**End of Potential Line (EOPL)**

We introduce the problem EndOfPotentialLine and the corresponding complexity class EOPL of all problems that can be reduced to it in polynomial time. This class captures problems that admit a single combinatorial proof of their joint membership in the complexity classes PPAD of fixpoint problems and PLS of local search problems. EOPL is a combinatorially-defined alternative to the class CLS (for Continuous Local Search), which was introduced in with the goal of capturing the complexity of some well-known problems in PPAD $\cap$ PLS that have resisted, in some cases for decades, attempts to put them in polynomial time. Two of these are Contraction, the problem of finding a fixpoint of a contraction map, and P-LCP, the problem of solving a P-matrix Linear Complementarity Problem. We show that EndOfPotentialLine is in CLS via a two-way reduction to EndOfMeteredLine. The latter was defined in to show query and cryptographic lower bounds for CLS. Our two main results are to show that both PL-Contraction (Piecewise-Linear Contraction) and P-LCP are in EOPL. Our reductions imply that the promise versions of PL-Contraction and P-LCP are in the promise class UniqueEOPL, which corresponds to the case of a single potential line. This also shows that simple-stochastic, discounted, mean-payoff, and parity games are in EOPL. Using the insights from our reduction for PL-Contraction, we obtain the first polynomial-time algorithms for finding fixed points of contraction maps in fixed dimension for any $\ell_p$ norm, where previously such algorithms were only known for the $\ell_2$ and $\ell_\infty$ norms. Our reduction from P-LCP to EndOfPotentialLine allows a technique of Aldous to be applied, which in turn gives the fastest-known randomized algorithm for the P-LCP. … **Multi-Lane Capsule Network (MLCN)**

We introduce Multi-Lane Capsule Networks (MLCN), which are a separable and resource efficient organization of Capsule Networks (CapsNet) that allows parallel processing, while achieving high accuracy at reduced cost. A MLCN is composed of a number of (distinct) parallel lanes, each contributing to a dimension of the result, trained using the routing-by-agreement organization of CapsNet. Our results indicate similar accuracy with a much reduced cost in number of parameters for the Fashion-MNIST and Cifar10 datsets. They also indicate that the MLCN outperforms the original CapsNet when using a proposed novel configuration for the lanes. MLCN also has faster training and inference times, being more than two-fold faster than the original CapsNet in the same accelerator. … **Orthogonal Matching Pursuit**

In text classification, the problem of overfitting arises due to the high dimensionality, making regularization essential. Although classic regularizers provide sparsity, they fail to return highly accurate models. On the contrary, state-of-the-art group-lasso regularizers provide better results at the expense of low sparsity. In this paper, we apply a greedy variable selection algorithm, called Orthogonal Matching Pursuit, for the text classification task. We also extend standard group OMP by introducing overlapping group OMP to handle overlapping groups of features. Empirical analysis verifies that both OMP and overlapping GOMP constitute powerful regularizers, able to produce effective and super-sparse models. Code and data are available at: https://…tring_G0\string_0DlcGkq6tQb2zqAaca\string dl\string=0 . …

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Mar 2019

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