand a

map µ:A→[0,1] called the membership function. One is to interpret the

value µ(x)for x∈Ato be the membership strength of xto the set A. us

membership of a set is no longer a bi-valent true or false property as in

classical set theory, but a fuzzy property taking values in the unit interval.

We wish to formalize this in terms of category theory.

Let Ibe the unit interval (0,1] ⊆Rwith topology given by intervals

of the form [0, a)for a∈(0,1]. e category of open sets (with morphisms

given by inclusions) can be imbued with a Grothendieck topology in the

natural way for any poset category.

Denition 3. A presheaf Pon Iis a functor from Iop to Sets. A fuzzy set

is a presheaf on Isuch that all maps P(a≤b)are injections.

Presheaves on Iform a category with morphisms given by natural

transformations. We can thus form a category of fuzzy sets by simply re-

stricting to the sub-category of presheaves that are fuzzy sets. We note that

such presheaves are trivially sheaves under the Grothendieck topology on

I. As one might expect, limits (including products) of such sheaves are

well dened, but care must be taken to dene colimits (and coproducts) of

sheaves. To link to the classical approach to fuzzy sets one can think of a

section P([0, a)) as the set of all elements with membership strength at

least a. We can now dene the category of fuzzy sets.

Denition 4. e category Fuzz of fuzzy sets is the full subcategory of

sheaves on Ispanned by fuzzy sets.

7

With this categorical presentation in hand, dening fuzzy simplicial

sets is simply a maer of considering presheaves of ∆valued in the cate-

gory of fuzzy sets rather than the category of sets.

Denition 5. e category of fuzzy simplicial sets sFuzz is the category

with objects given by functors from ∆op to Fuzz, and morphisms given by

natural transformations.

Alternatively, a fuzzy simplicial set can be viewed as a sheaf over ∆×I,

where ∆is given the trivial topology and ∆×Ihas the product topology.

We will use ∆n

<a to denote the sheaf given by the representable functor of

the object ([n],[0, a)). e importance of this fuzzy (sheaed) version of

simplicial sets is their relationship to metric spaces. We begin by consider-

ing the larger category of extended-pseudo-metric spaces.

Denition 6. An extended-pseudo-metric space (X, d)is a set Xand a

map d:X×X→R≥0∪ {∞} such that

1. d(x, y)>0, and x=yimplies d(x, y)=0;

2. d(x, y) = d(y, x); and

3. d(x, z)6d(x, y) + d(y, z)or d(x, z) = ∞.

e category of extended-pseudo-metric spaces EPMet has as objects extended-

pseudo-metric spaces and non-expansive maps as morphisms. We denote the

subcategory of nite extended-pseudo-metric spaces FinEPMet.

e choice of non-expansive maps in Denition 6 is due to Spivak, but

we note that it closely mirrors the work of Carlsson and Memoli in [13] on

topological methods for clustering as applied to nite metric spaces. is

choice is signicant since pure isometries are too strict and do not provide

large enough Hom-sets.

In [52] Spivak constructs a pair of adjoint functors, Real and Sing be-

tween the categories sFuzz and EPMet. ese functors are the natural ex-

tension of the classical realization and singular set functors from algebraic

topology. e functor Real is dened in terms of standard fuzzy simplices

∆n

<a as

Real(∆n

<a),((t0, . . . , tn)∈Rn+1 |

n

X

i=0

ti=−log(a), ti≥0)

similarly to the classical realization functor |·|. e metric on Real(∆n

<a)

is simply inherited from Rn+1. A morphism ∆n

<a →∆m

<b exists only if

8

a≤b, and is determined by a ∆morphism σ: [n]→[m]. e action of

Real on such a morphism is given by the map

(x0, x1, . . . , xn)7→ log(b)

log(a)

X

i0∈σ−1(0)

xi0,X

i0∈σ−1(1)

xi0,..., X

i0∈σ−1(m)

xi0

.

Such a map is clearly non-expansive since 0≤a≤b≤1implies that

log(b)/log(a)≤1.

We then extend this to a general simplicial set Xvia colimits, dening

Real(X),colim

∆n

<a→XReal(∆n

<a).

Since the functor Real preserves colimits, it follows that there exists a

right adjoint functor. Again, analogously to the classical case, we nd the

right adjoint, denoted Sing, is dened for an extended pseudo metric space

Yin terms of its action on the category ∆×I:

Sing(Y) : ([n],[0, a)) 7→ homEPMet(Real(∆n

<a), Y ).

For our case we are only interested in nite metric spaces. To corre-

spond with this we consider the subcategory of bounded fuzzy simplicial

sets Fin-sFuzz. We therefore use the analogous adjoint pair FinReal and

FinSing. Formally we dene the nite fuzzy realization functor as follows:

Denition 7. Dene the functor FinReal :Fin-sFuzz →FinEPMet by

seing

FinReal(∆n

<a),({x1, x2, . . . , xn}, da),

where

da(xi, xj) =

−log(a)if i6=j,

0otherwise

.

and then dening

FinReal(X),colim

∆n

<a→XFinReal(∆n

<a).

Similar to Spivak’s construction, the action of FinReal on a map ∆n

<a →

∆m

<b, where a≤bdened by σ: ∆n→∆m, is given by

({x1, x2, . . . , xn}, da)7→ ({xσ(1), xσ(2), . . . , xσ(n)}, db),

which is a non-expansive map since a≤bimplies da≥db.

Since FinReal preserves colimits it admits a right adjoint, the fuzzy sin-

gular set functor FinSing. We can then dene the (nite) fuzzy singular set

functor in terms of the action of its image on ∆×I, analogously to Sing.

9

Denition 8. Dene the functor FinSing :FinEPMet →Fin-sFuzz by

FinSing(Y) : ([n],[0, a)) 7→ homFinEPMet(FinReal(∆n

<a), Y ).

We then have the following theorem.

eorem 1. e functors FinReal :Fin-sFuzz →FinEPMet and FinSing :

FinEPMet →Fin-sFuzz form an adjunction with FinReal the le adjoint

and FinSing the right adjoint.

e proof of this is by construction. Appendix B provides a full proof

of the theorem.

With the necessary theoretical background in place, the means to han-

dle the family of incompatible metric spaces described above becomes clear.

Each metric space in the family can be translated into a fuzzy simplicial

set via the fuzzy singular set functor, distilling the topological information

while still retaining metric information in the fuzzy structure. Ironing out

the incompatibilities of the resulting family of fuzzy simplicial sets can be

done by simply taking a (fuzzy) union across the entire family. e result

is a single fuzzy simplicial set which captures the relevant topological and

underlying metric structure of the manifold M.

It should be noted, however, that the fuzzy singular set functor applies

to extended-pseudo-metric spaces, which are a relaxation of traditional

metric spaces. e results of Lemma 1 only provide accurate approxima-

tions of geodesic distance local to Xifor distances measured from Xi–

the geodesic distances between other pairs of points within the neighbor-

hood of Xiare not well dened. In deference to this lack of information we

dene distances between Xjand Xkin the extended-pseudo metric space

local to Xi(where i6=jand i6=k) to be innite (local neighborhoods of

Xjand Xkwill provide suitable approximations).

For real data it is safe to assume that the manifold Mis locally con-

nected. In practice this can be realized by measuring distance in the extended-

pseudo-metric space local to Xias geodesic distance beyond the nearest

neighbor of Xi. Since this sets the distance to the nearest neighbor to be

equal to 0 this is only possible in the more relaxed seing of extended-

pseudo-metric spaces. It ensures, however, that each 0-simplex is the face

of some 1-simplex with fuzzy membership strength 1, meaning that the

resulting topological structure derived from the manifold is locally con-

nected. We note that this has a similar practical eect to the truncated

similarity approach of Lee and Verleysen [33], but derives naturally from

the assumption of local connectivity of the manifold.

10

Combining all of the above we can dene the fuzzy topological repre-

sentation of a dataset.

Denition 9. Let X={X1, . . . , XN}be a dataset in Rn. Let {(X, di)}i=1...N

be a family of extended-pseudo-metric spaces with common carrier set Xsuch

that

di(Xj, Xk) =

dM(Xj, Xk)−ρif i=jor i=k,

∞otherwise ,

where ρis the distance to the nearest neighbor of Xiand dMis geodesic

distance on the manifold M, either known apriori, or approximated as per

Lemma 1.

e fuzzy topological representation of Xis

n

[

i=1

FinSing((X, di)).

e (fuzzy set) union provides the means to merge together the dier-

ent metric spaces. is provides a single fuzzy simplicial set as the global

representation of the manifold formed by patching together the many local

representations.

Given the ability to construct such topological structures, either from

a known manifold, or by learning the metric structure of the manifold, we

can perform dimension reduction by simply nding low dimensional rep-

resentations that closely match the topological structure of the source data.

We now consider the task of nding such a low dimensional representation.

2.3 Optimizing a low dimensional representation

Let Y={Y1, . . . , YN} ⊆ Rdbe a low dimensional (dn) representation

of Xsuch that Yirepresents the source data point Xi. In contrast to the

source data where we want to estimate a manifold on which the data is

uniformly distributed, a target manifold for Yis chosen apriori (usually this

will simply be Rditself, but other choices such as d-spheres or d-tori are

certainly possible) . erefore we know the manifold and manifold metric

apriori, and can compute the fuzzy topological representation directly. Of

note, we still want to incorporate the distance to the nearest neighbor as per

the local connectedness requirement. is can be achieved by supplying a

parameter that denes the expected distance between nearest neighbors in

the embedded space.

11

Given fuzzy simplicial set representations of Xand Y, a means of com-

parison is required. If we consider only the 1-skeleton of the fuzzy sim-

plicial sets we can describe each as a fuzzy graph, or, more specically, a

fuzzy set of edges. To compare two fuzzy sets we will make use of fuzzy set

cross entropy. For these purposes we will revert to classical fuzzy set no-

tation. at is, a fuzzy set is given by a reference set Aand a membership

strength function µ:A→[0,1]. Comparable fuzzy sets have the same

reference set. Given a sheaf representation Pwe can translate to classical

fuzzy sets by seing A=Sa∈(0,1] P([0, a)) and µ(x) = sup{a∈(0,1] |

x∈P([0, a))}.

Denition 10. e cross entropy Cof two fuzzy sets (A, µ)and (A, ν)is

dened as

C((A, µ),(A, ν)) ,X

a∈Aµ(a) log µ(a)

ν(a)+ (1 −µ(a)) log 1−µ(a)

1−ν(a).

Similar to t-SNE we can optimize the embedding Ywith respect to fuzzy

set cross entropy Cby using stochastic gradient descent. However, this re-

quires a dierentiable fuzzy singular set functor. If the expected minimum

distance between points is zero the fuzzy singular set functor is dieren-

tiable for these purposes, however for any non-zero value we need to make

a dierentiable approximation (chosen from a suitable family of dieren-

tiable functions).

is completes the algorithm: by using manifold approximation and

patching together local fuzzy simplicial set representations we construct a

topological representation of the high dimensional data. We then optimize

the layout of data in a low dimensional space to minimize the error between

the two topological representations.

We note that in this case we restricted aention to comparisons of the

1-skeleton of the fuzzy simplicial sets. One can extend this to `-skeleta by

dening a cost function C`as

C`(X, Y ) =

`

X

i=1

λiC(Xi, Yi),

where Xidenotes the fuzzy set of i-simplices of Xand the λiare suit-

ably chosen real valued weights. While such an approach will capture the

overall topological structure more accurately, it comes at non-negligible

computational cost due to the increasingly large numbers of higher dimen-

sional simplices. For this reason current implementations restrict to the

1-skeleton at this time.

12

3 A Computational View of UMAP

To understand what computations the UMAP algorithm is actually making

from a practical point of view, a less theoretical and more computational

description may be helpful for the reader. is description of the algorithm

lacks the motivation for a number of the choices made. For that motivation

please see Section 2.

e theoretical description of the algorithm works in terms of fuzzy

simplicial sets. Computationally this is only tractable for the one skeleton

which can ultimately be described as a weighted graph. is means that,

from a practical computational perspective, UMAP can ultimately be de-

scribed in terms of, construction of, and operations on, weighted graphs.

In particular this situates UMAP in the class of k-neighbour based graph

learning algorithms such as Laplacian Eigenmaps, Isomap and t-SNE.

As with other k-neighbour graph based algorithms, UMAP can be de-

scribed in two phases. In the rst phase a particular weighted k-neighbour

graph is constructed. In the second phase a low dimensional layout of this

graph is computed. e dierences between all algorithms in this class

amount to specic details in how the graph is constructed and how the

layout is computed. e theoretical basis for UMAP as described in Section

2 provides novel approaches to both of these phases, and provides clear

motivation for the choices involved.

Finally, since t-SNE is not usually described as a graph based algorithm,

a direct comparison of UMAP with t-SNE, using the similarity/probability

notation commonly used to express the equations of t-SNE, is given in the

Appendix C.

In section 2 we made a few basic assumptions about our data. From

these assumptions we made use of category theory to derive the UMAP

algorithms. at said, all these derivations assume these axioms to be true.

1. ere exists a manifold on which the data would be uniformly dis-

tributed.

2. e underlying manifold of interest is locally connected.

3. Preserving the topological structure of this manifold is the primary

goal.

e topological theory of Section 2 is driven by these axioms, particularly

the interest in modelling and preserving topological structure. In particular

Section 2.1 highlights the underlying motivation, in terms of topological

theory, of representing a manifold as a k-neighbour graph.

13

As highlighted in Appendix C any algorithm that aempts to use a

mathematical structure akin to a k-neighbour graph to approximate a man-

ifold must follow a similar basic structure.

•Graph Construction

1. Construct a weighted k-neighbour graph

2. Apply some transform on the edges to ambient local distance.

3. Deal with the inherent asymmetry of the k-neighbour graph.

•Graph Layout

1. Dene an objective function that preserves desired characteris-

tics of this k-neighbour graph.

2. Find a low dimensional representation which optimizes this ob-

jective function.

Many dimension reduction algorithms can be broken down into these

steps because they are fundamental to a particular class of solutions. Choices

for each step must be either chosen through task oriented experimentation

or by selecting a set of believable axioms and building strong theoretical

arguments from these. Our belief is that basing our decisions on a strong

foundational theory will allow for a more extensible and generalizable al-

gorithm in the long run.

We theoretically justify using the choice of using a k-neighbour graph

to represent a manifold in Section 2.1. e choices for our kernel transform

an symmetrization function can be found in Section 2.2. Finally, the justi-

cations underlying our choices for our graph layout are outlined in Section

2.3.

3.1 Graph Construction

e rst phase of UMAP can be thought of as the construction of a weighted

k-neighbour graph. Let X={x1, . . . , xN}be the input dataset, with a

metric (or dissimilarity measure) d:X×X→R≥0. Given an input hyper-

parameter k, for each xiwe compute the set {xi1, . . . , xik}of the knearest

neighbors of xiunder the metric d. is computation can be performed via

any nearest neighbour or approximate nearest neighbour search algorithm.

For the purposes of our UMAP implemenation we prefer to use the nearest

neighbor descent algorithm of [18].

For each xiwe will dene ρiand σi. Let

ρi= min{d(xi, xij)|1≤j≤k, d(xi, xij)>0},

14

and set σito be the value such that

k

X

j=1

exp −max(0, d(xi, xij)−ρi)

σi= log2(k).

e selection of ρiderives from the local-connectivity constraint described

in Section 2.2. In particular it ensures that xiconnects to at least one other

data point with an edge of weight 1; this is equivalent to the resulting fuzzy

simplicial set being locally connected at xi. In practical terms this signif-

icantly improves the representation on very high dimensional data where

other algorithms such as t-SNE begin to suer from the curse of dimen-

sionality.

e selection of σicorresponds to (a smoothed) normalisation factor,

dening the Riemannian metric local to the point xias described in Section

2.1.

We can now dene a weighted directed graph ¯

G= (V, E , w). e

vertices Vof ¯

Gare simply the set X. We can then form the set of directed

edges E={(xi, xij)|1≤j≤k, 1≤i≤N}, and dene the weight

function wby seing

w((xi, xij)) = exp −max(0, d(xi, xij)−ρi)

σi.

For a given point xithere exists an induced graph of xiand outgoing edges

incident on xi. is graph is the 1-skeleton of the fuzzy simplicial set as-

sociated to the metric space local to xiwhere the local metric is dened

in terms of ρiand σi. e weight associated to the edge is the member-

ship strength of the corresponding 1-simplex within the fuzzy simplicial

set, and is derived from the adjunction of eorem 1 using the right adjoint

(nearest inverse) of the geometric realization of a fuzzy simplicial set. In-

tuitively one can think of the weight of an edge as akin to the probability

that the given edge exists. Section 2 demonstrates why this construction

faithfully captures the topology of the data. Given this set of local graphs

(represented here as a single directed graph) we now require a method to

combine them into a unied topological representation. We note that while

patching together incompatible nite metric spaces is challenging, by using

eorem 1 to convert to a fuzzy simplicial set representation, the combin-

ing operation becomes natural.

Let Abe the weighted adjacency matrix of ¯

G, and consider the sym-

metric matrix

B=A+A>−A◦A>,

15

where ◦is the Hadamard (or pointwise) product. is formula derives from

the use of the probabilistic t-conorm used in unioning the fuzzy simplicial

sets. If one interprets the value of Aij as the probability that the directed

edge from xito xjexists, then Bij is the probability that at least one of

the two directed edges (from xito xjand from xjto xi) exists. e UMAP

graph Gis then an undirected weighted graph whose adjacency matrix is

given by B. Section 2 explains this construction in topological terms, pro-

viding the justication for why this construction provides an appropriate

fuzzy topological representation of the data – that is, this construction cap-

tures the underlying geometric structure of the data in a faithful way.

3.2 Graph Layout

In practice UMAP uses a force directed graph layout algorithm in low di-

mensional space. A force directed graph layout utilizes of a set of aractive

forces applied along edges and a set of repulsive forces applied among ver-

tices. Any force directed layout algorithm requires a description of both the

aractive and repulsive forces. e algorithm proceeds by iteratively ap-

plying aractive and repulsive forces at each edge or vertex. is amounts

to a non-convex optimization problem. Convergence to a local minima is

guaranteed by slowly decreasing the aractive and repulsive forces in a

similar fashion to that used in simulated annealing.

In UMAP the aractive force between two vertices iand jat coordi-

nates yiand yjrespectively, is determined by:

−2abkyi−yjk2(b−1)

2

1 + kyi−yjk2

2

w((xi, xj)) (yi−yj)

where aand bare hyper-parameters.

Repulsive forces are computed via sampling due to computational con-

straints. us, whenever an aractive force is applied to an edge, one of

that edge’s vertices is repulsed by a sampling of other vertices. e repul-

sive force is given by

2b

+kyi−yjk2

21 + akyi−yjk2b

2(1 −w((xi, xj))) (yi−yj).

is a small number to prevent division by zero (0.001 in the current

implementation).

16

e algorithm can be initialized randomly but in practice, since the sym-

metric Laplacian of the graph Gis a discrete approximation of the Laplace-

Beltrami operator of the manifold, we can use a spectral layout to initialize

the embedding. is provides both faster convergence and greater stability

within the algorithm.

e forces described above are derived from gradients optimising the

edge-wise cross-entropy between the weighted graph G, and an equiva-

lent weighted graph Hconstructed from the points {yi}i=1..N . at is, we

are seeking to position points yisuch that the weighted graph induced by

those points most closely approximates the graph G, where we measure

the dierence between weighted graphs by the total cross entropy over all

the edge existence probabilities. Since the weighted graph Gcaptures the

topology of the source data, the equivalent weighted graph Hconstructed

from the points {yi}i=1..N matches the topology as closely as the optimiza-

tion allows, and thus provides a good low dimensional representation of the

overall topology of the data.

4 Implementation and Hyper-parameters

Having completed a theoretical description of the approach, we now turn

our aention to the practical realization of this theory. We begin by pro-

viding a more detailed description of the algorithm as implemented, and

then discuss a few implementation specic details. We conclude this sec-

tion with a discussion of the hyper-parameters for the algorithm and their

practical eects.

4.1 Algorithm description

In overview the UMAP algorithm is relatively straightforward (see Algo-

rithm 1). When performing a fuzzy union over local fuzzy simplicial sets

we have found it most eective to work with the probabilistic t-conorm (as

one would expect if treating membership strengths as a probability that the

simplex exists). e individual functions for constructing the local fuzzy

simplicial sets, determining the spectral embedding, and optimizing the

embedding with regard to fuzzy set cross entropy, are described in more

detail below.

e inputs to Algorithm 1 are: X, the dataset to have its dimension

reduced; n, the neighborhood size to use for local metric approximation;

d, the dimension of the target reduced space; min-dist, an algorithmic pa-

17

Algorithm 1 UMAP algorithm

function UMAP(X,n,d, min-dist, n-epochs)

#Construct the relevant weighted graph

for all x∈Xdo

fs-set[x]←LocalFuzzySimplicialSet(X,x,n)

top-rep ←Sx∈Xfs-set[x]#We recommend the probabilistic t-conorm

#Perform optimization of the graph layout

Y←SpectralEmbedding(top-rep, d)

Y←OptimizeEmbedding(top-rep, Y, min-dist, n-epochs)

return Y

rameter controlling the layout; and n-epochs, controlling the amount of

optimization work to perform.

Algorithm 2 describes the construction of local fuzzy simplicial sets.

To represent fuzzy simplicial sets we work with the fuzzy set images of [0]

and [1] (i.e. the 1-skeleton), which we denote as fs-set0and fs-set1. One can

work with higher order simplices as well, but the current implementation

does not. We can construct the fuzzy simplicial set local to a given point x

by nding the nnearest neighbors, generating the appropriate normalised

distance on the manifold, and then converting the nite metric space to a

simplicial set via the functor FinSing, which translates into exponential of

the negative distance in this case.

Rather than directly using the distance to the nth nearest neighbor as

the normalization, we use a smoothed version of knn-distance that xes

the cardinality of the fuzzy set of 1-simplices to a xed value. We selected

log2(n)for this purpose based on empirical experiments. is is described

briey in Algorithm 3.

Spectral embedding is performed by considering the 1-skeleton of the

global fuzzy topological representation as a weighted graph and using stan-

dard spectral methods on the symmetric normalized Laplacian. is pro-

cess is described in Algorithm 4.

e nal major component of UMAP is the optimization of the em-

bedding through minimization of the fuzzy set cross entropy. Recall that

18

Algorithm 2 Constructing a local fuzzy simplicial set

function LocalFuzzySimplicialSet(X,x,n)

knn, knn-dists ←ApproxNearestNeighbors(X,x,n)

ρ←knn-dists[1] # Distance to nearest neighbor

σ←SmoothKNNDist(knn-dists, n,ρ) # Smooth approximator to

knn-distance

fs-set0←X

fs-set1← {([x, y],0) |y∈X}

for all y∈knn do

dx,y ←max{0,dist(x, y)−ρ}/σ

fs-set1←fs-set1∪([x, y],exp(−dx,y ))

return fs-set

Algorithm 3 Compute the normalizing factor for distances σ

function SmoothKNNDist(knn-dists, n,ρ)

Binary search for σsuch that Pn

i=1 exp(−(knn-distsi−ρ)/σ) = log2(n)

return σ

Algorithm 4 Spectral embedding for initialization

function SpectralEmbedding(top-rep, d)

A←1-skeleton of top-rep expressed as a weighted adjacency matrix

D←degree matrix for the graph A

L←D1/2(D−A)D1/2

evec ←Eigenvectors of L(sorted)

Y←evec[1..d + 1] #0-base indexing assumed

return Y

19

fuzzy set cross entropy, with respect given membership functions µand ν,

is given by

C((A, µ),(A, ν)) = X

a∈A

µ(a) log µ(a)

ν(a)+ (1 −µ(a)) log 1−µ(a)

1−ν(a)

=X

a∈A

(µ(a) log(µ(a)) + (1 −µ(a)) log(1 −µ(a)))

−X

a∈A

(µ(a) log(ν(a)) + (1 −µ(a)) log(1 −ν(a))) .

(1)

e rst sum depends only on µwhich takes xed values during the op-

timization, thus the minimization of cross entropy depends only on the

second sum, so we seek to minimize

−X

a∈A

(µ(a) log(ν(a)) + (1 −µ(a)) log(1 −ν(a))) .

Following both [54] and [41], we take a sampling based approach to the

optimization. We sample 1-simplices with probability µ(a)and update ac-

cording to the value of ν(a), which handles the term µ(a) log(ν(a)). e

term (1 −µ(a)) log(1 −ν(a)) requires negative sampling – rather than

computing this over all potential simplices we randomly sample potential

1-simplices and assume them to be a negative example (i.e. with member-

ship strength 0) and update according to the value of 1−ν(a). In contrast

to [54] the above formulation provides a vertex sampling distribution of

P(xi) = P{a∈A|d0(a)=xi}1−µ(a)

P{b∈A|d0(b)6=xi}1−µ(b)

for negative samples, which can be reasonably approximated by a uniform

distribution for suciently large data sets.

It therefore only remains to nd a dierentiable approximation to ν(a)

for a given 1-simplex aso that gradient descent can be applied for opti-

mization. is is done as follows:

Denition 11. Dene Φ : Rd×Rd→[0,1], a smooth approximation of the

membership strength of a 1-simplex between two points in Rd, as

Φ(x,y) = 1 + a(kx−yk2

2)b−1,

20

where aand bare chosen by non-linear least squares ing against the curve

Ψ : Rd×Rd→[0,1] where

Ψ(x,y) = (1if kx−yk2≤min-dist

exp(−(kx−yk2−min-dist)) otherwise .

e optimization process is now executed by stochastic gradient de-

scent as given by Algorithm 5.

Algorithm 5 Optimizing the embedding

function OptimizeEmbedding(top-rep, Y, min-dist, n-epochs)

α←1.0

Fit Φfrom Ψdened by min-dist

for e←1,...,n-epochs do

for all ([a, b], p)∈top-rep1do

if Random( ) ≤pthen #Sample simplex with probability p

ya←ya+α· ∇(log(Φ))(ya, yb)

for i←1,...,n-neg-samples do

c←random sample from Y

ya←ya+α· ∇(log(1 −Φ))(ya, yc)

α←1.0−e/n-epochs

return Y

is completes the UMAP algorithm.

4.2 Implementation

Practical implementation of this algorithm requires (approximate) k-nearest-

neighbor calculation and ecient optimization via stochastic gradient de-

scent.

Ecient approximate k-nearest-neighbor computation can be achieved

via the Nearest-Neighbor-Descent algorithm of [18]. e error intrinsic in

a dimension reduction technique means that such approximation is more

than adequate for these purposes. While no theoretical complexity bounds

21

have been established for Nearest-Neighbor-Descent the authors of the

original paper report an empirical complexity of O(N1.14). A further ben-

et of Nearest-Neighbor-Descent is its generality; it works with any valid

dissimilarity measure, and is ecient even for high dimensional data.

In optimizing the embedding under the provided objective function, we

follow work of [54]; making use of probabilistic edge sampling and nega-

tive sampling [41]. is provides a very ecient approximate stochastic

gradient descent algorithm since there is no normalization requirement.

Furthermore, since the normalized Laplacian of the fuzzy graph represen-

tation of the input data is a discrete approximation of the Laplace-Betrami

operator of the manifold [?, see]]belkin2002laplacian, belkin2003laplacian,

we can provide a suitable initialization for stochastic gradient descent by

using the eigenvectors of the normalized Laplacian. e amount of opti-

mization work required will scale with the number of edges in the fuzzy

graph (assuming a xed negative sampling rate), resulting in a complexity

of O(kN ).

Combining these techniques results in highly ecient embeddings, which

we will discuss in Section 5. e overall complexity is bounded by the ap-

proximate nearest neighbor search complexity and, as mentioned above, is

empirically approximately O(N1.14). A reference implementation can be

found at https://github.com/lmcinnes/umap, and an R implementa-

tion can be found at https://github.com/jlmelville/uwot.

For simplicity these experiments were carried out on a single core ver-

sion of our algorithm. It should be noted that at the time of this publication

that both Nearest-Neighbour-Descent and SGD have been parallelized and

thus the python reference implementation can be signicantly accelerated.

Our intention in this paper was to introduce the underlying theory behind

our UMAP algorithm and we felt that parallel vs single core discussions

would distract from our intent.

4.3 Hyper-parameters

As described in Algorithm 1, the UMAP algorithm takes four hyper-parameters:

1. n, the number of neighbors to consider when approximating the local

metric;

2. d, the target embedding dimension;

3. min-dist, the desired separation between close points in the embed-

ding space; and

22

4. n-epochs, the number of training epochs to use when optimizing the

low dimensional representation.

e eects of the parameters dand n-epochs are largely self-evident, and

will not be discussed in further detail here. In contrast the eects of the

number of neighbors nand of min-dist are less clear.

One can interpret the number of neighbors nas the local scale at which

to approximate the manifold as roughly at, with the manifold estimation

averaging over the nneighbors. Manifold features that occur at a smaller

scale than within the nnearest-neighbors of points will be lost, while large

scale manifold features that cannot be seen by patching together locally at

charts at the scale of nnearest-neighbors may not be well detected. us

nrepresents some degree of trade-o between ne grained and large scale

manifold features — smaller values will ensure detailed manifold structure

is accurately captured (at a loss of the “big picture” view of the manifold),

while larger values will capture large scale manifold structures, but at a loss

of ne detail structure which will get averaged out in the local approxima-

tions. With smaller nvalues the manifold tends to be broken into many

small connected components (care needs to be taken with the spectral em-

bedding for initialization in such cases).

In contrast min-dist is a hyperparameter directly aecting the output,

as it controls the fuzzy simplicial set construction from the low dimensional

representation. It acts in lieu of the distance to the nearest neighbor used

to ensure local connectivity. In essence this determines how closely points

can be packed together in the low dimensional representation. Low values

on min-dist will result in potentially densely packed regions, but will likely

more faithfully represent the manifold structure. Increasing the value of

min-dist will force the embedding to spread points out more, assisting vi-

sualization (and avoiding potential overploing issues). We view min-dist

as an essentially aesthetic parameter, governing the appearance of the em-

bedding, and thus is more important when using UMAP for visualization.

In Figure 1 we provide examples of the eects of varying the hyper-

parameters for a toy dataset. e data is uniform random samples from a

3-dimensional color-cube, allowing for easy visualization of the original 3-

dimensional coordinates in the embedding space by using the correspond-

ing RGB colour. Since the data lls a 3-dimensional cube there is no local

manifold structure, and hence for such data we expect larger nvalues to be

more useful. Low values will interpret the noise from random sampling as

ne scale manifold structure, producing potentially spurious structure1.

1See the discussion of the constellation eect in Section 6

23

Figure 1: Variation of UMAP hyperparameters nand min-dist result in dierent

embeddings. e data is uniform random samples from a 3-dimensional color-

cube, allowing for easy visualization of the original 3-dimensional coordinates

in the embedding space by using the corresponding RGB colour. Low values of

nspuriously interpret structure from the random sampling noise – see Section 6

for further discussion of this phenomena.

24

In Figure 2 we provides examples of the same hyperparamter choices

as Figure 1, but for the PenDigits dataset2. In this case we expect small

to medium nvalues to be most eective, since there is signicant cluster

structure naturally present in the data. e min-dist parameter expands out

tightly clustered groups, allowing more of the internal structure of densely

packed clusters to be seen.

Finally, in Figure 3 we provide an equivalent example of hyperparame-

ter choices for the MNIST dataset3. Again, since this dataset is expected to

have signifcant cluster structure we expect medium sized values of nto be

most eective. We note that large values of min-dist result in the distinct

clusters being compressed together, making the distinctions between the

clusters less clear.

5 Practical Ecacy

While the strong mathematical foundations of UMAP were the motivation

for its development, the algorithm must ultimately be judged by its prac-

tical ecacy. In this section we examine the delity and performance of

low dimensional embeddings of multiple diverse real world data sets under

UMAP. e following datasets were considered:

Pen digits [1, 10] is a set of 1797 grayscale images of digits entered using

a digitiser tablet. Each image is an 8x8 image which we treat as a single 64

dimensional vector, assumed to be in Euclidean vector space.

COIL 20 [43] is a set of 1440 greyscale images consisting of 20 objects un-

der 72 dierent rotations spanning 360 degrees. Each image is a 128x128

image which we treat as a single 16384 dimensional vector for the purposes

of computing distance between images.

COIL 100 [44] is a set of 7200 colour images consisting of 100 objects un-

der 72 dierent rotations spanning 360 degrees. Each image consists of 3

128x128 intensity matrices (one for each color channel). We treat this as

a single 49152 dimensional vector for the purposes of computing distance

between images.

Mouse scRNA-seq [11] is proled gene expression data for 20,921 cells

from an adult mouse. Each sample consists of a vector of 26,774 measure-

ments.

Statlog (Shuttle) [35] is a NASA dataset consisting of various data associ-

ated to the positions of radiators in the space shule, including a timestamp.

2See Section 5 for a description of the PenDigits dataset

3See section 5 for details on the MNIST dataset

25

Figure 2: Variation of UMAP hyperparameters nand min-dist result in dier-

ent embeddings. e data is the PenDigits dataset, where each point is an 8x8

grayscale image of a hand-wrien digit.

26

Figure 3: Variation of UMAP hyperparameters nand min-dist result in dier-

ent embeddings. e data is the MNIST dataset, where each point is an 28x28

grayscale image of a hand-wrien digit.

27

e dataset has 58000 points in a 9 dimensional feature space.

MNIST [32] is a dataset of 28x28 pixel grayscale images of handwrien

digits. ere are 10 digit classes (0 through 9) and 70000 total images. is

is treated as 70000 dierent 784 dimensional vectors.

F-MNIST [63] or Fashion MNIST is a dataset of 28x28 pixel grayscale im-

ages of fashion items (clothing, footwear and bags). ere are 10 classes

and 70000 total images. As with MNIST this is treated as 70000 dierent

784 dimensional vectors.

Flow cytometry [51, 9] is a dataset of ow cytometry measurements of

CDT4 cells comprised of 1,000,000 samples, each with 17 measurements.

GoogleNews word vectors [41] is a dataset of 3 million words and phrases

derived from a sample of Google News documents and embedded into a 300

dimensional space via word2vec.

For all the datasets except GoogleNews we use Euclidean distance be-

tween vectors. For GoogleNews, as per [41], we use cosine distance (or

angular distance in t-SNE which does support non-metric distances, in con-

trast to UMAP).

5.1 alitative Comparison of Multiple Algorithms

We compare a number of algorithms–UMAP, t-SNE [60, 58], LargeVis [54],

Laplacian Eigenmaps [7], and Principal Component Analysis [27]–on the

COIL20 [43], MNIST [32], Fashion-MNIST [63], and GoogleNews [41] datasets.

e Isomap algorithm was also tested, but failed to complete in any reason-

able time for any of the datasets larger than COIL20.

e Multicore t-SNE package [57] was used for t-SNE. e reference

implementation [53] was used for LargeVis. e scikit-learn [10] imple-

mentations were used for Laplacian Eigenmaps and PCA. Where possible

we aempted to tune parameters for each algorithm to give good embed-

dings.

Historically t-SNE and LargeVis have oered a dramatic improvement

in nding and preserving local structure in the data. is can be seen qual-

itatively by comparing their embeddings to those generated by Laplacian

Eigenmaps and PCA in Figure 4. We claim that the quality of embeddings

produced by UMAP is comparable to t-SNE when reducing to two or three

dimensions. For example, Figure 4 shows both UMAP and t-SNE embed-

dings of the COIL20, MNIST, Fashion MNIST, and Google News datasets.

While the precise embeddings are dierent, UMAP distinguishes the same

structures as t-SNE and LargeVis.

28

Figure 4: A comparison of several dimension reduction algorithms. We note

that UMAP successfully reects much of the large scale global structure that is

well represented by Laplacian Eigenmaps and PCA (particularly for MNIST and

Fashion-MNIST), while also preserving the local ne structure similar to t-SNE

and LargeVis.

29

It can be argued that UMAP has captured more of the global and topo-

logical structure of the datasets than t-SNE [4, 62]. More of the loops in the

COIL20 dataset are kept intact, including the intertwined loops. Similarly

the global relationships among dierent digits in the MNIST digits dataset

are more clearly captured with 1 (red) and 0 (dark red) at far corners of

the embedding space, and 4,7,9 (yellow, sea-green, and violet) and 3,5,8 (or-

ange, chartreuse, and blue) separated as distinct clumps of similar digits.

In the Fashion MNIST dataset the distinction between clothing (dark red,

yellow, orange, vermilion) and footwear (chartreuse, sea-green, and violet)

is made more clear. Finally, while both t-SNE and UMAP capture groups of

similar word vectors, the UMAP embedding arguably evidences a clearer

global structure among the various word clusters.

5.2 antitative Comparison of Multiple Algorithms

We compare UMAP, t-SNE, LargeVis, Laplacian Eigenmaps and PCA em-

beddings with respect to the performance of a k-nearest neighbor clas-

sier trained on the embedding space for a variety of datasets. e k-

nearest neighbor classier accuracy provides a clear quantitative measure

of how well the embedding has preserved the important local structure of

the dataset. By varying the hyper-parameter kused in the training we

can also consider how structure preservation varies under transition from

purely local to non-local, to more global structure. e embeddings used

for training the kNN classier are for those datasets that come with dened

training labels: PenDigits, COIL-20, Shule, MNIST, and Fashion-MNIST.

We divide the datasets into two classes: smaller datasets (PenDigits and

COIL-20), for which a smaller range of kvalues makes sense, and larger

datasets, for which much larger values of kare reasonable. For each of

the small datasets a stratied 10-fold cross-validation was used to derive

a set of 10 accuracy scores for each embedding. For the Shule dataset a

10-fold cross-validation was used due to constraints imposed by class sizes

and the stratied sampling. For MNIST and Fashion-MNIST a 20-fold cross

validation was used, producing 20 accuracy scores.

In Table 1 we present the average accuracy across the 10-folds for the

PenDigits and COIL-20 datasets. UMAP performs at least as well as t-SNE

and LargeVis (given the condence bounds on the accuracy) for kin the

range 10 to 40, but for larger kvalues of 80 and 160 UMAP has signicantly

higher accuracy on COIL-20, and shows evidence of higher accuracy on

PenDigits. Figure 5 provides swarm plots of the accuracy results across the

COIL-20 and PenDigits datasets.

30

In Table 2 we present the average cross validation accuracy for the Shut-

tle, MNIST and Fashion-MNIST datasets. UMAP performs at least as well

as t-SNE and LargeVis (given the condence bounds on the accuracy) for k

in the range 100 to 400 on the Shule and MNIST datasets (but notably un-

derperforms on the Fashion-MNIST dataset), but for larger kvalues of 800

and 3200 UMAP has signicantly higher accuracy on the Shule dataset,

and shows evidence of higher accuracy on MNIST. For kvalues of 1600 and

3200 UMAP establishes comparable performance on Fashion-MNIST. Fig-

ure 6 provides swarm plots of the accuracy results across the Shule and

MNIST and Fashion-MNIST datasets.

k t-SNE UMAP LargeVis Eigenmaps PCA

COIL-20

10 0.934 (±0.115) 0.921 (±0.075) 0.888 (±0.092) 0.629 (±0.153) 0.667 (±0.179)

20 0.901 (±0.133) 0.907 (±0.064) 0.870 (±0.125) 0.605 (±0.185) 0.663 (±0.196)

40 0.857 (±0.125) 0.904 (±0.056) 0.833 (±0.106) 0.578 (±0.159) 0.620 (±0.230)

80 0.789 (±0.118) 0.899 (±0.058) 0.803 (±0.100) 0.565 (±0.119) 0.531 (±0.294)

160 0.609 (±0.067) 0.803 (±0.138) 0.616 (±0.066) 0.446 (±0.110) 0.375 (±0.111)

PenDigits

10 0.977 (±0.033) 0.973 (±0.044) 0.966 (±0.053) 0.778 (±0.113) 0.622 (±0.092)

20 0.973 (±0.033) 0.976 (±0.035) 0.973 (±0.044) 0.778 (±0.116) 0.633 (±0.082)

40 0.956 (±0.064) 0.954 (±0.060) 0.959 (±0.066) 0.778 (±0.112) 0.636 (±0.078)

80 0.948 (±0.060) 0.951 (±0.072) 0.949 (±0.072) 0.767 (±0.111) 0.643 (±0.085)

160 0.949 (±0.065) 0.951 (±0.085) 0.921 (±0.085) 0.747 (±0.108) 0.629 (±0.107)

Table 1: kNN Classier accuracy for varying values of kover the embedding

spaces of COIL-20 and PenDigits datasets. Average accuracy scores are given

over a 10-fold cross-validation for each of PCA, Laplacian Eigenmaps, LargeVis,

t-SNE and UMAP.

As evidenced by this comparison UMAP provides largely comparable

perfomance in embedding quality to t-SNE and LargeVis at local scales, but

performs markedly beer than t-SNE or LargeVis at non-local scales. is

bears out the visual qualitative assessment provided in Subsection 5.1.

5.3 Embedding Stability

Since UMAP makes use of both stochastic approximate nearest neighbor

search, and stochastic gradient descent with negative sampling for opti-

mization, the resulting embedding is necessarily dierent from run to run,

and under sub-sampling of the data. is is potentially a concern for a

31

k t-SNE UMAP LargeVis Eigenmaps PCA

Shule

100 0.994 (±0.002) 0.993 (±0.002) 0.992 (±0.003) 0.962 (±0.004) 0.833 (±0.013)

200 0.992 (±0.002) 0.990 (±0.002) 0.987 (±0.003) 0.957 (±0.006) 0.821 (±0.007)

400 0.990 (±0.002) 0.988 (±0.002) 0.976 (±0.003) 0.949 (±0.006) 0.815 (±0.007)

800 0.969 (±0.005) 0.988 (±0.002) 0.957 (±0.004) 0.942 (±0.006) 0.804 (±0.003)

1600 0.927 (±0.005) 0.981 (±0.002) 0.904 (±0.007) 0.918 (±0.006) 0.792 (±0.003)

3200 0.828 (±0.004) 0.957 (±0.005) 0.850 (±0.008) 0.895 (±0.006) 0.786 (±0.001)

MNIST

100 0.967 (±0.015) 0.967 (±0.014) 0.962 (±0.015) 0.668 (±0.016) 0.462 (±0.023)

200 0.966 (±0.015) 0.967 (±0.014) 0.962 (±0.015) 0.667 (±0.016) 0.467 (±0.023)

400 0.964 (±0.015) 0.967 (±0.014) 0.961 (±0.015) 0.664 (±0.016) 0.468 (±0.024)

800 0.963 (±0.016) 0.967 (±0.014) 0.961 (±0.015) 0.660 (±0.017) 0.468 (±0.023)

1600 0.959 (±0.016) 0.966 (±0.014) 0.947 (±0.015) 0.651 (±0.014) 0.467 (±0.0233)

3200 0.946 (±0.017) 0.964 (±0.014) 0.920 (±0.017) 0.639 (±0.017) 0.459 (±0.022)

Fashion-MNIST

100 0.818 (±0.012) 0.790 (±0.013) 0.808 (±0.014) 0.631 (±0.010) 0.564 (±0.018)

200 0.810 (±0.013) 0.785 (±0.014) 0.805 (±0.013) 0.624 (±0.013) 0.565 (±0.016)

400 0.801 (±0.013) 0.780 (±0.013) 0.796 (±0.013) 0.612 (±0.011) 0.564 (±0.017)

800 0.784 (±0.011) 0.767 (±0.014) 0.771 (±0.014) 0.600 (±0.012) 0.560 (±0.017)

1600 0.754 (±0.011) 0.747 (±0.013) 0.742 (±0.013) 0.580 (±0.014) 0.550 (±0.017)

3200 0.727 (±0.011) 0.730 (±0.011) 0.726 (±0.012) 0.542 (±0.014) 0.533 (±0.017)

Table 2: kNN Classier accuracy for varying values of kover the embedding

spaces of Shule, MNIST and Fashion-MNIST datasets. Average accuracy scores

are given over a 10-fold or 20-fold cross-validation for each of PCA, Laplacian

Eigenmaps, LargeVis, t-SNE and UMAP.

32

Figure 5: kNN Classier accuracy for varying values of kover the embedding

spaces of COIL-20 and PenDigits datasets. Accuracy scores are given for each

fold of a 10-fold cross-validation for each of PCA, Laplacian Eigenmaps, LargeVis,

t-SNE and UMAP. We note that UMAP produces competitive accuracy scores to

t-SNE and LargeVis for most cases, and outperforms both t-SNE and LargeVis for

larger kvalues on COIL-20.

variety of uses cases, so establishing some measure of how stable UMAP

embeddings are, particularly under sub-sampling, is of interest. In this sub-

section we compare the stability under subsampling of UMAP, LargeVis and

t-SNE (the three stochastic dimension reduction techniques considered).

To measure the stability of an embedding we make use of the nor-

malized Procrustes distance to measure the distance between two poten-

tially comparable distributions. Given two datasets X={x1, . . . , xN}and

Y={y1, . . . , yN}such that xicorresponds to yi, we can dene the Pro-

custes distance between the datasets dP(X, Y )in the following manner.

Determine Y0={y10, . . . , yN0}the optimal translation, uniform scaling,

and rotation of Ythat minimizes the squared error PN

i=1(xi−yi0)2, and

dene

dP(X, Y ) = v

u

u

t

N

X

i=1

(xi−yi0)2.

Since any measure that makes use of distances in the embedding space is

potentially sensitive to the extent or scale of the embedding, we normal-

ize the data before computing the Procrustes distance by dividing by the

average norm of the embedded dataset. In Figure 7 we visualize the re-

sults of using Procrustes alignment of embedding of sub-samples for both

33

Figure 6: kNN Classier accuracy for varying values of kover the embedding

spaces of Shule, MNIST and Fashion-MNIST datasets. Accuracy scores are

given for each fold of a 10-fold cross-validation for Shule, and 20-fold cross-

validation for MNIST and Fashion-MNIST, for each of PCA, Laplacian Eigen-

maps, LargeVis, t-SNE and UMAP. UMAP performs beer than the other algo-

rithms for large k, particularly on the Shule dataset. For Fashion-MNIST UMAP

provides slightly poorer accuracy than t-SNE and LargeVis at small scales, but is

competitive at larger kvalues.

34

(a) UMAP (b) t-SNE

Figure 7: Procrustes based alignment of a 10% subsample (red) against the full

dataset (blue) for the ow cytometry dataset for both UMAP and t-SNE.

UMAP and t-SNE, demonstrating how Procrustes distance can measure the

stability of the overall structure of the embedding.

Given a measure of distance between dierent embeddings we can ex-

amine stability under sub-sampling by considering the normalized Pro-

crustes distance between the embedding of a sub-sample, and the corre-

sponding sub-sample of an embedding of the full dataset. As the size of

the sub-sample increases the average distance per point between the sub-

sampled embeddings should decrease, potentially toward some asymptote

of maximal agreement under repeated runs. Ideally this asymptotic value

would be zero error, but for stochastic embeddings such as UMAP and t-

SNE this is not achievable.

We performed an empirical comparison of algorithms with respect to

stability using the Flow Cytometry dataset due its large size, interesting

structure, and low ambient dimensionality (aiding runtime performance

for t-SNE). We note that for a dataset this large we found it necessary to

increase the default n_iter value for t-SNE from 1000 to 1500 to ensure bet-

ter convergence. While this had an impact on the runtime, it signicantly

improved the Procrustes distance results by providing more stable and con-

sistent embeddings. Figure 8 provides a comparison between UMAP and

t-SNE, demonstrating that UMAP has signifcantly more stable results than

35

t-SNE. In particular, aer sub-sampling on 5% of the million data points, the

per point error for UMAP was already below any value achieved by t-SNE.

5.4 Computational Performance Comparisons

Benchmarks against the real world datasets were performed on a Macbook

Pro with a 3.1 GHz Intel Core i7 and 8GB of RAM for Table 3, and on a

server with Intel Xeon E5-2697v4 processors and 512GB of RAM for the

large scale benchmarking in Subsections 5.4.1, 5.4.2, and 5.4.3.

For t-SNE we chose MulticoreTSNE [57], which we believe to be the

fastest extant implementation of Barnes-Hut t-SNE at this time, even when

run in single core mode. It should be noted that MulticoreTSNE is a heav-

ily optimized implementation wrien in C++ based on Van der Maaten’s

bhtsne [58] code.

As a fast alternative approach to t-SNE we also consider the FIt-SNE

algorithm [37]. We used the reference implementation [36], which, like

MulticoreTNSE is an optimized C++ implementation. We also note that

FIt-SNE makes use of multiple cores.

LargeVis [54] was benchmarked using the reference implementation

[53]. It was run with default parameters including use of 8 threads on the 4-

core machine. e only exceptions were small datasets where we explicitly

set the -samples parameter to n_samples/100 as per the recommended

values in the documentation of the reference implementation.

e Isomap [55] and Laplacian Eigenmaps [7] implementations in scikit-

learn [10] were used. We suspect the Laplacian eigenmaps implementation

may not be well optimized for large datasets but did not nd a beer per-

forming implementation that provided comparable quality results. Isomap

failed to complete for the Shule, Fashion-MNIST, MNIST and Google-

News datasets, while Laplacian Eigenmaps failed to run for the Google-

News dataset.

To allow a broader range of algorithms to run some of the datasets

where subsampled or had their dimension reduced by PCA. e Flow Cy-

tometry dataset was benchmarked on a 10% sample and the GoogleNews

was subsampled down to 200,000 data points. Finally, the Mouse scRNA

dataset was reduced to 1,000 dimensions via PCA.

Timing were performed for the COIL20 [43], COIL100 [44], Shule [35],

MNIST [32], Fashion-MNIST [63], and GoogleNews [41] datasets. Results

can be seen in Table 3. UMAP consistently performs faster than any of

the other algorithms aside from on the very small Pendigits dataset, where

Laplacian Eigenmaps and Isomap have a small edge.

36

Figure 8: Comparison of average Procustes distance per point for t-SNE, LargeVis

and UMAP over a variety of sizes of subsamples from the full Flow Cytometry

dataset. UMAP sub-sample embeddings are very close to the full embedding even

for subsamples of 5% of the full dataset, outperforming the results of t-SNE and

LargeVis even when they use the full Flow Cytometry dataset.

37

UMAP FIt-SNE t-SNE LargeVis Eigenmaps Isomap

Pen Digits 9s 48s 17s 20s 2s 2s

(1797x64)

COIL20 12s 75s 22s 82s 47s 58s

(1440x16384)

COIL100 85s 2681s 810s 3197s 3268s 3210s

(7200x49152)

scRNA 28s 131s 258s 377s 470s 923s

(21086x1000)

Shuttle 94s 108s 714s 615s 133s –

(58000x9)

MNIST 87s 292s 1450s 1298s 40709s –

(70000x784)

F-MNIST 65s 278s 934s 1173s 6356s –

(70000x784)

Flow 102s 164s 1135s 1127s 30654s –

(100000x17)

Google News 361s 652s 16906s 5392s – –

(200000x300)

Table 3: Runtime of several dimension reduction algorithms on various datasets.

To allow a broader range of algorithms to run some of the datasets where sub-

sampled or had their dimension reduced by PCA. e Flow Cytometry dataset

was benchmarked on a 10% sample and the GoogleNews was subsampled down

to 200,000 data points. Finally, the Mouse scRNA dataset was reduced to 1,000

dimensions via PCA. e fastest runtime for each dataset has been bolded.

38

5.4.1 Scaling with Embedding Dimension

UMAP is signicantly more performant than t-SNE4when embedding into

dimensions larger than 2. is is particularly important when the intention

is to use the low dimensional representation for further machine learning

tasks such as clustering or anomaly detection rather than merely for visu-

alization. e computation performance of UMAP is far more ecient than

t-SNE, even for very small embedding dimensions of 6 or 8 (see Figure 9).

is is largely due to the fact that UMAP does not require global normali-

sation (since it represents data as a fuzzy topological structure rather than

as a probability distribution). is allows the algorithm to work without

the need for space trees —such as the quad-trees and oct-trees that t-SNE

uses [58]—. Such space trees scale exponentially in dimension, resulting

in t-SNE’s relatively poor scaling with respect to embedding dimension.

By contrast, we see that UMAP consistently scales well in embedding di-

mension, making the algorithm practical for a wider range of applications

beyond visualization.

5.4.2 Scaling with Ambient Dimension

rough a combination of the local-connectivity constraint and the approx-

imate nearest neighbor search, UMAP can perform eective dimension re-

duction even for very high dimensional data (see Figure 13 for an example

of UMAP operating directly on 1.8 million dimensional data). is stands in

contrast to many other manifold learning techniques, including t-SNE and

LargeVis, for which it is generally recommended to reduce the dimension

with PCA before applying these techniques (see [59] for example).

To compare runtime performance scaling with respect to the ambient

dimension of the data we chose to use the Mouse scRNA dataset, which

is high dimensional, but is also amenable to the use of PCA to reduce the

dimension of the data as a pre-processing step without losing too much

of the important structure5. We compare the performance of UMAP, FIt-

SNE, MulticoreTSNE, and LargeVis on PCA reductions of the Mouse scRNA

dataset to varying dimensionalities, and on the original dataset, in Figure

10.

While all the implementations tested show a signicant increase in run-

time with increasing dimension, UMAP is dramatically more ecient for

4Comparisons were performed against MulticoreTSNE as the current implementation of FIt-

SNE does not support embedding into any dimension larger than 2.

5In contrast to COIL100, on which PCA destroys much of the manifold structure

39

(a) A comparison of run time for UMAP,

t-SNE and LargeVis with respect to em-

bedding dimension on the Pen digits

dataset. We see that t-SNE scales worse

than exponentially while UMAP and

LargeVis scale linearly with a slope so

slight to be undetectable at this scale.

(b) Detail of scaling for embedding di-

mension of six or less. We can see that

UMAP and LargeVis are essentially at.

In practice they appear to scale linearly,

but the slope is essentially undetectable

at this scale.

Figure 9: Scaling performance with respect to embedding dimension of UMAP,

t-SNE and LargeVis on the Pen digits dataset.

40

Figure 10: Runtime performance scaling of UMAP, t-SNE, FIt-SNE and Largevis

with respect to the ambient dimension of the data. As the ambient dimension

increases beyond a few thousand dimensions the computational cost of t-SNE,

FIt-SNE, and LargeVis all increase dramatically, while UMAP continues to per-

form well into the tens-of-thousands of dimensions.

41

large ambient dimensions, easily scaling to run on the original unreduced

dataset. e ability to run manifold learning on raw source data, rather than

dimension reduced data that may have lost important manifold structure in

the pre-processing, is a signicant advantage. is advantage comes from

the local connectivity assumption which ensures good topological repre-

sentation of high dimensional data, particularly with smaller numbers of

near neighbors, and the eciency of the NN-Descent algorithm for approx-

imate nearest neighbor search even in high dimensions.

Since UMAP scales well with ambient dimension the python implemen-

tation also supports input in sparse matrix format, allowing scaling to ex-

tremely high dimensional data, such as the integer data shown in Figures

13 and 14.

5.4.3 Scaling with the Number of Samples

For dataset size performance comparisons we chose to compare UMAP with

FIt-SNE [37], a version of t-SNE that uses approximate nearest neighbor

search and a Fourier interpolation optimisation approach; MulticoreTSNE

[57], which we believe to be the fastest extant implementation of Barnes-

Hut t-SNE; and LargeVis [54]. It should be noted that FIt-SNE, MulticoreT-

SNE, and LargeVis are all heavily optimized implementations wrien in

C++. In contrast our UMAP implementation was wrien in Python — mak-

ing use of the numba [31] library for performance. MulticoreTSNE and

LargeVis were run in single threaded mode to make fair comparisons to

our single threaded UMAP implementation.

We benchmarked all four implementations using subsamples of the Google-

News dataset. e results can be seen in Figure 11. is demonstrates that

UMAP has superior scaling performance in comparison to Barnes-Hut t-

SNE, even when Barnes-Hut t-SNE is given multiple cores. Asymptotic

scaling of UMAP is comparable to that of FIt-SNE (and LargeVis). On this

dataset UMAP demonstrated somewhat faster absolute performance com-

pared to FIt-SNE, and was dramatically faster than LargeVis.

e UMAP embedding of the full GoogleNews dataset of 3 million word

vectors, as seen in Figure 12, was completed in around 200 minutes, as com-

pared with several days required for MulticoreTSNE, even using multiple

cores.

To scale even further we were inspired by the work of John Williamson

on embedding integers [61], as represented by (sparse) binary vectors of

their prime divisibility. is allows the generation of arbitrarily large, ex-

tremely high dimension datasets that still have meaningful structure to be

42

Figure 11: Runtime performance scaling of t-SNE and UMAP on various sized

sub-samples of the full Google News dataset. e lower t-SNE line is the wall

clock runtime for Multicore t-SNE using 8 cores.

43

Figure 12: Visualization of the full 3 million word vectors from the GoogleNews

dataset as embedded by UMAP.

44

explored. In Figures 13 and 14 we show an embedding of 30,000,000 data

samples from an ambient space of approximately 1.8 million dimensions.

is computation took approximately 2 weeks on a large memory SMP.

Note that despite the high ambient dimension, and vast amount of data,

UMAP is still able to nd and display interesting structure. In Figure 15 we

show local regions of the embedding, demonstrating the ne detail struc-

ture that was captured.

6 Weaknesses

While we believe UMAP to be a very eective algorithm for both visualiza-

tion and dimension reduction, most algorithms must make trade-os and

UMAP is no exception. In this section we will briey discuss those areas or

use cases where UMAP is less eective, and suggest potential alternatives.

For a number of use cases the interpretability of the reduced dimension

results is of critical importance. Similarly to most non-linear dimension re-

duction techniques (including t-SNE and Isomap), UMAP lacks the strong

interpretability of Principal Component Analysis (PCA) and related tech-

niques such a Non-Negative Matrix Factorization (NMF). In particular the

dimensions of the UMAP embedding space have no specic meaning, un-

like PCA where the dimensions are the directions of greatest variance in

the source data. Furthermore, since UMAP is based on the distance between

observations rather than the source features, it does not have an equivalent

of factor loadings that linear techniques such as PCA, or Factor Analysis

can provide. If strong interpretability is critical we therefore recommend

linear techniques such as PCA, NMF or pLSA.

One of the core assumptions of UMAP is that there exists manifold

structure in the data. Because of this UMAP can tend to nd manifold

structure within the noise of a dataset – similar to the way the human mind

nds structured constellations among the stars. As more data is sampled

the amount of structure evident from noise will tend to decrease and UMAP

becomes more robust, however care must be taken with small sample sizes

of noisy data, or data with only large scale manifold structure. Detecting

when a spurious embedding has occurred is a topic of further research.

UMAP is derived from the axiom that local distance is of more im-

portance than long range distances (similar to techniques like t-SNE and

LargeVis). UMAP therefore concerns itself primarily with accurately rep-

resenting local structure. While we believe that UMAP can capture more

global structure than these other techniques, it remains true that if global

45

Figure 13: Visualization of 30,000,000 integers as represented by binary vectors

of prime divisibility, colored by density of points.

46

Figure 14: Visualization of 30,000,000 integers as represented by binary vectors

of prime divisibility, colored by integer value of the point (larger values are green

or yellow, smaller values are blue or purple).

47

(a) Upper right spiral (b) Lower right spiral and starbursts

(c) Central cloud

Figure 15: Zooming in on various regions of the integer embedding reveals fur-

ther layers of ne structure have been preserved.

48

structure is of primary interest then UMAP may not be the best choice for

dimension reduction. Multi-dimensional scaling specically seeks to pre-

serve the full distance matrix of the data, and as such is a good candidate

when all scales of structure are of equal importance. PHATE [42] is a good

example of a hybrid approach that begins with local structure information

and makes use of MDS to aempt to preserve long scale distances as well. It

should be noted that these techniques are more computationally intensive

and thus rely on landmarking approaches for scalability.

It should also be noted that a signicant contributor to UMAP’s relative

global structure preservation is derived from the Laplacian Eigenmaps ini-

tialization (which, in turn, followed from the theoretical foundations). is

was noted in, for example, [29]. e authors of that paper demonstrate

that t-SNE, with similar initialization, can perform equivalently to UMAP

in a particular measure of global structure preservation. However, the ob-

jective function derived for UMAP (cross-entropy) is signicantly dierent

from that of t-SNE (KL-divergence), in how it penalizes failures to preserve

non-local and global structure, and is also a signicant contributor6.

It is worth noting that, in combining the local simplicial set structures,

pure nearest neighbor structure in the high dimensional space is not ex-

plicitly preserved. In particular it introduces so called ”reverse-nearest-

neighbors” into the classical knn-graph. is, combined with the fact that

UMAP is preserving topology rather than pure metric structures, mean that

UMAP will not perform as well as some methods on quality measures based

on metric structure preservation – particularly methods, such as MDS –

which are explicitly designed to optimize metric structure preservation.

UMAP aempts to discover a manifold on which your data is uniformly

distributed. If you have strong condence in the ambient distances of your

data you should make use of a technique that explicitly aempts to preserve

these distances. For example if your data consisted of a very loose structure

in one area of your ambient space and a very dense structure in another

region region UMAP would aempt to put these local areas on an even

footing.

Finally, to improve the computational eciency of the algorithm anum-

ber of approximations are made. is can have an impact on the results

of UMAP for small (less than 500 samples) dataset sizes. In particular the

use of approximate nearest neighbor algorithms, and the negative sampling

used in optimization, can result in suboptimal embeddings. For this reason

6e authors would like to thank Nikolay Oskolkov for his article (tSNE vs. UMAP: Global

Structure) which does an excellent job of highlighting these aspects from an empirical and the-

oretical basis.

49

we encourage users to take care with particularly small datasets. A slower

but exact implementation of UMAP for small datasets is a future project.

7 Future Work

Having established both relevant mathematical theory and a concrete im-

plementation, there still remains signicant scope for future developments

of UMAP.

A comprehensive empirical study which examines the impact of the

various algorithmic components, choices, and hyper-parameters of the al-

gorithm would be benecial. While the structure and choices of the algo-

rithm presented were derived from our foundational mathematical frame-

work, examining the impacts that these choices have on practical results

would be enlightening and a signicant contribution to the literature.

As noted in the weaknesses section there is a great deal of uncertainty

surrounding the preservation of global structure among the eld of man-

ifold learning algorithms. In particular this is hampered by the lack clear

objective measures, or even denitions, of global structure preservation.

While some metrics exist, they are not comprehensive, and are oen spe-

cic to various downstream tasks. A systematic study of both metrics of

non-local and global structure preservation, and performance of various

manifold learning algorithms with respect to them, would be of great ben-

et. We believe this would aid in beer understanding UMAP’s success in

various downstream tasks.

Making use of the fuzzy simplicial set representation of data UMAP can

potentially be extended to support (semi-)supervised dimension reduction,

and dimension reduction for datasets with heterogeneous data types. Each

data type (or prediction variables in the supervised case) can be seen as an

alternative view of the underlying structure, each with a dierent associ-

ated metric – for example categorical data may use Jaccard or Dice distance,

while ordinal data might use Manhaan distance. Each view and metric can

be used to independently generate fuzzy simplicial sets, which can then be

intersected together to create a single fuzzy simplicial set for embedding.

Extending UMAP to work with mixed data types would vastly increase the

range of datasets to which it can be applied. Use cases for (semi-)supervised

dimension reduction include semi-supervised clustering, and interactive la-

belling tools.

e computational framework established for UMAP allows for the po-

tential development of techniques to add new unseen data points into an

50

existing embedding, and to generate high dimensional representations of

arbitrary points in the embedded space. Furthermore, the combination of

supervision and the addition of new samples to an existing embedding pro-

vides avenues for metric learning. e addition of new samples to an ex-

isting embedding would allow UMAP to be used as a feature engineering

tool as part of a general machine learning pipeline for either clustering or

classication tasks. Pulling points back to the original high dimensional

space from the embedded space would potentially allow UMAP to be used

as a generative model similar to some use cases for autoencoders. Finally,

there are many use cases for metric learning; see [64] or [8] for example.

ere also remains signicant scope to develop techniques to both de-

tect and mitigate against potentially spurious embeddings, particularly for

small data cases. e addition of such techniques would make UMAP far

more robust as a tool for exploratory data analysis, a common use case

when reducing to two dimensions for visualization purposes.

Experimental versions of some of this work are already available in the

referenced implementations.

8 Conclusions

We have developed a general purpose dimension reduction technique that

is grounded in strong mathematical foundations. e algorithm imple-

menting this technique is demonstrably faster than t-SNE and provides

beer scaling. is allows us to generate high quality embeddings of larger

data sets than had previously been aainable. e use and eectiveness

of UMAP in various scientic elds demonstrates the strength of the algo-

rithm.

Acknowledgements e authors would like to thank Colin Weir, Rick

Jardine, Brendan Fong, David Spivak and Dmitry Kobak for discussion and

useful commentary on various dras of this paper.

A Proof of Lemma 1

Lemma 1. Let (M, g)be a Riemannian manifold in an ambient Rn, and let

p∈Mbe a point. If gis locally constant about pin an open neighbourhood

Usuch that gis a constant diagonal matrix in ambient coordinates, then in a

ball B⊆Ucentered at pwith volume πn/2

Γ(n/2+1) with respect to g, the geodesic

51

distance from pto any point q∈Bis 1

rdRn(p, q), where ris the radius of the

ball in the ambient space and dRnis the existing metric on the ambient space.

Proof. Let x1, . . . , xnbe the coordinate system for the ambient space. A

ball Bin Munder Riemannian metric ghas volume given by

ZBpdet(g)dx1∧ · · · ∧ dxn.

If Bis contained in U, then gis constant in Band hence pdet(g)is con-

stant and can be brought outside the integral. us, the volume of Bis

pdet(g)ZB

dx1∧... ∧dxn=pdet(g)πn/2rn

Γ(n/2 + 1),

where ris the radius of the ball in the ambient Rn. If we x the volume of

the ball to be πn/2

Γ(n/2+1) we arrive at the requirement that

det(g) = 1

r2n.

Now, since gis assumed to be diagonal with constant entries we can solve

for gitself as

gij =

1

r2if i=j,

0otherwise

.(2)

e geodesic distance on Munder gfrom pto q(where p, q ∈B) is dened

as

inf

c∈CZb

apg( ˙c(t),˙c(t))dt,

where Cis the class of smooth curves con Msuch that c(a) = pand

c(b) = q, and ˙cdenotes the rst derivative of con M. Given that gis as

dened in (2) we see that this can be simplied to

1

rinf

c∈CZb

a

hp˙c(t),˙c(t)idt

=1

rinf

c∈CZb

a

hk ˙c(t),˙c(t)kdt

=1

rdRn(p, q).

(3)

52

B Proof that FinReal and FinSing are adjoint

eorem 2. e functors FinReal :Fin-sFuzz →FinEPMet and FinSing :

FinEPMet →Fin-sFuzz form an adjunction with FinReal the le adjoint

and FinSing the right adjoint.

Proof. e adjunction is evident by construction, but can be made more

explicit as follows. Dene a functor F:∆×I→FinEPMet by

F([n],[0, a)) = ({x1, x2, . . . , xn}, da),

where

da(xi, xj) =

−log(a)if i6=j,

0otherwise .

Now FinSing can be dened in terms of Fas

FinSing(Y) : ([n],[0, a)) 7→ homFinEPMet(F([n],[0, a)), Y ).

where the face maps diare given by pre-composition with F di, and sim-

ilarly for degeneracy maps, at any given value of a. Furthermore post-

composition with Flevel-wise for each adenes maps of fuzzy simplicial

sets making FinSing a functor.

We now construct FinReal as the le Kan extension of Falong the

Yoneda embedding:

Fin-sFuzz

FinReal

((

∆×I

+

y

88

F

//FinEPMet

Explicitly this results in a denition of FinReal at a fuzzy simplicial set X

as a colimit:

FinReal(X) = colim

y([n],[0,a))→XF([n]).

Further, it follows from the Yoneda lemma that FinReal(∆n

<a)∼

=F([n],[0, a)),

and hence this denition as a le Kan extension agrees with Denition 7,

and the denition of FinSing above agrees with that of Denition 8. To see

that FinReal and FinSing are adjoint we note that

homFin-sFuzz(∆n

<a,FinSing(Y)) ∼

=FinSing(Y)n

<a

= homFinEPMet(F([n],[0, a)), Y )

∼

=homFinEPMet(FinReal(∆n

<a), Y )).

(4)

53

e rst isomorphism follows from the Yoneda lemma, the equality is by

construction, and the nal isomorphism follows by another application of

the Yoneda lemma. Since every simplicial set can be canonically expressed

as a colimit of standard simplices and FinReal commutes with colimits (as

it was dened via a colimit formula), it follows that FinReal is completely

determined by its image on standard simplices. As a result the isomor-

phism of equation (4) extends to the required isomorphism demonstrating

the adjunction.

C From t-SNE to UMAP

As an aid to implementation of UMAP and to illuminate the algorithmic

similarities with t-SNE and LargeVis, here we review the main equations

used in those methods, and then present the equivalent UMAP expressions

in a notation which may be more familiar to users of those other methods.

In what follows we are concerned with dening similarities between

two objects iand jin the high dimensional input space Xand low di-

mensional embedded space Y. ese are normalized and symmetrized in

various ways. In a typical implementation, these pair-wise quantities are

stored and manipulated as (potentially sparse) matrices. antities with

the subscript ij are symmetric, i.e. vij =vji. Extending the conditional

probability notation used in t-SNE, j|iindicates an asymmetric similarity,

i.e. vj|i6=vi|j.

t-SNE denes input probabilities in three stages. First, for each pair of

points, iand j, in X, a pair-wise similarity, vij, is calculated, Gaussian with

respect to the Euclidean distance between xiand xj:

vj|i= exp(− kxi−xjk2

2/2σ2

i)(5)

where σ2

iis the variance of the Gaussian.

Second, the similarities are converted into Nconditional probability

distributions by normalization:

pj|i=vj|i

Pk6=ivk|i

(6)

σiis chosen by searching for a value such that the perplexity of the proba-

bility distribution p·|imatches a user-specied value.

ird, these probability distributions are symmetrized and then further

normalized over the entire matrix of values to give a joint probability dis-

tribution:

54

pij =pj|i+pi|j

2N(7)

We note that this is a heuristic denition and not in accordance with stan-

dard relationship between conditional and joint probabilities that would be

expected under probability semantics usually used to describe t-SNE.

Similarities between pairs of points in the output space Yare dened

using a Student t-distribution with one degree of freedom on the squared

Euclidean distance:

wij =1 + kyi−yjk2

2−1(8)

followed by the matrix-wise normalization, to form qij:

qij =wij

Pk6=lwkl

(9)

e t-SNE cost is the Kullback-Leibler divergence between the two proba-

bility distributions:

Ct−SN E =X

i6=j

pij log pij

qij

(10)

this can be expanded into constant and non-constant contributions:

Ct−SN E =X

i6=j

pij log pij −pij log qij (11)

Because both pij and qij require calculations over all pairs of points, im-

proving the eciency of t-SNE algorithms has involved separate strategies

for approximating these quantities. Similarities in the high dimensions are

eectively zero outside of the nearest neighbors of each point due to the

calibration of the pj|ivalues to reproduce a desired perplexity. erefore an

approximation used in Barnes-Hut t-SNE is to only calculate vj|ifor nnear-

est neighbors of i, where nis a multiple of the user-selected perplexity and

to assume vj|i= 0 for all other j. Because the low dimensional coordinates

change with each iteration, a dierent approach is used to approximate

qij. In Barnes-Hut t-SNE and related methods this usually involves group-

ing together points whose contributions can be approximated as a single

point.

A further heuristic algorithm optimization technique employed by t-

SNE implementations is the use of early exaggeration where, for some num-

ber of initial iterations, the pij are multiplied by some constant greater than

55

1.0 (usually 12.0). In theoretical analyses of t-SNE such as [38] results are

obtained only under an early exaggeration regimen with either large con-

stant (of order of the number of samples), or in the limit of innite exagger-

ation. Further papers such as [37], and [28], suggest the option of using ex-

aggeration for all iterations rather than just early ones, and demonstrate the

utility of this. e eectiveness of these analyses and practical approaches

suggests that KL-divergence as a measure between probability distributions

is not what makes the t-SNE algorithm work, since, under exaggeration, the

pij are manifestly not a probability distribution. is is another example

of the probability semantics used to describe t-SNE are primarily descrip-

tive rather than foundational. None the less, t-SNE is highly eective and

clearly produces useful results on a very wide variety of tasks.

LargeVis uses a similar approach to Barnes-Hut t-SNE when approxi-

mating pij, but further improves eciency by only requiring approximate

nearest neighbors for each point. For the low dimensional coordinates,

it abandons normalization of wij entirely. Rather than use the Kullback-

Leibler divergence, it optimizes a likelihood function, and hence is maxi-

mized, not minimized:

CLV =X

i6=j

pij log wij +γX

i6=j

log (1 −wij )(12)

pij and wij are dened as in Barnes-Hut t-SNE (apart from the use of

approximate nearest neighbors for pij, and the fact that, in implementation,

LargeVis does not normalize the pij by N) and γis a user-chosen positive

constant which weights the strength of the the repulsive contributions (sec-

ond term) relative to the aractive contribution (rst term). Note also that

the rst term resembles the optimizable part of the Kullback-Leibler diver-

gence but using wij instead of qij. Abandoning calculation of qij is a crucial

change, because the LargeVis cost function is amenable to optimization via

stochastic gradient descent.

Ignoring specic denitions of vij and wij , the UMAP cost function,

the cross entropy, is:

CUMAP =X

i6=j

vij log vij

wij + (1 −vij) log 1−vij

1−wij (13)

Like the Kullback-Leibler divergence, this can be arranged into two con-

stant contributions (those containing vij only) and two optimizable contri-

butions (containing wij):

56

CUMAP =X

i6=j

vij log vij + (1 −vij) log (1 −vij)

−vij log wij −(1 −vij) log (1 −wij)

(14)

Ignoring the two constant terms, the UMAP cost function has a very

similar form to that of LargeVis, but without a γterm to weight the re-

pulsive component of the cost function, and without requiring matrix-wise

normalization in the high dimensional space. e cost function for UMAP

can therefore be optimized (in this case, minimized) with stochastic gradi-

ent descent in the same way as LargeVis.

Although the above discussion places UMAP in the same family of meth-

ods as t-SNE and LargeVis, it does not use the same denitions for vij and

wij. Using the notation established above, we now provide the equivalent

expressions for the UMAP similarities. In the high dimensional space, the

similarities vj|iare the local fuzzy simplicial set memberships, based on the

smooth nearest neighbors distances:

vj|i= exp[(−d(xi, xj)−ρi)/σi](15)

As with LargeVis, vj|iis calculated only for napproximate nearest neigh-

bors and vj|i= 0 for all other j.d(xi, xj)is the distance between xiand

xj, which UMAP does not require to be Euclidean. ρiis the distance to the

nearest neighbor of i.σiis the normalizing factor, which is chosen by Al-

gorithm 3 and plays a similar role to the perplexity-based calibration of σi

in t-SNE. Calculation of vj|iwith Equation 15 corresponds to Algorithm 2.

Symmetrization is carried out by fuzzy set union using the probabilistic

t-conorm and can be expressed as:

vij =vj|i+vi|j−vj|ivi|j(16)

Equation 16 corresponds to forming top-rep in Algorithm 1. Unlike t-SNE,

further normalization is not carried out.

e low dimensional similarities are given by:

wij =1 + akyi−yjk2b

2−1(17)

where aand bare user-dened positive values. e procedure for nding

them is given in Denition 11. Use of this procedure with the default values

in the UMAP implementation results in a≈1.929 and b≈0.7915. Seing

a= 1 and b= 1 results in the Student t-distribution used in t-SNE.

57

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