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.
Our approach is related to the local regression method @cite_24 @cite_15 . More recent related work is @cite_13 that uses smoothing techniques in high-dimensional lasso regression in the context of temporal data. Another recent approach proposed by @cite_19 achieves code locality by approximating data points using a linear combination of nearby basis points. The main difference is that traditional local regression techniques do not involve basis learning. In this work, we propose to learn the basis or dictionary along with the regression coefficients locally.
@cite_24 @cite_19 @cite_15 @cite_13
2386452692 1989060270 2123202508 2962726112
Lasso regression is a widely used technique in data mining for model selection and feature extraction. In many applications, it remains challenging to apply the regression model to large-scale problems that have massive data samples with high-dimensional features. One popular and promising strategy is to solve the Lasso problem in parallel. Parallel solvers run multiple cores in parallel on a shared memory system to speedup the computation, while the practical usage is limited by the huge dimension in the feature space. Screening is a promising method to solve the problem of high dimensionality by discarding the inactive features and removing them from optimization. However, when integrating screening methods with parallel solvers, most of solvers cannot guarantee the convergence on the reduced feature matrix. In this paper, we propose a novel parallel framework by parallelizing screening methods and integrating it with our proposed parallel solver. We propose two parallel screening algorithms: Parallel Strong Rule (PSR) and Parallel Dual Polytope Projection (PDPP). For the parallel solver, we proposed an Asynchronous Grouped Coordinate Descent method (AGCD) to optimize the regression problem in parallel on the reduced feature matrix. AGCD is based on a grouped selection strategy to select the coordinate that has the maximum descent for the objective function in a group of candidates. Empirical studies on the real-world datasets demonstrate that the proposed parallel framework has a superior performance compared to the state-of-the-art parallel solvers. We study the problem of estimating high-dimensional regression models regularized by a structured sparsity-inducing penalty that encodes prior structural information on either the input or output variables. We consider two widely adopted types of penalties of this kind as motivating examples: (1) the general overlapping-group-lasso penalty, generalized from the group-lasso penalty; and (2) the graph-guided-fused-lasso penalty, generalized from the fused-lasso penalty. For both types of penalties, due to their nonseparability and nonsmoothness, developing an efficient optimization method remains a challenging problem. In this paper we propose a general optimization approach, the smoothing proximal gradient (SPG) method, which can solve structured sparse regression problems with any smooth convex loss under a wide spectrum of structured sparsity-inducing penalties. Our approach combines a smoothing technique with an effective proximal gradient method. It achieves a convergence rate significantly faster than the standard first-order methods, subgradient methods, and is much more scalable than the most widely used interior-point methods. The efficiency and scalability of our method are demonstrated on both simulation experiments and real genetic data sets. We consider the problem of learning a coefficient vector xο ∈ RN from noisy linear observation y = Axo + ∈ Rn. 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. 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 model ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications. We consider the problem of fitting the parameters of a high-dimensional linear regression model. In the regime where the number of parameters p is comparable to or exceeds the sample size n, a successful approach uses an l1-penalized least squares estimator, known as Lasso. Unfortunately, unlike for linear estimators (e.g. ordinary least squares), no well-established method exists to compute confidence intervals or p-values on the basis of the Lasso estimator. Very recently, a line of work [8], [7], [13] has addressed this problem by constructing a debiased version of the Lasso estimator. We propose a special debiasing method that is well suited for random designs with sparse inverse covariance. Our approach improves over the state of the art in that it yields nearly optimal average testing power if sample size n asymptotically dominates s0(logp)2, with s0 being the sparsity level (number of non-zero coefficients). Earlier work achieved similar performances only for much larger sample size, namely it requires n to asymptotically dominates (s0 log p)2. We evaluate our method on synthetic data, and compare it with earlier proposals.