The Bernstein Function: A Unifying Framework of Nonconvex Penalization in Sparse Estimation

In this paper we study nonconvex penalization using Bernstein functions. Since the Bernstein function is concave and nonsmooth at the origin, it can induce a class of nonconvex functions for high-dimensional sparse estimation problems. We derive a threshold function based on the Bernstein penalty and give its mathematical properties in sparsity modeling. We show that a coordinate descent algorithm is especially appropriate for penalized regression problems with the Bernstein penalty. Additionally, we prove that the Bernstein function can be defined as the concave conjugate of a $\varphi$-divergence and develop a conjugate maximization algorithm for finding the sparse solution. Finally, we particularly exemplify a family of Bernstein nonconvex penalties based on a generalized Gamma measure and conduct empirical analysis for this family

TYPOLOGY OF NOTHING: HEIDEGGER, DAOISM AND BUDDHISM [abstract]

Parmenides expelled nonbeing from the realm of knowledge and forbade us to think or talk about it. But still there has been a long tradition of nay-sayings throughout the history of Western and Eastern philosophy. Are those philosophers talking about the same nonbeing or nothing? If not, how do their concepts of nothing differ from each other? Could there be different types of nothing? Surveying the traditional classifications of nothing or nonbeing in the East and West have led me to develop a typology of nothing that consists of three main types: 1) privative nothing, commonly known as absence; 2) negative nothing, the altogether not or absolute nothing; and finally 3) original nothing, the nothing that is equivalent to being. I will test my threefold typology of nothing by comparing the similarities and differences between the conceptions of nothing in Heidegger, Daoism and Buddhism. These are three of the very few philosophical strains that have launched themselves into the wonderland of negativity by developing respectively the concepts of nothing (Nichts), nothing (wu 無) and emptiness (sunyata). With this analysis, I hope that I will clarify some confusion in the understanding of nothing in Heidegger, Daoism and Buddhism, and shed light on the central philosophical issue of what there is not

Finite-dimensional subalgebras of the Virasoro algebra

We determine all two-dimensional Lie subalgebras of the centreless Virasoro algebra and complete the characterization of all finite dimensional Lie subalgebras of the complex Virasoro algebra

The Matrix Ridge Approximation: Algorithms and Applications

We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matrix factorization plus a ridge term. Moreover, we present probabilistic interpretations using a normal latent variable model and a Wishart model for this approximation approach. The idea behind the latent variable model in turn leads us to an efficient EM iterative method for handling the matrix ridge approximation problem. Finally, we illustrate the applications of the approximation approach in multivariate data analysis. Empirical studies in spectral clustering and Gaussian process regression show that the matrix ridge approximation with the EM iteration is potentially useful

Characterisation of matrix entropies

The notion of matrix entropy was introduced by Tropp and Chen with the aim of measuring the fluctuations of random matrices. It is a certain entropy functional constructed from a representing function with prescribed properties, and Tropp and Chen gave some examples. We give several abstract characterisations of matrix entropies together with a sufficient condition in terms of the second derivative of their representing function.Comment: Major revision. We found an error in the previous version that we cannot repair. It implies that we no longer can be certain that the sufficient condition of operator convexity of the second derivative of a matrix entropy is also necessary. We added more abstract characterisations of matrix entropies and improved the analysis of the concrete example