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Register a new userMulticlass Classification, Information, Divergence, and Surrogate Risk
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Added by
Khashayar Khosravi
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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We provide a unifying view of statistical information measures, multi-way Bayesian hypothesis testing, loss functions for multi-class classification problems, and multi-distribution $f$-divergences, elaborating equivalence results between all of these objects, and extending existing results for binary outcome spaces to more general ones. We consider a generalization of $f$-divergences to multiple distributions, and we provide a constructive equivalence between divergences, statistical information (in the sense of DeGroot), and losses for multiclass classification. A major application of our results is in multi-class classification problems in which we must both infer a discriminant function $gamma$---for making predictions on a label $Y$ from datum $X$---and a data representation (or, in the setting of a hypothesis testing problem, an experimental design), represented as a quantizer $mathsf{q}$ from a family of possible quantizers $mathsf{Q}$. In this setting, we characterize the equivalence between loss functions, meaning that optimizing either of two losses yields an optimal discriminant and quantizer $mathsf{q}$, complementing and extending earlier results of Nguyen et. al. to the multiclass case. Our results provide a more substantial basis than standard classification calibration results for comparing different losses: we describe the convex losses that are consistent for jointly choosing a data representation and minimizing the (weighted) probability of error in multiclass classification problems.
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We consider the problem of learning a coefficient vector x_0in R^N from noisy linear observation y=Ax_0+w in R^n. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator x. In this case, a popular approach consists in solving an L1-penalized least squares problem known as the LASSO or Basis Pursuit DeNoising (BPDN). For sequences of matrices A of increasing dimensions, with independent gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical models ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications.
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