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arXiv · 2603.03245

Testing the mixture model hypothesis via spectral gap

March T. Boedihardjo·Joe Kileel·Vandy Tombs
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Abstract

In this paper, we study the problem of testing whether or not a given probability measure $μ$ 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 $μ$, namely, the eigenvalues of the 4th moment operator $T_μ$ 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 $T_μ$ 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.

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March T. Boedihardjo, Joe Kileel, Vandy Tombs. 2026-05-22. Testing the mixture model hypothesis via spectral gap. https://arxiv.org/abs/2603.03245

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