The Curveball algorithm is a variation on well-known switch-based Markov chain approaches for uniformly sampling binary matrices with fixed row and column sums. Instead of a switch, the Curveball algorithm performs a so-called binomial trade in every iteration of the algorithm. Intuitively, this could lead to a better convergence rate for reaching the stationary (uniform) distribution in certain cases. Some experimental evidence for this has been given in the literature. In this note we give a spectral gap comparison between two switch-based chains and the Curveball chain. In particular, this comparison allows us to conclude that the Curveball Markov chain is rapidly mixing whenever one of the two switch chains is rapidly mixing. Our analysis directly extends to the case of sampling binary matrices with forbidden entries (under the assumption of irreducibility). This in particular captures the case of sampling simple directed graphs with given degrees. As a by-product of our analysis, we show that the switch Markov chain of the Kannan-Tet ali-Vempala conjecture only has non-negative eigenvalues if the sampled binary matrices have at least three columns. This shows that the Markov chain does not have to be made lazy, which is of independent interest. We also obtain an improved bound on the smallest eigenvalue for the switch Markov chain studied by Greenhill for uniformly sampling simple directed regular graphs.
The transition matrix of the Curveball Markov chain is a special case of a heat-bath Markov chain, as introduced by Dyer, Greenhill and Ullrich @cite_19 . Our work partially builds on @cite_19 in the sense that we compare a Markov chain, with a similar decomposition property as in the definition of a heat-bath chain, to its heat-bath variant. We explain these ideas in the next section.
@cite_19
1983481540 2071849960 2376316332 2057853309
We prove that heat-bath chains (which we define in a general setting) have no negative eigenvalues. Two applications of this result are presented: one to single-site heat-bath chains for spin systems and one to a heat-bath Markov chain for sampling contingency tables. Some implications of our main result for the analysis of the mixing time of heat-bath Markov chains are discussed. We also prove an alternative characterisation of heat-bath chains, and consider possible generalisations. We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so–called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information. Markov chain Monte Carlo (MCMC) methods have been used in many fields (physics, chemistry, biology, and computer science) for simulation, inference, and optimization. In many applications, Markov chains are simulated for sampling from target probabilities π(X) defined on graphs G. The graph vertices represent elements of the system, the edges represent spatial relationships, while X is a vector of variables on the vertices which often take discrete values called labels or colors. Designing efficient Markov chains is a challenging task when the variables are strongly coupled. Because of this, methods such as the single-site Gibbs sampler often experience suboptimal performance. A well-celebrated algorithm, the Swendsen–Wang (SW) method, can address the coupling problem. It clusters the vertices as connected components after turning off some edges probabilistically, and changes the color of one cluster as a whole. It is known to mix rapidly under certain conditions. Unfortunately, the SW method has limited ... A Markov model of evolution of characters on a phylogenetic tree consists of a tree topology together with a specification of probability transition matrices on the edges of the tree. Previous work has shown that, under mild conditions, the tree topology may be reconstructed, in the sense that the topology is identifiable from knowledge of the joint distribution of character states at pairs of terminal nodes of the tree. Also, the method of maximum likelihood is statistically consistent for inferring the tree topology. In this article we answer the analogous questions for reconstructing the full model, including the edge transition matrices. Under mild conditions, such full reconstruction is achievable, not by using pairs of terminal nodes, but rather by using triples of terminal nodes. The identifiability result generalizes previous results that were restricted either to characters having two states or to transition matrices having special structure. The proof develops matrix relationships that may be exploited to identify the model. We also use the identifiability result to prove that the method of maximum likelihood is consistent for reconstructing the full model.