On sparsity averaging

Recent developments in Carrillo et al. (2012) and Carrillo et al. (2013) introduced a novel regularization method for compressive imaging in the context of compressed sensing with coherent redundant dictionaries. The approach relies on the observation that natural images exhibit strong average sparsity over multiple coherent frames. The associated reconstruction algorithm, based on an analysis prior and a reweighted $\ell_1$ scheme, is dubbed Sparsity Averaging Reweighted Analysis (SARA). We review these advances and extend associated simulations establishing the superiority of SARA to regularization methods based on sparsity in a single frame, for a generic spread spectrum acquisition and for a Fourier acquisition of particular interest in radio astronomy.Comment: 4 pages, 3 figures, Proceedings of 10th International Conference on Sampling Theory and Applications (SampTA), Code available at https://github.com/basp-group/sopt, Full journal letter available at http://arxiv.org/abs/arXiv:1208.233

A diffusion strategy for distributed dictionary learning

International audienceWe consider the problem of a set of nodes which is required to collectively learn a common dictionary from noisy measurements. This distributed dictionary learning approach may be useful in several contexts including sensor networks. Dif-fusion cooperation schemes have been proposed to estimate a consensus solution to distributed linear regression. This work proposes a diffusion-based adaptive dictionary learning strategy. Each node receives measurements which may be shared or not with its neighbors. All nodes cooperate with their neighbors by sharing their local dictionary to estimate a common representa-tion. In a diffusion approach, the resulting algorithm corresponds to a distributed alternate optimization. Beyond dictionary learn-ing, this strategy could be adapted to many matrix factorization problems in various settings. We illustrate its efficiency on some numerical experiments, including the difficult problem of blind hyperspectral images unmixing

Empirical Bayes and Full Bayes for Signal Estimation

We consider signals that follow a parametric distribution where the parameter values are unknown. To estimate such signals from noisy measurements in scalar channels, we study the empirical performance of an empirical Bayes (EB) approach and a full Bayes (FB) approach. We then apply EB and FB to solve compressed sensing (CS) signal estimation problems by successively denoising a scalar Gaussian channel within an approximate message passing (AMP) framework. Our numerical results show that FB achieves better performance than EB in scalar channel denoising problems when the signal dimension is small. In the CS setting, the signal dimension must be large enough for AMP to work well; for large signal dimensions, AMP has similar performance with FB and EB.Comment: This work was presented at the Information Theory and Application workshop (ITA), San Diego, CA, Feb. 201

Sample Complexity of Bayesian Optimal Dictionary Learning

We consider a learning problem of identifying a dictionary matrix D (M times N dimension) from a sample set of M dimensional vectors Y = N^{-1/2} DX, where X is a sparse matrix (N times P dimension) in which the density of non-zero entries is 0<rho< 1. In particular, we focus on the minimum sample size P_c (sample complexity) necessary for perfectly identifying D of the optimal learning scheme when D and X are independently generated from certain distributions. By using the replica method of statistical mechanics, we show that P_c=O(N) holds as long as alpha = M/N >rho is satisfied in the limit of N to infinity. Our analysis also implies that the posterior distribution given Y is condensed only at the correct dictionary D when the compression rate alpha is greater than a certain critical value alpha_M(rho). This suggests that belief propagation may allow us to learn D with a low computational complexity using O(N) samples.Comment: 5pages, 5figure

A heuristic for sparse signal reconstruction

Compressive Sampling (CS) is a new method of signal acquisition and reconstruction from frequency data which do not follow the basic principle of the Nyquist-Shannon sampling theory. This new method allows reconstruction of the signal from substantially fewer measurements than those required by conventional sampling methods. We present and discuss a new, swarm based, technique for representing and reconstructing signals, with real values, in a noiseless environment. The method consists of finding an approximation of the l_0-norm based problem, as a combinatorial optimization problem for signal reconstruction. We also present and discuss some experimental results which compare the accuracy and the running time of our heuristic to the IHT and IRLS methods

Wavelet Features for Recognition of First Episode of Schizophrenia from MRI Brain Images

Machine learning methods are increasingly used in various fields of medicine, contributing to early diagnosis and better quality of care. These outputs are particularly desirable in case of neuropsychiatric disorders, such as schizophrenia, due to the inherent potential for creating a new gold standard in the diagnosis and differentiation of particular disorders. This paper presents a scheme for automated classification from magnetic resonance images based on multiresolution representation in the wavelet domain. Implementation of the proposed algorithm, utilizing support vector machines classifier, is introduced and tested on a dataset containing 104 patients with first episode schizophrenia and healthy volunteers. Optimal parameters of different phases of the algorithm are sought and the quality of classification is estimated by robust cross validation techniques. Values of accuracy, sensitivity and specificity over 71% are achieved