The need for function approximations arises in many branches of applied mathematics, and computer science in particular. In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way.
One can distinguish two major classes of function approximation problems: First, for known target functions approximation theory (q.v.) is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by simpler functions (for example, polynomials or rational functions).
Second, the target function, call it g, may be unknown; instead of an explicit formula, only a set of points of the form (x, g(x)) is provided. Depending on the structure of the domain and codomain of g, several techniques for approximating g may be applicable. For example, if g is an operation on the real numbers, techniques of interpolation, extrapolation, regression analysis, and curve fitting can be used. If the codomain of g is a finite set, one is dealing with a classification problem instead.
To some extent the different problems (regression, classification) have received a unified treatment in statistical learning theory, where they are viewed as supervised learning problems.
In (b)-(g), the function is shown as a dotted line, and the neural net approximation (based on the noisy samples shown as circles) is shown as the solid line.
In each case, the dotted line is the underlying function to be approximated, the solid line is the neural net output, and the open circles indicate the data points used for training.
Here, the underlying function to be approximated is a sine wave, and is a perturbation term, producing small "blips" or kinks near the peaks of the sine curve.
One can distinguish two major classes of functionapproximation problems: First, for known target functionsapproximation theory (q.v.) is the branch of numerical analysis that investigates how certain known function (for example, special functions) can be approximated by simpler functions (for example, polynomials or rational functions).
Functionapproximation is usually posed as an optimization problem as we are attempting to find a solution where the error is at a minimum.
The simplest example of functionapproximation is in the one dimensional case.
Feeds |
Contact