Testing the mixture model hypothesis via spectral gap

In this paper, we study the problem of testing whether or not a given probability measure $\mu$ 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 $\mu$, namely, the eigenvalues of the 4th moment operator $T_{\mu}$ 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 $T_{\mu}$ 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.