Thursday, November 20, 2014

Nonlinear Information-Theoretic Compressive Measurement Design

Compressive sensing, nonlinearity, classification, terrific !

We investigate design of general nonlinear functions for mapping high-dimensional data into a lower-dimensional (compressive) space. The nonlinear measurements are assumed contaminated by additive Gaussian noise. Depending on the application, we are either interested in recovering the high-dimensional data from the non-linear compressive measurements, or performing classification directly based on these measurements. The latter case corresponds to classification based on nonlinearly constituted and noisy features. The nonlinear measurement functions are designed based on constrained mutual-information optimization. New analytic results are developed for the gradient of mutual information in this setting, for arbitrary input-signal statistics. We make connections to kernel-based methods, such as the support vector machine. Encouraging results are presented on multiple datasets, for both signal recovery and classification. The nonlinear approach is shown to be particularly valuable in high-noise scenarios.

Some supplementary material

Also semi-relevant:

Compressive Classification of a Mixture of Gaussians: Analysis, Designs and Geometrical Interpretation by Hugo Reboredo, Francesco Renna, Robert Calderbank, Miguel R. D. Rodrigues

This paper derives fundamental limits on the performance of compressive classification when the source is a mixture of Gaussians. It provides an asymptotic analysis of a Bhattacharya based upper bound on the misclassification probability for the optimal Maximum-A-Posteriori (MAP) classifier that depends on quantities that are dual to the concepts of diversity-order and coding gain in multi-antenna communications. The diversity-order of the measurement system determines the rate at which the probability of misclassification decays with signal-to-noise ratio (SNR) in the low-noise regime. The counterpart of coding gain is the measurement gain which determines the power offset of the probability of misclassification in the low-noise regime. These two quantities make it possible to quantify differences in misclassification probability between random measurement and (diversity-order) optimized measurement. Results are presented for two-class classification problems first with zero-mean Gaussians then with nonzero-mean Gaussians, and finally for multiple-class Gaussian classification problems. The behavior of misclassification probability is revealed to be intimately related to certain fundamental geometric quantities determined by the measurement system, the source and their interplay. Numerical results, representative of compressive classification of a mixture of Gaussians, demonstrate alignment of the actual misclassification probability with the Bhattacharya based upper bound. The connection between the misclassification performance and the alignment between source and measurement geometry may be used to guide the design of dictionaries for compressive classification.

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