arXiv subjects

José Henrique de Morais Goulart

Publications and source records attributed to José Henrique de Morais Goulart.

13 recordsLinked to original sources

arXiv2026

Whitening Spherical Gaussian Mixtures in the Large-Dimensional Regime

Whitening is a classical technique in unsupervised learning that can facilitate estimation tasks by standardizing data. An important application is the estimation of latent variable models via the decomposition of tensors built from high-order moments. In particular, whitening orthogonalizes the means of a spherical Gaussian mixture model (GMM), thereby making the corresponding moment tensor orthogonally decomposable, hence easier to decompose. However, in the large-dimensional regime (LDR) where data are high-dimensional and scarce, the standard whitening matrix built from the sample covariance becomes ineffective because the latter is spectrally distorted. Consequently, whitened means of a spherical GMM are no longer orthogonal. Using random matrix theory, we derive exact limits for their dot products, which are generally nonzero in the LDR. As our main contribution, we then construct a corrected whitening matrix that restores asymptotic orthogonality, allowing for performance gains in spherical GMM estimation.

stat.ML↗

arXiv2025

Online simplex-structured matrix factorization

Simplex-structured matrix factorization (SSMF) is a common task encountered in signal processing and machine learning. Minimum-volume constrained unmixing (MVCU) algorithms are among the most widely used methods to perform this task. While MVCU algorithms generally perform well in an offline setting, their direct application to online scenarios suffers from scalability limitations due to memory and computational demands. To overcome these limitations, this paper proposes an approach which can build upon any off-the-shelf MVCU algorithm to operate sequentially, i.e., to handle one observation at a time. The key idea of the proposed method consists in updating the solution of MVCU only when necessary, guided by an online check of the corresponding optimization problem constraints. It only stores and processes observations identified as informative with respect to the geometrical constraints underlying SSMF. We demonstrate the effectiveness of the approach when analyzing synthetic and real datasets, showing that it achieves estimation accuracy comparable to the offline MVCU method upon which it relies, while significantly reducing the computational cost.

eess.SP↗

arXiv2024

On-the-fly spectral unmixing based on Kalman filtering

This work introduces an on-the-fly (i.e., online) linear unmixing method which is able to sequentially analyze spectral data acquired on a spectrum-by-spectrum basis. After deriving a sequential counterpart of the conventional linear mixing model, the proposed approach recasts the linear unmixing problem into a linear state-space estimation framework. Under Gaussian noise and state models, the estimation of the pure spectra can be efficiently conducted by resorting to Kalman filtering. Interestingly, it is shown that this Kalman filter can operate in a lower-dimensional subspace while ensuring the nonnegativity constraint inherent to pure spectra. This dimensionality reduction allows significantly lightening the computational burden, while leveraging recent advances related to the representation of essential spectral information. The proposed method is evaluated through extensive numerical experiments conducted on synthetic and real Raman data sets. The results show that this Kalman filter-based method offers a convenient trade-off between unmixing accuracy and computational efficiency, which is crucial for operating in an on-the-fly setting. To the best of the authors' knowledge, this is the first operational method which is able to solve the spectral unmixing problem efficiently in a dynamic fashion. It also constitutes a valuable building block for benefiting from acquisition and processing frameworks recently proposed in the microscopy literature, which are motivated by practical issues such as reducing acquisition time and avoiding potential damages being inflicted to photosensitive samples.

eess.SP↗

arXiv2024

Majorization-minimization for Sparse Nonnegative Matrix Factorization with the $β$-divergence

This article introduces new multiplicative updates for nonnegative matrix factorization with the $β$-divergence and sparse regularization of one of the two factors (say, the activation matrix). It is well known that the norm of the other factor (the dictionary matrix) needs to be controlled in order to avoid an ill-posed formulation. Standard practice consists in constraining the columns of the dictionary to have unit norm, which leads to a nontrivial optimization problem. Our approach leverages a reparametrization of the original problem into the optimization of an equivalent scale-invariant objective function. From there, we derive block-descent majorization-minimization algorithms that result in simple multiplicative updates for either $\ell_{1}$-regularization or the more "aggressive" log-regularization. In contrast with other state-of-the-art methods, our algorithms are universal in the sense that they can be applied to any $β$-divergence (i.e., any value of $β$) and that they come with convergence guarantees. We report numerical comparisons with existing heuristic and Lagrangian methods using various datasets: face images, an audio spectrogram, hyperspectral data, and song play counts. We show that our methods obtain solutions of similar quality at convergence (similar objective values) but with significantly reduced CPU times.

cs.LG↗

arXiv2024

Performance Gaps in Multi-view Clustering under the Nested Matrix-Tensor Model

We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically examined the performance of a tensor-based approach, which relies on finding a best rank-one approximation, a problem known to be computationally hard. A tractable alternative approach consists in computing instead the best rank-one (matrix) approximation of an unfolding of the observed tensor data, but its performance was hitherto unknown. We quantify here the performance gap between these two approaches, in particular by deriving the precise algorithmic threshold of the unfolding approach and demonstrating that it exhibits a BBP-type transition behavior. This work is therefore in line with recent contributions which deepen our understanding of why tensor-based methods surpass matrix-based methods in handling structured tensor data.

stat.ML↗

arXiv2023

On the Accuracy of Hotelling-Type Asymmetric Tensor Deflation: A Random Tensor Analysis

This work introduces an asymptotic study of Hotelling-type tensor deflation in the presence of noise, in the regime of large tensor dimensions. Specifically, we consider a low-rank asymmetric tensor model of the form $\sum_{i=1}^r β_i{\mathcal{A}}_i + {\mathcal{W}}$ where $β_i\geq 0$ and the ${\mathcal{A}}_i$'s are unit-norm rank-one tensors such that $\left| \langle {\mathcal{A}}_i, {\mathcal{A}}_j \rangle \right| \in [0, 1]$ for $i\neq j$ and ${\mathcal{W}}$ is an additive noise term. Assuming that the dominant components are successively estimated from the noisy observation and subsequently subtracted, we leverage recent advances in random tensor theory in the regime of asymptotically large tensor dimensions to analytically characterize the estimated singular values and the alignment of estimated and true singular vectors at each step of the deflation procedure. Furthermore, this result can be used to construct estimators of the signal-to-noise ratios $β_i$ and the alignments between the estimated and true rank-1 signal components.

stat.ML↗

arXiv2023

Hotelling Deflation on Large Symmetric Spiked Tensors

This paper studies the deflation algorithm when applied to estimate a low-rank symmetric spike contained in a large tensor corrupted by additive Gaussian noise. Specifically, we provide a precise characterization of the large-dimensional performance of deflation in terms of the alignments of the vectors obtained by successive rank-1 approximation and of their estimated weights, assuming non-trivial (fixed) correlations among spike components. Our analysis allows an understanding of the deflation mechanism in the presence of noise and can be exploited for designing more efficient signal estimation methods.

stat.ML↗

arXiv2023

Joint Majorization-Minimization for Nonnegative Matrix Factorization with the $β$-divergence

This article proposes new multiplicative updates for nonnegative matrix factorization (NMF) with the $β$-divergence objective function. Our new updates are derived from a joint majorization-minimization (MM) scheme, in which an auxiliary function (a tight upper bound of the objective function) is built for the two factors jointly and minimized at each iteration. This is in contrast with the classic approach in which a majorizer is derived for each factor separately. Like that classic approach, our joint MM algorithm also results in multiplicative updates that are simple to implement. They however yield a significant drop of computation time (for equally good solutions), in particular for some $β$-divergences of important applicative interest, such as the squared Euclidean distance and the Kullback-Leibler or Itakura-Saito divergences. We report experimental results using diverse datasets: face images, an audio spectrogram, hyperspectral data and song play counts. Depending on the value of $β$ and on the dataset, our joint MM approach can yield CPU time reductions from about $13\%$ to $78\%$ in comparison to the classic alternating scheme.

cs.LG↗

arXiv2022

COL0RME: COvariance-based $\ell_0$ super-Resolution Microscopy with intensity Estimation

Super-resolution light microscopy overcomes the physical barriers due to light diffraction, allowing for the observation of otherwise indistinguishable subcellular entities. However, the specific acquisition conditions required by state-of-the-art super-resolution methods to achieve adequate spatio-temporal resolution are often very challenging. Exploiting molecules fluctuations allows good spatio-temporal resolution live-cell imaging by means of common microscopes and conventional fluorescent dyes. In this work, we present the method COL0RME for COvariance-based $\ell_0$ super-Resolution Microscopy with intensity Estimation. It codifies the assumption of sparse distribution of the fluorescent molecules as well as the temporal and spatial independence between emitters via a non-convex optimization problem formulated in the covariance domain. In order to deal with real data, the proposed approach also estimates background and noise statistics. It also includes a final estimation step where intensity information is retrieved, which is valuable for biological interpretation and future applications to super-resolution imaging.

math.OC↗

arXiv2022

A Random Matrix Perspective on Random Tensors

Tensor models play an increasingly prominent role in many fields, notably in machine learning. In several applications, such as community detection, topic modeling and Gaussian mixture learning, one must estimate a low-rank signal from a noisy tensor. Hence, understanding the fundamental limits of estimators of that signal inevitably calls for the study of random tensors. Substantial progress has been recently achieved on this subject in the large-dimensional limit. Yet, some of the most significant among these results--in particular, a precise characterization of the abrupt phase transition (with respect to signal-to-noise ratio) that governs the performance of the maximum likelihood (ML) estimator of a symmetric rank-one model with Gaussian noise--were derived based of mean-field spin glass theory, which is not easily accessible to non-experts. In this work, we develop a sharply distinct and more elementary approach, relying on standard but powerful tools brought by years of advances in random matrix theory. The key idea is to study the spectra of random matrices arising from contractions of a given random tensor. We show how this gives access to spectral properties of the random tensor itself. For the aforementioned rank-one model, our technique yields a hitherto unknown fixed-point equation whose solution precisely matches the asymptotic performance of the ML estimator above the phase transition threshold in the third-order case. A numerical verification provides evidence that the same holds for orders 4 and 5, leading us to conjecture that, for any order, our fixed-point equation is equivalent to the known characterization of the ML estimation performance that had been obtained by relying on spin glasses. Moreover, our approach sheds light on certain properties of the ML problem landscape in large dimensions and can be extended to other models, such as asymmetric and non-Gaussian.

stat.ML↗

arXiv2022

COL0RME: Super-resolution microscopy based on sparse blinking/fluctuating fluorophore localization and intensity estimation

To overcome the physical barriers caused by light diffraction, super-resolution techniques are often applied in fluorescence microscopy. State-of-the-art approaches require specific and often demanding acquisition conditions to achieve adequate levels of both spatial and temporal resolution. Analyzing the stochastic fluctuations of the fluorescent molecules provides a solution to the aforementioned limitations, as sufficiently high spatio-temporal resolution for live-cell imaging can be achieved by using common microscopes and conventional fluorescent dyes. Based on this idea, we present COL0RME, a method for COvariance-based $\ell_0$ super-Resolution Microscopy with intensity Estimation, which achieves good spatio-temporal resolution by solving a sparse optimization problem in the covariance domain and discuss automatic parameter selection strategies. The method is composed of two steps: the former where both the emitters' independence and the sparse distribution of the fluorescent molecules are exploited to provide an accurate localization; the latter where real intensity values are estimated given the computed support. The paper is furnished with several numerical results both on synthetic and real fluorescence microscopy images and several comparisons with state-of-the art approaches are provided. Our results show that COL0RME outperforms competing methods exploiting analogously temporal fluctuations; in particular, it achieves better localization, reduces background artifacts and avoids fine parameter tuning.

math.OC↗

arXiv2021

On the Realization of Impulse Invariant Bilinear Volterra Kernels

As previously shown, the direct extension of the impulse invariance principle to Volterra kernels has to be modified in order to provide a condition for the exact modeling of mixed-signal chains. At first sight this would seem to seriously complicate the otherwise simple discrete-time realization of separable kernels (among which bilinear kernels are of particular importance). We show here, however, that this not the case. By defining a cascade operator, the structure of a generalized impulse invariance can be unveiled, leading to a realization without an inordinate increase in computational complexity.

eess.SP↗

arXiv2018

On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition

Under the action of the general linear group with tensor structure, the ranks of matrices $A$ and $B$ forming an $m \times n$ pencil $A + λB$ can change, but in a restricted manner. Specifically, with every pencil one can associate a pair of minimal ranks, which is unique up to a permutation. This notion can be defined for matrix pencils and, more generally, also for matrix polynomials of arbitrary degree. In this paper, we provide a formal definition of the minimal ranks, discuss its properties and the natural hierarchy it induces in a pencil space. Then, we show how the minimal ranks of a pencil can be determined from its Kronecker canonical form. For illustration, we classify the orbits according to their minimal ranks (under the action of the general linear group) in the case of real pencils with $m, n \le 4$. Subsequently, we show that real regular $2k \times 2k$ pencils having only complex-valued eigenvalues, which form an open positive-volume set, do not admit a best approximation (in the norm topology) on the set of real pencils whose minimal ranks are bounded by $2k-1$. Our results can be interpreted from a tensor viewpoint, where the minimal ranks of a degree-$(d-1)$ matrix polynomial characterize the minimal ranks of matrices constituting a block-term decomposition of an $m \times n \times d$ tensor into a sum of matrix-vector tensor products.

math.NA↗