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1. HV geometry for signal comparison

   https://doi.org/10.1090/qam/1672  

   Han, Ruiyu; Slepčev, Dejan; Yang, Yunan (June 2024, Quarterly of Applied Mathematics)

   In order to compare and interpolate signals, we investigate a Riemannian geometry on the space of signals. The metric allows discontinuous signals and measures both horizontal (thus providing many benefits of the Wasserstein metric) and vertical deformations. Moreover, it allows for signed signals, which overcomes the main deficiency of optimal transportation-based metrics in signal processing. We characterize the metric properties of the space of signals and establish the regularity and stability of geodesics. Furthermore, we introduce an efficient numerical scheme to compute the geodesics and present several experiments which highlight the nature of the metric. 

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2. On Eisenhart’s Type Theorem for Sub-Riemannian Metrics on Step $$2$$ Distributions with $$\mathrm{ad}$$-Surjective Tanaka Symbols

   https://doi.org/10.1134/S1560354724020023  

   Lin, Zaifeng; Zelenko, Igor (February 2024, Regular and Chaotic Dynamics)

   The classical result of Eisenhart states that, if a Riemannian metric admits a Riemannian metric that is not constantly proportional to and has the same (parameterized) geodesics as in a neighborhood of a given point, then is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step graded nilpotent Lie algebras, called -surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step distributions with -surjective Tanaka symbols. The class of ad-surjective step nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case. 

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3. Riemannian Metric Learning via Optimal Transport

   Scarvelis, Christopher; Solomon, Justin (May 2023, International Conference on Learning Representations)

   We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field and efficiently optimize our model's objective using a simple alternating scheme. Using this learned metric, we can non-linearly interpolate between probability measures and compute geodesics on the manifold. We show that metrics learned using our method improve the quality of trajectory inference on scRNA and bird migration data at the cost of little additional cross-sectional data. 

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4. Injectivity of the Heisenberg X-ray transform

   https://doi.org/10.1016/j.jfa.2020.108886  

   Flynn, Steven (March 2021, Journal of Functional Analysis)

   We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier transform, we prove an operator-valued incarnation of the Fourier Slice Theorem, and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the family of taming metrics $$g_\epsilon$$ approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all $$\epsilon > 0$$. This result gives a concrete example of an injective X-ray transform in a geometry with an abundance of conjugate points. 

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5. Bayesian Tracking of Video Graphs Using Joint Kalman Smoothing and Registration

   https://doi.org/10.1007/978-3-031-19833-5_26  

   Bal, A.B.; Mounir, R.; Aakur, S.; Sarkar, S.; Srivastava, A. (November 2022, European Conference on Computer Vision (ECCV))

   Avidan, S.; Brostow, G.; Cissé, M.; Farinella. G.M.; Hassner, T. (Ed.)

   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. 

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