n sets of vectors, where each set
k vectors in
R^d. I know there is some unknown linear transformation
W under which the distance matrix
kxk matrix) is approximately “the same” (i.e. has a low variance) among all sets
To illustrate, it might be the case that for each vector in each
S_i, the first
k/2 numbers are random noise, and the latter
k/2 are the same for each
i. In this case,
W would recover the last
k/2 elements from each vector. But in practice the structure of the vectors may be more complex than in this toy example.
Is there a method to learn
W – either using direct linear algebra methods, or through learning (e.g. by optimizing a specific loss on the distance matrices), without finding the naive solution of
W being the zero mapping, or other mapping that maps all vectors to the same vector?