Harm Derksen

Segmentation of Multivariate Mixed Data via Lossy Data Coding and Compression

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 data, subject to a given distortion. By analyzing the coding length/rate of mixed data, we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and rate-distortion theory. We show that a deterministic segmentation is approximately the (asymptotically) optimal solution for compressing mixed data. We propose a very simple and effective algorithm that depends on a single parameter, the allowable distortion. At any given distortion, the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation. Simulation results reveal intriguing phase-transition-like behaviors of the number of segments when changing the level of distortion or the amount of outliers. Finally, we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data.

Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data

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 cannot be described by a single model. In this paper we propose to use subspace arrangements—a union of multiple subspaces—for modeling mixed data: each subspace in the arrangement is used to model just a homogeneous subset of the data. Thus, multiple subspaces together can capture the heterogeneous structures within the data set. In this paper, we give a comprehensive introduction to a new approach for the estimation of subspace arrangements. This is known as generalized principal component analysis (GPCA). In particular, we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust, even when the given data are corrupted by noise or contaminated with outliers. This new method in many ways improves and generalizes extant methods for modeling or clustering mixed data. There have been successful applications of this new method to many real-world problems in computer vision, image processing, and system identification. In this paper, we will examine several of those representative applications. This paper is intended to be expository in nature. However, in order that this may serve as a more complete reference for both theoreticians and practitioners, we take the liberty of filling in several gaps between the theory and the practice in the existing literature.

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

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 ⊗ Vν if and only if the partitions λ, μ and ν satisfy an explicit set of inequalities. This gives new proofs of the results of Klyachko ([7, 3]) and Knutson and Tao ([8]). The proof is based on another general result about semi-invariants of quivers (Theorem 1). In the paper [10], Schofield defined a semi-invariant cW for each indecomposable representation W of Q. We show that the semi-invariants of this type span each weight space in SI(Q,α). This seems to be a fundamental fact, connecting semi-invariants and modules in a direct way. Given this fact, the results on sets of weights follow at once from the results in another paper of Schofield [11].

On the Littlewood-Richardson polynomials

We prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers, due to Domokos and Zubkov, Schofield and van den Bergh, and our earlier work. We also show that the dimensions of semi-invariants of weights nσ depend polynomially on n.  2002 Elsevier Science (USA). All rights reserved.

On the Canonical Decomposition of Quiver Representations

Kac introduced the notion of the canonical decomposition for a dimension vector of a quiver. Here we will give an efficient algorithm to compute the canonical decomposition. Our study of the canonical decomposition for quivers with three vertices gives us fractal-like pictures.

A sharp bound for the Castelnuovo–Mumford regularity of subspace arrangements

Abstract We show that the ideal of an arrangement of d linear subspaces of projective space is d -regular in the sense of Castelnuovo and Mumford, answering a question of B. Sturmfels. In particular, this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d .

Quantum automata and algebraic groups

We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.

Polynomial bounds for rings of invariants

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 only polynomially on the input data.

The kernel of a derivation

Let K be a field of characteristic 0. Nagata and Nowicki have shown that the kernel of a derivation on K[X1,…,Xn] is of finite type over K if n≤3. We construct a derivation of a polynomial ring in 32 variables which kernel is not of finite type over K. Furthermore we show that for every field extension L over K of finite transcendence degree, every intermediate field which is algebraically closed in L is the kernel of a K-derivation of L.

Segmentation of multivariate mixed data via lossy coding and compression

In this paper, based on ideas from lossy data coding and compression, we present a simple but surprisingly effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions or linear subspaces. The goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data, subject to a given distortion. We show that deterministic segmentation minimizes an upper bound on the (asymptotically) optimal solution. The proposed algorithm does not require any prior knowledge of the number or dimension of the groups, nor does it involve any parameter estimation. Simulation results reveal intriguing phase-transition behaviors of the number of segments when changing the level of distortion or the amount of outliers. Finally, we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data.