Scampi: a robust approximate message-passing framework for compressive imaging

Reconstruction of images from noisy linear measurements is a core problem in image processing, for which convex optimization methods based on total variation (TV) minimization have been the long-standing state-of-the-art. We present an alternative probabilistic reconstruction procedure based on approximate message-passing, Scampi, which operates in the compressive regime, where the inverse imaging problem is underdetermined. While the proposed method is related to the recently proposed GrAMPA algorithm of Borgerding, Schniter, and Rangan, we further develop the probabilistic approach to compressive imaging by introducing an expectation-maximizaiton learning of model parameters, making the Scampi robust to model uncertainties. Additionally, our numerical experiments indicate that Scampi can provide reconstruction performance superior to both GrAMPA as well as convex approaches to TV reconstruction. Finally, through exhaustive best-case experiments, we show that in many cases the maximal performance of both Scampi and convex TV can be quite close, even though the approaches are a prori distinct. The theoretical reasons for this correspondence remain an open question. Nevertheless, the proposed algorithm remains more practical, as it requires far less parameter tuning to perform optimally.Comment: Presented at the 2015 International Meeting on High-Dimensional Data Driven Science, Kyoto, Japa

Vector Approximate Message Passing for the Generalized Linear Model

The generalized linear model (GLM), where a random vector $\boldsymbol{x}$ is observed through a noisy, possibly nonlinear, function of a linear transform output $\boldsymbol{z}=\boldsymbol{Ax}$, arises in a range of applications such as robust regression, binary classification, quantized compressed sensing, phase retrieval, photon-limited imaging, and inference from neural spike trains. When $\boldsymbol{A}$ is large and i.i.d. Gaussian, the generalized approximate message passing (GAMP) algorithm is an efficient means of MAP or marginal inference, and its performance can be rigorously characterized by a scalar state evolution. For general $\boldsymbol{A}$, though, GAMP can misbehave. Damping and sequential-updating help to robustify GAMP, but their effects are limited. Recently, a "vector AMP" (VAMP) algorithm was proposed for additive white Gaussian noise channels. VAMP extends AMP's guarantees from i.i.d. Gaussian $\boldsymbol{A}$ to the larger class of rotationally invariant $\boldsymbol{A}$. In this paper, we show how VAMP can be extended to the GLM. Numerical experiments show that the proposed GLM-VAMP is much more robust to ill-conditioning in $\boldsymbol{A}$ than damped GAMP

Sparse Analysis Recovery via Iterative Cosupport Detection Estimation

Cosparse analysis model (CAM) provides a new signal processing paradigm for recovering cosparse signals with respect to a given analysis operator from the undersampled linear measurements in the context of emerging theory of compressed sensing (CS). The sparse analysis recovery/cosparse recovery is a key one brought up by this new paradigm. In this paper, we propose a new family of analysis pursuit algorithms for the sparse analysis recovery problem when the signals obey the cosparse analysis model, termed as iterative cosupport detection estimation (ICDE). ICDE is an algorithmic framework, which alternates between detecting a cosupport set of the unknown true signal and estimating the underlying signal by solving a truncated analysis pursuit problem on the detected cosupport. Further, we propose effective implementations of ICDE equipped with an efficient thresholding strategy for cosupport detection. Empirical performance comparisons show that ICDE is favorable in comparison with the state-of-the-art sparse analysis recovery algorithms. Source code of ICDE has been made publicly available on Github: https://github.com/songhp/ICDE.Beijing Natural Science Foundation (BNSF) under Grant No. 4194076, the Natural Science Foundation of Jiangsu Province under Grant No. BK20170558 and the China Scholarship Council (CSC, No. 202008320094)

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

On the Error in Phase Transition Computations for Compressed Sensing

Evaluating the statistical dimension is a common tool to determine the asymptotic phase transition in compressed sensing problems with Gaussian ensemble. Unfortunately, the exact evaluation of the statistical dimension is very difficult and it has become standard to replace it with an upper-bound. To ensure that this technique is suitable, [1] has introduced an upper-bound on the gap between the statistical dimension and its approximation. In this work, we first show that the error bound in [1] in some low-dimensional models such as total variation and $\ell_1$ analysis minimization becomes poorly large. Next, we develop a new error bound which significantly improves the estimation gap compared to [1]. In particular, unlike the bound in [1] that is not applicable to settings with overcomplete dictionaries, our bound exhibits a decaying behavior in such cases