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.
The cover tree was first introduced in @cite_14 as a way of performing approximate nearest neighbors in @math time for a point cloud with @math points in arbitrary metric spaces of a fixed dimension (though constants depend exponentially on the dimension). More recent work has shown that cover trees can be used for analysis of high dimensional point clouds with a low dimensional intrinsic structure @cite_16 . Similarly to @cite_16 , we use the cover tree to decompose the point cloud into geometrically simpler parts, but we encode more explicitly the stratified space structure in our representation, and we also autotune which nodes at which levels represent each part, as explained in .
@cite_14 @cite_16
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We present a tree data structure for fast nearest neighbor operations in general n-point metric spaces (where the data set consists of n points). The data structure requires O(n) space regardless of the metric's structure yet maintains all performance properties of a navigating net (Krauthgamer & Lee, 2004b). If the point set has a bounded expansion constant c, which is a measure of the intrinsic dimensionality, as defined in (Karger & Ruhl, 2002), the cover tree data structure can be constructed in O (c6n log n) time. Furthermore, nearest neighbor queries require time only logarithmic in n, in particular O (c12 log n) time. Our experimental results show speedups over the brute force search varying between one and several orders of magnitude on natural machine learning datasets. Large data sets arise in a wide variety of applications and are often modeled as samples from a probability distribution in high-dimensional space. It is sometimes assumed that the support of such probability distribution is well approximated by a set of low intrinsic dimension, perhaps even a low-dimensional smooth manifold. Samples are often corrupted by high-dimensional noise. We are interested in developing tools for studying the geometry of such high-dimensional data sets. In particular, we present here a multiscale transform that maps high-dimensional data as above to a set of multiscale coefficients that are compressible sparse under suitable assumptions on the data. We think of this as a geometric counterpart to multi-resolution analysis in wavelet theory: whereas wavelets map a signal (typically low dimensional, such as a one-dimensional time series or a two-dimensional image) to a set of multiscale coefficients, the geometric wavelets discussed here map points in a high-dimensional point cloud to a multiscale set of coefficients. The geometric multi-resolution analysis (GMRA) we construct depends on the support of the probability distribution, and in this sense it fits with the paradigm of dictionary learning or data-adaptive representations, albeit the type of representation we construct is in fact mildly nonlinear, as opposed to standard linear representations. Finally, we apply the transform to a set of synthetic and real-world data sets.