Dependence and the dimensionality reduction principle

Stone’s dimensionality reduction principle has been confirmed on several occasions for independent observations. When dependence is expressed with ϕ-mixing, a minimum distance estimate is proposed for a smooth projection pursuit regression-type function θ∈Я, that is either additive or multiplicative, in the presence of or without interactions. Upper bounds on theL ₁-risk and theL ₁-error of are obtained, under restrictions on the order of decay of the mixing coefficient. The bounds show explicitly the addive effect of ϕ-mixing on the error, and confirm the dimensionality reduction principle.     

Zeta functions of three-dimensional <Emphasis Type="Italic">p</Emphasis>-adic Lie algebras

We give an explicit formula for the subalgebra zeta function of a general three-dimensional Lie algebra over the p-adic integers . To this end, we associate to such a Lie algebra a ternary quadratic form over . The formula for the zeta function is given in terms of Igusa’s local zeta function associated to this form. We acknowledge support from the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation.     

Bruijning—Nagata and Hashimoto—Hattori characteristics of covering dimension revisited

New dimension functions for topological spaces are introduced in the spirit of Nagata’s approach. Expressions for the new functions in terms of covering dimension include the Bruijning—Nagata and Hashimoto—Hattori formulas.     

A new algorithm of non-Gaussian component analysis with radial kernel functions

We consider high-dimensional data which contains a linear low-dimensional non-Gaussian structure contaminated with Gaussian noise, and discuss a method to identify this non-Gaussian subspace. For this problem, we provided in our previous work a very general semi-parametric framework called non-Gaussian component analysis (NGCA). NGCA has a uniform probabilistic bound on the error of finding the non-Gaussian components and within this framework, we presented an efficient NGCA algorithm called Multi-index Projection Pursuit. The algorithm is justified as an extension of the ordinary projection pursuit (PP) methods and is shown to outperform PP particularly when the data has complicated non-Gaussian structure. However, it turns out that multi-index PP is not optimal in the context of NGCA. In this article, we therefore develop an alternative algorithm called iterative metric adaptation for radial kernel functions (IMAK), which is theoretically better justifiable within the NGCA framework. We demonstrate that the new algorithm tends to outperform existing methods through numerical examples.     

The group reduction for bounded cosine functions on UMD spaces

It is shown that if A generates a bounded cosine operator function on a UMD space X, then i(−A)^1/2 generates a bounded C ₀-group. The proof uses a transference principle for cosine functions.