Testing the mixture model hypothesis via spectral gap
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914593195950080 |
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| author | Boedihardjo, March T. Kileel, Joe Tombs, Vandy |
| author_facet | Boedihardjo, March T. Kileel, Joe Tombs, Vandy |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_03245 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Testing the mixture model hypothesis via spectral gap Boedihardjo, March T. Kileel, Joe Tombs, Vandy Probability Statistics Theory 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. |
| title | Testing the mixture model hypothesis via spectral gap |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2603.03245 |