Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations

In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomials to be real-valued. As applications of our results, we discuss the relation between nonnegative polynomials and sums of squares in the context of complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for complex tensors are introduced, extending properties from the Hermitian matrices. Finally, we discuss an important property for symmetric tensors, which states that the largest absolute value of eigenvalue of a symmetric real tensor is equal to its largest singular value; the result is known as Banach's theorem. We show that a similar result holds in the complex case as well

Ternary Weight Networks

We introduce ternary weight networks (TWNs) - neural networks with weights constrained to +1, 0 and -1. The Euclidian distance between full (float or double) precision weights and the ternary weights along with a scaling factor is minimized. Besides, a threshold-based ternary function is optimized to get an approximated solution which can be fast and easily computed. TWNs have stronger expressive abilities than the recently proposed binary precision counterparts and are thus more effective than the latter. Meanwhile, TWNs achieve up to 16$\times$ or 32$\times$ model compression rate and need fewer multiplications compared with the full precision counterparts. Benchmarks on MNIST, CIFAR-10, and large scale ImageNet datasets show that the performance of TWNs is only slightly worse than the full precision counterparts but outperforms the analogous binary precision counterparts a lot.Comment: 5 pages, 3 fitures, conferenc