Classical principal component analysis (PCA) may suffer from the sensitivity to outliers and noise. Therefore PCA based on $\ell_1$-norm and $\ell_p$-norm ($0 < p < 1$) have been studied. Among them, the ones based on $\ell_p$-norm seem to be most interesting from the robustness point of view. However, their numerical performance is not satisfactory. Note that, although T$\ell_1$-norm is similar to $\ell_p$-norm ($0 < p < 1$) in some sense, it has the stronger suppression effect to outliers and better continuity. So PCA based on T$\ell_1$-norm is proposed in this paper. Our numerical experiments have shown that its performance is superior than PCA-$\ell_p$ and $\ell_p$SPCA as well as PCA, PCA-$\ell_1$ obviously.
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 hard to apply PCA to the large-scale data with high dimensionality. 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 eigen vectors. Moreover, the L1-norm maximization base…
Hong Kong University of Science and Technology（香港科技大学）；The Hong Kong Polytechnic University（香港理工大学）
Dictionary learning is a classic representation learning method that has been widely applied in signal processing and data analytics. In this paper, we investigate a family of $\ell_p$-norm ($p>2,p \in \mathbb{N}$) maximization approaches for the complete dictionary learning problem from theoretical and algorithmic aspects. Specifically, we prove that the global maximizers of these formulations are very close to the true dictionary with high probability, even when Gaussian noise is present. Based on the generalized power method (GPM), an efficient algorithm is then developed for the $\ell_p$-b…
Principal Component Analysis (PCA) is a popular tool for dimensionality reduction and feature extraction in data analysis. There is a probabilistic version of PCA, known as Probabilistic PCA (PPCA). However, standard PCA and PPCA are not robust, as they are sensitive to outliers. To alleviate this problem, this paper introduces the Self-Paced Learning mechanism into PPCA, and proposes a novel method called Self-Paced Probabilistic Principal Component Analysis (SP-PPCA). Furthermore, we design the corresponding optimization algorithm based on the alternative search strategy and the expectation-…
University of California, Berkeley（加州大学伯克利分校）；Kuaishou Technology（快手科技）；Columbia University（哥伦比亚大学）
This paper considers the fundamental problem of learning a complete (orthogonal) dictionary from samples of sparsely generated signals. Most existing methods solve the dictionary (and sparse representations) based on heuristic algorithms, usually without theoretical guarantees for either optimality or complexity. The recent $\ell^1$-minimization based methods do provide such guarantees but the associated algorithms recover the dictionary one column at a time. In this work, we propose a new formulation that maximizes the $\ell^4$-norm over the orthogonal group, to learn the entire dictionary. W…
With the rapid development of information technologies, centralized data processing is subject to many limitations, such as computational overheads, communication delays, and data privacy leakage. Decentralized data processing over networked terminal nodes becomes an important technology in the era of big data. Dictionary learning is a powerful representation learning method to exploit the low-dimensional structure from the high-dimensional data. By exploiting the low-dimensional structure, the storage and the processing overhead of data can be effectively reduced. In this paper, we propose a …
School of Computer Science and Engineering, The Hebrew University of Jerusalem（耶路撒冷希伯来大学计算机科学与工程学院）；Google Research（谷歌研究院）
We consider Fair Principal Component Analysis (FPCA) and search for a low dimensional subspace that spans multiple target vectors in a fair manner. FPCA is defined as a non-concave maximization of the worst projected target norm within a given set. The problem arises in filter design in signal processing, and when incorporating fairness into dimensionality reduction schemes. The state of the art approach to FPCA is via semidefinite relaxation and involves a polynomial yet computationally expensive optimization. To allow scalability, we propose to address FPCA using naive sub-gradient descent. …
Lehigh University（理海大学）；University of Virginia（弗吉尼亚大学）
Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto manifolds, its effectiveness is limited to data with a single modality. In this paper, we present a novel Gaussian latent variable model that provides a unique way to integrate multiple PGA models into a maximum-likelihood framework. This leads to a well-defined mixture model of probabilistic principal geodesic analysis (MPPGA) on sub-populations, …
Singapore University of Technology and Design（新加坡科技设计大学）；University of Edinburgh（爱丁堡大学）
We propose a novel topic discovery algorithm for unlabeled images based on the bag-of-words (BoW) framework. We first extract a dictionary of visual words and subsequently for each image compute a visual word occurrence histogram. We view these histograms as rows of a large matrix from which we extract sparse principal components (PCs). Each PC identifies a sparse combination of visual words which co-occur frequently in some images but seldom appear in others. Each sparse PC corresponds to a topic, and images whose interference with the PC is high belong to that topic, revealing the common par…
Tensor completion and robust principal component analysis have been widely used in machine learning while the key problem relies on the minimization of a tensor rank that is very challenging. A common way to tackle this difficulty is to approximate the tensor rank with the $\ell_1-$norm of singular values based on its Tensor Singular Value Decomposition (T-SVD). Besides, the sparsity of a tensor is also measured by its $\ell_1-$norm. However, the $\ell_1$ penalty is essentially biased and thus the result will deviate. In order to sidestep the bias, we propose a novel non-convex tensor rank sur…
Even with the rise in popularity of over-parameterized models, simple dimensionality reduction and clustering methods, such as PCA and k-means, are still routinely used in an amazing variety of settings. A primary reason is the combination of simplicity, interpretability and computational efficiency. The focus of this article is on improving upon PCA and k-means, by allowing non-linear relations in the data and more flexible cluster shapes, without sacrificing the key advantages. The key contribution is a new framework for Principal Elliptical Analysis (PEA), defining a simple and computationa…
Principal Component Analysis (PCA) is a powerful tool in statistics and machine learning. While existing study of PCA focuses on the recovery of principal components and their associated eigenvalues, there are few precise characterizations of individual principal component scores that yield low-dimensional embedding of samples. That hinders the analysis of various spectral methods. In this paper, we first develop an $\ell_p$ perturbation theory for a hollowed version of PCA in Hilbert spaces which provably improves upon the vanilla PCA in the presence of heteroscedastic noises. Through a novel…
Mixtures of probabilistic principal component analysis (MPPCA) is a well-known mixture model extension of principal component analysis (PCA). Similar to PCA, MPPCA assumes the data samples in each mixture contain homoscedastic noise. However, datasets with heterogeneous noise across samples are becoming increasingly common, as larger datasets are generated by collecting samples from several sources with varying noise profiles. The performance of MPPCA is suboptimal for data with heteroscedastic noise across samples. This paper proposes a heteroscedastic mixtures of probabilistic PCA technique …
In this paper, a new method is proposed for sparse PCA based on the recursive divide-and-conquer methodology. The main idea is to separate the original sparse PCA problem into a series of much simpler sub-problems, each having a closed-form solution. By recursively solving these sub-problems in an analytical way, an efficient algorithm is constructed to solve the sparse PCA problem. The algorithm only involves simple computations and is thus easy to implement. The proposed method can also be very easily extended to other sparse PCA problems with certain constraints, such as the nonnegative spa…
We consider a family of vector dot products that can be implemented using sign changes and addition operations only. The dot products are energy-efficient as they avoid the multiplication operation entirely. Moreover, the dot products induce the $\ell_1$-norm, thus providing robustness to impulsive noise. First, we analytically prove that the dot products yield symmetric, positive semi-definite generalized covariance matrices, thus enabling principal component analysis (PCA). Moreover, the generalized covariance matrices can be constructed in an Energy Efficient (EEF) manner due to the multipl…
We introduce a variant of (sparse) PCA in which the set of feasible support sets is determined by a graph. In particular, we consider the following setting: given a directed acyclic graph $G$ on $p$ vertices corresponding to variables, the non-zero entries of the extracted principal component must coincide with vertices lying along a path in $G$. From a statistical perspective, information on the underlying network may potentially reduce the number of observations required to recover the population principal component. We consider the canonical estimator which optimally exploits the prior know…
We investigate the difference between using an $\ell_1$ penalty versus an $\ell_1$ constraint in generalized eigenvalue problems, such as principal component analysis and discriminant analysis. Our main finding is that an $\ell_1$ penalty may fail to provide very sparse solutions; a severe disadvantage for variable selection that can be remedied by using an $\ell_1$ constraint. Our claims are supported both by empirical evidence and theoretical analysis. Finally, we illustrate the advantages of an $\ell_1$ constraint in the context of discriminant analysis and principal component analysis.
Functional principal component analysis has become the most important dimension reduction technique in functional data analysis. Based on B-spline approximation, functional principal components (FPCs) can be efficiently estimated by the expectation-maximization (EM) and the geometric restricted maximum likelihood (REML) algorithms under the strong assumption of Gaussianity on the principal component scores and observational errors. When computing the solution, the EM algorithm does not exploit the underlying geometric manifold structure, while the performance of REML is known to be unstable. I…
Establishing a low-dimensional representation of the data leads to efficient data learning strategies. In many cases, the reduced dimension needs to be explicitly stated and estimated from the data. We explore the estimation of dimension in finite samples as a constrained optimization problem, where the estimated dimension is a maximizer of a penalized profile likelihood criterion within the framework of a probabilistic principal components analysis. Unlike other penalized maximization problems that require an "optimal" penalty tuning parameter, we propose a data-averaging procedure whereby th…
In this paper, we introduce a novel technique for constrained submodular maximization, inspired by barrier functions in continuous optimization. This connection not only improves the running time for constrained submodular maximization but also provides the state of the art guarantee. More precisely, for maximizing a monotone submodular function subject to the combination of a $k$-matchoid and $\ell$-knapsack constraint (for $\ell\leq k$), we propose a potential function that can be approximately minimized. Once we minimize the potential function up to an $ε$ error it is guaranteed that we hav…
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）；Expedia Group（亿客行集团）；University of Haifa（海法大学）
In this paper, we propose scalable methods for maximizing a regularized submodular function $f = g - \ell$ expressed as the difference between a monotone submodular function $g$ and a modular function $\ell$. Indeed, submodularity is inherently related to the notions of diversity, coverage, and representativeness. In particular, finding the mode of many popular probabilistic models of diversity, such as determinantal point processes, submodular probabilistic models, and strongly log-concave distributions, involves maximization of (regularized) submodular functions. Since a regularized function…
Forschungszentrum Jülich GmbH（于利希研究中心）；RWTH Aachen University（亚琛工业大学）；JARA-ENERGY（于利希-亚琛研究联盟能源组）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Neural networks-based learning of the distribution of non-dispatchable renewable electricity generation from sources such as photovoltaics (PV) and wind as well as load demands has recently gained attention. Normalizing flow density models are particularly well suited for this task due to the training through direct log-likelihood maximization. However, research from the field of image generation has shown that standard normalizing flows can only learn smeared-out versions of manifold distributions. Previous works on normalizing flow-based scenario generation do not address this issue, and the…
University of Oxford（牛津大学）；Five AI Ltd.（Five AI有限公司）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学（KAUST））
Randomized smoothing has recently emerged as an effective tool that enables certification of deep neural network classifiers at scale. All prior art on randomized smoothing has focused on isotropic $\ell_p$ certification, which has the advantage of yielding certificates that can be easily compared among isotropic methods via $\ell_p$-norm radius. However, isotropic certification limits the region that can be certified around an input to worst-case adversaries, i.e., it cannot reason about other "close", potentially large, constant prediction safe regions. To alleviate this issue, (i) we theore…
