DOCSLIB.ORG

Linear Manifold Regularization for Large Scale Semi-Supervised Learning

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Manifold Regularization for Large Scale Semi-Supervised Learning Linear Manifold Regularization for Large Scale Semi-supervised Learning Vikas Sindhwani [email protected] Partha Niyogi [email protected] Mikhail Belkin [email protected] Department of Computer Science, University of Chicago, Hyde Park, Chicago, IL 60637 Sathiya Keerthi [email protected] Yahoo! Research Labs, 210 S Delacey Avenue, Pasadena, CA 91105 Abstract tion (Belkin, Niyogi & Sindhwani, 2004; Sindhwani, The enormous wealth of unlabeled data in Niyogi and Belkin, 2005). In this framework, unla- many applications of machine learning is be- beled data is incorporated within a geometrically mo- ginning to pose challenges to the designers tivated regularizer. We specialize this framework for of semi-supervised learning methods. We linear classification, and focus on the problem of deal- are interested in developing linear classifica- ing with large amounts of data. tion algorithms to efficiently learn from mas- This paper is organized as follows: In section 2, we sive partially labeled datasets. In this paper, setup the problem of linear semi-supervised learning we propose Linear Laplacian Support Vector and discuss some prior work. The general Manifold Machines and Linear Laplacian Regularized Regularization framework is reviewed in section 3. In Least Squares as promising solutions to this section 4, we discuss our methods. Section 5 outlines problem. some research in progress. 1. Introduction 2. Linear Semi-supervised Learning The problem of linear semi-supervised clasification is A single web-crawl by search engines like Yahoo and setup as follows: We are given l labeled examples Google indexes billions of webpages. Only a very l l+u fxi; yigi=1 and u unlabeled examples fxigi=l+1, where small fraction of these web-pages can be hand-labeled d and assembled into topic directories. The remaining the patterns x 2 X ⊂ R are d-dimensional vectors web-pages form a massive collection of unlabeled doc- and the labels yi 2 {−1; +1g represent two classes. We are interested in developing tractable algorithms uments. Text categorization is among an increasing T range of applications that stand to immensely benefit to learn a linear classifier f(x) = sign(w x + b) from the development of large scale semi-supervised given by a weight vector w and a threshold b, in the learning methods. case where u is very large. We can hope for tractability when the size of the data matrix is still manageable, In this paper, we propose linear semi-supervised clas- e.g when dealing with low dimensional problems, or sification algorithms for dealing with large datasets. with very sparse high-dimensional problems (e.g text Linear methods have traditionally played a pivotal role categorization). in the development of machine learning algorithms. They have been pervasively deployed in information A number of recent efforts have considered semi- retrieval, data analysis and pattern recognition sys- supervised extensions of well-established supervised tems. methods e.g Support Vector Machines (SVM). Our algorithms are rooted in a general framework for Transductive SVMs (Joachims, 1999) implement the semi-supervised learning called Manifold Regulariza- following philosophy for using unlabeled data to choose a weight vector: Find a weight vector and a labeling Appearing in Proc. of the 22 st ICML Workshop on Learn- of the unlabeled examples so that the data (both la- ing with Partially Classified Training Data, Bonn, Ger- beled and unlabeled examples) is separated with max- many, August 2005. Copyright 2005 by the author(s). imum margin. This requires a joint optimization of 80 the SVM objective function over possible choices of ing rests on how much information unlabeled exam- binary-valued labels on the unlabeled data and the ples carry about the distribution of labels in the pat- weight vector. tern space. Frequently, unlabeled examples may help in identifying clusters or a low-dimensional manifold structure along which labels can be assumed to vary Transductive SVM: smoothly. These are the cluster and manifold assump- l tion respectively. 1 (w∗; b∗) = argmin wT w + C V (y ; wT x + b) 2 l X i i The Manifold regularization framework extends the w;b i=1 yl+1::: yl+u classical framework of regularization in Reproducing l+u Kernel Hilbert Spaces (RKHS) to exploit the geome- T +Cu X V (yi; w xi + b) try of the marginal distribution, as estimated by unla- i=l+1 beled data, when these assumptions are satisfied. Un- labeled data is incorporated via a regularization term where V (y; f) = (1 − yf)+ is the hinge loss and C ; C l u (in addition to the RKHS norm) whose role is to pe- are real-valued parameters that balance the hinge loss nalize functions for changing rapidly along a manifold between the labeled and unlabeled examples. Thus, or within a cluster. TSVM minimizes regularized hinge loss over choices of labels for the unlabeled examples. This optimiza- The empirical estimate of these underlying structures tion is implemented in (Joachims, 1999) by first using can be encoded as a graph whose vertices are the an inductive SVM to label the unlabeled data and then labeled and unlabeled data points and whose edge l+u iteratively solving SVM quadratic programs. At each weights fWij gi;j=1 represent appropriate pairwise sim- step labels are switched to improve the objective func- ilarity relationships between examples. Given this tion. This procedure is susceptible to local minima graph, one can bias learning in an RKHS towards and requires an unknown, possibly large number of functions that vary smoothly along the edges (along label switches before converging. Thus, this approach the underlying structure). The following optimization does not seem to be well-suited to large scale problems. problem is solved over an RKHS HK with kernel func- tion K(x; y) and loss function V : Semi-supervised SVMs (S3VM) (Bennett & Demirez, 1998) use a similar objective function optimized us- ing mixed integer programming { Assume both posi- Manifold Regularization: tive and negative labels for each labeled examples and l choose the label that incurs smaller total hinge loss: ∗ 1 2 f = argmin X V (xi; yi; f) + γAkfkK 2H l Semi-supervised SVM: f K i=1 + l l u 2 ∗ ∗ 1 T T +γ (f(x ) − f(x )) W (w ; b ) = argmin w w + Cl X V (yi; w xi + b) + I X i j ij 2 =1 w;b i=1 i;j l+u Here, γA; γI are regularization parameters that control T T Cu X min V (−1; w xi + b); V (+1; w xi + b) the RKHS norm and the intrinsic norm respectively. i=l+1 T Defining, f^ = [f(x1); : : : ; f(xl+u)] , and L as the The 1-norm of the weight vector is preferred in (Ben- Laplacian matrix of the graph, given by L = D − W nett & Demirez, 1998); the optimization found to be where D is the diagnonal degree matrix given by D l+u W intractable even for moderate amounts of unlabeled ii = Pj=1 ij , we re-write the above problem as: data. (Fung & Mangasarian, 2001) reformulate this l approach as a concave minimization problem which is ∗ 1 2 ^T ^ f = argmin X V (xi; yi; f) + γAkfkK + γI f Lf 2H l solved by a successive linear approximation algorithm. f K i=1 These methods are applied to relatively small sized One can also take powers of the Laplacian Lp to de- problems. fine the graph regularizer. For more on graph regu- larization, see (Belkin,Matveeva & Niyogi; Kondor & 3. Manifold Regularization Lafferty,2002). Before discussing our algorithms, we briefly discuss A version of the Representer theorem can be easily the intuition and algorithmic framework of Manifold proved that shows that the minimizer has the following Regularization. The success of semi-supervised learn- form: 81 Here Xl is the submatrix of X corresponding to la- + l u beled examples and Y is the vector of labels. This is f(x) = X αiK(xi; x) (1) a d × d system which can be easily solved when d is i=1 small. When d is large but feature vctors are highly This reduces the optimization problem in manifold sparse with d¯ number of non-zero entries, we can em- regularization into a finite dimensional problem of es- ploy Conjugate Gradient (CG) methods to solve this timating the (l + u) expansion coefficients αi. The system. CG techniques are Krylov methods that solve algorithms that estimate these coefficients and con- a system Ax = b by repeatedly multiplying a candi- struct the optimal function are called Laplacian RLS date solution z by A. In the case of linear Laplacian and Laplacian SVM for the squared loss and hinge RLS, we can construct the matrix-vector product in loss respectively. Both algorithms involve inverting the LHS of (3) in time O(n(d¯+pk)),where p (typically (l + u) × (l + u) dense Gram matrices so that the com- very small) is the power of the Laplacian matrix (if plexity of a naive implementation is O(l + u)3 which Lp is used as the graph regularizer), and k (typically is prohibitive for large datasets. Experiments showing small) is average number of entries per column in L. state-of-the-art semi-supervised learning performance This is done by using an intermediate vector Xw and with small to moderate number of unlabeled examples appropriately forming sparse matrix-vector products. is reported in (Sindhwani, Niyogi and Belkin, 2005). Thus, the algorithm can employ very well-developed methods for efficiently obtaining the solution. 4. Linear Manifold Regularization 4.2. Linear Laplacian SVM Linear manifold regularization specializes the above al- To solve the optimization problem 4 for the hinge gorithms to linear functions resulting in the following loss, we adopt a different strategy (which also works optimization problem: for Linear Laplacian RLS).
Load more

Recommended publications
  • ECS289: Scalable Machine Learning
    ECS289: Scalable Machine Learning Cho-Jui Hsieh UC Davis Nov 3, 2015 Outline PageRank Semi-supervised Learning Label Propagation Manifold regularization PageRank Ranking Websites Text based ranking systems (a dominated approach in the early 90s) Compute the similarity between query and websites (documents) Keywords are a very limited way to express a complex information Need to rank websites by \popularity", \authority", . PageRank: Developed by Brin and Page (1999) Determine the authority and popularity by hyperlinks PageRank Main idea: estimate the ranking of websites by the link structure. Topology of Websites Transform the hyperlinks to a directed graph: The adjacency matrix A such that Aij = 1 if page j points to page i Transition Matrix Normalize the adjacency matrix so that the matrix is a stochastic matrix (each column sum up to 1) Pij : probability that arriving at page i from page j P: a stochastic matrix or a transition matrix Random walk: step 1 Random walk through the transition matrix Start from [1; 0; 0; 0] (can use any initialization) Random walk: step 1 Random walk through the transition matrix xt+1 = Pxt Random walk: step 2 Random walk through the transition matrix Random walk: step 2 Random walk through the transition matrix xt+2 = Pxt+1 PageRank (convergence) P Start from an initial vector x with i xi = 1 (the initial distribution) For t = 1; 2;::: xt+1 = Pxt Each xt is a probability distribution (sums up to 1) Converges to a stationary distribution π such that π = Pπ if P satisfies the following two conditions: t 1 P is irreducible: for all i; j, there exists some t such that (P )i;j > 0 t 2 P is aperiodic: for all i; j, we have gcdft :(P )i;j > 0g = 1 π is the unique right eigenvector of P with eigenvalue 1.
    [Show full text]
  • Semi-Supervised Classification Using Local and Global Regularization
    Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008) Semi- supervised Classification Using Local and Global Regularization Fei Wang1, Tao Li2, Gang Wang3, Changshui Zhang1 1Department of Automation, Tsinghua University, Beijing, China 2School of Computing and Information Sciences, Florida International University, Miami, FL, USA 3Microsoft China Research, Beijing, China Abstract works concentrate on the derivation of different forms of In this paper, we propose a semi-supervised learning (SSL) smoothness regularizers, such as the ones using combinato- algorithm based on local and global regularization. In the lo- rial graph Laplacian (Zhu et al., 2003)(Belkin et al., 2006), cal regularization part, our algorithm constructs a regularized normalized graph Laplacian (Zhou et al., 2004), exponen- classifier for each data point using its neighborhood, while tial/iterative graph Laplacian (Belkin et al., 2004), local lin- the global regularization part adopts a Laplacian regularizer ear regularization (Wang & Zhang, 2006)and local learning to smooth the data labels predicted by those local classifiers. regularization (Wu & Sch¨olkopf, 2007), but rarely touch the We show that some existing SSL algorithms can be derived problem of how to derive a more efficient loss function. from our framework. Finally we present some experimental In this paper, we argue that rather than applying a global results to show the effectiveness of our method. loss function which is based on the construction of a global predictor using the whole data set, it would be more desir- Introduction able to measure such loss locally by building some local pre- Semi-supervised learning (SSL), which aims at learning dictors for different regions of the input data space.
    [Show full text]
  • Semi-Supervised Deep Learning Using Improved Unsupervised Discriminant Projection?
    Semi-Supervised Deep Learning Using Improved Unsupervised Discriminant Projection? Xiao Han∗, Zihao Wang∗, Enmei Tu, Gunnam Suryanarayana, and Jie Yang School of Electronics, Information and Electrical Engineering, Shanghai Jiao Tong University, China [email protected], [email protected], [email protected] Abstract. Deep learning demands a huge amount of well-labeled data to train the network parameters. How to use the least amount of labeled data to obtain the desired classification accuracy is of great practical significance, because for many real-world applications (such as medical diagnosis), it is difficult to obtain so many labeled samples. In this paper, modify the unsupervised discriminant projection algorithm from dimen- sion reduction and apply it as a regularization term to propose a new semi-supervised deep learning algorithm, which is able to utilize both the local and nonlocal distribution of abundant unlabeled samples to improve classification performance. Experiments show that given dozens of labeled samples, the proposed algorithm can train a deep network to attain satisfactory classification results. Keywords: Manifold Regularization · Semi-supervised learning · Deep learning. 1 Introduction In reality, one of the main difficulties faced by many machine learning tasks is manually tagging large amounts of data. This is especially prominent for deep learning, which usually demands a huge number of well-labeled samples. There- fore, how to use the least amount of labeled data to train a deep network has become an important topic in the area. To overcome this problem, researchers proposed that the use of a large number of unlabeled data can extract the topol- ogy of the overall data's distribution.
    [Show full text]
  • Hyper-Parameter Optimization for Manifold Regularization Learning
    Cassiano Ot´avio Becker Hyper-parameter Optimization for Manifold Regularization Learning Otimiza¸cao~ de Hiperparametros^ para Aprendizado do Computador por Regulariza¸cao~ em Variedades Campinas 2013 i ii Universidade Estadual de Campinas Faculdade de Engenharia Eletrica´ e de Computa¸cao~ Cassiano Ot´avio Becker Hyper-parameter Optimization for Manifold Regularization Learning Otimiza¸cao~ de Hiperparametros^ para Aprendizado do Computador por Regulariza¸cao~ em Variedades Master dissertation presented to the School of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Master in Electrical Engineering. Concentration area: Automation. Disserta¸c~ao de Mestrado apresentada ao Programa de P´os- Gradua¸c~ao em Engenharia El´etrica da Faculdade de Engenharia El´etrica e de Computa¸c~ao da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obten¸c~ao do t´ıtulo de Mestre em Engenharia El´etrica. Area´ de concentra¸c~ao: Automa¸c~ao. Orientador (Advisor): Prof. Dr. Paulo Augusto Valente Ferreira Este exemplar corresponde `a vers~ao final da disserta¸c~aodefendida pelo aluno Cassiano Ot´avio Becker, e orientada pelo Prof. Dr. Paulo Augusto Valente Ferreira. Campinas 2013 iii Ficha catalográfica Universidade Estadual de Campinas Biblioteca da Área de Engenharia e Arquitetura Rose Meire da Silva - CRB 8/5974 Becker, Cassiano Otávio, 1977- B388h BecHyper-parameter optimization for manifold regularization learning / Cassiano Otávio Becker. – Campinas, SP : [s.n.], 2013. BecOrientador: Paulo Augusto Valente Ferreira. BecDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação. Bec1. Aprendizado do computador. 2. Aprendizado semi-supervisionado. 3. Otimização matemática. 4.
    [Show full text]
  • Zero Shot Learning Via Multi-Scale Manifold Regularization
    Zero Shot Learning via Multi-Scale Manifold Regularization Shay Deutsch 1,2 Soheil Kolouri 3 Kyungnam Kim 3 Yuri Owechko 3 Stefano Soatto 1 1 UCLA Vision Lab, University of California, Los Angeles, CA 90095 2 Department of Mathematics, University of California Los Angeles 3 HRL Laboratories, LLC, Malibu, CA {shaydeu@math., soatto@cs.}ucla.edu {skolouri, kkim, yowechko,}@hrl.com Abstract performed in a transductive setting, using all unlabeled data where the classification and learning process on the transfer We address zero-shot learning using a new manifold data is entirely unsupervised. alignment framework based on a localized multi-scale While our suggested approach is similar in scope to other transform on graphs. Our inference approach includes a ”visual-semantic alignment” methods (see [5, 10] and ref- smoothness criterion for a function mapping nodes on a erences therein) it is, to the best of our knowledge, the first graph (visual representation) onto a linear space (semantic to use multi-scale localized representations that respect the representation), which we optimize using multi-scale graph global (non-flat) geometry of the data space and the fine- wavelets. The robustness of the ensuing scheme allows us scale structure of the local semantic representation. More- to operate with automatically generated semantic annota- over, learning the relationship between visual features and tions, resulting in an algorithm that is entirely free of man- semantic attributes is unified into a single process, whereas ual supervision, and yet improves the state-of-the-art as in most ZSL approaches it is divided into a number of inde- measured on benchmark datasets.
    [Show full text]
  • Semi-Supervised Deep Metric Learning Networks for Classification of Polarimetric SAR Data
    remote sensing Letter Semi-Supervised Deep Metric Learning Networks for Classification of Polarimetric SAR Data Hongying Liu , Ruyi Luo, Fanhua Shang * , Xuechun Meng, Shuiping Gou and Biao Hou Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, School of Artificial Intelligence, Xidian University, Xi’an 710071, China; [email protected] (H.L.); [email protected] (R.L.); [email protected] (X.M.); [email protected] (S.G.); [email protected] (B.H.) * Corresponding authors: [email protected] Received: 7 April 2020; Accepted: 14 May 2020; Published: 17 May 2020 Abstract: Recently, classification methods based on deep learning have attained sound results for the classification of Polarimetric synthetic aperture radar (PolSAR) data. However, they generally require a great deal of labeled data to train their models, which limits their potential real-world applications. This paper proposes a novel semi-supervised deep metric learning network (SSDMLN) for feature learning and classification of PolSAR data. Inspired by distance metric learning, we construct a network, which transforms the linear mapping of metric learning into the non-linear projection in the layer-by-layer learning. With the prior knowledge of the sample categories, the network also learns a distance metric under which all pairs of similarly labeled samples are closer and dissimilar samples have larger relative distances. Moreover, we introduce a new manifold regularization to reduce the distance between neighboring samples since they are more likely to be homogeneous. The categorizing is achieved by using a simple classifier. Several experiments on both synthetic and real-world PolSAR data from different sensors are conducted and they demonstrate the effectiveness of SSDMLN with limited labeled samples, and SSDMLN is superior to state-of-the-art methods.
    [Show full text]
  • Manifold Regularization and Semi-Supervised Learning: Some Theoretical Analyses
    Manifold Regularization and Semi-supervised Learning: Some Theoretical Analyses Partha Niyogi∗ January 22, 2008 Abstract Manifold regularization (Belkin, Niyogi, Sindhwani, 2004)isage- ometrically motivated framework for machine learning within which several semi-supervised algorithms have been constructed. Here we try to provide some theoretical understanding of this approach. Our main result is to expose the natural structure of a class of problems on which manifold regularization methods are helpful. We show that for such problems, no supervised learner can learn effectively. On the other hand, a manifold based learner (that knows the manifold or “learns” it from unlabeled examples) can learn with relatively few labeled examples. Our analysis follows a minimax style with an em- phasis on finite sample results (in terms of n: the number of labeled examples). These results allow us to properly interpret manifold reg- ularization and its potential use in semi-supervised learning. 1 Introduction A learning problem is specified by a probability distribution p on X × Y according to which labelled examples zi = (xi, yi) pairs are drawn and presented to a learning algorithm (estimation procedure). We are interested in an understanding of the case in which X = IRD, ∗Departments of Computer Science, Statistics, The University of Chicago 1 Y ⊂ IR but pX (the marginal distribution of p on X) is supported on some submanifold M ⊂ X. In particular, we are interested in understanding how knowledge of this submanifold may potentially help a learning algorithm1. To this end, we will consider two kinds of learning algorithms: 1. Algorithms that have no knowledge of the submanifold M but learn from (xi, yi) pairs in a purely supervised way.
    [Show full text]
  • A Graph Based Approach to Semi-Supervised Learning
    A graph based approach to semi-supervised learning Michael Lim 1 Feb 2011 Michael Lim A graph based approach to semi-supervised learning Two papers M. Belkin, P. Niyogi, and V Sindhwani. Manifold regularization: a geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research 1-48, 2006. M. Belkin, P. Niyogi. Towards a Theoretical Foundation for Laplacian Based Manifold Methods. Journal of Computer and System Sciences, 2007. Michael Lim A graph based approach to semi-supervised learning What is semi-supervised learning? Prediction, but with the help of unsupervised examples. Michael Lim A graph based approach to semi-supervised learning Practical reasons: unlabeled data cheap More natural model of human learning Why semi-supervised learning? Michael Lim A graph based approach to semi-supervised learning More natural model of human learning Why semi-supervised learning? Practical reasons: unlabeled data cheap Michael Lim A graph based approach to semi-supervised learning Why semi-supervised learning? Practical reasons: unlabeled data cheap More natural model of human learning Michael Lim A graph based approach to semi-supervised learning An example Michael Lim A graph based approach to semi-supervised learning An example Michael Lim A graph based approach to semi-supervised learning An example Michael Lim A graph based approach to semi-supervised learning An example Michael Lim A graph based approach to semi-supervised learning Semi-supervised learning framework 1 l labeled examples (x; y) generated
    [Show full text]
  • Cross-Modal and Multimodal Data Analysis Based on Functional Mapping of Spectral Descriptors and Manifold Regularization
    Journal of Machine Learning Research 1 (2021) 1-37 Submitted 4/21; Published 00/00 Cross-Modal and Multimodal Data Analysis Based on Functional Mapping of Spectral Descriptors and Manifold Regularization Maysam Behmanesh [email protected] Artificial Intelligence Department, Faculty of Computer Engineering, University of Isfahan, Iran Peyman Adibi [email protected] Artificial Intelligence Department, Faculty of Computer Engineering, University of Isfahan, Iran Jocelyn Chanussot [email protected] Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France Sayyed Mohammad Saeed Ehsani [email protected] Artificial Intelligence Department, Faculty of Computer Engineering, University of Isfahan, Iran Editor: Abstract Multimodal manifold modeling methods extend the spectral geometry-aware data analysis to learning from several related and complementary modalities. Most of these methods work based on two major assumptions: 1) there are the same number of homogeneous data sam- ples in each modality, and 2) at least partial correspondences between modalities are given in advance as prior knowledge. This work proposes two new multimodal modeling methods. The first method establishes a general analyzing framework to deal with the multimodal information problem for heterogeneous data without any specific prior knowledge. For this purpose, first, we identify the localities of each manifold by extracting local descriptors via spectral graph wavelet signatures (SGWS). Then, we propose a manifold regulariza- tion framework based on the functional mapping between SGWS descriptors (FMBSD) for finding the pointwise correspondences. The second method is a manifold regularized mul- timodal classification based on pointwise correspondences (M2CPC) used for the problem of multiclass classification of multimodal heterogeneous, which the correspondences be- tween modalities are determined based on the FMBSD method.
    [Show full text]
  • Classification by Discriminative Regularization
    Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence (2008) Classification by Discriminative Regularization 1¤ 2 3 1 1 Bin Zhang , Fei Wang , Ta-Hsin Li , Wen jun Yin , Jin Dong 1IBM China Research Lab, Beijing, China 2Department of Automation, Tsinghua University, Beijing, China 3IBM Watson Research Center, Yorktown Heights, NY, USA ¤Corresponding author, email: [email protected] Abstract Generally achieving a good classifier needs a sufficiently ² ^ Classification is one of the most fundamental problems large amount of training data points (such that may have a better approximation of ), however, in realJ world in machine learning, which aims to separate the data J from different classes as far away as possible. A com- applications, it is usually expensive and time consuming mon way to get a good classification function is to min- to get enough labeled points. imize its empirical prediction loss or structural loss. In Even if we have enough training data points, and the con- this paper, we point out that we can also enhance the ² structed classifier may fit very well on the training set, it discriminality of those classifiers by further incorporat- may not have a good generalization ability on the testing ing the discriminative information contained in the data set as a prior into the classifier construction process. In set. This is the problem we called overfitting. such a way, we will show that the constructed classifiers Regularization is one of the most important techniques will be more powerful, and this will also be validated by to handle the above two problems. First, for the overfitting the final empirical study on several benchmark data sets.
    [Show full text]
  • Approximate Manifold Regularization: Scalable Algorithm and Generalization Analysis
    Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19) Approximate Manifold Regularization: Scalable Algorithm and Generalization Analysis Jian Li1;2 , Yong Liu1 , Rong Yin1;2 and Weiping Wang1∗ 1Institute of Information Engineering, Chinese Academy of Sciences 2School of Cyber Security, University of Chinese Academy of Sciences flijian9026, liuyong, yinrong, [email protected] Abstract vised to approximate Laplacian graph by a line or spanning tree [Cesa-Bianchi et al., 2013] and improved by minimiz- Graph-based semi-supervised learning is one of ing tree cut (MTC) in [Zhang et al., 2016]. (2) Acceler- the most popular and successful semi-supervised ate operations associated with kernel matrix. Several dis- learning approaches. Unfortunately, it suffers from tributed approaches have been applied to semi-supervised high time and space complexity, at least quadratic learning [Chang et al., 2017], decomposing a large scale with the number of training samples. In this pa- problem into smaller ones. Anchor Graph regularization per, we propose an efficient graph-based semi- (Anchor) constructs an anchor graph with the training sam- supervised algorithm with a sound theoretical guar- ples and a few anchor points to approximate Laplacian graph antee. The proposed method combines Nystrom [Liu et al., 2010]. The work of [McWilliams et al., 2013; subsampling and preconditioned conjugate gradi- Rastogi and Sampath, 2017] applied random projections in- ent descent, substantially improving computational cluding Nystrom¨ methods and random features into mani- efficiency and reducing memory requirements. Ex- fold regularization. Gradient methods are introduced to solve tensive empirical results reveal that our method manifold regularization on the primal problem, such as pre- achieves the state-of-the-art performance in a short conditioned conjugate gradient [Melacci and Belkin, 2011], time even with limited computing resources.
    [Show full text]
  • A Geometric Viewpoint of Manifold Learning
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector Lin et al. Applied Informatics (2015) 2:3 DOI 10.1186/s40535-015-0006-6 REVIEW Open Access A geometric viewpoint of manifold learning Binbin Lin1*, Xiaofei He2 and Jieping Ye1 *Correspondence: [email protected] Abstract 1 Center for Evolutionary Medicine In many data analysis tasks, one is often confronted with very high dimensional data. and Informatics, The Biodesign Institute, Arizona State University, The manifold assumption, which states that the data is sampled from a submanifold 727 E. Tyler St., 85287 Tempe, AZ, embedded in much higher dimensional Euclidean space, has been widely adopted by USA Full list of author information is many researchers. In the last 15 years, a large number of manifold learning algorithms available at the end of the article have been proposed. Many of them rely on the evaluation of the geometrical and topological of the data manifold. In this paper, we present a review of these methods on a novel geometric perspective. We categorize these methods by three main groups: Laplacian-based, Hessian-based, and parallel field-based methods. We show the connection and difference between these three groups on their continuous and discrete counterparts. The discussion is focused on the problem of dimensionality reduction and semi-supervised learning. Keywords: Manifold learning; Semi-supervised learning; Geometry Review Introduction In many data analysis tasks, one is often confronted with very high dimensional data. There is a strong intuition that the data may have a lower dimensional intrinsic repre- sentation.
    [Show full text]