A Simple Method for Sparse Signal Recovery from Noisy Observations Using Kalman Filtering. Embedding Approximate Quasi-Norms for Improved Accuracy

In a primary research report we have introduced a simple method for recovering sparse signals from a series of noisy observations using a Kalman filter (KF). The new method utilizes a so-called pseudomeasurement technique for optimizing the convex minimization problem following from the theory of compressed sensing (CS). The CS-embedded KF approach has shown promising results when applied for reconstruction of linear Gaussian sparse processes. This report presents an improved version of the KF algorithm for the recovery of sparse signals. In this work we substitute the l1 norm by an approximate quasi-norm lp, 0 ≤ p < 1. This modification, which better suits the original combinatorial problem, greatly improves the accuracy of the resulting KF algorithm. This, however, involves the implementation of an extended KF (EKF) stage for properly computing the state statistics. Sparse Signal Recovery Consider an R-valued random discrete-time process {xk}k=1 that is sparse in some known orthonormal sparsity basis ψ ∈ R, that is zk = ψ xk, |supp(zk)| << n (1) where supp(zk) denotes the support of zk. Assume that zk evolves according to zk+1 = Azk + wk (2) where A ∈ R and {wk}k=1 is a zero-mean white Gaussian sequence with covariance Qk. The process xk is measured using a sequence of noisy observations given by yk = Hxk + ζk = H zk + ζk (3) where {ζk} ∞ k=1 is a zero-mean white Gaussian sequence with covariance Rk, and H := H ψ ∈ R with m < n. Letting y := [y1, . . . , yk], our problem is defined as follows. We are interested in a y -measurable estimator x̂k such that the minimum mean square error (MMSE) E [