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barycenter

Function barycenter tries to find appropriate weights to reconstruct the point x from a subset (y1, y2, ..., yn), where weights sum to one.

This is just a simple case of Equality Constrained Least Squares [1] with constrain dot(np.ones(n), x) = 1. In particular, the Q matrix from the QR decomposition of B is the Householder reflection of np.ones(n).

Purpose

This method was added to ease some computations in the future manifold module, namely in LLE. However, it is still to be shown that it is useful and efficient in that context.

Performance

The algorithm has to iterate over n_samples, which is the main bottleneck. It would be great to vectorize this loop. Also, the rank updates could probably be moved outside the loop.

Also, least squares solution could be computed more efficiently by a QR factorization, since probably we don't care about a minimum norm solution for the undertermined case.

The paper 'An introduction to Locally Linear Embeddings', Saul & Roweis solves the problem by the normal equation method over the covariance matrix. However, it does not degrade gracefully when the covariance is singular, requiring to explicitly add regularization.

Stability

Should be good as it uses SVD to solve the LS problem. TODO: explicit bounds.

API

The API is convenient to use from NeighborsBarycenter and kneighbors_graph, but might not be very easy to use directly due to the fact that Y must be a 3-D array.

It should be checked that it is usable in other contexts.

Footnotes

[1] Section 12.1.4 ('Equality Constrained Least Squares'), 'Matrix Computations' by Golub & Van Loan
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