Hilbertian Metrics and Positive Definite Kernels on Probability Measures

By Matthias Hein, Olivier Bousquet. 2005

231 citations recorded by the source corpus.

Subfields: Probabilistic & Bayesian Methods, Statistical Learning Theory

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Citations in the corpus

Asymmetric LSH (ALSH) for Sublinear Time Maximum Inner Product Search (MIPS) (2014, neutral): “Based on this key observation, we show an explicit construction of asymmetric hash function, leading to the first provably sublinear query time algorithmfor approximate similarity search with (unnormalized) inner product as the similarity.”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “In Hein and Bousquet (2005), an extension of a two-parameter family of Hilbertian metrics of Topsøe was used to define Hilbertian kernels on probability measures.”
Learning Theory for Distribution Regression (2014, methodology): “This category includes work on semigroup kernels (Cuturi et al., 2005), nonextensive information theoretical kernel constructions (Martins et al., 2009), and kernels based on Hilbertian metrics (Hein and Bousquet, 2005).”
Characteristic Kernels and Infinitely Divisible Distributions (2014, methodology): “The injective kernel mean map is important because it endows a metric in the set P(X ) of probability measures via the Hilbert space norm || · ||HX , i.e., the Hilbertian metric (Hein and Bousquet, 2005; Sriperumbudur et al., 2010).”
Learning from Distributions via Support Measure Machines (2012, methodology): “Jebara et al. (2004) proposed the probability product kernel (PPK) as a generalized inner product between two input objects. This kernel can be shown to be closely related to well-known kernels such as the Bhattacharyya kernel (Bhattacharyya 1943) and the exponential symmetrized Kullback-Leibler (KL”
Generative and Latent Mean Map Kernels (2010, background): “One particular kernel of Hein and Bousquet [8], Structural Kernel I, generalizes the kernel of Suquet.”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “Broadly, this relates to the work on Hilbertian metrics and positive definite kernels on probability measures by Hein and Bousquet (2005) and Fuglede and Topsøe (2003).”
On integral probability metrics, φ-divergences and binary classification (2009, background): “Similarly, in the case of φ -divergences, some functions of D φ are shown to be metrics on P λ (see Theorem 2 for the notation), for example, the square root of the variational distance, the square root of Hellinger’s distance, the square root of the Jensen-Shannon divergence [62]–[64], etc.”
Asymmetric Minwise Hashing (2014, background): “It is often the case that just the presence or absence information sufﬁces in practice [9, 19, 23, 28].”
Kernel-Based Methods for Hypothesis Testing: A Unified View (2013, unknown): “Hence, kernel-based methods can potentially be applied to any kind of data, ranging from data living in standard Euclidean spaces to data consisting of histograms, chains, trees, or graphs [27], [10].”
Asymmetric Minwise Hashing for Indexing Binary Inner Products and Set Containment (2015, background): “Thus in practice, often only the presence or absence information suffices [9, 20, 24].”
From Dependence to Causation (2016, background): “In a separate strand of research, there has been multiple proposals to learn from probability distributions (Jebara et al., 2004; Hein and Bousquet, 2005; Cuturi et al., 2005; Martins et al., 2009; Muandet et al., 2012; Póczos et al., 2013).”
Integrating structured biological data by Kernel Maximum Mean Discrepancy (2006, background): “For detailed discussions of the problem of defining kernels between distributions and sets, see (Cuturi et al., 2005; Hein and Bousquet, 2005).”
AdaGAN: Boosting Generative Models (2017, methodology): “As demonstrated in [16, 17], several commonly used symmetric f -divergences are Hilbertian metrics, which in particular means that their square root satisfies the triangle inequality.”
Nonextensive Information Theoretic Kernels on Measures (2009, background): “A metric d : X ×X → R is Hilbertian if there is some Hilbert space H and an isometry f : X → H such that d2(x, y) = 〈f(x)− f(y), f(x)− f(y)〉H holds for any x, y ∈ X (Hein and Bousquet, 2005).”
In Defense of Minhash over Simhash (2014, background): “It is often the case that just the presence or absence information suﬃces in practice [7, 14, 17, 23].”