A Class of New Metrics Based on Triangular Discrimination

Abstract

In the field of information theory, statistics and other application areas, the information-theoretic divergences are used widely. To meet the requirement of metric properties, we introduce a class of new metrics based on triangular discrimination which are bounded. Moreover, we obtain some sharp inequalities for the triangular discrimination and other information-theoretic divergences. Their asymptotic approximation properties are also involved.

On f-Divergences: Integral Representations, Local Behavior, and Inequalities

“…A list of some notable existing f -divergence inequalities is provided, e.g., in [ 22 ] Section 1 and [ 23 ] Section 3. State-of-the-art techniques which serve to derive bounds among f -divergences include: Moment inequalities which rely on log-convexity arguments ([ 22 ] Section 5.D, [ 24 , 25 , 26 , 27 , 28 ]); Inequalities which rely on a characterization of the exact locus of the joint range of f -divergences [ 29 ]; f -divergence inequalities via functional domination ([ 22 ] Section 3, [ 30 , 31 , 32 ]); Sharp f -divergence inequalities by using numerical tools for maximizing or minimizing an f -divergence subject to a finite number of constraints on other f -divergences [ 33 ]; Inequalities which rely on powers of f -divergences defining a distance [ 34 , 35 , 36 , 37 ]; Vajda and Pinsker-type inequalities for f -divergences ([ 4 , 10 , 13 , 22 ] Sections 6–7, [ 38 , 39 ]); Bounds among f -divergences when the relative information is bounded ([ 22 ] Sections 4–5, [ 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 ]), and reverse Pinsker inequalities ([ 22 ] Section 6, [ 40 , 48 ]); Inequalities which rely on the minimum of an ...…”

Information-Theoretic Regret Bounds for Bandits with Fixed Expert Advice

“…The squared Hellinger distance can be related to other measures of divergence. For instance, it is of the same order as the Jensen-Shannon divergence and the triangular discrimination [15], [16]. Thus, the bound of Theorem 6 can be stated, up to small constants, in terms of these measures as well.…”

Investigating the topology dependence of quark and gluon jets

“…In order to study the affect of jet radius on separation power, samples are generated for each topology with jet radii in the range 0.2 ≤ R ≤ 1.5 in steps of 0.1. 5 In the language of information theory, this is closely related to the χ 2 divergence; both are fdivergences [49][50][51] with f (u) = (u − 1) 2 /(u + 1) for the classifier separation and f (u) = (u − 1) 2 for the χ 2 divergence [52,53]. We are grateful to Ben Elder, who pointed out to us that this quantity has also been referred to as the triangular discriminator in the information theory literature [54].…”