We propose and analyze a novel framework for learning sparse representations, based on two statistical techniques: kernel smoothing and marginal regression. The proposed approach provides a flexible framework for incorporating feature similarity or temporal information present in data sets, via non-parametric kernel smoothing. We provide generalization bounds for dictionary learning using smooth sparse coding and show how the sample complexity depends on the L1 norm of kernel function used. Furthermore, we propose using marginal regression for obtaining sparse codes, which significantly improves the speed and allows one to scale to large dictionary sizes easily. We demonstrate the advantages of the proposed approach, both in terms of accuracy and speed by extensive experimentation on several real data sets. In addition, we demonstrate how the proposed approach could be used for improving semi-supervised sparse coding.
In contrast to previous sparse coding papers we propose to use marginal regression for learning the regression coefficients, which results in a significant computational speedup with no loss of accuracy. Marginal regression is a relatively old technique that has recently reemerged as a computationally faster alternative to lasso regression @cite_1 . See also @cite_6 for a statistical comparison of lasso regression and marginal regression.
@cite_1 @cite_6
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The lasso is an important method for sparse, high-dimensional regression problems, with efficient algorithms available, a long history of practical success, and a large body of theoretical results supporting and explaining its performance. But even with the best available algorithms, finding the lasso solutions remains a computationally challenging task in cases where the number of covariates vastly exceeds the number of data points. Marginal regression, where each dependent variable is regressed separately on each covariate, offers a promising alternative in this case because the estimates can be computed roughly two orders faster than the lasso solutions. The question that remains is how the statistical performance of the method compares to that of the lasso in these cases. In this paper, we study the relative statistical performance of the lasso and marginal regression for sparse, high-dimensional regression problems. We consider the problem of learning which coefficients are non-zero. Our main results are as follows: (i) we compare the conditions under which the lasso and marginal regression guarantee exact recovery in the fixed design, noise free case; (ii) we establish conditions under which marginal regression provides exact recovery with high probability in the fixed design, noise free, random coefficients case; and (iii) we derive rates of convergence for both procedures, where performance is measured by the number of coefficients with incorrect sign, and characterize the regions in the parameter space recovery is and is not possible under this metric. In light of the computational advantages of marginal regression in very high dimensional problems, our theoretical and simulations results suggest that the procedure merits further study. Meinshausen and Buhlmann [Ann. Statist. 34 (2006) 1436-1462] showed that, for neighborhood selection in Gaussian graphical models, under a neighborhood stability condition, the LASSO is consistent, even when the number of variables is of greater order than the sample size. Zhao and Yu [(2006) J. Machine Learning Research 7 2541-2567] formalized the neighborhood stability condition in the context of linear regression as a strong irrepresentable condition. That paper showed that under this condition, the LASSO selects exactly the set of nonzero regression coefficients, provided that these coefficients are bounded away from zero at a certain rate. In this paper, the regression coefficients outside an ideal model are assumed to be small, but not necessarily zero. Under a sparse Riesz condition on the correlation of design variables, we prove that the LASSO selects a model of the correct order of dimensionality, controls the bias of the selected model at a level determined by the contributions of small regression coefficients and threshold bias, and selects all coefficients of greater order than the bias of the selected model. Moreover, as a consequence of this rate consistency of the LASSO in model selection, it is proved that the sum of error squares for the mean response and the l α -loss for the regression coefficients converge at the best possible rates under the given conditions. An interesting aspect of our results is that the logarithm of the number of variables can be of the same order as the sample size for certain random dependent designs.