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Published at November 26

Divergence Inequalities with Applications in Ergodic Theory

cs.IT

math.IT

quant-ph

Released Date: November 26, 2024

Authors: Ian George¹, Alice Zheng¹, Akshay Bansal¹

Aff.: ¹Not provided

Arxiv: http://arxiv.org/abs/2411.17241v1

Divergence	$f(t)$	$L_{f}$	Conditions	Reference
KL-divergence	$t\ln t$	$4$		Proposition 10
Reverse KL-divergence	$-\ln t$	$4$		Proposition 11
Rényi’s information gain	$\frac{t^{\alpha}-1}{\alpha(\alpha-1)}$	$4$	$\alpha\in[-1,0]\cup[1,2]$	Proposition 12
Rényi’s information gain	$\frac{t^{\alpha}-1}{\alpha(\alpha-1)}$	$1$	otherwise	Proposition 12
Hellinger divergence	$\frac{t^{\alpha}-1}{\alpha-1}$	$4\alpha$	$\alpha\in[1,2]$	Corollary 13
Hellinger divergence	$\frac{t^{\alpha}-1}{\alpha-1}$	$\alpha$	$\alpha\in(0,1)\cup(2,+\infty)$	Corollary 13
Pearson $\chi^{2}$ -divergence	$t^{2}-1$	$8$		Corollary 14
Neyman $\chi^{2}$ -divergence	$\frac{1}{t}-1$	$8$		Corollary 15
Symmetric $\chi^{2}$ -divergence	$\frac{(t-1)^{2}(t+1)}{t}$	$16$		Proposition 16
Arithmetic-geometric mean	$\left(\frac{t+1}{2}\right)\ln\left(\frac{t+1}{2\sqrt{t}}\right)$	$1$		Proposition 17