Frozen 1‐RSB structure of the symmetric Ising perceptron
We prove, under an assumption on the critical points of a real‐valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1‐RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density....
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| Veröffentlicht in: | Random structures & algorithms 2024-07, Vol.64 (4), p.856-877 |
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| creator | Perkins, Will Xu, Changji |
| description | We prove, under an assumption on the critical points of a real‐valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1‐RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1‐RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model. |
| doi_str_mv | 10.1002/rsa.21202 |
| format | Article |
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Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1‐RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. 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Moreover, we prove this in a very strong form conjectured by Huang, Wong, and Kabashima: a typical solution of the model is isolated with high probability and the Hamming distance to all other solutions is linear in the dimension. The frozen 1‐RSB scenario is part of a recent and intriguing explanation of the performance of learning algorithms by Baldassi, Ingrosso, Lucibello, Saglietti, and Zecchina. We prove this structural result by comparing the symmetric Ising perceptron model to a planted model and proving a comparison result between the two models. Our main technical tool towards this comparison is an inductive argument for the concentration of the logarithm of number of solutions in the model.</description><subject>Algorithms</subject><subject>Critical point</subject><subject>frozen solutions</subject><subject>Ising model</subject><subject>learning algorithms</subject><subject>Machine learning</subject><subject>perceptron</subject><subject>planted model</subject><subject>solution space</subject><issn>1042-9832</issn><issn>1098-2418</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp10L9OwzAQBnALgUQpDLyBJSaGtOc_wTFbqVqoVAmphdmKnQukapNgJ0Jl4hF4Rp6ElLAy3Q2_7076CLlkMGIAfOxDOuKMAz8iAwY6ibhkyfFhlzzSieCn5CyEDQAowcWA3M599YElZd-fX6v1HQ2Nb13TeqRVTptXpGG_22HjC0cXoShfaI3eYd34qjwnJ3m6DXjxN4fkeT57mj5Ey8f7xXSyjByPFY_QxbllFjRjGrnMVKIdSMshF5lQueUqkxKFFVbJDGSWsDwF1CkDZa3s1JBc9XdrX721GBqzqVpfdi-NgFgqrW9i3anrXjlfheAxN7UvdqnfGwbmUI3pqjG_1XR23Nv3Yov7_6FZrSd94gdRDmT3</recordid><startdate>202407</startdate><enddate>202407</enddate><creator>Perkins, Will</creator><creator>Xu, Changji</creator><general>John Wiley & Sons, Inc</general><general>Wiley Subscription Services, Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>202407</creationdate><title>Frozen 1‐RSB structure of the symmetric Ising perceptron</title><author>Perkins, Will ; Xu, Changji</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c2572-ec5fb1b09119e24d789c04b20f3d37fb27d44e3b3b74d04d81fa0e9a107bb40f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Algorithms</topic><topic>Critical point</topic><topic>frozen solutions</topic><topic>Ising model</topic><topic>learning algorithms</topic><topic>Machine learning</topic><topic>perceptron</topic><topic>planted model</topic><topic>solution space</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Perkins, Will</creatorcontrib><creatorcontrib>Xu, Changji</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Random structures & algorithms</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Perkins, Will</au><au>Xu, Changji</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Frozen 1‐RSB structure of the symmetric Ising perceptron</atitle><jtitle>Random structures & algorithms</jtitle><date>2024-07</date><risdate>2024</risdate><volume>64</volume><issue>4</issue><spage>856</spage><epage>877</epage><pages>856-877</pages><issn>1042-9832</issn><eissn>1098-2418</eissn><abstract>We prove, under an assumption on the critical points of a real‐valued function, that the symmetric Ising perceptron exhibits the ‘frozen 1‐RSB’ structure conjectured by Krauth and Mézard in the physics literature; that is, typical solutions of the model lie in clusters of vanishing entropy density. 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| source | Wiley Online Library Journals Frontfile Complete |
| subjects | Algorithms Critical point frozen solutions Ising model learning algorithms Machine learning perceptron planted model solution space |
| title | Frozen 1‐RSB structure of the symmetric Ising perceptron |
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