Principal feature detection via φ-Sobolev inequalities
Bernoulli, ISSN: 1350-7265, Vol: 30, Issue: 4, Page: 2979-3003
2024
- 2Citations
- 1Captures
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Example: if you select the 1-year option for an article published in 2019 and a metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019. If you select the 3-year option for the same article published in 2019 and the metric category shows 90%, that means that the article or review is performing better than 90% of the other articles/reviews published in that journal in 2019, 2018 and 2017.
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Article Description
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.
Bibliographic Details
http://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=85198261796&origin=inward; http://dx.doi.org/10.3150/23-bej1702; https://projecteuclid.org/journals/bernoulli/volume-30/issue-4/Principal-feature-detection-via-%cf%95-Sobolev-inequalities/10.3150/23-BEJ1702.full; https://dx.doi.org/10.3150/23-bej1702; https://projecteuclid.org/access-suspended
Bernoulli Society for Mathematical Statistics and Probability
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