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librosa.dtw with subseq=True is not symmetric w.r.t. its inputs

See original GitHub issue

cc @stefan-balke

My expectation for subsequence matching is that implies that either X can be matched to a subsequence of Y or Y can be matched to a subsequence of X. librosa.dtw doesn’t exhibit this behavior:

In [1]: import librosa
In [2]: import numpy as np
In [3]: X = np.array([10., 10., 0., 1., 2., 3., 10., 10.]).reshape([1, -1])
In [4]: Y = np.arange(4).astype(float).reshape([1, -1])
In [5]: librosa.dtw(X, Y, subseq=True)
Out[5]:
(array([[ 10.,   9.,   8.,   7.],
        [ 20.,  18.,  16.,  14.],
        [ 20.,  19.,  18.,  17.],
        [ 21.,  19.,  19.,  19.],
        [ 23.,  20.,  19.,  20.],
        [ 26.,  22.,  20.,  19.],
        [ 36.,  31.,  28.,  26.],
        [ 46.,  40.,  36.,  33.]]), array([[7, 3],
        [6, 3],
        [5, 3],
        [4, 2],
        [3, 1],
        [2, 1],
        [1, 1],
        [0, 1]]))
In [6]: librosa.dtw(Y, X, subseq=True)
Out[6]:
(array([[ 10.,  10.,   0.,   1.,   2.,   3.,  10.,  10.],
        [ 19.,  19.,   1.,   0.,   1.,   3.,  12.,  19.],
        [ 27.,  27.,   3.,   1.,   0.,   1.,   9.,  17.],
        [ 34.,  34.,   6.,   3.,   1.,   0.,   7.,  14.]]), array([[3, 5],
        [2, 4],
        [1, 3],
        [0, 2]]))

This is in contrast to djitw’s behavior (my expected behavior, obviously 😉

In [7]: import djitw

In [8]: import scipy

In [9]: djitw.dtw(scipy.spatial.distance.cdist(X.T, Y.T))
Out[9]: (array([2, 3, 4, 5]), array([0, 1, 2, 3]), 0.0)

In [10]: djitw.dtw(scipy.spatial.distance.cdist(Y.T, X.T))
Out[10]: (array([0, 1, 2, 3]), array([2, 3, 4, 5]), 0.0)

What is the motivation for making it non-symmetric? We should at least document it more specifically, saying that subseq=True allows for a subsequence of Y to be matched to the entirety of X but not vice-versa.