Detecting Sparse Mixtures: Rate of Decay of Error Probability

TL;DR

This paper analyzes how quickly the error probability decreases when detecting sparse signals amidst noise, providing the first characterization of decay rates for both false alarms and misses across various distributions.

Contribution

It introduces the first comprehensive analysis of error decay rates in sparse mixture detection, applicable to a broad class of distributions and signal strengths.

Findings

Error probability decays sublinearly with the number of observations.

Decay rates are characterized by the $oldsymbol{ extit{ ext{χ}}^2}$-divergence for weak signals.

Decay rates can be independent of divergence for strong signals.

Abstract

We study the rate of decay of the probability of error for distinguishing between a sparse signal with noise, modeled as a sparse mixture, from pure noise. This problem has many applications in signal processing, evolutionary biology, bioinformatics, astrophysics and feature selection for machine learning. We let the mixture probability tend to zero as the number of observations tends to infinity and derive oracle rates at which the error probability can be driven to zero for a general class of signal and noise distributions via the likelihood ratio test. In contrast to the problem of detection of non-sparse signals, we see the log-probability of error decays sublinearly rather than linearly and is characterized through the $\chi^2$-divergence rather than the Kullback-Leibler divergence for "weak" signals and can be independent of divergence for "strong" signals. Our contribution is the first characterization of the rate of decay of the error probability for this problem for both the false alarm and miss probabilities.