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The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck

The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of op...

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Autor principal: Agmon, Shlomi
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2023
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10606645/
https://www.ncbi.nlm.nih.gov/pubmed/37895492
http://dx.doi.org/10.3390/e25101370
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author Agmon, Shlomi
author_facet Agmon, Shlomi
author_sort Agmon, Shlomi
collection PubMed
description The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory when input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB’s intimate relations with RD, we provide substantial insights into its solution structure, highlighting caveats in its finite-dimensional treatments. Sub-optimal solutions are seen to collide or exchange optimality at its bifurcations. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB’s first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, we not only detect IB bifurcations but also identify their type in order to handle them accordingly. Rather than approaching the IB’s optimal tradeoff curve from sub-optimal directions, the latter allows us to follow a solution’s trajectory along the optimal curve under mild assumptions. We thereby translate an understanding of IB bifurcations into a surprisingly accurate numerical algorithm.
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spelling pubmed-106066452023-10-28 The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck Agmon, Shlomi Entropy (Basel) Article The Information Bottleneck (IB) is a method of lossy compression of relevant information. Its rate-distortion (RD) curve describes the fundamental tradeoff between input compression and the preservation of relevant information embedded in the input. However, it conceals the underlying dynamics of optimal input encodings. We argue that these typically follow a piecewise smooth trajectory when input information is being compressed, as recently shown in RD. These smooth dynamics are interrupted when an optimal encoding changes qualitatively, at a bifurcation. By leveraging the IB’s intimate relations with RD, we provide substantial insights into its solution structure, highlighting caveats in its finite-dimensional treatments. Sub-optimal solutions are seen to collide or exchange optimality at its bifurcations. Despite the acceptance of the IB and its applications, there are surprisingly few techniques to solve it numerically, even for finite problems whose distribution is known. We derive anew the IB’s first-order Ordinary Differential Equation, which describes the dynamics underlying its optimal tradeoff curve. To exploit these dynamics, we not only detect IB bifurcations but also identify their type in order to handle them accordingly. Rather than approaching the IB’s optimal tradeoff curve from sub-optimal directions, the latter allows us to follow a solution’s trajectory along the optimal curve under mild assumptions. We thereby translate an understanding of IB bifurcations into a surprisingly accurate numerical algorithm. MDPI 2023-09-22 /pmc/articles/PMC10606645/ /pubmed/37895492 http://dx.doi.org/10.3390/e25101370 Text en © 2023 by the author. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Agmon, Shlomi
The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck
title The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck
title_full The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck
title_fullStr The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck
title_full_unstemmed The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck
title_short The Information Bottleneck’s Ordinary Differential Equation: First-Order Root Tracking for the Information Bottleneck
title_sort information bottleneck’s ordinary differential equation: first-order root tracking for the information bottleneck
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10606645/
https://www.ncbi.nlm.nih.gov/pubmed/37895492
http://dx.doi.org/10.3390/e25101370
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