Regularization by spectral filtering

Inverse problems | Mathematical analysis

Spectral regularization is any of a class of regularization techniques used in machine learning to control the impact of noise and prevent overfitting. Spectral regularization can be used in a broad range of applications, from deblurring images to classifying emails into a spam folder and a non-spam folder. For instance, in the email classification example, spectral regularization can be used to reduce the impact of noise and prevent overfitting when a machine learning system is being trained on a labeled set of emails to learn how to tell a spam and a non-spam email apart. Spectral regularization algorithms rely on methods that were originally defined and studied in the theory of ill-posed inverse problems (for instance, see) focusing on the inversion of a linear operator (or a matrix) that possibly has a bad condition number or an unbounded inverse. In this context, regularization amounts to substituting the original operator by a bounded operator called the "regularization operator" that has a condition number controlled by a regularization parameter, a classical example being Tikhonov regularization. To ensure stability, this regularization parameter is tuned based on the level of noise. The main idea behind spectral regularization is that each regularization operator can be described using spectral calculus as an appropriate filter on the eigenvalues of the operator that defines the problem, and the role of the filter is to "suppress the oscillatory behavior corresponding to small eigenvalues". Therefore, each algorithm in the class of spectral regularization algorithms is defined by a suitable filter function (which needs to be derived for that particular algorithm). Three of the most commonly used regularization algorithms for which spectral filtering is well-studied are Tikhonov regularization, Landweber iteration, and truncated singular value decomposition (TSVD). As for choosing the regularization parameter, examples of candidate methods to compute this parameter include the discrepancy principle, generalized cross validation, and the L-curve criterion. It is of note that the notion of spectral filtering studied in the context of machine learning is closely connected to the literature on function approximation (in signal processing). (Wikipedia).

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Regularization by spectral filtering

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