Compressed Sensing with uncertainty - the Bayesian estimation perspective

Abstract

The Compressed Sensing (CS) framework outperforms the sampling rate limits given by Shannon's theory. This gap is possible since it is assumed that the signal of interest admits a linear decomposition of few vectors in a given sparsifying Basis (Fourier, Wavelet, ...). Unfortunately in realistic operating systems, uncertain knowledge of the CS model is inevitable and must be evaluated. Typically, this uncertainty drastically degrades the estimation performance of sparse-based estimators in the low noise variance regime. In this work, the Off-Grid (OG) and Basis Mismatch (BM) problems are compared in a Bayesian estimation perspective. At first glance, we are tempted to think that these two acronyms stand for the same problem. However, by comparing their Bayesian Cramèr-Rao Bounds (BCRB) for the estimation of a L-sparse amplitude vector based on N measurements, it is shown that the BM problem has a lower BCRB than the CS one in a general context. To go further into the analysis we provide for i.i.d. Gaussian amplitudes and in the low noise variance regime an interesting closed-form expression of a normalized 2-norm criterion of the difference of the two BCRB matrices. Based on the analysis of this closed-form expression, we obtain two conclusions. Firstly, the two uncertainty problems cannot be confused for a non-zero mismatch error variance and with finite N and L. Secondly, the two problems turn to be similar for any mismatch error variance in the large system regime, i.e., for N, L → ∞ with constant aspect ratio N/L → ρ.

Compressed Sensing with Basis Mismatch: Performance Bounds and Sparse-Based Estimator

“…Note that the BM problem described on Fig. 1, is slightly different that the well-known off-grid (OG) problem and we can find in [36] a comparison of these two types of uncertainty. We can described the two problems in the following manner:…”

“…• The BMSE of the BiCE converges toward C S|U T S y given in (36). This remark is important because it means that the BiCE when there is no BM remains statistically efficient with respect to the projected measurement vector.…”

“…The approximated BMSEs of the BiCE are given in expression (45). In addition, the (biased) I-ECRB and the (unbiased) ECRB are in expressions (36) and (37), respectively. We fix N = 50, K = 200, L = 10, σ 2 = 10 −5 and s = 2 (Gaussian BM error).…”