Harm Derksen

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In this paper, based on ideas from lossy data coding and compression, we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions, which are allowed to be almost degenerate. The goal is to find the optimal segmentation that minimizes the overall coding length of the segmented(More)
Recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high-dimensional data that have very complicated structures. From both geometric and statistical points of view, such unsorted data are considered mixed as different parts of the data have significantly different structures which(More)
Σ(Q,α) is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities. In particular the set Σ(Q,α) is saturated, i.e., if nσ ∈ Σ(Q,α), then also σ ∈ Σ(Q,α). These results, when applied to a special quiver Q = Tn,n,n and to a special dimension vector, show that the GLn-module Vλ appears in Vμ(More)
Invariant theory has been a major subject of research in the 19th century. One of the highlights was Gordan’s famous theorem from 1868 showing that the invariants and covariants of binary forms have a finite basis. His method was constructive and led to explicit degree bounds for a system of generators (Jordan 1876/79). In 1890, Hilbert presented a very(More)
The vanishing ideal I of a subspace arrangement V1 ∪ V2 ∪ · · · ∪ Vm ⊆ V is an intersection I1 ∩ I2 ∩ · · · ∩ Im of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of the product ideal J = I1I2 · · · Im without any assumptions on the subspace arrangement. It(More)
Hilbert proved that invariant rings are finitely generated for linearly reductive groups acting rationally on a finite dimensional vector space. Popov gave an explicit upper bound for the smallest integer d such that the invariants of degree ≤ d generate the invariant ring. This bound has factorial growth. In this paper we will give a bound which depends(More)
We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family(More)
Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in [2]. Here we study the minimal free resolution of the invariant ring. Recently it was shown that if G is a finite linearly reductive group, then the ring of invariants is generated in degree ≤ |G| (see [5, 6, 3]). This extends the classical(More)
Finding the rank of a tensor is a problem that has many applications. Unfortunately it is often very difficult to determine the rank of a given tensor. Inspired by the heuristics of convex relaxation, we consider the nuclear norm instead of the rank of a tensor. We determine the nuclear norm of various tensors of interest. Along the way, we also do a(More)