Add Sakoe-Chiba band constraint

[TomDLT]

This PR adds a Sakoe-Chiba band constraint [1], which constrains the path to stay in a band around the diagonal. For now it works only on squared DTW problems.

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[1] Sakoe, H., & Chiba, S. (1978). Dynamic programming algorithm optimization for spoken word recognition. IEEE transactions on acoustics, speech, and signal processing, 26(1), 43-49.

[mblondel]

This looks very good. Do you use it for speed up or regularization?

I think you could also test the forward pass as follows: use gen_all_paths to generate all paths, filter out those outside of the band and compute the softmin among those. See

https://github.com/mblondel/soft-dtw/blob/master/sdtw/tests/test_soft_dtw.py#L32

for an example.

[TomDLT]

This looks very good. Do you use it for speed up or regularization?

Both, but I use it mostly for speed, since it goes from O(N * N) to O(N * B) where B is the band width.

I think you could also test the forward pass as follows: use gen_all_paths to generate all paths, filter out those outside of the band and compute the softmin among those.

Good point, will do.

[mblondel]

One last question before merge... Why do you need this restriction?

[TomDLT]

For simplicity.
I updated the code and removed the restriction.