A higher-dimensional homologically persistent skeleton

Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data — for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when data points split into good clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a large interval in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimum spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale

Distributive lattices determined by weighted double skeletons

Related to his S-glued sum construction, the skeleton S(L) of a finite lattice L was introduced by C. Herrmann in 1973. Our theorem asserts that if D is a finite distributive lattice and its second skeleton, S(S(D)), is the trivial lattice, then D is characterized by its weighted double skeleton, introduced by the second author in 2006. The assumption on the second skeleton is essential.Comment: 12 pages and 2 figure

The President as International Leader

In this thesis, we address issues associated with programming modern heterogeneous systems while focusing on a special kind of heterogeneous systems that include multicore CPUs and one or more GPUs, called GPU-based systems.We consider the skeleton programming approach to achieve high level abstraction for efficient and portable programming of these GPU-based systemsand present our work on SkePU library which is a skeleton library for these systems. We extend the existing SkePU library with a two-dimensional (2D) data type and skeleton operations and implement several new applications using newly made skeletons. Furthermore, we consider the algorithmic choice present in SkePU and implement support to specify and automatically optimize the algorithmic choice for a skeleton call, on a given platform. To show how to achieve performance, we provide a case-study on optimized GPU-based skeleton implementation for 2D stencil computations and introduce two metrics to maximize resource utilization on a GPU. By devising a mechanism to automatically calculate these two metrics, performance can be retained while porting an application from one GPU architecture to another. Another contribution of this thesis is implementation of the runtime support for the SkePU skeleton library. This is achieved with the help of the StarPUruntime system. By this implementation,support for dynamic scheduling and load balancing for the SkePU skeleton programs is achieved. Furthermore, a capability to do hybrid executionby parallel execution on all available CPUs and GPUs in a system, even for a single skeleton invocation, is developed. SkePU initially supported only data-parallel skeletons. The first task-parallel skeleton (farm) in SkePU is implemented with support for performance-aware scheduling and hierarchical parallel execution by enabling all data parallel skeletons to be usable as tasks inside the farm construct. Experimental evaluations are carried out and presented for algorithmic selection, performance portability, dynamic scheduling and hybrid execution aspects of our work

Dynamic distance-based shape features for gait recognition

We propose a novel skeleton-based approach to gait recognition using our Skeleton Variance Image. The core of our approach consists of employing the screened Poisson equation to construct a family of smooth distance functions associated with a given shape. The screened Poisson distance function approximation nicely absorbs and is relatively stable to shape boundary perturbations which allows us to define a rough shape skeleton. We demonstrate how our Skeleton Variance Image is a powerful gait cycle descriptor leading to a significant improvement over the existing state of the art gait recognition rate