We formulate sparse support recovery as a salient set identification problem and use information-theoretic analyses to characterize the recovery performance and sample complexity. We consider a very general model where we are not restricted to linear models or specific distributions. We state non-asymptotic bounds on recovery probability and a tight mutual information formula for sample complexity. We evaluate our bounds for applications such as sparse linear regression and explicitly characterize effects of correlation or noisy features on recovery performance. We show improvements upon previous work and identify gaps between the performance of recovery algorithms and fundamental information.
Furthermore, prior work relied heavily on the design of sampling matrices with special structures such as Gaussian ensembles and RIP matrices, which is a key difference from the setting we consider herein as for our purpose we do not always have the freedom to design the matrix @math . We do not make explicit assumptions about the structure of the sensing matrix, such as the restricted isometry property @cite_10 or incoherence properties @cite_32 , or about the distribution of the matrix elements, such as sub-Gaussianity. Also, the existing IT bounds, which are largely based on Gaussian ensembles, are limited to the linear CS model, and hence not suitable for the non-linear models we consider herein.
@cite_10 @cite_32
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It is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This technique known as sensing" or sampling" relies on properties of the sensing matrix such as the restricted isometry property. In this note, we establish new results about the accuracy of the reconstruction from undersampled measurements which improve on earlier estimates, and have the advantage of being more elegant. R 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.