Affine Hull Research Articles

Approximately harmonic projection: Theoretical analysis and an algorithm

Manifold learning have attracted considerable attention over the last decade. The most frequently used functional is the l 2-norm of the gradient of the function. In this paper, we consider the linear manifold learning problem by minimizing this functional with appropriate constraint. We provide theoretical analysis on both the functional and the constraint, which shows the affine hulls of the manifold and the connected components are essential to linear manifold learning problem. Based on the theoretical analysis, we introduce a novel linear manifold learning algorithm called approximately harmonic projection (AHP). Unlike canonical linear methods such as principal component analysis, our method is sensitive to the connected components. This makes our method especially applicable to data clustering. We conduct several experimental results on three real data sets to demonstrate the effectiveness of our proposed method.

Graph realizations associated with minimizing the maximum eigenvalue of the Laplacian

In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance of at least one. We prove three main results for a slightly generalized form of this embedding problem. First, given a set of vertices partitioning the graph into several or just one part, the barycenter of each part is embedded on the same side of the affine hull of the set as the origin. Second, there is an optimal realization of dimension at most the tree-width of the graph plus one and this bound is best possible in general. Finally, bipartite graphs possess a one dimensional optimal embedding.

Affine Independence in Vector Spaces

Affine Independence in Vector Spaces In this article we describe the notion of affinely independent subset of a real linear space. First we prove selected theorems concerning operations on linear combinations. Then we introduce affine independence and prove the equivalence of various definitions of this notion. We also introduce the notion of the affine hull, i.e. a subset generated by a set of vectors which is an intersection of all affine sets including the given set. Finally, we introduce and prove selected properties of the barycentric coordinates.

Properties of the far-field operator for a multilayered scatterer in acoustic scattering theory

In this paper, we consider the inverse scattering problem which is to determine the shape of a multilayered scatterer. The adjoint of the far-field operator is found. In scattering theory the boundary data are given by the traces of plane waves. The closure of the span of the images of these plane waves under the far-field operator is characterized. Furthermore, taking the affine hull instead of the span of the plane waves on the boundary, some results are obtained.

Manifold Based Local Classifiers: Linear and Nonlinear Approaches

In case of insufficient data samples in high-dimensional classification problems, sparse scatters of samples tend to have many ‘holes’—regions that have few or no nearby training samples from the class. When such regions lie close to inter-class boundaries, the nearest neighbors of a query may lie in the wrong class, thus leading to errors in the Nearest Neighbor classification rule. The K-local hyperplane distance nearest neighbor (HKNN) algorithm tackles this problem by approximating each class with a smooth nonlinear manifold, which is considered to be locally linear. The method takes advantage of the local linearity assumption by using the distances from a query sample to the affine hulls of query’s nearest neighbors for decision making. However, HKNN is limited to using the Euclidean distance metric, which is a significant limitation in practice. In this paper we reformulate HKNN in terms of subspaces, and propose a variant, the Local Discriminative Common Vector (LDCV) method, that is more suitable for classification tasks where the classes have similar intra-class variations. We then extend both methods to the nonlinear case by mapping the nearest neighbors into a higher-dimensional space where the linear manifolds are constructed. This procedure allows us to use a wide variety of distance functions in the process, while computing distances between the query sample and the nonlinear manifolds remains straightforward owing to the linear nature of the manifolds in the mapped space. We tested the proposed methods on several classification tasks, obtaining better results than both the Support Vector Machines (SVMs) and their local counterpart SVM-KNN on the USPS and Image segmentation databases, and outperforming the local SVM-KNN on the Caltech visual recognition database.

A study of the quadratic semi-assignment polytope

We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem. We introduce an isomorphic projection and transform the polytope to a tractable full-dimensional polytope. As a result, some basic polyhedral properties, such as the dimension, the affine hull, and the trivial facets, are obtained. Further, we present valid inequalities called cut- and clique-inequalities and give complete characterizations for them to be facet-defining. We also discuss a simultaneous lifting of the clique-type facets. Finally, we show an application of the quadratic semi-assignment problem to hub location problems with some computational experiences.

Dimensionality Reductions in ℓ2 that Preserve Volumes and Distance to Affine Spaces

Let X be a subset of n points of the Euclidean space, and let 0 < e < 1. A classical result of Johnson and Lindenstrauss [JL] states that there is a projection of X onto a subspace of dimension O(e-2 log n) with distortion ≤ 1+ e. We show a natural extension of the above result to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared with [JL], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show how to embed a subset of size n of the Euclidean space into a O(e-2 log n)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1 + es-1. Moreover, distances of points from affine hulls of sets of at most k - 1 points in the space do not change by more than a factor of 1 + e. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbour Search. In particular, it shows a first polylogarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [F], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.

On a convex operator for finite sets

Let S be a finite set with m elements in a real linear space and let J S be a set of m intervals in R . We introduce a convex operator co ( S , J S ) which generalizes the familiar concepts of the convex hull, conv S , and the affine hull, aff S , of S. We prove that each homothet of conv S that is contained in aff S can be obtained using this operator. A variety of convex subsets of aff S with interesting combinatorial properties can also be obtained. For example, this operator can assign a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For two types of families J S we give two different upper bounds for the number of vertices of the polytopes produced as co ( S , J S ) . Our motivation comes from a recent improvement of the well-known Gauss–Lucas theorem. It turns out that a particular convex set co ( S , J S ) plays a central role in this improvement.

On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids

Given A ≔ { a 1 , … , a m } ⊂ R d whose affine hull is R d , we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum-volume enclosing ellipsoid of A . In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of A , and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A . Our algorithm is a modification of the algorithm of Kumar and Yıldırım, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small “core set.” We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yıldırım or any increase in the bound on the size of the computed core set. In addition, the “dropping idea” used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique.

The Symmetric Quadratic Semi-Assignment Polytope

We deal with quadratic semi-assignment problems with symmetric distances. This symmetry reduces the number of variables in its mixed integer programming formulation. We investigate a polytope arising from the problem, and obtain some basic polyhedral properties, the dimension, the affine hull, and certain facets through an isomorphic projection. We also present a class of facets.

A converse to the Eidelheit theorem in real Hilbert spaces

We establish the following converse to the Eidelheit theorem: an unbounded closed and convex set of a real Hilbert space may be separated by a closed hyperplane from every other disjoint closed and convex set, if and only if it has a finite codimension and a non-empty interior with respect to its affine hull.

Binary Representation of Normalized Symmetric and Correlation Matrices

With a view toward the correlation matrices, it is shown that the normalized real symmetric matrices are the affine hull of the binary correlation matrices, while the convex hull is a proper subset of the correlation matrices. A number of ways to identify the correlation matrices in the affine hull are discussed.

The Affine Hull of a Binary Automaton is Computable in Polynomial Time

We present the class of binary automaton, a new representation for the subsets of Nm that naturally extends the NDD ([Wolper, P. and B. Boigelot, An automata-theoretic approach to Presburger arithmetic constraints, in: Proc. 2nd Int. Symp. Static Analysis (SAS'95), Glasgow, UK, Sep. 1995, Lecture Notes in Computer Science 983 (1995), pp. 21–32], [Boudet, A. and H. Comon, Diophantine equations, Presburger arithmetic and finite automata, in: Proc. 21st Int. Coll. on Trees in Algebra and Programming (CAAP'96), Linköping, Sweden, Apr. 1996, Lecture Notes in Computer Science 1059 (1996), pp. 30–43]). We prove that the affine hull of the set of vectors represented by a binary automaton is computable in polynomial time. As application, we show that the set of place invariants [Ciardo, G., Petri nets with marking-dependent arc cardinality: Properties and analysis, in: Proc. 15th Int. Conf. Application and Theory of Petri Nets (ICATPN'94), Zaragoza, Spain, June 1994, Lecture Notes in Computer Science 815 (1994), pp. 179–198] of a counter system (an extension of the Broadcast Protocols [Emerson, E. A. and K. S. Namjoshi, On model checking for non-deterministic infinite-state systems, in: Proc. 13th IEEE Symp. Logic in Computer Science (LICS'98), Indianapolis, IN, USA, June 1998 (1998), pp. 70–80], [Delzanno, G., Verification of consistency protocols via infinite-state symbolic model checking: A case study, in: Proc. IFIP Joint Int. Conf. Formal Description Techniques & Protocol Specification, Testing, and Verification (FORTE-PSTV'00), Pisa, Italy, Oct. 2000, IFIP Conference Proceedings 183 (2000), pp. 171–186], [Delzanno, G., Constraint-based verification of parameterized cache coherence protocols, Formal Methods in System Design. To appear], the Reset/Transfer Petri Nets [Dufourd, C., A. Finkel and P. Schnoebelen, Reset nets between decidability and undecidability, in: Proc. 25th Int. Coll. Automata, Languages, and Programming (ICALP'98), Aalborg, Denmark, July 1998, Lecture Notes in Computer Science 1443 (1998), pp. 103–115], [Ciardo, G., Petri nets with marking-dependent arc cardinality: Properties and analysis, in: Proc. 15th Int. Conf. Application and Theory of Petri Nets (ICATPN'94), Zaragoza, Spain, June 1994, Lecture Notes in Computer Science 815 (1994), pp. 179–198] and the linear systems [Finkel, A. and J. Leroux, How to compose Presburger-accelerations: Applications to broadcast protocols, in: Proc. 22nd Conf. Found. of Software Technology and Theor. Comp. Sci. (FST&TCS'2002), Kanpur, India, Dec. 2002, Lecture Notes in Computer Science 2556 (2002), pp. 145–156]), is computable in polynomial time.

A Squared Smoothing Newton Method for Nonsmooth Matrix Equations and Its Applications in Semidefinite Optimization Problems

We study a smoothing Newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases. This method, if specialized for solving semidefinite programs, needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy. We also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the Jacobian of the problem is positive definite on the affine hull of the critical cone at the solution. These results are based on the strong semismoothness and complete characterization of the B-subdifferential of a corresponding squared smoothing matrix function, which are of general theoretical interest.

Un Algorithme De Génération De Coupes Pour Le Problème De L’Affectation Quadratique

We address the Quadratic Assignment Problem following a polyhedral method. We consider the Quadratic Assignment Polytopc defined as the convex hull of the solutions of the linearized problem. Its dimension and a minimal description of its affine hull have been given by Padberg and Rijal (1996). Here we propose a large family of valid inequalities inducing facets. We show that the separation problem of this family is NP-complete. We propose a heuristic for the separation problem and a cutting plane algorithm based on this heuristic. Numerical results show the practical interest of this family of inequalities

Stability of the feasible set for linear inequality systems: A carrier index set approach

In this paper we consider a parametrized family of linear inequality systems whose coefficients depend continuously on a parameter ranging in an arbitrary metric space. We analyze the lower semicontinuity (lsc) of the feasible set mapping in terms of the so-called carrier index set, consisting of those indices whose associated inequalities are satisfied as equalities at every feasible point. This concept, which leads to a weakened Slater condition, allows us to characterize the lsc of the feasible set mapping in terms of certain convex combinations of the coefficient vectors associated with the carrier indices. This property entails the lsc of the carrier feasible set mapping, assigning to each parameter the affine hull of the feasible set, which is also analyzed in this paper. The last section is concerned with the semi-infinite case.

Extension theorems for functional equations with bisymmetric operations

In this paper we deal with the extension of the following functional equation¶¶\( f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K) \), (*)¶ where M is a k-variable operation on the image space Y, m1,..., mk are binary operations on X, \( K \subset X \) is closed under the operations m1,..., mk, and \( f : K \rightarrow Y \) is considered as an unknown function.¶ The main result of this paper states that if the operations m1,..., mk, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m1,..., mk)-affine hull K* of K, such that (*) holds over K*. (The set K* is defined as the smallest subset of X that contains K and is (m1,..., mk)-affine, i.e., if \( x \in X \), and there exists \( y \in K^* \) such that \( m_{1}(x, y), \ldots, m_{k}(x, y) \in K^* \) then \( x \in K^* \)). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.

On the Slater condition for the SDP relaxations of nonconvex sets

We prove that all results determining the dimension and the affine hull of feasible solutions of any combinatorial optimization problem, and various more general nonconvex optimization problems, directly imply the existence of the Slater points for a very wide class of semidefinite programming relaxations of these nonconvex problems. Our proofs are very concise, constructive and elementary.

Box-inequalities for quadratic assignment polytopes

Linear Programming based lower bounds have been considered both for the general as well as for the symmetric quadratic assignment problem several times in the recent years. Their quality has turned out to be quite good in practice. Investigations of the polytopes underlying the corresponding integer linear programming formulations (the non-symmetric and the symmetric quadratic assignment polytope) have been started during the last decade [34, 31, 21, 22]. They have lead to basic knowledge on these polytopes concerning questions like their dimensions, affine hulls, and trivial facets. However, no large class of (facet-defining) inequalities that could be used in cutting plane procedures had been found. We present in this paper the first such class of inequalities, the box inequalities, which have an interesting origin in some well-known hypermetric inequalities for the cut polytope. Computational experiments with a cutting plane algorithm based on these inequalities show that they are very useful with respect to the goal of solving quadratic assignment problems to optimality or to compute tight lower bounds. The most effective ones among the new inequalities turn out to be indeed facet-defining for both the non-symmetric as well as for the symmetric quadratic assignment polytope.

Median and center hyperplanes in Minkowski spaces—a unified approach

In this paper we will extend two known location problems from Euclidean n-space to all n-dimensional normed spaces, n⩾2. Let X be a finite set of weighted points whose affine hull is n-dimensional. Our first objective is to find a hyperplane minimizing (among all hyperplanes) the sum of weighted distances with respect to X . Such a hyperplane is called a median hyperplane with respect to X , and we will show that for all distance measures d derived from norms one of the median hyperplanes is the affine hull of n of the demand points. (This approach was already presented in the recent survey (Discrete Appl. Math. 89 (1998) 181), but without proofs. Here we give the complete proofs to all necessary lemmas.) On the other hand, we will prove that one of the hyperplanes minimizing (among all hyperplanes) the maximum weighted distance to some point from X has the same maximal distance to least n+1 affinely independent demand points (such a hyperplane is said to be a center hyperplane of X ). Both these results allow polynomially bounded algorithmical approaches to median and center hyperplanes and the respective distance sums or maximal distances for any fixed dimension n⩾2, and in particular we discuss the algorithms for both the problems in the case of polyhedral norms. Also two independence of norm results for optimal hyperplanes with fixed slope will be derived, and finally the considerations are even extended to gauges which are no longer combined with a norm.