Google Research

Online Learning in the Manifold of Low-Rank Matrices

  • Gal Chechik
  • Daphna Weinshall
  • Uri Shalit
Neural Information Processing Systems (NIPS 23), Curran Associates, Inc. (2011), pp. 2128-2136


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 × n, given rank-one gradients. We use this algorithm, LORETA, to learn a matrixform 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.

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