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SIAM J. MATRIX ANAL. APPL.©2023 Society for Industrial and Applied Mathematics
Vol. 44, No. 1, pp. 27-52
PARTIAL IDENTIFIABILITY FOR NONNEGATIVE MATRIX
FACTORIZATION∗
NICOLAS GILLIS†AND R´
OBERT RAJK ´
O‡
Abstract. Given a nonnegative matrix factorization, R, and a factorization rank, r, exact
nonnegative matrix factorization (exact NMF) decomposes Ras the product of two nonnegative
matrices, Cand Swith rcolumns, such as R=CS. A central research topic in the literature is the
conditions under which such a decomposition is unique/identifiable up to trivial ambiguities. In this
paper, we focus on partial identifiability, that is, the uniqueness of a subset of columns of Cand S.
We start our investigations with the data-based uniqueness (DBU) theorem from the chemometrics
literature. The DBU theorem analyzes all feasible solutions of exact NMF and relies on sparsity
conditions on Cand S. We provide a mathematically rigorous theorem of a recently published
restricted version of the DBU theorem, relying only on simple sparsity and algebraic conditions: it
applies to a particular solution of exact NMF (as opposed to all feasible solutions) and allows us
to guarantee the partial uniqueness of a single column of Cor S. Second, based on a geometric
interpretation of the restricted DBU theorem, we obtain a new partial identifiability result. This
geometric interpretation also leads us to another partial identifiability result in the case r= 3. Third,
we show how partial identifiability results can be used sequentially to guarantee the identifiability of
more columns of Cand S. We illustrate these results on several examples, including one from the
chemometrics literature.
Key words. nonnegative matrix factorization, uniqueness, identifiability, multivariate curve
resolution, window factor analysis, self-modeling curve resolution
MSC codes. 15A23
DOI. 10.1137/22M1507553
1. Introduction. Given a nonnegative matrix R∈Rm×n
+and a factorization
rank r, nonnegative matrix factorization (NMF) requires computing two nonnega-
tive matrices, C∈Rm×r
+and S∈Rn×r
+, such that CS≈R. NMF has become a
standard technique in unsupervised data analysis and has found numerous applica-
tions, e.g., in hyperspectral imaging, audio source separation, topic modeling, and
community detection, to cite a few; see, e.g., the books [8, 12] and the references
therein. An application where NMF has been particularly popular is multivariate
curve resolution (MCR) and self-modeling curve resolution, where the input matrix
Rrepresents the total response values from some chemical measurements of mixed
samples. An example is when we consider the evolution of the spectral profile of
a chemical reaction over time. More precisely, the ith row of Ris the cumulative
spectral content of the chemical reaction at the ith time step. An NMF of R, with
R(i, :) ≈C(i, :)Sfor all i, provides the spectral signature of the chemical compounds
in C, along with their proportion in the reaction over time in S. In general, the
∗Received by the editors July 6, 2022; accepted for publication (in revised form) October 12,
2022; published electronically January 25, 2023.
https://doi.org/10.1137/22M1507553
Funding: The work of the first author was supported by the Fonds de la Recherche Scientifique-
FNRS (F.R.S.-FNRS) and the Fonds Wetenschappelijk Onderzoek - Vlanderen (FWO) under EOS
project O005318F-RG47, and by the Francqui Foundation.
†Department of Mathematics and Operational Research, University of Mons, 7000 Mons, Belgium
(nicolas.gillis@umons.ac.be).
‡Institute of Mathematics and Informatics, Faculty of Sciences, University of P´ecs, Ifj´us´ag u. 6.
P´ecs, H-7624, Hungary (rajko@gamma.ttk.pte.hu).
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27
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28 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
FIG. 1. A three-component consecutive reaction example with the original composition in time
and the signal in wavelength profiles.
matrix Ccan be considered as the composition profile-matrix (each column of matrix
Cis a composition profile of a chemical, e.g., in a reaction in time), and the matrix
Sis the signal profile-matrix (each column of Swill be the spectrum of a chemical).
This model can cover most types of nonnegative measurement matrices and has been
used successfully in chemistry, physics, biology, engineering, and informatics [7, 8, 9,
12, 24, 27, 41]. We provide a consecutive reaction example in which the reactant X
forms an intermediate Yand the intermediate forms the product Zin two irreversible
first-order reactions: Xk1=20
−→ Yk2=3
−→ Z, where k1and k2are the first and the second
reaction rate constants, respectively. Figure 1 depicts the data matrix curves and
the original composition and signal profiles for the three components, Xin “navy
blue,” Yin “chocolate,” and Zin “gold tips” colors. See section 5.3 for another
example.
Uniqueness/Identifiability. A crucial question in many applications is the unique-
ness of a decomposition CSup to permutation and scaling, which is also known as
the identifiability of CS. In fact, uniqueness/identifiability (we will use both words
interchangeably without attributes) allow NMF to recover the groundtruth factors
that generated the data, such as the sources in audio source separation, the materials
in hyperspectral images, and the chemical components in a reaction; see the discus-
sions in [11] and [12, Chapter 4] and the references therein. To attack this question,
we focus in this paper on exact NMF (that is, an errorless reconstruction), defined as
follows.
Definition 1 (exact NMF of size r). Given a nonnegative matrix R∈Rm×n,
the decomposition CS,where C∈Rm×r
+and S∈Rn×r
+,is an exact NMF of Rof
size rif R=CS .
Let us formally define the full uniqueness/identifiability of an exact NMF.
Definition 2 (full identifiability of exact NMF). The exact NMF of R=CS
of size ris (fully) identifiable (also known as (a.k.a.) unique or essentially unique) if
and only if, for any other exact NMF of R=CSof size r, there exist a permutation
matrix Π∈{0,1}r×rand a nonsingular diagonal scaling matrix Dsuch that
C=CΠDand S=D−1ΠS
.
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PARTIAL IDENTIFIABILITY FOR NMF 29
In other words, any other exact NMF of R=CSof size rhas the form
(1) CS =
r
k=1
C(:, k)S(:, k)=
r
k=1
αkC(:, πk)
C(:,k)
1
αk
S(:, πk)
S(:,k)
for some permutation πof {1,2,...,r}and some positive scalars αk(1 ≤k≤r).
In the NMF literature, all works we are aware of have focused on the full iden-
tifiability of exact NMF, and actually this is simply referred to as the identifiability
of exact NMF. The chemometrics literature has been interested in the question of
partial identifiability: when all the chemical components are not identifiable, it asks
whether a subset of the profiles of these chemical components is identifiable. In the
chemometrics literature that studies the MCR problem, the following definitions are
used [35]:
•Full uniqueness: All profiles of all components are unique; that is, all columns
of Cand Sare identifiable. This coincides with Definition 2 above.
•Partial uniqueness: Both profiles of one or more, but not all, components are
unique.
•Fractional uniqueness: A single profile of a component is recovered uniquely,
while the others are not necessarily. This coincides with Definition 3 below.
•Nonuniqueness: No profile is identifiable; that is, a unique solution does not
exist even for a single profile. This definition would be different from the
NMF literature, where nonuniqueness means that at least one profile is not
identifiable.
In this paper, we will focus on full identifiability (Definition 2) and partial iden-
tifiability which we define as follows.
Definition 3 (partial identifiability in exact NMF). Let R=CS
be an exact
NMF of Rof size r. The kth column of Cis identifiable if and only if, for any other
exact NMF of R=CS of size r, there exist an index set jand a scalar α > 0such
that
C(:, j) = αC(:, k ).
Similarly, we can define the identifiability of the kth column of Susing symmetry,
which is referred to as the duality principle in the chemometrics literature [32], since
R=CS
if and only if R=SC
. We will focus in this paper on the partial
identifiability of the first factor, C, without loss of generality (w.l.o.g.) by symmetry
of the problem: any result that applies to Capplies to S.
Most results on the identifiability of exact NMF focus on the case r= rank(R),
as it is the most reasonable in most applications. We will also focus on this case in
this paper.
Contribution and outline of the paper. Although partial identifiability has been
considered in the chemometrics literature, there does not exist, to the best of our
knowledge, a detailed formal description (that is, a formal mathematical theorem) of
the assumptions needed to obtain such results, nor do rigorous proofs exist. The main
contribution of this paper is to provide several new theorems regarding the partial
identifiability of exact NMF.
The paper is organized as follows: In section 2, we briefly recall the geometric
interpretation of exact NMF on which our results and many identifiability results in
the literature rely on. In section 3, we review important results on the identifiability
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30 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
of exact NMF that will be useful in our discussions. Section 4 contains our main
contributions, namely,
•the restricted data-based uniqueness (DBU) theorem (Theorem 6), a partial
identifiability theorem for exact NMF;
•a geometric interpretation of the restricted DBU theorem (Lemma 1), which
will lead us to a new partial identifiability theorem for exact NMF
(Theorem 7);
•a new theorem allowing us to use any partial identifiability theorem sequen-
tially to guarantee the uniqueness of several columns of Cand S
(Theorem 8);
•a new partial identifiability theorem for exact NMF in the special case r= 3
(Theorem 9).
Finally, in section 5, we discuss the practical implications of our result, provide
an algorithm to automatically check partial identifiability in an exact NMF which
is available at https://gitlab.com/ngillis/nmf-partial-identifiability along with all the
examples presented in the paper, and illustrate the algorithm in an example from the
chemometrics literature. Note that we also provide small examples throughout the
paper to illustrate our theoretical results.
2. Preliminary: Geometric interpretation of exact NMF. Most results
on the identifiability of exact NMF rely on its geometric interpretation, including the
results of this paper. We therefore briefly recall it here for completeness.
For an exact NMF R=CS, we can assume w.l.o.g. that R,C, and Sare
column stochastic; that is, the entries in each column sum to one. Hence each column
of R,C, and Shas unit 1-norm (a.k.a. absolute sum norm, area norm, grid norm,
taxi cabnorm, Manhattan norm). The 1-norm coincides with the so-called Borgen
norm with z=ein the chemometrics literature, where eis the vector of all ones of
appropriate dimension [16, 33]. In fact, one can first remove zero columns and rows of
Rand remove the corresponding columns and rows of Sand C, respectively, which
do not bring any useful information, while it may lead to numerical problems [31].
Then one can normalize R=CS as follows:
(2) Rn(:, j) := R(:, j)
R(:, j)e=
r
k=1
C(:, k)
C(:, k)e
:=Cn(:,k)
C(:, k)e
R(:, j)eS(j, k )
:=Sn(j,k)
=
r
k=1
Cn(:, k)Sn(j, k).
Hence Rn=CnS
n,where Rnand Cnare column stochastic (that is, e=eRnand
eCn=e) by construction, while S
nis because
(3) e=eRn=eCnS
n=eS
n.
Let us therefore assume, w.l.o.g., that R,C, and Sare column stochastic. See the
chemometrics analogue using Borgen norms and closure in [30, 33]. This means that,
after normalization, the columns of Rbelong the convex hull of the columns of Cthat
are column stochastic since, for all j,
R(:, j) =
r
k=1
C(:, k)S(j, k) = C S(j, :),
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PARTIAL IDENTIFIABILITY FOR NMF 31
where S(j, :)∈∆ = {x|x≥0, ex= 1}, with ∆ being the probability simplex of
appropriate dimension. In the case r= rank(R), we must have col(R) = col(C), and
therefore
conv(R)⊆conv(C)⊆∆∩col(R),
where conv(R) = {x|x=Ry, y ∈∆}; see, e.g., [12, Chapter 2]. Hence exact
NMF reduces to finding a polytope (that is, a bounded polyhedron), conv(C) with
rvertices (the columns of C), nested between conv(R) and ∆ ∩col(R). This is the
so-called nested polytope problem (NPP) in computational geometry which is defined
as follows.
Definition 4 (NPP). Given a full-dimensional inner polytope defined by its ver-
tices {v1, v2,...,vn}, that is,
Pinn = conv([v1, v2,...,vn]) ⊆Rd,
a full-dimensional outer polytope defined by its facets1
Pout ={x∈Rd|F x +g≥0},where F∈Rm×rand g∈Rm,
such that Pinn ⊆Pout, and an integer p≥d+ 1, find a polytope, Pbet , with pvertices
nested between Pinn and Pout, that is, Pinn ⊆ Pbet ⊆ Pout.
The polytope conv(R) is typically not full-dimensional since m > r in most cases.
In fact, conv(R) has dimension rank(R)−1, and the NPP corresponding to the exact
NMF of Rsatisfies d= rank(R)−1. However, up to restricting the solution space to
the affine hull of R, the set conv(R) plays the role of Pinn in the NPP and ∆ ∩col(R)
the role of Pout.
Theorem 1 ([39]). The exact NMF problem with r= rank(R)is equivalent to
an NPP with d= rank(R)−1and p=d+ 1 and vice versa.
The equivalence between exact NMF and the NPP can be used to study the
identifiability of exact NMF. For example, for rank(R) = r= 2, the NPP is trivial
since Pinn and Pout are one-dimensional polytopes, that is, segments [23]; see also,
e.g., [34]. The exact NMF of Rwhen r= 2 is unique if and only if Pinn =Pout in
the corresponding NPP, which leads to necessary and sufficient conditions of R; see
section 3.
For r= 3, the NPP has dimension two and has been used extensively in the MCR
literature to study the identifiability of exact NMF; see, e.g., [6, 13, 36]. They refer
to the NPPs with feasible regions as Borgen–Rajk´o plots; see below for an example
of NPPs and also section 4. In this case, it is particularly useful to know how to
reduce an instance of exact NMF to the NPP and vice versa. Let us briefly recall
these reductions which we will use later in the paper.
From exact NMF to the NPP. Let Rbe an instance of exact NMF with r=
rank(R). First remove zero columns and rows of R, and normalize Rto become
column stochastic. Let Lbe the index set of rlinearly independent columns of Rso
that R=R(:,L)V≥0 for some V. Since Rand U=R(:,L) are column stochastic,
1A facet of a d-dimensional polytope is a (d−1)-dimensional face. The face of a polytope is the
intersection of that polytope with any closed half space whose boundary is disjoint from the interior
of the polytope. For the polytope Pout , each facet will have the form {x∈ Pout |F(i, :)x+gi= 0}
for some i.
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32 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
the entries in each column of Vsum to one by the same argument as in (3). We define
vj=V(1 : r−1, j) for j= 1,2,...,n whose convex hull forms Pinn, while
Pout ={x∈Rr−1|U(:,1 : r−1)x+U(:, r)(1 −ex)≥0}.
This NPP instance has a solution with rvertices if and only if Radmits an exact
NMF of size r[39].
From NPP to exact NMF. This reduction is particularly useful to construct ma-
trices coming from NPP problems in two dimensions. Given an NPP instance, the
matrix Ris constructed as follows: for all j= 1,2,...,n
R(:, j) = F vj+g, where R(:, j)≥0 since vj∈ Pinn ⊆ Pout .
The matrix Radmits an exact NMF of size rif and only if the NPP instance has a
solution with rvertices [39]. Observe that each row of Rcorresponds to a facet of
Pout and each column to a vertex of Pinn, while R(i, j) is the so-called slack of the
jth vertex with respect to the ith facet, namely, R(i, j) = F(i, :)vj+gi.
Example 1. Let us consider the NPP where Pout is the unit square [0,1]2defined
with the inequalities F x +g≥0,where
F=001−1
1−1 0 0
, g =0101,
while Pinn is the quadrilateral with the four vertices v1= (0.5,0), v2= (0,0.5),
v3= (0.25,0.75),and v4= (0.75,0.25); see Figure 2 for an illustration.
The matrix Rof the corresponding exact NMF problem is given by R(:, j) =
F vj+gfor all j, that is,
R=1
4
0231
4213
2013
2431
.
Looking at Figure 2, we observe that the unique nested triangle between Pinn and
Pout has the vertices s1= (0,0), s2= (1,0),and s3= (0,1), implying that Rhas a
unique exact NMF of size 3, given by
FIG. 2. Illustration of the NPP instance described in Example 1.
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PARTIAL IDENTIFIABILITY FOR NMF 33
R=1
4
0231
4213
2013
2431
=1
4
001
110
010
101
2200
2013
0231
.
The first factor, C, in the above decomposition is obtained using C(:, j) = F sj+gfor
j= 1,2,3.
3. Previous works on the identifiability of exact NMF. The conditions
that makes exact NMF identifiable have been studied extensively in the literature. In
this section, we briefly review some of the important works on the identifiability of
exact NMF.
3.1. Full identifiability. Let us first discuss some conditions under which exact
NMF is fully identifiable, as in Definition 2.
Necessary condition. Let us state a necessary condition for exact NMF to be fully
identifiable. The condition has been rediscovered several times and is relatively easy
to prove. It is based on the support of the columns of Cand S, the support being
the set of indices containing the nonzero entries.
Theorem 2. Let R=CS
be a fully identifiable exact NMF of Rof size r.
Then, the support of any column of C(resp., S) does not contain the support of any
other column of C(resp., S).
Proof. If a column of C, say C(:,1), contains the support of another column, say
C(:,2), then C(:,1) = C(:,1) −C(:,2) ≥0 for > 0 sufficiently small, which allows
us to construct another exact NMF. In fact, taking S(:,2) = S(:,2) + S(:,1) ≥0
and keeping the other columns untouched, that is, C(:, k ) = C(:, k) for all k= 1
and S(:, k) = S(:, k) for all k= 2, we obtain an exact NMF CS
which is not a
permutation and scaling of CS
.
Interestingly, this condition is also sufficient when r= 2, which is the only case for
which we have a necessary and sufficient condition for exact NMF to be identifiable.
For r= 2, this means that exact NMF is identifiable if and only if Cand Scontain
a 2-by-2 diagonal submatrix: each of the two columns of C(resp., S) must contain
a positive entry where the other column has a zero entry.
Sufficient condition based on separability. Several identifiability results for exact
NMF are based on the separability condition, defined as follows.
Definition 5 (separability). The matrix C∈Rm×rwith m≥ris separable if
there exists an index set Kof size rsuch that C(K,:) ∈Rr×ris a nonsingular diagonal
matrix.
Equivalently, the separable conditions requires that, for each k= 1,2,...,r, there
exists an index jsuch that C(j, :) = αe
(k)for some α > 0,where e(k)is the kth
unit vector (that is, the kth column of the identity matrix; recall that the notation
ewithout the subscript (k)is for the all-one vector, that is, the vector of all ones
of appropriate dimension). The separability condition was introduced in the NMF
literature by Donoho and Stodden [10]. We have the following result.
Theorem 3. Let R=CS
be an exact NMF of Rof size r. If Cand S
are
separable, then R=CS
is fully identifiable.
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34 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
It is difficult to trace back the origin of Theorem 3, and it does explicitly appear in
[10], although it can be derived from their result, where they relax the condition on C.
Other sufficient conditions based on separability have been proposed in the literature,
where Sis required to be separable, while there are some sparsity conditions on
C[21]. The results of this paper will be of this flavor but will focus on partial
identifiability.
Sufficiently scattered condition. The separability condition for both factors, C
and S, is rather strong and not satisfied by most data sets; see, e.g., the discussion
in [12, Chapter 4]. It can be relaxed to the following condition while retaining the
full identifiability.
Definition 6 (sufficiently scattered condition). The matrix C∈Rm×rwith
m≥rsatisfies the sufficiently scattered condition (SSC) if there is the following:
1. {x∈Rr
+|ex≥√r−1x2} ⊆ cone(C) = {x|x=Chfor h≥0}.
2. There does not exist any orthogonal matrix Qsuch that cone(C)⊆cone(Q),
except for permutation matrices. (An orthogonal matrix Qis a square matrix
such that QQ=I.)
Geometrically, separability requires that cone(C) is the nonnegative orthant,
while the SSC only requires cone(C) to contain the second-order (ice-cream) cone
tangent to every facet of the nonnegative orthant.
Theorem 4 ([17]). Let R=CS
be an exact NMF of Rof size r. If Cand
S
satisfy the SSC, then R=CS
is fully identifiable.
It is out of the scope of this paper to discuss in detail the geometric interpretation
of the SSC. An important issue with the SSC is that it is NP-hard to check in general
[17]. We refer the reader to [17], [11], and [12, Chapter 4] for more details. We will
briefly compare the SSC with our conditions in Remark 3.
Other full identifiability results for NMF are based on sparsity conditions; see the
recent paper [1] and the references therein.
3.2. Partial identifiability. In the MCR literature, the set of feasible solutions
(SFS) of exact NMF, a.k.a. the feasible regions (FRs), has been extensively studied,
especially in small dimensions (r= 3,4) [13]. Several algorithms exist; the best-known
one is the first developed for r= 2 by Lawton and Sylvestre [23] who introduced and
coined the special term self-modeling curve resolution (SMCR) for finding all feasible
solutions for a matrix decomposition with the nonnegativity constraint. In general, the
goal of MCR (and NMF) algorithms is to compute one set of particular profiles (that
is, generate one solution) without considering the fact that other profiles (solutions)
may exist with the same properties (namely, satisfying the same constraints and hav-
ing the same objective function value). Rerunning several times these algorithms with
different initializations can help to detect the nonuniqueness; however, in general, this
process cannot generate the SFS. For r= 3, after several randomized/approximate
trials, Borgen and Kowalski [6] published an analytical solution using the tangent and
the simplex rotation algorithms. These algorithms were found mathematically hard
to understand and implement for nonmathematicians, e.g., chemists; thus it was not
developed further, although being cited in the chemometrics literature for 20 years.
Rajk´o and Istv´an [36] revised Borgen's study and could enlighten the concepts based
on the geometry of the abstract space. Computational geometry tools (including
convex hulls, Fourier–Motzkin elimination, double-description) were used for devel-
oping the algorithm to draw Borgen–Rajk´o plots. The systematic grid search method
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PARTIAL IDENTIFIABILITY FOR NMF 35
was introduced to approximate the SFS/FRs numerically first for two-component
systems [40], and subsequently it was extended for three-component systems as well
[14]. Sawall et al. [38] developed the polygon inflation algorithm for three-component
systems as a faster and more accurate alternative to the grid search. The duality
concept was first used for calculating SFS/FRs for SMCR by Beyramysoltan, Ab-
dollahi, and Rajk´o [5]. For r= 4, the first attempt appeared in 2013 [15] using the
triangle enclosure method to approximate the boundary of the two-dimensional slices.
Later in 2016, Sawall et al. [37] introduced the polyhedron inflation method as the
generalization of the polygon inflation one. The chapter [37], and a subsequent paper
[29] from the same research group, provided the most comprehensive summary for
the SFS/FRs and related concepts up to now. See also [22] for a recent sampling
algorithm for larger values of rand [2] for an improved algorithm for the boundary
curve construction along with an implementation.
Necessary condition for partial identifiability. Interestingly, the necessary condi-
tion for the identifiability for exact NMF based on the supports of the columns of C
can be extended to the partial identifiability case. Note that this result is, to the best
of our knowledge, not present in the literature, although it follows directly from the
proof of Theorem 2.
Theorem 5. Let R=CS
be an exact NMF of Rof size r. If the kth column
of C(resp., S) is identifiable, then the support of the kth column of C(resp., S)
does not contain the support of any other column of C(resp., S).
Proof. This proof is similar to that of Theorem 2.
Sufficient conditions for partial identifiability: DBU and restricted DBU theorems.
In the paper [35], a partial identifiability result for NMF is presented and discussed;
it is called the DBU concept. Data-based means there that it does not only use the
estimated profiles but also the data generated by them and all feasible profiles. It
was formulated based on band solutions, that is, using not just a particular set of
estimated profiles (that is, a particular exact NMF solution) but all feasible solutions
based on SMCR (that is, the corresponding NPP with FRs a.k.a. Borgen–Rajk´o plots
[13]). Thus the SFS/FRs are needed to use the original DBU concept [35]. However,
as explained in section 3.2, there are working algorithms to get SFS/FRs only for up to
four-component systems. This fact inspired the development of the particular-profile
DBU or restricted DBU [20] that uses a particular solution. It was a step back to the
profile-based concept, which is also used in different ways by Maeder [25], Malinowski
[26], and Manne [28]. The concept was intended for practitioners (such as analytical
chemists) which is why both papers [20, 35] were published in Analytica Chimica
Acta; thus the rigorous mathematical descriptions are missing. In the following, the
lack of the formal descriptions and proofs will be remedied.
The idea [20] was restricted to analyze a particular solution relying on the fol-
lowing two conditions: Given R=CS
,where C∈Rm×r
+and S∈Rn×r
+with
rank(R) = r,
•(zero-region window) there exists a row of C, say the ith, such that C(i, k) =
0 and all feasible profiles C(i, p)>0 for all p=k;
•(selective window) there exists a row of S, say the jth, such that S(j, :) =
αe
(k)for some α > 0, that is, S(j, k)>0 and all feasible profiles S(j, p)=0
for all p=k.
Let us comment on the two conditions above:
1. Zero-region window: This condition means that C(:, k) contains an entry
equal to zero, where all other entries of Cin the same row are positive.
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36 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
Geometrically, this means that C(:, k) is the only column of Con some
facet of the nonnegative orthant.
2. Selective window: This condition means that there exists a column of R, say
the jth, such that R(:, j) = γC(:, k ) for some γ > 0. In other words, it means
that the kth column of Cappears, up to scaling, in the data set. This is
closely related to the separability condition in the NMF literature; see the
previous section 3.1. In fact, all columns of Csatisfy the selective window
condition if and only if S
is separable.
As was mentioned above, the restricted DBU concept in [20] does not have a
formal statement, nor a formal proof. Authors provide an informal one with expla-
nations, focusing on the intuitions behind their result, which is more suitable for
non–mathematically trained practitioners.
4. Partial identifiability theorems for exact NMF. In this section, we
propose a rigorous statement and proof for the restricted DBU concept from [20];
see Theorem 6 (section 4.1). In section 4.2, we provide a geometric interpretation
of Theorem 6. This leads us to a new partial identifiability result for exact NMF,
Theorem 7, in section 4.3. In section 4.4, we show how to apply Theorems 6 and 7
to allow the identifiability of more than one column of C; see Theorem 8. Finally, in
section 4.5, we use the geometric interpretation of Theorem 7 to obtain a new partial
identifiability result for the special case r= 3; see Theorem 9.
4.1. Restricted DBU theorem. Let us state and prove the restricted DBU
theorem.
Theorem 6 (restricted DBU theorem). Let R=CS
,where C∈Rm×r
+and
S∈Rn×r
+with rank(R) = r. The kth column of Cis identifiable if the following two
conditions hold:
•(Full-rank zero-region window (FRZRW)) Let I={i|C(i, k)=0}be the
complement of the support of the kth column of C. The submatrix of C
formed by the rows indexed by Ihas rank r−1, that is, rank(C(I,:)) = r−1.
•(Selective window) There exists a row of S, say the jth, such that S(j, :) =
αe
(k)for α > 0.
Proof. Let R=CSbe an exact NMF of Rof size r, that is, C∈Rm×r
+and
S∈Rn×r
+. We need to show that C(:, ) = βC(:, k) for some and some β > 0.
Since R=CS =CS
and since S(j, :) = αe
(k)for some α > 0 (selective window
condition), we have
(4) R(:, j) = CS(j, :)=αC(:, k ) = CS(j, :)=
r
p=1
C(:, p)S(j, p).
Let us denote the set of indices corresponding to columns of Cthat have zero elements
in Ias
P={p|C(I, p) = 0}⊆{1,2,...,r}
and P={1,2,...,r}\P as its complement. By nonnegativity of all the terms in (4),
S(j, p) = 0 for all p∈ P; otherwise a zero entry of C(:, k) is approximated by a
positive one since C(I, p)= 0 for p∈ P. Note that |P| ≥ 1; otherwise C(:, k) cannot
be reconstructed since C(:, k)= 0 as rank(C) = r. Below, we prove that |P| ≥r−1,
and hence |P|≤1. This will imply that |P| = 1, that is, P={}for some . Putting
this back into (4), this gives
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PARTIAL IDENTIFIABILITY FOR NMF 37
αC(:, k) =
p∈P={}
C(:, p)S(j, p) +
p∈P
C(:, p)S(j, p)
=0
=C(:, )S(j, ),
where S(j, )>0 since C(:, k)= 0 and α > 0. Finally, C(:,) = α
S(j,)C(:, k) which
completes the proof.
It remains to show that |P| ≥ r−1. For this, let us show that the rank of R(I,:)
is r−1. First, note that rank(R) = rank(CS) = rank(C) = rank(S) = rby the
conditions that rank(R) = r,R=CS
; both Cand Shave rcolumns. Then,
R(I,:) = C(I,:)S
=
p=k
C(I, p)S(:, p)=C(I,K)S(:,K),
where K={1,2,...,r}\{k}. By the FRZRW condition, rank(C(I,K)) = r−1, while
we have rank(S(:,K)) = r−1 since it is made of r−1 columns of Sthat have rank
r. Since both factors in the decomposition R(I,:) = C(I,K)S(:,K)have full rank
r−1, rank(R(I,:)) = r−1. Now, since R=CS, we also have
R(I,:) = C(I,:)S=C(I,P)S(:,P)+C(I,P)S(:,P)=C(I,P)S(:,P)
since C(I,P) = 0 by definition. As shown above, rank(R(I,:)) = r−1. This implies
that C(I,P) has at least r−1 columns, that is, |P|≥r−1.
Let us illustrate Theorem 6 in a simple example.
Example 2. Let us consider
R=
222
131
113
022
012
C
100
010
001
S
.
By Theorem 6, the first column of Cis uniquely identifiable since the two conditions
of Theorem 6 are satisfied.
1. (FRZRW) C(I,1) = 0 for I={4,5},while
rankC(I,K)= rank 2 2
1 2 = 2,where K={2,3}.
2. (Selective window) S(1,:) = e
(1) so that R(:,1) = C(:,1).
Remark 1. In the example above, Sis the identity matrix with n=r, which is
not realistic, is not a very interesting NMF decomposition (it is the trivial decompo-
sition, R=RI), and would be useless in practice. However, for our purpose, such
examples are enough. One could add any number of rows to Sand replace the iden-
tity matrix by a diagonal matrix to make it more realistic, but it would not change
our observations and discussions about the identifiability.
Remark 2. The strengthened FRZRW condition compared to the zero-region
window condition used in [20] comes from the fact that Theorem 6 provides a global
uniqueness result. The result in [20] implicitly focuses on locally unique (a.k.a. locally
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38 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
FIG. 3. Illustration of the NPP instance corresponding to the matrix Rin (5). The nested
polygon corresponding to the trivial factorization in (5),R=R I, is Pinn itself, while the nested
polygon corresponding to the factorization in (6) is denoted Pbet.
rigid) solutions; see [19] for more details on local uniqueness and rigidity of exact NMF
solutions. For example,
(5) R=
222
131
113
022
=C
100
010
001
=S
satisfies the zero-region window and selective window conditions for the first column
of C: the last row, [0,2,2], is a zero-region window, while the first row of Sis
e(1)and hence is a selective window. However, this first column is not uniquely
identifiable, up to scaling, as there exists another decomposition where that column
does not appear (up to scaling):
(6) R=
222
131
113
022
=
110
101
011
002
120
102
011
.
However, in the first factorization, in (5), the first column of Ris actually locally
unique: any nearby exact NMF factorization must contain R(:,1) as a column up
to scaling. Note that, in the second factorization above, in (6), all columns of the
first factor are locally partially identifiable; see Figure 3 for the corresponding NPP
instance.
An interesting direction of research would be to analyze conditions under which
solutions are partially locally unique.
Remark 3(FRZRW and SSC). It turns out that the FRZRW condition for each
column of Cis a necessary condition for the SSC. In fact, the SSC requires that
Chas at least r−1 zero per column, while the submatrix C(I,K) (using the same
notation as in the proof of Theorem 6) needs to contain the all-one vector in its
relative interior [12, Theorem 4.28] which requires that the rank of C(I,K) is equal to
r−1.
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PARTIAL IDENTIFIABILITY FOR NMF 39
Hence the SSC is stronger than the FRZRW condition. However, if both Cand
Ssatisfy the SSC, the exact NMF of R=CS is unique, which is not the case for
the FRZRW condition. For example, the following matrix
C
=
100110
010101
001011
111000
satisfies the FRZRW condition, but CC does not admit a unique NMF, e.g., CC
=
CC ,where
C=
110100
101010
011001
000111
;
see [12, Example 4.29].
In the next section, section 4.2, we show that Theorem 6 has a simple geomet-
ric interpretation in terms of the NPP. This will be used to obtain a new partial
identifiability result for exact NMF in section 4.3 (Theorem 7).
4.2. Geometric interpretation of the restricted DBU theorem (Theo-
rem 6). Let us consider an NPP with Pinn ⊆ Pout , and make the following simple
observation. If a vertex, v, of the inner polytope, Pinn , coincides with a vertex of
the outer polytope, Pout , then it has to belong to any nested polytope, Pbet. In fact,
since Pinn ⊆Pbet ⊆Pout and v∈Pinn ∩ Pout , we must have v∈Pbet.
It turns out the conditions of Theorem 6 (namely, the selective window and
FRZRW conditions) are equivalent to the condition that the inner and outer poly-
topes in the corresponding NPP share a vertex but written in algebraic terms. Let
us prove this equivalence. We will then use this geometric insight to provide a new
partial identifiability result in section 4.3.
In this section, we work on the outer polytope directly obtained from the reduction
from exact NMF to the NPP; see section 2. It is given by
C= col(C)∩∆ = {x|x=Cz ≥0, ex= 1}={C z |C z ≥0, ez= 1},
where Cis normalized to be column stochastic, so that x=Cz is column stochastic if
and only if ez= 1 since ex=eCz =ez. It will be useful to note that the facets
of Chave the form {Cz ∈C | (C z)i=C(i, :)z= 0}for some i.
Let us define the smallest dimensional face of the outer polytope, C, containing a
given point.
Definition 7 (minimal face of Ccontaining y). Given a column stochastic ma-
trix, C∈Rm×r
+, and the vector y∈C = col(C)∩∆, we define
FC(y) = {x∈C | supp(x)⊆supp(y)}.(7)
The set FC(y) can be characterized as follows:
FC(y) = {Cz |z∈Rr, C z ≥0,(Cz)i= 0 when yi= 0, z e= 1}.
This means that all the points in FC(y) have to belong to the same facets of Cas
y. Hence FC(y) is the face of Cof minimal dimension containing ybecause a face
of a polytope is obtained by intersecting a subset of its facets. Note that a vertex
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40 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
of a poltyope is a zero-dimensional face, and hence yis a vertex of Cif and only if
FC(y) = {y}.
Let us now prove that the FRZRW condition of Theorem 6 is equivalent to the fact
that C(:, k) is a vertex of C= col(C)∩∆, that is, FC(C(:, k)) = {C(:, k )}. Note that
the selective window assumption of Theorem 6 will require that the inner polytope
has a vertex corresponding to C(:, k).
Lemma 1. Given a nonsingular column stochastic matrix, C∈Rm×r
+, the FRZRW
condition on the kth column of Cis equivalent to the following geometric condition:
(8) FCC(:, k)=C(:, k).
Proof. Recall that Idenotes the set of indices corresponding to the zero entries
in C(:, k), and let us denote Iits complement which is the support of C(:, k).
⇒Assume the FRZRW condition holds. Let x=Cz ∈ FC(C(:, k)), that is,
Cz ≥0, (C z)i= 0 for i∈I,ze= 1. The condition (Cz)i= 0 for i∈ I can be written
as C(I,:)z= 0. Since C(I, k) = 0, by definition, this requires C(I,K)z(K) = 0,where
K={1,2,...,r}\{k}and the rank of C(I,K) is r−1, by the FRZRW condition, and
hence z(K) = 0. This implies that z=e(k)since ze= 1, and therefore (8) holds.
⇐Assume (8) holds. Since Cis nonsingular, (8) is equivalent to assuming that
the solution to the system
Cz ≥0, C (I,:)z= 0, ez= 1,
is unique and given by z=e(k). The SFS of the above system can be written as
Z={z|C(I,:)z≥0, C(I,K)z(K) = 0, ez= 1}.
Because of (8) and Cbeing nonsingular, Z={e(k)}. Since C(I, k )>0, by definition,
z=e(k)belongs to the relative interior of Z. Let us show that rank(C(I,K)) < r −1
implies that the relative interior of Zis made of more than one point, leading to a
contradiction, and hence rank(C(I,K)) = r−1 since |K|=r−1. Let y= 0 belong to
the kernel of C(I,K), that is, C(I,K)y= 0, with ey=β∈R. Let us define z∈Rr
as follows: z(K) = αy and z(k) = 1 −αβ so that ez= 1. For αsufficiently small,
we have C(I,:)z>0 since
C(I,:)z=αC(I,K)y+C(I, k )
>0
(1 −αβ),
and hence z∈Z while z=e(k).
Lemma 1 implies that, for r= 2, the conditions of Theorem 6 are necessary
and sufficient since the condition that Pinn and Pout have a vertex that coincides is
necessary and sufficient; see section 3.
4.3. New partial identifiability theorem for exact NMF. By Theorem 2,
for a column of Cto be identifiable, it has to belong to at least one facet of Cwhere the
other columns of Care not located. In fact, its support cannot contain the support of
any other column of C. Geometrically, this means that, for C(:,k ) to be identifiable,
a necessary condition is that FC(C(:, k)) is a face of dimension smaller than or equal
to r−2 (recall that Chas dimension r−1), where no other column of Cis located,
that is,
C(:, j)/∈ FCC(:, k)for all j=k.
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PARTIAL IDENTIFIABILITY FOR NMF 41
Inspired by this observation, we obtain a new sufficient condition for partial iden-
tifiability in the following theorem.
Theorem 7. Let R=CS
,where C∈Rm×r
+and S∈Rr×n
+with rank(R) = r.
W.l.o.g., assume R, Cand S
are column stochastic; see (3) and (2). The kth column
of Cis identifiable if it satisfies the selective window condition, and there exists a
subset, J, of r−1columns of R, namely, R(:,J), such that rank(R(:,J)) = r−1
and, for all j∈ J ,
(9) FCC(:, k)∩ FCR(:, j )=∅;
that is, the minimal face on which the kth columns of Clie on does not intersect the
minimal faces on which the columns of R(:,J)lie on.
Proof. Let R=CSbe another exact NMF of Rof size r, where, w.l.o.g., we
assume Cand Sare column stochastic. Let K={1,2,...,r}\{k}. We have
R=C(:, k)S(:, k)+C(:,K)S(:,K)=
r
j=1
C(:, j)S(:, j ).
Let us introduce the following terminology: given two nonnegative matrices, Aand
B, of the same dimension, we say that Atouches Bif there exists (i, j) such that
A(i, j)>0 and B(i, j )>0. Below, we show that (9) implies that, for j= 1,2,...,r,
it is not possible that C(:, j )S(:, j)touches C(:, k)S(:, k)while C(:, j )S(J, j)
touches R(:,J). Since R(:,J) has rank r−1, by the exclusion principle, exactly one
rank-one factor touches C(:, k)S(:, k),and hence it has to coincide with it.
Assume C(:, p)S(:, p)touches C(:, k)S(:, k)and C(:, p)S(J, p)touches R(:
,J) for some p. As rank(R(:,J)) = r−1, R(:, j)= 0 for all j∈J. Since C(:, p)S(:, p)
touches C(:, k)S(:, k), the support of C(:, p) is contained in the support of C(:, k ),
and hence C(:, p)∈ FC(C(:, k)). By (9), C(:, p)/∈ FC(R(:, j)) for all j∈ J ; that
is, the support of C(:, p) is not contained in the support of any column of R(:,J)
implying that it cannot touch any column of R(:,J), a contradiction.
The condition in Theorem 7 implies that the kth column of Scontains at least
r−1 entries equal to zero, namely, S(J, k) = 0, since (9) requires that the support
of R(:, j) for j∈ J does not contain the support of C(:, k ).
Example 3. Let us consider the NPP where Pout is the unit square [0,1]2as in
Example 1, while Pinn is the triangle with the vertices v1= (0.5,0), v2= (0.2,1), and
v3= (0.8,1); see Figure 4 for an illustration.
The matrix Rof the corresponding exact NMF problem is given by R(:, j) =
F vj+gfor all j, that is,
R=
011
100
0.5 0.2 0.8
0.5 0.8 0.2
=
0 1 1
1 0 0
0.501
0.510
1 0 0
0 0.8 0.2
0 0.2 0.8
.
The first column of Csatisfies the selective window assumption, while we observe
in Figure 4 that Theorem 7 applies using R(:,[2,3]) whose minimal faces do not
intersect with that of C(:,1) which is therefore identifiable. Note that the restricted
DBU Theorem 6 is not applicable to C(:,1) since it does not correspond to a vertex
of Pout.
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42 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
FIG. 4. Illustration of the NPP instance described in Example 3.
It is important to note the following:
•Theorem 7 does not subsume Theorem 6 which applies to a column of C
which is a vertex of Pout in which case the existence of a subset of col-
umns of Rsatisfying (9) is not necessary. For example, taking Pout as
the square in two dimensions, as above, and taking the vertices of Pinn as
v1= (0,0) (bottom left corner), v2= (0,0.5),and v3= (0.5,0), the conditions
of Theorem 7 do not apply to the first column of C(corresponding to v1
since the minimal faces of v2and of v3contain v1), while Theorem 6 does
apply.
•For condition (9) to be satisfied, a necessary, but not sufficient, condition is
that the supports of C(:, k) and R(:, j ) are not contained in one another. In
fact, this support condition implies that FC(C(:, k)) and FC(R(:, j )) are
distinct faces but not that their intersection is empty.
Theorem 7 can be directly used to obtain a full identifiability result.
Corollary 1. Let R=CS
,where C∈Rm×r
+and S∈Rr×n
+with rank(R) = r.
W.l.o.g., assume R, Cand S
are column stochastic. If
1. every column of Csatisfies the selective window assumption, that is, S
is
separable, and
2. the following holds for all k=j,
FCC(:, k)∩ FCC(:, j )=∅,
then (C, S)is (fully) identifiable.
Proof. On one hand, Theorem 7 applies to all columns of C, taking R(:,J) =
C(:,{1...,r}\{k}) for all k= 1,2,...,r, since S
is separable. On the other hand S
is identifiable since Cis identifiable and rank(C) = r.
For example, in two dimensions, when r= 3, full identifiability based on Corollary
1 requires that, in the NPP, the three vertices of Pinn corresponding to the three
columns of Care located on three nonadjacent edges of the polygon Pout . Note that
this requires Pout to have at least six edges, that is, to be an n-gon with n≥6. This
implies that Rneeds to have at least 6 rows.
Example 4. Let us take an example with r= 4, for which the NPP has dimension
three. Consider the outer polytope as the unit cube in dimension three, with Pout =
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PARTIAL IDENTIFIABILITY FOR NMF 43
FIG. 5. Geometric interpretation of Example 4that satisfies the conditions of Theorem 7.
[0,1]3, and take the vertices of Pinn as (0,0,0.75), (1, 0, 0.25), (0, 1, 0.25), and (1,
1, 0.75). This construction satisfies the conditions of Theorem 7 for all ksince the
vertices of Pinn are on (minimal) faces (namely, edges) that do not intersect; see
Figure 5 for an illustration.
The corresponding Ris given by
(0,0,0.75) (1,0,0.25) (0,1,0.25) (1,1,0.75)
x1≥00101
x2≥00011
x3≥0 0.75 0.25 0.25 0.75
x1≤11010
x2≤11100
x3≤1 0.25 0.75 0.75 0.25
and therefore has a unique exact NMF, R=RI. Note that no column of Rsatisfies
the FRZRW condition of Theorem 6 since Rhas only two zero entries per column:
Geometrically, no vertices of Pinn are a vertex of Pout.
4.3.1. Is it easy to check the conditions of Theorem 7?. Let R=CSbe
an exact NMF of size r= rank(R),where Rand Care column stochastic (w.l.o.g.).
A column of R, say the jth, fails to satisfy condition (9) if and only if there exists x
such that
x∈ FCC(:, k)∩ FCR(:, j ).
Such an xexists if the following linear system in variable z∈Rrhas a solution:
x=Cz ≥0, ze= 1,(Cz)i= 0 for all i∈Kk,j ={p|C(p, k) = 0 or R(p, j ) = 0}.
This is a linear system in rvariables, with O(m) equalities and inequalities. In our
implementation (see section 5.2), to avoid numerical issue, we rather solve the linear
optimization problem (which is always feasible)
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44 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
(10) min
z
i∈Kk,j
(Cz)isuch that Cz ≥0 and ez= 1
and check whether the optimal objective function value is below a given threshold
(we used 10−6).
4.4. Using partial identifiability theorems sequentially. In this section,
we provide a simple general framework to generalize partial identifiability theorems,
assuming a subset of columns of Cis already identifiable.
Theorem 8. Let R=CS,where C∈Rm×r+and S∈Rr×n+with
rank(R) = r. Assume pcolumns of Care identifiable for p∈{1,2,...,r−1}, say the
first pw.l.o.g.; that is, C(:, j)are identifiable for j= 1,2,...,p (Definition 3). Let J
be the index set corresponding to the columns of Rthat do not contain the support of
the first pcolumns of C.
If rank (S(J, p + 1 : r)) = r−pand if the (p+ 1)th column of Ccan be certified
to be identifiable in the exact NMF R(:,J) = C(:, p + 1 : r)S(J, p + 1 : r)of size
r−p, then C(:, p + 1) is identifiable in the exact NMF of Rof size r.
Proof. Let R=CSbe an exact NMF of Xof size rwith C∈Rm×r
+and
S∈Rn×r
+. W.l.o.g., C(:,1 : p) = C(:,1 : p)D, where Dis a diagonal matrix since the
first pcolumns of Care identifiable. We have
(11) R(:,J) = CS(J,:)=
r
q=p+1
C(:, q)S(J, q).
The last equality follows by construction: the columns of R(:,J) do not contain the
support of the columns of C(:,1 : p), which coincide with that of C(:,1 : p), implying
S(J, q) = 0 for all q≤p. The fact that rank(S(p+ 1 : r, J)) = r−pimplies that
rank(R(:,J)) = r−psince rank(C(: p+ 1 : r)) = r−pas rank(C) = r, and hence (11)
is an exact NMF of rank r−p. By assumption, C(:, p + 1) is identifiable in the exact
NMF (11) so that one of the columns of C(:, p + 1 : r) is equal to C(:, p + 1) up to
scaling.
Let us illustrate Theorem 8 in a simple example where all columns of Ccan be
certified to be identifiable, using Theorem 6 sequentially.
Example 5. Let
R=
0111
0123
0121
1012
1021
1101
1110
C
1000
0100
0010
0001
S
.
All columns of Care identifiable. The first one is by Theorem 6. The second one is
by combining Theorem 8 and Theorem 6: the last three columns of Rdo not belong
to the support of C(:,1); we have
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PARTIAL IDENTIFIABILITY FOR NMF 45
R(:,2 : 4) =
111
123
121
012
021
101
110
C(:,2:4)
100
010
001
S(2:4,2:4)
,
where rank(S(2 : 4,2 : 4)) = 3. We can therefore apply Theorem 6 to the above
exact NMF of size r−p= 3, which certifies the identifiability of C(:,2) (the selective
window and FRZRW conditions hold). One can certify the identifiability of the last
two columns of Cin the same way.
It is important to note that the conditions of Theorem 8 do not necessarily become
milder as pincreases. In practice, this means one needs to check p
p=0 r
pcases for
each column of Cnot identified yet. However, this can be implemented relatively
easily using recursion; see section 5.2 for the details. Let us illustrate this in another
example.
Example 6. Let
R=
0111
0123
0121
1012
1021
1001
1110
C
1011
0100
0010
0001
S
=
0111
0123
0121
1023
1032
1012
1121
.
As in Example 5, the first column of Cis identifiable. Now, we realize that Theorem
8 with p= 1 for the second column is not applicable: the last two columns of R
do contain the support of C(:,1) so that J={2}, and rank(S(p+ 1 : r, J) = 1 <
r−p−1 = 2. However, the second column of Csatisfies the conditions of Theorem
6 and hence is identifiable.
Note that the last two columns of Cdo not satisfy the selective window assump-
tion, and it turns out that they are not identifiable since another exact NMF is given
by R=RI.
4.5. Partial identifiability for exact NMF when r= 3. We now analyze
the case when r= 3, which is of particular interest in the MCR literature, by providing
a new condition for identifiability of two columns of C. Before that, let us show the
following lemma.
Lemma 2. Let R=CS
be an exact NMF of Rof size r= rank(R),where the
kth column of Csatisfies the selective window assumption. Let R=CSbe an exact
NMF of Rof size r. W.l.o.g., assume Cand Care column stochastic. If the kth
column of Cis not identified in C, that is, C(:, j )=C(:, k)for all j, then there
exists an index set Jwith |J|≥2such that
C(:, j)∈ FCC(:, k)for j∈ J .
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46 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
Proof. Since C(:, k) satisfies the selective window assumption, that is, C(:, k) =
αR(:, j) for some jand α > 0, we have C(:, k) = Cz for some z∈∆. The result then
follows from the two observations:
•Since C(:, k) = Cz, supp(C(:, j )) ⊆supp(C(:, k)) for all jsuch that for
zj>0. Therefore C(:, j )∈FC(C(:, k )) since col(C) = col(C) = col(R).
•Let J={j|zj>0}. If |J |= 1, C(: j) = C(:, k) for some j, a contradiction;
hence |J|≥2.
Theorem 9. Let R=CS
,where C∈Rm×3
+and S∈R3×n
+with rank(R) = 3,
and R,C,and S
are normalized to be column stochastic as in (2). Let us assume
that two columns of Csatisfy the selective window assumption, say the first and
second one w.l.o.g.. Also let the supports of C(:,1) and C(:,2) not be contained in
one another. Then, these two columns are identifiable if there exists a column of R,
say the jth, such that
if FC(C(:,2)) ∩FC(C(:,1)) = ∅,
(12) R(:, j)/∈convC(:,1),FCC(:,2)∪convC(:,2),FCC(:,1),
else
(13) R(:, j)/∈convFCC(:,1),FCC(:,2).
Proof. Let R=CSbe an exact NMF of Rof size r= 3. The proof mostly relies
on Lemma 2: if C(:, k) is not identified, then there are least two columns of Cin
FC(C(:, k)). Note that, for r= 3, Cis a polygon, and hence there are three types
of facets depending on their dimension: the two-dimensional polygon itself, C, one-
dimensional segments, and zero-dimensional vertices. By (12) or (13), FC(C(:, j ))
for j= 1,2 cannot be the polygon itself and hence is either segments or vertices.
Case 1: FC(C(:,2)) ∩ FC(C(:,1)) = ∅.Since Chas three columns, there
cannot be four columns of Cin FC(C(:, k)) for k∈ {1,2},and therefore C(:, k )
is identified for k= 1 or k= 2, say C(:,1) w.l.o.g.. Then, because of (12), C(:,2)
must also be identified; otherwise R(:, j) cannot be reconstructed. In fact, if C(:,2)
was not identified, the two columns of Cnot multiples of C(:,1) (which is identified)
must be on FC(C(:,2)), a contradiction between the fact that R(:, j) = CS(j, :)
and (12).
Case 2: FC(C(:,2)) ∩ FC(C(:,1)) =∅.The two facets FC(C(:,1)) and
FC(C(:,2)) intersect in a vertex. In fact, for r= 3, FC(C(:,1)) and FC(C(:,2))
are adjacent segments of Csince the support of C(:,1) does not contain and is not
contained in that of C(:,2). Moreover, by the same support condition, C(:,1) /∈
FC(C(:,2)) and vice versa. Therefore, if C(:,1) or C(:,2) is not identified, the
three columns of Cbelong to FC(C(:,1)) ∪ FC(C(:,2)), which is a contradiction
since R(:, j) does not belong to the convex hull of these sets (see (13)), and hence C
cannot be used to reconstruct R(:, j ).
Example 7. Let us construct two examples to illustrate the two cases in
Theorem 9. To do so, we use the equivalence of exact NMF with the NPP and
use the same outer polygon Pout = [0,1]2as in Example 1.
In the first case of Theorem 9, the two minimal faces containing C(:,1) and
C(:,2) do not intersect. For example, one can take the two points (0.5,0) and (0.5,1);
see Figure 6.
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PARTIAL IDENTIFIABILITY FOR NMF 47
FIG. 6. Geometric interpretation of the exact NMF problem: identifiability of the first two
columns of C, case 1of Theorem 9.
These two points correspond to
C(:,1) = F(0.5,0) + g= (0,1,0.5,0.5)and C(:,2) = F(0.5,1) + g= (1,0,0.5,0.5).
If a column of R=CS
does not belong to
convC(:,1),FCC(:,2)∪convC(:,2),FCC(:,1),
then both columns are identifiable. This is the case in Figure 6 with
R(:,3) = F(0.9,0.4) + g= (0.4,0.6,0.9,0.1).
In the second case of Theorem 9, the two minimal faces containing C(:,1) and
C(:,2) do intersect. For example, one can take the two points (0.5,0) and (0,0.5); see
Figure 7. These two points correspond to
C(:,1) = F(0.5,0) + g= (0,1,0.5,0.5)and C(:,2) = F(0,0.5) + g= (0.5,0.5,0,1).
If a column of R=CS
does not belong to
convFCC(:,1),FCC(:,2),
then both columns are identifiable.
This is the case in Figure 7, with
R(:,3) = F(0.75,0.75) + g= (0.75,0.25,0.75,0.25).
5. Applications of the new partial identifiability results. In this section,
we first discuss whether the conditions of our identifiability results are reasonable
in practice. Then we propose an algorithm, Algorithm 1, that combines our partial
identifiability results to certify the partial identifiability results for a given input
matrix R. Finally, we illustrate its use in an example from the chemometrics literature.
5.1. Are the conditions of our identifiability results reasonable?. All
our proposed identifiability results rely on two facts:
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48 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
FIG. 7. Geometric interpretation of the exact NMF problem: identifiability of the first two
columns of C, case 2of Theorem 9.
1. Some columns of Csatisfy the selective window assumption; this requires
some rows of Sto be unit vectors (up to scaling).
2. These columns of Cmust have some degree of sparsity. (Note that this is a
necessary condition for identifiability of exact NMF; see Theorem 5).
This implies some degree of sparsity in R=CSsince some columns of Rwill be
equal to the columns of Cthat have zero entries.
The selective window assumption is reasonable in many applications; see, e.g., the
discussion in [12, Chapter 7] about separability and the references therein. However,
sparsity is not necessarily natural in all applications where separability arises, e.g., in
blind hyperspectral unmixing, where spectral signatures are typically dense, and in
facial feature extraction, where facial images are dense. However, it is reasonable in
other applications. The following are examples:
•MCR: The spectral content of some sources/components can be high (overlap-
ping), while it is zero/small for others at some wavelength (selective window
assumption). Moreover, all components are not present at all time window
(sparsity); see an example in section 5.3.
•Topic modeling: The presence of anchor words, which are words associated to
a single topic, is a reasonable assumption [4] (selective window), while most
documents only discuss a few topics (sparsity). For example, using the widely
used data set tdt2˙top30 (9394 documents and 19528 words) we computed an
approximate exact NMF of the form R≈˜
R=CS for r∈ {1,2,...,100}using
a separable NMF algorithm, namely, the successive projection algorithm [3],
one of the most widely used ones. All decompositions ˜
R=CS obtained are
certified to be unique using the restricted DBU theorem (Theorem 6). (Here
we can only certify that the exact NMF of the approximation is identifiable
since there does not exist an exact NMF of Rfor a small r; in fact,2rank(R)≥
800. This is often the case in practice because of the noise and model
misfit.)
2We stopped the modified Gram–Schmidt with column pivoting at r= 800 after about 5 hours
on a standard laptop.
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PARTIAL IDENTIFIABILITY FOR NMF 49
Algorithm 1 Partial identifiability guarantees for Cin an exact NMF R=CSof
size rank(R).
Input: An exact NMF of R=CS , with C∈Rm×r
+and S∈Rn×r
+,where r=rank(R).
Output: A subset Kof the columns of Cthat are guaranteed to be identifiable
(Definition 3).
1: Normalize (R , C, S) so that they are column stochastic; see (2).
2: Initialize K=∅.
3: Let Lbe the set of columns of Cthat satisfies the selective window assumption,
that is, L={i|there exist kand α > 0 such that S(k, :) = αe
(i)}.
4: % Use Theorems 6and 7
5: for every index in k∈L\K do
6: if rank(C(I,:)) = r−1,where I={i|C(i, k) = 0}then
7: K←K∪{k}.
8: end if
9: if there exists Js.t. FC(C(:, k)) ∩FC(R(:, j)) = ∅for all j∈J, rank(R(:,J))
=r−1then
10: K←K∪{k}.
11: end if
12: end for
13: % Use Theorem 8combined Theorems 6and 7, recursively
14: i= 1
15: while i≤|K| do
16: P={1,2,...,r}\{K(i)}%K(i)is the ith element in the set K
17: Let Jbe the subset of columns of Rnot containing the support of C(:,K(i)).
18: if rank(S(J,P)) = r−1then
19: K= Algorithm 1(C(:,P), S(J,P))
20: K←K∪P(K),
21: end if
22: i←i+ 1
23: end while
24: if r= 3 then use Theorem 9 for pairs of indices in K.
In summary, our results will likely apply when Rcontains some columns with
sufficiently many zero entries, while the selective window assumption makes sense.
5.2. An algorithm to check partial identifiability. Relying on our new
theoretical results, we provide in this section an algorithm that provides partial
identifiability guarantees for the exact NMF of a given nonnegative matrix R; see
Algorithm 1. As for all the results of this paper, Algorithm 1 assumes rank(R) =
rank(C) = rwhich is reasonable in most real-world applications. Algorithm 1 is
available at https://gitlab.com/ngillis/nmf-partial-identifiability along with all the
examples presented in the paper (and two other ones).
Remark 4(use of Algorithm 1 for real-world data). NMF algorithms may return
Cand Swith many entries close to zero but not exactly zero (e.g., if the algorithm
has not converged). Therefore, to check whether your computed solution is close to
being (partially) identifiable, you can set these entries to zero using some threshold
strategy and then call Algorithm 1.
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50 NICOLAS GILLIS AND R ´
OBERT RAJK ´
O
FIG. 8. A five-component data set. On the left, the elution profiles of the chemical components
which are the columns of C. On the right, the spectra of the chemical components which are the
columns of S.
Another strategy is to weaken the sharp zero condition in the sense of generalized
Borgen plots [18]. This strategy is useful for experimental noisy data which may also
include a background subtraction resulting in small negative entries.
5.3. Numerical example from the chemometrics literature. Let us con-
sider the five-component data set from [20, section 4.3]; see Figure 8.
Algorithm 1 certifies that the first and third columns of Care identifiable and
the fifth column of S
[K,L] =check_partial_identif(C,S),
K=[1 3], L =5.
For example, for C, using the restricted DBU theorem, the elution profiles (that
is, the columns of C) that can be guaranteed to be identifiable are the first (A) and
third ones (C): they satisfy the selective window condition (first wavelengths for A,
last ones for C), while the FRZRW condition can be checked (the other elution profiles
have rank r−1 when restricted to the entries where the corresponding column of Cis
zero). Note that the first columns of Ssatisfy the selective window condition but not
the FRZRW condition because, when its spectrum is equal to zero, the spectrum of
other components also are (namely, all of them but C). These are the same conclusions
as in [20].
6. Conclusion. In this paper, we have provided the following partial identifia-
bility results for exact NMF:
•a rigorous description and proof of the restricted DBU theorem (Theorem 6);
•a new partial identifiability result based on the geometric interpretation of
the restricted DBU theorem (Theorem 7);
•a sequential approach to guarantee the identifiability of more factors (Theo-
rem 8).
Since this paper is, to the best of our knowledge, the first to rigorously investigate
partial identifiability of exact NMF, there is still a lot to be done. In particular, can
stronger partial identifiability theorems be obtained? For example, is it possible
to provide partial identifiability results for several components simultaneously under
weaker conditions? We have done this for the case r= 3 considering two components
at a time (see Theorem 9), and this idea can probably be generalized to larger r. In
particular, considering all factors allows one to relax the selective window assumption
(a.k.a. separability, which is rather strong) to the SSC; see Theorem 4.
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PARTIAL IDENTIFIABILITY FOR NMF 51
Acknowledgments. We thank the reviewers for their insightful comments that
helped us improve the paper.
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