The Polynomial Eigenvalue Problem is Well Conditioned for Random Inputs

We compute the exact expected value of the squared condition number for the polynomial eigenvalue problem, when the input matrices have entries coming from the standard complex Gaussian distribution, showing that in general this problem is quite well conditioned.The first author's work was partially supported by Agencia Nacional de Investigación e Innovación (ANII), Uruguay, and by CSIC group 618, Universidad de La República, Uruguay. The second author's work was partially supported by MTM2017-83816-P and MTM2017-90682-REDT from Spanish Ministry of Science MICINN and by 21.SI01.64658 from Universidad de Cantabria and Banco de Santander

Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions

We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford's trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions.Comment: Published in at http://dx.doi.org/10.1214/08-AOP445 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Pairwise MRF Calibration by Perturbation of the Bethe Reference Point

We investigate different ways of generating approximate solutions to the pairwise Markov random field (MRF) selection problem. We focus mainly on the inverse Ising problem, but discuss also the somewhat related inverse Gaussian problem because both types of MRF are suitable for inference tasks with the belief propagation algorithm (BP) under certain conditions. Our approach consists in to take a Bethe mean-field solution obtained with a maximum spanning tree (MST) of pairwise mutual information, referred to as the \emph{Bethe reference point}, for further perturbation procedures. We consider three different ways following this idea: in the first one, we select and calibrate iteratively the optimal links to be added starting from the Bethe reference point; the second one is based on the observation that the natural gradient can be computed analytically at the Bethe point; in the third one, assuming no local field and using low temperature expansion we develop a dual loop joint model based on a well chosen fundamental cycle basis. We indeed identify a subclass of planar models, which we refer to as \emph{Bethe-dual graph models}, having possibly many loops, but characterized by a singly connected dual factor graph, for which the partition function and the linear response can be computed exactly in respectively O(N) and $O(N^2)$ operations, thanks to a dual weight propagation (DWP) message passing procedure that we set up. When restricted to this subclass of models, the inverse Ising problem being convex, becomes tractable at any temperature. Experimental tests on various datasets with refined $L_0$ or $L_1$ regularization procedures indicate that these approaches may be competitive and useful alternatives to existing ones.Comment: 54 pages, 8 figure. section 5 and refs added in V

Scaling limits of k-ary growing trees

For each integer $k \geq 2$, we introduce a sequence of $k$-ary discrete trees constructed recursively by choosing at each step an edge uniformly among the present edges and grafting on "its middle" $k-1$ new edges. When $k=2$, this corresponds to a well-known algorithm which was first introduced by R\'emy. Our main result concerns the asymptotic behavior of these trees as $n$ becomes large: for all $k$, the sequence of $k$-ary trees grows at speed $n^{1/k}$ towards a $k$-ary random real tree that belongs to the family of self-similar fragmentation trees. This convergence is proved with respect to the Gromov-Hausdorff-Prokhorov topology. We also study embeddings of the limiting trees when $k$ varies

Compressed sensing and finite rate of innovation for efficient data acquisition of quantitative acoustic microscopy images

La microscopie acoustique quantitative (MAQ) est une modalité d'imagerie bien établie qui donne accès à des cartes paramétriques 2D représentatives des propriétés mécaniques des tissus à une échelle microscopique. Dans la plupart des études sur MAQ, l'échantillons est scanné ligne par ligne (avec un pas de 2µm) à l'aide d'un transducteur à 250 MHz. Ce type d'acquisition permet d'obtenir un cube de données RF 3D, avec deux dimensions spatiales et une dimension temporelle. Chaque signal RF correspondant à une position spatiale dans l'échantillon permet d'estimer des paramètres acoustiques comme par exemple la vitesse du son ou l'impédance. Le temps d'acquisition en MAQ est directement proportionnel à la taille de l'échantillon et peut aller de quelques minutes à quelques dizaines de minutes. Afin d'assurer des conditions d'acquisition stables et étant donnée la sensibilité des échantillons à ces conditions, diminuer le temps d'acquisition est un des grand défis en MAQ. Afin de relever ce défi, ce travail de thèse propose plusieurs solutions basées sur l'échantillonnage compressé (EC) et la théories des signaux ayant un faible nombre de degré de liberté (finite rate of innovation - FRI, en anglais). Le principe de l'EC repose sur la parcimonie des données, sur l'échantillonnage incohérent de celles-ci et sur les algorithmes d'optimisation numérique. Dans cette thèse, les phénomènes physiques derrière la MAQ sont exploités afin de créer des modèles adaptés aux contraintes de l'EC et de la FRI. Plus particulièrement, ce travail propose plusieurs pistes d'application de l'EC en MAQ : un schéma d'acquisition spatiale innovant, un algorithme de reconstruction d'images exploitant les statistiques des coefficients en ondelettes des images paramétriques, un modèle FRI adapté aux signaux RF et un schéma d'acquisition compressée dans le domaine temporel.Quantitative acoustic microscopy (QAM) is a well-accepted modality for forming 2D parameter maps making use of mechanical properties of soft tissues at microscopic scales. In leading edge QAM studies, the sample is raster-scanned (spatial step size of 2µm) using a 250 MHz transducer resulting in a 3D RF data cube, and each RF signal for each spatial location is processed to obtain acoustic parameters, e.g., speed of sound or acoustic impedance. The scanning time directly depends on the sample size and can range from few minutes to tens of minutes. In order to maintain constant experimental conditions for the sensitive thin sectioned samples, the scanning time is an important practical issue. To deal with the current challenge, we propose the novel approach inspired by compressed sensing (CS) and finite rate of innovation (FRI). The success of CS relies on the sparsity of data under consideration, incoherent measurement and optimization technique. On the other hand, the idea behind FRI is supported by a signal model fully characterized as a limited number of parameters. From this perspective, taking into account the physics leading to data acquisition of QAM system, the QAM data can be regarded as an adequate application amenable to the state of the art technologies aforementioned. However, when it comes to the mechanical structure of QAM system which does not support canonical CS measurement manners on the one hand, and the compositions of the RF signal model unsuitable to existing FRI schemes on the other hand, the advanced frameworks are still not perfect methods to resolve the problems that we are facing. In this thesis, to overcome the limitations, a novel sensing framework for CS is presented in spatial domain: a recently proposed approximate message passing (AMP) algorithm is adapted to account for the underlying data statistics of samples sparsely collected by proposed scanning patterns. In time domain, as an approach for achieving an accurate recovery from a small set of samples of QAM RF signals, we employ sum of sincs (SoS) sampling kernel and autoregressive (AR) model estimator. The spiral scanning manner, introduced as an applicable sensing technique to QAM system, contributed to the significant reduction of the number of spatial samples when reconstructing speed of sound images of a human lymph node. Furthermore, the scanning time was also hugely saved due to the merit of the mechanical movement of the proposed sensing pattern. Together with the achievement in spatial domain, the introduction of SoS kernel and AR estimator responsible for an innovation rate sampling and a parameter estimation respectively led to dramatic reduction of the required number of samples per RF signal compared to a conventional approach. Finally, we showed that both data acquisition frameworks based on the CS and FRI can be combined into a single spatio-temporal solution to maximize the benefits stated above

Breaking the waves: asymmetric random periodic features for low-bitrate kernel machines

Many signal processing and machine learning applications are built from evaluating a kernel on pairs of signals, e.g. to assess the similarity of an incoming query to a database of known signals. This nonlinear evaluation can be simplified to a linear inner product of the random Fourier features of those signals: random projections followed by a periodic map, the complex exponential. It is known that a simple quantization of those features (corresponding to replacing the complex exponential by a different periodic map that takes binary values, which is appealing for their transmission and storage), distorts the approximated kernel, which may be undesirable in practice. Our take-home message is that when the features of only one of the two signals are quantized, the original kernel is recovered without distortion; its practical interest appears in several cases where the kernel evaluations are asymmetric by nature, such as a client-server scheme. Concretely, we introduce the general framework of asymmetric random periodic features, where the two signals of interest are observed through random periodic features: random projections followed by a general periodic map, which is allowed to be different for both signals. We derive the influence of those periodic maps on the approximated kernel, and prove uniform probabilistic error bounds holding for all signal pairs from an infinite low-complexity set. Interestingly, our results allow the periodic maps to be discontinuous, thanks to a new mathematical tool, i.e. the mean Lipschitz smoothness. We then apply this generic framework to semi-quantized kernel machines (where only one signal has quantized features and the other has classical random Fourier features), for which we show theoretically that the approximated kernel remains unchanged (with the associated error bound), and confirm the power of the approach with numerical simulations

Compressive sensing using the modified entropy functional

Cataloged from PDF version of article.In most compressive sensing problems, 1 norm is used during the signal reconstruction process. In this article, a modified version of the entropy functional is proposed to approximate the 1 norm. The proposed modified version of the entropy functional is continuous, differentiable and convex. Therefore, it is possible to construct globally convergent iterative algorithms using Bregman’s row-action method for compressive sensing applications. Simulation examples with both 1D signals and images are presented. © 2013 Elsevier Inc. All rights reserved