The case for Fast Transforms as Averaging Machines

Are Fast Transforms like the FFT and Walsh Hadamard transform averaging machines?


Certainly there is an all sum term such as A+B+C+D+E+F+G in the diagram which can be considered an average especially after vector length renormalization where all the output terms would typically be multiplied by 1/sqr(width), 1/sqr(8) in this case, to leave vector magnitude unchanged by the transform.

For a different input, say {+1,+1,-1,-1,-1,-1,+1,+1] the output term A+B-C-D-E-F+G+H becomes 1+1+1+1+1+1+1+1.  For that input pattern that particular output term becomes an averaging operation.


Degrees of freedom

For standard averaging of n values many different combinations of values can result in the same average. There are many degrees of freedom.

As you start demanding more and more output terms of a fast transform be particular values the degrees of freedom in the input shrink. Until, when you demand all the outputs be particular values, then only one exact input vector will give that result.

To get around that you can:

  • Use a wider transform than the wanted output vector.
  • Accept say a 1% output error tolerance which will restore some degree of freedom.
If you accept either of those 2 conditions then local errors in the input to the transform may be organized such that they cancel out globally.

For example there might be too few local parameters per function in a system of parametric functions for local errors to be cancelled out.

However the errors could be cancelled out globally by mutual cooperation of the local functions if they were all interconnected in some way.
And a fast transform can provide that connectivity.

As a practical matter you may want to remove the spectral bias of the transform (ie. a strong respose to certain sine and cosine waves) in this particular application. As you only need the connectivity behavior. You could consider a fixed random pattern of sign flips applied to the output of the transform to remove the bias. Or a sub-random pattern of sign flips.


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