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- W4376167414 abstract "We investigate the approximation of high-dimensional target measures as low-dimensional updates of a dominating reference measure. This approximation class replaces the associated density with the composition of: (i) a feature map that identifies the leading principal components or features of the target measure, relative to the reference, and (ii) a low-dimensional profile function. When the reference measure satisfies a subspace $phi$-Sobolev inequality, we construct a computationally tractable approximation that yields certifiable error guarantees with respect to the Amari $alpha$-divergences. Our construction proceeds in two stages. First, for any feature map and any $alpha$-divergence, we obtain an analytical expression for the optimal profile function. Second, for linear feature maps, the principal features are obtained from eigenvectors of a matrix involving gradients of the log-density. Neither step requires explicit access to normalizing constants. Notably, by leveraging the $phi$-Sobolev inequalities, we demonstrate that these features universally certify approximation errors across the range of $alpha$-divergences $alpha in (0,1]$. We then propose an application to Bayesian inverse problems and provide an analogous construction with approximation guarantees that hold in expectation over the data. We conclude with an extension of the proposed dimension reduction strategy to nonlinear feature maps." @default.
- W4376167414 created "2023-05-12" @default.
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- W4376167414 date "2023-05-10" @default.
- W4376167414 modified "2023-09-27" @default.
- W4376167414 title "Principal Feature Detection via $Phi$-Sobolev Inequalities" @default.
- W4376167414 doi "https://doi.org/10.48550/arxiv.2305.06172" @default.
- W4376167414 hasPublicationYear "2023" @default.
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