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Ignat Domanov

Publications and source records attributed to Ignat Domanov.

11 recordsLinked to original sources
arXiv2019

From computation to comparison of tensor decompositions

Decompositions of higher-order tensors into sums of simple terms are ubiquitous. We show that in order to verify that two tensors are generated by the same (possibly scaled) terms it is not necessary to compute the individual decompositions. In general the explicit computation of such a decomposition may have high complexity and can be ill-conditioned. We now show that under some assumptions the verification can be reduced to a comparison of both the column and row spaces of the corresponding matrix representations of the tensors. We consider rank-1 terms as well as low multilinear rank terms (also known as block terms) and show that the number of the terms and their multilinear rank can be inferred as well. The comparison relies only on numerical linear algebra and can be done in a numerically reliable way. We also illustrate how our results can be applied to solve a multi-label classification problem that appears in the context of blind source separation.

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arXiv2019

On uniqueness and computation of the decomposition of a tensor into multilinear rank-$(1, L_{r},L_{r})$ terms

Canonical Polyadic Decomposition (CPD) represents a third-order tensor as the minimal sum of rank-1 terms. Because of its uniqueness properties the CPD has found many concrete applications in telecommunication, array processing, machine learning, etc. On the other hand, in several applications the rank-1 constraint on the terms is too restrictive. A multilinear rank-$(M,N,L)$ constraint (where a rank-1 term is the special case for which $M=N=L=1$) could be more realistic, while it still yields a decomposition with attractive uniqueness properties. In this paper we focus on the decomposition of a tensor $\mathcal T$ into a sum of multilinear rank-$(1,L_r,L_r)$ terms, $r=1,...,R$. This particular decomposition type has already found applications in wireless communication, chemometrics and the blind signal separation of signals that can be modelled as exponential polynomials and rational functions. We find conditions on the terms which guarantee that the decomposition is unique and can be computed by means of the eigenvalue decomposition of a matrix even in the cases where none of the factor matrices has full column rank. We consider both the case where the decomposition is exact and the case where the decomposition holds only approximately. We show that in both cases the number of the terms $R$ and their "sizes" $L_1,...,L_R$ do not have to be known a priori and can be estimated as well. The conditions for uniqueness are easy to verify, especially for terms that can be considered "generic". In particular, we obtain the following two generalizations of a well known result on generic uniqueness of the CPD (i.e., the case $L_1=...=L_R=1$): we show that the multilinear rank-$(1,L_r,L_r)$ decomposition of an $I\times J\times K$ tensor is generically unique if i) $L_1=...=L_R=:L$ and $R\leq\min((J-L)(K-L),I)$ or if ii) $\sum L_R\leq\min((I-1)(J-1),K)$ and $J\geq\max(L_i+L_j)$.

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arXiv2018

On the largest multilinear singular values of higher-order tensors

Let $σ_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times\dots \times I_N$ tensor $\mathcal T$. We prove that $$ σ_1^2+\dots+σ_{n-1}^2+σ_{n+1}^2+\dots+σ_{N}^2\leq (N-2)\|\mathcal T\|^2 + σ_n^2,\quad n=1,\dots,N, \qquad\qquad (1) $$ where $\|\cdot\|$ denotes the Frobenius norm. We also show that at least for the cubic tensors the inverse problem always has a solution. Namely, for each $σ_1,\dots,σ_N$ that satisfy (1) and the trivial inequalities $σ_1\geq \frac{1}{\sqrt{I}}\|\mathcal T\|,\dots, σ_N\geq \frac{1}{\sqrt{I}}\|\mathcal T\|$, there always exists an $I\times \dots\times I$ tensor whose largest multilinear singular values are equal to $σ_1,\dots,σ_N$. For $N=3$ we show that if the equality $σ_1^2+σ_2^2= \|\mathcal T\|^2 + σ_3^2$ in (1) holds, then $\mathcal T$ is necessarily equal to a sum of multilinear rank-$(L_1,1,L_1)$ and multilinear rank-$(1,L_2,L_2)$ tensors and we give a complete description of all its multilinear singular values. We establish a connection with honeycombs and eigenvalues of the sum of two Hermitian matrices. This seems to give at least a partial explanation of why results on the joint distribution of multilinear singular values are scarce.

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arXiv2016

Canonical polyadic decomposition of third-order tensors: relaxed uniqueness conditions and algebraic algorithm

Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-$1$ tensors. We find new mild deterministic conditions for the uniqueness of individual rank-$1$ tensors in CPD and present an algorithm to recover them. We call the algorithm "algebraic" because it relies only on standard linear algebra. It does not involve more advanced procedures than the computation of the null space of a matrix and eigen/singular value decomposition. Simulations indicate that the new conditions for uniqueness and the working assumptions for the algorithm hold for a randomly generated $I\times J\times K$ tensor of rank $R\geq K\geq J\geq I\geq 2$ if $R$ is bounded as $R\leq (I+J+K-2)/2 + (K-\sqrt{(I-J)^2+4K})/2$ at least for the dimensions that we have tested. This improves upon the famous Kruskal bound for uniqueness $R\leq (I+J+K-2)/2$ as soon as $I\geq 3$. In the particular case $R=K$, the new bound above is equivalent to the bound $R\leq(I-1)(J-1)$ which is known to be necessary and sufficient for the generic uniqueness of the CPD. An existing algebraic algorithm (based on simultaneous diagonalization of a set of matrices) computes the CPD under the more restrictive constraint $R(R-1)\leq I(I-1)J(J-1)/2$ (implying that $R<(J-\frac{1}{2})(I-\frac{1}{2})/\sqrt{2}+1$). On the other hand, optimization-based algorithms fail to compute the CPD in a reasonable amount of time even in the low-dimensional case $I=3$, $J=7$, $K=R=12$. By comparison, in our approach the computation takes less than $1$ sec. We demonstrate that, at least for $R\leq 24$, our algorithm can recover the rank-$1$ tensors in the CPD up to $R\leq(I-1)(J-1)$.

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arXiv2016

Generic uniqueness of a structured matrix factorization and applications in blind source separation

Algebraic geometry, although little explored in signal processing, provides tools that are very convenient for investigating generic properties in a wide range of applications. Generic properties are properties that hold "almost everywhere". We present a set of conditions that are sufficient for demonstrating the generic uniqueness of a certain structured matrix factorization. This set of conditions may be used as a checklist for generic uniqueness in different settings. We discuss two particular applications in detail. We provide a relaxed generic uniqueness condition for joint matrix diagonalization that is relevant for independent component analysis in the underdetermined case. We present generic uniqueness conditions for a recently proposed class of deterministic blind source separation methods that rely on mild source models. For the interested reader we provide some intuition on how the results are connected to their algebraic geometric roots.

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arXiv2015

Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL

We find conditions that guarantee that a decomposition of a generic third-order tensor in a minimal number of rank-$1$ tensors (canonical polyadic decomposition (CPD)) is unique up to permutation of rank-$1$ tensors. Then we consider the case when the tensor and all its rank-$1$ terms have symmetric frontal slices (INDSCAL). Our results complement the existing bounds for generic uniqueness of the CPD and relax the existing bounds for INDSCAL. The derivation makes use of algebraic geometry. We stress the power of the underlying concepts for proving generic properties in mathematical engineering.

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arXiv2013

Canonical polyadic decomposition of third-order tensors: reduction to generalized eigenvalue decomposition

Canonical Polyadic Decomposition (CPD) of a third-order tensor is decomposition in a minimal number of rank-$1$ tensors. We call an algorithm algebraic if it is guaranteed to find the decomposition when it is exact and if it only relies on standard linear algebra (essentially sets of linear equations and matrix factorizations). The known algebraic algorithms for the computation of the CPD are limited to cases where at least one of the factor matrices has full column rank. In the paper we present an algebraic algorithm for the computation of the CPD in cases where none of the factor matrices has full column rank. In particular, we show that if the famous Kruskal condition holds, then the CPD can be found algebraically.

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arXiv2013

On the Uniqueness of the Canonical Polyadic Decomposition of third-order tensors --- Part I: Basic Results and Uniqueness of One Factor Matrix

Canonical Polyadic Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of rank-1 tensors. We give an overview of existing results concerning uniqueness. We present new, relaxed, conditions that guarantee uniqueness of one factor matrix. These conditions involve Khatri-Rao products of compound matrices. We make links with existing results involving ranks and k-ranks of factor matrices. We give a shorter proof, based on properties of second compound matrices, of existing results concerning overall CPD uniqueness in the case where one factor matrix has full column rank. We develop basic material involving $m$-th compound matrices that will be instrumental in Part II for establishing overall CPD uniqueness in cases where none of the factor matrices has full column rank.

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arXiv2013

On the Uniqueness of the Canonical Polyadic Decomposition of third-order tensors --- Part II: Uniqueness of the overall decomposition

Canonical Polyadic (also known as Candecomp/Parafac) Decomposition (CPD) of a higher-order tensor is decomposition in a minimal number of rank-1 tensors. In Part I, we gave an overview of existing results concerning uniqueness and presented new, relaxed, conditions that guarantee uniqueness of one factor matrix. In Part II we use these results for establishing overall CPD uniqueness in cases where none of the factor matrices has full column rank. We obtain uniqueness conditions involving Khatri-Rao products of compound matrices and Kruskal-type conditions.

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