Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds by Cavalletti, F, Mondino, A

“…Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and, more generally, RCD*(K,N) spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds. …”

Turmell-Meter: A Device for Estimating the Subtalar and Talocrural Axes of the Human Ankle Joint by Applying the Product of Exponentials Formula by Óscar Agudelo-Varela, Julio Vargas-Riaño, Ángel Valera

“…Finally, we tested the device by capturing positions and fitting them into the bi-axial ankle model as a Riemannian manifold. The Turmell-meter is a hardware platform for human ankle joint axes estimation. …”
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Multiple Kernel Stein Spatial Patterns for the Multiclass Discrimination of Motor Imagery Tasks by Steven Galindo-Noreña, David Cárdenas-Peña, Álvaro Orozco-Gutierrez

“…The Stein kernel provides a parameterized similarity metric for covariance matrices that belong to a Riemannian manifold. Lastly, the multiple kernel learning assembles the similarities from each spectral decomposition into a single kernel matrix that feeds the classifier. …”
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A Computationally Efficient Gradient Algorithm for Downlink Training Sequence Optimization in FDD Massive MIMO Systems by Muntadher Alsabah, Marwah Abdulrazzaq Naser, Basheera M. Mahmmod, Sadiq H. Abdulhussain

“…To overcome this challenge, we propose a computationally efficient conjugate gradient-descent (CGD) algorithm based on the Riemannian manifold in order to optimize the DL training sequence at base station (BS), while improving the convergence rate to provide a fast CSI estimation for an FDD m-MIMO system. …”
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Canonical Diffeomorphisms of Manifolds Near Spheres by Wang, Bing, Zhao, Xinrui

“…Abstract For a given Riemannian manifold $$(M^n, g)$$ ( M n , g ) which is near standard sphere $$(S^n, g_{round})$$ ( S n , g round ) in the Gromov–Hausdorff topology and satisfies $$Rc \ge n-1$$ R c ≥ n - 1 , it is known by Cheeger–Colding theory that M is diffeomorphic to $$S^n$$ S n . …”
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Calibrations and minimal Lagrangian submanifolds by Goldstein, Edward, 1977-

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Homotopy properties of horizontal loop spaces and applications to closed sub-riemannian geodesics by Lerario, A, Mondino, A

“…In the general case we prove that if $(M, \Delta)$ is a compact sub-riemannian manifold, each non trivial homotopy class in $\pi_1(M)$ can be represented by a closed sub-riemannian geodesic. …”

Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity by Mondino, A, Templeton-Browne, A

“…We prove rigidity results involving the Hawking mass for surfaces immersed in a 3-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (respectively, with scalar curvature bounded below by $-6$ ). …”

Smoothness of Ito maps and diffusion processes on path spaces by Li, X, Lyons, T

“…As a corollary in the particular case where p = 1, we obtain that the development from the space of finite 1-variation paths on Rd to the space of finite 1-variation paths on a d-dimensional compact Riemannian manifold is a smooth bijection. © 2006 Elsevier Masson SAS. …”

Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid by Shkoller, S

“…We establish the existence of three new subgroups of the group of volume-preserving diffeomorphisms of a compact n-dimensional (n ≥ 2) Riemannian manifold which are associated with the Dirichlet, Neumann, and Mixed type boundary conditions that arise in second-order elliptic PDEs. …”

Measure of Similarity between GMMs Based on Autoencoder-Generated Gaussian Component Representations by Vladimir Kalušev, Branislav Popović, Marko Janev, Branko Brkljač, Nebojša Ralević

“…As the autoencoder training cost function, the Frobenious norm between the input and output layers of such network is used and combined with regularizer terms in the form of various pieces of information, as well as the Riemannian manifold-based distances between SPD representatives corresponding to the computed autoencoder feature maps. …”
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Fall Detection of Elderly People Using the Manifold of Positive Semidefinite Matrices by Abdessamad Youssfi Alaoui, Youness Tabii, Rachid Oulad Haj Thami, Mohamed Daoudi, Stefano Berretti, Pietro Pala

“…Specifically, our approach involves four steps: (1) the body skeleton is detected by V2V-PoseNet in each frame; (2) joints of skeleton are first mapped into the Riemannian manifold of positive semidefinite matrices of fixed-rank 2 to build time-parameterized trajectories; (3) a temporal warping is performed on the trajectories, providing a (dis-)similarity measure between them; (4) finally, a pairwise proximity function SVM is used to classify them into fall or non-fall, incorporating the (dis-)similarity measure into the kernel function. …”
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3M_BANTOR: A regression framework for multitask and multisession brain network distance metrics by Chal E. Tomlinson, Paul J. Laurienti, Robert G. Lyday, Sean L. Simpson

“…A novel strategy is implemented to simulate symmetric positive-definite (SPD) connection matrices, allowing for the testing of metrics on the Riemannian manifold. Via simulation studies, we assess all approaches for estimation and inference while comparing them with existing multivariate distance matrix regression (MDMR) methods. …”
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An inverse scattering approach for geometric body generation: a machine learning perspective by Jinghong Li, Hongyu Liu, Wing-Yan Tsui, Xianchao Wang

“…The proposed method is in sharp difference from the existing methodologies in the literature, which usually treat the human body as a suitable Riemannian manifold and the generation is based on non-Euclidean approximation and interpolation. …”
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Twistor Bundle of a Neutral Kähler Surface by Włodzimierz Jelonek

“…We prove that an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>O</mi><mrow><mo>+</mo><mo>,</mo><mo>+</mo></mrow></msup><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-oriented four-dimensional neutral semi-Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula> admits a complex structure <i>J</i> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>Ω</mo><mi>J</mi></msub><mo>∈</mo><msup><mo>⋀</mo><mo>−</mo></msup><mi>M</mi></mrow></semantics></math></inline-formula>, such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>J</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a neutral-Kähler manifold if and only if the twistor bundle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>Z</mi><mn>1</mn></msup><mrow><mo>(</mo><mi>M</mi><mo>)</mo></mrow><mo>,</mo><msub><mi>g</mi><mi>c</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> admits a vertical Killing vector field.…”
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Intelligent Reflecting Surface Assisted Secure Transmission in UAV-MIMO Communication Systems by Tianhao Cheng, Buhong Wang, Zhen Wang, Kunrui Cao, Runze Dong, Jiang Weng

“…The alternating direction method of multipliers (ADMM), majorization-minimization (MM), and Riemannian manifold gradient (RCG) algorithms are presented, respectively, to solve the IRS phase shift matrix, and then the performance of the three algorithms is compared. …”
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The Scalar Curvature of a Riemannian Almost Paracomplex Manifold and Its Conformal Transformations by Vladimir Rovenski, Josef Mikeš, Sergey Stepanov

“…A Riemannian almost paracomplex manifold is a 2<i>n</i>-dimensional Riemannian manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>M</mi><mo>,</mo><mi>g</mi><mo>)</mo></mrow></semantics></math></inline-formula>, whose structural group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula> is reduced to the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo><mo>×</mo><mi>O</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>. …”
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SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group by Rosen, David M., Carlone, Luca, Bandeira, Afonso S., Leonard, John J.

“…We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. …”
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Beyond convexity—Contraction and global convergence of gradient descent by Wensing, Patrick M, Slotine, Jean-Jacques

“…More broadly, contraction analysis provides new insights for the case of geodesically-convex optimization, wherein non-convex problems in Euclidean space can be transformed to convex ones posed over a Riemannian manifold. In this case, natural gradient descent converges to a unique equilibrium if it is contracting in any metric, with geodesic convexity of the cost corresponding to contraction in the natural metric. …”
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A Riemannian perspective on matrix recovery and constrained optimization by Goyens, F

“…</p> <p>We formulate the rank minimization of the feature matrix as the minimization of a smooth cost function on a Riemannian manifold, which then allows us to employ the theoretically rigorous Riemannian optimization framework, that calls on differential geometry tools to construct feasible iterative algorithms on specific Riemannian manifolds. …”