Abstract
This paper studies the classification of high-dimensional Gaussian signals from low-dimensional noisy, linear measurements. In particular, it provides upper bounds (sufficient conditions) on the number of measurements required to drive the probability of misclassification to zero in the low-noise regime, both for random measurements and designed ones. Such bounds reveal two important operational regimes that are a function of the characteristics of the source: i) when the number of classes is less than or equal to the dimension of the space spanned by signals in each class, reliable classification is possible in the low-noise regime by using a one-vs-all measurement design; ii) when the dimension of the spaces spanned by signals in each class is lower than the number of classes, reliable classification is guaranteed in the low-noise regime by using a simple random measurement design. Simulation results both with synthetic and real data show that our analysis is sharp, in the sense that it is able to gauge the number of measurements required to drive the misclassification probability to zero in the low-noise regime.
Highlights
C OMPRESSIVE SENSING (CS) is an emerging paradigm that offers the means to simultaneously sense and compress a signal without significant loss of information [3]–[5]Manuscript received January 14, 2016; revised May 27, 2016 and July 10, 2016; accepted July 18, 2016
We show how our upper bound on the minimum number of measurements required for the phase transition compares to those associated with state-of-the-art measurement designs such as information discriminant analysis (IDA) methods [21] and methods based on the maximization of Shannon mutual information and quadratic Renyi entropy [14]
It is instructive to discuss the impact of model mismatch on the classification performance of the maximum a posteriori (MAP) classifier (2) in practical application scenarios
Summary
C OMPRESSIVE SENSING (CS) is an emerging paradigm that offers the means to simultaneously sense and compress a signal without significant loss of information [3]–[5]Manuscript received January 14, 2016; revised May 27, 2016 and July 10, 2016; accepted July 18, 2016. The associate editor coordinating the review of this manuscript and approving it for publication was Dr John McAllister. Renna was carried out in part when he was in the Department of Electronic and Electrical Engineering of University College London, and was supported by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant 655282. This paper was presented in part at the 2013 IEEE International Symposium on Information Theory [1] and the 2013 IEEE Global Conference on Signal and Information Processing [2]
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