# #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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