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Pré-Publication, Document De Travail Année : 2024

Generalized multi-view model: Adaptive density estimation under low-rank constraints

Résumé

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 $\beta$-Hölder 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 $\min((K/n)^{\beta/(2\beta+1)}, n^{-\beta/(2\beta+2)})$ up to logarithmic factors, and it is adaptive to the unknown support as well as to the smoothness $\beta$ and to the unknown number of separable components $K$. We present efficient algorithms for computing our estimators.
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Dates et versions

hal-04557030 , version 1 (25-04-2024)

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  • HAL Id : hal-04557030 , version 1

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Julien Chhor, Olga Klopp, Alexandre Tsybakov. Generalized multi-view model: Adaptive density estimation under low-rank constraints. 2024. ⟨hal-04557030⟩
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