Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
problem Rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.
method Survey and observation on Cheeger-Yau inequality on RCD spaces.
result Observations on the Cheeger-Yau inequality and its applications.
The paper proves conjectures and classifies metrics on 3D manifolds.
problem Proving conjectures and classifying metrics on 3D manifolds with specific curvature conditions.
method Analytical proofs and classification theorems.
result Critical metrics on 3D manifolds are isometric to geodesic balls in space forms.
The paper splits manifolds using infinity harmonic functions with linear growth.
problem Splitting manifolds with specific harmonic functions.
method Analyzes manifolds with non-negative Ricci or sectional curvature, focusing on infinity harmonic functions with linear growth.
result Extends Savin's theorem to surfaces with non-negative sectional curvature.
Non-negative L1-approximating polynomials for Gaussian distributions are proven for certain classes of sets.
problem Existence of non-negative L1-approximating polynomials for Gaussian distributions. method Proving the existence of degree-k non-negative polynomials that approximate indicator functions of sets with Gaussian surface area in L1-norm. result Proves the existence of non-negative L1-approximating polynomials for certain classes of sets with Gaussian surface area. SOS programming verifies MTW tensor non-negativity for optimal transport maps.
problem Verifying MTW tensor non-negativity for general cost functions is difficult.
method Sum-of-Squares (SOS) programming for verifying and approximating MTW non-negativity.
result SOS programming provides certificates and approximations of MTW non-negativity.
Motivated by applications in hyperspectral imaging we investigate methods for approximating a high-dimensional non-negative matrix Y by a product of two lower-dimensional, non-negative matrices K and X. This so-called non-negative matrix factorization is based…
In this paper, we study open complete metric spaces with non-negative curvature. Among other things, we establish an extension of Perelman's soul theorem for possibly singular spaces: "Let X be a complete, non-compact, finite dimensional Alexandrov space with non-negative curvature. Suppose that X has no boundary and h…
For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on concentration of the measure under positive Ollivier curvature.
Sharp Hardy inequalities on Riemannian submanifolds with non-negative curvature.
problem Establishing Hardy inequalities for submanifolds in Riemannian geometry.
method Analyzing distance functions and using Riemannian submanifolds with non-negative curvature.
result Sharp weighted Hardy inequalities valid for compact and non-compact submanifolds, even in compact ambient manifolds.
Global existence and convergence proved for Kazdan-Warner equation with non-negative prescribed function.
problem Existence and convergence of solutions to the Kazdan-Warner equation on a closed Riemann surface.
method Global existence and convergence proved using additional assumptions on the prescribed function and the geometry of the surface.
result Global existence and convergence of solutions proved under specific conditions.
Paper proposes robust risk measures for non-negative risks with partial information.
problem Tackles robustness of distortion risk measures under distributional uncertainty.
method Introduces new uncertainty sets and derives closed-form expressions for risk maximization.
result Derives closed-form expressions for risk maximization over uncertainty sets.
New bounds on Bartnik mass for surfaces with non-negative first eigenvalue.
problem Bounding Bartnik mass for surfaces with spectral non-negativity condition.
method Proving upper bound on Bartnik mass using spectral non-negativity condition.
result Bounded above by √(|S²|_g/16π) under spectral non-negativity.
The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.
problem Proving a sharp mean value inequality for non-negative superharmonic functions.
method Develops a new sharp mean value inequality and an explicit formula for weighted scalar curvature.
result The new inequality removes the radius restriction of Schoen-Yau's result and provides an explicit formula for integral of weighted scalar curvature.
We study the sparse non-negative least squares (S-NNLS) problem. S-NNLS occurs naturally in a wide variety of applications where an unknown, non-negative quantity must be recovered from linear measurements. We present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior on the spars…
Unified framework for non-negative matrices and tensors using Wasserstein loss.
problem Finding low-dimensional representations of high-dimensional datasets with non-negative constraints.
method Unified mathematical framework with a smoothed Wasserstein loss, convex dual formulation for efficient computation.
result Efficient solution for non-negative matrix and tensor factorisations with Wasserstein loss.
In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non…
We prove nonexistence of nonconstant local minimizers for a class of functionals, which typically appears in the scalar two-phase field model, over a smooth N-dimensional Riemannian manifold without boundary with non-negative Ricci curvature. Conversely for a class of surfaces possessing a simple closed geodesic along …
Non-negative matrix factorization is a popular tool for decomposing data into feature and weight matrices under non-negativity constraints. It enjoys practical success but is poorly understood theoretically. This paper proposes an algorithm that alternates between decoding the weights and updating the features, and sho…
In this paper, we prove that the Lp essential spectra of the Laplacian on functions are [0,+∞) on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative…
A common challenge in estimating parameters of probability density functions is the intractability of the normalizing constant. While in such cases maximum likelihood estimation may be implemented using numerical integration, the approach becomes computationally intensive. The score matching method of Hyvärinen [2005] …
Study finds geodesic networks for surfaces with convex boundary.
problem Finding geodesic networks for surfaces with convex boundary.
method Investigates free boundary geodesic networks in surfaces with non-negative sectional curvature and convex boundary.
result Existence of a geodesic network realizing the first width of a surface with non-negative sectional curvature and strictly convex boundary.
A theoretical framework for non-negative matrix factorization based on generalized dual Kullback-Leibler divergence, which includes members of the exponential family of models, is proposed. A family of algorithms is developed using this framework and its convergence proven using the Expectation-Maximization algorithm. …
This work shows MLPs can approximate monotonic functions without bounded activations.
problem Optimizing MLPs with monotonic constraints and bounded activations.
method Generalized theoretical results showing MLPs with non-negative weights and saturating activations are universal approximators.
result MLPs with non-negative weights and saturating activations are universal approximators for monotonic functions.
Study growth rates of harmonic functions on curved surfaces.
problem Understanding the growth rates of harmonic functions on curved surfaces.
method Gradient estimate and frequency analysis on complete surfaces and manifolds with non-negative curvature.
result Existence and properties of nonconstant polynomial growth harmonic functions on manifolds with maximal volume growth.
Study shows bound on Uryson width for specific 3D manifolds.
problem Bounding Uryson width for 3D manifolds with non-negative Ricci curvature and strictly mean convex boundary.
method Proved existence of a Morse function with uniform diameter bounds on level sets.
result Upper bound on Uryson width for the specified 3D manifolds.
We study minimal graphic functions on complete Riemannian manifolds $\Si$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with …
Let (M,g) be a complete non-compact Riemannian manifold. We consider operators of the form Δg+V, where Δg is the non-negative Laplacian associated with the metric g, and V a locally integrable function. Let ρ:(M^,g^)→(M,g) be a Riemannian covering, with Laplacian Δg^ and potenti…
Non-negative matrix factorization (NMF) is a dimensionality reduction technique which tends to produce a sparse representation of data. Commonly, the error between the actual and recreated matrices is used as an objective function, but this method may not produce the type of representation we desire as it allows for th…
A novel method relaxes binary constraints to non-negative spheres for multi-matching and clustering.
problem Optimization problems over binary matrices with injectivity constraints.
method Non-negative spherical relaxation followed by conditional power iteration.
result Automatic adjustment of the continuous parameter related to universe size.
Several rigidity results are proved for critical points of natural Riemannian functionals on the space of metrics on 3-manifolds. Two of these results are as follows. Let (N, g) be a complete Riemannian 3-manifold, satisfying one of the following variational conditions: (i) (N, g) has non-negative scalar curvature and …
DeepMP improves non-negative sparse recovery performance.
problem Recovering non-negative sparse signals with high coherence.
method Reformulated non-negative matching pursuit as a deep neural network.
result DeepMP yields significant improvement in exact recovery performance.
Sharp inequality in spaces with non-negative Ricci curvature.
problem Proving a sharp isoperimetric inequality in metric measure spaces.
method Using volume entropy in non-compact metric measure spaces with non-negative synthetic Ricci curvature.
result Proved a sharp dimension-free isoperimetric inequality.
Odd GKM-manifolds with non-negative curvature split cohomology.
problem Understanding cohomology of odd-dimensional GKM-manifolds.
method Proving cohomology splitting for specific manifolds.
result Cohomology splits for GKM3 manifolds of non-negative curvature. The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.
problem Proving isoperimetric inequalities in manifolds with small negative Ricci curvature.
method Expanding on the ABP method, the paper uses the elliptic Kato constant to control the non-negativity of the Ricci-tensor and applies techniques from Li-Tam and Kasue.
result Sharp isoperimetric inequalities in the limit are proven in the presence of small negative curvature.
Every connected, weighted graph with non-negative curvature has exactly two ends.
problem Characterizing the structure of connected, weighted graphs with non-negative curvature.
method Extremal Lipschitz extensions, variational principle, study of harmonic functions.
result Every salami has exactly two ends and no vertices with positive curvature.
The study examines the geometry of Q-curvature and its associated functions.
problem Investigating the properties of Q-curvature and its associated functions. method Analyzing a complete and conformal metric g=e2u∣dx∣2 on Rn with non-negative nth-order Q-curvature and non-negative scalar curvature. result The growth rate of kth elementary symmetric function of Ricci curvature over geodesic ball of radius r is at most polynomial in r with order n−2k for all 1≤k≤2n−2. Non-negative curvature affects Markov chains' mixing and expansion properties.
problem Understanding the behavior of Markov chains with non-negative curvature.
method Analyzing conductance, displacement, and cutoff phenomenon in sparse Markov chains.
result Non-negatively curved Markov chains exhibit specific, non-standard behavior in terms of mixing and expansion.
Formal manifolds with non-negative Ricci curvature have formal covers.
problem Formality of manifolds with non-negative Ricci curvature.
method Study of universal covers and formal properties.
result Closed non-orientable manifolds with non-negative Ricci curvature are formal.
Paper finds properties of special geometric shapes in higher dimensions.
problem Understanding translating solitons for curvature flows.
method Analyzes properties of γ-translators using maximum principles. result Uniqueness of strictly convex γ-translators with a single end. In this paper we study non-negatively curved and rationally elliptic GKM4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifol…
Proves solution uniqueness for biomembrane shape prediction.
problem Proving solution uniqueness for the genus one Canham variational problem.
method Combining numeric analytic continuation and singularity analysis to prove non-negativity of a sequence.
result Proves positivity of the sequence, leading to solution uniqueness.
Study Kähler metrics on complex tori with almost non-negative scalar curvature.
problem Stability of Kähler metrics on complex tori.
method Proved convergence of non-collapsing subsequence of Kähler metrics to flat torus.
result Kähler metrics with almost non-negative scalar curvature on complex tori converge to flat torus.
Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.
problem Establishing a Fenchel-Willmore inequality for submanifolds in manifolds with non-negative Ricci curvature.
method Analyzing submanifolds in manifolds with non-negative intermediate Ricci curvature and Euclidean volume growth.
result Sharp Fenchel-Willmore inequality for submanifolds in manifolds with non-negative intermediate Ricci curvature.
The study examines symmetries in spaces with positive or non-negative curvature.
problem Understanding symmetries in spaces with curvature constraints.
method Survey of existing results for Riemannian manifolds with specified curvature properties and symmetries.
result Results on symmetries in spaces with curvature bounds.
We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show tha…
There is a growing body of literature showing that deep neural networks are vulnerable to adversarial input modification. Recently this work has been extended from image classification to malware classification over boolean features. In this paper we present several new methods for training restricted networks in this …
Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.
problem Extending non-existence theorem of positive scalar curvature to product manifolds.
method Introduced intermediate curvature and studied rigidity conditions.
result Rigidity when intermediate curvature is non-negative in low dimensions.
Study Hilbert's projective metric for bounded growth functions leading to Sinkhorn's algorithm convergence.
problem Optimal transport in unbounded settings with heavy-tailed distributions.
method Hilbert's projective metric for integrable functions of bounded growth, kernel integral operators as contractions.
result Exponential convergence of Sinkhorn's algorithm for light-tailed marginal distributions.