Convolution

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Main Idea

The convolution is an Operator mapping from two functions to another, on the same domain. For two functions $f$ and $g$ both mapping ${\mathbb{R}}^d\to \mathbb{R}$ it is defined as

This is well-defined for a variety of conditions, such as

• Compactly Supported continuous functions $f$ and $g$ admit a compactly supported and continuous convolution,
• Square Integrable functions and supported on an interval of the form $[a,\mathrm{\infty })$ (not sure what this means),
• $f$ and $g$ are both Integrable functions ($L^1({\mathbb{R}}^d)$), making the resulting convolution also integrable (see below for more details).

The function $g$ is often referred to as the Kernel. In many applications, this operation is discretized, such as in Convolutional Neural Networks, which are actually cross-correlations ($f(x)\star g(x)=f(-x)\ast g(x)$). The discrete convolution for $d=1$ can be given as

(possibly off by a constant based on $\mathrm{\Delta }x$), also requiring boundary conditions for $i$ to have a finite range.

Properties

• Under some restriction on the input functions, convolution is Bilinear, which thus admits many properties
• Commutative, so $f\ast g=g\ast f$ (along with other nice properties).
• Translation Invariance