Identification of autoregressive models with exogenous input (ARX) is a classical problem in system identification. This article considers the errors-in-variables (EIV) ARX model identification problem, where input measurements are also corrupted with noise. The recently proposed DIPCA technique solves the EIV identification problem but is only applicable to white measurement errors. We propose a novel identification algorithm based on a modified Dynamic Iterative Principal Components Analysis (DIPCA) approach for identifying the EIV-ARX model for single-input, single-output (SISO) systems where the output measurements are corrupted with coloured noise consistent with the ARX model. Most of the existing methods assume important parameters like input-output orders, delay, or noise-variances to be known. This work's novelty lies in the joint estimation of error variances, process order, delay, and model parameters. The central idea used to obtain all these parameters in a theoretically rigorous manner is based on transforming the lagged measurements using the appropriate error covariance matrix, which is obtained using estimated error variances and model parameters. Simulation studies on two systems are presented to demonstrate the efficacy of the proposed algorithm.
Principal component analysis (PCA) is often used to reduce the dimension of data by selecting a few orthonormal vectors that explain most of the variance structure of the data. L1 PCA uses the L1 norm to measure error, whereas the conventional PCA uses the L2 norm. For the L1 PCA problem minimizing the fitting error of the reconstructed data, we propose an exact reweighted and an approximate algorithm based on iteratively reweighted least squares. We provide convergence analyses, and compare their performance against benchmark algorithms in the literature. The computational experiment shows th…
Robust PCA, the problem of PCA in the presence of outliers has been extensively investigated in the last few years. Here we focus on Robust PCA in the outlier model where each column of the data matrix is either an inlier or an outlier. Most of the existing methods for this model assumes either the knowledge of the dimension of the lower dimensional subspace or the fraction of outliers in the system. However in many applications knowledge of these parameters is not available. Motivated by this we propose a parameter free outlier identification method for robust PCA which a) does not require th…
Microsoft Corporation（微软公司）；Conduent Labs India（Conduent印度实验室）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；University of California Irvine（加利福尼亚大学尔湾分校）；Xerox Research Centre India（施乐印度研究中心）
The robust PCA problem, wherein, given an input data matrix that is the superposition of a low-rank matrix and a sparse matrix, we aim to separate out the low-rank and sparse components, is a well-studied problem in machine learning. One natural question that arises is that, as in the inductive setting, if features are provided as input as well, can we hope to do better? Answering this in the affirmative, the main goal of this paper is to study the robust PCA problem while incorporating feature information. In contrast to previous works in which recovery guarantees are based on the convex rela…
We consider the problem of estimating multiple principal components using the recently-proposed Sparse and Functional Principal Components Analysis (SFPCA) estimator. We first propose an extension of SFPCA which estimates several principal components simultaneously using manifold optimization techniques to enforce orthogonality constraints. While effective, this approach is computationally burdensome so we also consider iterative deflation approaches which take advantage of existing fast algorithms for rank-one SFPCA. We show that alternative deflation schemes can more efficiently extract sign…
Subspace clustering (SC) is a popular method for dimensionality reduction of high-dimensional data, where it generalizes Principal Component Analysis (PCA). Recently, several methods have been proposed to enhance the robustness of PCA and SC, while most of them are computationally very expensive, in particular, for high dimensional large-scale data. In this paper, we develop much faster iterative algorithms for SC, incorporating robustness using a {\em non-squared} $\ell_2$-norm objective. The known implementations for optimizing the objective would be costly due to the alternative optimizatio…
We present three generalisations of Kernel Principal Components Analysis (KPCA) which incorporate knowledge of the class labels of a subset of the data points. The first, MV-KPCA, penalises within class variances similar to Fisher discriminant analysis. The second, LSKPCA is a hybrid of least squares regression and kernel PCA. The final LR-KPCA is an iteratively reweighted version of the previous which achieves a sigmoid loss function on the labeled points. We provide a theoretical risk bound as well as illustrative experiments on real and toy data sets.
Technion-Israel Institute of Technology（以色列理工学院）；Tsinghua University（清华大学）；Eindhoven University of Technology（埃因霍温理工大学）
Contrast enhanced ultrasound is a radiation-free imaging modality which uses encapsulated gas microbubbles for improved visualization of the vascular bed deep within the tissue. It has recently been used to enable imaging with unprecedented subwavelength spatial resolution by relying on super-resolution techniques. A typical preprocessing step in super-resolution ultrasound is to separate the microbubble signal from the cluttering tissue signal. This step has a crucial impact on the final image quality. Here, we propose a new approach to clutter removal based on robust principle component anal…
The University of Queensland（昆士兰大学）；University of Texas at Arlington（德克萨斯大学阿灵顿分校）
Principal component analysis (PCA) has been widely applied to dimensionality reduction and data pre-processing for different applications in engineering, biology and social science. Classical PCA and its variants seek for linear projections of the original variables to obtain a low dimensional feature representation with maximal variance. One limitation is that it is very difficult to interpret the results of PCA. In addition, the classical PCA is vulnerable to certain noisy data. In this paper, we propose a convex sparse principal component analysis (CSPCA) algorithm and apply it to feature a…
We consider the problem of outlier robust PCA (OR-PCA) where the goal is to recover principal directions despite the presence of outlier data points. That is, given a data matrix $M^*$, where $(1-α)$ fraction of the points are noisy samples from a low-dimensional subspace while $α$ fraction of the points can be arbitrary outliers, the goal is to recover the subspace accurately. Existing results for \OR-PCA have serious drawbacks: while some results are quite weak in the presence of noise, other results have runtime quadratic in dimension, rendering them impractical for large scale applications…
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-…
We present an algorithm for L1-norm kernel PCA and provide a convergence analysis for it. While an optimal solution of L2-norm kernel PCA can be obtained through matrix decomposition, finding that of L1-norm kernel PCA is not trivial due to its non-convexity and non-smoothness. We provide a novel reformulation through which an equivalent, geometrically interpretable problem is obtained. Based on the geometric interpretation of the reformulated problem, we present a fixed-point type algorithm that iteratively computes a binary weight for each observation. As the algorithm requires only inner pr…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Katholieke Universiteit Leuven（鲁汶大学）
Motivated by learning problems including max-norm regularized matrix completion and clustering, robust PCA and sparse inverse covariance selection, we propose a novel optimization algorithm for minimizing a convex objective which decomposes into three parts: a smooth part, a simple non-smooth Lipschitz part, and a simple non-smooth non-Lipschitz part. We use a time variant smoothing strategy that allows us to obtain a guarantee that does not depend on knowing in advance the total number of iterations nor a bound on the domain.
Stanford University（斯坦福大学）；MIT（麻省理工学院）；Microsoft Research, New England（新英格兰微软研究院）
We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvemen…
Sparse principal component analysis (PCA) involves nonconvex optimization for which the global solution is hard to obtain. To address this issue, one popular approach is convex relaxation. However, such an approach may produce suboptimal estimators due to the relaxation effect. To optimally estimate sparse principal subspaces, we propose a two-stage computational framework named "tighten after relax": Within the 'relax' stage, we approximately solve a convex relaxation of sparse PCA with early stopping to obtain a desired initial estimator; For the 'tighten' stage, we propose a novel algorithm…
As a widely used method in machine learning, principal component analysis (PCA) shows excellent properties for dimensionality reduction. It is a serious problem that PCA is sensitive to outliers, which has been improved by numerous Robust PCA (RPCA) versions. However, the existing state-of-the-art RPCA approaches cannot easily remove or tolerate outliers by a non-iterative manner. To tackle this issue, this paper proposes Angular Embedding (AE) to formulate a straightforward RPCA approach based on angular density, which is improved for large scale or high-dimensional data. Furthermore, a trimm…
LTCI, Télécom ParisTech, Université Paris Saclay（巴黎萨克雷大学巴黎高等电信学院LTCI实验室）
An autoencoder is a neural network which data projects to and from a lower dimensional latent space, where this data is easier to understand and model. The autoencoder consists of two sub-networks, the encoder and the decoder, which carry out these transformations. The neural network is trained such that the output is as close to the input as possible, the data having gone through an information bottleneck : the latent space. This tool bears significant ressemblance to Principal Component Analysis (PCA), with two main differences. Firstly, the autoencoder is a non-linear transformation, contra…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；UT Austin（德克萨斯大学奥斯汀分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Web3 Foundation（Web3基金会）
We study high-dimensional sparse estimation tasks in a robust setting where a constant fraction of the dataset is adversarially corrupted. Specifically, we focus on the fundamental problems of robust sparse mean estimation and robust sparse PCA. We give the first practically viable robust estimators for these problems. In more detail, our algorithms are sample and computationally efficient and achieve near-optimal robustness guarantees. In contrast to prior provable algorithms which relied on the ellipsoid method, our algorithms use spectral techniques to iteratively remove outliers from the d…
Deep learning models hold state of the art performance in many fields, yet their design is still based on heuristics or grid search methods that often result in overparametrized networks. This work proposes a method to analyze a trained network and deduce an optimized, compressed architecture that preserves accuracy while keeping computational costs tractable. Model compression is an active field of research that targets the problem of realizing deep learning models in hardware. However, most pruning methodologies tend to be experimental, requiring large compute and time intensive iterations o…
We consider the Principal Component Analysis problem for large tensors of arbitrary order $k$ under a single-spike (or rank-one plus noise) model. On the one hand, we use information theory, and recent results in probability theory, to establish necessary and sufficient conditions under which the principal component can be estimated using unbounded computational resources. It turns out that this is possible as soon as the signal-to-noise ratio $β$ becomes larger than $C\sqrt{k\log k}$ (and in particular $β$ can remain bounded as the problem dimensions increase). On the other hand, we analyze s…
Hong Kong University of Science and Technology（香港科技大学）
The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty term into the objective function that encourages a sparse solution. However, the resulting methods achieve sparsity at the expense of sacrificing the orthogonality property. In this paper, we develop a new method to estimate dominant sparse eigenvectors without trading off their or…
sparseGeoHOPCA: A Geometric Solution to Sparse Higher-Order PCA Without Covariance Estimation
Department of Electrical Engineering and Centre for Intelligent Multidimensional Data Analysis（电气工程系和智能多维数据分析中心）；City University of Hong Kong（香港城市大学）；State Key Laboratory of Public Big Data（公共大数据国家重点实验室）；School of Mathematics and Statistics（数学与统计学学院）；School of Mathematics（数学学院）
University of Electronic Science and Technology of China（电子科技大学）
Principal Component Analysis (PCA) has been widely used for dimensionality reduction and feature extraction. Robust PCA (RPCA), under different robust distance metrics, such as l1-norm and l2, p-norm, can deal with noise or outliers to some extent. However, real-world data may display structures that can not be fully captured by these simple functions. In addition, existing methods treat complex and simple samples equally. By contrast, a learning pattern typically adopted by human beings is to learn from simple to complex and less to more. Based on this principle, we propose a novel method cal…
Halıcıoğlu Data Science Institute, UCSD（加州大学圣迭戈分校哈利乔格鲁数据科学学院）；Computer Science and Engineering Department, UCSD（加州大学圣迭戈分校计算机科学与工程系）；Electrical and Computer Engineering Department, UCSD（加州大学圣迭戈分校电气与计算机工程系）
Spectral clustering algorithms typically require a priori selection of input parameters such as the number of clusters, a scaling parameter for the affinity measure, or ranges of these values for parameter tuning. Despite efforts for automating the process of spectral clustering, the task of grouping data in multi-scale and higher dimensional spaces is yet to be explored. This study presents a spectral clustering heuristic algorithm that obviates the need for an input by estimating the parameters from the data itself. Specifically, it introduces the heuristic of iterative eigengap search with …
Department of Computer Science, University College London（伦敦大学学院计算机科学系）；Huawei Noah’s Ark Lab（华为诺亚方舟实验室）；Huawei R&D U.K.（华为英国研发中心）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）
The Schrödinger equation is at the heart of modern quantum mechanics. Since exact solutions of the ground state are typically intractable, standard approaches approximate Schrödinger equation as forms of nonlinear generalized eigenvalue problems $F(V)V = SVΛ$ in which $F(V)$, the matrix to be decomposed, is a function of its own top-$k$ smallest eigenvectors $V$, leading to a "self-consistency problem". Traditional iterative methods heavily rely on high-quality initial guesses of $V$ generated via domain-specific heuristics methods based on quantum mechanics. In this work, we eliminate such a …
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；University of Chinese Academy of Sciences（中国科学院大学）
In dynamic magnetic resonance (MR) imaging, low-rank plus sparse (L+S) decomposition, or robust principal component analysis (PCA), has achieved stunning performance. However, the selection of the parameters of L+S is empirical, and the acceleration rate is limited, which are common failings of iterative compressed sensing MR imaging (CS-MRI) reconstruction methods. Many deep learning approaches have been proposed to address these issues, but few of them use a low-rank prior. In this paper, a model-based low-rank plus sparse network, dubbed L+S-Net, is proposed for dynamic MR reconstruction. I…
University of Wisconsin, Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）；Microsoft Research（微软研究院）；Stanford University（斯坦福大学）
We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. Specifically, we are given a set $T$ of $n$ points in $\mathbb{R}^d$ and a parameter $0< α<\frac 1 2$ such that an $α$-fraction of the points in $T$ are i.i.d. samples from a well-behaved distribution $\mathcal{D}$ and the remaining $(1-α)$-fraction are arbitrary. The goal is to output a small list of vectors, at least one of which is close to the mean of $\mathcal{D}$. We develop new algorithms for list-decodable mean estimation, achieving nearly-optimal statistical guarantees, wi…
National University of Singapore（新加坡国立大学）；Arizona State University（亚利桑那州立大学）；Samsung Electronics（三星电子）；Georgia Institute of Technology（佐治亚理工学院）
Multivariate functional data from a complex system are naturally high-dimensional and have complex cross-correlation structure. The complexity of data structure can be observed as that (1) some functions are strongly correlated with similar features, while some others may have almost no cross-correlations with quite diverse features; and (2) the cross-correlation structure may also change over time due to the system evolution. With this regard, this paper presents a dynamic subspace learning method for multivariate functional data modeling. In particular, we consider different functions come f…
The virtual reference feedback tuning (VRFT) is a non-iterative data-driven (DD) method employed to tune a controller's parameters aiming to achieve a prescribed closed-loop performance. In its most common formulation, the parameters of a linearly parametrized controller are estimated by solving a least squares (LS) problem, which in the presence of noise leads to a biased estimate of the controller's parameters. To eliminate this bias, an instrumental variable (IV) variant of the method is usual, at the cost of increasing significantly the estimate's variance. In the present work, we propose …
