Sparse non Gaussian component analysis by semidefinite programming

Diederichs, Elmar; Juditsky, Anatoli; Nemirovski, Arkadi; Spokoiny, Vladimir

doi:10.1007/s10994-013-5331-1

Cited by 9 publications

(9 citation statements)

References 25 publications

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“…Finally, we illustrate our test and present a sequential procedure that assesses the rank of a covariance operator. The problem of covariance rank estimation is adressed in several domains: functional regression [9,7], classification [41] and dimension reduction methods such as PCA, Kernel PCA and Non-Gaussian Component Analysis [3,12,13] where the dimension of the kept subspace is a crucial problem. 3 Here is the outline of the paper.…”

Section: Introductionmentioning

confidence: 99%

A one-sample test for normality with kernel methods

Kellner¹,

Célisse²

2019

Bernoulli

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We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure.

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Section: Introductionmentioning

confidence: 99%

A one-sample test for normality with kernel methods

Kellner¹,

Célisse²

2019

Bernoulli

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“…In particular, [BKS + 06] provide an algorithm based on Stein's characterization of the Gaussian random variable and [DJNS13] use semi-definite programming. We remark that while the NGCA problem resembles ICA problem, the algorithms for the latter do not seem to be directly applicable to the former.…”

Section: Related Workmentioning

confidence: 99%

Non-Gaussian component analysis using entropy methods

Goyal

Shetty

2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Non-Gaussian component analysis (NGCA) is a problem in multidimensional data analysis which, since its formulation in 2006, has attracted considerable attention in statistics and machine learning. In this problem, we have a random variable X in n-dimensional Euclidean space. There is an unknown subspace Γ of the n-dimensional Euclidean space such that the orthogonal projection of X onto Γ is standard multidimensional Gaussian and the orthogonal projection of X onto Γ ⊥ , the orthogonal complement of Γ, is non-Gaussian, in the sense that all its one-dimensional marginals are different from the Gaussian in a certain metric defined in terms of moments. The NGCA problem is to approximate the non-Gaussian subspace Γ ⊥ given samples of X.Vectors in Γ ⊥ correspond to 'interesting' directions, whereas vectors in Γ correspond to the directions where data is very noisy. The most interesting applications of the NGCA model is for the case when the magnitude of the noise is comparable to that of the true signal, a setting in which traditional noise reduction techniques such as PCA don't apply directly. NGCA is also related to dimension reduction and to other data analysis problems such as ICA. NGCA-like problems have been studied in statistics for a long time using techniques such as projection pursuit.We give an algorithm that takes polynomial time in the dimension n and has an inverse polynomial dependence on the error parameter measuring the angle distance between the non-Gaussian subspace and the subspace output by the algorithm. Our algorithm is based on relative entropy as the contrast function and fits under the projection pursuit framework. The techniques we develop for analyzing our algorithm maybe of use for other related problems.

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“…Instead, the loadings matrix (the matrix which performs the dimension reduction) can be calculated using a smaller, representative dataset, and then the transformation quickly calculated for the rest of the data. PCA was introduced for Gaussian distributed data and has been extended significantly over the years to address different research problems (see for example Tipping and Bishop (1999), Collins et al (2002), Ding andHe (2004), de Leeuw (2006) and Diederichs et al (2013) among others). The need to develop a generalised version of PCA to address application to the exponential family of distributions (which encompasses a wide variety of models for real-world data) was recognised in Landgraf and Lee (2015) who developed Generalised Principal Component Analysis (GPCA).…”

Section: Introductionmentioning

confidence: 99%

Sparse Generalised Principal Component Analysis

Smallman

Artemiou

Morgan

2018

Pattern Recognition

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In this paper, we develop a sparse method for unsupervised dimension reduction for data from an exponential-family distribution. Our idea extends previous work on Generalised Principal Component Analysis by adding L 1 and SCAD penalties to introduce sparsity. We demonstrate the significance and advantages of our method with synthetic and real data examples. We focus on the application to text data which is high-dimensional and non-Gaussian by nature and discuss the potential advantages of our methodology in achieving dimension reduction.

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Sparse non Gaussian component analysis by semidefinite programming

Cited by 9 publications

References 25 publications

A one-sample test for normality with kernel methods

A one-sample test for normality with kernel methods

Non-Gaussian component analysis using entropy methods

Sparse Generalised Principal Component Analysis

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