Generalized multi-view model: Adaptive density estimation under low-rank constraints
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913558683451392 |
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| author | Chhor, Julien Klopp, Olga Tsybakov, Alexandre |
| author_facet | Chhor, Julien Klopp, Olga Tsybakov, Alexandre |
| contents | We study the problem of bivariate discrete or continuous probability density estimation under low-rank constraints.For discrete distributions, we assume that the two-dimensional array to estimate is a low-rank probability matrix. In the continuous case, we assume that the density with respect to the Lebesgue measure satisfies a generalized multi-view model, meaning that it is $β$-H{ö}lder and can be decomposed as a sum of $K$ components, each of which is a product of one-dimensional functions. In both settings, we propose estimators that achieve, up to logarithmic factors, the minimax optimal convergence rates under such low-rank constraints. In the discrete case, the proposed estimator is adaptive to the rank $K$. In the continuous case, our estimator converges with the $L_1$ rate $\min((K/n)^{β/(2β+1)}, n^{-β/(2β+2)})$ up to logarithmic factors, and it is adaptive to the unknown support as well as to the smoothness $β$ and to the unknown number of separable components $K$. We present efficient algorithms for computing our estimators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized multi-view model: Adaptive density estimation under low-rank constraints Chhor, Julien Klopp, Olga Tsybakov, Alexandre Statistics Theory We study the problem of bivariate discrete or continuous probability density estimation under low-rank constraints.For discrete distributions, we assume that the two-dimensional array to estimate is a low-rank probability matrix. In the continuous case, we assume that the density with respect to the Lebesgue measure satisfies a generalized multi-view model, meaning that it is $β$-H{ö}lder and can be decomposed as a sum of $K$ components, each of which is a product of one-dimensional functions. In both settings, we propose estimators that achieve, up to logarithmic factors, the minimax optimal convergence rates under such low-rank constraints. In the discrete case, the proposed estimator is adaptive to the rank $K$. In the continuous case, our estimator converges with the $L_1$ rate $\min((K/n)^{β/(2β+1)}, n^{-β/(2β+2)})$ up to logarithmic factors, and it is adaptive to the unknown support as well as to the smoothness $β$ and to the unknown number of separable components $K$. We present efficient algorithms for computing our estimators. |
| title | Generalized multi-view model: Adaptive density estimation under low-rank constraints |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2404.17209 |