|Stochastic Systems Group|
Professor Vivek Goyal
Estimation of signals from noisy or partial data is a fundamental problem in communications and signal processing. This talk provides an overview of recent results in two scenarios.
The first part of the talk gives an analysis of estimation in linear state-space systems with Markov dynamics. The main result is a bound on the minimum achievable estimation error for such "jump linear systems." A simple estimator that achieves this bound can be found through a convex optimization with linear matrix inequality (LMI) constraints. One application of this framework provides bounds for tracking a time-varying signal from noisy observations with losses. Another application is a method for making predictive quantization robust to communication losses.
Another estimation problem considers signals sparsely represented with respect to overcomplete sets of vectors. A lower bound, based on rate-distortion theory, on the ability of sparse approximation to fit a white Gaussian signal is given. This bound provides a quantitative explanation for sparse approximation rejecting noise. Several further results apply to large random frames. Easy to compute estimates for the probability of recovering the correct subspace and for the MSE are given. The asymptotic behaviors of these estimates reveal an SNR threshold behavior.
Both parts are from the thesis work of Alyson Fletcher, performed under joint supervision with Kannan Ramchandran at UC-Berkeley.
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