The Bregman chord divergence

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.Comment: 10 page

Co-occurrence Vectors from Corpora vs. Distance Vectors from Dictionaries

A comparison was made of vectors derived by using ordinary co-occurrence statistics from large text corpora and of vectors derived by measuring the inter-word distances in dictionary definitions. The precision of word sense disambiguation by using co-occurrence vectors from the 1987 Wall Street Journal (20M total words) was higher than that by using distance vectors from the Collins English Dictionary (60K head words + 1.6M definition words). However, other experimental results suggest that distance vectors contain some different semantic information from co-occurrence vectors.Comment: 6 pages, appeared in the Proc. of COLING94 (pp. 304-309)

Hierarchies in Dictionary De

A dictionary defines words in terms of other words. Definitions can tell you the meanings of words you don't know, but only if you know the meanings of the defining words. How many words do you need to know (and which ones) in order to be able to learn all the rest from definitions? We reduced dictionaries to their "grounding kernels" (GKs), about 10% of the dictionary, from which all the other words could be defined. The GK words turned out to have psycholinguistic correlates: they were learned at an earlier age and more concrete than the rest of the dictionary. But one can compress still more: the GK turns out to have internal structure, with a strongly connected "kernel core" (KC) and a surrounding layer, from which a hierarchy of definitional distances can be derived, all the way out to the periphery of the full dictionary. These definitional distances, too, are correlated with psycholinguistic variables (age of acquisition, concreteness, imageability, oral and written frequency) and hence perhaps with the ``mental lexicon" in each of our heads

Approximation algorithms for wavelet transform coding of data streams

This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance between f and its representation. The first known algorithms for finding provably approximate representations minimizing general $\ell_p$ distances (including $\ell_\infty$) under a wide variety of compactly supported wavelet bases are presented in this paper. For the Haar basis, a polynomial time approximation scheme is demonstrated. These algorithms are applicable in the one-pass sublinear-space data stream model of computation. They generalize naturally to multiple dimensions and weighted norms. A universal representation that provides a provable approximation guarantee under all p-norms simultaneously; and the first approximation algorithms for bit-budget versions of the problem, known as adaptive quantization, are also presented. Further, it is shown that the algorithms presented here can be used to select a basis from a tree-structured dictionary of bases and find a B-term representation of the given function that provably approximates its best dictionary-basis representation.Comment: Added a universal representation that provides a provable approximation guarantee under all p-norms simultaneousl

Gems from Johnson\u27s Dictionary

English lexicographers, those harmless drudges as Dr. Johnson called them, go all the way back to the English Expositour (1617) and maybe farther, to lists of hard words compiled by curious logophiles. But Ursa Major himself is surely the dean of all dictionary-makers. Here are some of Dr. Johnson\u27s own definitions. His famous definition of network as any thing reticulated or decussated, at equal distances, with interstices between the intersections at least is intellibible to those who know some Latin ( I do not love Latin originals he said under ferry, but his sesquipedalian vocabulary often denies this), but see what you can do to produce the words still in current speech that the good doctor explained like this

Vector Bosons in the Randall-Sundrum 2 and Lykken-Randall models and unparticles

Unparticle behavior is shown to be realized in the Randall-Sundrum 2 (RS 2) and the Lykken-Randall (LR) brane scenarios when brane-localized Standard Model currents are coupled to a massive vector field living in the five-dimensional warped background of the RS 2 model. By the AdS/CFT dictionary these backgrounds exhibit certain properties of the unparticle CFT at large N_c and strong 't Hooft coupling. Within the RS 2 model we also examine and contrast in detail the scalar and vector position-space correlators at intermediate and large distances. Unitarity of brane-to-brane scattering amplitudes is seen to imply a necessary and sufficient condition on the positivity of the bulk mass, which leads to the well-known unitarity bound on vector operators in a CFT.Comment: 60 pages, 8 figure