Matrix analysis
Topics in matrix analysis
Learning Multiple Tasks with Kernel Methods
The Journal of Machine Learning Research
Bounds for Linear Multi-Task Learning
The Journal of Machine Learning Research
Uncovering shared structures in multiclass classification
Proceedings of the 24th international conference on Machine learning
Learning Similarity with Operator-valued Large-margin Classifiers
The Journal of Machine Learning Research
Convex multi-task feature learning
Machine Learning
A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization
The Journal of Machine Learning Research
Exact Matrix Completion via Convex Optimization
Foundations of Computational Mathematics
When Is There a Representer Theorem? Vector Versus Matrix Regularizers
The Journal of Machine Learning Research
Metric and kernel learning using a linear transformation
The Journal of Machine Learning Research
Regularization techniques for learning with matrices
The Journal of Machine Learning Research
Finite rank kernels for multi-task learning
Advances in Computational Mathematics
Regularizers for structured sparsity
Advances in Computational Mathematics
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In this paper, we study the problem of learning a matrix W from a set of linear measurements. Our formulation consists in solving an optimization problem which involves regularization with a spectral penalty term. That is, the penalty term is a function of the spectrum of the covariance of W. Instances of this problem in machine learning include multi-task learning, collaborative filtering and multi-view learning, among others. Our goal is to elucidate the form of the optimal solution of spectral learning. The theory of spectral learning relies on the von Neumann characterization of orthogonally invariant norms and their association with symmetric gauge functions. Using this tool we formulate a representer theorem for spectral regularization and specify it to several useful example, such as Schatten p-norms, trace norm and spectral norm, which should proved useful in applications.