We study an extention of total variation denoising over images to over Cartesian power graphs and its applications to estimating non-parametric network models. The power graph fused lasso (PGFL) segments a matrix by exploiting a known graphical structure, @math , over the rows and columns. Our main results shows that for any connected graph, under subGaussian noise, the PGFL achieves the same mean-square error rate as 2D total variation denoising for signals of bounded variation. We study the use of the PGFL for denoising an observed network @math , where we learn the graph @math as the @math -nearest neighborhood graph of an estimated metric over the vertices. We provide theoretical and empirical results for estimating graphons, a non-parametric exchangeable network model, and compare to the state of the art graphon estimation methods.
There is an extensive body of literature on solving the fused lasso, . Algorithms for solving the fused lasso can be divided into two categories: solvers for a fixed @math , and path algorithms that find the solution for every @math within a range. The fused lasso for a fixed @math has a quadratic program dual form, and some popular algorithms for this are the projected Newton algorithm of @cite_7 @cite_22 , first-order primal-dual algorithm @cite_11 , and split-Bregman iteration @cite_9 . Some path algorithms include the generalized lasso path algorithm of @cite_31 , and a max-flow version for the fused lasso in @cite_2 . If applied directly to C2-power graphs, these methods would have computational and memory complexity that scale with the number of dyads, @math .
@cite_22 @cite_7 @cite_9 @cite_2 @cite_31 @cite_11
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We present a path algorithm for the generalized lasso problem. This problem penalizes the @math norm of a matrix D times the coefficient vector, and has a wide range of applications, dictated by the choice of D. Our algorithm is based on solving the dual of the generalized lasso, which greatly facilitates computation of the path. For @math (the usual lasso), we draw a connection between our approach and the well-known LARS algorithm. For an arbitrary D, we derive an unbiased estimate of the degrees of freedom of the generalized lasso fit. This estimate turns out to be quite intuitive in many applications. The fused lasso, also known as total-variation denoising, is a locally-adaptive function estimator over a regular grid of design points. In this paper, we extend the fused lasso to settings in which the points do not occur on a regular grid, leading to a new approach for non-parametric regression. This approach, which we call the @math -nearest neighbors ( @math -NN) fused lasso, involves (i) computing the @math -NN graph of the design points; and (ii) performing the fused lasso over this @math -NN graph. We show that this procedure has a number of theoretical advantages over competing approaches: specifically, it inherits local adaptivity from its connection to the fused lasso, and it inherits manifold adaptivity from its connection to the @math -NN approach. We show that excellent results are obtained in a simulation study and on an application to flu data. For completeness, we also study an estimator that makes use of an @math -graph rather than a @math -NN graph, and contrast this with the @math -NN fused lasso. The Lasso is a very well-known penalized regression model, which adds an L1 penalty with parameter λ1 on the coefficients to the squared error loss function. The Fused Lasso extends this model by also putting an L1 penalty with parameter λ2 on the difference of neighboring coefficients, assuming there is a natural ordering. In this article, we develop a path algorithm for solving the Fused Lasso Signal Approximator that computes the solutions for all values of λ1 and λ2. We also present an approximate algorithm that has considerable speed advantages for a moderate trade-off in accuracy. In the Online Supplement for this article, we provide proofs and further details for the methods developed in the article. In this paper, we propose a new stochastic alternating direction method of multipliers (ADMM) algorithm, which incrementally approximates the full gradient in the linearized ADMM formulation. Besides having a low per-iteration complexity as existing stochastic ADMM algorithms, the proposed algorithm improves the convergence rate on convex problems from @math to @math , where @math is the number of iterations. This matches the convergence rate of the batch ADMM algorithm, but without the need to visit all the samples in each iteration. Experiments on the graph-guided fused lasso demonstrate that the new algorithm is significantly faster than state-of-the-art stochastic and batch ADMM algorithms. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . The fused lasso was proposed recently to enable recovery of high-dimensional patterns which are piece-wise constant on a graph, by penalizing the ‘1-norm of dierences of measurements at vertices that share an edge. While there have been some attempts at coming up with ecient algorithms for solving the fused lasso optimization, a theoretical analysis of its performance is mostly lacking except for the simple linear graph topology. In this paper, we investigate sparsistency of fused lasso for general graph structures, i.e. its ability to correctly recover the exact support of piece-wise constant graphstructured patterns asymptotically (for largescale graphs). To emphasize this distinction over previous work, we will refer to it as Edge Lasso. We focus on the (structured) normal means setting, and our results provide necessary and sucient conditions on the graph properties as well as the signal-to-noise ratio needed to ensure sparsistency. We examplify our results using simple graph-structured patterns, and demonstrate that in some cases fused lasso is sparsistent at very weak signal-to-noise ratios (scaling as p (logn)=jAj, where n is the number of vertices in the graph and A is the smallest set of vertices with constant activation). In other cases, it performs no better than thresholding the dierence The lasso is a popular tool for sparse linear regression, especially for problems in which the number of variables p exceeds the number of observations n. But when p > n, the lasso criterion is not strictly convex, and hence it may not have a unique minimum. An important question is: when is the lasso solution well-defined (unique)? We review results from the literature, which show that if the predictor variables are drawn from a continuous probability distribution, then there is a unique lasso solution with probability one, regardless of the sizes of n and p. We also show that this result extends easily to l1 penalized minimization problems over a wide range of loss functions. A second important question is: how can we manage the case of non-uniqueness in lasso solutions? In light of the aforementioned result, this case really only arises when some of the predictor variables are discrete, or when some post-processing has been performed on continuous predictor measurements. Though we certainly cannot claim to provide a complete answer to such a broad question, we do present progress towards understanding some aspects of nonuniqueness. First, we extend the LARS algorithm for computing the lasso solution path to cover the non-unique case, so that this path algorithm works for any predictor matrix. Next, we derive a simple method for computing the component-wise uncertainty in lasso solutions of any given problem instance, based on linear programming. Finally, we review results from the literature on some of the unifying properties of lasso solutions, and also point out particular forms of solutions that have distinctive properties. Summary. The lasso penalizes a least squares regression by the sum of the absolute values (L1-norm) of the coefficients. The form of this penalty encourages sparse solutions (with many coefficients equal to 0). We propose the ‘fused lasso’, a generalization that is designed for problems with features that can be ordered in some meaningful way. The fused lasso penalizes the L1-norm of both the coefficients and their successive differences. Thus it encourages sparsity of the coefficients and also sparsity of their differences—i.e. local constancy of the coefficient profile. The fused lasso is especially useful when the number of features p is much greater than N, the sample size. The technique is also extended to the ‘hinge’ loss function that underlies the support vector classifier.We illustrate the methods on examples from protein mass spectroscopy and gene expression data. We develop a novel online learning algorithm for the group lasso in order to efficiently find the important explanatory factors in a grouped manner. Different from traditional batch-mode group lasso algorithms, which suffer from the inefficiency and poor scalability, our proposed algorithm performs in an online mode and scales well: at each iteration one can update the weight vector according to a closed-form solution based on the average of previous subgradients. Therefore, the proposed online algorithm can be very efficient and scalable. This is guaranteed by its low worst-case time complexity and memory cost both in the order of O(d), where d is the number of dimensions. Moreover, in order to achieve more sparsity in both the group level and the individual feature level, we successively extend our online system to efficiently solve a number of variants of sparse group lasso models. We also show that the online system is applicable to other group lasso models, such as the group lasso with overlap and graph lasso. Finally, we demonstrate the merits of our algorithm by experimenting with both synthetic and real-world datasets. Sparse-Group Lasso (SGL) has been shown to be a powerful regression technique for simultaneously discovering group and within-group sparse patterns by using a combination of the @math and @math norms. However, in large-scale applications, the complexity of the regularizers entails great computational challenges. In this paper, we propose a novel Two-Layer Feature REduction method (TLFre) for SGL via a decomposition of its dual feasible set. The two-layer reduction is able to quickly identify the inactive groups and the inactive features, respectively, which are guaranteed to be absent from the sparse representation and can be removed from the optimization. Existing feature reduction methods are only applicable for sparse models with one sparsity-inducing regularizer. To our best knowledge, TLFre is the first one that is capable of dealing with multiple sparsity-inducing regularizers. Moreover, TLFre has a very low computational cost and can be integrated with any existing solvers. We also develop a screening method---called DPC (DecomPosition of Convex set)---for the nonnegative Lasso problem. Experiments on both synthetic and real data sets show that TLFre and DPC improve the efficiency of SGL and nonnegative Lasso by several orders of magnitude. We study , a widely used technique for eliciting structured sparsity. In particular, we propose efficient algorithms for computing prox-operators for @math -norm TV. The most important among these is @math -norm TV, for whose prox-operator we present a new geometric analysis which unveils a hitherto unknown connection to taut-string methods. This connection turns out to be remarkably useful as it shows how our geometry guided implementation results in efficient weighted and unweighted 1D-TV solvers, surpassing state-of-the-art methods. Our 1D-TV solvers provide the backbone for building more complex (two or higher-dimensional) TV solvers within a modular proximal optimization approach. We review the literature for an array of methods exploiting this strategy, and illustrate the benefits of our modular design through extensive suite of experiments on (i) image denoising, (ii) image deconvolution, (iii) four variants of fused-lasso, and (iv) video denoising. To underscore our claims and permit easy reproducibility, we provide all the reviewed and our new TV solvers in an easy to use multi-threaded C++, Matlab and Python library. 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. A very fast noniterative algorithm is proposed for denoising or smoothing one-dimensional discrete signals, by solving the total variation regularized least-squares problem or the related fused lasso problem. A C code implementation is available on the web page of the author. We investigate the relation of two fundamental tools in machine learning and signal processing, that is the support vector machine (SVM) for classification, and the Lasso technique used in regression. We show that the resulting optimization problems are equivalent, in the following sense. Given any instance of an L2-loss soft-margin (or hard-margin) SVM, we construct a Lasso instance having the same optimal solutions, and vice versa. As a consequence, many existing optimization algorithms for both SVMs and Lasso can also be applied to the respective other problem instances. Also, the equivalence allows for many known theoretical insights for SVM and Lasso to be translated between the two settings. One such implication gives a simple kernelized version of the Lasso, analogous to the kernels used in the SVM setting. Another consequence is that the sparsity of a Lasso solution is equal to the number of support vectors for the corresponding SVM instance, and that one can use screening rules to prune the set of support vectors. Furthermore, we can relate sublinear time algorithms for the two problems, and give a new such algorithm variant for the Lasso. We also study the regularization paths for both methods. The lasso is a popular technique for simultaneous estimation and variable selection. Lasso variable selection has been shown to be consistent under certain conditions. In this work we derive a necessary condition for the lasso variable selection to be consistent. Consequently, there exist certain scenarios where the lasso is inconsistent for variable selection. We then propose a new version of the lasso, called the adaptive lasso, where adaptive weights are used for penalizing different coefficients in the l1 penalty. We show that the adaptive lasso enjoys the oracle properties; namely, it performs as well as if the true underlying model were given in advance. Similar to the lasso, the adaptive lasso is shown to be near-minimax optimal. Furthermore, the adaptive lasso can be solved by the same efficient algorithm for solving the lasso. We also discuss the extension of the adaptive lasso in generalized linear models and show that the oracle properties still hold under mild regularity conditions. As a bypro... Purpose – The purpose of this paper is to suggest an alternative to the widely used Times Cited criterion for analysing citation networks. The approach involves taking account of the natures of the papers that cite a given paper, so as to differentiate between papers that attract the same number of citations.Design methodology approach – ArticleRank is an algorithm that has been derived from Google's PageRank algorithm to measure the influence of journal articles. ArticleRank is applied to two datasets – a citation network based on an early paper on webometrics, and a self‐citation network based on the 19 most cited papers in the Journal of Documentation – using citation data taken from the Web of Knowledge database.Findings – ArticleRank values provide a different ranking of a set of papers from that provided by the corresponding Times Cited values, and overcomes the inability of the latter to differentiate between papers with the same numbers of citations. The difference in rankings between Times Cited ... Entropic regularization is quickly emerging as a new standard in optimal transport (OT). It enables to cast the OT computation as a differentiable and unconstrained convex optimization problem, which can be efficiently solved using the Sinkhorn algorithm. However, entropy keeps the transportation plan strictly positive and therefore completely dense, unlike unregularized OT. This lack of sparsity can be problematic in applications where the transportation plan itself is of interest. In this paper, we explore regularizing the primal and dual OT formulations with a strongly convex term, which corresponds to relaxing the dual and primal constraints with smooth approximations. We show how to incorporate squared @math -norm and group lasso regularizations within that framework, leading to sparse and group-sparse transportation plans. On the theoretical side, we bound the approximation error introduced by regularizing the primal and dual formulations. Our results suggest that, for the regularized primal, the approximation error can often be smaller with squared @math -norm than with entropic regularization. We showcase our proposed framework on the task of color transfer. We propose a new stochastic alternating direction method of multipliers (ADMM) algorithm, which incrementally approximates the full gradient in the linearized ADMM formulation. Besides having a low per-iteration complexity as existing stochastic ADMM algorithms, it improves the convergence rate on convex problems from O(1 √T) to O(1 T ), where T is the number of iterations. This matches the convergence rate of the batch ADMM algorithm, but without the need to visit all the samples in each iteration. Experiments on the graph-guided fused lasso demonstrate that the new algorithm is significantly faster than state-of-the-art stochastic and batch ADMM algorithms. Convex sparsity-promoting regularizations are ubiquitous in modern statistical learning. By construction, they yield solutions with few non-zero coefficients, which correspond to saturated constraints in the dual optimization formulation. Working set (WS) strategies are generic optimization techniques that consist in solving simpler problems that only consider a subset of constraints, whose indices form the WS. Working set methods therefore involve two nested iterations: the outer loop corresponds to the definition of the WS and the inner loop calls a solver for the subproblems. For the Lasso estimator a WS is a set of features, while for a Group Lasso it refers to a set of groups. In practice, WS are generally small in this context so the associated feature Gram matrix can fit in memory. Here we show that the Gauss-Southwell rule (a greedy strategy for block coordinate descent techniques) leads to fast solvers in this case. Combined with a working set strategy based on an aggressive use of so-called Gap Safe screening rules, we propose a solver achieving state-of-the-art performance on sparse learning problems. Results are presented on Lasso and multi-task Lasso estimators. Compressed sensing (CS) lowers the number of measurements required for reconstruction and estimation of signals that are sparse when expanded over a proper basis. Traditional CS approaches deal with time-invariant sparse signals, meaning that, during the measurement process, the signal of interest does not exhibit variations. However, many signals encountered in practice are varying with time as the observation window increases (e.g., video imaging, where the signal is sparse and varies between different frames). The present paper develops CS algorithms for time-varying signals, based on the least-absolute shrinkage and selection operator (Lasso) that has been popular for sparse regression problems. The Lasso here is tailored for smoothing time-varying signals, which are modeled as vector valued discrete time series. Two algorithms are proposed: the Group-Fused Lasso, when the unknown signal support is time-invariant but signal samples are allowed to vary with time; and the Dynamic Lasso, for the general class of signals with time-varying amplitudes and support. Performance of these algorithms is compared with a sparsity-unaware Kalman smoother, a support-aware Kalman smoother, and the standard Lasso which does not account for time variations. The numerical results amply demonstrate the practical merits of the novel CS algorithms.