On the optimality of shape and data representation in the spectral domain

A proof of the optimality of the eigenfunctions of the Laplace-Beltrami operator (LBO) in representing smooth functions on surfaces is provided and adapted to the field of applied shape and data analysis. It is based on the Courant-Fischer min-max principle adapted to our case. % The theorem we present supports the new trend in geometry processing of treating geometric structures by using their projection onto the leading eigenfunctions of the decomposition of the LBO. Utilisation of this result can be used for constructing numerically efficient algorithms to process shapes in their spectrum. We review a couple of applications as possible practical usage cases of the proposed optimality criteria. % We refer to a scale invariant metric, which is also invariant to bending of the manifold. This novel pseudo-metric allows constructing an LBO by which a scale invariant eigenspace on the surface is defined. We demonstrate the efficiency of an intermediate metric, defined as an interpolation between the scale invariant and the regular one, in representing geometric structures while capturing both coarse and fine details. Next, we review a numerical acceleration technique for classical scaling, a member of a family of flattening methods known as multidimensional scaling (MDS). There, the optimality is exploited to efficiently approximate all geodesic distances between pairs of points on a given surface, and thereby match and compare between almost isometric surfaces. Finally, we revisit the classical principal component analysis (PCA) definition by coupling its variational form with a Dirichlet energy on the data manifold. By pairing the PCA with the LBO we can handle cases that go beyond the scope defined by the observation set that is handled by regular PCA

Robust Temporally Coherent Laplacian Protrusion Segmentation of 3D Articulated Bodies

In motion analysis and understanding it is important to be able to fit a suitable model or structure to the temporal series of observed data, in order to describe motion patterns in a compact way, and to discriminate between them. In an unsupervised context, i.e., no prior model of the moving object(s) is available, such a structure has to be learned from the data in a bottom-up fashion. In recent times, volumetric approaches in which the motion is captured from a number of cameras and a voxel-set representation of the body is built from the camera views, have gained ground due to attractive features such as inherent view-invariance and robustness to occlusions. Automatic, unsupervised segmentation of moving bodies along entire sequences, in a temporally-coherent and robust way, has the potential to provide a means of constructing a bottom-up model of the moving body, and track motion cues that may be later exploited for motion classification. Spectral methods such as locally linear embedding (LLE) can be useful in this context, as they preserve "protrusions", i.e., high-curvature regions of the 3D volume, of articulated shapes, while improving their separation in a lower dimensional space, making them in this way easier to cluster. In this paper we therefore propose a spectral approach to unsupervised and temporally-coherent body-protrusion segmentation along time sequences. Volumetric shapes are clustered in an embedding space, clusters are propagated in time to ensure coherence, and merged or split to accommodate changes in the body's topology. Experiments on both synthetic and real sequences of dense voxel-set data are shown. This supports the ability of the proposed method to cluster body-parts consistently over time in a totally unsupervised fashion, its robustness to sampling density and shape quality, and its potential for bottom-up model constructionComment: 31 pages, 26 figure

On the Sample Complexity of Subspace Learning

A large number of algorithms in machine learning, from principal component analysis (PCA), and its non-linear (kernel) extensions, to more recent spectral embedding and support estimation methods, rely on estimating a linear subspace from samples. In this paper we introduce a general formulation of this problem and derive novel learning error estimates. Our results rely on natural assumptions on the spectral properties of the covariance operator associated to the data distribu- tion, and hold for a wide class of metrics between subspaces. As special cases, we discuss sharp error estimates for the reconstruction properties of PCA and spectral support estimation. Key to our analysis is an operator theoretic approach that has broad applicability to spectral learning methods.Comment: Extendend Version of conference pape

Unified Framework for Spectral Dimensionality Reduction, Maximum Variance Unfolding, and Kernel Learning By Semidefinite Programming: Tutorial and Survey

This is a tutorial and survey paper on unification of spectral dimensionality reduction methods, kernel learning by Semidefinite Programming (SDP), Maximum Variance Unfolding (MVU) or Semidefinite Embedding (SDE), and its variants. We first explain how the spectral dimensionality reduction methods can be unified as kernel Principal Component Analysis (PCA) with different kernels. This unification can be interpreted as eigenfunction learning or representation of kernel in terms of distance matrix. Then, since the spectral methods are unified as kernel PCA, we say let us learn the best kernel for unfolding the manifold of data to its maximum variance. We first briefly introduce kernel learning by SDP for the transduction task. Then, we explain MVU in detail. Various versions of supervised MVU using nearest neighbors graph, by class-wise unfolding, by Fisher criterion, and by colored MVU are explained. We also explain out-of-sample extension of MVU using eigenfunctions and kernel mapping. Finally, we introduce other variants of MVU including action respecting embedding, relaxed MVU, and landmark MVU for big data.Comment: To appear as a part of an upcoming textbook on dimensionality reduction and manifold learning. v2: corrected some typo

From which world is your graph?

Discovering statistical structure from links is a fundamental problem in the analysis of social networks. Choosing a misspecified model, or equivalently, an incorrect inference algorithm will result in an invalid analysis or even falsely uncover patterns that are in fact artifacts of the model. This work focuses on unifying two of the most widely used link-formation models: the stochastic blockmodel (SBM) and the small world (or latent space) model (SWM). Integrating techniques from kernel learning, spectral graph theory, and nonlinear dimensionality reduction, we develop the first statistically sound polynomial-time algorithm to discover latent patterns in sparse graphs for both models. When the network comes from an SBM, the algorithm outputs a block structure. When it is from an SWM, the algorithm outputs estimates of each node's latent position.Comment: To appear in NIPS 201

High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation

The ratio between two probability density functions is an important component of various tasks, including selection bias correction, novelty detection and classification. Recently, several estimators of this ratio have been proposed. Most of these methods fail if the sample space is high-dimensional, and hence require a dimension reduction step, the result of which can be a significant loss of information. Here we propose a simple-to-implement, fully nonparametric density ratio estimator that expands the ratio in terms of the eigenfunctions of a kernel-based operator; these functions reflect the underlying geometry of the data (e.g., submanifold structure), often leading to better estimates without an explicit dimension reduction step. We show how our general framework can be extended to address another important problem, the estimation of a likelihood function in situations where that function cannot be well-approximated by an analytical form. One is often faced with this situation when performing statistical inference with data from the sciences, due the complexity of the data and of the processes that generated those data. We emphasize applications where using existing likelihood-free methods of inference would be challenging due to the high dimensionality of the sample space, but where our spectral series method yields a reasonable estimate of the likelihood function. We provide theoretical guarantees and illustrate the effectiveness of our proposed method with numerical experiments.Comment: With supplementary materia

Diffusion Maps for dimensionality reduction and visualization of meteorological data

This is the author’s version of a work that was accepted for publication in Neurocomputing. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Neurocomputing, VOL 163, (2015) DOI 10.1016/j.neucom.2014.08.090The growing interest in big data problems implies the need for unsupervised methods for data visualization and dimensionality reduction. Diffusion Maps (DM) is a recent technique that can capture the lower dimensional geometric structure underlying the sample patterns in a way which can be made to be independent of the sampling distribution. Moreover, DM allows us to define an embedding whose Euclidean metric relates to the sample's intrinsic one which, in turn, enables a principled application of k-means clustering. In this work we give a self-contained review of DM and discuss two methods to compute the DM embedding coordinates to new out-of-sample data. Then, we will apply them on two meteorological data problems that involve time and spatial compression of numerical weather forecasts and show how DM is capable to, first, greatly reduce the initial dimension while still capturing relevant information in the original data and, also, how the sample-derived DM embedding coordinates can be extended to new patterns.The authors acknowledge partial support from Spain's grant TIN2010-21575-C02-01 and the UAM{ADIC Chair for Machine Learning. The first author is also supported by an FPI{UAM grant and kindly thanks the Applied Mathematics Department of Yale University for receiving her during her visits