On the critical group of matrices

Given a graph G with a distinguished vertex s, the critical group of (G,s) is the cokernel of their reduced Laplacian matrix L(G,s). In this article we generalize the concept of the critical group to the cokernel of any matrix with entries in a commutative ring with identity. In this article we find diagonal matrices that are equivalent to some matrices that generalize the reduced Laplacian matrix of the path, the cycle, and the complete graph over an arbitrary commutative ring with identity. We are mainly interested in those cases when the base ring is the ring of integers and some subrings of matrices. Using these equivalent diagonal matrices we calculate the critical group of the m-cones of the l-duplications of the path, the cycle, and the complete graph. Also, as byproduct, we calculate the critical group of another matrices, as the m-cones of the l-duplication of the bipartite complete graph with m vertices in each partition, the bipartite complete graph with 2m vertices minus a matching.Comment: 18 pages, 5 figure

Exact and Efficient Simulation of Concordant Computation

Concordant computation is a circuit-based model of quantum computation for mixed states, that assumes that all correlations within the register are discord-free (i.e. the correlations are essentially classical) at every step of the computation. The question of whether concordant computation always admits efficient simulation by a classical computer was first considered by B. Eastin in quant-ph/1006.4402v1, where an answer in the affirmative was given for circuits consisting only of one- and two-qubit gates. Building on this work, we develop the theory of classical simulation of concordant computation. We present a new framework for understanding such computations, argue that a larger class of concordant computations admit efficient simulation, and provide alternative proofs for the main results of quant-ph/1006.4402v1 with an emphasis on the exactness of simulation which is crucial for this model. We include detailed analysis of the arithmetic complexity for solving equations in the simulation, as well as extensions to larger gates and qudits. We explore the limitations of our approach, and discuss the challenges faced in developing efficient classical simulation algorithms for all concordant computations.Comment: 16 page

Conditional Restricted Boltzmann Machines for Structured Output Prediction

Conditional Restricted Boltzmann Machines (CRBMs) are rich probabilistic models that have recently been applied to a wide range of problems, including collaborative filtering, classification, and modeling motion capture data. While much progress has been made in training non-conditional RBMs, these algorithms are not applicable to conditional models and there has been almost no work on training and generating predictions from conditional RBMs for structured output problems. We first argue that standard Contrastive Divergence-based learning may not be suitable for training CRBMs. We then identify two distinct types of structured output prediction problems and propose an improved learning algorithm for each. The first problem type is one where the output space has arbitrary structure but the set of likely output configurations is relatively small, such as in multi-label classification. The second problem is one where the output space is arbitrarily structured but where the output space variability is much greater, such as in image denoising or pixel labeling. We show that the new learning algorithms can work much better than Contrastive Divergence on both types of problems