In this paper, we give an almost complete classication 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
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.
Penghang YinUniversity of California at Los AngelesShuai ZhangUniversity of California at IrvineJiancheng LyuUniversity of California at Los AngelesStanley OsherUniversity of California at IrvineYingyong QiUniversity of California at IrvineJack XinUniversity of California at Irvine
We propose BinaryRelax, a simple two-phase algorithm, for training deep neural networks with quantized weights. The set constraint that characterizes the quantization of weights is not imposed until the late stage of training, and a sequence of pseudo quantized weights is maintained. Specifically, we relax the hard constraint into a continuous regularizer via Moreau envelope, which turns out to be the squared Euclidean distance to the set of quantized weights. The pseudo quantized weights are obtained by linearly interpolating between the float weights and their quantizations. A continuation strategy is adopted to push the weights towards the quantized state by gradually increasing the regularization parameter. In the second phase, exact quantization scheme with a small learning rate is invoked to guarantee fully quantized weights. We test BinaryRelax on the benchmark CIFAR- 10 and CIFAR-100 color image datasets to demonstrate the superiority of the relaxed quantization approach and the improved accuracy over the state-of-the-art training methods. Finally, we prove the convergence of BinaryRelax under an approximate orthogonality condition.
Bao WangDepartment of Mathematics, UCLAPenghang YinDepartment of Mathematics, UCLAAndrea L. BertozziDepartment of Mathematics, UCLAP. Jeffrey BrantinghamDepartment of Anthropology, UCLAStanley J. OsherDepartment of Mathematics, UCLAJack XinDepartment of Mathematics, UCLA&UCI
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].