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.

This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of the corresponding Markov matrix is optimal under a certain mean squared error criterion. Furthermore, assuming that data points are random samples from a density p(x) = e −U(x) we identify these eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck operator in a potential 2U(x) with reflecting boundary conditions. Finally, applying known results regarding the eigenvalues and eigenfunctions of the continuous Fokker-Planck operator, we provide a mathematical justification for the success of spectral clustering and dimensional reduction algorithms based on these first few eigenvectors. This analysis elucidates, in terms of the characteristics of diffusion processes, many empirical findings regarding spectral clustering algorithms.

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality, at the dimension of the true underlying manifold. This observation suggests to explore non-local manifold learning algorithms which attempt to discover shared structure in the tangent planes at different positions. A criterion for such an algorithm is proposed and experiments estimating a tangent plane prediction function are presented, showing its advantages with respect to local manifold learning algorithms: it is able to generalize very far from training data (on learning handwritten character image rotations), where a local non-parametric method fails. 1

In recent years, a variety of nonlinear dimensionality reduction techniques have been proposed, many of which rely on the evaluation of local properties of the data. The paper presents a review and systematic comparison of these techniques. The performances of the techniques are investigated on artificial and natural tasks. The results of the experiments reveal that nonlinear techniques perform well on selected artificial tasks, but do not outperform the traditional PCA on real-world tasks. The paper explains these results by identifying weaknesses of current nonlinear techniques, and suggests how the performance of nonlinear dimensionality reduction techniques may be improved.

Sparse grids were recently introduced for classification and regression problems. In this article we apply the sparse grid approach to semi-supervised classification. We formulate the semi-supervised learning problem by a regularization approach. Here, besides a regression formulation for the labeled data, an additional term is involved which is based on the graph Laplacian for an adjacency graph of all, labeled and unlabeled data points. It reflects the intrinsic geometric structure of the data distribution. We discretize the resulting problem in function space by the sparse grid method and solve the arising equations using the so-called combination technique. In contrast to recently proposed kernel based methods which currently scale cubic in regard to the number of overall data, our method scales only linear, provided that a sparse graph Laplacian is used. This allows to deal with huge data sets which involve millions of points. We show experimental results with the new approach. 1.

In Quantum Mechanics the transition from a deterministic description to a probabilistic one is done using a simple rule termed the Born rule. This rule states that the probability of an outcome (a) given a state (Ψ) is the square of their inner products ((a ⊤ Ψ) 2). In this paper, we will explore the use of the Born-rule-based probabilities for clustering, feature selection, classification, and for comparison between sets. We show how these probabilities lead to existing and new algebraic algorithms for which no other complete probabilistic justification is known.

We describe a semi-supervised regression algorithm that learns to transform one time series into another time series given examples of the transformation. This algorithm is applied to tracking, where a time series of observations from sensors is transformed to a time series describing the pose of a target. Instead of defining and implementing such transformations for each tracking task separately, our algorithm learns a memoryless transformation of time series from a few example input-output mappings. The algorithm searches for a smooth function that fits the training examples and, when applied to the input time series, produces a time series that evolves according to assumed dynamics. The learning procedure is fast and lends itself to a closed-form solution. It is closely related to nonlinear system identification and manifold learning techniques. We demonstrate our algorithm on the tasks of tracking RFID tags from signal strength measurements, recovering the pose of rigid objects, deformable bodies, and articulated bodies from video sequences. For these tasks, this algorithm requires significantly fewer examples compared to fully-supervised regression algorithms or semi-supervised learning algorithms that do not take the dynamics of the output time series into account.

Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early work in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the low-pass filtering approach to mesh smoothing. Over the past fifteen years, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and high-performance computing. This paper aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background is provided. Existing works covered are classified according to different criteria: the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used. Despite much empirical success, there still remain many open questions pertaining to the spectral approach. These are discussed as we conclude the survey and provide our perspective on possible future research.

Metric and kernel learning arise in several machine learning applications. However, most existing metric learning algorithms are limited to learning metrics over low-dimensional data, while existing kernel learning algorithms are often limited to the transductive setting and do not generalize to new data points. In this paper, we study the connections between metric learning and kernel learning that arise when studying metric learning as a linear transformation learning problem. In particular, we propose a general optimization framework for learning metrics via linear transformations, and analyze in detail a special case of our framework—that of minimizing the LogDet divergence subject to linear constraints. We then propose a general regularized framework for learning a kernel matrix, and show it to be equivalent to our metric learning framework. Our theoretical connections between metric and kernel learning have two main consequences: 1) the learned kernel matrix parameterizes a linear transformation kernel function and can be applied inductively to new data points, 2) our result yields a constructive method for kernelizing most existing Mahalanobis metric learning formulations. We demonstrate our learning approach by applying it to large-scale real world problems in computer vision, text mining and semi-supervised kernel dimensionality reduction. Keywords: divergence metric learning, kernel learning, linear transformation, matrix divergences, logdet 1.

Abstract. Human body is an articulated object with high degrees of freedom. Despite the high dimensionality of the configuration space, many human motion activities lie intrinsically on low dimensional manifolds. Although the intrinsic body configuration manifolds might be very low in dimensionality, the resulting appearance manifolds are challenging to model given various aspects that affects the appearance such as the shape and appearance of the person performing the motion, or variation in the view point, or illumination. Our objective is to learn representations for the shape and the appearance of moving (dynamic) objects that support tasks such as synthesis, pose recovery, reconstruction, and tracking. We studied various approaches for representing global deformation manifolds that preserve their geometric structure. Given such representations, we can learn generative models for dynamic shape and appearance. We also address the fundamental question of separating style and content on nonlinear manifolds representing dynamic objects. We learn factorized generative models that explicitly decompose the intrinsic body configuration (content) as a function of time from the appearance/shape (style factors) of the person performing the action as time-invariant parameters. We show results on pose recovery, body tracking, gait recognition, as well as facial expression tracking and recognition. 1