Recurrent Kernel Networks
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Large-Scale Kernel Ranksvm
Large-scale Kernel RankSVM Tzu-Ming Kuo∗ Ching-Pei Leey Chih-Jen Linz Abstract proposed [11, 23, 5, 1, 14]. However, for some tasks that Learning to rank is an important task for recommendation the feature set is not rich enough, nonlinear methods systems, online advertisement and web search. Among may be needed. Therefore, it is important to develop those learning to rank methods, rankSVM is a widely efficient training methods for large kernel rankSVM. used model. Both linear and nonlinear (kernel) rankSVM Assume we are given a set of training label-query- have been extensively studied, but the lengthy training instance tuples (yi; qi; xi); yi 2 R; qi 2 S ⊂ Z; xi 2 n time of kernel rankSVM remains a challenging issue. In R ; i = 1; : : : ; l, where S is the set of queries. By this paper, after discussing difficulties of training kernel defining the set of preference pairs as rankSVM, we propose an efficient method to handle these (1.1) P ≡ f(i; j) j q = q ; y > y g with p ≡ jP j; problems. The idea is to reduce the number of variables from i j i j quadratic to linear with respect to the number of training rankSVM [10] solves instances, and efficiently evaluate the pairwise losses. Our setting is applicable to a variety of loss functions. Further, 1 T X min w w + C ξi;j general optimization methods can be easily applied to solve w;ξ 2 (i;j)2P the reformulated problem. Implementation issues are also (1.2) subject to wT (φ (x ) − φ (x )) ≥ 1 − ξ ; carefully considered. -
A Kernel Method for Multi-Labelled Classification
A kernel method for multi-labelled classification Andre´ Elisseeff and Jason Weston BIOwulf Technologies, 305 Broadway, New York, NY 10007 andre,jason ¡ @barhilltechnologies.com Abstract This article presents a Support Vector Machine (SVM) like learning sys- tem to handle multi-label problems. Such problems are usually decom- posed into many two-class problems but the expressive power of such a system can be weak [5, 7]. We explore a new direct approach. It is based on a large margin ranking system that shares a lot of common proper- ties with SVMs. We tested it on a Yeast gene functional classification problem with positive results. 1 Introduction Many problems in Text Mining or Bioinformatics are multi-labelled. That is, each point in a learning set is associated to a set of labels. Consider for instance the classification task of determining the subjects of a document, or of relating one protein to its many effects on a cell. In either case, the learning task would be to output a set of labels whose size is not known in advance: one document can for instance be about food, meat and finance, although another one would concern only food and fat. Two-class and multi-class classification or ordinal regression problems can all be cast into multi-label ones. This makes the latter quite attractive but at the same time it gives a warning: their generality hides their difficulty to solve them. The number of publications is not going to contradict this statement: we are aware of only a few works about the subject [4, 5, 7] and they all concern text mining applications. -
Deep Kernel Learning
Deep Kernel Learning Andrew Gordon Wilson∗ Zhiting Hu∗ Ruslan Salakhutdinov Eric P. Xing CMU CMU University of Toronto CMU Abstract (1996), who had shown that Bayesian neural net- works with infinitely many hidden units converged to Gaussian processes with a particular kernel (co- We introduce scalable deep kernels, which variance) function. Gaussian processes were subse- combine the structural properties of deep quently viewed as flexible and interpretable alterna- learning architectures with the non- tives to neural networks, with straightforward learn- parametric flexibility of kernel methods. ing procedures. Where neural networks used finitely Specifically, we transform the inputs of a many highly adaptive basis functions, Gaussian pro- spectral mixture base kernel with a deep cesses typically used infinitely many fixed basis func- architecture, using local kernel interpolation, tions. As argued by MacKay (1998), Hinton et al. inducing points, and structure exploit- (2006), and Bengio (2009), neural networks could ing (Kronecker and Toeplitz) algebra for automatically discover meaningful representations in a scalable kernel representation. These high-dimensional data by learning multiple layers of closed-form kernels can be used as drop-in highly adaptive basis functions. By contrast, Gaus- replacements for standard kernels, with ben- sian processes with popular kernel functions were used efits in expressive power and scalability. We typically as simple smoothing devices. jointly learn the properties of these kernels through the marginal likelihood of a Gaus- Recent approaches (e.g., Yang et al., 2015; Lloyd et al., sian process. Inference and learning cost 2014; Wilson, 2014; Wilson and Adams, 2013) have (n) for n training points, and predictions demonstrated that one can develop more expressive costO (1) per test point. -
Density Based Data Clustering
California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 3-2015 Density Based Data Clustering Rayan Albarakati California State University - San Bernardino Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Other Computer Engineering Commons Recommended Citation Albarakati, Rayan, "Density Based Data Clustering" (2015). Electronic Theses, Projects, and Dissertations. 134. https://scholarworks.lib.csusb.edu/etd/134 This Project is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of Graduate Studies 3-2015 Density Based Data Clustering Rayan Albarakati Follow this and additional works at: http://scholarworks.lib.csusb.edu/etd This Project is brought to you for free and open access by the Office of Graduate Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected], [email protected]. DESNITY BASED DATA CLUSTERING A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Science in Computer Science by Rayan Albarakati March 2015 DESNITY BASED DATA CLUSTERING A Project Presented to the Faculty of California State University, San Bernardino by Rayan Albarakati March 2015 Approved by: Haiyan Qiao, Advisor, School of Computer Date Science and Engineering Owen J.Murphy Krestin Voigt © 2015 Rayan Albarakati ABSTRACT Data clustering is a data analysis technique that groups data based on a measure of similarity. -
A User's Guide to Support Vector Machines
A User's Guide to Support Vector Machines Asa Ben-Hur Jason Weston Department of Computer Science NEC Labs America Colorado State University Princeton, NJ 08540 USA Abstract The Support Vector Machine (SVM) is a widely used classifier. And yet, obtaining the best results with SVMs requires an understanding of their workings and the various ways a user can influence their accuracy. We provide the user with a basic understanding of the theory behind SVMs and focus on their use in practice. We describe the effect of the SVM parameters on the resulting classifier, how to select good values for those parameters, data normalization, factors that affect training time, and software for training SVMs. 1 Introduction The Support Vector Machine (SVM) is a state-of-the-art classification method introduced in 1992 by Boser, Guyon, and Vapnik [1]. The SVM classifier is widely used in bioinformatics (and other disciplines) due to its high accuracy, ability to deal with high-dimensional data such as gene ex- pression, and flexibility in modeling diverse sources of data [2]. SVMs belong to the general category of kernel methods [4, 5]. A kernel method is an algorithm that depends on the data only through dot-products. When this is the case, the dot product can be replaced by a kernel function which computes a dot product in some possibly high dimensional feature space. This has two advantages: First, the ability to generate non-linear decision boundaries using methods designed for linear classifiers. Second, the use of kernel functions allows the user to apply a classifier to data that have no obvious fixed-dimensional vector space representation. -
The Forgetron: a Kernel-Based Perceptron on a Fixed Budget
The Forgetron: A Kernel-Based Perceptron on a Fixed Budget Ofer Dekel Shai Shalev-Shwartz Yoram Singer School of Computer Science & Engineering The Hebrew University, Jerusalem 91904, Israel oferd,shais,singer @cs.huji.ac.il { } Abstract The Perceptron algorithm, despite its simplicity, often performs well on online classification problems. The Perceptron becomes especially effec- tive when it is used in conjunction with kernels. However, a common dif- ficulty encountered when implementing kernel-based online algorithms is the amount of memory required to store the online hypothesis, which may grow unboundedly. In this paper we describe and analyze a new in- frastructure for kernel-based learning with the Perceptron while adhering to a strict limit on the number of examples that can be stored. We first describe a template algorithm, called the Forgetron, for online learning on a fixed budget. We then provide specific algorithms and derive a uni- fied mistake bound for all of them. To our knowledge, this is the first online learning paradigm which, on one hand, maintains a strict limit on the number of examples it can store and, on the other hand, entertains a relative mistake bound. We also present experiments with real datasets which underscore the merits of our approach. 1 Introduction The introduction of the Support Vector Machine (SVM) [7] sparked a widespread interest in kernel methods as a means of solving (binary) classification problems. Although SVM was initially stated as a batch-learning technique, it significantly influenced the develop- ment of kernel methods in the online-learning setting. Online classification algorithms that can incorporate kernels include the Perceptron [6], ROMMA [5], ALMA [3], NORMA [4] and the Passive-Aggressive family of algorithms [1]. -
Kernel Methods Through the Roof: Handling Billions of Points Efficiently
Kernel methods through the roof: handling billions of points efficiently Giacomo Meanti Luigi Carratino MaLGa, DIBRIS MaLGa, DIBRIS Università degli Studi di Genova Università degli Studi di Genova [email protected] [email protected] Lorenzo Rosasco Alessandro Rudi MaLGa, DIBRIS, IIT & MIT INRIA - École Normale Supérieure Università degli Studi di Genova PSL Research University [email protected] [email protected] Abstract Kernel methods provide an elegant and principled approach to nonparametric learning, but so far could hardly be used in large scale problems, since naïve imple- mentations scale poorly with data size. Recent advances have shown the benefits of a number of algorithmic ideas, for example combining optimization, numerical linear algebra and random projections. Here, we push these efforts further to develop and test a solver that takes full advantage of GPU hardware. Towards this end, we designed a preconditioned gradient solver for kernel methods exploiting both GPU acceleration and parallelization with multiple GPUs, implementing out-of-core variants of common linear algebra operations to guarantee optimal hardware utilization. Further, we optimize the numerical precision of different operations and maximize efficiency of matrix-vector multiplications. As a result we can experimentally show dramatic speedups on datasets with billions of points, while still guaranteeing state of the art performance. Additionally, we make our software available as an easy to use library1. 1 Introduction Kernel methods provide non-linear/non-parametric extensions of many classical linear models in machine learning and statistics [45, 49]. The data are embedded via a non-linear map into a high dimensional feature space, so that linear models in such a space effectively define non-linear models in the original space. -
Kernel Methods As Before We Assume a Space X of Objects and a Feature Map Φ : X → RD
Kernel Methods As before we assume a space X of objects and a feature map Φ : X → RD. We also assume training data hx1, y1i,... hxN , yN i and we define the data matrix Φ by defining Φt,i to be Φi(xt). In this section we assume L2 regularization and consider only training algorithms of the following form. N ∗ X 1 2 w = argmin Lt(w) + λ||w|| (1) w 2 t=1 Lt(w) = L(yt, w · Φ(xt)) (2) We have written Lt = L(yt, w · Φ) rather then L(mt(w)) because we now want to allow the case of regression where we have y ∈ R as well as classification where we have y ∈ {−1, 1}. Here we are interested in methods for solving (1) for D >> N. An example of D >> N would be a database of one thousand emails where for each email xt we have that Φ(xt) has a feature for each word of English so that D is roughly 100 thousand. We are interested in the case where inner products of the form K(x, y) = Φ(x)·Φ(y) can be computed efficiently independently of the size of D. This is indeed the case for email messages with feature vectors with a feature for each word of English. 1 The Kernel Method We can reduce ||w|| while holding L(yt, w · Φ(xt)) constant (for all t) by remov- ing any the component of w orthoganal to all vectors Φ(xt). Without loss of ∗ generality we can therefore assume that w is in the span of the vectors Φ(xt). -
3.2 the Periodogram
3.2 The periodogram The periodogram of a time series of length n with sampling interval ∆is defined as 2 ∆ n In(⌫)= (Xt X)exp( i2⇡⌫t∆) . n − − t=1 X In words, we compute the absolute value squared of the Fourier transform of the sample, that is we consider the squared amplitude and ignore the phase. Note that In is periodic with period 1/∆andthatIn(0) = 0 because we have centered the observations at the mean. The centering has no e↵ect for Fourier frequencies ⌫ = k/(n∆), k =0. 6 By mutliplying out the absolute value squared on the right, we obtain n n n 1 ∆ − I (⌫)= (X X)(X X)exp( i2⇡⌫(t s)∆)=∆ γ(h)exp( i2⇡⌫h). n n t − s − − − − t=1 s=1 h= n+1 X X X− Hence the periodogram is nothing else than the Fourier transform of the estimatedb acf. In the following, we assume that ∆= 1 in order to simplify the formula (although for applications the value of ∆in the original time scale matters for the interpretation of frequencies). By the above result, the periodogram seems to be the natural estimator of the spectral density 1 s(⌫)= γ(h)exp( i2⇡⌫h). − h= X1 However, a closer inspection shows that the periodogram has two serious shortcomings: It has large random fluctuations, and also a bias which can be large. We first consider the bias. Using the spectral representation, we see that up to a term which involves E(X ) X t − 2 1 n 2 i2⇡(⌫ ⌫0)t i⇡(n+1)(⌫ ⌫0) In(⌫)= e− − Z(d⌫0) = n e− − Dn(⌫ ⌫0)Z(d⌫0) . -
An Algorithmic Introduction to Clustering
AN ALGORITHMIC INTRODUCTION TO CLUSTERING APREPRINT Bernardo Gonzalez Department of Computer Science and Engineering UCSC [email protected] June 11, 2020 The purpose of this document is to provide an easy introductory guide to clustering algorithms. Basic knowledge and exposure to probability (random variables, conditional probability, Bayes’ theorem, independence, Gaussian distribution), matrix calculus (matrix and vector derivatives), linear algebra and analysis of algorithm (specifically, time complexity analysis) is assumed. Starting with Gaussian Mixture Models (GMM), this guide will visit different algorithms like the well-known k-means, DBSCAN and Spectral Clustering (SC) algorithms, in a connected and hopefully understandable way for the reader. The first three sections (Introduction, GMM and k-means) are based on [1]. The fourth section (SC) is based on [2] and [5]. Fifth section (DBSCAN) is based on [6]. Sixth section (Mean Shift) is, to the best of author’s knowledge, original work. Seventh section is dedicated to conclusions and future work. Traditionally, these five algorithms are considered completely unrelated and they are considered members of different families of clustering algorithms: • Model-based algorithms: the data are viewed as coming from a mixture of probability distributions, each of which represents a different cluster. A Gaussian Mixture Model is considered a member of this family • Centroid-based algorithms: any data point in a cluster is represented by the central vector of that cluster, which need not be a part of the dataset taken. k-means is considered a member of this family • Graph-based algorithms: the data is represented using a graph, and the clustering procedure leverage Graph theory tools to create a partition of this graph. -
Knowledge-Based Systems Layer-Constrained Variational
Knowledge-Based Systems 196 (2020) 105753 Contents lists available at ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys Layer-constrained variational autoencoding kernel density estimation model for anomaly detectionI ∗ Peng Lv b, Yanwei Yu a,b, , Yangyang Fan c, Xianfeng Tang d, Xiangrong Tong b a Department of Computer Science and Technology, Ocean University of China, Qingdao, Shandong 266100, China b School of Computer and Control Engineering, Yantai University, Yantai, Shandong 264005, China c Shanghai Key Lab of Advanced High-Temperature Materials and Precision Forming, Shanghai Jiao Tong University, Shanghai 200240, China d College of Information Sciences and Technology, The Pennsylvania State University, University Park, PA 16802, USA article info a b s t r a c t Article history: Unsupervised techniques typically rely on the probability density distribution of the data to detect Received 16 October 2019 anomalies, where objects with low probability density are considered to be abnormal. However, Received in revised form 5 March 2020 modeling the density distribution of high dimensional data is known to be hard, making the problem of Accepted 7 March 2020 detecting anomalies from high-dimensional data challenging. The state-of-the-art methods solve this Available online 10 March 2020 problem by first applying dimension reduction techniques to the data and then detecting anomalies Keywords: in the low dimensional space. Unfortunately, the low dimensional space does not necessarily preserve Anomaly detection the density distribution of the original high dimensional data. This jeopardizes the effectiveness of Variational autoencoder anomaly detection. In this work, we propose a novel high dimensional anomaly detection method Kernel density estimation called LAKE. -
Lecture 2: Density Estimation 2.1 Histogram
STAT 535: Statistical Machine Learning Autumn 2019 Lecture 2: Density Estimation Instructor: Yen-Chi Chen Main reference: Section 6 of All of Nonparametric Statistics by Larry Wasserman. A book about the methodologies of density estimation: Multivariate Density Estimation: theory, practice, and visualization by David Scott. A more theoretical book (highly recommend if you want to learn more about the theory): Introduction to Nonparametric Estimation by A.B. Tsybakov. Density estimation is the problem of reconstructing the probability density function using a set of given data points. Namely, we observe X1; ··· ;Xn and we want to recover the underlying probability density function generating our dataset. 2.1 Histogram If the goal is to estimate the PDF, then this problem is called density estimation, which is a central topic in statistical research. Here we will focus on the perhaps simplest approach: histogram. For simplicity, we assume that Xi 2 [0; 1] so p(x) is non-zero only within [0; 1]. We also assume that p(x) is smooth and jp0(x)j ≤ L for all x (i.e. the derivative is bounded). The histogram is to partition the set [0; 1] (this region, the region with non-zero density, is called the support of a density function) into several bins and using the count of the bin as a density estimate. When we have M bins, this yields a partition: 1 1 2 M − 2 M − 1 M − 1 B = 0; ;B = ; ; ··· ;B = ; ;B = ; 1 : 1 M 2 M M M−1 M M M M In such case, then for a given point x 2 B`, the density estimator from the histogram will be n number of observations within B` 1 M X p (x) = × = I(X 2 B ): bM n length of the bin n i ` i=1 The intuition of this density estimator is that the histogram assign equal density value to every points within 1 Pn the bin.