Rate-Distortion via Markov Chain Monte Carlo

We propose an approach to lossy source coding, utilizing ideas from Gibbs sampling, simulated annealing, and Markov Chain Monte Carlo (MCMC). The idea is to sample a reconstruction sequence from a Boltzmann distribution associated with an energy function that incorporates the distortion between the source and reconstruction, the compressibility of the reconstruction, and the point sought on the rate-distortion curve. To sample from this distribution, we use a `heat bath algorithm': Starting from an initial candidate reconstruction (say the original source sequence), at every iteration, an index i is chosen and the i-th sequence component is replaced by drawing from the conditional probability distribution for that component given all the rest. At the end of this process, the encoder conveys the reconstruction to the decoder using universal lossless compression. The complexity of each iteration is independent of the sequence length and only linearly dependent on a certain context parameter (which grows sub-logarithmically with the sequence length). We show that the proposed algorithms achieve optimum rate-distortion performance in the limits of large number of iterations, and sequence length, when employed on any stationary ergodic source. Experimentation shows promising initial results. Employing our lossy compressors on noisy data, with appropriately chosen distortion measure and level, followed by a simple de-randomization operation, results in a family of denoisers that compares favorably (both theoretically and in practice) with other MCMC-based schemes, and with the Discrete Universal Denoiser (DUDE).Comment: 35 pages, 16 figures, Submitted to IEEE Transactions on Information Theor

Estimation of the Rate-Distortion Function

Motivated by questions in lossy data compression and by theoretical considerations, we examine the problem of estimating the rate-distortion function of an unknown (not necessarily discrete-valued) source from empirical data. Our focus is the behavior of the so-called "plug-in" estimator, which is simply the rate-distortion function of the empirical distribution of the observed data. Sufficient conditions are given for its consistency, and examples are provided to demonstrate that in certain cases it fails to converge to the true rate-distortion function. The analysis of its performance is complicated by the fact that the rate-distortion function is not continuous in the source distribution; the underlying mathematical problem is closely related to the classical problem of establishing the consistency of maximum likelihood estimators. General consistency results are given for the plug-in estimator applied to a broad class of sources, including all stationary and ergodic ones. A more general class of estimation problems is also considered, arising in the context of lossy data compression when the allowed class of coding distributions is restricted; analogous results are developed for the plug-in estimator in that case. Finally, consistency theorems are formulated for modified (e.g., penalized) versions of the plug-in, and for estimating the optimal reproduction distribution.Comment: 18 pages, no figures [v2: removed an example with an error; corrected typos; a shortened version will appear in IEEE Trans. Inform. Theory

Recovery from Linear Measurements with Complexity-Matching Universal Signal Estimation

We study the compressed sensing (CS) signal estimation problem where an input signal is measured via a linear matrix multiplication under additive noise. While this setup usually assumes sparsity or compressibility in the input signal during recovery, the signal structure that can be leveraged is often not known a priori. In this paper, we consider universal CS recovery, where the statistics of a stationary ergodic signal source are estimated simultaneously with the signal itself. Inspired by Kolmogorov complexity and minimum description length, we focus on a maximum a posteriori (MAP) estimation framework that leverages universal priors to match the complexity of the source. Our framework can also be applied to general linear inverse problems where more measurements than in CS might be needed. We provide theoretical results that support the algorithmic feasibility of universal MAP estimation using a Markov chain Monte Carlo implementation, which is computationally challenging. We incorporate some techniques to accelerate the algorithm while providing comparable and in many cases better reconstruction quality than existing algorithms. Experimental results show the promise of universality in CS, particularly for low-complexity sources that do not exhibit standard sparsity or compressibility.Comment: 29 pages, 8 figure

Bayesian plug & play methods for inverse problems in imaging.

Thèse de Doctorat de Mathématiques Appliquées (Université de Paris)Tesis de Doctorado en Ingeniería Eléctrica (Universidad de la República)This thesis deals with Bayesian methods for solving ill-posed inverse problems in imaging with learnt image priors. The first part of this thesis (Chapter 3) concentrates on two particular problems, namely joint denoising and decompression and multi-image super-resolution. After an extensive study of the noise statistics for these problem in the transformed (wavelet or Fourier) domain, we derive two novel algorithms to solve this particular inverse problem. One of them is based on a multi-scale self-similarity prior and can be seen as a transform-domain generalization of the celebrated non-local bayes algorithm to the case of non-Gaussian noise. The second one uses a neural-network denoiser to implicitly encode the image prior, and a splitting scheme to incorporate this prior into an optimization algorithm to find a MAP-like estimator. The second part of this thesis concentrates on the Variational AutoEncoder (VAE) model and some of its variants that show its capabilities to explicitly capture the probability distribution of high-dimensional datasets such as images. Based on these VAE models, we propose two ways to incorporate them as priors for general inverse problems in imaging : • The first one (Chapter 4) computes a joint (space-latent) MAP estimator named Joint Posterior Maximization using an Autoencoding Prior (JPMAP). We show theoretical and experimental evidence that the proposed objective function satisfies a weak bi-convexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other non-convex MAP approaches which more often get stuck in spurious local optima. • The second one (Chapter 5) develops a Gibbs-like posterior sampling algorithm for the exploration of posterior distributions of inverse problems using multiple chains and a VAE as image prior. We showhowto use those samples to obtain MMSE estimates and their corresponding uncertainty.Cette thèse traite des méthodes bayésiennes pour résoudre des problèmes inverses mal posés en imagerie avec des distributions a priori d’images apprises. La première partie de cette thèse (Chapitre 3) se concentre sur deux problèmes partic-uliers, à savoir le débruitage et la décompression conjoints et la super-résolutionmulti-images. Après une étude approfondie des statistiques de bruit pour ces problèmes dans le domaine transformé (ondelettes ou Fourier), nous dérivons deuxnouveaux algorithmes pour résoudre ce problème inverse particulie. L’un d’euxest basé sur une distributions a priori d’auto-similarité multi-échelle et peut êtrevu comme une généralisation du célèbre algorithme de Non-Local Bayes au cas dubruit non gaussien. Le second utilise un débruiteur de réseau de neurones pourcoder implicitement la distribution a priori, et un schéma de division pour incor-porer cet distribution dans un algorithme d’optimisation pour trouver un estima-teur de type MAP. La deuxième partie de cette thèse se concentre sur le modèle Variational Auto Encoder (VAE) et certaines de ses variantes qui montrent ses capacités à capturer explicitement la distribution de probabilité d’ensembles de données de grande dimension tels que les images. Sur la base de ces modèles VAE, nous proposons deuxmanières de les incorporer comme distribution a priori pour les problèmes inverses généraux en imagerie: •Le premier (Chapitre 4) calcule un estimateur MAP conjoint (espace-latent) nommé Joint Posterior Maximization using an Autoencoding Prior (JPMAP). Nous montrons des preuves théoriques et expérimentales que la fonction objectif proposée satisfait une propriété de bi-convexité faible qui est suffisante pour garantir que notre schéma d’optimisation converge vers un pointstationnaire. Les résultats expérimentaux montrent également la meilleurequalité des solutions obtenues par notre approche JPMAP par rapport à d’autresapproches MAP non convexes qui restent le plus souvent bloquées dans desminima locaux. •Le second (Chapitre 5) développe un algorithme d’échantillonnage a poste-riori de type Gibbs pour l’exploration des distributions a posteriori de problèmes inverses utilisant des chaînes multiples et un VAE comme distribution a priori. Nous montrons comment utiliser ces échantillons pour obtenir desestimations MMSE et leur incertitude correspondante.En esta tesis se estudian métodos bayesianos para resolver problemas inversos mal condicionados en imágenes usando distribuciones a priori entrenadas. La primera parte de esta tesis (Capítulo 3) se concentra en dos problemas particulares, a saber, el de eliminación de ruido y descompresión conjuntos, y el de superresolución a partir de múltiples imágenes. Después de un extenso estudio de las estadísticas del ruido para estos problemas en el dominio transformado (wavelet o Fourier),derivamos dos algoritmos nuevos para resolver este problema inverso en particular. Uno de ellos se basa en una distribución a priori de autosimilitud multiescala y puede verse como una generalización al dominio wavelet del célebre algoritmo Non-Local Bayes para el caso de ruido no Gaussiano. El segundo utiliza un algoritmo de eliminación de ruido basado en una red neuronal para codificar implícitamente la distribución a priori de las imágenes y un esquema de relajación para incorporar esta distribución en un algoritmo de optimización y así encontrar un estimador similar al MAP. La segunda parte de esta tesis se concentra en el modelo Variational AutoEncoder (VAE) y algunas de sus variantes que han mostrado capacidad para capturar explícitamente la distribución de probabilidad de conjuntos de datos en alta dimensión como las imágenes. Basándonos en estos modelos VAE, proponemos dos formas de incorporarlos como distribución a priori para problemas inversos genéricos en imágenes : •El primero (Capítulo 4) calcula un estimador MAP conjunto (espacio imagen y latente) llamado Joint Posterior Maximization using an Autoencoding Prior (JPMAP). Mostramos evidencia teórica y experimental de que la función objetivo propuesta satisface una propiedad de biconvexidad débil que es suficiente para garantizar que nuestro esquema de optimización converge a un punto estacionario. Los resultados experimentales también muestran la mayor calidad de las soluciones obtenidas por nuestro enfoque JPMAP con respecto a otros enfoques MAP no convexos que a menudo se atascan en mínimos locales espurios. •El segundo (Capítulo 5) desarrolla un algoritmo de muestreo tipo Gibbs parala exploración de la distribución a posteriori de problemas inversos utilizando múltiples cadenas y un VAE como distribución a priori. Mostramos cómo usar esas muestras para obtener estimaciones de MMSE y su correspondiente incertidumbr