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1. About every convex set in any generic Riemannian manifold

   https://doi.org/10.1515/crelle-2021-0058  

   Lytchak, Alexander; Petrunin, Anton (October 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We give a necessary condition on a geodesic in a Riemannian manifold that can run in some convex hypersurface.As a corollary, we obtain peculiar properties that hold true for every convex set in any generic Riemannian manifold ( M , g ) {(M,g)} .For example, if a convex set in ( M , g ) {(M,g)} is bounded by a smooth hypersurface, then it is strictly convex. 

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2. The moduli space of marked generalized cusps in real projective manifolds

   https://doi.org/10.1090/ecgd/367  

   Ballas, Samuel; Cooper, Daryl; Leitner, Arielle (August 2022, Conformal Geometry and Dynamics of the American Mathematical Society)

   ln this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to M × [ 0 , ∞ ) M\times [0,\infty ) where M M is a closed Euclidean manifold. These are classified by Ballas, Cooper, and Leitner [J. Topol. 13 (2020), pp. 1455-1496]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of π 1 M \pi _1M . It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on M M , and the fiber is a closed cone in the space of cubic differentials. For 3 3 -dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus. 

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3. Approximately low-rank recovery from noisy and local measurements by convex program

   https://doi.org/10.1093/imaiai/iaad013  

   Lee, Kiryung; Sharma, Rakshith Srinivasa; Junge, Marius; Romberg, Justin (May 2023, Information and Inference: A Journal of the IMA)

   Abstract Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the $$\ell _1$$ norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the nuclear norm. We provide a tensor-norm-constrained estimator for the recovery of approximately low-rank matrices from local measurements corrupted with noise. A tensor-norm regularizer is designed to adapt to the local structure. We derive statistical analysis of the estimator over matrix completion and decentralized sketching by applying Maurey’s empirical method to tensor products of Banach spaces. The estimator provides a near-optimal error bound in a minimax sense and admits a polynomial-time algorithm for these applications. 

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4. Grassmannian optimization for online tensor completion and tracking with the t-svd

   https://doi.org/10.1109/TSP.2022.3164837  

   Gilman, Kyle; Tarzanagh, Davoud Ataee; Balzano, Laura (January 2022, IEEE transactions on signal processing)

   We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a lowtubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling. 

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5. Min–max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

   https://doi.org/10.1007/s00208-020-02096-0  

   Guang, Qiang; Li, Martin Man-chun; Wang, Zhichao; Zhou, Xin (April 2021, Mathematische Annalen)

   null (Ed.)

   Abstract For any smooth Riemannian metric on an $$(n+1)$$ ( n + 1 ) -dimensional compact manifold with boundary $$(M,\partial M)$$ ( M , ∂ M ) where $$3\le (n+1)\le 7$$ 3 ≤ ( n + 1 ) ≤ 7 , we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the $$C^\infty $$ C ∞ Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M . If $$\partial M$$ ∂ M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting. 

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