#StackBounty: #machine-learning #distance #metric #linear-algebra Finding linear transformation under which distance matrices are similar

Bounty: 100

I have n sets of vectors, where each set S_i contains k vectors in R^d. I know there is some unknown linear transformation W under which the distance matrix D_i (a kxk matrix) is approximately “the same” (i.e. has a low variance) among all sets S_i.

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?

Thanks


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