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**21 - 24**of**24**### massachusetts institute of technology, cambridge, ma 02139 usa — www.csail.mit.eduMultiscale Geometric Methods for Data Sets I: Multiscale SVD, Noise and Curvature

, 2012

"... Large data sets are often modeled as being noisy samples from probability distributions µ in R D, with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider set ..."

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Large data sets are often modeled as being noisy samples from probability distributions µ in R D, with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider sets that are locally well approximated by k-dimensional planes, with k ≪ D, with k-dimensional manifolds isometrically embedded in R D being a special case. Samples from µ are furthermore corrupted by D-dimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem well-suited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r, with a fixed center and varying r, and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M. As an application, we present an algorithm for estimating the intrinsic dimension of M. 1

### Subspace Clustering

"... Abstract We present a new approach to rigid-body mo-tion segmentation from two views. We use a previously de-veloped nonlinear embedding of two-view point correspon-dences into a 9-dimensional space and identify the differ-ent motions by segmenting lower-dimensional subspaces. In order to overcome n ..."

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Abstract We present a new approach to rigid-body mo-tion segmentation from two views. We use a previously de-veloped nonlinear embedding of two-view point correspon-dences into a 9-dimensional space and identify the differ-ent motions by segmenting lower-dimensional subspaces. In order to overcome nonuniform distributions along the subspaces, whose dimensions are unknown, we suggest the novel concept of global dimension and its minimization for clustering subspaces with some theoretical motivation. We propose a fast projected gradient algorithm for minimiz-ing global dimension and thus segmenting motions from 2-views. We develop an outlier detection framework around the proposed method, and we present state-of-the-art results on outlier-free and outlier-corrupted two-view data for seg-menting motion.

### 1Spectral Clustering on Multiple Manifolds

"... Abstract—Spectral clustering is a large family of grouping methods which partition data using eigenvectors of an affin-ity matrix derived from the data. Though spectral clustering methods have been successfully applied to a large number of challenging clustering scenarios, it is noteworthy that they ..."

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Abstract—Spectral clustering is a large family of grouping methods which partition data using eigenvectors of an affin-ity matrix derived from the data. Though spectral clustering methods have been successfully applied to a large number of challenging clustering scenarios, it is noteworthy that they will fail when there are significant intersections among different clusters. In this paper, based on the analysis that spectral clustering methods are able to work well when the affinity values of the points belonging to different clusters are relatively low, we propose a new method, called SMMC (Spectral Multi-Manifold Clustering), which is able to handle intersections. In our model, the data are assumed to lie on or close to multiple smooth low-dimensional manifolds, where some data manifolds are separated but some are intersected. Then, local geometric information of the sampled data is incorporated to construct a suitable affinity matrix. Finally, spectral method is applied to this affinity matrix to group the data. Extensive experiments on synthetic as well as real data sets demonstrate the promising performance of SMMC. Index Terms—Clustering, spectral clustering, manifold cluster-ing, local tangent space. I.

### Multiscale Estimation of Intrinsic Dimensionality of Point Cloud Data and Multiscale Analysis of Dynamic Graphs

, 2010

"... 1.2 Multiscale SVD for Dimension Estimation.................. 6 ..."

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