human: Given a hyperedge-centered hypergraph: <hypergraph>, where the center hyperedge is induced by one source paper and its member nodes are the cited papers.

Details of the hypergraph:
Center hyperedge (induced by source paper "belief propagation bethe approximation and polynomials"):
  Abstract: Factor graphs are important models for succinctly representing probability distributions in machine learning, coding theory, and statistical physics. Several computational problems, such as computing marginals and partition functions, arise naturally when working with factor graphs. Belief propagation is a widely deployed iterative method for solving these problems. However, despite its significant empirical success, not much is known about the correctness and efficiency of belief propagation. #R##N#Bethe approximation is an optimization-based framework for approximating partition functions. While it is known that the stationary points of the Bethe approximation coincide with the fixed points of belief propagation, in general, the relation between the Bethe approximation and the partition function is not well understood. It has been observed that for a few classes of factor graphs, the Bethe approximation always gives a lower bound to the partition function, which distinguishes them from the general case, where neither a lower bound, nor an upper bound holds universally. This has been rigorously proved for permanents and for attractive graphical models. #R##N#Here we consider bipartite normal factor graphs and show that if the local constraints satisfy a certain analytic property, the Bethe approximation is a lower bound to the partition function. We arrive at this result by viewing factor graphs through the lens of polynomials. In this process, we reformulate the Bethe approximation as a polynomial optimization problem. Our sufficient condition for the lower bound property to hold is inspired by recent developments in the theory of real stable polynomials. We believe that this way of viewing factor graphs and its connection to real stability might lead to a better understanding of belief propagation and factor graphs in general.

Includes members:
  Member "a generalization of permanent inequalities and applications in counting and optimization":
    also connected through other hyperedges:
      - Hyperedge HE-1 (induced by source paper "gauges loops and polynomials for partition functions of graphical models")
  Member "real stable polynomials and matroids optimization and counting":
    also connected through other hyperedges:
      - Hyperedge HE-2 (induced by source paper "dynamic sampling from graphical models")
  Member "effective resistance reducing flows spectrally thin trees and asymmetric tsp":
    also connected through other hyperedges:
      - Hyperedge HE-3 (induced by source paper "a spectral approach to network design")
  Member "the bethe permanent of a nonnegative matrix":
    also connected through other hyperedges:
      - Hyperedge HE-4 (induced by source paper "gauges loops and polynomials for partition functions of graphical models")
  Member "entropy optimization and counting":
    also connected through other hyperedges:
      - Hyperedge HE-5 (induced by source paper "fair distributions from biased samples a maximum entropy optimization framework")
  Member "counting in graph covers a combinatorial characterization of the bethe entropy function":
    also connected through other hyperedges:
      - Hyperedge HE-6 (induced by source paper "pseudocodeword based decoding of quantum stabilizer codes")
  Member "the bethe partition function of log supermodular graphical models":
    also connected through other hyperedges:
      - Hyperedge HE-7 (induced by source paper "lifted hybrid variational inference")
  Member "partition functions of normal factor graphs":
    also connected through other hyperedges:
      - Hyperedge HE-8 (induced by source paper "codes on graphs models for elementary algebraic topology and statistical physics")

Question: Please tell me which class the source paper belongs to. The 40 classes are: cs.NA(Numerical Analysis), cs.MM(Multimedia), cs.LO(Logic in Computer Science), cs.CY(Computers and Society), cs.CR(Cryptography and Security), cs.DC(Distributed, Parallel, and Cluster Computing), cs.HC(Human-Computer Interaction), cs.CE(Computational Engineering, Finance, and Science), cs.NI(Networking and Internet Architecture), cs.CC(Computational Complexity), cs.AI(Artificial Intelligence), cs.MA(Multiagent Systems), cs.GL(General Literature), cs.NE(Neural and Evolutionary Computing), cs.SC(Symbolic Computation), cs.AR(Hardware Architecture), cs.CV(Computer Vision and Pattern Recognition), cs.GR(Graphics), cs.ET(Emerging Technologies), cs.SY(Systems and Control), cs.CG(Computational Geometry), cs.OH(Other Computer Science), cs.PL(Programming Languages), cs.SE(Software Engineering), cs.LG(Machine Learning), cs.SD(Sound), cs.SI(Social and Information Networks), cs.RO(Robotics), cs.IT(Information Theory), cs.PF(Performance), cs.CL(Computation and Language), cs.IR(Information Retrieval), cs.MS(Mathematical Software), cs.FL(Formal Languages and Automata Theory), cs.DS(Data Structures and Algorithms), cs.OS(Operating Systems), cs.GT(Computer Science and Game Theory), cs.DB(Databases), cs.DL(Digital Libraries), cs.DM(Discrete Mathematics). Directly output the class name.

gpt: cs.LG(Machine Learning)