1 Similarity Metrics Many recommendation algorithms employ some form of similarity metric in the generation of ratings predictions. Similarity metrics are often associated with some form of distance measure. Definition 1.0.1. Let δ be a function δ : R × R → R. Let x,y, z ∈ R. Then δ is a distance measure if it satisfies the following four properties. (d1) δ(x,y) ≥ 0 (no negative distances). (d2...

Sparse support recovery (SSR) is an important part of the compressive sensing (CS). Most of the current SSR methods are with the full information measurements. But in practice the amplitude part of the measurements may be seriously destroyed. The corrupted measurements mismatch the current SSR algorithms, which leads to serious performance degeneration. This paper considers the problem of SSR w...

We want to use a variety of sparseness measured applied to ‘the minimal `1 norm representation’ of a music signal in an over-complete dictionary as features for automatic classification of music. Unfortunately, the process of computing the optimal `1 norm representation is rather slow, and we therefore investigate the use of matching pursuit, alternating projection, and Moore-Penrose inverse fo...

We study the question of identity testing for structured distributions. More precisely, given samples from a structured distribution q over [n] and an explicit distribution p over [n], we wish to distinguish whether q = p versus q is at least ε-far from p, in L1 distance. In this work, we present a unified approach that yields new, simple testers, with sample complexity that is information-theo...

Given a set P of n point sites in the plane, the city Voronoi diagram subdivides the plane into the Voronoi regions of the sites, with respect to the city metric. This metric is induced by quickest paths according to the Manhattan metric and an accelerating transportation network that consists of c non-intersecting axis-parallel line segments. We describe an algorithm that constructs the city V...

Given k labelings of a finite d-dimensional cubical grid, define the combined distance between two labels to be the sum of the l1-distance between the two labels in each labeling. We want to construct k labelings which maximize the minimum combined distance between any two labels. When d = 1, this can be interpreted as placing n non-attacking rooks in a k-dimensional chessboard of size n in suc...

Let Ω be a bounded planar domain which is convex (although not necessarily strictly convex) with area A. We prove that, for each real number H satisfying AH2 < 2 , with = ( p 5 1)=2, there exists a graph on Ω with constant mean curvature H and boundary @Ω. This existence theorem is deduced as a consequence of an L estimate for compact constant mean curvature surfaces with planar boundary, in te...

We propose a one-norm support vector machine (SVM) formulation as an alternative to the well-known formulation that uses parameter C in order to balance the two inherent objective functions of the problem. Our formulation is motivated by the ǫ-constraint approach that is used in bicriteria optimization and we propose expressing the objective of minimizing total empirical error as a constraint w...

Social networks support efficient decentralized search: people can collectively construct short paths to a specified target in the network. Rank-based friendship—where the probability that person u befriends person v is inversely proportional to the number of people who are closer to u than v is—is an empirically validated model of acquaintanceship that provably results in efficient decentraliz...

The notion of an L1–norm density estimator process indexed by a class of kernels is introduced. Then a functional central limit theorem and a Glivenko–Cantelli theorem are established for this process. While assembling the necessary machinery to prove these results, a body of Poissonization techniques and restricted chaining methods is developed, which is useful for studying weak convergence of...