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DOI: 10.4230/LIPIcs.SEA.2024.25

Convex Relaxation for the Generalized Maximum-Entropy Sampling Problem

Authors: Gabriel Ponte, Marcia Fampa, and Jon Lee

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract

The generalized maximum-entropy sampling problem (GMESP) is to select an order-s principal submatrix from an order-n covariance matrix, to maximize the product of its t greatest eigenvalues, 0 < t ≤ s < n. It is a problem that specializes to two fundamental problems in statistical design theory: (i) maximum-entropy sampling problem (MESP); (ii) binary D-optimality (D-Opt). In the general case, it is motivated by a selection problem in the context of PCA (principal component analysis). We introduce the first convex-optimization based relaxation for GMESP, study its behavior, compare it to an earlier spectral bound, and demonstrate its use in a branch-and-bound scheme. We find that such an approach is practical when s-t is very small.

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Gabriel Ponte, Marcia Fampa, and Jon Lee. Convex Relaxation for the Generalized Maximum-Entropy Sampling Problem. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ponte_et_al:LIPIcs.SEA.2024.25,
  author =	{Ponte, Gabriel and Fampa, Marcia and Lee, Jon},
  title =	{{Convex Relaxation for the Generalized Maximum-Entropy Sampling Problem}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{25:1--25:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.25},
  URN =		{urn:nbn:de:0030-drops-203901},
  doi =		{10.4230/LIPIcs.SEA.2024.25},
  annote =	{Keywords: maximum-entropy sampling, D-optimality, convex relaxation, branch-and-bound, integer nonlinear optimization, principal component analysis}
}

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Convex Relaxation for the Generalized Maximum-Entropy Sampling Problem

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