Function-Train: A continuous analogue of the tensor-train decomposition

We describe a new function approximation framework based on a continuous extension of the tensor-train decomposition. The approximation, termed a function-train (FT), results in a tensor-train structure whose cores are univariate functions. An advantage of the FT over discrete approaches is that it produces an adaptive approximation of tensor fibers that is not tied to any tensorized discretization procedure; indeed, the algorithm can be coupled with any univariate linear or nonlinear approximation procedure. Furthermore, the representation of low-rank functions in FT format enables efficient continuous computation: we can add, multiply, integrate, and differentiate functions in polynomial time with respect to dimension. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. Our contributions include creating an algorithm for finding the maximum volume submatrix of a matrix-valued function, a maximum-volume based cross approximation algorithm to obtain skeleton decompositions of vector-valued functions, a cross approximation algorithm for converting black-box functions into FT format, and a continuous rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate the benefits of our approach by integrating high-dimensional and discontinuous functions, and apply it to a variety of approximation tasks.

Similar Publications

In this dissertation we study the tractability of the information-based complexity $n(\varepsilon,d)$ for $d$-variate function approximation problems. In the deterministic setting for many unweighted problems the curse of dimensionality holds, that means, for some fixed error tolerance $\varepsilon>0$ the complexity $n(\varepsilon,d)$ grows exponentially in $d$. For integration problems one can usually break the curse with the standard Monte Carlo method. Read More

Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network flow problems. Regularization typically is needed to ensure uniqueness for the linear ground distance case and to improve optimization convergence; state-of-the-art techniques employ entropic regularization on the transportation matrix. Read More

In this paper we extend the application of Hamiltonian Boundary Value Methods (HBVMs), a class of energy-conserving Runge-Kutta methods for Hamiltonian problems, to the numerical solution of Hamiltonian systems with holonomic constraints. The extension is obtained through a straightforward use of the so called line integral approach on which the methods rely, which is here used also for dealing with the constraints. As a result, the modified methods conserve both the Hamiltonian and the constraints, as is shown in their analysis. Read More

We consider the multilinear pagerank problem studied in [Gleich, Lim and Yu, Multilinear Pagerank, 2015], which is a system of quadratic equations with stochasticity and nonnegativity constraints. We use the theory of quadratic vector equations to prove several properties of its solutions and suggest new numerical algorithms. In particular, we prove the existence of a certain minimal solution, which does not always coincide with the stochastic one that is required by the problem. Read More

We study the exponent of the exponential rate of convergence in terms of the number of degrees of freedom for various non-standard $hp$-finite element spaces employing reduced cardinality basis. More specifically, we show that serendipity finite element methods and discontinuous Galerkin finite element methods with total degree $\mathcal{P}_p$ basis have a faster exponential convergence with respect to the number of degrees of freedom than their counterparts employing the tensor product $\mathcal{Q}_p$ basis for quadrilateral/hexahedral elements, for piecewise analytic problems under $p$-refinement. The above results are proven by using a new $p$-optimal error bound for the $L^2$-orthogonal projection onto the total degree $\mathcal{P}_p$ basis, and for the $H^1$-projection onto the serendipity finite element space over tensor product elements with dimension $d\geq2$. Read More

In this article, we consider discrete schemes for a fractional diffusion equation involving a tempered fractional derivative in time. We present a semi-discrete scheme by using the local discontinuous Galerkin (LDG) discretization in the spatial variables. We prove that the semi-discrete scheme is unconditionally stable in $L^2$ norm and convergence with optimal convergence rate $\mathcal{O}(h^{k+1})$. Read More

This paper introduces a new generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. Read More

In this article we explore an algorithm for diffeomorphic random sampling of nonuniform probability distributions on Riemannian manifolds. The algorithm is based on optimal information transport (OIT)---an analogue of optimal mass transport (OMT). Our framework uses the deep geometric connections between the Fisher-Rao metric on the space of probability densities and the right-invariant information metric on the group of diffeomorphisms. Read More

This paper considers the problem of decentralized optimization with a composite objective containing smooth and non-smooth terms. To solve the problem, a proximal-gradient scheme is studied. Specifically, the smooth and nonsmooth terms are dealt with by gradient update and proximal update, respectively. Read More

We propose an optimal transportation approach to price European options under the Stein-Stein stochastic volatility model by using the flow that transports the set of particles from a prior to a posterior distribution. We also propose to direct the flow to a rarely visited areas of the state space by using a mutation and a reweighing algorithm. We demonstrate the efficiency of our approach on a simple example for which the closed form formula is available. Read More