Sparse Gaussian mixture detection: Low complexity, high performance tests via quantization

We study the problem of testing between a sparse signal in noise, modeled as a mixture distribution, versus pure noise, with a Gaussian signal and noise of same variance, but differing means as the mixture proportion tends to zero. We construct a simple new adaptive test based on quantizing data with sample size-dependent quantizers and prove its consistency. The proposed test has almost linear time complexity and sublinear space complexity, which is better than existing tests, and in particular, the celebrated Higher Criticism test. Moreover, our numerical results show that the proposed test is competitive with commonly used tests even with a small number of quantizer levels.