Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by (i) discussing the variety of mechanisms that allow it to be surprisingly efficient;(ii) addressing the issue of conditioning; (iii) presenting algorithms for solving linear systems within this framework; and (iv) demonstrating methods for dealing with antisymmetric functions, as arise in the multiparticle Schrödinger equation in quantum mechanics.Numerical examples are given.1. Introduction. The computational complexity of most algorithms in dimension d grows exponentially in d. Even simply accessing a vector in dimension d requires M d operations, where M is the number of entries in each direction. This effect has been dubbed the curse of dimensionality [1, p. 94], and it is the single greatest impediment to computing in higher dimensions. In [3] we introduced a strategy for performing numerical computations in high dimensions with greatly reduced cost, while maintaining the desired accuracy. In the present work, we extend and develop these techniques in a number of ways. In particular, we address the issue of conditioning, describe how to solve linear systems, and show how to deal with antisymmetric functions. We present numerical examples for each of these algorithms.

When an algorithm in dimension one is extended to dimension d, in nearly every case its computational cost is taken to the power d. This fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality. For numerical analysis in dimension d, we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy. Basic linear algebra operations can be performed in this representation using one-dimensional operations, thus bypassing the exponential scaling with respect to the dimension. Although not all operators and algorithms may be compatible with this representation, we believe that many of the most important ones are. We prove that the multiparticle Schrö dinger operator, as well as the inverse Laplacian, can be represented very efficiently in this form. We give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the groundstate eigenfunction for a simplified Schrö dinger operator with 30 particles. We conjecture and provide numerical evidence that functions of operators inherit this property, in which case numerical operator calculus in higher dimensions becomes feasible.I n almost all problems that arise from physics there is an underlying physical dimension, and in almost every case the algorithm to solve the problem will have computational complexity that grows exponentially in the physical dimension. In other words, when an algorithm in dimension one is extended to dimension d, its computational cost is taken to the power d. In this paper we present an approach that, in several important cases, allows one-dimensional algorithms to be extended to d dimensions without their computational complexity growing exponentially in d. In moderate dimensions (d ϭ 2, 3, 4) our approach greatly accelerates a number of algorithms. In higher dimensions, such as those arising from the multiparticle Schrödinger equation, where the wave function for p particles has d ϭ 3p variables, our approach makes algorithms feasible that would be unthinkable in a traditional approach.As an example of the exponential growth in d, consider ordinary matrix-matrix multiplication. In dimension d a matrix has (N 2 ) d entries, and matrix-matrix multiplication takes (N 3 ) d operations. Using a ''fast'' one-dimensional algorithm does not help: a banded matrix has (bN) d entries, and matrix-matrix multiplication takes (b 2 N) d operations. This fundamental difficulty is the single greatest impediment to solving many realworld problems and has been dubbed the curse of dimensionality (1).In problems in physics where the underlying assumptions permit, separation of variables has been the most successful approach for avoiding the high cost of working in d dimensions. Instead of trying to find a d-dimensional function that solves the given equation (e.g., the multiparticle Schrödinger equation), one only considers functions that can be represented as a product:[1...

We present an algorithm for learning (or estimating) a function of many variables from scattered data. The function is approximated by a sum of separable functions, following the paradigm of separated representations. The central fitting algorithm is linear in both the number of data points and the number of variables and, thus, is suitable for large data sets in high dimensions. We present numerical evidence for the utility of these representations. In particular, we show that our method outperforms other methods on several benchmark data sets.

ABSTRACT. Spherical harmonics arise on the sphere S 2 in the same way that the (Fourier) exponential functions {e ikO }k~Z arise on the circle. Spherical harmonic series have many of the same wonderful properties as Fourier series, but have lacked one important thing: a numerically stable fast transform analogous to the Fast Fourier Transform ( FFT). Without a fast transform, evaluating (or expanding in) spherical harmonic series on the computer is slow--for large computations prohibitively slow. This paper provides a fast transform. For a grid of O(N 2) points on the sphere, a direct calculation has computational complexity O(N4), but a simple separation of variables and FFT reduce it to 0 (N 3) time.Here we present algorithms with times O(N 5/2 log N) and (9(N2(log N)2).The problem quickly reduces to the fast application of matrices of associated Legendre functions of certain orders. The essential insight is that although these matrices are dense and oscillatory, locally they can be represented efficiently in trigonometric series.