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1. On the Reconstruction of Geodesic Subspaces of ℝ^N

   https://doi.org/10.1142/S0218195922500066  

   Fasy, Brittany Terese ; Komendarczyk, Rafal ; Majhi, Sushovan ; Wenk, Carola ( January 2022 , International Journal of Computational Geometry & Applications)

   We consider the topological and geometric reconstruction of a geodesic subspace of [Formula: see text] both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radius as our sampling parameters for the reconstruction problem. For a geodesic subspace with finite distortion and positive convexity radius, we guarantee a correct computation of its homotopy and homology groups from the sample. This technique provides alternative sampling conditions to the existing and commonly used conditions based on weak feature size and [Formula: see text]–reach, and performs better under certain types of perturbations of the geodesic subspace. For geodesic subspaces of [Formula: see text], we also devise an algorithm to output a homotopy equivalent geometric complex that has a very small Hausdorff distance to the unknown underlying space.
2. Metric Thickenings and Group Actions

   https://doi.org/10.1142/S1793525320500569  

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

   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.
3. TAaCGH Suite for Detecting Cancer—Specific Copy Number Changes Using Topological Signatures

   https://doi.org/10.3390/e24070896  

   Aslam, Jai ; Ardanza-Trevijano, Sergio ; Xiong, Jingwei ; Arsuaga, Javier ; Sazdanovic, Radmila ( July 2022 , Entropy)

   Copy number changes play an important role in the development of cancer and are commonly associated with changes in gene expression. Persistence curves, such as Betti curves, have been used to detect copy number changes; however, it is known these curves are unstable with respect to small perturbations in the data. We address the stability of lifespan and Betti curves by providing bounds on the distance between persistence curves of Vietoris–Rips filtrations built on data and slightly perturbed data in terms of the bottleneck distance. Next, we perform simulations to compare the predictive ability of Betti curves, lifespan curves (conditionally stable) and stable persistent landscapes to detect copy number aberrations. We use these methods to identify significant chromosome regions associated with the four major molecular subtypes of breast cancer: Luminal A, Luminal B, Basal and HER2 positive. Identified segments are then used as predictor variables to build machine learning models which classify patients as one of the four subtypes. We find that no single persistence curve outperforms the others and instead suggest a complementary approach using a suite of persistence curves. In this study, we identified new cytobands associated with three of the subtypes: 1q21.1-q25.2, 2p23.2-p16.3, 23q26.2-q28 with the Basalmore »subtype, 8p22-p11.1 with Luminal B and 2q12.1-q21.1 and 5p14.3-p12 with Luminal A. These segments are validated by the TCGA BRCA cohort dataset except for those found for Luminal A.« less
4. GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes

   Simon Zhang, Mengbai Xiao ( January 2020 , Proceedings of Symposium of Computational Geometry (SoCG 2020))

   The computation of Vietoris-Rips persistence barcodes is both execution-intensive and memory-intensive. In this paper, we study the computational structure of Vietoris-Rips persistence barcodes, and identify several unique mathematical properties and algorithmic opportunities with connections to the GPU. Mathematically and empirically, we look into the properties of apparent pairs, which are independently identifiable persistence pairs comprising up to 99% of persistence pairs. We give theoretical upper and lower bounds of the apparent pair rate and model the average case. We also design massively parallel algorithms to take advantage of the very large number of simplices that can be processed independently of each other. Having identified these opportunities, we develop a GPU-accelerated software for computing Vietoris-Rips persistence barcodes, called Ripser++. The software achieves up to 30x speedup over the total execution time of the original Ripser and also reduces CPU-memory usage by up to 2.0x. We believe our GPU-acceleration based efforts open a new chapter for the advancement of topological data analysis in the post-Moore's Law era.
5. Comparing the Fidelity of Contemporary Pointing with Controller Interactions on Performance of Personal Space Target Selection

   https://doi.org/10.1109/ISMAR55827.2022.00056  

   Babu, Sabarish V. ; Huang, Hsiao-Chuan ; Teather, Robert J. ; Chuang, Jung-Hong ( October 2022 , 2022 IEEE International Symposium on Mixed and Augmented Reality (ISMAR))

   The goal of this research is to provide much needed empirical data on how the fidelity of popular hand gesture tracked based pointing metaphors versus commodity controller based input affects the efficiency and speed-accuracy tradeoff in users’ spatial selection in personal space interactions in VR. We conduct two experiments in which participants select spherical targets arranged in a circle in personal space, or near-field within their maximum arms reach distance, in VR. Both experiments required participants to select the targets with either a VR controller or with their dominant hand’s index finger, which was tracked with one of two popular contemporary tracking methods. In the first experiment, the targets are arranged in a flat circle in accordance with the ISO 9241-9 Fitts’ law standard, and the simulation selected random combinations of 3 target amplitudes and 3 target widths. Targets were placed centered around the users’ eye level, and the arrangement was placed at either 60%, 75%, or 90% depth plane of the users’ maximum arm’s reach. In experiment 2, the targets varied in depth randomly from one depth plane to another within the same configuration of 13 targets within a trial set, which resembled button selection task in hierarchical menusmore »in differing depth planes in the near-field. The study was conducted using the HTC Vive head-mounted display, and used either a VR controller (HTC Vive), low-fidelity virtual pointing (Leap Motion), or a high-fidelity virtual pointing (tracked VR glove) conditions. Our results revealed that low-fidelity pointing performed worse than both high-fidelity pointing and the VR controller. Overall, target selection performance was found to be worse in depth planes closer to the maximum arms reach, as compared to middle and nearer distances.« less