Extended Rank-One Theorem to special metric spaces.
problem Extending a theorem to new types of spaces.
method Applied to a new class of metric measure spaces.
result Rank-One Theorem proven for spaces.
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In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.
Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
Rank One Hilbert Geometriesmath.GT
We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for Hilbert geometries and characterize them as being equivalent to contracting elem…
The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.
Classifies foliations on specific symmetric spaces.
problem Classifying foliations on symmetric spaces of rank one.
method Orbit equivalence classification.
result Polar homogeneous foliations classified.
We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…
Improved stability for matrix recovery from rank-one measurements.
problem Phase retrieval problem of recovering rank-one positive semidefinite matrices.
method Developed a smoothing Newton method based on Bures-Wasserstein gradient descent.
result Superlinear convergence with rigorous guarantees and stable implementation.
Classifies rank-one submanifolds in Euclidean space.
problem Classifying submanifolds with singularities.
method Associate degree to ruled submanifolds and analyze singularities.
result An open and dense subset of rank-one submanifolds is the union of cylindrical, conical, and tangent regions.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
New insights into compact rank-one ECS manifolds, proving they are bundles over circles.
problem Understanding the structure of compact rank-one ECS manifolds.
method Analyzing the properties of pseudo-Riemannian manifolds with parallel Weyl tensor.
result Compact rank-one ECS manifolds are bundles over the circle with specific leaf structures.
Compact rank one symmetric spaces are rigid under certain curvature conditions.
problem Rigidity of compact rank one symmetric spaces under curvature constraints.
method Examined compact symmetric spaces with metric of rank one, and another metric with sectional curvature bounded by 0 to 1.
result If equals outside a convex subset, then is isometric with .
Closed ray affine manifoldsmath.DG
Study closed manifolds with rank one ray structures, proving completeness or covering properties.
problem Characterize closed manifolds with specific affine structures.
method Analyze the developing map and automorphism group properties.
result Closed manifolds with rank one ray structures are either complete or cover the complement of an affine subspace.
In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.
We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.
We show that cocompact lattices in rank one simple Lie groups of non-compact type distinct from SO(2m,1) (m>0) contain surface subgroups.
Researchers describe the metric structure of compact ECS manifolds.
problem Understanding the metric structure of compact rank-one ECS manifolds.
method Analyzing pseudo-Riemannian manifolds with nonzero parallel Weyl tensor.
result Compact rank-one ECS manifolds are either translational or noncompact.
Stochastic Rank-One Bandits (Katarya et al, (2017a,b)) are a simple framework for regret minimization problems over rank-one matrices of arms. The initially proposed algorithms are proved to have logarithmic regret, but do not match the existing lower bound for this problem. We close this gap by first proving that rank…
Novel algorithm for Markov decision processes using rank-one approximation.
problem Solving planning and learning problems of Markov decision processes.
method Policy iteration with rank-one approximation of transition probability matrix.
result The proposed algorithm consistently outperforms first-order algorithms and their accelerated versions.
Rank-one measurements limit feasible sets for low-rank PSD matrices.
problem Feasibility of PSD matrices under rank-one measurements.
method Characterization of feasible sets for PSD matrices given rank-one projections.
result Radius of feasible sets determines singleton solution sets for low-rank matrices.
In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter g…
New random walk results on rank one symmetric spaces.
problem Analyzing random walks on noncompact rank one symmetric spaces.
method Unified algebraic framework using Möbius addition and harmonic analysis of spherical functions.
result Renormalized walk converges to heat kernel on Laplace-Beltrami operator.
We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.
We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z_2-coefficients.
The Besson-Courtois-Gallot theorem is proven for noncompact finite volume Riemannian manifolds. In particular, no bounded geometry assumptions are made. This proves the minimal entropy conjecture for nonuniform rank one lattices.
Study shows generative priors improve rank-one matrix recovery with optimal sample complexity.
problem Recovering a rank-one signal matrix from noisy data with additional prior information.
method Analysis of a nonlinear least squares objective with a favorable global optimization landscape.
result Established optimal sample complexity for generative priors in rank-one matrix recovery.
The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.
problem Recovering tensors with low symmetric rank from symmetric rank-one measurements.
method Covering numbers argument, Carbery-Wright inequality, orthogonal polynomials, Fano's inequality.
result Near-optimal sample complexity bounds for log-concave distributions.
Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.
Gradient descent solves rank-one matrix estimation problem with detailed time evolution analysis.
problem Estimating a rank-one symmetric matrix corrupted by noise.
method Gradient descent on a sphere, using local versions of the semi-circle law.
result Explicit formulas for the time evolution of the estimator and cost function, revealing phase transitions.
We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.
In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.
We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds.
New rigidity theorem for product of lattices.
problem Understanding quasi-isometry of product lattices.
method Demonstrated rigidity for product of non-uniform rank one lattice and nilpotent lattice.
result Any quasi-isometric group is an extension of a non-uniform rank one lattice by a nilpotent lattice.
Rank-one partitioning: formalization, illustrative examples, and a new cluster enhancing strategycs.LG
New method unifies and formalizes data partitioning using a single vector.
problem Data partitioning and clustering methods.
method Rank-one matrix factorization and denoising of piecewise constant signals.
result Demonstrates robustness of denoising step in partitioning.
Completes the space of vector-valued one-forms on manifolds.
problem Metric incompleteness of the space of full-ranked one-forms.
method Distance equality and quotient structures.
result Concrete description of the metric completion of the space of full-ranked one-forms.
We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. F…
Study shows how fast a specific matrix completion method works.
problem Completing a rank-one matrix from a subset of revealed entries.
method Alternating minimization approach for matrix completion.
result Polynomial upper bound on convergence rate.
Let be a Hadamard manifold, and a non-elementary discrete group of isometries of which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold to the behavior of the Poincar{é} series of . Precisely, the aim of this paper is to extend the so-called…
This work presents a novel approach to train invertible linear layers by adding rank-one perturbations.
problem Training invertible linear layers during optimization with gradient-based methods is challenging.
method Train rank-one perturbations and add them to weight matrices infrequently, keeping track of inverses and determinants.
result Invertible linear layers improve mixing and mode separation in normalizing flows.
There has been growing interest in extending traditional vector-based machine learning techniques to their tensor forms. An example is the support tensor machine (STM) that utilizes a rank-one tensor to capture the data structure, thereby alleviating the overfitting and curse of dimensionality problems in the conventio…
A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetr…
We give an explicit description of all complete -invariant Ricci-flat Kähler metrics on the tangent bundle $T(G/K)\cong G^\bbC/K^\bbC$ of rank-one Riemannian symmetric spaces of compact type, in terms of associated vector-functions.
Estimates the probability of a random symmetric tensor being close to rank-one.
problem Estimating the probability of a random symmetric tensor being close to rank-one.
method Using Weyl's tube formula and techniques from Random Matrix theory, we study metric invariants of the real Veronese variety.
result Explicit formula for the reach and curvature coefficients of the real Veronese variety with respect to the Bombieri-Weyl metric.
This paper sets fundamental limits for rank-one matrix estimation with varying noise levels.
problem Estimating a rank-one matrix from Gaussian observations with different noise levels across blocks.
method Novel reduction from heterogeneous noise to homogeneous noise, proving asymptotic error bounds.
result Asymptotically exact formulas for minimum mean-squared error in estimating rank-one matrix and factors.
Study on rank-one ECS manifolds, focusing on dilational type.
problem Characterizing ECS manifolds with specific properties.
method Analyzing properties of pseudo-Riemannian manifolds with parallel Weyl tensor.
result Generic compact rank-one ECS manifolds are either translational or locally homogeneous.
We prove minimal entropy rigidity for complete, finite volume manifolds locally isometric to a product of rank one symmetric spaces of dimension at least 3: the locally symmetric metric uniquely minimizes (normalized) entropy among all Riemannian metrics. The corresponding theorem is true for maps into these spaces as …
In this paper we study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods unify most of the existing methods used in the existing literature for proving this …
After introducing the different boundary geometries of rank one symmetric spaces, we state and prove Fried's theorem in the general setting of all those geometries: a closed manifold with a similarity structure is either complete or the developing map is a covering onto the Heisenberg-type space deprived of a point.