A complete set of filters @math for the optimal-depth @math -input sorting network problem is such that if there exists an @math -input sorting network of depth @math then there exists one of the form @math for some @math . Previous work on the topic presents a method for finding complete set of filters @math and @math that consists only of networks of depths one and two respectively, whose outputs are minimal and representative up to permutation and reflection. We present a novel practical approach for finding a complete set of filters @math containing only networks of depth three whose outputs are minimal and representative up to permutation and reflection. In previous work, we have developed a highly efficient algorithm for finding extremal sets ( i.e. outputs of comparator networks; itemsets; ) up to permutation. In this paper we present a modification to this algorithm that identifies the representative itemsets up to permutation and reflection. Hence, the here presented practical approach is the successful combination of known theory and practice that we apply to the domain of sorting networks. For all @math , we empirically compute the complete set of filters @math of the representative minimal up to permutation and reflection @math -input networks of depth three.
--- we took an existing algorithm for finding minimal itemsets up to permutation which a dataset (a collection of itemsets). We present a modification which is linear (in terms of the number of itemsets) in time and space to find the ones which are minimal up to permutation and reflection. --- this is a direct improvement of Bundala's technique of not considering all of the @math inputs to determine that no @math -input sorting network of depth @math exists. We take one step further to find the minimal up to permutation and reflection itemsets after applying the input set reduction (described in Experiment 3 @cite_6 ). --- we experimentally evaluated the modified algorithm to find the three layered @math -input comparator networks whose outputs (itemsets) are minimal up to permutation and reflection. The set @math is generated by applying all network levels to @math and then finding the minimal up to permutation and reflection ones, whereas @math is derived by applying all levels to all itemsets in the set @math and then reducing.
@cite_6
1769744536 2952149393 2766371994 2036804629
A complete set of filters @math for the optimal-depth @math -input sorting network problem is such that if there exists an @math -input sorting network of depth @math then there exists one of the form @math for some @math . Previous work on the topic presents a method for finding complete set of filters @math and @math that consists only of networks of depths one and two respectively, whose outputs are minimal and representative up to permutation and reflection. We present a novel practical approach for finding a complete set of filters @math containing only networks of depth three whose outputs are minimal and representative up to permutation and reflection. In previous work, we have developed a highly efficient algorithm for finding extremal sets ( i.e. outputs of comparator networks; itemsets; ) up to permutation. In this paper we present a modification to this algorithm that identifies the representative itemsets up to permutation and reflection. Hence, the here presented practical approach is the successful combination of known theory and practice that we apply to the domain of sorting networks. For all @math , we empirically compute the complete set of filters @math of the representative minimal up to permutation and reflection @math -input networks of depth three. In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order @math for all pairs @math , where @math is a constant and @math for all @math and @math . It is assumed that the errors are independent. Given the status of the queries the goal is to find the maximum likelihood order. In other words, the goal is find a permutation @math that minimizes the number of pairs @math where @math . The problem so defined is the feedback arc set problem on distributions of inputs, each of which is a tournament obtained as a noisy perturbations of a linear order. Note that when @math and @math is large, it is impossible to recover the original order @math . It is known that the weighted feedback are set problem on tournaments is NP-hard in general. Here we present an algorithm of running time @math and sampling complexity @math that with high probability solves the noisy sorting without re-sampling problem. We also show that if @math is an optimal solution of the problem then it is close'' to the original order. More formally, with high probability it holds that @math and @math . Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts. We consider the problem of learning a one-hidden-layer neural network: we assume the input @math is from Gaussian distribution and the label @math , where @math is a nonnegative vector in @math with @math , @math is a full-rank weight matrix, and @math is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function @math whose landscape is guaranteed to have the following properties: 1. All local minima of @math are also global minima. 2. All global minima of @math correspond to the ground truth parameters. 3. The value and gradient of @math can be estimated using samples. With these properties, stochastic gradient descent on @math provably converges to the global minimum and learn the ground-truth parameters. We also prove finite sample complexity result and validate the results by simulations. When faced with a complex task, is it better to be systematic or to proceed by making random adjustments? We study aspects of this problem in the context of generating random elements of a finite group. For example, suppose we want to fill n empty spaces with zeros and ones such that the probability of configuration x = (x1, . . . , xn) is θ n−|x|(1− θ)|x|, with |x| the number of ones in x. A systematic scan approach works left to right, filling each successive place with a θ coin toss. A random scan approach picks places at random, and a given site may be hit many times before all sites are hit. The systematic approach takes order n steps and the random approach takes order 1 4n log n steps. Realistic versions of this toy problem arise in image analysis and Ising-like simulations, where one must generate a random array by a Monte Carlo Markov chain. Systematic updating and random updating are competing algorithms that are discussed in detail in Section 2. There are some successful analyses for random scan algorithms, but the intuitively appealing systematic scan algorithms have resisted analysis. Our main results show that the binary problem just described is exceptional; for the examples analyzed in this paper, systematic and random scans converge in about the same number of steps. Let W be a finite Coxeter group generated by simple reflections s1, s2, . . . , sn, where s2 i = id. For example, W may be the permutation group Sn+1 with si = (i, i + 1). The length function (w) is the smallest k such that w = si1si2 · · · sik . Fix 0 < θ ≤ 1 and define a probability distribution on W by π(w) = θ − (w) PW (θ−1) , where PW(θ −1) = ∑