We propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution. We focus on a semi-supervised framework that incorporates labeled and unlabeled data in a general-purpose learner. Some transductive graph learning algorithms and standard methods including Support Vector Machines and Regularized Least Squares can be obtained as special cases. We utilize properties of Reproducing Kernel Hilbert spaces to prove new Representer theorems that provide theoretical basis for the algorithms. As a result (in contrast to purely graph-based approaches) we obtain a natural out-of-sample extension to novel examples and so are able to handle both transductive and truly semi-supervised settings. We present experimental evidence suggesting that our semi-supervised algorithms are able to use unlabeled data effectively. Finally we have a brief discussion of unsupervised and fully supervised learning within our general framework. 1.

We provide evidence that non-linear dimensionality reduction, clustering and data set parameterization can be solved within one and the same framework. The main idea is to define a system of coordinates with an explicit metric that reflects the connectivity of a given data set and that is robust to noise. Our construction, which is based on a Markov random walk on the data, offers a general scheme of simultaneously reorganizing and subsampling graphs and arbitrarily shaped data sets in high dimensions using intrinsic geometry. We show that clustering in embedding spaces is equivalent to compressing operators. The objective of data partitioning and clustering is to coarse-grain the random walk on the data while at the same time preserving a diffusion operator for the intrinsic geometry or connectivity of the data set up to some accuracy. We show that the quantization distortion in diffusion space bounds the error of compression of the operator, thus giving a rigorous justification for k-means clustering in diffusion space and a precise measure of the performance of general clustering algorithms.

We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. In this case our construction can be viewed as a far-reaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the non-standard wavelet representation of Calderón-Zygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.

This paper introduces a novel spectral framework for solving Markov decision processes (MDPs) by jointly learning representations and optimal policies. The major components of the framework described in this paper include: (i) A general scheme for constructing representations or basis functions by diagonalizing symmetric diffusion operators (ii) A specific instantiation of this approach where global basis functions called proto-value functions (PVFs) are formed using the eigenvectors of the graph Laplacian on an undirected graph formed from state transitions induced by the MDP (iii) A three-phased procedure called representation policy iteration comprising of a sample collection phase, a representation learning phase that constructs basis functions from samples, and a final parameter estimation phase that determines an (approximately) optimal policy within the (linear) subspace spanned by the (current) basis functions. (iv) A specific instantiation of the RPI framework using least-squares policy iteration (LSPI) as the parameter estimation method (v) Several strategies for scaling the proposed approach to large discrete and continuous state spaces, including the Nyström extension for out-of-sample interpolation of eigenfunctions, and the use of Kronecker sum factorization to construct compact eigenfunctions in product spaces such as factored MDPs (vi) Finally, a series of illustrative discrete and continuous control tasks, which both illustrate the concepts and provide a benchmark for evaluating the proposed approach. Many challenges remain to be addressed in scaling the proposed framework to large MDPs, and several elaboration of the proposed framework are briefly summarized at the end.

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.

A deformation invariant representation of surfaces, the GPS embedding, is introduced using the eigenvalues and eigenfunctions of the Laplace-Beltrami differential operator. Notably, since the definition of the GPS embedding completely avoids the use of geodesic distances, and is based on objects of global character, the obtained representation is robust to local topology changes. The GPS embedding captures enough information to handle various shape processing tasks as shape classification, segmentation, and correspondence. To demonstrate the practical relevance of the GPS embedding, we introduce a deformation invariant shape descriptor called G2-distributions, and demonstrate their discriminative power, invariance under natural deformations, and robustness.

A central problem in data analysis is the low dimensional representation of high dimensional data, and the concise description of its underlying geometry and density. In the analysis of large scale simulations of complex dynamical systems, where the notion of time evolution comes into play, important problems are the identification of slow variables and dynamically meaningful reaction coordinates that capture the long time evolution of the system. In this paper we provide a unifying view of these apparently different tasks, by considering a family of di#usion maps, defined as the embedding of complex (high dimensional) data onto a low dimensional Euclidian space, via the eigenvectors of suitably defined random walks defined on the given datasets. Assuming that the data is randomly sampled from an underlying general probability distribution p(x) = e -U(x) , we show that as the number of samples goes to infinity, the eigenvectors of each di#usion map converge to the eigenfunctions of a corresponding di#erential operator defined on the support of the probability distribution. Di#erent normalizations of the Markov chain on the graph lead to di#erent limiting di#erential operators.

(will be inserted by the editor) A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching

Data fusion and multi-cue data matching are fundamental tasks of high-dimensional data analysis. In this paper, we apply the recently introduced diffusion framework to address these tasks. Our contribution is three-fold. First, we present the Laplace-Beltrami approach for computing density invariant embeddings which are essential for integrating different sources of data. Second, we describe a refinement of the Nyström extension algorithm called “geometric harmonics”. We also explain how to use this tool for data assimilation. Finally, we introduce a multi-cue data matching scheme based on nonlinear spectral graphs alignment. The effectiveness of the presented schemes is validated by applying it to the problems of lip-reading and image sequence alignment.

This paper exploits the properties of the commute time between nodes of a graph for the purposes of clustering and embedding, and explores its applications to image segmentation and multi-body motion tracking. Our starting point is the lazy random walk on the graph, which is determined by the heatkernel of the graph and can be computed from the spectrum of the graph Laplacian. We characterize the random walk using the commute time (i.e. the expected time taken for a random walk to travel between two nodes and return) and show how this quantity may be computed from the Laplacian spectrum using the discrete Green’s function. Our motivation is that the commute time can be anticipated to be a more robust measure of the proximity of data than the raw proximity matrix. In this paper, we explore two applications of the commute time. The first is to develop a method for image segmentation using the eigenvector corresponding to the smallest eigenvalue of the commute time matrix. We show that our commute time segmentation method has the property of enhancing the intra-group coherence while weakening inter-group coherence and is superior to the normalized cut. The second application is to develop a robust multi-body motion tracking method using an embedding based on the commute time. Our embedding procedure preserves commute time, and is closely akin to kernel PCA, the Laplacian eigenmap and the diffusion map. We illustrate the results both on synthetic image sequences and real world video sequences, and compare our results with several alternative methods.