Real-time crime forecasting is important. However, accurate prediction of when and where the next crime will happen is difficult. No known physical model provides a reasonable approximation to such a complex system. Historical crime data are sparse in both space and time and the signal of interests is weak. In this work, we first present a proper representation of crime data. We then adapt the spatial temporal residual network on the well represented data to predict the distribution of crime in Los Angeles at the scale of hours in neighborhood-sized parcels. These experiments as well as comparisons with several existing approaches to prediction demonstrate the superiority of the proposed model in terms of accuracy. Finally, we present a ternarization technique to address the resource consumption issue for its deployment in real world. This work is an extension of our short conference proceeding paper [Wang et al, Arxiv 1707.03340].

entangled states. We construct sets of fewer than d orthogonal maximally entangled states which are not distinguished by one-way local operations and classical communication (LOCC) in the Hilbert space of d ⊗ d. The proof, based on the Fourier transform of an additive group, is very simple but quite effective. Simultaneously, our results give a general unified upper bound for the minimum number of one-way LOCC indistinguishable maximally entangled states. This improves previous results which only showed sets of N  d − 2 such states. Finally, our results also show that previous conjectures in Zhang et al. [Z.-C. Zhang, Q.-Y. Wen, F. Gao, G.-J. Tian, and T.-Q. Cao, Quant. Info. Proc. 13, 795 (2014)] are indeed correct.

In this paper, we give an almost complete classication of toric surface codes of dimension less than or equal to 7, according to monomially equivalence. This is a natural extension of our previous work [YZ], [LYZZ]. More pairs of monomially equivalent toric codes constructed from non-equivalent lattice polytopes are discovered. A new phenomenon appears, that is, the monomially non-equivalence of two toric codes C P(10)7and CP(19)7 can be discerned on Fq, for all q  8, except q = 29. This sudden break seems to be strange and interesting. Moreover, the parameters, such as the numbers of codewords with dierent weights, depends on q heavily. More meticulous analyses have been made to have the possible distinct families of reducible polynomials.

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We study the minimization problem of a non-convex sparsity promoting penalty func- tion, the transformed l1 (TL1), and its application in compressed sensing (CS). The TL1 penalty inter- polates l0 and l1 norms through a nonnegative parameter a ∈ (0, +∞), similar to lp with p ∈ (0, 1], and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of l0 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l0 norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized con- strained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an l1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value a = 1, and compare DCATL1 with other CS algorithms on two classes of sensing ma- trices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on l1/2 norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on l1 minus l2 penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also com- petitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.