More Like this

1. Curvature Sets Over Persistence Diagrams

   https://doi.org/10.1007/s00454-024-00634-0  

   Gómez, Mario; Mémoli, Facundo (April 2024, Discrete & Computational Geometry)

   Abstract We study a family of invariants of compact metric spaces that combines the Curvature Sets defined by Gromov in the 1980 s with Vietoris–Rips Persistent Homology. For given integers$$k\ge 0$$ $k\ge 0$and$$n\ge 1$$ $n\ge 1$we consider the dimensionkVietoris–Rips persistence diagrams ofallsubsets of a given metric space with cardinality at mostn. We call these invariantspersistence setsand denote them as$${\textbf{D}}_{n,k}^{\textrm{VR}}$$ $D_{n,k}^{\text{VR}}$. We first point out that this family encompasses the usual Vietoris–Rips diagrams. We then establish that (1) for certain range of values of the parametersnandk, computing these invariants is significantly more efficient than computing the usual Vietoris–Rips persistence diagrams, (2) these invariants have very good discriminating power and, in many cases, capture information that is imperceptible through standard Vietoris–Rips persistence diagrams, and (3) they enjoy stability properties analogous to those of the usual Vietoris–Rips persistence diagrams. We precisely characterize some of them in the case of spheres and surfaces with constant curvature using a generalization of Ptolemy’s inequality. We also identify a rich family of metric graphs for which$${\textbf{D}}_{4,1}^{\textrm{VR}}$$ $D_{4,1}^{\text{VR}}$fully recovers their homotopy type by studying split-metric decompositions. Along the way we prove some useful properties of Vietoris–Rips persistence diagrams using Mayer–Vietoris sequences. These yield a geometric algorithm for computing the Vietoris–Rips persistence diagram of a spaceXwith cardinality$$2k+2$$ $2k+2$with quadratic time complexity as opposed to the much higher cost incurred by the usual algebraic algorithms relying on matrix reduction. 

   more » « less
2. Ephemeral persistence features and the stability of filtered chain complexes

   https://doi.org/10.20382/jocg.v15i2a8  

   Zhou, Ling; Mémoli, Facundo (January 2025, Journal of computational geometry)

   We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes points with zero persistence which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics that are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes, thus confirming that these ephemeral points strengthen the standard VR barcode. 

   more » « less
3. SEP-Graph: finding shortest execution paths for graph processing under a hybrid framework on GPU

   Hao Wang, Liang Geng (January 2019, Proceedings of 24th ACM SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming (PPoPP 2019))

   In general, the performance of parallel graph processing is determined by three pairs of critical parameters, namely synchronous or asynchronous execution mode (Sync or Async), Push or Pull communication mechanism (Push or Pull), and Data-driven or Topology-driven traversing scheme (DD or TD), which increases the complexity and sophistication of programming and system implementation of GPU. Existing graph-processing frameworks mainly use a single combination in the entire execution for a given application, but we have observed their variable and suboptimal performance. In this paper, we present SEP-Graph, a highly efficient software framework for graph-processing on GPU. The hybrid execution mode is automatically switched among three pairs of parameters, with an objective to achieve the shortest execution time in each iteration. We also apply a set of optimizations to SEP-Graph, considering the characteristics of graph algorithms and underlying GPU architectures. We show the effectiveness of SEP-Graph based on our intensive and comparative performance evaluation on NVIDIA 1080, P100, and V100 GPUs. Compared with existing and representative GPU graph-processing framework Groute and Gunrock, SEP-Graph can reduce execution time up to 45.8 times and 39.4 times. 

   more » « less
4. Protecting Real-Time GPU Kernels on Integrated CPU-GPU SoC Platforms

   https://doi.org/10.4230/LIPIcs.ECRTS.2018.19  

   Ali, Waqar; Yun, Heechul (July 2018, Leibniz international proceedings in informatics)

   Integrated CPU-GPU architecture provides excellent acceleration capabilities for data parallel applications on embedded platforms while meeting the size, weight and power (SWaP) requirements. However, sharing of main memory between CPU applications and GPU kernels can severely affect the execution of GPU kernels and diminish the performance gain provided by GPU. For example, in the NVIDIA Jetson TX2 platform, an integrated CPU-GPU architecture, we observed that, in the worst case, the GPU kernels can suffer as much as 3X slowdown in the presence of co-running memory intensive CPU applications. In this paper, we propose a software mechanism, which we call BWLOCK++, to protect the performance of GPU kernels from co-scheduled memory intensive CPU applications. 

   more » « less
5. Metric Thickenings and Group Actions

   https://doi.org/10.1142/S1793525320500569  

   Adams, Henry; Heim, Mark; Peterson, Chris (January 2021, Journal of Topology and Analysis)

   null (Ed.)

   Let [Formula: see text] be a group acting properly and by isometries on a metric space [Formula: see text]; it follows that the quotient or orbit space [Formula: see text] is also a metric space. We study the Vietoris–Rips and Čech complexes of [Formula: see text]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. 

   more » « less