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Online Learning in The Manifold of Low-Rank Matrices
Uri Shalit · Daphna Weinshall · Gal Chechik

Wed Dec 08 03:20 PM -- 03:25 PM (PST) @ Regency Ballroom

When learning models that are represented in matrix forms, enforcing
a low-rank constraint can dramatically improve the memory and run
time complexity, while providing a natural regularization of the
model. However, naive approaches for minimizing functions over the
set of low-rank matrices are either prohibitively time
consuming (repeated singular value decomposition of the matrix) or
numerically unstable (optimizing a factored representation of the
low rank matrix). We build on recent advances in optimization over
manifolds, and describe an iterative online learning procedure, consisting of a
gradient step, followed by a second-order retraction back to the
manifold. While the ideal retraction is hard to compute, and so is
the projection operator that approximates it, we describe another
second-order retraction that can be computed efficiently, with run
time and memory complexity of O((n+m)k) for a rank-k
matrix of dimension m x n, given rank one gradients. We use
this algorithm, LORETA, to learn a matrix-form similarity measure over pairs of
documents represented as high dimensional vectors. LORETA
improves the mean average precision over a passive-
aggressive approach in a factorized model, and also improves over
a full model trained over pre-selected features using the same
memory requirements. LORETA also showed consistent improvement over
standard methods in a large (1600 classes) multi-label image classification task.

Author Information

Uri Shalit (Technion)
Daphna Weinshall (Hebrew Univeristy of Jerusalem)
Gal Chechik (NVIDIA, BIU)

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