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We study approximations of the partition function of dense graphical models. Partition functions of graphical models play a fundamental role is statistical physics, in statistics and in machine learning. Two of the main methods for…

Machine Learning · Computer Science 2018-02-21 Vishesh Jain , Frederic Koehler , Elchanan Mossel

Probabilistic graphical models are a key tool in machine learning applications. Computing the partition function, i.e., normalizing constant, is a fundamental task of statistical inference but it is generally computationally intractable,…

Machine Learning · Statistics 2020-01-29 Sungsoo Ahn , Michael Chertkov , Adrian Weller , Jinwoo Shin

We study the problem of approximating the Ising model partition function with complex parameters on bounded degree graphs. We establish a deterministic polynomial-time approximation scheme for the partition function when the interactions…

Quantum Physics · Physics 2019-07-12 Ryan L. Mann , Michael J. Bremner

We study the problem of approximating the partition function of the ferromagnetic Ising model in graphs and hypergraphs. Our first result is a deterministic approximation scheme (an FPTAS) for the partition function in bounded degree graphs…

Data Structures and Algorithms · Computer Science 2018-12-26 Jingcheng Liu , Alistair Sinclair , Piyush Srivastava

Probabilistic graphical models have emerged as a powerful modeling tool for several real-world scenarios where one needs to reason under uncertainty. A graphical model's partition function is a central quantity of interest, and its…

Artificial Intelligence · Computer Science 2021-05-25 Durgesh Agrawal , Yash Pote , Kuldeep S Meel

We investigate quantum computational complexity of calculating partition functions of Ising models. We construct a quantum algorithm for an additive approximation of Ising partition functions on square lattices. To this end, we utilize the…

Quantum Physics · Physics 2014-08-13 Akira Matsuo , Keisuke Fujii , Nobuyuki Imoto

We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…

Numerical Analysis · Mathematics 2015-03-19 Adam M. Oberman

We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the "winding" technology devised by…

Data Structures and Algorithms · Computer Science 2021-03-17 Martin Dyer , Marc Heinrich , Mark Jerrum , Haiko Müller

While the Ising model remains essential to understand physical phenomena, its natural connection to combinatorial reasoning makes it also one of the best models to probe complex systems in science and engineering. We bring a computational…

Computational Physics · Physics 2022-12-27 Shaan A. Nagy , Roger Paredes , Jeffrey M. Dudek , Leonardo Dueñas-Osorio , Moshe Y. Vardi

Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of…

Machine Learning · Statistics 2014-11-03 Hidekazu Oiwa , Ryohei Fujimaki

We present an algorithm to approximate partition functions of 3-body classical Ising models on two-dimensional lattices of arbitrary genus, in the real-temperature regime. Even though our algorithm is purely classical, it is designed by…

Quantum Physics · Physics 2014-02-05 M. Van den Nest , W. Dür

In a series of recent works, we have generalised the consistency results in the stochastic block model literature to the case of uniform and non-uniform hypergraphs. The present paper continues the same line of study, where we focus on…

Machine Learning · Computer Science 2017-05-18 Debarghya Ghoshdastidar , Ambedkar Dukkipati

In this paper we present a new approach for tightening upper bounds on the partition function. Our upper bounds are based on fractional covering bounds on the entropy function, and result in a concave program to compute these bounds and a…

Machine Learning · Computer Science 2012-10-19 Tamir Hazan , Jian Peng , Amnon Shashua

Undirected graphical models have many applications in such areas as machine learning, image processing, and, recently, psychology. Psychopathology in particular has received a lot of attention, where symptoms of disorders are assumed to…

Methodology · Statistics 2019-05-29 Lourens Waldorp , Maarten Marsman

We design a recursive algorithm to compute the partition function of the Ising model, summed over cubic maps with fixed size and genus. The algorithm runs in polynomial time, which is much faster than methods based on a Tutte-like, or…

Combinatorics · Mathematics 2025-09-15 Mireille Bousquet-Mélou , Ariane Carrance , Baptiste Louf

In this paper we relate the partition function to the max-statistics of random variables. In particular, we provide a novel framework for approximating and bounding the partition function using MAP inference on randomly perturbed models. As…

Machine Learning · Computer Science 2012-07-03 Tamir Hazan , Tommi Jaakkola

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…

Numerical Analysis · Mathematics 2008-04-11 Néstor E. Aguilera , Pedro Morin

We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation…

Computer Vision and Pattern Recognition · Computer Science 2011-12-06 Jan Lellmann , Frank Lenzen , Christoph Schnörr

We study the complexity of approximating the partition function of dense Ising models in the critical regime. Recent work of Chen, Chen, Yin, and Zhang (FOCS 2025) established fast mixing at criticality, and even beyond criticality in a…

Computational Complexity · Computer Science 2026-03-24 Andreas Galanis , Daniel Stefankovic , Eric Vigoda

The breakthrough ideas in the modern proximal splitting methodologies allow us to express the set of all minimizers of a superposition of multiple nonsmooth convex functions as the fixed point set of computable nonexpansive operators. In…

Optimization and Control · Mathematics 2022-07-01 Isao Yamada , Masao Yamagishi
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