Estimating Gaussian mixtures using sparse polynomial moment systems
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909220530552832 |
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| author | Lindberg, Julia Améndola, Carlos Rodriguez, Jose Israel |
| author_facet | Lindberg, Julia Améndola, Carlos Rodriguez, Jose Israel |
| contents | The method of moments is a classical statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. In addition, we show that a generic Gaussian $k$-mixture model is identifiable from its first $3k+2$ moments. Using these results, we present a homotopy algorithm that performs parameter recovery for high dimensional Gaussian mixture models where the number of paths tracked scales linearly in the dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_15675 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Estimating Gaussian mixtures using sparse polynomial moment systems Lindberg, Julia Améndola, Carlos Rodriguez, Jose Israel Methodology Algebraic Geometry Statistics Theory 62R01, 65H14, 65H10, 62H30, 62F10 The method of moments is a classical statistical technique for density estimation that solves a system of moment equations to estimate the parameters of an unknown distribution. A fundamental question critical to understanding identifiability asks how many moment equations are needed to get finitely many solutions and how many solutions there are. We answer this question for classes of Gaussian mixture models using the tools of polyhedral geometry. In addition, we show that a generic Gaussian $k$-mixture model is identifiable from its first $3k+2$ moments. Using these results, we present a homotopy algorithm that performs parameter recovery for high dimensional Gaussian mixture models where the number of paths tracked scales linearly in the dimension. |
| title | Estimating Gaussian mixtures using sparse polynomial moment systems |
| topic | Methodology Algebraic Geometry Statistics Theory 62R01, 65H14, 65H10, 62H30, 62F10 |
| url | https://arxiv.org/abs/2106.15675 |