This paper considers the problem of estimating the principal eigenvector of a covariance matrix from independent and identically distributed data samples in streaming settings. The streaming rate of data in many contemporary applications can be high enough that a single processor cannot finish an iteration of existing methods for eigenvector estimation before a new sample arrives. This paper formulates and analyzes a distributed variant of the classical Krasulina's method (D-Krasulina) that can keep up with the high streaming rate of data by distributing the computational load across multiple processing nodes. The analysis shows that---under appropriate conditions---D-Krasulina converges to the principal eigenvector in an order-wise optimal manner; i.e., after receiving $M$ samples across all nodes, its estimation error can be $O(1/M)$. In order to reduce the network communication overhead, the paper also develops and analyzes a mini-batch extension of D-Krasulina, which is termed DM-Krasulina. The analysis of DM-Krasulina shows that it can also achieve order-optimal estimation error rates under appropriate conditions, even when some samples have to be discarded within the network due to communication latency. Finally, experiments are performed over synthetic and real-world data to validate the convergence behaviors of D-Krasulina and DM-Krasulina in high-rate streaming settings.
George Washington University（乔治华盛顿大学）；National Institute of Standards and Technology（美国国家标准与技术研究院）
The movement of large quantities of data during the training of a Deep Neural Network presents immense challenges for machine learning workloads. To minimize this overhead, especially on the movement and calculation of gradient information, we introduce streaming batch principal component analysis as an update algorithm. Streaming batch principal component analysis uses stochastic power iterations to generate a stochastic k-rank approximation of the network gradient. We demonstrate that the low rank updates produced by streaming batch principal component analysis can effectively train convolut…
Kernel principal component analysis (KPCA) provides a concise set of basis vectors which capture non-linear structures within large data sets, and is a central tool in data analysis and learning. To allow for non-linear relations, typically a full $n \times n$ kernel matrix is constructed over $n$ data points, but this requires too much space and time for large values of $n$. Techniques such as the Nyström method and random feature maps can help towards this goal, but they do not explicitly maintain the basis vectors in a stream and take more space than desired. We propose a new approach for s…
Sequential or online dimensional reduction is of interests due to the explosion of streaming data based applications and the requirement of adaptive statistical modeling, in many emerging fields, such as the modeling of energy end-use profile. Principal Component Analysis (PCA), is the classical way of dimensional reduction. However, traditional Singular Value Decomposition (SVD) based PCA fails to model data which largely deviates from Gaussian distribution. The Bregman Divergence was recently introduced to achieve a generalized PCA framework. If the random variable under dimensional reductio…
Neural network models of early sensory processing typically reduce the dimensionality of streaming input data. Such networks learn the principal subspace, in the sense of principal component analysis (PCA), by adjusting synaptic weights according to activity-dependent learning rules. When derived from a principled cost function these rules are nonlocal and hence biologically implausible. At the same time, biologically plausible local rules have been postulated rather than derived from a principled cost function. Here, to bridge this gap, we derive a biologically plausible network for subspace …
Target Corporation（塔吉特公司）；University of Michigan（密歇根大学）；Rutgers University–New Brunswick（罗格斯大学新伯朗士威分校）
Emerging applications of machine learning in numerous areas involve continuous gathering of and learning from streams of data. Real-time incorporation of streaming data into the learned models is essential for improved inference in these applications. Further, these applications often involve data that are either inherently gathered at geographically distributed entities or that are intentionally distributed across multiple machines for memory, computational, and/or privacy reasons. Training of models in this distributed, streaming setting requires solving stochastic optimization problems in a…
We study the statistical and computational aspects of kernel principal component analysis using random Fourier features and show that under mild assumptions, $O(\sqrt{n} \log n)$ features suffices to achieve $O(1/ε^2)$ sample complexity. Furthermore, we give a memory efficient streaming algorithm based on classical Oja's algorithm that achieves this rate.
In this paper we propose a new algorithm for streaming principal component analysis. With limited memory, small devices cannot store all the samples in the high-dimensional regime. Streaming principal component analysis aims to find the $k$-dimensional subspace which can explain the most variation of the $d$-dimensional data points that come into memory sequentially. In order to deal with large $d$ and large $N$ (number of samples), most streaming PCA algorithms update the current model using only the incoming sample and then dump the information right away to save memory. However the informat…
Principal components analysis (PCA) is a widely used dimension reduction technique with an extensive range of applications. In this paper, an online distributed algorithm is proposed for recovering the principal eigenspaces. We further establish its rate of convergence and show how it relates to the number of nodes employed in the distributed computation, the effective rank of the data matrix under consideration, and the gap in the spectrum of the underlying population covariance matrix. The proposed algorithm is illustrated on low-rank approximation and $\boldsymbol{k}$-means clustering tasks…
Principal Component Analysis (PCA) is a fundamental data preprocessing tool in the world of machine learning. While PCA is often thought of as a dimensionality reduction method, the purpose of PCA is actually two-fold: dimension reduction and uncorrelated feature learning. Furthermore, the enormity of the dimensions and sample size in the modern day datasets have rendered the centralized PCA solutions unusable. In that vein, this paper reconsiders the problem of PCA when data samples are distributed across nodes in an arbitrarily connected network. While a few solutions for distributed PCA exi…
Institute for Interdisciplinary Information Sciences（交叉信息研究院）；Tsinghua University（清华大学）
We study the robust principal component analysis (RPCA) problem in a distributed setting. The goal of RPCA is to find an underlying low-rank estimation for a raw data matrix when the data matrix is subject to the corruption of gross sparse errors. Previous studies have developed RPCA algorithms that provide stable solutions with fast convergence. However, these algorithms are typically hard to scale and cannot be implemented distributedly, due to the use of either SVD or large matrix multiplication. In this paper, we propose the first distributed robust principal analysis algorithm based on co…
Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are uncorrelated. Furthermore, the ever-increasing volume of data in the modern world often requires storage of data samples across multiple machines, which precludes the use of centralized PCA algorithms. This paper focuses on the dual objective of PCA, namely, dimensionality reduction and decorrelation of features, but in a distributed setting. This requires estimati…
Oja's algorithm has been the cornerstone of streaming methods in Principal Component Analysis (PCA) since it was first proposed in 1982. However, Oja's algorithm does not have a standardized choice of learning rate (step size) that both performs well in practice and truly conforms to the online streaming setting. In this paper, we propose a new learning rate scheme for Oja's method called AdaOja. This new algorithm requires only a single pass over the data and does not depend on knowing properties of the data set a priori. AdaOja is a novel variation of the Adagrad algorithm to Oja's algorithm…
New York University（纽约大学）；Princeton University（普林斯顿大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
The growing size of modern data sets brings many challenges to the existing statistical estimation approaches, which calls for new distributed methodologies. This paper studies distributed estimation for a fundamental statistical machine learning problem, principal component analysis (PCA). Despite the massive literature on top eigenvector estimation, much less is presented for the top-$L$-dim ($L>1$) eigenspace estimation, especially in a distributed manner. We propose a novel multi-round algorithm for constructing top-$L$-dim eigenspace for distributed data. Our algorithm takes advantage of …
Principal component analysis (PCA) has been a prominent tool for high-dimensional data analysis. Online algorithms that estimate the principal component by processing streaming data are of tremendous practical and theoretical interests. Despite its rich applications, theoretical convergence analysis remains largely open. In this paper, we cast online PCA into a stochastic nonconvex optimization problem, and we analyze the online PCA algorithm as a stochastic approximation iteration. The stochastic approximation iteration processes data points incrementally and maintains a running estimate of t…
Principal Subspace Analysis (PSA) -- and its sibling, Principal Component Analysis (PCA) -- is one of the most popular approaches for dimensionality reduction in signal processing and machine learning. But centralized PSA/PCA solutions are fast becoming irrelevant in the modern era of big data, in which the number of samples and/or the dimensionality of samples often exceed the storage and/or computational capabilities of individual machines. This has led to the study of distributed PSA/PCA solutions, in which the data are partitioned across multiple machines and an estimate of the principal s…
Technion - Israel Institute of Technology（以色列理工学院）；Weizmann Institute of Science（魏茨曼科学研究学院）；Toyotoa Technological Institute at Chicago（芝加哥丰田技术研究所）
We study the fundamental problem of Principal Component Analysis in a statistical distributed setting in which each machine out of $m$ stores a sample of $n$ points sampled i.i.d. from a single unknown distribution. We study algorithms for estimating the leading principal component of the population covariance matrix that are both communication-efficient and achieve estimation error of the order of the centralized ERM solution that uses all $mn$ samples. On the negative side, we show that in contrast to results obtained for distributed estimation under convexity assumptions, for the PCA object…
Princeton University（普林斯顿大学）；Tencent AI Lab（腾讯人工智能实验室）
In this paper, we propose to adopt the diffusion approximation tools to study the dynamics of Oja's iteration which is an online stochastic gradient descent method for the principal component analysis. Oja's iteration maintains a running estimate of the true principal component from streaming data and enjoys less temporal and spatial complexities. We show that the Oja's iteration for the top eigenvector generates a continuous-state discrete-time Markov chain over the unit sphere. We characterize the Oja's iteration in three phases using diffusion approximation and weak convergence tools. Our t…
Columbia University（哥伦比亚大学）；Colorado State University（科罗拉多州立大学）；Stony Brook University（石溪大学）；Brookhaven National Laboratory（布鲁克海文国家实验室）
We validate the use of matrix factorization for the automatic identification of relevant components from atomic pair distribution function (PDF) data. We also present a newly developed software infrastructure for analyzing the PDF data arriving in streaming manner. We then apply two matrix factorization techniques, Principal Component Analysis (PCA) and Non-negative Matrix Factorization (NMF), to study simulated and experiment datasets in the context of in situ experiment.
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…
Generalized principal component analysis (GLM-PCA) facilitates dimension reduction of non-normally distributed data. We provide a detailed derivation of GLM-PCA with a focus on optimization. We also demonstrate how to incorporate covariates, and suggest post-processing transformations to improve interpretability of latent factors.
Principal component analysis (PCA) is fundamental to statistical machine learning. It extracts latent principal factors that contribute to the most variation of the data. When data are stored across multiple machines, however, communication cost can prohibit the computation of PCA in a central location and distributed algorithms for PCA are thus needed. This paper proposes and studies a distributed PCA algorithm: each node machine computes the top $K$ eigenvectors and transmits them to the central server; the central server then aggregates the information from all the node machines and conduct…
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…
University of Michigan（密歇根大学）；Carnegie Mellon University（卡内基梅隆大学）；Harvard University（哈佛大学）
For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data problem, such that only a small fraction of data attributes are observed. These complications impose significant, and unconventional, constraints on the problem of streaming Principal Component Analysis (PCA) and subspace tracking, which is an essential building block for many infer…
Fan et al. [$\mathit{Annals}$ $\mathit{of}$ $\mathit{Statistics}$ $\textbf{47}$(6) (2019) 3009-3031] constructed a distributed principal component analysis (PCA) algorithm to reduce the communication cost between multiple servers significantly. However, their algorithm's guarantee is only for sub-Gaussian data. Spurred by this deficiency, this paper enhances the effectiveness of their distributed PCA algorithm by utilizing robust covariance matrix estimators of Minsker [$\mathit{Annals}$ $\mathit{of}$ $\mathit{Statistics}$ $\textbf{46}$(6A) (2018) 2871-2903] and Ke et al. [$\mathit{Statistical…
We study the problem of recovering the subspace spanned by the first $k$ principal components of $d$-dimensional data under the streaming setting, with a memory bound of $O(kd)$. Two families of algorithms are known for this problem. The first family is based on the framework of stochastic gradient descent. Nevertheless, the convergence rate of the family can be seriously affected by the learning rate of the descent steps and deserves more serious study. The second family is based on the power method over blocks of data, but setting the block size for its existing algorithms is not an easy tas…
Hong Kong Baptist University（香港浸会大学）；Xidian University（西安电子科技大学）
Despite enormous research interest and rapid application of federated learning (FL) to various areas, existing studies mostly focus on supervised federated learning under the horizontally partitioned local dataset setting. This paper will study the unsupervised FL under the vertically partitioned dataset setting. Accordingly, we propose the federated principal component analysis for vertically partitioned dataset (VFedPCA) method, which reduces the dimensionality across the joint datasets over all the clients and extracts the principal component feature information for downstream data analysis…
Sparse principal component analysis (PCA) is an important technique for dimensionality reduction of high-dimensional data. However, most existing sparse PCA algorithms are based on non-convex optimization, which provide little guarantee on the global convergence. Sparse PCA algorithms based on a convex formulation, for example the Fantope projection and selection (FPS), overcome this difficulty, but are computationally expensive. In this work we study sparse PCA based on the convex FPS formulation, and propose a new algorithm that is computationally efficient and applicable to large and high-d…
University of Science and Technology of China（中国科学技术大学）；Adobe Research（奥多比研究院）；Stonybrook University（石溪大学）；University of Washington（华盛顿大学）
Principal Component Analysis (PCA) is a widely used technique in machine learning, data analysis and signal processing. With the increase in the size and complexity of datasets, it has become important to develop low-space usage algorithms for PCA. Streaming PCA has gained significant attention in recent years, as it can handle large datasets efficiently. The kernel method, which is commonly used in learning algorithms such as Support Vector Machines (SVMs), has also been applied in PCA algorithms. We propose a streaming algorithm for Kernel PCA problems based on the traditional scheme by Oja.…
Oja's rule [Oja, Journal of mathematical biology 1982] is a well-known biologically-plausible algorithm using a Hebbian-type synaptic update rule to solve streaming principal component analysis (PCA). Computational neuroscientists have known that this biological version of Oja's rule converges to the top eigenvector of the covariance matrix of the input in the limit. However, prior to this work, it was open to prove any convergence rate guarantee. In this work, we give the first convergence rate analysis for the biological version of Oja's rule in solving streaming PCA. Moreover, our convergen…
Flatiron Institute（弗莱提伦研究所）；NYU Langone Medical Center（纽约大学朗格尼医学中心）
Big data problems frequently require processing datasets in a streaming fashion, either because all data are available at once but collectively are larger than available memory or because the data intrinsically arrive one data point at a time and must be processed online. Here, we introduce a computationally efficient version of similarity matching, a framework for online dimensionality reduction that incrementally estimates the top K-dimensional principal subspace of streamed data while keeping in memory only the last sample and the current iterate. To assess the performance of our approach, …
Correlated anomaly detection (CAD) from streaming data is a type of group anomaly detection and an essential task in useful real-time data mining applications like botnet detection, financial event detection, industrial process monitor, etc. The primary approach for this type of detection in previous researches is based on principal score (PS) of divided batches or sliding windows by computing top eigenvalues of the correlation matrix, e.g. the Lanczos algorithm. However, this paper brings up the phenomenon of principal score degeneration for large data set, and then mathematically and practic…
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Minnesota（明尼苏达大学）；The Chinese University of Hong Kong（香港中文大学）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Distributed learning has become a critical enabler of the massively connected world envisioned by many. This article discusses four key elements of scalable distributed processing and real-time intelligence --- problems, data, communication and computation. Our aim is to provide a fresh and unique perspective about how these elements should work together in an effective and coherent manner. In particular, we {provide a selective review} about the recent techniques developed for optimizing non-convex models (i.e., problem classes), processing batch and streaming data (i.e., data types), over th…
This paper presents a methodology for creating streaming, distributed inference algorithms for Bayesian nonparametric (BNP) models. In the proposed framework, processing nodes receive a sequence of data minibatches, compute a variational posterior for each, and make asynchronous streaming updates to a central model. In contrast to previous algorithms, the proposed framework is truly streaming, distributed, asynchronous, learning-rate-free, and truncation-free. The key challenge in developing the framework, arising from the fact that BNP models do not impose an inherent ordering on their compon…
EPFL（洛桑联邦理工学院）；American University of Beirut（贝鲁特美国大学）；Université de Nice Sophia-Antipolis（尼斯索菲亚安提波利斯大学）；Northwestern Polytechnical University（西北工业大学）
The problem of learning simultaneously several related tasks has received considerable attention in several domains, especially in machine learning with the so-called multitask learning problem or learning to learn problem [1], [2]. Multitask learning is an approach to inductive transfer learning (using what is learned for one problem to assist in another problem) and helps improve generalization performance relative to learning each task separately by using the domain information contained in the training signals of related tasks as an inductive bias. Several strategies have been derived with…
Wayne State University（韦恩州立大学）；Rutgers University–New Brunswick（罗格斯大学新布朗斯维克分校）
Motivated by machine learning applications in networks of sensors, internet-of-things (IoT) devices, and autonomous agents, we propose techniques for distributed stochastic convex learning from high-rate data streams. The setup involves a network of nodes---each one of which has a stream of data arriving at a constant rate---that solve a stochastic convex optimization problem by collaborating with each other over rate-limited communication links. To this end, we present and analyze two algorithms---termed distributed stochastic approximation mirror descent (D-SAMD) and accelerated distributed …
KTH Royal Institute of Technology（瑞典皇家理工学院）；Hive Streaming AB（Hive Streaming AB公司）；Maastricht University（马斯特里赫特大学）
Live video streaming has become a mainstay as a standard communication solution for several enterprises worldwide. To efficiently stream high-quality live video content to a large amount of offices, companies employ distributed video streaming solutions which rely on prior knowledge of the underlying evolving enterprise network. However, such networks are highly complex and dynamic. Hence, to optimally coordinate the live video distribution, the available network capacity between viewers has to be accurately predicted. In this paper we propose a graph representation learning technique on weigh…
The principal support vector machines method (Li et al., 2011) is a powerful tool for sufficient dimension reduction that replaces original predictors with their low-dimensional linear combinations without loss of information. However, the computational burden of the principal support vector machines method constrains its use for massive data. To address this issue, we in this paper propose two distributed estimation algorithms for fast implementation when the sample size is large. Both the two distributed sufficient dimension reduction estimators enjoy the same statistical efficiency as mergi…
Indian Institute of Technology Delhi（德里印度理工学院）；University of Vienna（维也纳大学）；University of Washington（华盛顿大学）
Fueled by massive data, important decision making is being automated with the help of algorithms, therefore, fairness in algorithms has become an especially important research topic. In this work, we design new streaming and distributed algorithms for the fair $k$-center problem that models fair data summarization. The streaming and distributed models of computation have an attractive feature of being able to handle massive data sets that do not fit into main memory. Our main contributions are: (a) the first distributed algorithm; which has provably constant approximation ratio and is extremel…
Hamad Bin Khalifa University（哈马德·本·哈利法大学）；Qatar University（卡塔尔大学）；Comsats Institute of Information Technology（COMSATS信息技术学院）
Crowdsourced live video streaming (livecast) services such as Facebook Live, YouNow, Douyu and Twitch are gaining more momentum recently. Allocating the limited resources in a cost-effective manner while maximizing the Quality of Service (QoS) through real-time delivery and the provision of the appropriate representations for all viewers is a challenging problem. In our paper, we introduce a machine-learning based predictive resource allocation framework for geo-distributed cloud sites, considering the delay and quality constraints to guarantee the maximum QoS for viewers and the minimum cost …
In this paper, we study the problems of principal Generalized Eigenvector computation and Canonical Correlation Analysis in the stochastic setting. We propose a simple and efficient algorithm, Gen-Oja, for these problems. We prove the global convergence of our algorithm, borrowing ideas from the theory of fast-mixing Markov chains and two-time-scale stochastic approximation, showing that it achieves the optimal rate of convergence. In the process, we develop tools for understanding stochastic processes with Markovian noise which might be of independent interest.
Google Inc.（谷歌公司）；Carnegie Mellon University（卡内基梅隆大学）；Singapore Mgt. University（新加坡管理大学）
Traditional graph-based semi-supervised learning (SSL) approaches, even though widely applied, are not suited for massive data and large label scenarios since they scale linearly with the number of edges $|E|$ and distinct labels $m$. To deal with the large label size problem, recent works propose sketch-based methods to approximate the distribution on labels per node thereby achieving a space reduction from $O(m)$ to $O(\log m)$, under certain conditions. In this paper, we present a novel streaming graph-based SSL approximation that captures the sparsity of the label distribution and ensures …
Community detection is an important research topic in graph analytics that has a wide range of applications. A variety of static community detection algorithms and quality metrics were developed in the past few years. However, most real-world graphs are not static and often change over time. In the case of streaming data, communities in the associated graph need to be updated either continuously or whenever new data streams are added to the graph, which poses a much greater challenge in devising good community detection algorithms for maintaining dynamic graphs over streaming data. In this pap…
FiLark: a streaming-first software framework for end-to-end exploration, annotation, and algorithm integration in distributed acoustic sensing
organization= State Key Laboratory of Ocean Sensing \& Ocean College, Zhejiang University , city= Zhoushan , postcode= 316021 , country= China；organization= College of Information Science；Electronic Engineering, Zhejiang University , city= Hangzhou , country= China；organization= College of Computer Science；Technology, Zhejiang University , city= Hangzhou , country= China
We consider the problems of robust PAC learning from distributed and streaming data, which may contain malicious errors and outliers, and analyze their fundamental complexity questions. In particular, we establish lower bounds on the communication complexity for distributed robust learning performed on multiple machines, and on the space complexity for robust learning from streaming data on a single machine. These results demonstrate that gaining robustness of learning algorithms is usually at the expense of increased complexities. As far as we know, this work gives the first complexity result…
Low-tubal-rank tensor approximation has been proposed to analyze large-scale and multi-dimensional data. However, finding such an accurate approximation is challenging in the streaming setting, due to the limited computational resources. To alleviate this issue, this paper extends a popular matrix sketching technique, namely Frequent Directions, for constructing an efficient and accurate low-tubal-rank tensor approximation from streaming data based on the tensor Singular Value Decomposition (t-SVD). Specifically, the new algorithm allows the tensor data to be observed slice by slice, but only …
Cross Layer Optimization and Distributed Reinforcement Learning for Wireless 360° Video Streaming
King Fahd University of Petroleum and Minerals (KFUPM)（法赫德国王石油与矿产大学）；Telefónica Research（西班牙电信研究院）；University of the Basque Country (UPV/EHU)（巴斯克大学）；University of Oulu（奥卢大学）；Purdue University（普渡大学）
