In binary classification, kernel-free linear or quadratic support vector machines are proposed to avoid dealing with difficulties such as finding appropriate kernel functions or tuning their hyper-parameters. Furthermore, Universum data points, which do not belong to any class, can be exploited to embed prior knowledge into the corresponding models so that the generalization performance is improved. In this paper, we design novel kernel-free Universum quadratic surface support vector machine models. Further, we propose the L1 norm regularized version that is beneficial for detecting potential sparsity patterns in the Hessian of the quadratic surface and reducing to the standard linear models if the data points are (almost) linearly separable. The proposed models are convex such that standard numerical solvers can be utilized for solving them. Nonetheless, we formulate a least squares version of the L1 norm regularized model and next, design an effective tailored algorithm that only requires solving one linear system. Several theoretical properties of these models are then reported/proved as well. We finally conduct numerical experiments on both artificial and public benchmark data sets to demonstrate the feasibility and effectiveness of the proposed models.
Kernel-Free Universum Quadratic Surface Twin Support Vector Machines for Imbalanced Data
Jan Evangelista Purkyně University（詹·埃瓦尔德·普日科夫大学）；Charles University（查尔斯大学）；American University（美国阿美利克大学）；University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Northeastern University（东北大学）
University of Guilan（吉兰大学）；Charles University（查理大学）；University of Bojnord（博季诺尔德大学）
Multi-task learning (MTL) has emerged as a promising topic of machine learning in recent years, aiming to enhance the performance of numerous related learning tasks by exploiting beneficial information. During the training phase, most of the existing multi-task learning models concentrate entirely on the target task data and ignore the non-target task data contained in the target tasks. To address this issue, Universum data, that do not correspond to any class of a classification problem, may be used as prior knowledge in the training model. This study looks at the challenge of multi-task lear…
L0-Regularized Quadratic Surface Support Vector Machines
Carlson School of Management, University of Minnesota（明尼苏达大学卡尔森管理学院）；Department of Industrial and Systems Engineering, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校工业与系统工程系）
Douglas-Rachford Splitting for Group-Sparse Feedback Linear-Quadratic Control
Department of Applied Mathematics, The Hong Kong Polytechnic University（应用数学系，香港理工大学）；College of Artificial Intelligence, Nankai University（人工智能学院，南开大学）
Department of Applied Mathematics, The Hong Kong Polytechnic University（应用数学系，香港理工大学）；College of Artificial Intelligence, Nankai University（人工智能学院，南开大学）
We consider the problem of the recovery of a k-sparse vector from compressed linear measurements when data are corrupted by a quantization noise. When the number of measurements is not sufficiently large, different $k$-sparse solutions may be present in the feasible set, and the classical l1 approach may be unsuccessful. For this motivation, we propose a non-convex quadratic programming method, which exploits prior information on the magnitude of the non-zero parameters. This results in a more efficient support recovery. We provide sufficient conditions for successful recovery and numerical si…
University of Minnesota（明尼苏达大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）
We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal Newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees. Specifically, by leveraging a sophisticated characterization of sparse modeling structures/assumptions (i.e., local restricted strong convexity and Hessian smoothness), we prove that within each stage of convex relaxation, our proposed algorithm achieves (local) …
Delft University of Technology（代尔夫特理工大学）；Stanford University（斯坦福大学）；VinAI Research（VinAI研究院）；Umea University（于默奥大学）
We propose a novel approximation hierarchy for cardinality-constrained, convex quadratic programs that exploits the rank-dominating eigenvectors of the quadratic matrix. Each level of approximation admits a min-max characterization whose objective function can be optimized over the binary variables analytically, while preserving convexity in the continuous variables. Exploiting this property, we propose two scalable optimization algorithms, coined as the "best response" and the "dual program", that can efficiently screen the potential indices of the nonzero elements of the original program. We…
Quadratic regression involves modeling the response as a (generalized) linear function of not only the features $x^{j_1}$ but also of quadratic terms $x^{j_1}x^{j_2}$. The inclusion of such higher-order "interaction terms" in regression often provides an easy way to increase accuracy in already-high-dimensional problems. However, this explodes the problem dimension from linear $O(p)$ to quadratic $O(p^2)$, and it is common to look for sparse interactions (typically via heuristics). In this paper, we provide a new algorithm - Interaction Hard Thresholding (IntHT) which is the first one to prova…
TU Munich（慕尼黑工业大学）；University of Bonn（波恩大学）；Halmstad University（哈尔姆斯塔德大学）
We address the non-convex optimisation problem of finding a sparse matrix on the Stiefel manifold (matrices with mutually orthogonal columns of unit length) that maximises (or minimises) a quadratic objective function. Optimisation problems on the Stiefel manifold occur for example in spectral relaxations of various combinatorial problems, such as graph matching, clustering, or permutation synchronisation. Although sparsity is a desirable property in such settings, it is mostly neglected in spectral formulations since existing solvers, e.g. based on eigenvalue decomposition, are unable to acco…
The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to recent state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modific…
IBM Research（IBM研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We consider support recovery in the quadratic logistic regression setting - where the target depends on both p linear terms $x_i$ and up to $p^2$ quadratic terms $x_i x_j$. Quadratic terms enable prediction/modeling of higher-order effects between features and the target, but when incorporated naively may involve solving a very large regression problem. We consider the sparse case, where at most $s$ terms (linear or quadratic) are non-zero, and provide a new faster algorithm. It involves (a) identifying the weak support (i.e. all relevant variables) and (b) standard logistic regression optimiz…
University of California, Berkeley（加州大学伯克利分校）；University of Pennsylvania（宾夕法尼亚大学）
In this work, we propose a robust approach to design distributed controllers for unknown-but-sparse linear and time-invariant systems. By leveraging modern techniques in distributed controller synthesis and structured linear inverse problems as applied to system identification, we show that near-optimal distributed controllers can be learned with sub-linear sample complexity and computed with near-linear time complexity, both measured with respect to the dimension of the system. In particular, we provide sharp end-to-end guarantees on the stability and the performance of the designed distribut…
The covariance matrix of a $p$-dimensional random variable is a fundamental quantity in data analysis. Given $n$ i.i.d. observations, it is typically estimated by the sample covariance matrix, at a computational cost of $O(np^{2})$ operations. When $n,p$ are large, this computation may be prohibitively slow. Moreover, in several contemporary applications, the population matrix is approximately sparse, and only its few large entries are of interest. This raises the following question, at the focus of our work: Assuming approximate sparsity of the covariance matrix, can its large entries be dete…
Sparse Partial Optimal Transport via Quadratic Regularization
Department of Computer Science, Ho Chi Minh University of Science（胡志明大学计算机科学系）；School of Science, Aalto University（艾尔沃斯大学科学学院）；Department of Science, Lycée Français Alexandre Yersin de Hanoi（河内亚历山大·耶尔辛法语中学科学系）；Department of Math, High School for the Gifted（天才高中数学系）；Department of Electrical and Computer Engineering, University of Massachusetts Amherst（马萨诸塞大学阿姆赫斯特分校电子与计算机工程系）；Department of Computer Science, Georgia Institute of Technology（佐治亚理工学院计算机科学系）；Department of Computer Science, Binh Duong University（平杜ong大学计算机科学系）
In this paper, we propose an optimization-based sparse learning approach to identify the set of most influential reactions in a chemical reaction network. This reduced set of reactions is then employed to construct a reduced chemical reaction mechanism, which is relevant to chemical interaction network modeling. The problem of identifying influential reactions is first formulated as a mixed-integer quadratic program, and then a relaxation method is leveraged to reduce the computational complexity of our approach. Qualitative and quantitative validation of the sparse encoding approach demonstra…
While the self-attention mechanism has been widely used in a wide variety of tasks, it has the unfortunate property of a quadratic cost with respect to the input length, which makes it difficult to deal with long inputs. In this paper, we present a method for accelerating and structuring self-attentions: Sparse Adaptive Connection (SAC). In SAC, we regard the input sequence as a graph and attention operations are performed between linked nodes. In contrast with previous self-attention models with pre-defined structures (edges), the model learns to construct attention edges to improve task-spec…
LORIA（洛林计算机研究与应用实验室）；Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；University of California, Berkeley（加州大学伯克利分校）；Linköping University（林雪平大学）
This paper deals with sparse phase retrieval, i.e., the problem of estimating a vector from quadratic measurements under the assumption that few components are nonzero. In particular, we consider the problem of finding the sparsest vector consistent with the measurements and reformulate it as a group-sparse optimization problem with linear constraints. Then, we analyze the convex relaxation of the latter based on the minimization of a block l1-norm and show various exact recovery and stability results in the real and complex cases. Invariance to circular shifts and reflections are also discuss…
Transformers-based models, such as BERT, have been one of the most successful deep learning models for NLP. Unfortunately, one of their core limitations is the quadratic dependency (mainly in terms of memory) on the sequence length due to their full attention mechanism. To remedy this, we propose, BigBird, a sparse attention mechanism that reduces this quadratic dependency to linear. We show that BigBird is a universal approximator of sequence functions and is Turing complete, thereby preserving these properties of the quadratic, full attention model. Along the way, our theoretical analysis re…
University of Leicester（莱斯特大学）；Institut Curie（居里研究所）；PSL Research University（巴黎文理研究大学）；Mines Paris Tech（巴黎高科矿业学院）；Inserm（法国国家健康与医学研究院）
Most of machine learning approaches have stemmed from the application of minimizing the mean squared distance principle, based on the computationally efficient quadratic optimization methods. However, when faced with high-dimensional and noisy data, the quadratic error functionals demonstrated many weaknesses including high sensitivity to contaminating factors and dimensionality curse. Therefore, a lot of recent applications in machine learning exploited properties of non-quadratic error functionals based on $L_1$ norm or even sub-linear potentials corresponding to quasinorms $L_p$ ($0<p<1$). …
Gaussian processes are powerful models for probabilistic machine learning, but are limited in application by their $O(N^3)$ inference complexity. We propose a method for deriving parametric families of kernel functions with compact spatial support, which yield naturally sparse kernel matrices and enable fast Gaussian process inference via sparse linear algebra. These families generalize known compactly-supported kernel functions, such as the Wendland polynomials. The parameters of this family of kernels can be learned from data using maximum likelihood estimation. Alternatively, we can quickly…
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Pennsylvania State University（宾夕法尼亚州立大学）；Stanford University（斯坦福大学）；University of California（加利福尼亚大学）
We present a novel approach for recovering a sparse signal from cross-correlated data. Cross-correlations naturally arise in many fields of imaging, such as optics, holography and seismic interferometry. Compared to the sparse signal recovery problem that uses linear measurements, the unknown is now a matrix formed by the cross correlation of the unknown signal. Hence, the bottleneck for inversion is the number of unknowns that grows quadratically. The main idea of our proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal of the unknown matrix, whose …
