Random Projections and the WHT

If you randomly flip the signs of input data to the Walsh Hadamard transform, the data takes on a random distribution. After the WHT the data will have a Gaussian distribution. 

You are applying a random projection to the input data that spreads every single element in the input data to all the output elements, in a random way.  This results in fair distribution of information about the input. Sub-sampling the output you can recreate all the input (with information loss) by inverting the random projection.

Random permutation can be used in place of (or in conjunction with) random sign flipping. 

You can repeat the process a number of times for better quality or more consistent mathematical behavior.

Clockwise image, random projection, truncated random projection, inverse random projection.

While it looks like a lot of information is lost by truncation special restoration methods can be applied that give a greatly improved result. This is because the actual amount of information in a natural image is not very high and the loss by truncation is less than it might appear.