It comes naturally when one learns SVD and FT and recognize both are powerful in same applications such as image compression. So I need to set aside some time to compare SVD and FT deeply.

Both the SVD and the fourier transform are characterized by some unitary transform (i.e. a change of basis) (1). Once in some new basis, you can make a choice to remove parts of the basis which are sensitive to small changes in the data (2). For both SVD and DFT these components are removed by some “windowing function” which select a viable window of either singular values or frequencies.

Assertion (1): For the SVD a matrix A=UΣVTA=UΣVT where UU and VV are unitary transforms into either the row or column space of AA. The Fourier transform is also a unitary transform into new orthogonal basis composed of sins and cos functions – omega or frequency domains. (https://en.wikipedia.org/wiki/Fourier_transform#Uncertainty_principle)

Assertion (2): For the SVD “parts of the basis sensitive to small changes” means the singular vectors (basis vectors or eigen vectors) which correspond to small singular values. For Fourier Transform, those with smaller coefficients are removed.

“Assuming your data is vector-valued time series, SVD and Fourier analysis give different information. (Singular vectors obtained by) SVD will essentially give you the “dominant” noise components without any useful time information.

On the other hand, doing Fourier analysis on a bunch of different scalar time series will give you dominant temporal frequencies of the noise components, but will not give you the “vector” noise components associated with those frequencies. ” — cited from a web search, I do not agree but placed here for future modification.