Moment Identifiability of Homoscedastic Gaussian Mixtures
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866908868887445504 |
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| author | Agostini, Daniele Améndola, Carlos Ranestad, Kristian |
| author_facet | Agostini, Daniele Améndola, Carlos Ranestad, Kristian |
| contents | We consider the problem of identifying a mixture of Gaussian distributions with same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander-Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two component setting we provide a closed form solution for parameter recovery based on moments up to order four, while in the one dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1905_05141 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Moment Identifiability of Homoscedastic Gaussian Mixtures Agostini, Daniele Améndola, Carlos Ranestad, Kristian Statistics Theory Algebraic Geometry 62R01, 62F10, 13P25, 14N07, 14Q15 We consider the problem of identifying a mixture of Gaussian distributions with same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander-Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two component setting we provide a closed form solution for parameter recovery based on moments up to order four, while in the one dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves. |
| title | Moment Identifiability of Homoscedastic Gaussian Mixtures |
| topic | Statistics Theory Algebraic Geometry 62R01, 62F10, 13P25, 14N07, 14Q15 |
| url | https://arxiv.org/abs/1905.05141 |