Principal feature detection via ϕ-Sobolev inequalities
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Author(s)
Li, Matthew TC
Marzouk, Youssef
Zahm, Olivier
Date Issued
November 2024
Journal
Bernoulli
Publisher
Bernoulli Society for Mathematical Statistics and Probability
Citation
Matthew T.C. Li. Youssef Marzouk. Olivier Zahm. "Principal feature detection via ϕ-Sobolev inequalities." Bernoulli 30 (4) 2979 - 3003, November 2024.
Version
Final published version
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 ϕ-Sobolev inequality, we construct a computationally tractable approximation that yields certifiable error guarantees with respect to the Amari α-divergences. Our construction proceeds in two stages. First, for any feature map and any α-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 ϕ-Sobolev inequalities, we demonstrate that these features universally certify approximation errors across the range of α-divergences
α
∈
(
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.
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DOI of Published Version
https://doi.org/10.3150/23-bej1702
