Statistical properties of kernel principal component analysis

By Gilles Blanchard, Olivier Bousquet, Laurent Zwald. 2004

172 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistics and Probability

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Many classical learning algorithms—such as the perceptron (Rosenblatt 1958), support vector machine (SVM) (Cortes and Vapnik 1995), and principal component analysis (PCA) (Pearson 1901, Hotelling 1933)—employ data instances, e.g., x,x′ ∈ X , only through an inner product 〈x,x′〉, which basically is a”
Testing for Homogeneity with Kernel Fisher Discriminant Analysis (2007, methodology): “all make frequent use of tensor product notations. The tensor product operator u⊗vfor u,v∈ H is deﬁned for all f∈ H as (u⊗v)f= hv,fiH u. We now introduce the mean element and covariance operator (see Blanchard et al., 2007). If R k1/2(x,x)P(dx) <∞, the mean element µP is deﬁned as the unique elem”
Geometry on Probability Spaces (2009, background): “Some analysis related to Proposition 1 can be found in [19, 14, 5, 12].”
Statistical performance of support vector machines (2008, result): “If we compare this to our main result Theorem 2, in setting (S1) , we see that our complexity penalty is always of smaller order (up to a a constant factor, and to the relation between empirical and true spectrum, which we do not cover here, but is studied e.g. in [8, 29]).”
On Relevant Dimensions in Kernel Feature Spaces (2008, background): “Note that the problem considered here is significantly different from the problem studying the performance of kernel PCA itself (see, for example, Blanchard et al., 2007; Shawe-Taylor et al., 2005; Mika, 2002).”
Kernel Change-point Analysis (2008, methodology): “This is described in the operator-theoretic framework, developed for the statistical analysis of kernel-based learning and testing algorithms in [10, 11].”
A Survey of Modern Questions and Challenges in Feature Extraction (2015, methodology): “In this case argmin f ∈F L(f, x 1 , ..., x m ) is the rank-d Kernel PCA (KPCA) algorithm (Schölkopf et al., 1997; Blanchard et al., 2007), which is the projection onto the top-d eigenspace of the empirical covariance operator in H K .”
Statistical analysis of coupled time series with Kernel Cross-Spectral Density operators (2013, methodology): “• the Mean Element (see [3, 1]): μi = E [Xi] , • the Cross-covariance operator (see [6]): Cij = Cov [Xi,Xj ] = E[Xi ⊗Xj ]− μi ⊗ μj , where we use the tensor product notation f ⊗ g∗ to represent the rank one operator defined by f ⊗g∗ = 〈 g, .”
Measuring Statistical Dependence with Hilbert-Schmidt Norms (2005, background): “Our presentation follows [21, 12], the main difference being that we deal with cross-covariance operators rather than the covariance operators.”
Tail inequalities for sums of random matrices that depend on the intrinsic dimension (2012, unknown): “A related result was given in [29] for Frobenius norm error ‖Σ̂n − Σ‖F rather than spectral norm error.”
The Fast Convergence of Incremental PCA (2013, background): “There is an extensive line of work analyzing PCA from the statistical perspective, in which the convergence of various estimators is characterized under certain conditions, including generative models of the data [5] and various assumptions on the covariance matrix spectrum [14, 4] and eigenvalue sp”
Kernel-Based Methods for Hypothesis Testing: A Unified View (2013, unknown): “Then, the eigenfunctions of the corresponding covariance operator can be expressed analytically using Hermite polynomials [40].”
A Fast, Consistent Kernel Two-Sample Test (2009, methodology): “from which it follows that the covariance operator must be Hilbert-Schmidt: this is guaranteed by the assumption ∑∞ l=1 λ 1/2 l < ∞ (see also [7]).”