Testing the mixture model hypothesis via spectral gap

Abstract

In this paper, we study the problem of testing whether or not a given probability measure $\hat{I}¼$ on ${\mathbb{R}}^d$ can be decomposed as a mixture of two probability measures whose second order statistics are significantly different. We call this the problem of testing the mixture model hypothesis. To tackle it, we introduce a new set of computable orthogonal invariants of $\hat{I}¼$, namely, the eigenvalues of the 4th moment operator $T_{\hat{I}}¼$ associated with the measure. We prove that the largest eigenvalue is always an outlier eigenvalue. Further, we show how the first and second largest eigenvalues of $T_{\hat{I}}¼$ give nonasymptotic bounds for this problem and give a complete resolution of the asymptotic version of the problem under the $L^8$-$L^2$ equivalence assumption.