Decoupling Multivariate Polynomials Using First-Order Information

We present a method to decompose a set of multivariate real polynomials into linear combinations of univariate polynomials in linear forms of the input variables. The method proceeds by collecting the first-order information of the polynomials in a set of operating points, which is captured by the Jacobian matrix evaluated at the operating points. The polyadic canonical decomposition of the three-way tensor of Jacobian matrices directly returns the unknown linear relations, as well as the necessary information to reconstruct the univariate polynomials. The conditions under which this decoupling procedure works are discussed, and the method is illustrated on several numerical examples

Tensor-based framework for training flexible neural networks

Activation functions (AFs) are an important part of the design of neural networks (NNs), and their choice plays a predominant role in the performance of a NN. In this work, we are particularly interested in the estimation of flexible activation functions using tensor-based solutions, where the AFs are expressed as a weighted sum of predefined basis functions. To do so, we propose a new learning algorithm which solves a constrained coupled matrix-tensor factorization (CMTF) problem. This technique fuses the first and zeroth order information of the NN, where the first-order information is contained in a Jacobian tensor, following a constrained canonical polyadic decomposition (CPD). The proposed algorithm can handle different decomposition bases. The goal of this method is to compress large pretrained NN models, by replacing subnetworks, {\em i.e.,} one or multiple layers of the original network, by a new flexible layer. The approach is applied to a pretrained convolutional neural network (CNN) used for character classification.Comment: 26 pages, 13 figure

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube $[0,1]^s$. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost $1/N$, where $N$ is the number of points, independently of dimension) to so-called “product and order dependent†(POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets. doi:10.1017/S144618111200007

RG Flow from ϕ4\phi^4 Theory to the 2D Ising Model

We study 1+1 dimensional $\phi^4$ theory using the recently proposed method of conformal truncation. Starting in the UV CFT of free field theory, we construct a complete basis of states with definite conformal Casimir, $\mathcal{C}$. We use these states to express the Hamiltonian of the full interacting theory in lightcone quantization. After truncating to states with $\mathcal{C} \leq \mathcal{C}_{\max}$, we numerically diagonalize the Hamiltonian at strong coupling and study the resulting IR dynamics. We compute non-perturbative spectral densities of several local operators, which are equivalent to real-time, infinite-volume correlation functions. These spectral densities, which include the Zamolodchikov $C$-function along the full RG flow, are calculable at any value of the coupling. Near criticality, our numerical results reproduce correlation functions in the 2D Ising model.Comment: 31+12 page

A Conformal Truncation Framework for Infinite-Volume Dynamics

We present a new framework for studying conformal field theories deformed by one or more relevant operators. The original CFT is described in infinite volume using a basis of states with definite momentum, $P$, and conformal Casimir, $\mathcal{C}$. The relevant deformation is then considered using lightcone quantization, with the resulting Hamiltonian expressed in terms of this CFT basis. Truncating to states with $\mathcal{C} \leq \mathcal{C}_{\max}$, one can numerically find the resulting spectrum, as well as other dynamical quantities, such as spectral densities of operators. This method requires the introduction of an appropriate regulator, which can be chosen to preserve the conformal structure of the basis. We check this framework in three dimensions for various perturbative deformations of a free scalar CFT, and for the case of a free $O(N)$ CFT deformed by a mass term and a non-perturbative quartic interaction at large-$N$. In all cases, the truncation scheme correctly reproduces known analytic results. We also discuss a general procedure for generating a basis of Casimir eigenstates for a free CFT in any number of dimensions.Comment: 48+37 pages, 17 figures; v2: references added, small clarification