On The Jensen–Shannon Symmetrization Of . - Frank Nielsen
On the Jensen–Shannon Symmetrization ofDistances Relying on Abstract MeansFrank NielsenSony Computer Science Laboratories, Inchttps://franknielsen.github.io/Paper: https://www.mdpi.com/1099-4300/21/5/485July 2020Code: https://franknielsen.github.io/M-JS/
Unbounded Kullback-Leibler divergence (KLD)Also called relative entropy:Cross-entropy:Shannon’s entropy:(self cross-entropy)Reverse KLD:(KLD forward KLD)
Symmetrizations of the KLDJeffreys’ divergence (twice the arithmetic mean of oriented KLDs):Resistor average divergence (harmonic mean of forward reverse KLD)Question: Role and extensions of the mean?
Bounded Jensen-Shannon divergence (JSD)(Shannon entropy h isstrictly concave, JSD 0)JSD is bounded:Proof:: Square root of the JSD is a metric distance (moreover Hilbertian)
Invariant f-divergences, symmetrized f-divergencesConvex generator f, strictly convex at 1with f(1) 0 (standard when f’(1) 0, f’’(1) 1)f-divergences are said invariant in information geometry because theysatisfy coarse-graining (data processing inequality)f-divergences can always be symmetrized: Reverse f-divergence forJeffreys f-generator:Jensen-Shannon f-generator:
Statistical distances vs parameter vector distancesA statistical distance D between two parametric distributions of a samefamily (eg., Gaussian family) amount to a parameter distance P:For example, the KLD between two densities of a same exponential familyamounts to a reverse Bregman divergence for the Bregman cumulant generator:From a smooth C3 parameter distance ( contrast function),we can build a dualistic information-geometric structure
Skewed Jensen-Bregman divergencesJS-kind symmetrization of the parameter Bregman divergence:Notation for the linear interpolation:
J-Symmetrization and JS-SymmetrizationJ-symmetrization of a statistical/parameter distance D:JS-symmetrization of a statistical/parameter distance D:Example: J-symmetrization and JS-symmetrization of f-divergences:Conjugate f-generator:
Generalized Jensen-Shannon divergences:Role of abstract weighted means, generalized mixturesQuasi-arithmetic weighted means for a strictly increasing function h:When M AArithmetic mean,Normalizer Z is 1
Definitions: M-JSD and M-JS symmetrizationsFor generic distance D (not necessarily KLD):
Generic definition: (M,N)-JS symmetrizationConsider two abstract means M and N:The main advantage of (M,N)-JSD is to get closed-form formulafor distributions belonging to given parametric familiesby carefully choosing the M-mean.For example, geometric mean for exponential families,or harmonic mean for Cauchy or t-Student families, etc.
(A,G)-Jensen-Shannon divergence for exponential familiesExponential family:Natural parameter space:Geometric statistical mixture:Normalization coefficient:Jensen parameter divergence:
(A,G)-Jensen-Shannon divergence for exponential familiesClosed-form formula the KLD between two geometric mixtures in term of aBregman divergence between interpolated parameters:
Example: Multivariate Gaussian exponential familyFamily of Normal distributions:Canonical factorization:Sufficient statistics:Cumulant function/log-normalizer:
Example: Multivariate Gaussian exponential familyDual moment parameterization:Conversions between ordinary/natural/expectation parameters:Dual potential function ( negative differential Shannon entropy):
More examples: Abstract means and M-mixtureshttps://www.mdpi.com/1099-4300/21/5/485
Summary: Generalized Jensen-Shannon divergences Jensen-Shannon divergence (JSD) is a bounded symmetrization of the KullbackLeibler divergence (KLD). Jeffreys divergence (JD) is an unbounded symmetrizationof KLD. Both JSD and JD are invariant f-divergences. Although KLD and JD between Gaussians (or densities of a same exponentialfamily) admits closed-form formulas, the JSD between Gaussians does not have aclosed expression, and these distances need to be approximated in applications.(machine learning, eg., deep learning in GANs) The skewed Jensen-Shannon divergence is based on statistical arithmetic mixtures.We define generic statistical M-mixtures based on an abstract mean, and defineaccordingly the M-Jensen-Shannon divergence, and the (M,N)-JSD. When M G is the geometric weighted mean, we obtain closed-form formula forthe G-Jensen-Shannon divergence between Gaussian distributions. Applicationsto machine learning (eg, deep learning GANs) nielsen.github.io/M-JS/
Summary: Generalized Jensen -Shannon divergences Jensen-Shannon divergence (JSD) is a . bounded symmetrization. of the Kullback-Leibler divergence (KLD). Jeffreysdivergence (JD) is an unbounded symmetrization of KLD. Both JSD and JD are invariant f-divergences. Although KLD and
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