Curse of dimensionality and integration

The curse of dimensionality refers to problems whose difficulty increases exponentially with dimension. For example, suppose you want to estimate the integral of a function of one variable by evaluating it at 10 points. If you take the analogous approach to integrating a function of two variables, you need a grid of 100 points. For a function of three variables, you need a cube of 1000 points, and so forth.

You cannot estimate high-dimensional integrals this way. For a function of 100 variables, using a lattice with just two points in each direction would require 2100 points.

There are much more efficient ways to approximate integrals than simply adding up values at grid points, assuming your integrand is smooth. But when applying any of these methods to multi-dimensional integrals, the computational effort increases exponentially with dimension.

The methods that first come to mind don’t scale well with dimension, but that doesn’t necessarily mean there aren’t any methods that do scale well.

Are there numerical integration methods whose computational requirement scale slower than exponentially with dimension? In general, no. You can beat the curse of dimension for special integrals. And if you’re content with probabilistic bounds rather than deterministic bounds, you can get around the curse by using Monte Carlo integration, sort of [1].

If you want worst-case error bounds, you cannot escape the curse of dimensionality [2]. You can require that the functions you’re integrating have so many derivatives and that these derivatives are bounded by some constant, but the amount of computation necessary to guarantee that the error is below a specified threshold increases exponentially with dimension.

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[1] In one sense Monte Carlo methods are independent of dimension. But in an important sense they scale poorly with dimension. See a resolution of this tension here.

[2] The curse of dimensionality for numerical integration of smooth functions. A. Hinrichs, E. Novak, M. Ullrich and H. Wo┼║niakowski. Mathematics of Computation, Vol. 83, No. 290 (November 2014), pp. 2853-2863