Confliction of the Convexity and Metric Properties in f-Divergences

By M. Khosravifard, D. Fooladivanda, T. Gulliver. 2007

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On the empirical estimation of integral probability metrics (2012, neutral): “S |p− q| dλ (a) = Dφ(P,Q), where (a) follows from simple algebra after substituting φ in Dφ(P,Q) (see [23]).”
On integral probability metrics, φ-divergences and binary classification (2009, background): “The proof of Lemma 3 uses the following result from [44], which is quite easy to prove.”
On Measures of Entropy and Information (2015, unknown): “= Cf(A;B), f(t) = ∣∣t− 1∣∣ The only f-divergence (33) which is also a metric [67].”
Proposal : Scalable , Active and Flexible Learning on Distributions (2015, unknown): “In fact, the only metric f -divergences are multiples of the total variation distance, discussed shortly (Khosravifard et al. 2007) — though e.g. the Hellinger distance is the square of a metric.”
Information, Divergence and Risk for Binary Experiments (2009, background): “Their only common point is when the respective convex functions satisfy f(t) = φ(t) = t ln t − at + b (for a, b ∈ R) in which case both If and Bφ are the KL divergence.”
On the chi square and higher-order chi distances for approximating f-divergences (2013, background): “TABLE I SOME COMMON f -DIVERGENCESIf WITH CORRESPONDING GENERATORS : EXCEPT THE TOTAL VARIATION, f -DIVERGENCES ARE NOT METRIC[5].”