We study the basic computational problem of detecting approximate stationary points for continuous piecewise affine (PA) functions. Our contributions span multiple aspects, including complexity, regularity, and algorithms. Specifically, we show that testing first-order approximate stationarity concepts, as defined by commonly used generalized subdifferentials, is computationally intractable unless P = NP. To facilitate computability, we consider a polynomial-time solvable relaxation by abusing the convex subdifferential sum rule and establish a tight characterization of its exactness. Furthermore, addressing an open issue motivated by the need to terminate the subgradient method in finite time, we introduce the first oracle-polynomial-time algorithm to detect so-called near-approximate stationary points for PA functions.

Feature selection plays an important role in a wide range of applications. Most sparsity-based feature selection methods solve a relaxed$\ell _{2,p}$-norm ($0 \lt p \leq 1$) regularized problem, which often results in a sub-optimal feature subset and requires extensive effort to tune regularization parameters. Optimizing the non-convex$\ell _{2,0}$-norm constrained problem remains an open challenge. Existing optimization algorithms for solving the$\ell _{2,0}$-norm constrained problem often rely on specific data distribution assumptions and cannot guarantee global convergence. In this article, we propose an unsupervised discriminative feature selection method using$\ell _{2,0}$-norm constrained sparse projection (SPDFS) to address these challenges. Specifically, building on the principle of supervised linear discriminant analysis, fuzzy membership learning and$\ell _{2,0}$-norm constrained projection learning are jointly performed to learn a feature-wise sparse projection for unsupervised discriminative feature selection. More importantly, we follow two optimization strategies to address the NP-hard nature of the problem: a non-iterative algorithm with a globally optimal solution is derived for a special case, and an iterative algorithm with both ascent property and approximation guarantee is employed for the general case. Additionally, we explore the relationship between our model and its potential variants. Experimental results on both synthetic and real-world datasets demonstrate the superiority of the proposed method over several state-of-the-art methods in data clustering and text classification tasks. The code is available at:https://github.com/xiadongcs/SPDFS.

The principal subspace estimation is directly connected to dimension reduction and is important when there is more than one principal component of interest. In this article, we introduce two new algorithms to solve the feature-sparsity constrained PCA problem (FSPCA) for the principal subspace estimation task, which performs feature selection and PCA simultaneously. Existing optimization methods for FSPCA require data distribution assumptions and are lack of global convergence guarantee. Though the general FSPCA problem is NP-hard, we show that, for a low-rank covariance, FSPCA can be solved globally (Algorithm 1). Then, we propose another strategy (Algorithm 2) to solve FSPCA for the general covariance by iteratively building a carefully designed proxy. We prove (data-dependent) approximation bound and regular stationary convergence guarantees for the new algorithms. For the spectrum of covariance with exponential/Zipf's distribution, we provide exponential/posynomial approximation bounds. Constructive examples and numerical results are provided to demonstrate the tightness of our results. Experimental results show the promising performance and efficiency of the new algorithms compared with the state-of-the-arts on both synthetic and real-world datasets.

This paper is concerned with the algorithmic aspects of sharper stationarity of a nonconvex, nonsmooth, Clarke irregular machine learning model. We study the SVM problem with a $\rho$-margin loss function, which is the margin theory generalization bound of SVM introduced in the learning theory textbook by Mohri et al. [2018], and has been extensively studied in operations research, statistics, and machine learning communities. However, due to its nonconvex, nonsmooth, and irregular nature, none of the existing optimization methods can efficiently compute a d(irectional)-stationary point, which turns out to be also a local minimum, for the $\rho$-margin loss SVM problem. After a detailed discussion of various nonsmooth stationarity notions, we propose a highly efficient nonconvex semi-proximal ADMM-based scheme that provably computes d-stationary points and enjoys a local linear convergence rate. We report concrete examples to demonstrate the necessity of our assumptions. Numerical results verify the effectiveness of the new algorithm and complement our theoretical results.

In convex optimization, the problem of finding near-stationary points has not been adequately studied yet, unlike other optimality measures such as the function value. Even in the deterministic case, the optimal method (OGM-G, due to Kim and Fessler (2021)) has just been discovered recently. In this work, we conduct a systematic study of algorithmic techniques for finding near-stationary points of convex finite-sums. Our main contributions are several algorithmic discoveries: (1) we discover a memory-saving variant of OGM-G based on the performance estimation problem approach (Drori and Teboulle, 2014); (2) we design a new accelerated SVRG variant that can simultaneously achieve fast rates for minimizing both the gradient norm and function value; (3) we propose an adaptively regularized accelerated SVRG variant, which does not require the knowledge of some unknown initial constants and achieves near-optimal complexities. We put an emphasis on the simplicity and practicality of the new schemes, which could facilitate future work.

We report a practical finite-time algorithmic scheme to compute approximately stationary points for nonconvex nonsmooth Lipschitz functions. In particular, we are interested in two kinds of approximate stationarity notions for nonconvex nonsmooth problems, i.e., Goldstein approximate stationarity (GAS) and near-approximate stationarity (NAS). For GAS, our scheme removes the unrealistic subgradient selection oracle assumption in (Zhang et al., 2020, Assumption 1) and computes GAS with the same finite-time complexity. For NAS, Davis & Drusvyatskiy (2019) showed that $\rho$-weakly convex functions admit finite-time computation, while Tian & So (2021) provided the matching impossibility results of dimension-free finite-time complexity for first-order methods. Complement to these developments, in this paper, we isolate a new class of functions that could be Clarke irregular (and thus not weakly convex anymore) and show that our new algorithmic scheme can compute NAS points for functions in that class within finite time. To demonstrate the wide applicability of our new theoretical framework, we show that $\rho$-margin SVM, $1$-layer, and $2$-layer ReLU neural networks, all being Clarke irregular, satisfy our new conditions.

In this article, we propose a novel feature selection approach via explicitly addressing the long-standing subspace sparsity issue. Leveraging$\ell _{2,1}$-norm regularization for feature selection is the major strategy in existing methods, which, however, confronts sparsity limitation and parameter-tuning trouble. To circumvent this problem, employing the$\ell _{2,0}$-norm constraint to improve the sparsity of the model has gained more attention recently whereas, optimizing the subspace sparsity constraint is still an unsolved problem, which only can acquire an approximate solution and without convergence proof. To address the above challenges, we innovatively propose a novel subspace sparsity discriminative feature selection (S2DFS) method which leverages a subspace sparsity constraint to avoid tuning parameters. In addition, the trace ratio formulated objective function extremely ensures the discriminability of selected features. Most important, an efficient iterative optimization algorithm is presented to explicitly solve the proposed problem with a closed-form solution and strict convergence proof. To the best of our knowledge, such an optimization algorithm of solving the subspace sparsity issue is first proposed in this article, and a general formulation of the optimization algorithm is provided for improving the extensibility and portability of our method. Extensive experiments conducted on several high-dimensional text and image datasets demonstrate that the proposed method outperforms related state-of-the-art methods in pattern classification and image retrieval tasks.

In this article, we propose a novel feature selection approach, named unsupervised feature selection with constrained$\ell _{2,0}$-norm (row-sparsity constrained) and optimized graph (RSOGFS), which unifies feature selection and similarity matrix construction into a general framework instead of independently performing the two-stage process; thus, the similarity matrix preserving the local manifold structure of data can be determined adaptively. Unlike those sparse learning-based feature selection methods that can only solve the relaxation or approximation problems by introducing sparsity regularization term into the objective function, the proposed method directly tackles the original$\ell _{2,0}$-norm constrained problem to achieve group feature selection. Two optimization strategies are provided to solve the original sparse constrained problem. The convergence and approximation guarantees for the new algorithms are rigorously proved, and the computational complexity and parameter determination are theoretically analyzed. Experimental results on real-world data sets show that the proposed method for solving a nonconvex problem is superior to the state of the arts for solving the relaxed or approximate convex problems.

Principal Component Analysis (PCA) is one of the most important unsupervised methods to handle high-dimensional data. However, due to the high computational complexity of its eigen-decomposition solution, it is hard to apply PCA to the large-scale data with high dimensionality, e.g., millions of data points with millions of variables. Meanwhile, the squared L2-norm based objective makes it sensitive to data outliers. In recent research, the L1-norm maximization based PCA method was proposed for efficient computation and being robust to outliers. However, this work used a greedy strategy to solve the eigenvectors. Moreover, the L1-norm maximization based objective may not be the correct robust PCA formulation, because it loses the theoretical connection to the minimization of data reconstruction error, which is one of the most important intuitions and goals of PCA. In this paper, we propose to maximize the L21-norm based robust PCA objective, which is theoretically connected to the minimization of reconstruction error. More importantly, we propose the efficient non-greedy optimization algorithms to solve our objective and the more general L21-norm maximization problem with theoretically guaranteed convergence. Experimental results on real world data sets show the effectiveness of the proposed method for principal component analysis.

In this paper, we first propose a novel Structured Sparse Subspace Learning S^3L module to address the long-standing subspace sparsity issue. Elicited by proposed module, we design a new discriminative feature selection method, named Subspace Sparsity Discriminant Feature Selection S^2DFS which enables the following new functionalities: 1) Proposed S^2DFS method directly joints trace ratio objective and structured sparse subspace constraint via L2,0-norm to learn a row-sparsity subspace, which improves the discriminability of model and overcomes the parameter-tuning trouble with comparison to the methods used L2,1-norm regularization; 2) An alternative iterative optimization algorithm based on the proposed S^3L module is presented to explicitly solve the proposed problem with a closed-form solution and strict convergence proof. To our best knowledge, such objective function and solver are first proposed in this paper, which provides a new though for the development of feature selection methods. Extensive experiments conducted on several high-dimensional datasets demonstrate the discriminability of selected features via S^2DFS with comparison to several related SOTA feature selection methods. Source matlab code: https://github.com/StevenWangNPU/L20-FS.

This paper presents new algorithms to solve the feature-sparsity constrained PCA problem (FSPCA), which performs feature selection and PCA simultaneously. Existing optimization methods for FSPCA require data distribution assumptions and are lack of global convergence guarantee. Though the general FSPCA problem is NP-hard, we show that, for a low-rank covariance, FSPCA can be solved globally (Algorithm 1). Then, we propose another strategy (Algorithm 2) to solve FSPCA for the general covariance by iteratively building a carefully designed proxy. We prove (data-dependent) approximation bound and convergence guarantees for the new algorithms. For the spectrum of covariance with exponential/Zipf's distribution, we provide exponential/posynomial approximation bound. Experimental results show the promising performance and efficiency of the new algorithms compared with the state-of-the-arts on both synthetic and real-world datasets.

Semi-supervised learning models for multiview data are important in image classification tasks, since heterogeneous features are easy to obtain and semi-supervised schemes are economical and effective. To model the view importance, conventional graph-based multiview learning models learn a linear combination of views while assuming a priori weights distribution. In this paper, we present a novel structural regularized semi-supervised model for multiview data, termed Adaptive MUltiview SEmi-supervised model (AMUSE). Our new model learns weights from a priori graph structure, which is more reasonable than weight regularization. Theoretical analysis reveals the significant difference between AMUSE and the prior arts. An efficient optimization algorithm is provided to solve the new model. Experimental results on six real-world data sets demonstrate the effectiveness of the structural regularized weights learning scheme.

Learning a set of weights to combine views linearly forms a series of popular schemes in multi-view learning. Three weight learning paradigms, i.e., Norm Regularization (NR), Exponential Decay (ED), and p-th Root Loss (pRL), are widely used in the literature, while the relations between them and the limiting behaviors of them are not well understood yet. In this paper, we present a Unified Paradigm (UP) that contains the aforementioned three popular paradigms as special cases. Specifically, we extend the domain of hyper-parameters of NR from positive to real numbers and show this extension bridges NR, ED, and pRL. Besides, we provide detailed discussion on the weights sparsity, hyper-parameter setting, and counterintuitive limiting behavior of these paradigms. Furthermore, we show the generality of our technique with examples in Multi-Task Learning and Fuzzy Clustering. Our results may provide insights to understand existing algorithms better and inspire research on new weight learning schemes. Numerical results support our theoretical analysis.

In this paper, we make a multiview extension of the spectral rotation technique raised in single view spectral clustering research. Since spectral rotation is closely related to the Procrustes Analysis for points matching, we point out that classical Procrustes Average approach can be used for multiview clustering. Besides, we show that direct applying Procrustes Average (PA) in multiview tasks may not be optimal theoretically and empirically, since it does not take the clustering capacity differences of different views into consideration. Other than that, we propose an Adaptively Weighted Procrustes (AWP) approach to overcome the aforementioned deficiency. Our new AWP weights views with their clustering capacities and forms a weighted Procrustes Average problem accordingly. The optimization algorithm to solve the new model is computational complexity analyzed and convergence guaranteed. Experiments on five real-world datasets demonstrate the effectiveness and efficiency of the new models.