I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces

<p>This thesis comprises three apparently very independent parts. However, there is a unity behind I would like to sketch very briefly.</p> <p>Formally graphs are in the background of most chapters and so is the duality local versus global. The first section is concerned with gl...

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Main Author: Baldi, Pierre
Format: Others
Language:en
Published: 1986
Online Access:https://thesis.library.caltech.edu/11440/2/Baldi_P_1986.pdf
Baldi, Pierre (1986) I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0bwx-nk73. https://resolver.caltech.edu/CaltechTHESIS:04052019-110135296 <https://resolver.caltech.edu/CaltechTHESIS:04052019-110135296>
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spelling ndltd-CALTECH-oai-thesis.library.caltech.edu-114402021-04-17T05:02:14Z https://thesis.library.caltech.edu/11440/ I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces Baldi, Pierre <p>This thesis comprises three apparently very independent parts. However, there is a unity behind I would like to sketch very briefly.</p> <p>Formally graphs are in the background of most chapters and so is the duality local versus global. The first section is concerned with globally coloring graphs under some local assumptions. Algorithmically it is an intrinsically difficult task and neural networks, the topic of the second part can be used to approach intractable problems. Simple local interactions with emergent collective behavior are one of the essential features of these networks. Their current models are similar to some of those encountered in statistical mechanics, like spin glasses. In the third part, we study ultrametricity, a concept recently rediscovered by theoretical physicists in the analysis of spin-glasses. Ultrametricity can be expressed as a local constraint on the shape of each triangle of the given metric space.</p> <p>Unless otherwise stated, results in the first and second part are essentially original. Since the third part represents a joint work with Michael Aschbacher, Eric Baum and Richard Wilson, I should perhaps try to outline my contribution though paternity of collective results is somewhat fuzzy. While working on neural networks and spin glasses Eric and I got interested in ultrametricity. Several of us had found an initial polynomial upper bound, but the final results of "n + 1" was first reached independently by Michael and Richard. I think I obtained the theorems: 4.5, 6.1, 6.3 (using an idea of Eric), 6.4, 6.5, 6.6, 6.7 (with Richard and helpful references from Bruce Rothschild and Olga Taussky) and participated in some other results.</p> 1986 Thesis NonPeerReviewed application/pdf en other https://thesis.library.caltech.edu/11440/2/Baldi_P_1986.pdf Baldi, Pierre (1986) I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0bwx-nk73. https://resolver.caltech.edu/CaltechTHESIS:04052019-110135296 <https://resolver.caltech.edu/CaltechTHESIS:04052019-110135296> https://resolver.caltech.edu/CaltechTHESIS:04052019-110135296 CaltechTHESIS:04052019-110135296 10.7907/0bwx-nk73
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description <p>This thesis comprises three apparently very independent parts. However, there is a unity behind I would like to sketch very briefly.</p> <p>Formally graphs are in the background of most chapters and so is the duality local versus global. The first section is concerned with globally coloring graphs under some local assumptions. Algorithmically it is an intrinsically difficult task and neural networks, the topic of the second part can be used to approach intractable problems. Simple local interactions with emergent collective behavior are one of the essential features of these networks. Their current models are similar to some of those encountered in statistical mechanics, like spin glasses. In the third part, we study ultrametricity, a concept recently rediscovered by theoretical physicists in the analysis of spin-glasses. Ultrametricity can be expressed as a local constraint on the shape of each triangle of the given metric space.</p> <p>Unless otherwise stated, results in the first and second part are essentially original. Since the third part represents a joint work with Michael Aschbacher, Eric Baum and Richard Wilson, I should perhaps try to outline my contribution though paternity of collective results is somewhat fuzzy. While working on neural networks and spin glasses Eric and I got interested in ultrametricity. Several of us had found an initial polynomial upper bound, but the final results of "n + 1" was first reached independently by Michael and Richard. I think I obtained the theorems: 4.5, 6.1, 6.3 (using an idea of Eric), 6.4, 6.5, 6.6, 6.7 (with Richard and helpful references from Bruce Rothschild and Olga Taussky) and participated in some other results.</p>
author Baldi, Pierre
spellingShingle Baldi, Pierre
I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces
author_facet Baldi, Pierre
author_sort Baldi, Pierre
title I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces
title_short I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces
title_full I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces
title_fullStr I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces
title_full_unstemmed I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces
title_sort i. on a family of generalized colorings. ii. some contributions to the theory of neural networks. iii. embeddings of ultrametric spaces
publishDate 1986
url https://thesis.library.caltech.edu/11440/2/Baldi_P_1986.pdf
Baldi, Pierre (1986) I. On a Family of Generalized Colorings. II. Some Contributions to the Theory of Neural Networks. III. Embeddings of Ultrametric Spaces. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0bwx-nk73. https://resolver.caltech.edu/CaltechTHESIS:04052019-110135296 <https://resolver.caltech.edu/CaltechTHESIS:04052019-110135296>
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