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 trimmed AE (TAE) is introduced to deal with data with large scale outliers. Extensive experiments on both synthetic and real-world datasets with vector-level or pixel-level outliers demonstrate that the proposed AE/TAE outperforms the state-of-the-art RPCA based methods.
Knowledge distillation (KD) is one of the most useful techniques for light-weight neural networks. Although neural networks have a clear purpose of embedding datasets into the low-dimensional space, the existing knowledge was quite far from this purpose and provided only limited information. We argue that good knowledge should be able to interpret the embedding procedure. This paper proposes a method of generating interpretable embedding procedure (IEP) knowledge based on principal component analysis, and distilling it based on a message passing neural network. Experimental results show that t…
Shanghai Institute for Advanced Communication and Data Science（上海先进通信与数据科学研究院）；Shanghai University（上海大学）
Learning a good speaker embedding is important for many automatic speaker recognition tasks, including verification, identification and diarization. The embeddings learned by softmax are not discriminative enough for open-set verification tasks. Angular based embedding learning target can achieve such discriminativeness by optimizing angular distance and adding margin penalty. We apply several different popular angular margin embedding learning strategies in this work and explicitly compare their performance on Voxceleb speaker recognition dataset. Observing the fact that encouraging inter-cla…
Ensembling word embeddings to improve distributed word representations has shown good success for natural language processing tasks in recent years. These approaches either carry out straightforward mathematical operations over a set of vectors or use unsupervised learning to find a lower-dimensional representation. This work compares meta-embeddings trained for different losses, namely loss functions that account for angular distance between the reconstructed embedding and the target and those that account normalized distances based on the vector length. We argue that meta-embeddings are bett…
Yale University（耶鲁大学）；Yale University School of Medicine（耶鲁大学医学院）
t-distributed Stochastic Neighborhood Embedding (t-SNE) is a method for dimensionality reduction and visualization that has become widely popular in recent years. Efficient implementations of t-SNE are available, but they scale poorly to datasets with hundreds of thousands to millions of high dimensional data-points. We present Fast Fourier Transform-accelerated Interpolation-based t-SNE (FIt-SNE), which dramatically accelerates the computation of t-SNE. The most time-consuming step of t-SNE is a convolution that we accelerate by interpolating onto an equispaced grid and subsequently using the…
Deep metric learning has attracted much attention in recent years, due to seamlessly combining the distance metric learning and deep neural network. Many endeavors are devoted to design different pair-based angular loss functions, which decouple the magnitude and direction information for embedding vectors and ensure the training and testing measure consistency. However, these traditional angular losses cannot guarantee that all the sample embeddings are on the surface of the same hypersphere during the training stage, which would result in unstable gradient in batch optimization and may influ…
In contrast to classical techniques for exploratory analysis of high-dimensional data sets, such as principal component analysis (PCA), neighbor embedding (NE) techniques tend to better preserve the local structure/topology of high-dimensional data. However, the ability to preserve local structure comes at the expense of interpretability: Techniques such as t-Distributed Stochastic Neighbor Embedding (t-SNE) or Uniform Manifold Approximation and Projection (UMAP) do not give insights into which input variables underlie the topological (cluster) structure seen in the corresponding embedding. We…
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…
Sharif University of Technology（谢里夫理工大学）；University of Tehran（德黑兰大学）；University of Western Australia（西澳大学）
Cardiovascular Disease (CVD) is considered as one of the principal causes of death in the world. Over recent years, this field of study has attracted researchers' attention to investigate heart sounds' patterns for disease diagnostics. In this study, an approach is proposed for normal/abnormal heart sound classification on the Physionet challenge 2016 dataset. For the first time, a fixed-length feature vector; called i-vector; is extracted from each heart sound using Mel Frequency Cepstral Coefficient (MFCC) features. Afterwards, Principal Component Analysis (PCA) transform and Variational Aut…
SAS Institute（SAS研究所）；Shanghai Jiao Tong University（上海交通大学）
Robust PCA methods are typically batch algorithms which requires loading all observations into memory before processing. This makes them inefficient to process big data. In this paper, we develop an efficient online robust principal component methods, namely online moving window robust principal component analysis (OMWRPCA). Unlike existing algorithms, OMWRPCA can successfully track not only slowly changing subspace but also abruptly changed subspace. By embedding hypothesis testing into the algorithm, OMWRPCA can detect change points of the underlying subspaces. Extensive simulation studies d…
Carnegie Mellon University（卡内基梅隆大学）；Georgia Institute of Technology（佐治亚理工学院）；Princeton University（普林斯顿大学）；IBM Research（IBM研究院）
We study the distributed computing setting in which there are multiple servers, each holding a set of points, who wish to compute functions on the union of their point sets. A key task in this setting is Principal Component Analysis (PCA), in which the servers would like to compute a low dimensional subspace capturing as much of the variance of the union of their point sets as possible. Given a procedure for approximate PCA, one can use it to approximately solve $\ell_2$-error fitting problems such as $k$-means clustering and subspace clustering. The essential properties of an approximate dist…
Graph Neural Networks (GNNs) have become a popular approach for various applications, ranging from social network analysis to modeling chemical properties of molecules. While GNNs often show remarkable performance on public datasets, they can struggle to learn long-range dependencies in the data due to over-smoothing and over-squashing tendencies. To alleviate this challenge, we propose PCAPass, a method which combines Principal Component Analysis (PCA) and message passing for generating node embeddings in an unsupervised manner and leverages gradient boosted decision trees for classification …
University of Pittsburgh Medical Center（匹兹堡大学医学中心）；University of Pittsburgh（匹兹堡大学）；Weill Cornell Medicine（威尔康奈尔医学院）
We consider the problem of interpretable network representation learning for samples of network-valued data. We propose the Principal Component Analysis for Networks (PCAN) algorithm to identify statistically meaningful low-dimensional representations of a network sample via subgraph count statistics. The PCAN procedure provides an interpretable framework for which one can readily visualize, explore, and formulate predictive models for network samples. We furthermore introduce a fast sampling-based algorithm, sPCAN, which is significantly more computationally efficient than its counterpart, bu…
Locally Linear Embedding (LLE) is a nonlinear spectral dimensionality reduction and manifold learning method. It has two main steps which are linear reconstruction and linear embedding of points in the input space and embedding space, respectively. In this work, we propose two novel generative versions of LLE, named Generative LLE (GLLE), whose linear reconstruction steps are stochastic rather than deterministic. GLLE assumes that every data point is caused by its linear reconstruction weights as latent factors. The proposed GLLE algorithms can generate various LLE embeddings stochastically wh…
Principal Component Analysis (PCA) is a ubiquitous tool with many applications in machine learning including feature construction, subspace embedding, and outlier detection. In this paper, we present an algorithm for computing the top principal components of a dataset with a large number of rows (examples) and columns (features). Our algorithm leverages both structured and unstructured random projections to retain good accuracy while being computationally efficient. We demonstrate the technique on the winning submission the KDD 2010 Cup.
Word embeddings are often used in natural language processing as a means to quantify relationships between words. More generally, these same word embedding techniques can be used to quantify relationships between features. In this paper, we first consider multiple different word embedding techniques within the context of malware classification. We use hidden Markov models to obtain embedding vectors in an approach that we refer to as HMM2Vec, and we generate vector embeddings based on principal component analysis. We also consider the popular neural network based word embedding technique known…
Carnegie Mellon University（卡内基梅隆大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）；IBM Research（IBM研究院）
Kernel Principal Component Analysis (KPCA) is a key machine learning algorithm for extracting nonlinear features from data. In the presence of a large volume of high dimensional data collected in a distributed fashion, it becomes very costly to communicate all of this data to a single data center and then perform kernel PCA. Can we perform kernel PCA on the entire dataset in a distributed and communication efficient fashion while maintaining provable and strong guarantees in solution quality? In this paper, we give an affirmative answer to the question by developing a communication efficient a…
Numerous dimensionality reduction problems in data analysis involve the recovery of low-dimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the number of samples in the neighborhood (sampling density) affect the performance of learning algorithms. In this work, we propose a theoretical analysis of local sampling conditions for the estimation of …
Carnegie Mellon University（卡内基梅隆大学）；University of Utah（犹他大学）
Word embeddings have become a staple of several natural language processing tasks, yet much remains to be understood about their properties. In this work, we analyze word embeddings in terms of their principal components and arrive at a number of novel and counterintuitive observations. In particular, we characterize the utility of variance explained by the principal components as a proxy for downstream performance. Furthermore, through syntactic probing of the principal embedding space, we show that the syntactic information captured by a principal component does not correlate with the amount…
University of Bergen（卑尔根大学）；Norwegian Computing Center（挪威计算中心）
There has been an intense recent activity in embedding of very high dimensional and nonlinear data structures, much of it in the data science and machine learning literature. We survey this activity in four parts. In the first part we cover nonlinear methods such as principal curves, multidimensional scaling, local linear methods, ISOMAP, graph based methods and diffusion mapping, kernel based methods and random projections. The second part is concerned with topological embedding methods, in particular mapping topological properties into persistence diagrams and the Mapper algorithm. Another t…
Dimensionality Reduction of Collective Motion by Principal Manifolds
Department of Mathematics, Clarkson University, Potsdam, NY-13699, USA.（数学系，克林顿大学，Potsdam分校）；Department of Mechanical and Aerospace Engineering, New York University Polytechnic School of Engineering, Brooklyn, NY-11201, USA.（机械与航空航天工程系，纽约大学理工学院，布鲁克林分校）
Adversarial training (AT) methods have been found to be effective against adversarial attacks on deep neural networks. Many variants of AT have been proposed to improve its performance. Pang et al. [1] have recently shown that incorporating hypersphere embedding (HE) into the existing AT procedures enhances robustness. We observe that the existing AT procedures are not designed for the HE framework, and thus fail to adequately learn the angular discriminative information available in the HE framework. In this paper, we propose integrating HE into AT with regularization terms that exploit the r…
The Hong Kong University of Science and Technology（香港科技大学）；Alibaba Group（阿里巴巴集团）；Huazhong University of Science and Technology（华中科技大学）；Nanjing University（南京大学）
Technische Universität Dresden（德累斯顿工业大学）；Tsinghua University（清华大学）
Analysis of 'big data' characterized by high-dimensionality such as word vectors and complex networks requires often their representation in a geometrical space by embedding. Recent developments in machine learning and network geometry have pointed out the hyperbolic space as a useful framework for the representation of this data derived by real complex physical systems. In the hyperbolic space, the radial coordinate of the nodes characterizes their hierarchy, whereas the angular distance between them represents their similarity. Several studies have highlighted the relationship between the an…
Learning embeddings of non-linear PDEs: the Burgers' equation
Departament de Física Quántica i Astrofí sica, Universitat de Barcelona（巴塞罗那大学量子物理与天体物理系）；Institut de Ciencies del Cosmos (ICC), Universitat de Barcelona（巴塞罗那大学天体物理研究所）；Institucio Catalana de Recerca i Estudis Avancats (ICREA)（加泰罗尼亚高级研究与高级教育研究所）
The State University of New York at Buffalo（纽约州立大学布法罗分校）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；University of Technology, Sydney（悉尼科技大学）
Many scientific datasets are of high dimension, and the analysis usually requires visual manipulation by retaining the most important structures of data. Principal curve is a widely used approach for this purpose. However, many existing methods work only for data with structures that are not self-intersected, which is quite restrictive for real applications. A few methods can overcome the above problem, but they either require complicated human-made rules for a specific task with lack of convergence guarantee and adaption flexibility to different tasks, or cannot obtain explicit structures of …
Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）
Correspondence analysis (CA) is a multivariate statistical tool used to visualize and interpret data dependencies by finding maximally correlated embeddings of pairs of random variables. CA has found applications in fields ranging from epidemiology to social sciences; however, current methods do not scale to large, high-dimensional datasets. In this paper, we provide a novel interpretation of CA in terms of an information-theoretic quantity called the principal inertia components. We show that estimating the principal inertia components, which consists in solving a functional optimization prob…
Angular Regularization for Positive-Unlabeled Learning on the Hypersphere
Athena RC Democritus University of Thrace（阿塔尼亚RC德摩克利特大学）；Athena RC University Campus Kimmeria（阿塔尼亚RC大学校园基米里亚）；Democritus University of Thrace Dept. Production and Management Engineering（德摩克利特大学生产与管理工程系）
