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Register a new userA Factor-Graph Approach to Algebraic Topology, With Applications to Kramers--Wannier Duality
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Added by
Ali Al-Bashabsheh
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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No Arabic abstract
Algebraic topology studies topological spaces with the help of tools from abstract algebra. The main focus of this paper is to show that many concepts from algebraic topology can be conveniently expressed in terms of (normal) factor graphs. As an application, we give an alternative proof of a classical duality result of Kramers and Wannier, which expresses the partition function of the two-dimensional Ising model at a low temperature in terms of the partition function of the two-dimensional Ising model at a high temperature. Moreover, we discuss analogous results for the three-dimensional Ising model and the Potts model.
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Network graphs have become a popular tool to represent complex systems composed of many interacting subunits; especially in neuroscience, network graphs are increasingly used to represent and analyze functional interactions between neural sources. Interactions are often reconstructed using pairwise bivariate analyses, overlooking their multivariate nature: it is neglected that investigating the effect of one source on a target necessitates to take all other sources as potential nuisance variables into account; also combinations of sources may act jointly on a given target. Bivariate analyses produce networks that may contain spurious interactions, which reduce the interpretability of the network and its graph metrics. A truly multivariate reconstruction, however, is computationally intractable due to combinatorial explosion in the number of potential interactions. Thus, we have to resort to approximative methods to handle the intractability of multivariate interaction reconstruction, and thereby enable the use of networks in neuroscience. Here, we suggest such an approximative approach in the form of an algorithm that extends fast bivariate interaction reconstruction by identifying potentially spurious interactions post-hoc: the algorithm flags potentially spurious edges, which may then be pruned from the network. This produces a statistically conservative network approximation that is guaranteed to contain non-spurious interactions only. We describe the algorithm and present a reference implementation to test its performance. We discuss the algorithm in relation to other approximative multivariate methods and highlight suitable application scenarios. Our approach is a tractable and data-efficient way of reconstructing approximative networks of multivariate interactions. It is preferable if available data are limited or if fully multivariate approaches are computationally infeasible.
This paper presents a novel successive factor-graph permutation (SFP) scheme that significantly improves the error-correction performance of Reed-Muller (RM) codes under successive-cancellation list (SCL) decoding. In particular, we perform maximum-likelihood decoding on the symmetry group of RM codes to carefully select a good factor-graph permutation on the fly. We further propose an SFP-aided fast SCL decoding that significantly reduces the decoding latency while preserving the error-correction performance of the code. The simulation results show that for the third and fourth order RM codes of length $256$, the proposed decoder reduces up to $85%$ of the memory consumption, $78%$ of the decoding latency, and more than $99%$ of the computational complexity of the state-of-the-art recursive projection-aggregation decoder at the frame error rate of $10^{-3}$.
This paper takes a rate-distortion approach to understanding the information-theoretic laws governing cache-aided communications systems. Specifically, we characterise the optimal tradeoffs between the delivery rate, cache capacity and reconstruction distortions for a single-user problem and some special cases of a two-user problem. Our analysis considers discrete memoryless sources, expected- and excess-distortion constraints, and separable and f-separable distortion functions. We also establish a strong converse for separable-distortion functions, and we show that los
A new method to measure nonlinear dependence between two variables is described using mutual information to analyze the separate linear and nonlinear components of dependence. This technique, which gives an exact value for the proportion of linear dependence, is then compared with another common test for linearity, the Brock, Dechert and Scheinkman (BDS) test.
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information.
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