Solomon and Elkin (SIAM J Discret Math 28(3):1173–1198, 2014) constructed a shortcutting scheme for weighted trees which results in 1-spanner the tree metric induced by input tree. The spanner has logarithmic lightness, diameter, linear number of edges bounded degree (provided degree). This been applied series papers devoted to designing degree, low-diameter, low-weight $$(1+\epsilon )$$ -spann...

An O(n log n) heuristic for the Euclidean Steiner Minimal Tree (ESMT) problem is presented. The algorithm is based on a decomposition approach which first partitions the vertex set into triangles via the Delaunay triangulation, then "recomposes" the suboptimal Steiner Minimal Tree (SMT) according to the Voronoi diagram and Minimum Spanning Tree (MST) of the point set. The ESMT algorithm was imp...

The manifold of Symmetric Positive Definite (SPD) matrices has been successfully used for data representation in image set classification. By endowing the SPD manifold with LogEuclidean Metric, existing methods typically work on vector-forms of SPD matrix logarithms. This however not only inevitably distorts the geometrical structure of the space of SPD matrix logarithms but also brings low eff...

Let P be a set (called points), Q be a set (called queries) and a function f : P×Q→ [0,∞) (called cost). For an error parameter > 0, a set S ⊆ P with a weight function w : P → [0,∞) is an ε-coreset if ∑ s∈S w(s)f(s, q) approximates ∑ p∈P f(p, q) up to a multiplicative factor of 1 ± ε for every given query q ∈ Q. Coresets are used to solve fundamental problems in machine learning of streaming an...

SpectralNet is a graph clustering method that uses neural network to find an embedding separates the data. So far it was only used with k-nn graphs, which are usually constructed using distance metric (e.g., Euclidean distance). graphs restrict points have fixed number of neighbors regardless local statistics around them. We proposed new similarity based on random projection trees (rpTrees). Ou...

For all 0 < t ≤ 1, we define a locally Euclidean metric ρt on R . These metrics are invariant under Euclidean isometries and, if t increases to 1, converge to the Euclidean metric dE . This research is motivated by expanding universe. key words. locally Euclidean metric PACS number(s). 98.80.Jk Mathematics Subject Classifications (2000). 85A40, 57M50 1 The metric ρt Let dE denote the Euclidean ...

In this paper, we present a new metric to estimate the perceived difference in contrast between an original image and a reproduction. This metric, named weighted-level framework EE (WLF-DEE), implements a multilevel filtering based on the difference of Gaussians model proposed by Tadmor and Tolhurst (2000) and the new Euclidean color difference formula in log-compressed OSA-UCS space proposed b...

Continuous word representations have been remarkably useful across NLP tasks but remain poorly understood. We ground word embeddings in semantic spaces studied in the cognitive-psychometric literature, taking these spaces as the primary objects to recover. To this end, we relate log co-occurrences of words in large corpora to semantic similarity assessments and show that co-occurrences are inde...

In the previous lecture notes, we saw that any metric (X, d) with |X| = n can be embedded into R 2 n) under any the `1 metric (actually, the same embedding works for any `p metic), with distortion O(log n). Here, we describe an extremely useful approach for reducing the dimensionality of a Euclidean (`2) metric, while incurring very little distortion. Such dimension reduction is useful for a nu...

The metric sketching problem is defined as follows. Given a metric on n points, and > 0, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up to a 1 + distortion. In this paper we consider metrics induced by `2 and `1 norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by Φ >...