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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Accurate tree-cover estimates are useful in deriving above-ground biomass density estimates from very high resolution (VHR) satellite imagery data. Numerous algorithms have been designed to perform tree-cover delineation in high-to-coarse-resolution satellite imagery, but most of them do not scale to terabytes of data, typical in these VHR data sets. In this paper, we present an automated probabilistic framework for the segmentation and classification of 1-m VHR data as obtained from the National Agriculture Imagery Program (NAIP) for deriving tree-cover estimates for the whole of Continental United States, using a high-performance computing architecture. The results from the classification and segmentation algorithms are then consolidated into a structured prediction framework using a discriminative undirected probabilistic graphical model based on conditional random field, which helps in capturing the higher order contextual dependence relations between neighboring pixels. Once the final probability maps are generated, the framework is updated and retrained by incorporating expert knowledge through the relabeling of misclassified image patches. This leads to a significant improvement in the true positive rates and reduction in false positive rates (FPRs). The tree-cover maps were generated for the state of California, which covers a total of 11 095 NAIP tiles and spans a total geographical area of 163 696 sq. miles. Our framework produced correct detection rates of around 88 for fragmented forests and 74 for urban tree-cover areas, with FPRs lower than 2 for both regions. Comparative studies with the National Land-Cover Data algorithm and the LiDAR high-resolution canopy height model showed the effectiveness of our algorithm for generating accurate high-resolution tree-cover maps. The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Renyi random graph G(n, c n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Renyi random graph G(n, 1 n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube 0, 1 n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3 2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4. Abstract For simple random walk on a finite tree, the cover time is the time taken to visit every vertex. For the balanced b-ary tree of height m, the cover time is shown to be asymptotic to 2m 2 b m + 1 ( log b) (b − 1) as m → ∞ . On the uniform random labeled tree on n vertices, we give a convincing heuristic argument that the mean time to cover and return to the root is asymptotic to 6(2π) 1 2 n 3 2 , and prove a weak O(n 3 2 ) upper bound. The argument rests upon a recursive formula for cover time of trees generated by a simple branching process. 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.