SVD vs Fourier analysis for data. Fourier analysis is useful for analysis in the frequency domain. SVD on the other hand is useful for analysis of data, and expressing noise in the data. I have a problem that needs extensive data analysis, it is in the area medicine. This could be generalized to other problems.  
The problem is that of gene expression, in case of long term gene mutation. Using Fourier analysis we can get a time series analysis of the genes(and thereby get noisy gene expression), and as time progresses, the changes in a particular organ. On the other hand, we could use Singular Value Decomposition, and the noisy gene expresses itself. This, is just an outline of the problem. Both SVD, and Fourier lend themselves to solve the problem that of expressing noisy genes. Is there any comparison of the two techniques, why one would be preferred over another qualitatively, or references that one can use for the problem of gene expression, thanks in anticipation.   
 A: When you say SVD, you probably mean something like the Karhunen–Loève decomposition, or maybe just a corresponding Arnoldi process. (I also regard Prony's method as something similar.) That wikipedia article includes analogies and comparisons with the Fourier transform, both for aiding understanding and highlighting the strengths of the decomposition.
When I work with Fourier analysis, I normally also have a window function, especially if transient behavior is involved. (For periodic behavior, I sometimes have a low pass filter instead.) In the cases where the Karhunen–Loève decomposition provides advantages, I normally compute it by going into the Fourier basis (without window function) first, because this can drastically reduce the size of the problem. It also allows me to interpret the resulting decomposition as a series of window functions (or low pass filters), which makes it easier to visualize and compare with a direct Fourier approach.
I was tempted more than once to also use an Arnoldi process (or Prony's method) instead of a Fourier transform with window function. It would have been optimal and more automatic, but the Fourier transform with window function was already good enough. Besides priorities, maybe it was also a bit of mistrust from my side against a mostly automatic Arnoldi process.
A: 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.
