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Orthogonal basis decompositions

Fourier and friends

Transforming something (let us assume, discrete experimental data rather than abstract mathematical objects) into a new basis, as in the Fourier or Laplace or z-transforms, usually with a goal of handling some inconveniently curvy data as linear data plus a non-local curvy transformation. This is big in, e.g. signal processing and time series analysis.

Other names: Spectral methods, Hilbert space methods, basis expansions, orthogonal projection methods.

An orthogonal basis requires you to have a domain and inner product specified. If you choose a new inner product, you need a different base to be orthogonal in general. If you use the machinery of an implicit Hilbert space for a given inner product without troubling to define an explicit form for the “feature space”, you are doing the kernel trick.

Todo: discuss orthogonal measurements in terms of isometry properties.

The discrete qualifier is ambiguous; Here what I mean is that we are finding orthogonal bases for signal vectors of countable, often finite, length, e.g. if we are looking at spaces of 1 second of audio at 44kHz, then we consider spaces \(\mathbb{R}^{44100}\). If we are looking at RGB webcam images we might consider \(\mathbb{R}_+^{640\times 480 \times 3}\).

This all gets weird and irrelevant for my current purposes if we start considering transforms of arbitrary objects over uncountable groups.

Orthogonal basis zoo

For related ideas, see also

Miscellaneous stuff to remember