We consider compressed sensing (CS) using partially coherent sensing matrices. Most of CS theory to date is focused on incoherent sensing, that is, columns from the sensing matrix are highly uncorrelated. However, sensing systems with naturally occurring correlations arise in many applications, such as signal detection, motion detection and radar. Moreover, in these applications it is often not necessary to know the support of the signal exactly, but instead small errors in the support and signal are tolerable. In this paper, we focus on @math -tol -erant recovery, in which support set reconstructions are considered accurate when their locations match the true locations within @math indices. Despite the abundance of work utilizing incoherent sensing matrices, for @math -tol -erant recovery we suggest that coherence is actually . This is especially true for situations with only a few and very noisy measurements as we demonstrate via numerical simulations. As a first step towards the theory of tolerant coherent sensing we introduce the notions of @math -co -herence and @math -tol -erant recovery. We then provide some theoretical arguments for a greedy algorithm applicable to @math -tol -erant recovery of signals with sufficiently spread support.
Along a different line of work, @cite_7 shows that mild coherence in the sensing matrix can be allowed when the signal is modeled as random. In this case, accurate recovery is still possible when the coherence scales like @math . Here again, in this setting the goal is exact recovery and the coherence is something that needs to be overcome, not something that aids in recovery.
@cite_7
2115447612
We consider the fundamental problem of estimating the mean of a vector y=Xβ+z, where X is an n×p design matrix in which one can have far more variables than observations, and z is a stochastic error term -— the so-called “p>n” setup. When β is sparse, or, more generally, when there is a sparse subset of covariates providing a close approximation to the unknown mean vector, we ask whether or not it is possible to accurately estimate Xβ using a computationally tractable algorithm. We show that, in a surprisingly wide range of situations, the lasso happens to nearly select the best subset of variables. Quantitatively speaking, we prove that solving a simple quadratic program achieves a squared error within a logarithmic factor of the ideal mean squared error that one would achieve with an oracle supplying perfect information about which variables should and should not be included in the model. Interestingly, our results describe the average performance of the lasso; that is, the performance one can expect in an vast majority of cases where Xβ is a sparse or nearly sparse superposition of variables, but not in all cases. Our results are nonasymptotic and widely applicable, since they simply require that pairs of predictor variables are not too collinear.