Integral Probability Metrics and Their Generating Classes of Functions

By Alfred Müller. 1997

Subfields: Probabilistic & Bayesian Methods

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Citations in the corpus

Generalization and Equilibrium in Generative Adversarial Nets (GANs) (2017, background): “This is also known as integral probability metrics[Müller, 1997].”
Wasserstein GAN (2017, background): “• When F is the set of all measurable functions bounded between -1 and 1 (or all continuous functions between -1 and 1), we retrieve dF (Pr,Pθ) = δ(Pr,Pθ) the total variation distance [14].”
DeepMDP: Learning Continuous Latent Space Models for Representation Learning (2019, methodology): “The Total Variation is also the Norm-MMD generated from the set of functions with absolute value bounded by 1 (Müller, 1997).”
Demystifying MMD GANs (2018, methodology): “The maximum mean discrepancy is a metric on probability measures (Gretton et al., 2012), which falls within the family of integral probability metrics (Müller, 1997); this family includes the Wasser-stein and Kolmogorov metrics, but not for instance the KL or χ 2 divergences.”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “When P 6= Q, an unbiased empirical estimate M̂MD 2 u[H ,X,Y] given in (3.33) converges in distribution to a Gaussian distribution √ m(M̂MD 2 u[H ,X,Y]−MMD2[H ,P,Q]) p→ N (0, σ2XY ) where σ2XY = 4(Ev1 [(Ev2h(v1,v2)) 2]− [Ev1,v2h(v1,v2)]2) uniformly at rate 1/ √ m. Sriperumbudur et al. (2012) studied ”
A Kernel Test of Goodness of Fit (2016, methodology): “rnel Hilbert Space (RKHS). To 1The bounded Lipschitz functions give rise to the Wasserstein integral probability metric. By contrast, the KolmogorovSmirnov test uses functions of bounded variation 1 (Müller, 1997). Multivariate generalisations of the K-S test exist, however the computational cost of”
Hypothesis testing using pairwise distances and associated kernels (2012, background): “An alternative interpretation of (8) is as an integral probability metric (Müller, 1997): see Gretton et al. (2012) for details.”
On the empirical estimation of integral probability metrics (2012, background): “The goal of this paper is to study the empirical estimation of a popular family of distance measures on probabilities, the integral probability metrics (IPM) [29]—also called probability metrics with a ζ-structure [55]—defined as”
MOPO: Model-based Offline Policy Optimization (2020, background): “We ﬁrst note that if F is a set of functions mapping S to R that contains V πM , then, where d F is the integral probability metric (IPM) [46] deﬁned by F .”
GenDICE: Generalized Offline Estimation of Stationary Values (2020, neutral): “Based on the definition of the IPM, these divergences directly lead to min-max optimizations similar to Equation 13 with the identity function as φ∗ (·) and different feasible sets for the dual functions.”
Information-Theoretic Considerations in Batch Reinforcement Learning (2019, background): “The proof shows that if φ is not a bisimulation, we can find f ∈ F either to witness the reward error or the transition error, and in the latter case, the choice of f achieves the maximum discrepancy in an integral probability metric (Müller, 1997) interpretation of the bisimulation condition on tra”
Distributional Reinforcement Learning with Quantile Regression (2017, methodology): “The p -Wasserstein distance is characterized as the L p metric on inverse cumulative distribution functions (inverse CDFs) (M¨uller 1997).”
Continuous Doubly Constrained Batch Reinforcement Learning (2021, methodology): “In principle, various f -divergences [10] or Integral Probability Metrics [42] could employed in D(·, ·).”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “In addition to the general application domains discussed earlier for metrics on probabilities, IPMs have been used in proving central limit theorems using Stein’s method (Stein, 1972; Barbour and Chen, 2005), and are popular in empirical process theory (van der Vaart and Wellner, 1996).”
Stabilizing Training of Generative Adversarial Networks through Regularization (2017, methodology): “As a remedy, it has been suggested to use an alternative family of distance functions known as integral probability metrics Müller (1997); Sriperumbudur et al. (2009).”
The Cramer Distance as a Solution to Biased Wasserstein Gradients (2017, background): “Then This is a special case of the celebrated Monge-Kantorovich duality (Rachev et al., 2013), and is the integral probability metric (IPM) with function class F ∞ (Müller, 1997).”