Stratification Learning: Detecting Mixed Density and Dimensionality in High Dimensional Point Clouds (PREPRINT)

Haro, Gloria; Randall, Grégory; Sapiro, Guillermo

doi:10.21236/ada478351

Cited by 32 publications

(38 citation statements)

References 12 publications

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“…Some of them emphasize modeling of the underlying flats and then use the models to infer the clusters (see e.g., Independent Component Analysis (Hyvärinen and Oja 2000), Subspace Separation (Kanatani 2001(Kanatani , 2002, Generalized Principal Component Analysis (GPCA) (Vidal et al 2005;Ma et al 2008)). A few others address the clustering part and then use its output to estimate the parameters of the underlying flats (see e.g., Multi-way Clustering algorithms (Agarwal et al 2005(Agarwal et al , 2006Govindu 2005;Shashua et al 2006), Tensor Voting (Medioni et al 2000), k-Manifolds (Souvenir and Pless 2005), Grassmann Clustering (Gruber and Theis 2006), Poisson Mixture Model (Haro et al 2006)). There are also algorithms that iterate between the two components of data clustering and subspace modeling (see e.g., Mixtures of PPCA (MoPPCA) (Tipping and Bishop 1999), K-Subspaces (Ho et al 2003)/kPlanes (Bradley and Mangasarian 2000;Tseng 1999)).…”

Section: Introductionmentioning

confidence: 99%

Spectral Curvature Clustering (SCC)

2008

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This paper presents novel techniques for improving the performance of a multi-way spectral clustering framework (Govindu Lerman, 2007, preprint in the supplementary webpage) for segmenting affine subspaces. Specifically, it suggests an iterative sampling procedure to improve the uniform sampling strategy, an automatic scheme of inferring the tuning parameter from the data, a precise initialization procedure for K-means, as well as a simple strategy for isolating outliers. The resulting algorithm, Spectral Curvature Clustering (SCC), requires only linear storage and takes linear running time in the size of the data. It is supported by theory which both justifies its successful performance and guides our practical choices. We compare it with other existing methods on a few artificial instances of affine subspaces. Application of the algorithm to several real-world problems is also discussed.

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Section: Introductionmentioning

confidence: 99%

Spectral Curvature Clustering (SCC)

2008

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“…First, if T → ∞, k → ∞, and k/T → 0, then we can approximate the binomial process by a Poisson process. Second, the density ρ(x t ) is considered constant inside the sphere, a valid assumption for small 1 We should mention that in [15] we compared the original framework (with no regularization or noise modelling as here developed), with a two step approach, where we first estimate the local dimensionality per point using the original Levina-Bickel approach, and then cluster following the information bottleneck approach [34]. This has been shown not only to be less elegant and mathematically funded than the approach here presented, but mush less robust, even when compared to the non-regularized and noise-transparent formulation.…”

Section: Local Intrinsic Dimension Estimationmentioning

confidence: 99%

“…In [15], we proposed to study a stratification by extending the Levina and Bickel's technique. Instead of modeling each point and its local ball of radius R as a Poisson process and computing the maximum likelihood (ML) for each ball separately, all the possible balls are considered at the same time in the ML function.…”

Section: Translation Poisson Mixture Model (Tpmm)mentioning

confidence: 99%

“…This technique automatically gives a soft clustering according to dimensionality and density, with an estimation of both quantities for each class. A preliminary version of this work was presented in [15] and a regularized version together with asymptotic results in [16]. These techniques are particular cases of the more general Translated Poisson model introduced in this paper in order to handle noise.…”

Section: Introductionmentioning

confidence: 99%

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Translated Poisson Mixture Model for Stratification Learning (PREPRINT)

Haro¹,

Randal²,

Sapiro³

2007

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. AbstractA framework for the regularized and robust estimation of non-uniform dimensionality and density in high dimensional noisy data is introduced in this work. This leads to learning stratifications, that is, mixture of manifolds representing different characteristics and complexities in the data set. The basic idea relies on modeling the high dimensional sample points as a process of Translated Poisson mixtures, with regularizing restrictions, leading to a model which includes the presence of noise. The Translated Poisson distribution is useful to model a noisy counting process, and it is derived from the noise-induced translation of a regular Poisson distribution. By maximizing the log-likelihood of the process counting the points falling into a local ball, we estimate the local dimension and density. We show that the sequence of all possible local counting in a point cloud formed by samples of a stratification can be modeled by a mixture of different Translated Poisson distributions, thus allowing the presence of mixed dimensionality and densities in the same data set. With this statistical model, the parameters which best describe the data, estimated via expectation maximization, divide the points in different classes according to both dimensionality and density, together with an estimation of these quantities for each class. Theoretical asymptotic results for the model are presented as well. The presentation of the theoretical framework is complemented with artificial and real examples showing the importance of regularized stratification learning in high dimensional data analysis in general and computer vision and image analysis in particular.

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“…Similar to [18], we take a deliberate departure from the manifold assumption. While their methods are statistical in nature, we use local homology to recognize locations where the assumption is violated.…”

Section: Introductionmentioning

confidence: 99%

Inferring Local Homology from Sampled Stratified Spaces

Bendich¹,

Cohen-Steiner²,

Edelsbrunner³

et al. 2007

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)

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We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a 1-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample.

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Stratification Learning: Detecting Mixed Density and Dimensionality in High Dimensional Point Clouds (PREPRINT)

Cited by 32 publications

References 12 publications

Spectral Curvature Clustering (SCC)

Spectral Curvature Clustering (SCC)

Translated Poisson Mixture Model for Stratification Learning (PREPRINT)

Inferring Local Homology from Sampled Stratified Spaces

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