arXiv subjects

Peter Bürgisser

Publications and source records attributed to Peter Bürgisser.

42 records · Page 3Linked to original sources

arXiv2012

Nonvanishing of Kronecker coefficients for rectangular shapes

We prove that for any partition $(λ_1,...,λ_{d^2})$ of size $\ell d$ there exists $k\ge 1$ such that the tensor square of the irreducible representation of the symmetric group $S_{k\ell d}$ with respect to the rectangular partition $(k\ell,...,k\ell)$ contains the irreducible representation corresponding to the stretched partition $(kλ_1,...,kλ_{d^2})$. We also prove a related approximate version of this statement in which the stretching factor $k$ is effectively bounded in terms of $d$. This investigation is motivated by questions of geometric complexity theory.

math.GR↗

arXiv2012

A coordinate-free condition number for convex programming

We introduce and analyze a natural geometric version of Renegar's condition number R for the homogeneous convex feasibility problem associated with a regular cone C subseteq R^n. Let Gr_{n,m} denote the Grassmann manifold of m-dimensional linear subspaces of R^n and consider the projection distance d_p(W_1,W_2) := ||Pi_{W_1} - Pi_{W_2}|| (spectral norm) between W_1 and W_2 in Gr_{n,m}, where Pi_{W_i} denotes the orthogonal projection onto W_i. We call C_G(W) := max {d_p(W,W')^{-1} | W' \in Sigma_m} the Grassmann condition number of W in Gr_{n,m}, where the set of ill-posed instances Sigma_m subset Gr_{n,m} is defined as the set of linear subspaces touching C. We show that if W = im(A^T) for a matrix A in R^{m\times n}, then C_G(W) \le R(A) \le C_G(W) kappa(A), where kappa(A) =||A|| ||A^\dagger|| denotes the matrix condition number. This extends work by Belloni and Freund in Math. Program. 119:95-107 (2009). Furthermore, we show that C_G(W) can as well be characterized in terms of the Riemannian distance metric on Gr_{n,m}. This differential geometric characterization of C_G(W) is the starting point of the sequel [arXiv:1112.2603] to this paper, where the first probabilistic analysis of Renegar's condition number for an arbitrary regular cone C is achieved.

math.OC↗

arXiv2012

Intrinsic volumes of symmetric cones

We compute the intrinsic volumes of the cone of positive semidefinite matrices over the real numbers, over the complex numbers, and over the quaternions, in terms of integrals related to Mehta's integral. Several applications for the probabilistic analysis of semidefinite programming are given.

math.OC↗

arXiv2010

Even Partitions in Plethysms

We prove that for all natural numbers k,n,d with k <= d and every partition lambda of size kn with at most k parts there exists an irreducible GL(d, C)-representation of highest weight 2*lambda in the plethysm Sym^k(Sym^(2n) (C^d)). This gives an affirmative answer to a conjecture by Weintraub (J. Algebra, 129 (1):103-114, 1990). Our investigation is motivated by questions of geometric complexity theory and uses ideas from quantum information theory.

math.GR↗

arXiv2010

Coverage processes on spheres and condition numbers for linear programming

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,α)$ be the probability that $n$ spherical caps of angular radius $α$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,α)$ in the case $α\in[π/2,π]$ and an upper bound for $p(n,m,α)$ in the case $α\in [0,π/2]$ which tends to $p(n,m,π/2)$ when $α\toπ/2$. In the case $α\in[0,π/2]$ this yields upper bounds for the expected number of spherical caps of radius $α$ that are needed to cover $S^m$. Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem $\exists x\in\mathbb{R}^{m+1}Ax\le0,x\ne0$ where $A\in\mathbb{R}^{n\times(m+1)}$ is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to $A$ being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all $n>m$, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.

math.PR↗

arXiv2010

Robust Smoothed Analysis of a Condition Number for Linear Programming

We perform a smoothed analysis of the GCC-condition number C(A) of the linear programming feasibility problem \exists x\in\R^{m+1} Ax < 0. Suppose that \bar{A} is any matrix with rows \bar{a_i} of euclidean norm 1 and, independently for all i, let a_i be a random perturbation of \bar{a_i} following the uniform distribution in the spherical disk in S^m of angular radius \arcsinσand centered at \bar{a_i}. We prove that E(\ln C(A)) = O(mn / σ). A similar result was shown for Renegar's condition number and Gaussian perturbations by Dunagan, Spielman, and Teng [arXiv:cs.DS/0302011]. Our result is robust in the sense that it easily extends to radially symmetric probability distributions supported on a spherical disk of radius \arcsinσ, whose density may even have a singularity at the center of the perturbation. Our proofs combine ideas from a recent paper of Bürgisser, Cucker, and Lotz (Math. Comp. 77, No. 263, 2008) with techniques of Dunagan et al.

math.OC↗