We present a novel approach for adaptive, differentiable parameterization of large-scale random fields. If the approach is coupled with any gradient-based optimization algorithm, it can be applied to a variety of optimization problems, including history matching. The developed technique is based on principal component analysis (PCA) but modifies a purely data-driven basis of principal components considering objective function behavior. To define an efficient encoding, Gradient-Sensitive PCA uses an objective function gradient with respect to model parameters. We propose computationally efficient implementations of the technique, and two of them are based on stationary perturbation theory (SPT). Optimality, correctness, and low computational costs of the new encoding approach are tested, verified, and discussed. Three algorithms for optimal parameter decomposition are presented and applied to an objective of 2D synthetic history matching. The results demonstrate improvements in encoding quality regarding objective function minimization and distributional patterns of the desired field. Possible applications and extensions are proposed.
PCA is often used in anomaly detection and statistical process control tasks. For bivariate data, we prove that the minor projection (the least varying projection) of the PCA-rotated data is the most sensitive to distributional changes, where sensitivity is defined by the Hellinger distance between distributions before and after a change. In particular, this is almost always the case if only one parameter of the bivariate normal distribution changes, i.e., the change is sparse. Simulations indicate that the minor projections are the most sensitive for a large range of changes and pre-change se…
College of Information Sciences and Technology（信息科学与技术学院）；School of Electrical Engineering and Computer Science（电气工程与计算机科学学院）；The Pennsylvania State University（宾夕法尼亚州立大学）；Microsoft Bing（微软必应）
It has been shown that dimension reduction methods such as PCA may be inherently prone to unfairness and treat data from different sensitive groups such as race, color, sex, etc., unfairly. In pursuit of fairness-enhancing dimensionality reduction, using the notion of Pareto optimality, we propose an adaptive first-order algorithm to learn a subspace that preserves fairness, while slightly compromising the reconstruction loss. Theoretically, we provide sufficient conditions that the solution of the proposed algorithm belongs to the Pareto frontier for all sensitive groups; thereby, the optimal…
Flatiron Institute（熨斗研究院）；NYU Langone Medical School（纽约大学朗格尼医学院）
Finding informative low-dimensional representations that can be computed efficiently in large datasets is an important problem in data analysis. Recently, contrastive Principal Component Analysis (cPCA) was proposed as a more informative generalization of PCA that takes advantage of contrastive learning. However, the performance of cPCA is sensitive to hyper-parameter choice and there is currently no online algorithm for implementing cPCA. Here, we introduce a modified cPCA method, which we denote cPCA*, that is more interpretable and less sensitive to the choice of hyper-parameter. We derive …
Online dimension reduction is a common method for high-dimensional streaming data processing. Online principal component analysis, online sliced inverse regression, online kernel principal component analysis and other methods have been studied in depth, but as far as we know, online supervised nonlinear dimension reduction methods have not been fully studied. In this article, an online kernel sliced inverse regression method is proposed. By introducing the approximate linear dependence condition and dictionary variable sets, we address the problem of increasing variable dimensions with the sam…
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 consider principal component analysis (PCA) in decomposable Gaussian graphical models. We exploit the prior information in these models in order to distribute its computation. For this purpose, we reformulate the problem in the sparse inverse covariance (concentration) domain and solve the global eigenvalue problem using a sequence of local eigenvalue problems in each of the cliques of the decomposable graph. We demonstrate the application of our methodology in the context of decentralized anomaly detection in the Abilene backbone network. Based on the topology of the network, we propose an…
University of Tsukuba（筑波大学）；The Institute of Statistical Mathematics（统计数理研究所）
Principal component analysis (PCA) is a widely used method for data processing, such as for dimension reduction and visualization. Standard PCA is known to be sensitive to outliers, and thus, various robust PCA methods have been proposed. It has been shown that the robustness of many statistical methods can be improved using mode estimation instead of mean estimation, because mode estimation is not significantly affected by the presence of outliers. Thus, this study proposes a modal principal component analysis (MPCA), which is a robust PCA method based on mode estimation. The proposed method …
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…
This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the exi…
This paper considers the sparse generalized eigenvalue problem (SGEP), which aims to find the leading eigenvector with at most $k$ nonzero entries. SGEP naturally arises in many applications in machine learning, statistics, and scientific computing, for example, the sparse principal component analysis (SPCA), the sparse discriminant analysis (SDA), and the sparse canonical correlation analysis (SCCA). In this paper, we focus on the development of a three-stage algorithm named {\em inverse-free truncated Rayleigh-Ritz method} ({\em IFTRR}) to efficiently solve SGEP. In each iteration of IFTRR, …
Carnegie Mellon University（卡内基梅隆大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）
Performance of nuclear threat detection systems based on gamma-ray spectrometry often strongly depends on the ability to identify the part of measured signal that can be attributed to background radiation. We have successfully applied a method based on Principal Component Analysis (PCA) to obtain a compact null-space model of background spectra using PCA projection residuals to derive a source detection score. We have shown the method's utility in a threat detection system using mobile spectrometers in urban scenes (Tandon et al 2012). While it is commonly assumed that measured photon counts f…
This is a tutorial and survey paper on various methods for Sufficient Dimension Reduction (SDR). We cover these methods with both statistical high-dimensional regression perspective and machine learning approach for dimensionality reduction. We start with introducing inverse regression methods including Sliced Inverse Regression (SIR), Sliced Average Variance Estimation (SAVE), contour regression, directional regression, Principal Fitted Components (PFC), Likelihood Acquired Direction (LAD), and graphical regression. Then, we introduce forward regression methods including Principal Hessian Dir…
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…
Thales Land and Air Systems（泰雷兹地面与空中系统公司）；Université Paris-Saclay（巴黎萨克雷大学）；CentraleSupélec（中央理工-高等电力学院）；Dataiku（Dataiku公司）
Machine Learning models are prone to fail when test data are different from training data, a situation often encountered in real applications known as distribution shift. While still valid, the training-time knowledge becomes less effective, requiring a test-time adaptation to maintain high performance. Following approaches that assume batch-norm layer and use their statistics for adaptation, we propose a Test-Time Adaptation with Principal Component Analysis (TTAwPCA), which presumes a fitted PCA and adapts at test time a spectral filter based on the singular values of the PCA for robustness …
Understanding and predicting the electric consumption patterns in the short-, mid- and long-term, at the distribution and transmission level, is a fundamental asset for smart grids infrastructure planning, dynamic network reconfiguration, dynamic energy pricing and savings, and thus energy efficiency. This work introduces the Functional Principal Component Analysis (FPCA) as a versatile method to both investigate and predict, at different level of spatial aggregation, the consumption patterns. The method was applied to a unique and sensitive dataset that includes electric consumption and contr…
The estimation of the covariance matrix is an initial step in many multivariate statistical methods such as principal components analysis and factor analysis, but in many practical applications the dimensionality of the sample space is large compared to the number of samples, and the usual maximum likelihood estimate is poor. Typically, improvements are obtained by modelling or regularization. From a practical point of view, these methods are often computationally heavy and rely on approximations. As a fast substitute, we propose an easily calculable maximum a posteriori (MAP) estimator based …
Motivation: Although principal component analysis is frequently applied to reduce the dimensionality of matrix data, the method is sensitive to noise and bias and has difficulty with comparability and interpretation. These issues are addressed by improving the fidelity to the study design. Principal axes and the components for variables are found through the arrangement of the training data set, and the centers of data are found according to the design. By using both the axes and the center, components for an observation that belong to various studies can be separately estimated. Both of the c…
In this article, we consider the problem of recovering the underlying trajectory when the longitudinal data are sparsely and irregularly observed and noise-contaminated. Such data are popularly analyzed with functional principal component analysis via the Principal Analysis by Conditional Estimation (PACE) method. The PACE method may sometimes be numerically unstable because it involves the inverse of the covariance matrix. We propose a sparse orthonormal approximation (SOAP) method as an alternative. It estimates the optimal empirical basis functions in the best approximation framework rather…
Principal component analysis (PCA) is an important tool in exploring data. The conventional approach to PCA leads to a solution which favours the structures with large variances. This is sensitive to outliers and could obfuscate interesting underlying structures. One of the equivalent definitions of PCA is that it seeks the subspaces that maximize the sum of squared pairwise distances between data projections. This definition opens up more flexibility in the analysis of principal components which is useful in enhancing PCA. In this paper we introduce scales into PCA by maximizing only the sum …
Due to the demand for tackling the problem of streaming data with high dimensional covariates, we propose an online sparse sliced inverse regression (OSSIR) method for online sufficient dimension reduction. The existing online sufficient dimension reduction methods focus on the case when the dimension $p$ is small. In this article, we show that our method can achieve better statistical accuracy and computation speed when the dimension $p$ is large. There are two important steps in our method, one is to extend the online principal component analysis to iteratively obtain the eigenvalues and eig…
CSIE Department, National Taiwan University（台湾大学计算机科学与信息工程学系（CSIE Department, National Taiwan University）/ 台湾大学（National Taiwan University））
We study multi-label classification (MLC) with three important real-world issues: online updating, label space dimensional reduction (LSDR), and cost-sensitivity. Current MLC algorithms have not been designed to address these three issues simultaneously. In this paper, we propose a novel algorithm, cost-sensitive dynamic principal projection (CS-DPP) that resolves all three issues. The foundation of CS-DPP is an online LSDR framework derived from a leading LSDR algorithm. In particular, CS-DPP is equipped with an efficient online dimension reducer motivated by matrix stochastic gradient, and e…
Classical game-theoretic approaches for multi-agent systems in both the forward policy design problem and the inverse reward learning problem often make strong rationality assumptions: agents perfectly maximize expected utilities under uncertainties. Such assumptions, however, substantially mismatch with observed humans' behaviors such as satisficing with sub-optimal, risk-seeking, and loss-aversion decisions. In this paper, we investigate the problem of bounded risk-sensitive Markov Game (BRSMG) and its inverse reward learning problem for modeling human realistic behaviors and learning human …
University of Washington（华盛顿大学）；University of California, Berkeley（加州大学伯克利分校）
We address the problem of inverse reinforcement learning in Markov decision processes where the agent is risk-sensitive. In particular, we model risk-sensitivity in a reinforcement learning framework by making use of models of human decision-making having their origins in behavioral psychology, behavioral economics, and neuroscience. We propose a gradient-based inverse reinforcement learning algorithm that minimizes a loss function defined on the observed behavior. We demonstrate the performance of the proposed technique on two examples, the first of which is the canonical Grid World example a…
Graduate School of Informatics, Kyoto University（京都大学信息学研究生院）；ATR Cognitive Mechanisms Laboratories（ATR认知机制实验室）
We present a novel algorithm (Principal Sensitivity Analysis; PSA) to analyze the knowledge of the classifier obtained from supervised machine learning techniques. In particular, we define principal sensitivity map (PSM) as the direction on the input space to which the trained classifier is most sensitive, and use analogously defined k-th PSM to define a basis for the input space. We train neural networks with artificial data and real data, and apply the algorithm to the obtained supervised classifiers. We then visualize the PSMs to demonstrate the PSA's ability to decompose the knowledge acqu…
University of California, San Diego（加州大学圣地亚哥分校）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
Principal components analysis (PCA) is a standard tool for identifying good low-dimensional approximations to data in high dimension. Many data sets of interest contain private or sensitive information about individuals. Algorithms which operate on such data should be sensitive to the privacy risks in publishing their outputs. Differential privacy is a framework for developing tradeoffs between privacy and the utility of these outputs. In this paper we investigate the theory and empirical performance of differentially private approximations to PCA and propose a new method which explicitly opti…
Carnegie Mellon University（卡内基梅隆大学）；University of Michigan（密歇根大学）；Xi’an Jiaotong University（西安交通大学）
One typical assumption in inverse reinforcement learning (IRL) is that human experts act to optimize the expected utility of a stochastic cost with a fixed distribution. This assumption deviates from actual human behaviors under ambiguity. Risk-sensitive inverse reinforcement learning (RS-IRL) bridges such gap by assuming that humans act according to a random cost with respect to a set of subjectively distorted distributions instead of a fixed one. Such assumption provides the additional flexibility to model human's risk preferences, represented by a risk envelope, in safe-critical tasks. Howe…
Given a data matrix $\mathbf{A} \in \mathbb{R}^{n \times d}$, principal component projection (PCP) and principal component regression (PCR), i.e. projection and regression restricted to the top-eigenspace of $\mathbf{A}$, are fundamental problems in machine learning, optimization, and numerical analysis. In this paper we provide the first algorithms that solve these problems in nearly linear time for fixed eigenvalue distribution and large n. This improves upon previous methods which have superlinear running times when both the number of top eigenvalues and inverse gap between eigenspaces is l…
The literature on Inverse Reinforcement Learning (IRL) typically assumes that humans take actions in order to minimize the expected value of a cost function, i.e., that humans are risk neutral. Yet, in practice, humans are often far from being risk neutral. To fill this gap, the objective of this paper is to devise a framework for risk-sensitive IRL in order to explicitly account for a human's risk sensitivity. To this end, we propose a flexible class of models based on coherent risk measures, which allow us to capture an entire spectrum of risk preferences from risk-neutral to worst-case. We …
