Fredholm factorization of Wiener-Hopf scalar and matrix kernels

A general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape

Prime Factorization in the Duality Computer

We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat's method in classical computing. All these algorithms are polynomial in the input size.Comment: 4 page

Deep factorization for speech signal

Various informative factors mixed in speech signals, leading to great difficulty when decoding any of the factors. An intuitive idea is to factorize each speech frame into individual informative factors, though it turns out to be highly difficult. Recently, we found that speaker traits, which were assumed to be long-term distributional properties, are actually short-time patterns, and can be learned by a carefully designed deep neural network (DNN). This discovery motivated a cascade deep factorization (CDF) framework that will be presented in this paper. The proposed framework infers speech factors in a sequential way, where factors previously inferred are used as conditional variables when inferring other factors. We will show that this approach can effectively factorize speech signals, and using these factors, the original speech spectrum can be recovered with a high accuracy. This factorization and reconstruction approach provides potential values for many speech processing tasks, e.g., speaker recognition and emotion recognition, as will be demonstrated in the paper.Comment: Accepted by ICASSP 2018. arXiv admin note: substantial text overlap with arXiv:1706.0177