Linear low-rank approximation and nonlinear dimensionality reduction

We present our recent work on both linear and nonlinear data reduction methods and algorithms: for the linear case we discuss results on structure analysis of SVD of column-partitioned matrices and sparse low-rank approximation; for the nonlinear case we investigate methods for nonlinear dimensionality reduction and manifold learning. The problems we address have attracted great deal of interest in data mining and machine learning.     

Convex Euclidean distance embedding for collaborative position localization with NLOS mitigation

One of the challenging problems in collaborative position localization arises when the distance measurements contain non-line-of-sight (NLOS) biases. Convex optimization has played a major role in modelling such problems and numerical algorithm developments. One of the successful examples is the semi-definite programming (SDP), which translates Euclidean distances into the constraints of positive semidefinite matrices, leading to a large number of constraints in the case of NLOS biases. In this paper, we propose a new convex optimization model that is built upon the concept of Euclidean distance matrix (EDM). The resulting EDM optimization has an advantage that its Lagrangian dual problem is well structured and hence is conducive to algorithm developments. We apply a recently proposed 3-block alternating direction method of multipliers to the dual problem and tested the algorithm on some real as well as simulated data of large scale. In particular, the EDM model significantly outperforms the existing SDP model and several others.     

Demand sensitivity to space-price competition with Manhattan and Euclidean representations of distance

This paper provides extensive evidence from a simulation model supporting our claim that it is not appropriate to use the Euclidean metric in a competitive system where the Manhattan metric would provide a more accurate representation of distances. The Euclidean metric has the property of biasing firms' demands by a distortion of their sensitivity to competitive strategies and, therefore, generates an excessive level of competition.     

Analysis of an alignment algorithm for nonlinear dimensionality reduction

The goal of dimensionality reduction or manifold learning for a given set of high-dimensional data points, is to find a low-dimensional parametrization for them. Usually it is easy to carry out this parametrization process within a small region to produce a collection of local coordinate systems. Alignment is the process to stitch those local systems together to produce a global coordinate system and is done through the computation of a partial eigendecomposition of a so-called alignment matrix. In this paper, we present an analysis of the alignment process, giving conditions under which the null space of the alignment matrix recovers the global coordinate system up to an affine transformation. We also propose a post-processing step that can determine the global coordinate system up to a rigid motion. This in turn shows that Local Tangent Space Alignment method (LTSA) can recover a locally isometric embedding up to a rigid motion. AMS subject classification (2000)  65F15, 62H30, 15A18     

Minimal faithful representations of reductive Lie algebras

Let G and H be Lie groups with Lie algebras and . Let G be connected. We prove that a Lie algebra homomorphism is exact if and only if it is completely positive. The main resource is a corresponding theorem about representations on Hilbert spaces. This article summarizes the main results of [1]. Received: 6 December 2005     

Minimal Euclidean representations of graphs

A simple graph G is representable in a real vector space of dimension m, if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values, α and β, with distance α if the vertices are adjacent and distance β otherwise. The Euclidean representation number of G is the smallest dimension in which G is representable. In this note, we bound the Euclidean representation number of a graph using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

Some remarks on the degrees of faithful irreducible representations

We study a class of finite p-groups admitting faithful irreducible complex representations of distinct degrees. In particular, for every prime p, we produce an example of such a p-group having minimal order p ⁸.Dedicated to A. Wagner on the occasion of his 60th birthdayLavoro eseguito nell'ambito dei finanziamenti M.U.R.S.T.     

Minimal dimensions of faithful representations for abelian Lie superalgebras

This paper determines the minimal dimensions of faithful representations for abelian Lie superalgebras of finite dimensions over an algebraically closed field of characteristic zero. In particular, we also obtain the maximal dimensions of abelian subalgebras for the general linear Lie superalgebras.     

Path loss prediction in urban environment using learning machines and dimensionality reduction techniques

Path loss prediction is a crucial task for the planning of networks in modern mobile communication systems. Learning machine-based models seem to be a valid alternative to empirical and deterministic methods for predicting the propagation path loss. As learning machine performance depends on the number of input features, a good way to get a more reliable model can be to use techniques for reducing the dimensionality of the data. In this paper we propose a new approach combining learning machines and dimensionality reduction techniques. We report results on a real dataset showing the efficiency of the learning machine-based methodology and the usefulness of dimensionality reduction techniques in improving the prediction accuracy.     

Minimal faithful representations of abelian Jordan algebras and Lie superalgebras

Over an algebraically closed field of characteristic zero, all the abelian subalgebras of the maximal dimension are classified for any special Jordan algebra. As a consequence, the minimal dimension of the faithful representations of any finite-dimensional abelian Jordan algebra is determined and the minimal faithful representations are classified for the so-called nice abelian Jordan algebras. The same is done for the purely odd Lie superalgebras.     

Minimal degrees of faithful quasi-permutation representations for direct products of p-groups

In [2], the algorithms of c(G), q(G) and p(G), the minimal degrees of faithful quasi-permutation and permutation representations of a finite group G are given. The main purpose of this paper is to consider the relationship between these minimal degrees of non-trivial p-groups H and K with the group H×K.     

Isometries of the Space of Convex Bodies in Euclidean Space

The isometries of the space of convex bodies of E^d with respectto the Hausdorff-metric are precisely the mappings of the formC i(C) + D where i is a rigid motion of E^d and D a fixed convexbody.     

Note on faithful representations and a local property of Lie groups

Let be any analytic group, let be a maximal toroid of the radical of , and let be a maximal semisimple analytic subgroup of . If is the Lie algebra of , is the radical of , and is the center of , we show that has a faithful representation if and only if (i) , and (ii) has a faithful representation.