Testing for Homogeneity with Kernel Fisher Discriminant Analysis

By Zaïd Harchaoui, Francis R. Bach, Eric Moulines. 2007

126 citations recorded by the source corpus.

We propose to investigate test statistics for testing homogeneity based on kernel Fisher discriminant analysis. Asymptotic null distributions under null hypothesis are derived, and consistency against fixed alternatives is assessed. Finally, experimental evidence of the performance of the proposed approach on both artificial and real datasets is provided.

Subfields: Artificial Intelligence, Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Interpretable Distribution Features with Maximum Testing Power (2016, methodology): “…=1 drawn independently and identically (i.i.d.) from P , and Y := { y i } ni =1 ⊂ R d from Q. Non-parametric two-sample tests based on RKHS distances (Eric et al., 2008; Fromont et al., 2012; Gretton et al., 2012a) or energy distances (Székely and Rizzo, 2004; Baringhaus and Franz, 2004) have as…”
Fast Two-Sample Testing with Analytic Representations of Probability Measures (2015, methodology): “Tests may also be based on quantities other than distances, an example being the Kernel Fisher Discriminant (KFD) [12], the estimation of which still requires calculating the RKHS norm of a difference of mean embeddings, with normalization by an inverse covariance operator.”
A Wild Bootstrap for Degenerate Kernel Tests (2014, methodology): “Statistical tests based on distribution embeddings into reproducing kernel Hilbert spaces have been applied in many contexts, including two sample testing [18, 15, 32], tests of independence [17, 33, 4], tests of conditional independence [14, 33], and tests for higher order (Lancaster) interactions ”
On the Decreasing Power of Kernel and Distance Based Nonparametric Hypothesis Tests in High Dimensions (2014, neutral): “(2012a); Eric et al. (2008); Gretton et al.”
Testing Hypotheses by Regularized Maximum Mean Discrepancy (2013, methodology): “This is relevant for many applications, for instance, schema matching in databases [9], and speaker identification [13].”
A Kernel Method for the Two-Sample-Problem (2006, methodology): “Finally, we note that Harchaoui et al. (2008) recently proposed a modification of the kernel MMD statistic in Lemma 4, by scaling the feature space mean distance using the inverse within-sample covariance operator, thus employing the kernel Fisher discriminant as a statistic for testing homogeneity.”
B-tests: Low Variance Kernel Two-Sample Tests (2013, methodology): “Kernel test statistics include the maximum mean discrepancy [10, 6] (of which the energy distance is an example [18, 2, 22]), which is the distance between expected features of P and Q in the RKHS; the kernel Fisher discriminant [12], which is the distance between expected feature maps normalized by”
Self-concordant analysis for logistic regression (2009, methodology): “Such quantity is an extension of the one used by [30] in the context of kernel Fisher discriminant analysis used as a test for homogeneity.”
A Fast, Consistent Kernel Two-Sample Test (2009, methodology): “We anticipate that our approach may also be generalized to kernel independence tests [15], and to homogeneity tests based on the kernel Fisher discriminant [18].”
Optimal kernel choice for large-scale two-sample tests (2012, methodology): “A number of recent approaches to two-sample testing have made use of mappings of the distributions to a reproducing kernel Hilbert space (RKHS); or have sought out RKHS functions with large amplitude where the probability mass of p and q differs most [8, 10, 15, 17, 7].”
A review of change point detection methods (2018, methodology): “In the context of single change point detection, examples of such tests include rankbased procedures [38, 94, 97], kernel Fisher discriminants [62, 63, 103], density ratio [74], etc.”
B-test: A Non-parametric, Low Variance Kernel Two-sample Test (2013, background): “These include the maximum mean discrepancy [9, 6] (of which the energy distance is an example [15, 2, 19]), which is the distance between expected features of P and Q in the RHKS; the kernel Fisher discriminant [11], which is the distance between expected feature maps normalized by the feature space”
Kernel Change-point Analysis (2008, methodology): “Following [9], we make the following assumptions on all the covariance operators Σ considered in this paper: (B1) the eigenvalues {λp(Σ)}p≥1 satisfy ∑∞ p=1 λ 1/2 p (Σ) < ∞, (B2) there are infinitely many strictly positive eigenvalues {λp(Σ)}p≥1 of Σ.”
A More Powerful Two-Sample Test in High Dimensions using Random Projection (2011, methodology): “In Section 4 we provide performance comparisons with ROC curves on synthetic data against a broader collection of methods, including some recent kernel-based and non-parametric approaches such as MMD [11], KFDA [12], and TreeRank [10].”