Computer Science – Information Theory
Scientific paper
2011-12-08
Computer Science
Information Theory
Submitted to the IEEE Transactions on Signal Processing on December 1, 2011
Scientific paper
Concentration of Measure (CoM) inequalities are a useful tool for the analysis of randomized linear operators. In this work, we derive such inequalities for randomized compressive Toeplitz matrices, i.e., Toeplitz matrices that have fewer rows than columns and are populated with entries drawn from an i.i.d. Gaussian random sequence. These inequalities show that the norm of a high-dimensional signal mapped by a compressive Toeplitz matrix to a low-dimensional space concentrates around its mean with a tail probability bound that decays exponentially in the dimension of the range space divided by a quantity which is a function of the signal. This implies that the CoM inequalities for compressive Toeplitz matrices are non-uniform and signal-dependent. To this end, we also study the behavior of the introduced quantity. For example, we show that for the class of sparse signals, the introduced quantity is bounded by the sparsity level of the signal. However, this bound is highly pessimistic for most sparse signals and we show that if a random distribution is imposed on the non-zero entries of the signal, the typical value of the quantity is bounded by a term that scales logarithmically in the ambient dimension. Moreover, we extend our analysis for signals that are sparse in a generic orthobasis. To this end, we introduce the notion of the Fourier coherence of an arbitrary orthobasis and state our generic results based on this measure. Compressive Toeplitz matrices arise in problems involving the analysis of high-dimensional dynamical systems from consecutive convolution-based measurements. As applications of the CoM inequalities, we consider Compressive System Identification (CSI) and Compressive Binary Detection (CBD) problems and discuss the CoM inequalities in such applications.
Sanandaji Borhan M.
Vincent Tyrone L.
Wakin Michael B.
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