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.
MLPCA has been used before ( @cite_15 , @cite_0 , @cite_9 ) to analyze data, but not, to our knowledge, in combination with the cover tree.
@cite_0 @cite_9 @cite_15
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We introduce a method called multi-scale local shape analysis, or MLSA, for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine learning algorithms operating on the dataset. Using synthetic and real dataset examples, we demonstrate significant performance improvement of classification algorithms constructed for these datasets with correspondingly augmented features. Creation and selection of relevant features for machine learning applications (including image classification) is typically a process requiring significant involvement of domain knowledge. It is thus desirable to cover at least part of that process with semi-automated techniques capable of discovering and visualizing those geometric characteristics of images that are potentially relevant to the classification objective. In this work, we propose to utilize multi-scale singular value decomposition (MSVD) along with approximate nearest neighbors algorithm: both have been recently realized using the randomized approach, and can be efficiently run on large, high-dimensional datasets (sparse or dense). We apply this technique to create a multi-scale view of every point in a publicly available set of LIDAR data of riparian images, with classification objective being separating ground from vegetation. We perform “centralized MSVD” for every point and its neighborhood generated by an approximate nearest neighbor algorithm. After completion of this procedure, the original set of 3-dimensional data is augmented by 36 dimensions generated by MSVD (in three different scales), which is then processed using a novel discretization pre-processing method and the SVM classification algorithm with RBF kernel. The result is two times better that the one previously obtained (in terms of its classification error level). The generic nature of the MSVD mechanism and standard mechanisms used for classification (SVM) suggest a wider utility of the proposed approach for other problems as well.