Bounds on the Number of Measurements for Reliable Compressive Classification

TL;DR

This paper establishes bounds on the minimum number of measurements needed for reliable classification of high-dimensional Gaussian signals in low-noise conditions, using either random or designed measurement schemes.

Contribution

It provides new upper bounds on measurements for reliable classification, revealing operational regimes based on class and signal space dimensions.

Findings

Reliable classification is achievable with fewer measurements in low-noise regimes.

One-vs-all measurement design is effective when class count is within signal space dimension.

Random measurements suffice when class spaces are lower-dimensional than the number of classes.

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.