Zigzag Persistent Homology in Matrix Multiplication Time

Abstract

We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two $n\times n$ matrices in $M(n)$ time, our algorithm runs in $O(M(n) log n)$ time if $M(n) = O(n^2)$, and $O(M(n))$ time otherwise, for a sequence of n additions and deletions. In particular, the running time is $O(n^2.376)$, by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes $O(n^3)$ time in the worst case.

Full text

Page 1

apport ?
?
de recherche?
SN 0249-6399
ISRN INRIA/RR--7393--FR+ENG
Thème COM
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Zigzag Persistent Homology in Matrix Multiplication
Time
Nikola Milosavljevi´ c — Dmitriy Morozov — Primož Škraba
N° 7393
September 2010
inria-00520171, version 1 - 22 Sep 2010

Page 2

inria-00520171, version 1 - 22 Sep 2010

Page 3

Centre de recherche INRIA Saclay – Île-de-France
Parc Orsay Université
4, rue Jacques Monod, 91893 ORSAY Cedex
Téléphone : +33 1 72 92 59 00
Zigzag Persistent Homology in Matrix Multiplication Time
Nikola Milosavljevi´ c∗, Dmitriy Morozov†, Primoˇ zˇSkraba‡
Th` eme COM — Syst` emes communicants
´Equipe-Projet Geometrica
Rapport de recherche n° 7393 — September 2010 — 18 pages
Abstract: We present a new algorithm for computing zigzag persistent homology, an algebraic
structure which encodes changes to homology groups of a simplicial complex over a sequence of
simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n
matrices in M(n) time, our algorithm runs in O(M(n)logn) time if M(n) = O(n2), and O(M(n))
time otherwise, for a sequence of n additions and deletions. In particular, the running time is
O(n2.376), by result of Coppersmith and Winograd. The fastest previously known algorithm for
this problem takes O(n3) time in the worst case.
Key-words:Zig-zag persistence, fast matrix multiplication, complexity
∗nikolam@mpi-inf.mpg.de
†dmitriy@mrzv.org
‡primoz.skraba@inria.fr
inria-00520171, version 1 - 22 Sep 2010

Page 4

Calcul de l’homologie persistante des zigzags par
multiplication rapide de matrices
R´ esum´ e :
structure algbrique qui code les changements survenant dans l’homologie d’un complexe simplicial
lors d’une squence d’insertions et de suppressions de simplexes. Sous l’hypothse qu’il existe des
algorithmes capables de multiplier deux matrices n × n en temps M(n), notre algorithme tourne
en temps O(M(n))logn si M(n) = O(n2) et O(M(n)) sinon, pour une squence de n additions
et suppressions. En particulier, le temps de calcul est O(n2.376) par le rsultat de Coppersmith et
Winograd. L’algorithme le plus rapide connu jusqu’ prsent pour ce problme prenait un temps O(n3)
dans le cas le pire.
Nous prsentons un algorithme pour calculer l’homologie persistante des zigzags, une
Mots-cl´ es : L’homologie persistante des zigzags, multiplication rapide de matrices, complexiti´ e
inria-00520171, version 1 - 22 Sep 2010

Page 5

Zigzag Persistent Homology in Matrix Multiplication Time3
1Introduction
Motivation.
tant tool in such wide-ranging domains as computational biology, geometric processing, machine
learning, scientific visualization, and sensor networks. Its success rests on two pillars: a solid
theoretical foundation [2], including the celebrated stability result [3], and the availability of fast
algorithms [4].
A more recent development is zigzag persistence [5], which is a generalization of ordinary per-
sistence built on algebraic insights. Together with an efficient algorithm in the homological setting,
zigzag persistence has already resolved open questions in the theoretical study of persistence [6].
In addition to the algebraic generality, it brings an appealing property for algorithm design: one is
not constrained to study growing families of spaces as in ordinary persistence, but instead is free
to choose whether to grow or shrink the space in question on demand. This flexibility has led to
a technique for computing ordinary persistent homology of a real-valued function using space that
depends only on the size of the largest levelset, rather than the entire domain [6].
Despite the growing interest in persistence, the complexity of its computation is not well under-
stood. Both ordinary and zigzag persistence have algorithms that are cubic in the worst case, but
little is known about their optimality. In this paper, we present an algorithm to compute zigzag
persistent homology (and therefore also ordinary persistent homology) that, in the worst case, runs
in the time it takes to multiply two matrices (unless that time is O(n2), in which case our algorithm
is O(n2logn)).
Since its introduction a decade ago, persistent homology [1] has become an impor-
Related Work.
some time [7, 8]. The sequential algorithm has the worst case bound of O(n3), which has been the
bound for most persistence computations. The computational complexity equivalence of Gaussian
elimination and matrix multiplication [9] was one of the factors in motivating this work. However,
most of the past work on speeding up homology computations has been based on combinatorial
techniques rather than matrix computations.
The first sub-cubic algorithm incrementally computes the Betti numbers of subcomplexes of
triangulations of the three sphere S3[10]. The running time is nα(n), where n is the number
of simplices and α(·) is the inverse Ackermann function. A different approach was given in [11]
to computing Betti numbers using combinatorial Laplacians. It uses the power method on the
Laplacian matrix, hence computing it via matrix-vector multiplication. However, the number of
multiplications depends on the eigenvalues of the matrix and hence is not easily bounded.
The introduction of persistent homology [1, 2], zigzag persistence [5, 6], and other variants [12,
13] came with algorithms based on the sequential Gaussian elimination and so had running time
of O(n3). In [14], an example complex is given where persistence takes cubic time. However,
experimental results show close to linear behavior, most often attributed to the sparsity of the
involved matrices.
In the literature, most work to speed up homology computation has had a different flavor, with
efforts being made to reduce the size of the input complex using combinatorial operations which
preserve the homology [15], most often simplicial collapses. In certain cases, notably 2-manifolds,
it is possible to create an optimal order of collapses to obtain the homology [16]. In the end of this
procedure we end up only with critical cells, allowing us to simply read off the homology. However,
it was also shown that in the general case, there may not exist a way to collapse the complex to the
minimal number of cells [17]. Furthermore, it was shown to be NP-hard to find an order which will
The computation of homology through Gaussian elimination has been known for
RR n° 7393
inria-00520171, version 1 - 22 Sep 2010