Related papers: Beyond Log-Supermodularity: Lower Bounds and the B…
Sudderth, Wainwright, and Willsky have conjectured that the Bethe approximation corresponding to any fixed point of the belief propagation algorithm over an attractive, pairwise binary graphical model provides a lower bound on the true…
Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition…
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…
For various classes of graphical models it has been observed that the ratio of the partition sum to its Bethe approximation is often close to being the square of the ratio of the partition sum to its degree-2 Bethe approximation. This is of…
In this thesis, we leverage finite graph covers to analyze the SPA and the Bethe partition function for both S-FGs and DE-FGs. There are two main contributions in this thesis. The first main contribution concerns a special class of S-FGs…
The Bethe approximation is a well-known approximation of the partition function used in statistical physics. Recently, an equality relating the partition function and its Bethe approximation was obtained for graphical models with binary…
We consider the Potts model in a magnetic field on an arbitrary graph $G$. Using a formula of F. Y. Wu for the partition function $Z$ of this model as a sum over spanning subgraphs of $G$, we prove some properties of $Z$ concerning…
For a graph $G=(V,E)$, $k\in \mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as \[ {\bf Z}(G;k,w):=\sum_{\phi:V\to [k]}\prod_{\substack{uv\in E \\ \phi(u)=\phi(v)}}w, \] where…
We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees…
We consider the Potts model with $q$ colors on a sequence of weighted graphs with adjacency matrices $A_n$, allowing for both positive and negative weights. Under a mild regularity condition the mean-field prediction for the log partition…
We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an…
We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter…
The submodular partitioning problem asks to minimize, over all partitions $P$ of a ground set $V$, the sum of a given submodular function $f$ over the parts of $P$. The problem has seen considerable work in approximability, as it…
It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different…
We present exact results on the partition function of the $q$-state Potts model on various families of graphs $G$ in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set…
Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions…
Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase…
The Bethe approximation, or loopy belief propagation algorithm is a successful method for approximating partition functions of probabilistic models associated with a graph. Chertkov and Chernyak derived an interesting formula called Loop…
The permanent of a non-negative matrix appears naturally in many information processing scenarios. Because of the intractability of the permanent beyond small matrices, various approximation techniques have been developed in the past. In…
For standard factor graphs (S-FGs) with non-negative real-valued local functions, Vontobel provided a combinatorial characterization of the Bethe approximation of the partition function, also known as the Bethe partition function, using…