Joint measures and cross-covariance operators

By C. Baker. 1973

Subfields: Probabilistic & Bayesian Methods

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “The MMD test statistic for testing independence can then be written as M̂MD(H ⊗ G ,PXY ,PX ⊗ PY ) = ‖µ̂PXY − µ̂PX⊗PY ‖2H ⊗G , (3.39) where µ̂PXY and µ̂PX⊗PY denote the empirical estimates of PXY and PX⊗PY , respectively.”
Domain Adaptation with Conditional Transferable Components (2016, background): “According to its deﬁnition (Baker, 1973), , where Σ ·· is the covariance or cross-covariance operator.”
On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions (2015, background): “It should not be confused with the (non-centered) covariance operator C (Baker, 1973), which is a self-adjoint operator on a different space, namely the RKHS F , deﬁned by g, Cf F = X f ( x ) g ( x ) dρ ( x ) .”
Kernel Mean Shrinkage Estimators (2014, neutral): “operator ΣYX : HX → HY such that 〈g,ΣYX f〉HY = EXY [(f(X)− EX [f(X)])(g(Y )− EY [g(Y )])] = Cov(f(X), g(Y )) holds for all f ∈ HX and g ∈ HY (Baker, 1973; Fukumizu et al., 2004).”
Hilbert Space Embeddings of Predictive State Representations (2013, background): “Given a joint distribution P [X,Y ] over two variables X on X and Y on Y, the uncentered covariance operator CXY is the linear operator which satisfies [17] 〈f, CXY g〉F = EXY [f(X)g(Y )] ∀f ∈ F , g ∈ G (7) Both µXY and CXY represent the distribution P [X,Y ].”
Kernel Adaptive Metropolis-Hastings (2013, background): “Next, the covariance operatorCP : Hk → Hk for a probability measureP is given byCP = ´ k(·, x) ⊗ k(·, x) dP (x) − µP ⊗ µP (Baker, 1973; Fukumizu et al., 2004), where fora, b, c ∈ Hk the tensor product is defined as(a ⊗ b)c = 〈b, c〉Hk a.”
Kernel Bayes' Rule (2010, positive): “By the decomposition Ĉ (n) Y X = Ĉ (n)1/2 Y Y Ŵ (n) Y XĈ (n)1/2 XX with ‖Ŵ (n) YX‖ ≤ 1 [2], we have ‖Ĉ YX ( Ĉ (n) XX + εnI −1‖ = Op(ε n ), which implies the first term is of Op(n ε −1/2 n ).”
Kernel dimension reduction in regression (2009, background): “The random element kX (·,X) has strong order two, and E[k(·,X)] equals mX , where mX is given by (4).”
Feature Selection via Dependence Maximization (2012, methodology): “This allows us to compute a measure of dependence between x and y simply by taking expectations over a set of kernel functions k and l with respect to the joint and marginal distributions in x and y without the need to perform density estimation (as may be needed for entropy based methods).”
Nonparametric Tree Graphical Models via Kernel Embeddings (2010, background): “Following (Baker, 1973; Fukumizu et al., 2004), we define the uncentered covariance operator CXX : F → F such that for all f ∈ F ,”
Supervised principal component analysis: Visualization, classification and regression on subspaces and submanifolds (2011, background): “According to [28,20], this operator can be defined as”
Kernel Dimensionality Reduction for Supervised Learning (2003, methodology): “ΣXY can be decomposed as ΣXY = Σ 1/2 XXV Σ 1/2 Y Y for a bounded operatorV (Theorem 1, [10]).”
Hilbert Space Embeddings of Hidden Markov Models (2010, methodology): “Given a joint distribution P(X,Y ) over two variables X on X and Y on Y1, the uncentered covariance operator CXY is (Baker, 1973) CXY := EXY [ϕ(X)⊗ φ(Y )], (15) where ⊗ denotes tensor product.”
Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces (2004, background): “(1987), Baker (1973), they are more easily defined on reproducing kernel Hilbert spaces (RKHS).”
Kernel Bayes' rule: Bayesian inference with positive definite kernels (2013, unknown): “By the decomposition Ĉ (n) Y X = Ĉ (n)1/2 YY Ŵ (n) Y X Ĉ (n)1/2 XX with ‖Ŵ (n) Y X ‖ ≤ 1 (Baker, 1973), we have ‖Ĉ Y X(Ĉ (n) XX + εnI) −1‖ ≤ ‖Ĉ YY Ŵ (n) Y X (Ĉ (n) XX + εnI) −1/2‖= Op(ε n ),”
A Hilbert-Schmidt Dependence Maximization Approach to Unsupervised Structure Discovery (2008, background): “We know from (Baker, 1973) that the covariance operator can be decomposed as Cxy = C1/2xx VxyC 1/2 yy , where Vxy is the normalized cross-covariance operator (its maximum singular value is bounded by 1).”
Consistency of the group Lasso and multiple kernel learning (2007, unknown): “Given a random variable X on X with bounded second order moment, that is, such that E k ( X ; X ) < , we can deﬁne the covariance operator as the bounded linear operator The operator S XX is auto-adjoint , non-negative and Hilbert-Schmidt , that is, for any orthonormal basis ( e p ) p > 1 of F , the”
Central Subspace Dimensionality Reduction Using Covariance Operators (2011, unknown): “covariance operator theorems [2], [7], [27].”
Nonlinear sufficient dimension reduction for functional data (2017, unknown): “See also Baker (1973).”