Optimization of the Sherrington-Kirkpatrick Hamiltonian
Let A be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing the quadratic form associated to A over binary vectors. In the language of statistical physics, this amounts to finding the ground state of the S...
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| Vydané v: | Proceedings / annual Symposium on Foundations of Computer Science s. 1417 - 1433 |
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01.11.2019
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| ISSN: | 2575-8454 |
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| Abstract | Let A be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing the quadratic form associated to A over binary vectors. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any ε > 0, outputs a feasible solution whose value is at least (1 - ε) of the optimum, with probability converging to one as the dimension n of the matrix diverges. The algorithm's time complexity is of order n 2 . It is a message-passing algorithm, but the specific structure of its update rules is new. As a side result, we prove that, at (low) non-zero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations. |
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| AbstractList | Let A be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing the quadratic form associated to A over binary vectors. In the language of statistical physics, this amounts to finding the ground state of the Sherrington-Kirkpatrick model of spin glasses. The asymptotic value of this optimization problem was characterized by Parisi via a celebrated variational principle, subsequently proved by Talagrand. We give an algorithm that, for any ε > 0, outputs a feasible solution whose value is at least (1 - ε) of the optimum, with probability converging to one as the dimension n of the matrix diverges. The algorithm's time complexity is of order n 2 . It is a message-passing algorithm, but the specific structure of its update rules is new. As a side result, we prove that, at (low) non-zero temperature, the algorithm constructs approximate solutions of the Thouless-Anderson-Palmer equations. |
| Author | Montanari, Andrea |
| Author_xml | – sequence: 1 givenname: Andrea surname: Montanari fullname: Montanari, Andrea organization: Stanford University |
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| Snippet | Let A be a symmetric random matrix with independent and identically distributed Gaussian entries above the diagonal. We consider the problem of maximizing the... |
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| StartPage | 1417 |
| SubjectTerms | Approximation algorithms Complexity theory Glass Message passing message passing algorithms Optimization Physics replica symmetry breaking Sherrington-Kirkpatrick Signal processing algorithms spin glasses |
| Title | Optimization of the Sherrington-Kirkpatrick Hamiltonian |
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