Generalized Hadamard Product and the Derivatives of Spectral Functions

In this work we propose a generalization of the Hadamard product between two matrices to a tensor-valued, multi-linear product between k matrices for any $k \ge 1$. A multi-linear dual operator to the generalized Hadamard product is presented. It is a natural generalization of the Diag x operator, that maps a vector $x \in \R^n$ into the diagonal matrix with x on its main diagonal. Defining an action of the $n \times n$ orthogonal matrices on the space of k-dimensional tensors, we investigate its interactions with the generalized Hadamard product and its dual. The research is motivated, as illustrated throughout the paper, by the apparent suitability of this language to describe the higher-order derivatives of spectral functions and the tools needed to compute them. For more on the later we refer the reader to [14] and [15], where we use the language and properties developed here to study the higher-order derivatives of spectral functions.Comment: 24 page

The higher-order derivatives of spectral functions

AbstractWe are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices

Completely monotone and Bernstein functions with convexity properties on their measures

The concepts of completely monotone and Bernstein functions have been introduced near one hundred years ago. They find wide applications in areas ranging from stochastic L\\u27{e}vy processes and complex analysis to monotone operator theory. They have well-known Bernstein and L\\u27{e}vy-Khintchine integral representations through which there are one-to-one correspondences between them and Radon measures on $[0,\infty)$ or $(0,\infty)$, respectively. In this thesis, we investigate subclasses of completely monotone and Bernstein functions with various convexity properties on their measures. These subclasses have intriguing applications in probability theories and convex analysis. The convexity properties we investigate include convexity, harmonic convexity and $\beta$-convexity of the cumulative distribution functions. We characterize measures with various convexity properties to obtain results analogous to the classical P\\u27{o}lya\u27s Theorem. Then we apply these characterizations of the measures to derive integral representations for these classes of completely monotone and Bernstein functions that are variants of the classical Bernstein and L\\u27{e}vy-Khintchine integral representations. To explore the connections among completely monotone and Bernstein functions with various convexity properties on their measures, we investigate the characterizations and obtain various necessary and sufficient conditions for a completely monotone or Bernstein function to belong to one of the subclasses. We also identify maps that transform completely monotone and Bernstein functions into one with certain convexity properties on their measures. Interesting parallels between completely monotone and Bernstein functions are observed. For example, the transformation that turn a Bernstein function into one having L\\u27{e}vy measure with harmonically concave tail is the same as the transformation that turns a completely monotone function into one having harmonically convex measure. To help understand these analogies, a criteria for completely monotone and Bernstein function to have measures with $\beta$-convexity property is obtained.That generalizes the conditions for both convexity and harmonic convexity. Let $\mathcal{H}_{CM}$ be the set of all Bernstein functions $h$, such that $f\circ h$ is the Laplace transform of a harmonically convex measure for {\it any} completely monotone function $f$. Similarly, let $\mathcal{H}_{BF}$ be the set of all Bernstein functions $h$, such that $g\circ h$ has L\\u27{e}vy measure with harmonically concave tail for {\it any} Bernstein function $g$. Surprisingly, we show that $\mathcal{H}_{CM} = \mathcal{H}_{BF}$ and are non-empty. For example we prove that $x^\alpha$ is in $\mathcal{H}_{BF}$ for any $\alpha \in (0, {2}/{3}]$. In other words, the Bernstein function $x \mapsto x^\alpha$ is a transformation that deforms the measure of any Bernstein (resp. completely monotone) function into one that not only has a continuous distribution function on $(0,\infty)$ but also a convenient concavity (reps. convexity) property. We give necessary and sufficient condition for a Bernstein function to be in $\mathcal{H}_{BF}$ in terms of its convolution semigroups of sub-probability measures. However, it is not well-understood what are the functions that ``generate\u27\u27 this set. We hope to investigate such issues in the future

Clarke Generalized Jacobian of the Projection onto the Cone of Positive Semidefinite Matrices

International audienceThis paper studies the differentiability properties of the projection onto the cone of positive semidefinite matrices. In particular, the expression of the Clarke generalized Jacobian of the projection at any symmetric matrix is given