We propose a flexible and multi-scale method for organizing, visualizing, and understanding point cloud datasets sampled from or near stratified spaces. The first part of the algorithm produces a cover tree for a dataset using an adaptive threshold that is based on multi-scale local principal component analysis. The resulting cover tree nodes reflect the local geometry of the space and are organized via a scaffolding graph. In the second part of the algorithm, the goals are to uncover the strata that make up the underlying stratified space using a local dimension estimation procedure and topological data analysis, as well as to ultimately visualize the results in a simplified spine graph. We demonstrate our technique on several synthetic examples and then use it to visualize song structure in musical audio data.
Many papers ( @cite_1 , for example) have analyzed stratified spaces as mixtures of general manifolds, and one @cite_4 as a union of flats. Using techniques derived mostly from Gaussian Mixture Models in the former, and Grassmanian methods in the latter, they prove theoretical guarantees about when a point belongs to a maximal stratum, but they do not say much about the singularities. Other work @cite_20 uses topological data analysis to make theoretically-backed inferences about the singularities themselves; however, the algorithms are too slow to be practical.
@cite_20 @cite_1 @cite_4
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The objective of this paper is to show that point cloud data can under certain circumstances be clustered by strata in a plausible way. For our purposes, we consider a stratified space to be a collection of manifolds of different dimensions which are glued together in a locally trivial manner inside some Euclidean space. To adapt this abstract definition to the world of noise, we first define a multi-scale notion of stratified spaces, providing a stratification at different scales which are indexed by a radius parameter. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata. We prove a correctness guarantee for this clustering method under certain topological conditions. We then provide a probabilistic guarantee for the clustering for the point sample setting -- we provide bounds on the minimum number of sample points required to state with high probability which points belong to the same strata. Finally, we give an explicit algorithm for the clustering. A convenient way of dealing with image sets is to represent them as points on Grassmannian manifolds. While several recent studies explored the applicability of discriminant analysis on such manifolds, the conventional formalism of discriminant analysis suffers from not considering the local structure of the data. We propose a discriminant analysis approach on Grassmannian manifolds, based on a graph-embedding framework. We show that by introducing within-class and between-class similarity graphs to characterise intra-class compactness and inter-class separability, the geometrical structure of data can be exploited. Experiments on several image datasets (PIE, BANCA, MoBo, ETH-80) show that the proposed algorithm obtains considerable improvements in discrimination accuracy, in comparison to three recent methods: Grassmann Discriminant Analysis (GDA), Kernel GDA, and the kernel version of Affine Hull Image Set Distance. We further propose a Grassmannian kernel, based on canonical correlation between subspaces, which can increase discrimination accuracy when used in combination with previous Grassmannian kernels. Abstract This paper mainly focuses on dimensional reduction of fused dataset of holistic and geometrical face features vectors by solving singularity problem of linear discriminant analysis and maximizing the Fisher ratio in nonlinear subspace region with the preservation of local discriminative features. The combinational feature vector space is projected into low dimensional subspace using proposed Kernel Locality Preserving Symmetrical Weighted Fisher Discriminant Analysis (KLSWFDA) method. Matching score level fusion technique has been applied on projected subspace and combinational entire Gabor subspace is framed. Euclidean distance metric (L2) and support vector machine (SVM) classifier has been implemented to recognize and classify the expressions. Performance of proposed approach is evaluated and compared with state of art approaches. Experimental results on JAFFE, YALE and FD expression database demonstrate the effectiveness of the proposed approach. This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object. It is now widely recognised that this kind of topological analysis can give qualitative information about data sets which is not readily available by other means. In particular, it can be an aid to visualisation of high dimensional data. Standard simplicial complexes for approximating the topological type of the underlying space (such as Cech, Rips, or a-shape) produce simplicial complexes whose vertex set has the same size as the underlying set of point cloud data. Such constructions are sometimes still tractable, but are wasteful (of computing resources) since the homotopy types of the underlying objects are generally realisable on much smaller vertex sets. We obtain smaller complexes by choosing a set of 'landmark' points from our data set, and then constructing a "witness complex" on this set using ideas motivated by the usual Delaunay complex in Euclidean space. The key idea is that the remaining (non-landmark) data points are used as witnesses to the existence of edges or simplices spanned by combinations of landmark points. Our construction generalises the topology-preserving graphs of Martinetz and Schulten [MS94] in two directions. First, it produces a simplicial complex rather than a graph. Secondly it actually produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology [ELZ00, ZC04]. We find that in addition to the complexes being smaller, they also provide (in a precise sense) a better picture of the homology, with less noise, than the full scale constructions using all the data points. We illustrate the use of these complexes in qualitatively analyzing a data set of 3 × 3 pixel patches studied by David [LPM03].