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We study the approximation capacity of deep ReLU recurrent neural networks (RNNs) and explore the convergence properties of nonparametric least squares regression using RNNs. We derive upper bounds on the approximation error of RNNs for H\"older smooth functions, in the sense that the output at each time step of an RNN can approximate a H\"older function that depends only on past and current information, termed a past-dependent function. This allows a carefully constructed RNN to simultaneously approximate a sequence of past-dependent H\"older functions. We apply these approximation results to derive non-asymptotic upper bounds for the prediction error of the empirical risk minimizer in regression problem. Our error bounds achieve minimax optimal rate under both exponentially $\beta$-mixing and i.i.d. data assumptions, improving upon existing ones. Our results provide statistical guarantees on the performance of RNNs.

Learning a data representation for downstream supervised learning tasks under unlabeled scenario is both critical and challenging. In this paper, we propose a novel unsupervised transfer learning approach using adversarial contrastive training (ACT). Our experimental results demonstrate outstanding classification accuracy with both fine-tuned linear probe and K-NN protocol across various datasets, showing competitiveness with existing state-of-the-art self-supervised learning methods. Moreover, we provide an end-to-end theoretical guarantee for downstream classification tasks in a misspecified, over-parameterized setting, highlighting how a large amount of unlabeled data contributes to prediction accuracy. Our theoretical findings suggest that the testing error of downstream tasks depends solely on the efficiency of data augmentation used in ACT when the unlabeled sample size is sufficiently large. This offers a theoretical understanding of learning downstream tasks with a small sample size.

Deep nonparametric regression, characterized by the utilization of deep neural networks to learn target functions, has emerged as a focus of research attention in recent years. Despite considerable progress in understanding convergence rates, the absence of asymptotic properties hinders rigorous statistical inference. To address this gap, we propose a novel framework that transforms the deep estimation paradigm into a platform conducive to conditional mean estimation, leveraging the conditional diffusion model. Theoretically, we develop an end-to-end convergence rate for the conditional diffusion model and establish the asymptotic normality of the generated samples. Consequently, we are equipped to construct confidence regions, facilitating robust statistical inference. Furthermore, through numerical experiments, we empirically validate the efficacy of our proposed methodology.

Machine learning is a rapidly advancing field with diverse applications across various domains. One prominent area of research is the utilization of deep learning techniques for solving partial differential equations(PDEs). In this work, we specifically focus on employing a three-layer tanh neural network within the framework of the deep Ritz method(DRM) to solve second-order elliptic equations with three different types of boundary conditions. We perform projected gradient descent(PDG) to train the three-layer network and we establish its global convergence. To the best of our knowledge, we are the first to provide a comprehensive error analysis of using overparameterized networks to solve PDE problems, as our analysis simultaneously includes estimates for approximation error, generalization error, and optimization error. We present error bound in terms of the sample size $n$ and our work provides guidance on how to set the network depth, width, step size, and number of iterations for the projected gradient descent algorithm. Importantly, our assumptions in this work are classical and we do not require any additional assumptions on the solution of the equation. This ensures the broad applicability and generality of our results.

This paper aims to conduct a comprehensive theoretical analysis of current diffusion models. We introduce a novel generative learning methodology utilizing the Schr{\"o}dinger bridge diffusion model in latent space as the framework for theoretical exploration in this domain. Our approach commences with the pre-training of an encoder-decoder architecture using data originating from a distribution that may diverge from the target distribution, thus facilitating the accommodation of a large sample size through the utilization of pre-existing large-scale models. Subsequently, we develop a diffusion model within the latent space utilizing the Schr{\"o}dinger bridge framework. Our theoretical analysis encompasses the establishment of end-to-end error analysis for learning distributions via the latent Schr{\"o}dinger bridge diffusion model. Specifically, we control the second-order Wasserstein distance between the generated distribution and the target distribution. Furthermore, our obtained convergence rates effectively mitigate the curse of dimensionality, offering robust theoretical support for prevailing diffusion models.

We present theoretical convergence guarantees for ODE-based generative models, specifically flow matching. We use a pre-trained autoencoder network to map high-dimensional original inputs to a low-dimensional latent space, where a transformer network is trained to predict the velocity field of the transformation from a standard normal distribution to the target latent distribution. Our error analysis demonstrates the effectiveness of this approach, showing that the distribution of samples generated via estimated ODE flow converges to the target distribution in the Wasserstein-2 distance under mild and practical assumptions. Furthermore, we show that arbitrary smooth functions can be effectively approximated by transformer networks with Lipschitz continuity, which may be of independent interest.

Continuous normalizing flows (CNFs) are a generative method for learning probability distributions, which is based on ordinary differential equations. This method has shown remarkable empirical success across various applications, including large-scale image synthesis, protein structure prediction, and molecule generation. In this work, we study the theoretical properties of CNFs with linear interpolation in learning probability distributions from a finite random sample, using a flow matching objective function. We establish non-asymptotic error bounds for the distribution estimator based on CNFs, in terms of the Wasserstein-2 distance. The key assumption in our analysis is that the target distribution satisfies one of the following three conditions: it either has a bounded support, is strongly log-concave, or is a finite or infinite mixture of Gaussian distributions. We present a convergence analysis framework that encompasses the error due to velocity estimation, the discretization error, and the early stopping error. A key step in our analysis involves establishing the regularity properties of the velocity field and its estimator for CNFs constructed with linear interpolation. This necessitates the development of uniform error bounds with Lipschitz regularity control of deep ReLU networks that approximate the Lipschitz function class, which could be of independent interest. Our nonparametric convergence analysis offers theoretical guarantees for using CNFs to learn probability distributions from a finite random sample., Comment: 60 pages, 3 tables, and 3 figures

We introduce an ordinary differential equation (ODE) based deep generative method for learning conditional distributions, named Conditional F\"ollmer Flow. Starting from a standard Gaussian distribution, the proposed flow could approximate the target conditional distribution very well when the time is close to 1. For effective implementation, we discretize the flow with Euler's method where we estimate the velocity field nonparametrically using a deep neural network. Furthermore, we also establish the convergence result for the Wasserstein-2 distance between the distribution of the learned samples and the target conditional distribution, providing the first comprehensive end-to-end error analysis for conditional distribution learning via ODE flow. Our numerical experiments showcase its effectiveness across a range of scenarios, from standard nonparametric conditional density estimation problems to more intricate challenges involving image data, illustrating its superiority over various existing conditional density estimation methods., Comment: The original title of this paper is "Deep Conditional Generative Learning: Model and Error Analysis"

We propose SDORE, a semi-supervised deep Sobolev regressor, for the nonparametric estimation of the underlying regression function and its gradient. SDORE employs deep neural networks to minimize empirical risk with gradient norm regularization, allowing computation of the gradient norm on unlabeled data. We conduct a comprehensive analysis of the convergence rates of SDORE and establish a minimax optimal rate for the regression function. Crucially, we also derive a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift. These theoretical findings offer valuable prior guidance for selecting regularization parameters and determining the size of the neural network, while showcasing the provable advantage of leveraging unlabeled data in semi-supervised learning. To the best of our knowledge, SDORE is the first provable neural network-based approach that simultaneously estimates the regression function and its gradient, with diverse applications including nonparametric variable selection and inverse problems. The effectiveness of SDORE is validated through an extensive range of numerical simulations and real data analysis.

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with $\mathcal{C}$-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for $\mathcal{C}$-mixing sequences and the neural network approximation theory for the H\"{o}lder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design., Comment: Full version of the paper accepted to the 38th Annual AAAI Conference on Artificial Intelligence (AAAI 2024)

Gaussian denoising has emerged as a powerful method for constructing simulation-free continuous normalizing flows for generative modeling. Despite their empirical successes, theoretical properties of these flows and the regularizing effect of Gaussian denoising have remained largely unexplored. In this work, we aim to address this gap by investigating the well-posedness of simulation-free continuous normalizing flows built on Gaussian denoising. Through a unified framework termed Gaussian interpolation flow, we establish the Lipschitz regularity of the flow velocity field, the existence and uniqueness of the flow, and the Lipschitz continuity of the flow map and the time-reversed flow map for several rich classes of target distributions. This analysis also sheds light on the auto-encoding and cycle consistency properties of Gaussian interpolation flows. Additionally, we study the stability of these flows in source distributions and perturbations of the velocity field, using the quadratic Wasserstein distance as a metric. Our findings offer valuable insights into the learning techniques employed in Gaussian interpolation flows for generative modeling, providing a solid theoretical foundation for end-to-end error analyses of learning Gaussian interpolation flows with empirical observations., Comment: 51 pages, 6 figures, 1 table

We propose a general approach to evaluating the performance of robust estimators based on adversarial losses under misspecified models. We first show that adversarial risk is equivalent to the risk induced by a distributional adversarial attack under certain smoothness conditions. This ensures that the adversarial training procedure is well-defined. To evaluate the generalization performance of the adversarial estimator, we study the adversarial excess risk. Our proposed analysis method includes investigations on both generalization error and approximation error. We then establish non-asymptotic upper bounds for the adversarial excess risk associated with Lipschitz loss functions. In addition, we apply our general results to adversarial training for classification and regression problems. For the quadratic loss in nonparametric regression, we show that the adversarial excess risk bound can be improved over those for a general loss., Comment: 27 pages, 3 tables

We study the properties of differentiable neural networks activated by rectified power unit (RePU) functions. We show that the partial derivatives of RePU neural networks can be represented by RePUs mixed-activated networks and derive upper bounds for the complexity of the function class of derivatives of RePUs networks. We establish error bounds for simultaneously approximating $C^s$ smooth functions and their derivatives using RePU-activated deep neural networks. Furthermore, we derive improved approximation error bounds when data has an approximate low-dimensional support, demonstrating the ability of RePU networks to mitigate the curse of dimensionality. To illustrate the usefulness of our results, we consider a deep score matching estimator (DSME) and propose a penalized deep isotonic regression (PDIR) using RePU networks. We establish non-asymptotic excess risk bounds for DSME and PDIR under the assumption that the target functions belong to a class of $C^s$ smooth functions. We also show that PDIR achieves the minimax optimal convergence rate and has a robustness property in the sense it is consistent with vanishing penalty parameters even when the monotonicity assumption is not satisfied. Furthermore, if the data distribution is supported on an approximate low-dimensional manifold, we show that DSME and PDIR can mitigate the curse of dimensionality., Comment: 78 pages, 20 figures, and 6 tables. arXiv admin note: text overlap with arXiv:2207.10442

We propose a penalized nonparametric approach to estimating the quantile regression process (QRP) in a nonseparable model using rectifier quadratic unit (ReQU) activated deep neural networks and introduce a novel penalty function to enforce non-crossing of quantile regression curves. We establish the non-asymptotic excess risk bounds for the estimated QRP and derive the mean integrated squared error for the estimated QRP under mild smoothness and regularity conditions. To establish these non-asymptotic risk and estimation error bounds, we also develop a new error bound for approximating $C^s$ smooth functions with $s >0$ and their derivatives using ReQU activated neural networks. This is a new approximation result for ReQU networks and is of independent interest and may be useful in other problems. Our numerical experiments demonstrate that the proposed method is competitive with or outperforms two existing methods, including methods using reproducing kernels and random forests, for nonparametric quantile regression., Comment: 44 pages, 10 figures, 6 tables

This paper studies the approximation capacity of ReLU neural networks with norm constraint on the weights. We prove upper and lower bounds on the approximation error of these networks for smooth function classes. The lower bound is derived through the Rademacher complexity of neural networks, which may be of independent interest. We apply these approximation bounds to analyze the convergences of regression using norm constrained neural networks and distribution estimation by GANs. In particular, we obtain convergence rates for over-parameterized neural networks. It is also shown that GANs can achieve optimal rate of learning probability distributions, when the discriminator is a properly chosen norm constrained neural network.

In this paper, we consider recovering $n$ dimensional signals from $m$ binary measurements corrupted by noises and sign flips under the assumption that the target signals have low generative intrinsic dimension, i.e., the target signals can be approximately generated via an $L$-Lipschitz generator $G: \mathbb{R}^k\rightarrow\mathbb{R}^{n}, k\ll n$. Although the binary measurements model is highly nonlinear, we propose a least square decoder and prove that, up to a constant $c$, with high probability, the least square decoder achieves a sharp estimation error $\mathcal{O} (\sqrt{\frac{k\log (Ln)}{m}})$ as long as $m\geq \mathcal{O}( k\log (Ln))$. Extensive numerical simulations and comparisons with state-of-the-art methods demonstrated the least square decoder is robust to noise and sign flips, as indicated by our theory. By constructing a ReLU network with properly chosen depth and width, we verify the (approximately) deep generative prior, which is of independent interest.

In this work, we consider the algorithm to the (nonlinear) regression problems with $\ell_0$ penalty. The existing algorithms for $\ell_0$ based optimization problem are often carried out with a fixed step size, and the selection of an appropriate step size depends on the restricted strong convexity and smoothness for the loss function, hence it is difficult to compute in practical calculation. In sprite of the ideas of support detection and root finding \cite{HJK2020}, we proposes a novel and efficient data-driven line search rule to adaptively determine the appropriate step size. We prove the $\ell_2$ error bound to the proposed algorithm without much restrictions for the cost functional. A large number of numerical comparisons with state-of-the-art algorithms in linear and logistic regression problems show the stability, effectiveness and superiority of the proposed algorithms.

We derive nearly sharp bounds for the bidirectional GAN (BiGAN) estimation error under the Dudley distance between the latent joint distribution and the data joint distribution with appropriately specified architecture of the neural networks used in the model. To the best of our knowledge, this is the first theoretical guarantee for the bidirectional GAN learning approach. An appealing feature of our results is that they do not assume the reference and the data distributions to have the same dimensions or these distributions to have bounded support. These assumptions are commonly assumed in the existing convergence analysis of the unidirectional GANs but may not be satisfied in practice. Our results are also applicable to the Wasserstein bidirectional GAN if the target distribution is assumed to have a bounded support. To prove these results, we construct neural network functions that push forward an empirical distribution to another arbitrary empirical distribution on a possibly different-dimensional space. We also develop a novel decomposition of the integral probability metric for the error analysis of bidirectional GANs. These basic theoretical results are of independent interest and can be applied to other related learning problems., Comment: Corresponding authors: Yunfei Yang (yyangdc@connect.ust.hk), Jian Huang (jian-huang@uiowa.edu), Yuling Jiao (yulingjiaomath@whu.edu.cn)

We propose a relative entropy gradient sampler (REGS) for sampling from unnormalized distributions. REGS is a particle method that seeks a sequence of simple nonlinear transforms iteratively pushing the initial samples from a reference distribution into the samples from an unnormalized target distribution. To determine the nonlinear transforms at each iteration, we consider the Wasserstein gradient flow of relative entropy. This gradient flow determines a path of probability distributions that interpolates the reference distribution and the target distribution. It is characterized by an ODE system with velocity fields depending on the density ratios of the density of evolving particles and the unnormalized target density. To sample with REGS, we need to estimate the density ratios and simulate the ODE system with particle evolution. We propose a novel nonparametric approach to estimating the logarithmic density ratio using neural networks. Extensive simulation studies on challenging multimodal 1D and 2D mixture distributions and Bayesian logistic regression on real datasets demonstrate that the REGS outperforms the state-of-the-art sampling methods included in the comparison.

Recovering sparse signals from observed data is an important topic in signal/imaging processing, statistics and machine learning. Nonconvex penalized least squares have been attracted a lot of attentions since they enjoy nice statistical properties. Computationally, coordinate descent (CD) is a workhorse for minimizing the nonconvex penalized least squares criterion due to its simplicity and scalability. In this work, we prove the linear convergence rate to CD for solving MCP/SCAD penalized least squares problems.

This paper studies how well generative adversarial networks (GANs) learn probability distributions from finite samples. Our main results establish the convergence rates of GANs under a collection of integral probability metrics defined through H\"older classes, including the Wasserstein distance as a special case. We also show that GANs are able to adaptively learn data distributions with low-dimensional structures or have H\"older densities, when the network architectures are chosen properly. In particular, for distributions concentrated around a low-dimensional set, we show that the learning rates of GANs do not depend on the high ambient dimension, but on the lower intrinsic dimension. Our analysis is based on a new oracle inequality decomposing the estimation error into the generator and discriminator approximation error and the statistical error, which may be of independent interest.

The goal of supervised representation learning is to construct effective data representations for prediction. Among all the characteristics of an ideal nonparametric representation of high-dimensional complex data, sufficiency, low dimensionality and disentanglement are some of the most essential ones. We propose a deep dimension reduction approach to learning representations with these characteristics. The proposed approach is a nonparametric generalization of the sufficient dimension reduction method. We formulate the ideal representation learning task as that of finding a nonparametric representation that minimizes an objective function characterizing conditional independence and promoting disentanglement at the population level. We then estimate the target representation at the sample level nonparametrically using deep neural networks. We show that the estimated deep nonparametric representation is consistent in the sense that its excess risk converges to zero. Our extensive numerical experiments using simulated and real benchmark data demonstrate that the proposed methods have better performance than several existing dimension reduction methods and the standard deep learning models in the context of classification and regression.

We propose a \textbf{uni}fied \textbf{f}ramework for \textbf{i}mplicit \textbf{ge}nerative \textbf{m}odeling (UnifiGem) with theoretical guarantees by integrating approaches from optimal transport, numerical ODE, density-ratio (density-difference) estimation and deep neural networks. First, the problem of implicit generative learning is formulated as that of finding the optimal transport map between the reference distribution and the target distribution, which is characterized by a totally nonlinear Monge-Amp\`{e}re equation. Interpreting the infinitesimal linearization of the Monge-Amp\`{e}re equation from the perspective of gradient flows in measure spaces leads to the continuity equation or the McKean-Vlasov equation. We then solve the McKean-Vlasov equation numerically using the forward Euler iteration, where the forward Euler map depends on the density ratio (density difference) between the distribution at current iteration and the underlying target distribution. We further estimate the density ratio (density difference) via deep density-ratio (density-difference) fitting and derive explicit upper bounds on the estimation error. Experimental results on both synthetic datasets and real benchmark datasets support our theoretical findings and demonstrate the effectiveness of UnifiGem.

Screening and working set techniques are important approaches to reducing the size of an optimization problem. They have been widely used in accelerating first-order methods for solving large-scale sparse learning problems. In this paper, we develop a new screening method called Newton screening (NS) which is a generalized Newton method with a built-in screening mechanism. We derive an equivalent KKT system for the Lasso and utilize a generalized Newton method to solve the KKT equations. Based on this KKT system, a built-in working set with a relatively small size is first determined using the sum of primal and dual variables generated from the previous iteration, then the primal variable is updated by solving a least-squares problem on the working set and the dual variable updated based on a closed-form expression. Moreover, we consider a sequential version of Newton screening (SNS) with a warm-start strategy. We show that NS possesses an optimal convergence property in the sense that it achieves one-step local convergence. Under certain regularity conditions on the feature matrix, we show that SNS hits a solution with the same signs as the underlying true target and achieves a sharp estimation error bound with high probability. Simulation studies and real data analysis support our theoretical results and demonstrate that SNS is faster and more accurate than several state-of-the-art methods in our comparative studies., Comment: This paper has fatal mistakes

Feature selection is important for modeling high-dimensional data, where the number of variables can be much larger than the sample size. In this paper, we develop a support detection and root finding procedure to learn the high dimensional sparse generalized linear models and denote this method by GSDAR. Based on the KKT condition for $\ell_0$-penalized maximum likelihood estimations, GSDAR generates a sequence of estimators iteratively. Under some restricted invertibility conditions on the maximum likelihood function and sparsity assumption on the target coefficients, the errors of the proposed estimate decays exponentially to the optimal order. Moreover, the oracle estimator can be recovered if the target signal is stronger than the detectable level. We conduct simulations and real data analysis to illustrate the advantages of our proposed method over several existing methods, including Lasso and MCP.

Sparse phase retrieval plays an important role in many fields of applied science and thus attracts lots of attention. In this paper, we propose a \underline{sto}chastic alte\underline{r}nating \underline{m}inimizing method for \underline{sp}arse ph\underline{a}se \underline{r}etrieval (\textit{StormSpar}) algorithm which {emprically} is able to recover $n$-dimensional $s$-sparse signals from only $O(s\,\mathrm{log}\, n)$ number of measurements without a desired initial value required by many existing methods. In \textit{StormSpar}, the hard-thresholding pursuit (HTP) algorithm is employed to solve the sparse constraint least square sub-problems. The main competitive feature of \textit{StormSpar} is that it converges globally requiring optimal order of number of samples with random initialization. Extensive numerical experiments are given to validate the proposed algorithm.

To address the challenges in learning deep generative models (e.g.,the blurriness of variational auto-encoder and the instability of training generative adversarial networks, we propose a novel deep generative model, named Wasserstein-Wasserstein auto-encoders (WWAE). We formulate WWAE as minimization of the penalized optimal transport between the target distribution and the generated distribution. By noticing that both the prior $P_Z$ and the aggregated posterior $Q_Z$ of the latent code Z can be well captured by Gaussians, the proposed WWAE utilizes the closed-form of the squared Wasserstein-2 distance for two Gaussians in the optimization process. As a result, WWAE does not suffer from the sampling burden and it is computationally efficient by leveraging the reparameterization trick. Numerical results evaluated on multiple benchmark datasets including MNIST, fashion- MNIST and CelebA show that WWAE learns better latent structures than VAEs and generates samples of better visual quality and higher FID scores than VAEs and GANs.

We propose a general framework to learn deep generative models via \textbf{V}ariational \textbf{Gr}adient Fl\textbf{ow} (VGrow) on probability spaces. The evolving distribution that asymptotically converges to the target distribution is governed by a vector field, which is the negative gradient of the first variation of the $f$-divergence between them. We prove that the evolving distribution coincides with the pushforward distribution through the infinitesimal time composition of residual maps that are perturbations of the identity map along the vector field. The vector field depends on the density ratio of the pushforward distribution and the target distribution, which can be consistently learned from a binary classification problem. Connections of our proposed VGrow method with other popular methods, such as VAE, GAN and flow-based methods, have been established in this framework, gaining new insights of deep generative learning. We also evaluated several commonly used divergences, including Kullback-Leibler, Jensen-Shannon, Jeffrey divergences as well as our newly discovered `logD' divergence which serves as the objective function of the logD-trick GAN. Experimental results on benchmark datasets demonstrate that VGrow can generate high-fidelity images in a stable and efficient manner, achieving competitive performance with state-of-the-art GANs.

We propose a semismooth Newton algorithm for pathwise optimization (SNAP) for the LASSO and Enet in sparse, high-dimensional linear regression. SNAP is derived from a suitable formulation of the KKT conditions based on Newton derivatives. It solves the semismooth KKT equations efficiently by actively and continuously seeking the support of the regression coefficients along the solution path with warm start. At each knot in the path, SNAP converges locally superlinearly for the Enet criterion and achieves an optimal local convergence rate for the LASSO criterion, i.e., SNAP converges in one step at the cost of two matrix-vector multiplication per iteration. Under certain regularity conditions on the design matrix and the minimum magnitude of the nonzero elements of the target regression coefficients, we show that SNAP hits a solution with the same signs as the regression coefficients and achieves a sharp estimation error bound in finite steps with high probability. The computational complexity of SNAP is shown to be the same as that of LARS and coordinate descent algorithms per iteration. Simulation studies and real data analysis support our theoretical results and demonstrate that SNAP is faster and accurate than LARS and coordinate descent algorithms.

We develop a primal dual active set with continuation algorithm for solving the \ell^0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly. Under certain conditions on the sensing matrix, i.e., mutual incoherence property or restricted isometry property, and the noise level, the finite step global convergence of the algorithm is established. Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and the convergence analysis.

In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including $\ell^0$, bridge, smoothly clipped absolute deviation, capped $\ell^1$ and minimax concavity penalty. First we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinate-wise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined using the primal and dual variables together. Further, this relation lends itself to an iterative algorithm of active set type which at each step involves first updating the primal variable only on the active set and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primal dual active set method is shown to converge globally to the underlying regression target under certain regularity conditions. Extensive numerical experiments with both simulated and real data demonstrate its superior performance in efficiency and accuracy compared with the existing sparse recovery methods.