Abstract One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space. Drawing on the correspondence between the graph Laplacian, the Laplace Beltrami operator on the manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for representing the high dimensional data. The algorithm provides a computationally efficient approach to non-linear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Some potential applications and illustrative examples are discussed. 1 Introduction In many areas of artificial intelligence, information retrieval and data mining, one is often confronted with intrinsically low dimensional data lying in a very high dimensional space. Consider, for example, gray scale images of an object taken under fixed lighting conditions with a moving camera. Each such image would typically be represented by a brightness value at each pixel. If there were n 2

An approach to semi-supervised learning is proposed that is based on a Gaussian random field model. Labeled and unlabeled data are represented as vertices in a weighted graph, with edge weights encoding the similarity between instances. The learning

We consider the general problem of learning from labeled and unlabeled data, which is often called semi-supervised learning or transductive inference. A principled approach to semi-supervised learning is to design a classifying function which is sufficiently smooth with respect to the intrinsic structure collectively revealed by known labeled and unlabeled points. We present a simple algorithm to obtain such a smooth solution. Our method yields encouraging experimental results on a number of classification problems and demonstrates effective use of unlabeled data. 1

We review the literature on semi-supervised learning, which is an area in machine learning and more generally, artificial intelligence. There has been a whole spectrum of interesting ideas on how to learn from both labeled and unlabeled data, i.e. semi-supervised learning. This document is a chapter excerpt from the author’s doctoral thesis (Zhu, 2005). However the author plans to update the online version frequently to incorporate the latest development in the field. Please obtain the latest version at http://www.cs.wisc.edu/~jerryzhu/pub/ssl_survey.pdf

Human action in video sequences can be seen as silhouettes of a moving torso and protruding limbs undergoing articulated motion. We regard human actions as three-dimensional shapes induced by the silhouettes in the space-time volume. We adopt a recent approach [14] for analyzing 2D shapes and generalize it to deal with volumetric space-time action shapes. Our method utilizes properties of the solution to the Poisson equation to extract space-time features such as local space-time saliency, action dynamics, shape structure and orientation. We show that these features are useful for action recognition, detection and clustering. The method is fast, does not require video alignment and is applicable in (but not limited to) many scenarios where the background is known. Moreover, we demonstrate the robustness of our method to partial occlusions, non-rigid deformations, significant changes in scale and viewpoint, high irregularities in the performance of an action, and low quality video. Index Terms Action representation, action recognition, space-time analysis, shape analysis, poisson equation

Accrue Software, Inc. Clustering is a division of data into groups of similar objects. Representing the data by fewer clusters necessarily loses certain fine details, but achieves simplification. It models data by its clusters. Data modeling puts clustering in a historical perspective rooted in mathematics, statistics, and numerical analysis. From a machine learning perspective clusters correspond to hidden patterns, the search for clusters is unsupervised learning, and the resulting system represents a data concept. From a practical perspective clustering plays an outstanding role in data mining applications such as scientific data exploration, information retrieval and text mining, spatial database applications, Web analysis, CRM, marketing, medical diagnostics, computational biology, and many others. Clustering is the subject of active research in several fields such as statistics, pattern recognition, and machine learning. This survey focuses on clustering in data mining. Data mining adds to clustering the complications of very large datasets with very many attributes of different types. This imposes unique

Consistency is a key property of statistical algorithms, when the data is drawn from some underlying probability distribution. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of a popular family of spectral clustering algorithms, which cluster the data with the help of eigenvectors of graph Laplacian matrices. We show that one of the two of major classes of spectral clustering (normalized clustering) converges under some very general conditions, while the other (unnormalized), is only consistent under strong additional assumptions, which, as we demonstrate, are not always satisfied in real data. We conclude that our analysis provides strong evidence for the superiority of normalized spectral clustering in practical applications. We believe that methods used in our analysis will provide a basis for future exploration of Laplacian-based methods in a statistical setting.

Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) -- a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.

Spectral graph theoretic methods have recently shown great promise for the problem of image segmentation. However, due to the computational demands of these approaches, applications to large problems such as spatiotemporal data and high resolution imagery have been slow to appear. The contribution of this paper is a method that substantially reduces the computational requirements of grouping algorithms based on spectral partitioning making it feasible to apply them to very large grouping problems. Our approach is based on a technique for the numerical solution of eigenfunction problems knownas the Nyström method. This method allows one to extrapolate the complete grouping solution using only a small number of "typical" samples. In doing so, we leverage the fact that there are far fewer coherent groups in a scene than pixels.

One key element in understanding the molecular machinery of the cell is to understand the meaning, or function, of each protein encoded in the genome. A very successful means of inferring the function of a previously unannotated protein is via sequence similarity with one or more proteins whose functions are already known. Currently, one of the most powerful such homology detection methods is the SVM-Fisher method of Jaakkola, Diekhans and Haussler (ISMB 2000). This method combines a generative, profile hidden Markov model (HMM) with a discriminative classification algorithm known as a support vector machine (SVM). The current work presents an alternative method for SVMbased protein classification. The method, SVM-pairwise, uses a pairwise sequence similarity algorithm such as SmithWaterman in place of the HMM in the SVM-Fisher method. The resulting algorithm, when tested on its ability to recognize previously unseen families from the SCOP database, yields significantly better remote protein homology detection than SVM-Fisher, profile HMMs and PSI-BLAST.