R2D2-Net improves Bayesian neural networks by preventing over-shrinkage of important weights.
problem Bayesian neural networks struggle with choosing appropriate priors, leading to over-shrinkage or poor predictive performance.
method Proposes R2D2-Net with an R^2-induced Dirichlet Decomposition prior and variational Gibbs inference algorithm.
result R2D2-Net effectively shrinks irrelevant coefficients while preventing key features from over-shrinkage.
New priors improve robustness and interpretability in penalized regression.
problem Improper priors in penalized regression lead to suboptimal solutions.
method Developed non-zero priors inspired by human decision heuristics.
result Robust priors yield excellent worst-case performance across various tasks.
Symmetries in shrinking Ricci solitons spread outward.
problem Understanding symmetries in shrinking Ricci solitons.
method Propagating approximate symmetries to larger scales.
result Symmetries in shrinking Ricci solitons spread outward.
In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton R4 or the round cyl…
GS-B3SE improves label shift estimation by smoothing priors on a graph.
problem Label shift adaptation when source and target distributions share conditional but not marginal probabilities.
method Graph-Smoothed Bayesian Black-Box Shift Estimator (GS-B3SE) places Laplacian-Gaussian priors on log-priors and confusion-matrix columns tied by a label-similarity graph. result GS-B3SE produces a tractable posterior with HMC or Newton-CG schemes, proving identifiability, contraction, and robustness. Study on shrinking solitons of generalized Ricci flow.
problem Characterizing shrinking solitons in generalized Ricci flow.
method Analyzing gradient shrinking solitons and pluriclosed solitons on compact manifolds.
result First non-trivial shrinking generalized soliton constructed.
Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.
problem Classifying shrinking Ricci solitons without curvature or volume restrictions.
method Proving the sharp Li-Yau equality for conjugate heat kernel on shrinking Ricci solitons.
result Several estimates and classification of four-dimensional, non-compact shrinking Ricci solitons.
Paper classifies 3D breathers and generalizes Ricci soliton results.
problem Classifying and understanding Ricci solitons and breathers.
method Developed a condition for the existence of asymptotic shrinking gradient Ricci solitons.
result Every complete shrinking Ricci soliton with bounded Ricci curvature is gradient.
5D shrinking solitons with bounded curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons.
method Proving rigidity for solitons with bounded curvature.
result 5D shrinking gradient Ricci solitons with bounded curvature are rigid.
The edge partition model (EPM) is a fundamental Bayesian nonparametric model for extracting an overlapping structure from binary matrix. The EPM adopts a gamma process (ΓP) prior to automatically shrink the number of active atoms. However, we empirically found that the model shrinkage of the EPM does not typically wo…
Paper proves rigidity for Ricci solitons with specific conditions.
problem Understanding the properties of Ricci solitons under various conditions.
method Analyzes shrinking and expanding Ricci solitons with specific constraints.
result Compact shrinking Ricci solitons are Einstein if the potential function is controlled.
New Gaussian priors for neural networks improve scalability and Bayesian inference stability.
problem Scalability and stability issues in Bayesian neural network inference.
method Introduces a new Gaussian neural network prior with decreasing variance in network width, enabling stable MCMC sampling.
result The new prior enables stable MCMC sampling for Bayesian neural network inference, improving scalability and stability.
We give two new proofs of Perelman's theorem that shrinking breathers of Ricci flow on closed manifolds are gradient Ricci solitons, using the fact that the singularity models of type I solutions are shrinking gradient Ricci solitons and the fact that non-collapsed type I ancient solutions have rescaled limits being sh…
New proof of shrinking gradient Ricci soliton rigidity.
problem Rigidity of shrinking gradient Ricci solitons.
method Maximum principle, maximum curvature condition.
result Shrinking gradient Ricci soliton with constant scalar curvature is isometric to a finite quotient of R^2 x S^2.
We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-nonc…
Bayesian pliable lasso with horseshoe prior models interactions in GLMs with missing data.
problem Modeling interactions in sparse regression problems with missing responses.
method Bayesian pliable lasso with hierarchical horseshoe prior for sparsity and uncertainty quantification.
result Advantages over existing methods in recovering complex interaction patterns under incomplete data.
Complete shrinking soliton found on a specific complex surface.
problem Classifying complete shrinking gradient Kähler-Ricci solitons in two complex dimensions.
method Proved existence of a unique soliton with bounded scalar curvature on a specific blowup.
result Complete classification of such solitons in two complex dimensions.
In this paper, we will give a local version of the Hamilton-Ivey type pinching estimate of the gradient shrinking soliton with vanishing Weyl tensor, and then give a complete classification on gradient shrinking solitons with vanishing Weyl tensor.
The study proves compactness and existence of entropy minimizers for self-shrinking surfaces.
problem Understanding entropy in higher-codimension mean curvature flow.
method Measure-theoretical techniques and rigidity results for self-shrinkers.
result Existence of entropy minimizers and improved rigidity results.
We introduce a new dynamical system for sequentially observed multivariate count data. This model is based on the gamma--Poisson construction---a natural choice for count data---and relies on a novel Bayesian nonparametric prior that ties and shrinks the model parameters, thus avoiding overfitting. We present an effici…
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
Bayesian optimisation is improved by incorporating expert prior through space warping.
problem Cold start phase in expensive function optimisation.
method Prior distribution warps the search space around high probability regions of function optimum.
result Improves optimisation performance through acquisition agnostic approach.
We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.
Let (Y,d) be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, Y is a smooth manifold satisfying a shrinking Ricci soliton equation.
Sharp upper diameter limit found for Ricci solitons.
problem Bounding the diameter of compact shrinking Ricci solitons.
method Used a sharp logarithmic Sobolev inequality and Vitali-type covering argument.
result Sharp upper diameter bound established in terms of scalar curvature and entropy.
A new method shrinks a complex structure without much change.
problem Understanding the geometry of Bing's wild involution.
method Producing a counterintuitive construction to shrink the Bing decomposition without much change.
result A method to shrink a complex structure (Bing's decomposition) without much change.
The study proves rotationally symmetric property of certain shrinking gradient Yamabe solitons.
problem Understanding the rotational symmetry of specific shrinking gradient Yamabe solitons.
method Analyzing nontrivial complete shrinking gradient Yamabe solitons with bounded scalar curvature.
result The assumption of bounded scalar curvature and strict inequality at some point is necessary and sufficient for rotational symmetry.
Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.
problem Analyzing Schrödinger heat kernel on gradient shrinking Ricci solitons.
method Deriving sharp Gaussian upper bounds for the Schrödinger heat kernel.
result Sharp upper and lower bounds for eigenvalues of the Schrödinger operator.
We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in Rn. In addtion, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.
The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
problem Characterizing constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
method Analyzing properties of hypersurfaces in specific ambient spaces (shrinking Ricci solitons).
result Conditions for a constant weighted mean curvature hypersurface to be a level set of the potential function.
Simply-connected shrinking Kähler-Ricci solitons are proven.
problem Simply-connectedness of shrinking Kähler-Ricci solitons.
method Using results by Sun-Zhang and Wylie.
result Shrinking Kähler-Ricci solitons are simply-connected.
We prove that there does not exist non-constant positive f-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the Lp(p≥1 or 0<p≤1) Liouville theorems on the complete gradient shrinking Ricci solitons.
We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the 4-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\…
The study examines four-dimensional gradient Ricci solitons and their properties.
problem Characterizing four-dimensional complete gradient shrinking Ricci solitons.
method Proving properties and providing curvature estimates for solitons under specific conditions.
result Conditions for four-dimensional complete gradient shrinking Ricci solitons.
Paper proves genus of surfaces decreases in mean curvature flow.
problem Understanding the evolution of surfaces under mean curvature flow.
method Analyzes mean curvature flow of compact surfaces in 3-manifolds with specific curvature conditions.
result Genus of the regular set decreases over time.
It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.
Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.
problem Finding topological obstructions to compact gradient shrinking Ricci solitons in dimension four.
method Discussion of background material, introduction of new problem, exploration of limitations of current results.
result Introduction of new problem and limitations of current results in extending Hitchin-Thorpe inequality.
5D shrinking Ricci solitons with constant scalar curvature are rigid.
problem Characterizing 5D shrinking gradient Ricci solitons with constant scalar curvature.
method Proving rigidity by showing they are finite quotients of a known space.
result 5D shrinking gradient Ricci solitons with constant scalar curvature are rigid.
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
We prove that all entire smooth strictly convex self-shrinking solutions on Rn to the Hessian quotient flows must be quadratic. This generalizes the rigidity theorem for entire self-shrinking solutions to the Lagrangian mean curvature flow in pseudo-Euclidean space due to Ding-Xin \cite{DX}. Moreover, we sh…
As of today, there are very few known complete shrinking Ricci solitons in dimension 4, and all examples discovered so far are Kähler and/or Einstein. In this note, we prove that any four dimensional J-invariant gradient shrinking Ricci solitons satisfy a differential form identity relating Kählerity annd Einstein-ness…
In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional…
In this paper we introduce entropy-stability and F-stability for homothetically shrinking Yang-Mills solitons, employing entropy and second variation of F-functional respectively. For a homothetically shrinking soliton which does not descend, we prove that entropy-stability implies F-stability. These stabil…
BS-NAS broadens and shrinks search space for optimal neural architectures.
problem Suboptimal channel numbers and model averaging effects in One-Shot NAS methods.
method Broadening with spring block for channel search, shrinking with underperforming operations removal, evolutionary algorithm for optimal architecture search.
result BS-NAS achieves state-of-the-art performance on ImageNet.
Alternative proof for 4D shrinking Ricci solitons with constant scalar curvature.
problem Proving the structure of four-dimensional shrinking gradient Ricci solitons with constant scalar curvature.
method Analyzing the asymptotic geometry at infinity.
result Alternative proof that such solitons are finite quotients of R^2 x S^2.
The study shows ends of shrinking gradient ρ-Einstein solitons are non-parabolic.
problem Characterizing the ends of shrinking gradient ρ-Einstein solitons. method Proving non-parabolicity of ends and connectivity at infinity for specific conditions.
result Gradient shrinking ρ-Einstein solitons have non-parabolic ends under certain conditions. Paper proves properties of minimal hypersurfaces in specific solitons.
problem Characterizing minimal hypersurfaces in shrinking gradient Ricci solitons.
method Analyzes stable minimal hypersurfaces with specific curvature conditions.
result Minimal hypersurfaces in these solitons have zero second fundamental form and normal Ricci curvature.
In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic, then the manifold is isometric to the Euclidean sphere. As a consequence, we have s…