Learn More
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)
The vanishing ideal I of a subspace arrangement V 1 ∪ V 2 ∪ · · · ∪ V m ⊆ V is an intersection I 1 ∩ I 2 ∩ · · · ∩ I m 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 = I 1 I 2 · · · I m without any assumptions on the subspace(More)
Let ρ: G → GL(V) be a representation of a group G on a vector space V of dimension n < ∞. For simplicity, we assume that the base field k is algebraically closed and of characteristic zero. As usual, the group G acts linearly on the k-algebra O(V) of polynomial functions on V , the coordinate ring of V. Of special interest is the subalgebra of invariant(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 study the left-right action of SL n × SL n on m-tuples of n × n matrices with entries in an infinite field K. We show that invariants of degree n 2 − n define the null cone. Consequently, invariants of degree ≤ n 6 generate the ring of invariants if char(K) = 0. We also prove that for m ≫ 0, invariants of degree at least n⌊ √ n + 1⌋ are required to(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)
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. Résumé Nous(More)