$L_p$-nested symmetric distributions

Abstract

Tractable generalizations of the Gaussian distribution play an important role for the analysis of high-dimensional data. One very general super-class of Normal distributions is the class of $\nu$-spherical distributions whose random variables can be represented as the product $\x = r\cdot \u$ of a uniformly distribution random variable $\u$ on the $1$-level set of a positively homogeneous function $\nu$ and arbitrary positive radial random variable $r$. Prominent subclasses of $\nu$-spherical distributions are spherically symmetric distributions ($\nu(\x)=\|\x\|_2$) which have been further generalized to the class of $L_p$-spherically symmetric distributions ($\nu(\x)=\|\x\|_p$). Both of these classes contain the Gaussian as a special case. In general, however, $\nu$-spherical distributions are computationally intractable since, for instance, the normalization constant or fast sampling algorithms are unknown for an arbitrary $\nu$. In this paper we introduce a new subclass of $\nu$-spherical distributions by choosing $\nu$ to be a nested cascade of $L_p$-norms. This class is still computationally tractable, but includes all the aforementioned subclasses as a special case. We derive a general expression for $L_p$-nested symmetric distributions as well as the uniform distribution on the $L_p$-nested unit sphere, including an explicit expression for the normalization constant. We state several general properties of $L_p$-nested symmetric distributions, investigate its marginals, maximum likelihood fitting and discuss its tight links to well known machine learning methods such as Independent Component Analysis (ICA), Independent Subspace Analysis (ISA) and mixed norm regularizers. Finally, we derive a fast and exact sampling algorithm for arbitrary $L_p$-nested symmetric distributions, and introduce the Nested Radial Factorization algorithm (NRF), which is a form of non-linear ICA.