arXiv subjects

Konstantin Usevich

Publications and source records attributed to Konstantin Usevich.

At least 19 recordsLinked to original sources

arXiv2026

Tensor-based Multi-layer Decoupling

The decoupling of multivariate functions is a powerful modeling paradigm for learning multivariate input-output relations from data. For the single-layer case, established CPD-based methods are available, but the multi-layer case remained largely unexplored. This work introduces a tensor-based framework for multi-layer decoupling, which is based on ParaTuck-type tensor decompositions and constrained optimization. We provide theoretical justification behind the considered tensor decompositions and parameterizations. Furthermore, we formulate a structured coupled matrix-tensor factorization that incorporates both Jacobian and function evaluations, together with a bilevel optimization approach for adaptively balancing first- and zeroth-order information. The feasibility of the proposed methodology is illustrated on synthetic systems, a nonlinear system identification benchmark and neural network compression.

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arXiv2025

Coupled tensor models for probability mass function estimation: Part I, Principles and algorithms

In this article, a Probability Mass Function (PMF) estimation method which tames the curse of dimensionality is proposed. This method, called Partial Coupled Tensor Factorization of 3D marginals or PCTF3D, has for principle to partially couple order-3 data projections -- seen as order-3 tensors -- to obtain a tensor decomposition of the probability mass tensor. The novelty of PCTF3D relies on partial coupling which consists in choosing a subset of 3D marginals. The choice of marginals is then formulated with hypergraphs. After presenting possible coupling strategies, some numerical experiments and an application of the method are proposed. This article is the first of a two-part article. While this first article focuses on a new algorithmic framework for PMF estimation, the second studies uniqueness properties of the model introduced in this article.

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arXiv2025

Coupled tensor models for probability mass function estimation: Part II, Uniqueness of the model

In this paper, uniqueness properties of a coupled tensor model are studied. This new coupled tensor model is used in a new method called Partial Coupled Tensor Factorization of 3D marginals or PCTF3D. This method performs estimation of probability mass functions by coupling 3D marginals, seen as order-3 tensors. The core novelty of PCTF3D's approach (detailed in the part I article) relies on the partial coupling which consists on the choice of 3D marginals to be coupled. Tensor methods are ubiquitous in many applications of statistical learning, with their biggest advantage of having strong uniqueness properties. In this paper, the uniqueness properties of PCTF3D's constrained coupled low-rank model is assessed. While probabilistic constraints of the coupled model are handled properly, it is shown that uniqueness highly depends on the coupling used in PCTF3D. After proposing a Jacobian algorithm providing maximum recoverable rank, different coupling strategies presented in the Part I article are examined with respect to their uniqueness properties. Finally, an identifiability bound is given for a so-called Cartesian coupling which permits enhancing sufficient bounds of the literature.

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arXiv2025

Low-Rank Tensor Decompositions for the Theory of Neural Networks

The groundbreaking performance of deep neural networks (NNs) promoted a surge of interest in providing a mathematical basis to deep learning theory. Low-rank tensor decompositions are specially befitting for this task due to their close connection to NNs and their rich theoretical results. Different tensor decompositions have strong uniqueness guarantees, which allow for a direct interpretation of their factors, and polynomial time algorithms have been proposed to compute them. Through the connections between tensors and NNs, such results supported many important advances in the theory of NNs. In this review, we show how low-rank tensor methods--which have been a core tool in the signal processing and machine learning communities--play a fundamental role in theoretically explaining different aspects of the performance of deep NNs, including their expressivity, algorithmic learnability and computational hardness, generalization, and identifiability. Our goal is to give an accessible overview of existing approaches (developed by different communities, ranging from computer science to mathematics) in a coherent and unified way, and to open a broader perspective on the use of low-rank tensor decompositions for the theory of deep NNs.

cs.LG↗

arXiv2025

Identifiability of Deep Polynomial Neural Networks

Polynomial Neural Networks (PNNs) possess a rich algebraic and geometric structure. However, their identifiability -- a key property for ensuring interpretability -- remains poorly understood. In this work, we present a comprehensive analysis of the identifiability of deep PNNs, including architectures with and without bias terms. Our results reveal an intricate interplay between activation degrees and layer widths in achieving identifiability. As special cases, we show that architectures with non-increasing layer widths are generically identifiable under mild conditions, while encoder-decoder networks are identifiable when the decoder widths do not grow too rapidly compared to the activation degrees. Our proofs are constructive and center on a connection between deep PNNs and low-rank tensor decompositions, and Kruskal-type uniqueness theorems. We also settle an open conjecture on the dimension of PNN's neurovarieties, and provide new bounds on the activation degrees required for it to reach the expected dimension.

cs.LG↗

arXiv2024

Personalized Coupled Tensor Decomposition for Multimodal Data Fusion: Uniqueness and Algorithms

Coupled tensor decompositions (CTDs) perform data fusion by linking factors from different datasets. Although many CTDs have been already proposed, current works do not address important challenges of data fusion, where: 1) the datasets are often heterogeneous, constituting different "views" of a given phenomena (multimodality); and 2) each dataset can contain personalized or dataset-specific information, constituting distinct factors that are not coupled with other datasets. In this work, we introduce a personalized CTD framework tackling these challenges. A flexible model is proposed where each dataset is represented as the sum of two components, one related to a common tensor through a multilinear measurement model, and another specific to each dataset. Both the common and distinct components are assumed to admit a polyadic decomposition. This generalizes several existing CTD models. We provide conditions for specific and generic uniqueness of the decomposition that are easy to interpret. These conditions employ uni-mode uniqueness of different individual datasets and properties of the measurement model. Two algorithms are proposed to compute the common and distinct components: a semi-algebraic one and a coordinate-descent optimization method. Experimental results illustrate the advantage of the proposed framework compared with the state of the art approaches.

cs.LG↗

arXiv2024

Computing asymptotic eigenvectors and eigenvalues of perturbed symmetric matrices

Computing the eigenvectors and eigenvalues of a perturbed matrix can be remarkably difficult when the unperturbed matrix has repeated eigenvalues. In this work we show how the limiting eigenvectors and eigenvalues of a symmetric matrix $K(\varepsilon)$ as $\varepsilon \to 0$ can be obtained relatively easily from successive Schur complements, provided that the entries scale in different orders of $\varepsilon$. If the matrix does not directly exhibit this structure, we show that putting the matrix into a ``generalised kernel form'' can be very informative. The resulting formulas are much simpler than classical expressions obtained from complex integrals involving the resolvent. We apply our results to the problem of computing the eigenvalues and eigenvectors of kernel matrices in the ``flat limit'', a problem that appears in many applications in statistics and approximation theory. In particular, we prove a conjecture from [SIAM J. Matrix Anal. Appl., 2021, 42(1):17--57] which connects the eigenvectors of kernel matrices to multivariate orthogonal polynomials.

math.NA↗

arXiv2023

On factorization of rank-one auto-correlation matrix polynomials

This article characterizes the rank-one factorization of auto-correlation matrix polynomials. We establish a sufficient and necessary uniqueness condition for uniqueness of the factorization based on the greatest common divisor (GCD) of multiple polynomials. In the unique case, we show that the factorization can be carried out explicitly using GCDs. In the non-unique case, the number of non-trivially different factorizations is given and all solutions are enumerated.

math.NA↗

arXiv2023

The barycenter in free nilpotent Lie groups and its application to iterated-integrals signatures

We establish the well-definedness of the barycenter (in the sense of Buser and Karcher) for every integrable measure on the free nilpotent Lie group of step $L$ (over $\mathbb{R}^d$). We provide two algorithms for computing it, using methods from Lie theory (namely, the Baker-Campbell-Hausdorff formula) and from the theory of Gr\"obner bases of modules. Our main motivation stems from measures induced by iterated-integrals signatures, and we calculate the barycenter for the signature of the Brownian motion.

math.RA↗

arXiv2023

A lifting approach to ParaTuck-2 tensor decompositions

The ParaTuck-2 decomposition (PT2D) of third-order tensor is a two-layer generalization of the well-known canonical polyadic decomposition (CPD).While being more flexible than the CPD, the PT2D also possesses similar uniqueness properties.In this paper, we show than under the best known uniqueness conditions, the exact PT2D can be computed by an algebraic algorithm (i.e., can the PT2D problems can be reduced to computing nullspaces and eigenvalues of certain matrices).We do so by lifting the slices of the tensor to higher-dimensional space, which also allows for refining the existing uniqueness conditions.The algorithms are developed for general PT2D and its symmetric version (DEDICOM), which leads to an algebraic algorithm for another generalization of the CPD, the PARAFAC2 decomposition.Our methods are also applicable in the approximation scenario, as shown by the numerical experiments.

math.NA↗

arXiv2022

Coupled CP tensor decomposition with shared and distinct components for multi-task fMRI data fusion

Discovering components that are shared in multiple datasets, next to dataset-specific features, has great potential for studying the relationships between different subjects or tasks in functional Magnetic Resonance Imaging (fMRI) data. Coupled matrix and tensor factorization approaches have been useful for flexible data fusion, or decomposition to extract features that can be used in multiple ways. However, existing methods do not directly recover shared and dataset-specific components, which requires post-processing steps involving additional hyperparameter selection. In this paper, we propose a tensor-based framework for multi-task fMRI data fusion, using a partially constrained canonical polyadic (CP) decomposition model. Differently from previous approaches, the proposed method directly recovers shared and dataset-specific components, leading to results that are directly interpretable. A strategy to select a highly reproducible solution to the decomposition is also proposed. We evaluate the proposed methodology on real fMRI data of three tasks, and show that the proposed method finds meaningful components that clearly identify group differences between patients with schizophrenia and healthy controls.

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arXiv2022

Polarimetric phase retrieval: uniqueness and algorithms

This work introduces a novel Fourier phase retrieval model, called polarimetric phase retrieval that enables a systematic use of polarization information in Fourier phase retrieval problems. We provide a complete characterization of uniqueness properties of this new model by unraveling equivalencies with a peculiar polynomial factorization problem. We introduce two different but complementary categories of reconstruction methods. The first one is algebraic and relies on the use of approximate greatest common divisor computations using Sylvester matrices. The second one carefully adapts existing algorithms for Fourier phase retrieval, namely semidefinite positive relaxation and Wirtinger-Flow, to solve the polarimetric phase retrieval problem. Finally, a set of numerical experiments permits a detailed assessment of the numerical behavior and relative performances of each proposed reconstruction strategy. We further highlight a reconstruction strategy that combines both approaches for scalable, computationally efficient and asymptotically MSE optimal performance.

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arXiv2022

Hankel low-rank approximation and completion in time series analysis and forecasting: a brief review

In this paper we offer a review and bibliography of work on Hankel low-rank approximation and completion, with particular emphasis on how this methodology can be used for time series analysis and forecasting. We begin by describing possible formulations of the problem and offer commentary on related topics and challenges in obtaining globally optimal solutions. Key theorems are provided, and the paper closes with some expository examples.

math.NA↗

arXiv2022

Gaussian Process Regression in the Flat Limit

Gaussian process (GP) regression is a fundamental tool in Bayesian statistics. It is also known as kriging and is the Bayesian counterpart to the frequentist kernel ridge regression. Most of the theoretical work on GP regression has focused on a large-$n$ asymptotics, i.e. as the amount of data increases. Fixed-sample analysis is much more difficult outside of simple cases, such as locations on a regular grid. In this work we perform a fixed-sample analysis that was first studied in the context of approximation theory by Driscoll & Fornberg (2002), called the ``flat limit''. In flat-limit asymptotics, the goal is to characterise kernel methods as the length-scale of the kernel function tends to infinity, so that kernels appear flat over the range of the data. Surprisingly, this limit is well-defined, and displays interesting behaviour: Driscoll & Fornberg showed that radial basis interpolation converges in the flat limit to polynomial interpolation, if the kernel is Gaussian. Subsequent work showed that this holds true in the multivariate setting as well, but that kernels other than the Gaussian may have (polyharmonic) splines as the limit interpolant. Leveraging recent results on the spectral behaviour of kernel matrices in the flat limit, we study the flat limit of Gaussian process regression. Results show that Gaussian process regression tends in the flat limit to (multivariate) polynomial regression, or (polyharmonic) spline regression, depending on the kernel. Importantly, this holds for both the predictive mean and the predictive variance, so that the posterior predictive distributions become equivalent. Our results have practical consequences: for instance, they show that optimal GP predictions in the sense of leave-one-out loss may occur at very large length-scales, which would be invisible to current implementations because of numerical difficulties.

math.ST↗

arXiv2021

Robust Eigenvectors of Symmetric Tensors

The tensor power method generalizes the matrix power method to higher order arrays, or tensors. Like in the matrix case, the fixed points of the tensor power method are the eigenvectors of the tensor. While every real symmetric matrix has an eigendecomposition, the vectors generating a symmetric decomposition of a real symmetric tensor are not always eigenvectors of the tensor. In this paper we show that whenever an eigenvector is a generator of the symmetric decomposition of a symmetric tensor, then (if the order of the tensor is sufficiently high) this eigenvector is robust, i.e., it is an attracting fixed point of the tensor power method. We exhibit new classes of symmetric tensors whose symmetric decomposition consists of eigenvectors. Generalizing orthogonally decomposable tensors, we consider equiangular tight frame decomposable and equiangular set decomposable tensors. Our main result implies that such tensors can be decomposed using the tensor power method.

math.NA↗

arXiv2021

Low-rank tensor recovery for Jacobian-based Volterra identification of parallel Wiener-Hammerstein systems

We consider the problem of identifying a parallel Wiener-Hammerstein structure from Volterra kernels. Methods based on Volterra kernels typically resort to coupled tensor decompositions of the kernels. However, in the case of parallel Wiener-Hammerstein systems, such methods require nontrivial constraints on the factors of the decompositions. In this paper, we propose an entirely different approach: by using special sampling (operating) points for the Jacobian of the nonlinear map from past inputs to the output, we can show that the Jacobian matrix becomes a linear projection of a tensor whose rank is equal to the number of branches. This representation allows us to solve the identification problem as a tensor recovery problem.

eess.SP↗

arXiv2021

Determinantal Point Processes in the Flat Limit

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix. In this paper, we study the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every other point, in a sense). We show that these limiting processes are best described in the formalism of extended L-ensembles and partial projection DPPs, and the exact limit depends mostly on the smoothness of the kernel function. In some cases, the limiting process is even universal, meaning that it does not depend on specifics of the kernel function, but only on its degree of smoothness. Since flat-limit DPPs are still repulsive processes, this implies that practically useful families of DPPs exist that do not require a spatial length-scale parameter.

math.PR↗

arXiv2021

Extended L-ensembles: a new representation for Determinantal Point Processes

Determinantal point processes (DPPs) are a class of repulsive point processes, popular for their relative simplicity. They are traditionally defined via their marginal distributions, but a subset of DPPs called "L-ensembles" have tractable likelihoods and are thus particularly easy to work with. Indeed, in many applications, DPPs are more naturally defined based on the L-ensemble formulation rather than through the marginal kernel. The fact that not all DPPs are L-ensembles is unfortunate, but there is a unifying description. We introduce here extended L-ensembles, and show that all DPPs are extended L-ensembles (and vice-versa). Extended L-ensembles have very simple likelihood functions, contain L-ensembles and projection DPPs as special cases. From a theoretical standpoint, they fix some pathologies in the usual formalism of DPPs, for instance the fact that projection DPPs are not L-ensembles. From a practical standpoint, they extend the set of kernel functions that may be used to define DPPs: we show that conditional positive definite kernels are good candidates for defining DPPs, including DPPs that need no spatial scale parameter. Finally, extended L-ensembles are based on so-called ``saddle-point matrices'', and we prove an extension of the Cauchy-Binet theorem for such matrices that may be of independent interest.

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