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We propose a novel framework for the statistical analysis of genus-zero 4D surfaces, i.e., 3D surfaces that deform and evolve over time. This problem is particularly challenging due to the arbitrary parameterizations of these surfaces and their varying deformation speeds, necessitating effective spatiotemporal registration. Traditionally, 4D surfaces are discretized, in space and time, before computing their spatiotemporal registrations, geodesics, and statistics. However, this approach may result in suboptimal solutions and, as we demonstrate in this paper, is not necessary. In contrast, we treat 4D surfaces as continuous functions in both space and time. We introduce Dynamic Spherical Neural Surfaces (D-SNS), an efficient smooth and continuous spatiotemporal representation for genus-0 4D surfaces. We then demonstrate how to perform core 4D shape analysis tasks such as spatiotemporal registration, geodesics computation, and mean 4D shape estimation, directly on these continuous representations without upfront discretization and meshing. By integrating neural representations with classical Riemannian geometry and statistical shape analysis techniques, we provide the building blocks for enabling full functional shape analysis. We demonstrate the efficiency of the framework on 4D human and face datasets. The source code and additional results are available at https://4d-dsns.github.io/DSNS/. 

ABSTRACT The International Pulsar Timing Array (IPTA)’s second data release (IPTA DR2) combines decades of observations of 65 millisecond pulsars from 7 radio telescopes. IPTA data sets should be the most sensitive data sets to nanohertz gravitational waves (GWs), but take years to assemble, often excluding valuable recent data. To address this, we introduce the IPTA ‘Lite’ analysis, where a Figure of Merit is used to select an optimal PTA data set to analyse for each pulsar, enabling immediate access to new data and preliminary results prior to full combination. We test the capabilities of the Lite analysis using IPTA DR2, finding that ‘DR2 Lite’ can be used to detect the common red noise process with an amplitude of $$A = 4.8^{+1.8}_{-1.8} \times 10^{-15}$$ at $$\gamma = 13/3$$. This amplitude is slightly large in comparison to the combined analysis, and likely biased high as DR2 Lite is more sensitive to systematic errors from individual pulsars than the full data set. Furthermore, although there is no strong evidence for Hellings-Downs correlations in IPTA DR2, we still find the full data set is better at resolving Hellings-Downs correlations than DR2 Lite. Alongside the Lite analysis, we also find that analysing a subset of pulsars from IPTA DR2, available at a hypothetical ‘early’ stage of combination (EDR2), yields equally competitive results as the full data set. Looking ahead, the Lite method will enable rapid synthesis of the latest PTA data, offering preliminary GW constraints before the superior full data set combinations are available. 

Graph-based representations are becoming increasingly popular for representing and analyzing video data, especially in object tracking and scene understanding applications. Accordingly, an essential tool in this approach is to generate statistical inferences for graphical time series associated with videos. This paper develops a Kalman-smoothing method for estimating graphs from noisy, cluttered, and incomplete data. The main challenge here is to find and preserve the registration of nodes (salient detected objects) across time frames when the data has noise and clutter due to false and missing nodes. First, we introduce a quotient-space representation of graphs that incorporates temporal registration of nodes, and we use that metric structure to impose a dynamical model on graph evolution. Then, we derive a Kalman smoother, adapted to the quotient space geometry, to estimate dense, smooth trajectories of graphs. We demonstrate this framework using simulated data and actual video graphs extracted from the Multiview Extended Video with Activities (MEVA) dataset. This framework successfully estimates graphs despite the noise, clutter, and missed detections. 

Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, statistical testing, and modeling. We demonstrate the efficacy of this framework using datasets taken from several problem areas, including letters, biochemical structures, and social networks.