On Spectral Learning
On Spectral Learning
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DOI：
10.5555/1756006.1756037
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发表时间：
2010-03
期刊：
影响因子：
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通讯作者：
Andreas Argyriou;C. Micchelli;M. Pontil
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作者：
Andreas Argyriou;C. Micchelli;M. Pontil
在本文中，我们研究从一组线性测量中学习矩阵\(W\)的问题。我们的公式在于解决一个优化问题，该问题涉及带有谱惩罚项的正则化。也就是说，惩罚项是\(W\)的协方差谱的函数。机器学习中此问题的实例包括多任务学习、协同过滤和多视图学习等。我们的目标是阐明谱学习最优解的形式。谱学习理论依赖于正交不变范数的冯·诺依曼特征及其与对称规范函数的关联。利用这个工具，我们为谱正则化制定了一个表示定理，并将其应用于几个有用的例子，例如沙滕\(p\)-范数、迹范数和谱范数，这在应用中应该是有用的。
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.