Approximating Local Homology from Samples

Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current reconstruction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence diagrams used for estimating local homology, can be approximated using families of Vietoris-Rips complexes, whose simple constructions are robust in any dimension. To the best of our knowledge, our results, for the first time, make applications based on local homology, such as stratification learning, feasible in high dimensions.Comment: 23 pages, 14 figure

On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

Characterizing and approximating eigenvalue sets of symmetric interval matrices

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guaran- tee exactness. We illustrate our approach by several examples and numerical experiments

Approximating L^2-signatures by their compact analogues

:Let G be a group together with an descending nested sequence of normal subgroups G=G_0, G_1, G_2 G_3, ... of finite index [G:G_k] such the intersection of the G_k-s is the trivial group. Let (X,Y) be a compact 4n-dimensional Poincare' pair and p: (\bar{X},\bar{Y}) \to (X,Y) be a G-covering, i.e. normal covering with G as deck transformation group. We get associated $G/_k$-coverings (X_k,Y_k) \to (X,Y). We prove that sign^{(2)}(\bar{X},\bar{Y}) = lim_{k\to\infty} \frac{sign(X_k,Y_k)}{[G : G_k]}, where sign or sign^{(2)} is the signature or L^2-signature, respectively, and the convergence of the right side for any such sequence (G_k)_k is part of the statement

Averaging and linear programming in some singularly perturbed problems of optimal control

The paper aims at the development of an apparatus for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems (that is, problems of optimal control of SP systems) considered on the infinite time horizon. We mostly focus on problems with time discounting criteria but a possibility of the extension of results to periodic optimization problems is discussed as well. Our consideration is based on earlier results on averaging of SP control systems and on linear programming formulations of optimal control problems. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional (ID) linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with two numerical examples.Comment: 53 pages, 10 figure

Compacting points-to sets through object clustering

Inclusion-based set constraint solving is the most popular technique for whole-program points-to analysis whereby an analysis is typically formulated as repeatedly resolving constraints between points-to sets of program variables. The set union operation is central to this process. The number of points-to sets can grow as analyses become more precise and input programs become larger, resulting in more time spent performing unions and more space used storing these points-to sets. Most existing approaches focus on improving scalability of precise points-to analyses from an algorithmic perspective and there has been less research into improving the data structures behind the analyses. Bit-vectors as one of the more popular data structures have been used in several mainstream analysis frameworks to represent points-to sets. To store memory objects in bit-vectors, objects need to mapped to integral identifiers. We observe that this object-to-identifier mapping is critical for a compact points-to set representation and the set union operation. If objects in the same points-to sets (co-pointees) are not given numerically close identifiers, points-to resolution can cost significantly more space and time. Without data on the unpredictable points-to relations which would be discovered by the analysis, an ideal mapping is extremely challenging. In this paper, we present a new approach to inclusion-based analysis by compacting points-to sets through object clustering. Inspired by recent staged analysis where an auxiliary analysis produces results approximating a more precise main analysis, we formulate points-to set compaction as an optimisation problem solved by integer programming using constraints generated from the auxiliary analysis's results in order to produce an effective mapping. We then develop a more approximate mapping, yet much more efficiently, using hierarchical clustering to compact bit-vectors. We also develop an improved representation of bit-vectors (called core bit-vectors) to fully take advantage of the newly produced mapping. Our approach requires no algorithmic change to the points-to analysis. We evaluate our object clustering on flow sensitive points-to analysis using 8 open-source programs (>3.1 million lines of LLVM instructions) and our results show that our approach can successfully improve the analysis with an up to 1.83× speed up and an up to 4.05× reduction in memory usage