arXiv subjects

Herwig Wendt

Publications and source records attributed to Herwig Wendt.

At least 19 recordsLinked to original sources

arXiv2025

Bayesian Multifractal Image Segmentation

Multifractal analysis (MFA) provides a framework for the global characterization of image textures by describing the spatial fluctuations of their local regularity based on the multifractal spectrum. Several works have shown the interest of using MFA for the description of homogeneous textures in images. Nevertheless, natural images can be composed of several textures and, in turn, multifractal properties associated with those textures. This paper introduces an unsupervised Bayesian multifractal segmentation method to model and segment multifractal textures by jointly estimating the multifractal parameters and labels on images, at the pixel-level. For this, a computationally and statistically efficient multifractal parameter estimation model for wavelet leaders is firstly developed, defining different multifractality parameters for different regions of an image. Then, a multiscale Potts Markov random field is introduced as a prior to model the inherent spatial and scale correlations (referred to as cross-scale correlations) between the labels of the wavelet leaders. A Gibbs sampling methodology is finally used to draw samples from the posterior distribution of the unknown model parameters. Numerical experiments are conducted on synthetic multifractal images to evaluate the performance of the proposed segmentation approach. The proposed method achieves superior performance compared to traditional unsupervised segmentation techniques as well as modern deep learning-based approaches, showing its effectiveness for multifractal image segmentation.

eess.IV↗

arXiv2025

Efficient Ptychography Reconstruction using the Hessian operator

X-ray ptychography is a powerful and robust coherent imaging method providing access to the complex object and probe (illumination). Ptychography reconstruction is typically performed using first-order methods due to their computational efficiency. Higher-order methods, while potentially more accurate, are often prohibitively expensive in terms of computation. In this study, we present a mathematical framework for reconstruction using second-order information, derived from an efficient computation of the bilinear Hessian and Hessian operator. The formulation is provided for Gaussian based models, enabling the simultaneous reconstruction of the object, probe, and object positions. Synthetic data tests, along with experimental near-field ptychography data processing, demonstrate a ten-fold reduction in computation time compared to first-order methods. The derived formulas for computing the Hessians, along with the strategies for incorporating them into optimization schemes, are well-structured and easily adaptable to various ptychography problem formulations.

physics.optics↗

arXiv2025

Optimization Methods for Joint Eigendecomposition

Joint diagonalization, the process of finding a shared set of approximate eigenvectors for a collection of matrices, arises in diverse applications such as multidimensional harmonic analysis or quantum information theory. This task is typically framed as an optimization problem: minimizing a non-convex function that quantifies off-diagonal matrix elements across possible bases. In this work, we introduce a suite of efficient algorithms designed to locate local minimizers of this functional. Our methods leverage the Hessian's structure to bypass direct computation of second-order derivatives, evaluating it as either an operator or bilinear form - a strategy that remains computationally feasible even for large-scale applications. Additionally, we demonstrate that this Hessian-based information enables precise estimation of parameters, such as step-size, in first-order optimization techniques like Gradient Descent and Conjugate Gradient, and the design of second-order methods such as (Quasi-)Newton. The resulting algorithms for joint diagonalization outperform existing techniques, and we provide comprehensive numerical evidence of their superior performance.

math.OC↗

arXiv2025

The bilinear Hessian for large scale optimization

Second order information is useful in many ways in smooth optimization problems, including for the design of step size rules and descent directions, or the analysis of the local properties of the objective functional. However, the computation and storage of the Hessian matrix using second order partial derivatives is prohibitive in many contexts, and in particular in large scale problems. In this work, we propose a new framework for computing and presenting second order information in analytic form. The key novel insight is that the Hessian for a problem can be worked with efficiently by computing its bilinear form or operator form using Taylor expansions, instead of introducing a basis and then computing the Hessian matrix. Our new framework is suited for high-dimensional problems stemming e.g. from imaging applications, where computation of the Hessian matrix is unfeasible. We also show how this can be used to implement Newton's step rule, Daniel's Conjugate Gradient rule, or Quasi-Newton schemes, without explicit knowledge of the Hessian matrix, and illustrate our findings with a simple numerical experiment.

math.OC↗

arXiv2025

A spectral clustering-type algorithm for the consistent estimation of the Hurst distribution in moderately high dimensions

Scale invariance (fractality) is a prominent feature of the large-scale behavior of many stochastic systems. In this work, we construct an algorithm for the statistical identification of the Hurst distribution (in particular, the scaling exponents) undergirding a high-dimensional fractal system. The algorithm is based on wavelet random matrices, modified spectral clustering and a model selection step for picking the value of the clustering precision hyperparameter. In a moderately high-dimensional regime where the dimension, the sample size and the scale go to infinity, we show that the algorithm consistently estimates the Hurst distribution. Monte Carlo simulations show that the proposed methodology is efficient for realistic sample sizes and outperforms another popular clustering method based on mixed-Gaussian modeling. We apply the algorithm in the analysis of real-world macroeconomic time series to unveil evidence for cointegration.

stat.ME↗

arXiv2024

On joint eigen-decomposition of matrices

The problem of approximate joint diagonalization of a collection of matrices arises in a number of diverse engineering and signal processing problems. This problem is usually cast as an optimization problem, and it is the main goal of this publication to provide a theoretical study of the corresponding cost-functional. As our main result, we prove that this functional tends to infinity in the vicinity of rank-deficient matrices with probability one, thereby proving that the optimization problem is well posed. Secondly, we provide unified expressions for its higher-order derivatives in multilinear form, and explicit expressions for the gradient and the Hessian of the functional in standard form, thereby opening for new improved numerical schemes for the solution of the joint diagonalization problem. A special section is devoted to the important case of self-adjoint matrices.

math.NA↗

arXiv2024

On high-dimensional wavelet eigenanalysis

In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. We show that the $r$ largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge in probability to scale-invariant functions in the high-dimensional limit. By contrast, the remaining $p-r$ eigenvalues remain bounded in probability. Under additional assumptions, we show that the $r$ largest log-eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference.

math.ST↗

arXiv2024

In-Flight Estimation of Instrument Spectral Response Functions Using Sparse Representations

Accurate estimates of Instrument Spectral Response Functions (ISRFs) are crucial in order to have a good characterization of high resolution spectrometers. Spectrometers are composed of different optical elements that can induce errors in the measurements and therefore need to be modeled as accurately as possible. Parametric models are currently used to estimate these response functions. However, these models cannot always take into account the diversity of ISRF shapes that are encountered in practical applications. This paper studies a new ISRF estimation method based on a sparse representation of atoms belonging to a dictionary. This method is applied to different high-resolution spectrometers in order to assess its reproducibility for multiple remote sensing missions. The proposed method is shown to be very competitive when compared to the more commonly used parametric models, and yields normalized ISRF estimation errors less than 1%.

math.NA↗

arXiv2024

Multivariate selfsimilarity: Multiscale eigen-structures for selfsimilarity parameter estimation

Scale-free dynamics, formalized by selfsimilarity, provides a versatile paradigm massively and ubiquitously used to model temporal dynamics in real-world data. However, its practical use has mostly remained univariate so far. By contrast, modern applications often demand multivariate data analysis. Accordingly, models for multivariate selfsimilarity were recently proposed. Nevertheless, they have remained rarely used in practice because of a lack of available robust estimation procedures for the vector of selfsimilarity parameters. Building upon recent mathematical developments, the present work puts forth an efficient estimation procedure based on the theoretical study of the multiscale eigenstructure of the wavelet spectrum of multivariate selfsimilar processes. The estimation performance is studied theoretically in the asymptotic limits of large scale and sample sizes, and computationally for finite-size samples. As a practical outcome, a fully operational and documented multivariate signal processing estimation toolbox is made freely available and is ready for practical use on real-world data. Its potential benefits are illustrated in epileptic seizure prediction from multi-channel EEG data.

stat.ME↗

arXiv2024

On the empirical spectral distribution of large wavelet random matrices based on mixed-Gaussian fractional measurements in moderately high dimensions

In this paper, we characterize the convergence of the (rescaled logarithmic) empirical spectral distribution of wavelet random matrices. We assume a moderately high-dimensional framework where the sample size $n$, the dimension $p(n)$ and, for a fixed integer $j$, the scale $a(n)2^j$ go to infinity in such a way that $\lim_{n \rightarrow \infty}p(n)\cdot a(n)/n = \lim_{n \rightarrow \infty} o(\sqrt{a(n)/n})= 0$. We suppose the underlying measurement process is a random scrambling of a sample of size $n$ of a growing number $p(n)$ of fractional processes. Each of the latter processes is a fractional Brownian motion conditionally on a randomly chosen Hurst exponent. We show that the (rescaled logarithmic) empirical spectral distribution of the wavelet random matrices converges weakly, in probability, to the distribution of Hurst exponents.

math.PR↗

arXiv2022

Wavelet eigenvalue regression in high dimensions

In this paper, we construct the wavelet eigenvalue regression methodology in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. Building upon the asymptotic and large scale properties of wavelet random matrices in high dimensions, the wavelet eigenvalue regression is shown to be consistent and, under additional assumptions, asymptotically Gaussian in the estimation of the fractal structure of the system. We further construct a consistent estimator of the effective dimension $r$ of the system that significantly increases the robustness of the methodology. The estimation performance over finite samples is studied by means of simulations.

math.ST↗

arXiv2021

Learning grounded word meaning representations on similarity graphs

This paper introduces a novel approach to learn visually grounded meaning representations of words as low-dimensional node embeddings on an underlying graph hierarchy. The lower level of the hierarchy models modality-specific word representations through dedicated but communicating graphs, while the higher level puts these representations together on a single graph to learn a representation jointly from both modalities. The topology of each graph models similarity relations among words, and is estimated jointly with the graph embedding. The assumption underlying this model is that words sharing similar meaning correspond to communities in an underlying similarity graph in a low-dimensional space. We named this model Hierarchical Multi-Modal Similarity Graph Embedding (HM-SGE). Experimental results validate the ability of HM-SGE to simulate human similarity judgements and concept categorization, outperforming the state of the art.

cs.CL↗

arXiv2021

Graph Constrained Data Representation Learning for Human Motion Segmentation

Recently, transfer subspace learning based approaches have shown to be a valid alternative to unsupervised subspace clustering and temporal data clustering for human motion segmentation (HMS). These approaches leverage prior knowledge from a source domain to improve clustering performance on a target domain, and currently they represent the state of the art in HMS. Bucking this trend, in this paper, we propose a novel unsupervised model that learns a representation of the data and digs clustering information from the data itself. Our model is reminiscent of temporal subspace clustering, but presents two critical differences. First, we learn an auxiliary data matrix that can deviate from the initial data, hence confer more degrees of freedom to the coding matrix. Second, we introduce a regularization term for this auxiliary data matrix that preserves the local geometrical structure present in the high-dimensional space. The proposed model is efficiently optimized by using an original Alternating Direction Method of Multipliers (ADMM) formulation allowing to learn jointly the auxiliary data representation, a nonnegative dictionary and a coding matrix. Experimental results on four benchmark datasets for HMS demonstrate that our approach achieves significantly better clustering performance then state-of-the-art methods, including both unsupervised and more recent semi-supervised transfer learning approaches.

cs.CV↗

arXiv2020

Learning event representations for temporal segmentation of image sequences by dynamic graph embedding

Recently, self-supervised learning has proved to be effective to learn representations of events suitable for temporal segmentation in image sequences, where events are understood as sets of temporally adjacent images that are semantically perceived as a whole. However, although this approach does not require expensive manual annotations, it is data hungry and suffers from domain adaptation problems. As an alternative, in this work, we propose a novel approach for learning event representations named Dynamic Graph Embedding (DGE). The assumption underlying our model is that a sequence of images can be represented by a graph that encodes both semantic and temporal similarity. The key novelty of DGE is to learn jointly the graph and its graph embedding. At its core, DGE works by iterating over two steps: 1) updating the graph representing the semantic and temporal similarity of the data based on the current data representation, and 2) updating the data representation to take into account the current data graph structure. The main advantage of DGE over state-of-the-art self-supervised approaches is that it does not require any training set, but instead learns iteratively from the data itself a low-dimensional embedding that reflects their temporal and semantic similarity. Experimental results on two benchmark datasets of real image sequences captured at regular time intervals demonstrate that the proposed DGE leads to event representations effective for temporal segmentation. In particular, it achieves robust temporal segmentation on the EDUBSeg and EDUBSeg-Desc benchmark datasets, outperforming the state of the art. Additional experiments on two Human Motion Segmentation benchmark datasets demonstrate the generalization capabilities of the proposed DGE.

cs.LG↗

arXiv2019

Enhancing temporal segmentation by nonlocal self-similarity

Temporal segmentation of untrimmed videos and photo-streams is currently an active area of research in computer vision and image processing. This paper proposes a new approach to improve the temporal segmentation of photo-streams. The method consists in enhancing image representations by encoding long-range temporal dependencies. Our key contribution is to take advantage of the temporal stationarity assumption of photostreams for modeling each frame by its nonlocal self-similarity function. The proposed approach is put to test on the EDUB-Seg dataset, a standard benchmark for egocentric photostream temporal segmentation. Starting from seven different (CNN based) image features, the method yields consistent improvements in event segmentation quality, leading to an average increase of F-measure of 3.71% with respect to the state of the art.

eess.IV↗

arXiv2018

A Generalized Multifractal Formalism for the Estimation of Nonconcave Multifractal Spectra

Multifractal analysis has become a powerful signal processing tool that characterizes signals or images via the fluctuations of their pointwise regularity, quantified theoretically by the so-called multifractal spectrum. The practical estimation of the multifractal spectrum fundamentally relies on exploiting the scale dependence of statistical properties of appropriate multiscale quantities, such as wavelet leaders, that can be robustly computed from discrete data. Despite successes of multifractal analysis in various real-world applications, current estimation procedures remain essentially limited to providing concave upper-bound estimates, while there is a priori no reason for the multifractal spectrum to be a concave function. This work addresses this severe practical limitation and proposes a novel formalism for multifractal analysis that enables nonconcave multifractal spectra to be estimated in a stable way. The key contributions reside in the development and theoretical study of a generalized multifractal formalism to assess the multiscale statistics of wavelet leaders, and in devising a practical algorithm that permits this formalism to be applied to real-world data, allowing for the estimation of nonconcave multifractal spectra. Numerical experiments are conducted on several synthetic multifractal processes as well as on a real-world remote-sensing image and demonstrate the benefits of the proposed multifractal formalism over the state of the art.

math.FA↗

arXiv2018

A Quasi-Newton algorithm on the orthogonal manifold for NMF with transform learning

Nonnegative matrix factorization (NMF) is a popular method for audio spectral unmixing. While NMF is traditionally applied to off-the-shelf time-frequency representations based on the short-time Fourier or Cosine transforms, the ability to learn transforms from raw data attracts increasing attention. However, this adds an important computational overhead. When assumed orthogonal (like the Fourier or Cosine transforms), learning the transform yields a non-convex optimization problem on the orthogonal matrix manifold. In this paper, we derive a quasi-Newton method on the manifold using sparse approximations of the Hessian. Experiments on synthetic and real audio data show that the proposed algorithm out-performs state-of-the-art first-order and coordinate-descent methods by orders of magnitude. A Python package for fast TL-NMF is released online at https://github.com/pierreablin/tlnmf.

stat.ML↗

arXiv2017

Nonnegative Matrix Factorization with Transform Learning

Traditional NMF-based signal decomposition relies on the factorization of spectral data, which is typically computed by means of short-time frequency transform. In this paper we propose to relax the choice of a pre-fixed transform and learn a short-time orthogonal transform together with the factorization. To this end, we formulate a regularized optimization problem reminiscent of conventional NMF, yet with the transform as additional unknown parameters, and design a novel block-descent algorithm enabling to find stationary points of this objective function. The proposed joint transform learning and factorization approach is tested for two audio signal processing experiments, illustrating its conceptual and practical benefits.

cs.LG↗