2020/08/07 by Marco Mondelli, Christos Thrampoulidis, Mondelli, Marco +3 · 1 citation
Computer Science · Engineering · #Blind Source Separation Techniques #Distributed Sensor Networks and Detection Algorithms #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Sparse and Compressive Sensing Techniques #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2008.03326
openalex publication_date 2020/08/07 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We study the problem of recovering an unknown signal boldsymbol x given\nmeasurements obtained from a generalized linear model with a Gaussian sensing\nmatrix. Two popular solutions are based on a linear estimator \ boldsymbol\nx rm L and a spectral estimator \ boldsymbol x rm s. The former\nis a data-dependent linear combination of the columns of the measurement\nmatrix, and its analysis is quite simple. The latter is the principal\neigenvector of a data-dependent matrix, and a recent line of work has studied\nits performance. In this paper, we show how to optimally combine\n\ boldsymbol x rm L and \ boldsymbol x rm s. At the heart\nof our analysis is the exact characterization of the joint empirical\ndistribution of ( boldsymbol x, \ boldsymbol x rm L, \ boldsymbol\nx rm s) in the high-dimensional limit. This allows us to compute the\nBayes-optimal combination of \ boldsymbol x rm L and\n\ boldsymbol x rm s, given the limiting distribution of the signal\n boldsymbol x. When the distribution of the signal is Gaussian, then the\nBayes-optimal combination has the form \θ\ boldsymbol x rm\nL+\ boldsymbol x rm s and we derive the optimal combination\ncoefficient. In order to establish the limiting distribution of ( boldsymbol\nx, \ boldsymbol x rm L, \ boldsymbol x rm s), we design and\nanalyze an Approximate Message Passing (AMP) algorithm whose iterates give\n\ boldsymbol x rm L and approach \ boldsymbol x rm s.\nNumerical simulations demonstrate the improvement of the proposed combination\nwith respect to the two methods considered separately.\n