McDonald, AM; Pontil, M; Stamos, D; (2016) Fitting Spectral Decay with the k-Support Norm. In: Gretton, A and Robert, CC, (eds.) Proceedings of the 19th International Conference on Artificial Intelligence and Statistics. (pp. pp. 1061-1069). Journal of Machine Learning Research

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Abstract

The spectral kk-support norm enjoys good estimation properties in low rank matrix learning problems, empirically outperforming the trace norm. Its unit ball is the convex hull of rank kk matrices with unit Frobenius norm. In this paper we generalize the norm to the spectral (k,p)(k,p)-support norm, whose additional parameter pp can be used to tailor the norm to the decay of the spectrum of the underlying model. We characterize the unit ball and we explicitly compute the norm. We further provide a conditional gradient method to solve regularization problems with the norm, and we derive an efficient algorithm to compute the Euclidean projection on the unit ball in the case p=∞p=∞. In numerical experiments, we show that allowing pp to vary significantly improves performance over the spectral kk-support norm on various matrix completion benchmarks, and better captures the spectral decay of the underlying model.

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