The paper extends a prediction method to curved spaces.
problem Prediction with expert advice in curved spaces.
method Geometric considerations and adaptation of online to batch conversion.
result Extension of exponentially weighted average forecaster to non-positive curvature spaces.
Deep learning predicts curvature of 2D interfaces in level-set method.
problem Estimating curvature in level-set method for complex interfaces.
method Deep learning using feed-forward neural networks trained on synthetic data.
result Deep learning models approximate curvature with comparable precision to traditional methods.
Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.
problem Enhance graph representation learning by leveraging spectral and curvature signals.
method Proposes Spectro-Riemannian Graph Neural Networks (CUSP) that combines spectral and curvature insights.
result Empirical evaluation shows CUSP outperforms state-of-the-art models by up to 5.3%.
Graph networks struggle with multi-task learning due to varying property loss surface curvatures.
problem Graph networks underperform in multi-task learning for crystal and molecule properties.
method Assessed curvature of property loss surfaces via spectral properties of Hessians, matrix-free using randomized numerical linear algebra.
result Varying curvature of property loss surfaces explains graph networks' multi-task learning inefficiency.
GOIMDA selects inputs to maximize expected influence on a goal functional, reducing data acquisition needs.
problem Challenges in active data acquisition for learning and optimization tasks in deep neural networks.
method GOIMDA uses inverse curvature and goal gradient to select inputs maximizing expected influence on a specified goal functional.
result GOIMDA achieves target performance with fewer labeled samples or function evaluations compared to baselines.
Paper proves Gromov's conjecture on manifolds with certain group properties.
problem Gromov's conjecture on positive scalar curvature and simplicial volume.
method Proves conjecture under a fundamental group decay property.
result Proves Gromov's conjecture for manifolds with a weakened rapid decay property.
Curvature penalties improve interpretability of KANs without sacrificing accuracy.
problem Pathologically high-curvature oscillations in KANs activations make them hard to interpret.
method Derived a curvature penalty and proved an upper bound on model curvature.
result KANs with curvature penalties achieve substantially smoother activations while maintaining accuracy.
Verified numerics prove existence of a curvature solution with known symmetries.
problem Existence of a curvature solution for the Nirenberg problem.
method Verified numerics and computer assistance.
result Existence of a genuine solution with known symmetry groups.
The doubling conjecture for positive scalar curvature is proven under certain conditions.
problem Determining when a manifold with a specific boundary condition admits positive scalar curvature.
method Surgery techniques for positive scalar and mean curvature, and existence of area-minimizing hypersurfaces.
result The doubling conjecture holds true for manifolds with certain split conditions on fundamental groups.
This study examines how removing edges from complete graphs affects Ollivier Ricci curvature.
problem Conditions under which Ollivier Ricci curvature changes sign after edge removal.
method Defined and analyzed graphs obtained by removing matching, vertex incident, and cycle edges from complete graphs.
result Ollivier Ricci curvature remains positive or zero for graphs formed by removing edges from complete graphs.
New research shows hyperbolic embeddings are useful for global consistency tasks in graphs.
problem The usefulness of hyperbolic representations in graph learning tasks.
method Computed hyperbolic embeddings for node classification and link prediction tasks, addressing optimization issues at zero curvature.
result Hyperbolic embeddings are more effective for tasks requiring global consistency, while Euclidean models are superior for other tasks.
Many real-world sequential decision-making problems can be formulated as optimal control with high-dimensional observations and unknown dynamics. A promising approach is to embed the high-dimensional observations into a lower-dimensional latent representation space, estimate the latent dynamics model, then utilize this…
We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in Rn+1 with O(1)×O(n) symmetry. We show they all have unique asymptotics as t→−∞ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer …
We develop techniques for classifying the nonnegatively curved left-invariant metrics on a compact Lie group G. We prove rigidity theorems for general G and a partial classification for G=SO(4). Our approach is to reduce the general question to an infinitesimal version; namely, to classify the directions one can move a…
The paper examines curvature and stability in quasi-geostrophic motions using spherical harmonics.
problem Analyzing the curvature and stability of quasi-geostrophic motions.
method Utilizing spherical harmonics and structure constants, the curvature of the L2 metric on the central extension is computed. result A lower bound for weather prediction error in a simplified model is suggested.
Closed Riemannian manifolds with positive mixed sectional curvature
problem Constructing closed Riemannian manifolds with positive mixed sectional curvature
method Explicit construction using totally geodesic foliations
result Positive mixed sectional curvature alone does not imply the Ferus--Adams estimate on closed manifolds
New entropy functionals for curved spaces help predict shape behavior.
problem Understanding entropy behavior in curved spaces.
method Introduced new entropy functionals for submanifolds of Cartan-Hadamard manifolds.
result Obtained sharp lower bounds on these entropies for certain closed hypersurfaces and observed a novel rigidity phenomenon.
The article confirms a conjecture for solvmanifolds with complex commutator.
problem Confirming a conjecture about compact Hermitian manifolds with constant holomorphic sectional curvature.
method Analyzing solvmanifolds with complex commutator, extending results on nilmanifolds.
result The conjecture is confirmed for all solvmanifolds with complex commutator.
ITF improves DSR but inflates curvature, while marginal likelihood reduces it, affecting QoIs.
problem Curvature mismatch between teacher forcing and marginal likelihood in chaotic dynamical systems.
method Comparing objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs.
result Curvature inflation by ITF and reduction by marginal likelihood affect dynamical quantities of interest.
Study proper sampling for X-ray transforms on simple surfaces.
problem Proper discretizing and sampling issues related to geodesic X-ray transforms on simple surfaces.
method Provide minimal sampling rates for faithful reconstruction, quantify sampling quality, and predict artifacts.
result Minimal sampling rates and artifact prediction for geodesic X-ray transforms on simple surfaces.
Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group S3. Using the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a privileged right invariant Killing field of constant length. Each such Riemannian metric p…
Paper uses Ricci curvature to measure and forecast China's stock market stability.
problem Measuring and predicting systemic stability of China's stock market.
method Geometric measure derived from discrete Ricci curvature applied to financial networks.
result Ricci curvature effectively captures market stability and predicts future trends.
Study of congestion in negative curvature manifolds using fair-division algorithms.
problem Estimating and predicting the size and location of congestion core in negative curvature manifolds.
method Introducing a novel fair-division algorithm to estimate congestion core.
result Demonstrated the effectiveness of fair-division algorithms in estimating congestion core.
Recently manifold learning algorithm for dimensionality reduction attracts more and more interests, and various linear and nonlinear, global and local algorithms are proposed. The key step of manifold learning algorithm is the neighboring region selection. However, so far for the references we know, few of which propos…
Gu and Zhu have shown that Type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on Sm, for all m≥3. In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit.
We simplify Volterra process predictions by reducing dimensionality and using a tailored deep learning model.
problem Predicting the conditional law of Volterra processes with stochastic volatility is challenging due to high dimensionality and non-smoothness.
method We developed a stable dimension reduction technique onto a low-dimensional statistical manifold of non-positive curvature and introduced a sequentially deep learning model tailored to this geometry.
result Our model can approximate the conditional law of Volterra processes with approximation rates achievable only with very large networks.
The Kähler-Ricci flow yields bounded diameter and Ricci curvature for minimal models.
problem Estimating the diameter and Ricci curvature of long-time solutions of the Kähler-Ricci flow.
method Analyzing the semi-ample canonical line bundle and using Perelman's estimates.
result Uniform bounds on diameter and Ricci curvature for long-time solutions.
Billiard trajectories in curved spaces have predictable travel times.
problem Understanding travel times in billiard trajectories on curved surfaces.
method Analyzing geodesic flows and sectional curvature to prove time-preserving conjugacy.
result Billiard trajectories with almost identical obstacles have identical shapes.
Survey on recent breakthrough linking curvature and Kobayashi hyperbolicity.
problem Connecting curvature properties to Kobayashi hyperbolicity in complex geometry.
method Detailed analysis of Wu-Yau theorem and its proof.
result A compact complex manifold with negative holomorphic sectional curvature admits a Kähler metric with negative Ricci curvature.
Sharp distance estimates for compact spin manifolds using Dirac operator.
problem Metric inequalities for compact spin manifolds with lower bounds on scalar and mean curvatures.
method Using the Dirac operator technique with spectral estimates and local boundary conditions.
result Optimal estimates for Riemannian bands and long neck problem solutions.
Sandpile Economics explains how economies can be prone to large crises from small shocks.
problem Capitalist economies' recurrent crises disproportionate to shocks.
method Formal framework interpreting instability as geometric fragility of production networks.
result Curvature of production networks predicts medium-run output dynamics and resilience.
Transformer models are compared to curved spacetime in General Relativity.
problem Understanding the geometric structure of Transformer models.
method Geometric analogy to General Relativity, focusing on attention and curvature effects.
result Transformer token embeddings exhibit curvature, affecting their evolution over layers.
This paper tackles denoising of complex measures using optimal transport and curvature analysis.
problem Denoising of complex, possibly non-log-concave measures.
method Score function and optimal transport theory to revert Langevin diffusion chains.
result The difficulty of denoising depends on the curvature complexity of the initial measure at specific SNR scales.
New CRB derived for curved models using extrinsic geometry.
problem Estimate curved statistical families accurately.
method Vector generalization of CRB with curvature correction using SDP and SOS relaxations.
result Directional curvature correction provides more accurate estimation.
Study of cosmic microwave background polarization using spin random fields.
problem Detecting deviations from Gaussianity and anisotropies in cosmic fields.
method Explicit formula for Lipschitz-Killing curvatures of spin spherical random fields.
result Coherent with asymptotic results, providing new metric expressions.
New algorithm helps escape saddle points in optimization problems.
problem Optimizing smooth non-convex functions to avoid saddle points.
method Perturbed Saddle-escape Descent (PSD) algorithm with explicit constants.
result PSD finds approximate second-order stationary points efficiently.
New framework uses geometry of embeddings to predict robustness.
problem Monitoring robustness in models without OOD labels.
method Constructs graphs from embeddings, measures spectral complexity and curvature.
result Representation geometry predicts robustness reliably.
New edge features improve GNN performance in biological datasets.
problem Inefficient use of edge features in GNNs.
method Self-supervised and unsupervised learning for new edge features, incorporating Forman-Ricci curvature.
result Improved node classification performance over baseline GNN models.
Lipid necks, large curvature bridges, are shown to be metastable.
problem Understanding the energetically prohibitive yet ubiquitous lipid necks in cell membranes.
method Geometric triality approach to demonstrate metastability.
result Lipid necks can exist for finite but potentially long times without stabilizing mechanisms.
The paper reveals that baselines significantly impact RL algorithms' convergence.
problem Understanding the true impact of baselines on policy optimization.
method Theoretical analysis of bandit and RL problems, focusing on natural policy gradient and EXP3.
result Baselines can determine algorithm convergence, contradicting traditional optimization theory.
A new spline method for manifold learning using Hessian-based curvature penalties.
problem Learning manifolds with curvature penalties in high dimensions.
method Generalizes thin-plate splines to flat manifolds using Hessian matrices, minimizing square error with curvature constraints.
result Existence and uniqueness of the spline solution, expressed as Green's functions and Hessian approximations.
Study develops curvature for contact-sequence networks, revealing temporal dynamics.
problem Lack of geometric analysis for temporal network sequences.
method Develops Forman--Ricci curvature on spatiotemporal prism complexes.
result Two curvature variants disagree on 56-67% of temporal edges.
Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with π1(M)=Z4timesZ/3, so that the index invariant in the KO-theory of the reduced C∗-algebra of π1(M) is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifolds cannot carry a metric of pos…
Effective training of deep neural networks suffers from two main issues. The first is that the parameter spaces of these models exhibit pathological curvature. Recent methods address this problem by using adaptive preconditioning for Stochastic Gradient Descent (SGD). These methods improve convergence by adapting to th…
Study optimizes CANN for actuarial tasks using RSM.
problem Optimizing hyperparameters for neural networks in actuarial science.
method Factorial design and response surface methodology (RSM).
result Reduced hyperparameter optimization from 288 to 188, achieving near-optimal performance.
This paper establishes minimax rates for online regression with arbitrary classes of functions and general losses. We show that below a certain threshold for the complexity of the function class, the minimax rates depend on both the curvature of the loss function and the sequential complexities of the class. Above this…
A new method improves adversarial robustness and interpretability with reduced training time.
problem Adversarial attacks on deep neural networks.
method A novel regularizer incorporating first and second order information via a quadratic approximation to the adversarial loss.
result Single iteration of the proposed regularizer achieves stronger robustness than prior methods.
A geometric account explains why 'The Dress' is ambiguous, predicting observable signatures in image processing.
problem Understanding and predicting ambiguity in image processing, particularly in intrinsic image decomposition.
method Geometric analysis of intrinsic image decomposition, focusing on the discontinuous switch in prior-mode sections.
result Predicted signatures in albedo Jacobian and Fernet curvature can be observed in various models and datasets.