Fast approximation algorithms for fractional packing and covering problems
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
A decision-theoretic generalization of on-line learning and an application to boosting
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems
Theoretical Computer Science
Learning a kernel matrix for nonlinear dimensionality reduction
ICML '04 Proceedings of the twenty-first international conference on Machine learning
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Predictive low-rank decomposition for kernel methods
ICML '05 Proceedings of the 22nd international conference on Machine learning
Efficient algorithms for online decision problems
Journal of Computer and System Sciences - Special issue: Learning theory 2003
Learning low-rank kernel matrices
ICML '06 Proceedings of the 23rd international conference on Machine learning
The weighted majority algorithm
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Logarithmic regret algorithms for online convex optimization
COLT'06 Proceedings of the 19th annual conference on Learning Theory
Decoding by linear programming
IEEE Transactions on Information Theory
Accurate low-rank matrix recovery from a small number of linear measurements
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
Online learning in the embedded manifold of low-rank matrices
The Journal of Machine Learning Research
Efficient protocols for distributed classification and optimization
ALT'12 Proceedings of the 23rd international conference on Algorithmic Learning Theory
Low-rank matrix completion using alternating minimization
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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Minimum rank problems arise frequently in machine learning applications and are notoriously difficult to solve due to the non-convex nature of the rank objective. In this paper, we present the first online learning approach for the problem of rank minimization of matrices over polyhedral sets. In particular, we present two online learning algorithms for rank minimization - our first algorithm is a multiplicative update method based on a generalized experts framework, while our second algorithm is a novel application of the online convex programming framework (Zinkevich, 2003). In the latter, we flip the role of the decision maker by making the decision maker search over the constraint space instead of feasible points, as is usually the case in online convex programming. A salient feature of our online learning approach is that it allows us to give provable approximation guarantees for the rank minimization problem over polyhedral sets. We demonstrate the effectiveness of our methods on synthetic examples, and on the real-life application of low-rank kernel learning.