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We propose a multilinear algebra framework to solve systems of polynomial equations with simple roots. We translate connections between univariate polynomial root-finding, eigenvalue decompositions...

Convolutional Neural Networks (CNNs) with U-shaped architectures have dominated medical image segmentation, which is crucial for various clinical purposes. However, the inherent locality of convolution makes CNNs fail to fully exploit global context, essential for better recognition of some structures, e.g., brain lesions. Transformers have recently proven promising performance on vision tasks, including semantic segmentation, mainly due to their capability of modeling long-range dependencies. Nevertheless, the quadratic complexity of attention makes existing Transformer-based models use self-attention layers only after somehow reducing the image resolution, which limits the ability to capture global contexts present at higher resolutions. Therefore, this work introduces a family of models, dubbed Factorizer, which leverages the power of low-rank matrix factorization for constructing an end-to-end segmentation model. Specifically, we propose a linearly scalable approach to context modeling, formulating Nonnegative Matrix Factorization (NMF) as a differentiable layer integrated into a U-shaped architecture. The shifted window technique is also utilized in combination with NMF to effectively aggregate local information. Factorizers compete favorably with CNNs and Transformers in terms of accuracy, scalability, and interpretability, achieving state-of-the-art results on the BraTS dataset for brain tumor segmentation and ISLES'22 dataset for stroke lesion segmentation. Highly meaningful NMF components give an additional interpretability advantage to Factorizers over CNNs and Transformers. Moreover, our ablation studies reveal a distinctive feature of Factorizers that enables a significant speed-up in inference for a trained Factorizer without any extra steps and without sacrificing much accuracy. The code and models are publicly available at https://github.com/pashtari/factorizer., Comment: 31 pages, 8 figures, 4 tables

Multicomponent alloys show intricate microstructure evolution, providing materials engineers with a nearly inexhaustible variety of solutions to enhance material properties. Multicomponent microstructure evolution simulations are indispensable to exploit these opportunities. These simulations, however, require the handling of high-dimensional and prohibitively large data sets of thermodynamic quantities, of which the size grows exponentially with the number of elements in the alloy, making it virtually impossible to handle the effects of four or more elements. In this paper, we introduce the use of tensor completion for high-dimensional data sets in materials science as a general and elegant solution to this problem. We show that we can obtain an accurate representation of the composition dependence of high-dimensional thermodynamic quantities, and that the decomposed tensor representation can be evaluated very efficiently in microstructure simulations. This realization enables true multicomponent thermodynamic and microstructure modeling for alloy design. ispartof: Npj Computational Materials vol:6 issue:1 status: published

© 2018 Elsevier B.V. Double coupled canonical polyadic decomposition (DC-CPD) decomposes multiple tensors with coupling in the first two modes, into minimal number of rank-1 tensors that also admit the double coupling structure. It has a particular interest in joint blind source separation (J-BSS) applications. In a preceding paper, we proposed an algebraic algorithm for underdetermined DC-CPD, of which the factor matrices in the first two modes of each tensor may have more columns than rows. It uses a pairwise coupled rank-1 detection mapping to transform a possibly underdetermined DC-CPD into an overdetermined DC-CPD, which can be solved algebraically via generalized eigenvalue decomposition (GEVD). In this paper, we generalize the pairwise or second-order coupled rank-1 detection mapping to an arbitrary order K≥2. Based on this generalized coupled rank-1 detection mapping, we propose a broad framework for the algebraic computation of DC-CPD, which consists of a series of algorithms with more relaxed working assumptions, each corresponding to a fixed order K≥2. Deterministic and generic uniqueness conditions are provided. We will show through analysis and numerical results that our new uniqueness conditions for DC-CPD are more relaxed than the existing results for DC-CPD and CPD. We will further show, through simulation results, the performance of the proposed algebraic DC-CPD framework in approximate DC-CPD and a J-BSS application, in comparison with existing DC-CPD and CPD algorithms. ispartof: SIGNAL PROCESSING-IMAGE COMMUNICATION vol:73 pages:22-36 status: published

ispartof: Signal Processing vol:198 status: published

© 2019 Society for Industrial and Applied Mathematics Decomposing tensors into simple terms is often an essential step toward discovering and understanding underlying processes or toward compressing data. However, storing the tensor and computing its decomposition is challenging in a large-scale setting. Though in many cases a tensor is structured, it can be represented using few parameters: a sparse tensor is determined by the positions and values of its nonzeros, a polyadic decomposition by its factor matrices, a Tensor Train by its core tensors, a Hankel tensor by its generating vector, etc. The complexity of tensor decomposition algorithms can be reduced significantly in terms of time and memory if these efficient representations are exploited directly. Only a few core operations such as norms and inner products need to be specialized to achieve this, thereby avoiding the explicit construction of multiway arrays. To improve the interpretability of tensor models, constraints are often imposed or multiple datasets are fused through joint factorizations. While imposing these constraints prohibits the use of traditional compression techniques, our framework allows constraints and compression, as well as other efficient representations, to be handled trivially as the underlying optimization variables do not change. To illustrate this, large-scale nonnegative tensor factorization is performed using MLSVD and Tensor Train compression. We also show how vector and matrix data can be analyzed using tensorization while keeping a vector or matrix complexity through the concept of implicit tensorization, as illustrated for Hankelization and Löwnerization. The concepts and numerical properties are extensively investigated through experiments. ispartof: SIAM Journal of Scientific Computing vol:41 issue:2 pages:A789-A815 status: published

© 1994-2012 IEEE. Higher order tensors and their decompositions are well-known tools in signal processing. More specifically, tensors admitting decompositions with structured factor matrices arise in various applications, such as telecommunications or convolutive independent component analysis. These applications motivate the development of efficient algorithms for structured tensor decompositions. In this letter, we develop a method for canonical polyadic decompositions (CPDs) with block-circulant factor matrices by extending a method for circulant factors. Block-circulant factor matrices arise in the identification of homogeneous Wiener-Hammerstein models. By transforming the data tensor to the frequency domain, the available structure is exploited using simple element-wise divisions. This approach reformulates the structured CPD as an unstructured CPD of lower rank. The resulting CPD of a possibly incomplete tensor can then be solved algebraically or using optimization-based methods. ispartof: IEEE Signal Processing Letters vol:25 issue:6 pages:798-802 status: published

Many real-life signals can be described in terms of much fewer parameters than the actual number of samples. Such compressible signals can often be represented very compactly with low-rank matrix and tensor models. The authors have adopted this strategy to enable large-scale instantaneous blind source separation. In this paper, we generalize the approach to the blind identification of large-scale convolutive systems. In particular, we apply the same idea to the system coefficients of finite impulse response systems. This allows us to reformulate blind system identification as a structured tensor decomposition. The tensor is obtained by applying a deterministic tensorization technique called segmentation on the observed output data. Exploiting the low-rank structure of the system coefficients enables a unique identification of the system and estimation of the inputs. We obtain a new type of deterministic uniqueness conditions. Moreover, the compactness of the low-rank models allows one to solve large-scale problems. We illustrate our method for direction-of-arrival estimation in large-scale antenna arrays and neural spike sorting in high-density microelectrode arrays.

Several applications in biomedical data processing, telecommunications, or chemometrics can be tackled by computing a structured tensor decomposition. In this paper, we focus on tensor decompositions with two or more block-Hankel factors, which arise in blind multiple-input-multiple-output (MIMO) convolutive system identification. By assuming statistically independent inputs, the blind system identification problem can be reformulated as a Hankel structured tensor decomposition. By capitalizing on the available block-Hankel and tensorial structure, a relaxed uniqueness condition for this structured decomposition is obtained. This condition is easy to check, yet very powerful. The uniqueness condition also forms the basis for two subspace-based algorithms, able to blindly identify linear underdetermined MIMO systems with finite impulse response.

© 2016 Elsevier Inc. 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×J×K tensor of rank R≥K≥J≥I≥2 if R is bounded as R≤(I+J+K−2)/2+(K−(I−J)2+4K)/2 at least for the dimensions that we have tested. This improves upon the famous Kruskal bound for uniqueness R≤(I+J+K−2)/2 as soon as I≥3. In the particular case R=K, the new bound above is equivalent to the bound R≤(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)≤I(I−1)J(J−1)/2 (implying that R

© 2015 IEEE. For the analysis of large-scale datasets one often assumes simple structures. In the case of tensors, a decomposition in a sum of rank-1 terms provides a compact and informative model. Finding this decomposition is intrinsically more difficult than its matrix counterpart. Moreover, for large-scale tensors, computational difficulties arise due to the curse of dimensionality. The randomized block sampling canonical polyadic decomposition method presented here combines increasingly popular ideas from randomization and stochastic optimization to tackle the computational problems. Instead of decomposing the full tensor at once, updates are computed from small random block samples. Using step size restriction the decomposition can be found up to near optimal accuracy, while reducing the computation time and number of data accesses significantly. The scalability is illustrated by the decomposition of a synthetic 8 TB tensor and a real life 12.5 GB tensor in a few minutes on a standard laptop. ispartof: IEEE Journal of Selected Topics in Signal Processing vol:10 issue:2 pages:284-295 status: published

© 2015 IEEE. We present structured data fusion (SDF) as a framework for the rapid prototyping of knowledge discovery in one or more possibly incomplete data sets. In SDF, each data set - stored as a dense, sparse, or incomplete tensor - is factorized with a matrix or tensor decomposition. Factorizations can be coupled, or fused, with each other by indicating which factors should be shared between data sets. At the same time, factors may be imposed to have any type of structure that can be constructed as an explicit function of some underlying variables. With the right choice of decomposition type and factor structure, even well-known matrix factorizations such as the eigenvalue decomposition, singular value decomposition and QR factorization can be computed with SDF. A domain specific language (DSL) for SDF is implemented as part of the software package Tensorlab, with which we offer a library of tensor decompositions and factor structures to choose from. The versatility of the SDF framework is demonstrated by means of four diverse applications, which are all solved entirely within Tensorlab's DSL. ispartof: IEEE Journal of Selected Topics in Signal Processing vol:9 issue:4 pages:586-600 status: published

© 2014 Society for Industrial and Applied Mathematics. Finding the real solutions of a bivariate polynomial system is a central problem in robotics, computer modeling and graphics, computational geometry, and numerical optimization. We propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition. In contrast to existing eigen-based solvers, the proposed algorithm does not depend on Gröbner bases or normal sets, nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution. The method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables. Solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and Newton-Raphson refinement. Numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the same time significantly faster and more accurate than competing double precision solvers. ispartof: SIAM Journal on Numerical Analysis vol:52 issue:4 pages:1551-1572 status: published

The canonical polyadic and rank-$(L_r,L_r,1)$ block term decomposition (CPD and BTD, respectively) are two closely related tensor decompositions. The CPD and, recently, BTD are important tools in psychometrics, chemometrics, neuroscience, and signal processing. We present a decomposition that generalizes these two and develop algorithms for its computation. Among these algorithms are alternating least squares schemes, several general unconstrained optimization techniques, and matrix-free nonlinear least squares methods. In the latter we exploit the structure of the Jacobian's Gramian to reduce computational and memory cost. Combined with an effective preconditioner, numerical experiments confirm that these methods are among the most efficient and robust currently available for computing the CPD, rank-$(L_r,L_r,1)$ BTD, and their generalized decomposition.

Although CPA (canonical/parallel factor analysis) has a unique solution, the actual computation can be made more robust by incorporating extra constraints. In several applications, the factors in one mode are known to be statistically independent. On the other hand, in Independent Component Analysis (ICA), it often makes sense to impose a Khatri-Rao structure on the mixing vectors. In this paper, we propose a new algorithm to impose independence constraints in CPA. Our algorithm implements the algebraic CPA structure and the property of statistical independence simultaneously. Numerical experiments show that our method outperforms in several cases pure CPA, pure ICA, and tensor ICA, a previously proposed method for combining ICA and CPA. We also present a strategy for imposing full or partial symmetry in CPA.

International audience; Canonical Polyadic Decomposition (CPD) of a higher-order tensor is an important tool in mathematical engineering. In many applications at least one of the matrix factors is constrained to be column-wise orthonormal. We first derive a relaxed condition that guarantees uniqueness of the CPD under this constraint. Second, we give a simple proof of the existence of the optimal low-rank approximation of a tensor in the case that a factor matrix is column-wise orthonormal. Third, we derive numerical algorithms for the computation of the constrained CPD. In particular, orthogonality-constrained versions of the CPD methods based on simultaneous matrix diagonalization and alternating least squares are presented. Numerical experiments are reported.

We present a new necessary and sufficient condition for essential uniqueness of the decomposition of a third-order tensor in rank-$(L_r,L_r,1)$ terms. We derive a new deterministic technique for blind signal separation that relies on this decomposition. The method assumes that the signals can be modeled as linear combinations of exponentials or, more generally, as exponential polynomials. The results are illustrated by means of numerical experiments.

Independent Component Analysis (ICA) aims at blindly decomposing a linear mixture of independent sources. It has lots of applications in diverse research areas. In some applications, there is prior knowledge on the sources and/or the mixing vectors. This prior knowledge can be incorporated in the computation of the independent sources. In this paper we provide an algorithm for so-called spatially constrained ICA (scICA). The algorithm deals with the situation when one mixing vector is exactly known. Also the generalization to more mixing vectors is discussed. Numerical experiments are reported that allow us to assess the improvement in accuracy that can be achieved with these algorithms compared to fully blind ICA and to a previously proposed constrained algorithm. We illustrate the approach with a biomedical application.

Higher-order tensors are used in many application fields, such as statistics, signal processing, and scientific computing. Efficient and reliable algorithms for manipulating these multi-way arrays are thus required. In this paper, we focus on the best rank-$(R_1,R_2,R_3)$ approximation of third-order tensors. We propose a new iterative algorithm based on the trust-region scheme. The tensor approximation problem is expressed as a minimization of a cost function on a product of three Grassmann manifolds. We apply the Riemannian trust-region scheme, using the truncated conjugate-gradient method for solving the trust-region subproblem. Making use of second order information of the cost function, superlinear convergence is achieved. If the stopping criterion of the subproblem is chosen adequately, the local convergence rate is quadratic. We compare this new method with the well-known higher-order orthogonal iteration method and discuss the advantages over Newton-type methods. Read More: http://epubs.siam.org/doi/abs/10.1137/090764827

This paper presents a high-precision method for the estimation of the parameters of a signal, modelled as a finite sum of complex damped exponentials, whose poles may be close. It is well known that increasing the sampling rate increases the accuracy of traditional methods, like total least squares (TLSs)-ESPRIT and HTLS, in the case of closely spaced exponentials. However, in the case of long sequences, the computational complexity makes these methods hard to apply. To overcome this problem, the HTLSDstack and the HTLSDsum methods make use of decimated sequences. In this paper, we propose a higher order counterpart of HTLSDstack, based on concepts from multilinear algebra. Basically, it consists of decimating an oversampled signal by a factor D, making sure that no aliasing occurs. Each sequence is arranged in a Hankel matrix and the D Hankel matrices are stacked in a third-order tensor. A dimensionality reduction algorithm is applied to the tensor and the resulting subspace is used to retrieve the parameters of interest. The dimensionality reduction is possibly unsymmetric in the different modes of the tensor, which allows us to take into account ill conditioning when poles are close. The new method yields better results than HTLSDstack while its complexity stays fairly low compared to non-decimative methods. Copyright © 2009 John Wiley & Sons, Ltd.

In this paper, we introduce an enhanced line search algorithm to accelerate the convergence of the alternating least squares (ALS) algorithm, which is often used to decompose a tensor in a sum of contributions. This scheme can be used for the computation in the complex case of the Parallel Factor model or the more general block component model. We then illustrate the performance of the algorithm in the context of blind separation-equalization of convolutive DS-CDMA mixtures.

In this paper we study a generalization of Kruskal's permutation lemma to partitioned matrices. We define the k'-rank of partitioned matrices as a generalization of the k-rank of matrices. We derive a lower-bound on the k'-rank of Khatri-Rao products of partitioned matrices. We prove that Khatri-Rao products of partitioned matrices are generically full column rank.

In this paper we introduce a new class of tensor decompositions. Intuitively, we decompose a given tensor block into blocks of smaller size, where the size is characterized by a set of mode-$n$ ranks. We study different types of such decompositions. For each type we derive conditions under which essential uniqueness is guaranteed. The parallel factor decomposition and Tucker's decomposition can be considered as special cases in the new framework. The paper sheds new light on fundamental aspects of tensor algebra.

In this paper we derive alternating least squares algorithms for the computation of the block term decompositions introduced in Part II. We show that degeneracy can also occur for block term decompositions.

In this paper we present new deterministic tensor-based techniques for the blind separation of a mixture of DS-CDMA signals received by an antenna array. First, we show that the blind receiver follows from a simultaneous matrix decomposition. We present a new, relaxed, bound on the number of users that can be allowed at the same time. We further derive two algorithms that jointly exploit the CDMA structure and the constant modulus property of the transmitted signals.

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a continuous distribution, and the third component matrix has full column rank. In this context, we obtain almost sure sufficient uniqueness conditions for the Candecomp/Parafac and Indscal models separately, involving only the order of the three-way array and the number of components in the decomposition. Both uniqueness conditions are closer to necessity than the classical uniqueness condition by Kruskal.

In this paper we develop a technique for detecting and retrieving the common harmonics in a multi-channel setup. The different (complex) signals are arranged in a set of Hankel matrices. First, we compute the singular value decomposition (SVD) of these matrices in order to denoise and normalize the relevant subspaces. Then a second SVD is applied to detect and select the common subspace. The poles of the harmonics are computed from the common subspace using the total least squares (TLS) technique. This method is very flexible, robust and outperforms existing subspace-based methods. Moreover, it can be applied to an arbitrary number of channels.

In this paper we review a multilinear generalization of the singular value decomposition and the best rank-( R 1 , R 2 ,…, R N ) approximation of higher-order tensors. We show that they are important tools for dimensionality reduction in higher-order signal processing. We discuss applications in independent component analysis, simultaneous matrix diagonalization and subspace variants of algorithms based on higher-order statistics.

In this paper we perform a first-order perturbation analysis of the least squares approximation of a given higher-order tensor by a tensor having prespecified n-mode ranks. This work generalizes the classical first-order perturbation analysis of the matrix singular value decomposition. We will show that there are important differences between the matrix and the higher-order tensor case. We subsequently address (1) the best rank-1 approximation of supersymmetric tensors, (2) the best rank-(R 1 , R 2 , R 3 ) approximation of arbitrary tensors and (3) the best rank-(R 1 , R 2 , R 3 ) approximation of arbitrary tensors.

The canonical decomposition of higher-order tensors is a key tool in multilinear algebra. First we review the state of the art. Then we show that, under certain conditions, the problem can be rephrased as the simultaneous diagonalization, by equivalence or congruence, of a set of matrices. Necessary and sufficient conditions for the uniqueness of these simultaneous matrix decompositions are derived. In a next step, the problem can be translated into a simultaneous generalized Schur decomposition, with orthogonal unknowns [A.-J. van der Veen and A. Paulraj, IEEE Trans. Signal Process., 44 (1996), pp. 1136--1155]. A first-order perturbation analysis of the simultaneous generalized Schur decomposition is carried out. We discuss some computational techniques (including a new Jacobi algorithm) and illustrate their behavior by means of a number of numerical experiments.

We address the problem of blind identification of a convolutive Multiple-Input Multiple-Output (MIMO) system with more inputs than outputs, and in particular, the 3-input 2-output case. We assume that the inputs are temporally white, non-Gaussian distributed, and spatially independent. Solutions for the scalar MIMO case, within scaling and permutation ambiguities, have been proposed in the past, based on the canonical decomposition of tensors constructed from higher-order cross-cumulants of the system output. In this paper, we look at the problem in the frequency domain, where, for each frequency we construct a number of tensors based on cross-polyspectra of the output. These tensors lead to the system frequency response within frequency dependent scaling and permutation ambiguities. We propose ways to resolve these ambiguities, and show that it is possible to obtain the system response within a scalar and a linear phase.

In this paper, we show that the restricted singular value decomposition of a matrix triplet $A\in \R^{n \times m}, B\in \R^{n \times l}, C\in \R^{p \times m}$ can be computed by means of the cosine-sine decomposition. In the first step, the matrices A, B, C are reduced to a lower-dimensional matrix triplet ${\cal A}, {\cal B}, {\cal C}$, in which ${\cal B}$ and ${\cal C}$ are nonsingular, using orthogonal transformations such as the QR-factorization with column pivoting and the URV decomposition. In the second step, the components of the restricted singular value decomposition of A, B, C are derived from the singular value decomposition of ${\cal B}^{-1}{\cal A}{\cal C}^{-1}$. Instead of explicitly forming the latter product, a link with the cosine-sine decomposition, which can be computed by Van Loan's method, is exploited. Some numerical examples are given to show the performance of the presented method.

SUMMARY This paper is an introduction to the concept of independent component analysis (ICA) which has recently been developed in the area of signal processing. ICA is a variant of principal component analysis (PCA) in which the components are assumed to be mutually statistically independent instead of merely uncorrelated. The stronger condition allows one to remove the rotational invariance of PCA, i.e. ICA provides a meaningful unique bilinear decomposition of two-way data that can be considered as a linear mixture of a number of independent source signals. The discipline of multilinear algebra offers some means to solve the ICA problem. In this paper we briefly discuss four orthogonal tensor decompositions that can be interpreted in terms of higher-order generalizations of the symmetric eigenvalue decomposition. Copyright © 2000 John Wiley & Sons, Ltd.

We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries a ect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors. ispartof: SIAM Journal on Matrix Analysis and Applications vol:21 issue:4 pages:1253-1278 status: published

Long-term electroencephalographic (EEG) recordings are important in the presurgical evaluation of refractory partial epilepsy for the delineation of the ictal onset zones. In this paper, we introduce a new concept for an automatic, fast, and objective localisation of the ictal onset zone in ictal EEG recordings. Canonical decomposition of ictal EEG decomposes the EEG in atoms. One or more atoms are related to the seizure activity. A single dipole was then fitted to model the potential distribution of each epileptic atom. In this study, we performed a simulation study in order to estimate the dipole localisation error. Ictal dipole localisation was very accurate, even at low signal-to-noise ratios, was not affected by seizure activity frequency or frequency changes, and was minimally affected by the waveform and depth of the ictal onset zone location. Ictal dipole localisation error using 21 electrodes was around 10.0 mm and improved more than tenfold in the range of 0.5–1.0 mm using 148 channels. In conclusion, our simulation study of canonical decomposition of ictal scalp EEG allowed a robust and accurate localisation of the ictal onset zone.

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de~Groote, who showed that for the multiplication of $2\times2$ matrices with $7$ active multiplications, all algorithms are essentially equivalent to Strassen's algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices, (e.g., $2\times3$ by $3\times2$ or $3\times3$ by $3\times3$ matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.