Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing

This paper focuses on the estimation of low-complexity signals when they are observed through $M$ uniformly quantized compressive observations. Among such signals, we consider 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of these two models. In this context, we prove the estimation efficiency of a variant of Basis Pursuit Denoise, called Consistent Basis Pursuit (CoBP), enforcing consistency between the observations and the re-observed estimate, while promoting its low-complexity nature. We show that the reconstruction error of CoBP decays like $M^{-1/4}$ when all parameters but $M$ are fixed. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a set of small intrinsic "dimension", as measured by the Gaussian mean width, and (ii) they share the same quantized (dithered) random projections. By solving CoBP with a proximal algorithm, we provide some extensive numerical observations that confirm the theoretical bound as $M$ is increased, displaying even faster error decay than predicted. The same phenomenon is observed in the special, yet important case of 1-bit CS.Comment: Keywords: Quantized compressed sensing, quantization, consistency, error decay, low-rank, sparsity. 10 pages, 3 figures. Note abbout this version: title change, typo corrections, clarification of the context, adding a comparison with BPD

Discrimination on the Grassmann Manifold: Fundamental Limits of Subspace Classifiers

We present fundamental limits on the reliable classification of linear and affine subspaces from noisy, linear features. Drawing an analogy between discrimination among subspaces and communication over vector wireless channels, we propose two Shannon-inspired measures to characterize asymptotic classifier performance. First, we define the classification capacity, which characterizes necessary and sufficient conditions for the misclassification probability to vanish as the signal dimension, the number of features, and the number of subspaces to be discerned all approach infinity. Second, we define the diversity-discrimination tradeoff which, by analogy with the diversity-multiplexing tradeoff of fading vector channels, characterizes relationships between the number of discernible subspaces and the misclassification probability as the noise power approaches zero. We derive upper and lower bounds on these measures which are tight in many regimes. Numerical results, including a face recognition application, validate the results in practice.Comment: 19 pages, 4 figures. Revised submission to IEEE Transactions on Information Theor

Sensitivity analysis of geomechanical parameters in a two-way coupling reservoir simulation

Currently reservoir simulators model geomechanical effects such as compaction, subsidence, fault reactivation, breach of the seal integrity, etc. using only the rock compressibility to change the pore volume. However, rock compressibility as a scalar quantity is unfit to represent the true rock mechanics in the reservoir. In order to accurately represent geomechanical effects in a reservoir simulation, a two-way coupling simulation of the stress analyzer and the reservoir simulator was done. Based on the poroelasticity theory during the production or depletion of the reservoir the porosity and permeability changed due to the stress or pore pressure changes. A sensitivity analysis was carried out in order to understand how geomechanical parameters impact reservoir performance under a certain set of assumptions. In the material modeling steps elastic and plastic rocks were created for the simulation. The Mohr-Coulomb failure criterion was implemented for the yield criteria of the materials. Sensitivity analysis study enabled us to understand why stress changes, rock deformation and rock failure occur during the depletion of the reservoir. Engineers will also be able to prevent disasters because different ranges of rock mechanics were identified as key factors in vertical displacement and stress changes in the reservoir. Overall, geomechanical parameters directly affecting reservoir performance will be identified and therefore improved due to this study --Abstract, page iii

Small Width, Low Distortions: Quantized Random Embeddings of Low-complexity Sets

Under which conditions and with which distortions can we preserve the pairwise-distances of low-complexity vectors, e.g., for structured sets such as the set of sparse vectors or the one of low-rank matrices, when these are mapped in a finite set of vectors? This work addresses this general question through the specific use of a quantized and dithered random linear mapping which combines, in the following order, a sub-Gaussian random projection in $\mathbb R^M$ of vectors in $\mathbb R^N$, a random translation, or "dither", of the projected vectors and a uniform scalar quantizer of resolution $\delta>0$ applied componentwise. Thanks to this quantized mapping we are first able to show that, with high probability, an embedding of a bounded set $\mathcal K \subset \mathbb R^N$ in $\delta \mathbb Z^M$ can be achieved when distances in the quantized and in the original domains are measured with the $\ell_1$- and $\ell_2$-norm, respectively, and provided the number of quantized observations $M$ is large before the square of the "Gaussian mean width" of $\mathcal K$. In this case, we show that the embedding is actually "quasi-isometric" and only suffers of both multiplicative and additive distortions whose magnitudes decrease as $M^{-1/5}$ for general sets, and as $M^{-1/2}$ for structured set, when $M$ increases. Second, when one is only interested in characterizing the maximal distance separating two elements of $\mathcal K$ mapped to the same quantized vector, i.e., the "consistency width" of the mapping, we show that for a similar number of measurements and with high probability this width decays as $M^{-1/4}$ for general sets and as $1/M$ for structured ones when $M$ increases. Finally, as an important aspect of our work, we also establish how the non-Gaussianity of the mapping impacts the class of vectors that can be embedded or whose consistency width provably decays when $M$ increases.Comment: Keywords: quantization, restricted isometry property, compressed sensing, dimensionality reduction. 31 pages, 1 figur