Concentration of Measure Inequalities for Toeplitz Matrices With Applications

• Published in: IEEE transactions on signal processing Vol. 61; no. 1; pp. 109 - 117
• Main Authors: Sanandaji, B. M., Vincent, T. L., Wakin, M. B.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.01.2013; Institute of Electrical and Electronics Engineers
• Subjects: Applied sciences; Compressed sensing; Compressive sensing; compressive Toeplitz matrices; concentration of measure inequalities; Covariance matrix; Detection, estimation, filtering, equalization, prediction; Eigenvalues and eigenfunctions; Exact sciences and technology; Information, signal and communications theory; Linear matrix inequalities; Sampling, quantization; Sensors; Signal and communications theory; Signal, noise; Sparse matrices; Symmetric matrices; Telecommunications and information theory; Compressive sensing; Tail probability; compressive Toeplitz matrices; Signal processing; concentration of measure inequalities; Random distribution; Toeplitz matrix; Compressed sensing
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2012.2222384

We derive Concentration of Measure (CoM) inequalities for randomized Toeplitz matrices. These inequalities show that the norm of a high-dimensional signal mapped by a 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. For the class of sparse signals, the introduced quantity is bounded by the sparsity level of the signal. However, we observe that 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. As an application of the CoM inequalities, we consider Compressive Binary Detection (CBD).