On the Convexity and Reliability of the Bethe Free Energy Approximation

• URL: http://arxiv.org/abs/2405.15514v1
• Date: Fri, 24 May 2024 12:57:40 GMT
• Title: On the Convexity and Reliability of the Bethe Free Energy Approximation
• Authors: Harald Leisenberger, Christian Knoll, Franz Pernkopf,
• Abstract summary: We analyze when the Bethe approximation is reliable and how this can be verified. As a practical contribution we propose $textttBETHE-MIN$, a projected quasi-Newton method to efficiently find a minimum of the Bethe free energy.
• Score: 12.02055630441676
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Bethe free energy approximation provides an effective way for relaxing NP-hard problems of probabilistic inference. However, its accuracy depends on the model parameters and particularly degrades if a phase transition in the model occurs. In this work, we analyze when the Bethe approximation is reliable and how this can be verified. We argue and show by experiment that it is mostly accurate if it is convex on a submanifold of its domain, the 'Bethe box'. For verifying its convexity, we derive two sufficient conditions that are based on the definiteness properties of the Bethe Hessian matrix: the first uses the concept of diagonal dominance, and the second decomposes the Bethe Hessian matrix into a sum of sparse matrices and characterizes the definiteness properties of the individual matrices in that sum. These theoretical results provide a simple way to estimate the critical phase transition temperature of a model. As a practical contribution we propose $\texttt{BETHE-MIN}$, a projected quasi-Newton method to efficiently find a minimum of the Bethe free energy.

Related papers

• Indistinguishability of identical bosons from a quantum information theory perspective [0.0]
  We present a general theory of indistinguishability of identical bosons in experiments consisting of passive linear optics followed by particle number detection. We identify the expectation value of the projector onto the $N$-particle symmetric subspace as an operationally meaningful measure of indistinguishability. We show that these states are diagonal in the computational basis up to a permutationally invariant unitary.
  arXiv  Detail & Related papers  (2023-07-13T08:45:51Z)
• Exact solution of the Bose Hubbard model with unidirectional hopping [4.430341888774933]
  A one-dimensional Bose Hubbard model with unidirectional hopping is shown to be exactly solvable. We prove the integrability of the model and derive the Bethe ansatz equations. The exact eigenvalue spectrum can be obtained by solving these equations.
  arXiv  Detail & Related papers  (2023-04-30T09:50:51Z)
• The kinetic Hamiltonian with position-dependent mass [0.0]
  We examine in a systematic way the most relevant orderings of pure kinetic Hamiltonians for five different position-dependent mass profiles. As a result of the non-commutativity between momentum and position operators, a diversity of effective potentials is generated. We obtain analytically the full-spectrum of energies and solutions in the twenty-five cases considered.
  arXiv  Detail & Related papers  (2023-03-04T21:23:42Z)
• Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes [52.92110730286403]
  It is commonly believed that the marginal likelihood should be reminiscent of cross-validation metrics and that both should deteriorate with larger input dimensions. We prove that by tuning hyper parameters, the performance, as measured by the marginal likelihood, improves monotonically with the input dimension. We also prove that cross-validation metrics exhibit qualitatively different behavior that is characteristic of double descent.
  arXiv  Detail & Related papers  (2022-10-14T08:09:33Z)
• Statistical Efficiency of Score Matching: The View from Isoperimetry [96.65637602827942]
  We show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated. We formalize these results both in the sample regime and in the finite regime.
  arXiv  Detail & Related papers  (2022-10-03T06:09:01Z)
• Fermionic approach to variational quantum simulation of Kitaev spin models [50.92854230325576]
  Kitaev spin models are well known for being exactly solvable in a certain parameter regime via a mapping to free fermions. We use classical simulations to explore a novel variational ansatz that takes advantage of this fermionic representation. We also comment on the implications of our results for simulating non-Abelian anyons on quantum computers.
  arXiv  Detail & Related papers  (2022-04-11T18:00:01Z)
• Nonparametric Conditional Local Independence Testing [69.31200003384122]
  Conditional local independence is an independence relation among continuous time processes. No nonparametric test of conditional local independence has been available. We propose such a nonparametric test based on double machine learning.
  arXiv  Detail & Related papers  (2022-03-25T10:31:02Z)
• Assessment of the variational quantum eigensolver: application to the Heisenberg model [0.0]
  We present and analyze large-scale simulation results of a hybrid quantum-classical variational method to calculate the ground state energy of the anti-ferromagnetic Heisenberg model. We predict that a fully functional quantum computer with 100 qubits can calculate the ground state energy with a relatively small error.
  arXiv  Detail & Related papers  (2022-01-13T16:49:04Z)
• Fixed Point Quantum Monte Carlo [0.0]
  We present a new approach to the study of equilibrium properties in many-body quantum physics. Our method takes inspiration from Density Matrix Quantum Monte Carlo and incorporates new crucial features. We benchmark our method by applying it to two case-studies in condensed matter physics, show its accuracy and further discuss its efficiency.
  arXiv  Detail & Related papers  (2022-01-04T23:52:14Z)
• PSD Representations for Effective Probability Models [117.35298398434628]
  We show that a recently proposed class of positive semi-definite (PSD) models for non-negative functions is particularly suited to this end. We characterize both approximation and generalization capabilities of PSD models, showing that they enjoy strong theoretical guarantees. Our results open the way to applications of PSD models to density estimation, decision theory and inference.
  arXiv  Detail & Related papers  (2021-06-30T15:13:39Z)
• The Variational Method of Moments [65.91730154730905]
  conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables. Motivated by a variational minimax reformulation of OWGMM, we define a very general class of estimators for the conditional moment problem. We provide algorithms for valid statistical inference based on the same kind of variational reformulations.
  arXiv  Detail & Related papers  (2020-12-17T07:21:06Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.