Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
New formulas for geodesics on Stiefel and flag manifolds using trust-region method.
problem Computing geodesics and logarithms on Stiefel and flag manifolds.
method Closed-form geodesic formulas, trust-region solver, Fréchet derivatives.
result Efficient computation of geodesic distance and logarithm map.
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.
We show that there is no bi-Lipschitz homeomorphism of R2 that maps a spiral with a sub-exponential decay of winding radii to an unwinded arc. This result is sharp as shows an example of a logarithmic spiral.
New MD algorithms using Tempesta logarithms for machine learning.
problem Optimization in machine learning with tailored hyperparameters.
method Developed Mirror Descent algorithms using Tempesta multi-parametric logarithms.
result Wide and flexible family of Mirror Descent and mirror-less updates.
We investigate the connections between the differential-geometric properties of the exponential map from the space of real skew symmetric matrices onto the group of real special orthogonal matrices and the manifold of real orthogonal matrices equipped with the Riemannian structure induced by the Frobenius metric.
Inspired by results of Eskin and Mirzakhani counting closed geodesics of length ≤L in the moduli space of a fixed closed surface, we consider a similar question in the Out(Fr) setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping…
Paper studies statistical manifolds with logarithmic divergences.
problem Understanding statistical manifolds induced by logarithmic divergences.
method Constructs dual foliation of the statistical manifold.
result Extends dual foliation of a dually flat manifold.
Deviation inequalities and limit laws for random walks on metric spaces.
problem Understanding random walks on metric spaces with contracting isometries.
method Adapting Gouëzel's pivotal time construction to establish deviation inequalities.
result Exponential bounds and limit laws for random walks on mapping class groups and CAT(0) spaces.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012b) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, and if and only if the opti…
The paper finds optimal strategies for hedging in incomplete markets using derivatives.
problem Optimal static hedging in incomplete markets with two underlying assets and vanilla options.
method Formulated as a utility maximization problem, solved through variational methods and fixed point analysis.
result Semi-analytical solutions for exponential, power/logarithmic, and quadratic utilities, with convergence to a fixed point for exponential utility.
New uncertainty principle for Schrödinger equations on hyperbolic manifolds.
problem Uncertainty principle for Schrödinger equations on hyperbolic manifolds.
method General strategy of Escauriaza-Kenig-Ponce-Vega, new Carleman estimates, logarithmic convexity, new mollifier and weight function.
result Similar rigidity phenomenon as in Euclidean space persists in hyperbolic geometry.
Generalizes CNNs for Lie group equivariance across various data types.
problem Equivariance to transformations like rotations for non-image data.
method Constructs equivariant convolutional layers for Lie groups.
result Models conserve linear and angular momentum in Hamiltonian systems.
Consider the one-parameter generalizations of the logarithmic and exponential functions which are obtained from the integration of non-symmetrical hyperboles. These generalizations coincide to the one obtained in the context of non-extensive thermostatistics. We show that these functions are suitable to describe and un…
A novel hyperbolic graph attention network for non-Euclidean graph data.
problem Non-Euclidean graph data requires specialized models to capture its unique properties.
method Employed gyrovector spaces to transform features and hyperbolic proximity attention mechanism for aggregation. Novel acceleration strategy using logarithmic and exponential mappings.
result Demonstrated superior performance on real-world datasets compared to state-of-the-art methods.
Algorithm approximates functions into manifolds with curvature bounds.
problem Approximating functions into manifolds with lower curvature bounds.
method Algorithm using manifold exponential and logarithm, with error bounds based on sectional curvature.
result Error bounds for nonnegative sectional curvature are similar to linear space approximations.
Study robust utility maximization with uncertain continuous semimartingales.
problem Maximizing utility in continuous time under model uncertainty.
method Duality and conjugate problems for logarithmic, exponential, and power utilities.
result Existence of optimal portfolios for various utilities.
In this paper, a Bayesian inference technique based on Taylor series approximation of the logarithm of the likelihood function is presented. The proposed approximation is devised for the case, where the prior distribution belongs to the exponential family of distributions. The logarithm of the likelihood function is li…
Hyperbolic GNNs improve graph data learning.
problem Learning from graph-structured data.
method Proposes a novel GNN architecture for Riemannian manifolds.
result Hyperbolic GNNs lead to substantial improvements on benchmark datasets.
Canonical correlation analysis (CCA) is a technique to find statistical dependencies between a pair of multivariate data. However, its application to high dimensional data is limited due to the resulting time complexity. While the conventional CCA algorithm requires polynomial time, we have developed an algorithm that …
New algorithm reduces online logistic regression regret without exponential constant.
problem Improper learning in online logistic regression with logarithmic regret.
method Regularized empirical risk minimization with surrogate losses.
result Regret scaling as O(B log(Bn)) with low computational complexity.
In the paper, the author studies properties of three functions relating to the exponential function and the existence of partitions of unity, including accurate and explicit computation of their derivatives, analyticity, complete monotonicity, logarithmically complete monotonicity, absolute monotonicity, and the like.
The study improves bounds on the number of closed geodesics and logarithmic improvements in the Weyl law.
problem Estimating the number of closed geodesics and improving logarithmic bounds in the Weyl law.
method Study of non-degeneracy properties of nearly closed orbits for predominant sets of metrics.
result Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics.
Greedy pruning reduces neural networks by a logarithmic number of tickets, improving accuracy.
problem Pruning large neural networks to reduce size while maintaining accuracy.
method Greedy optimization-based pruning method with exponential decay guarantee.
result The discrepancy between pruned and original networks decays exponentially with network size.
We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler mea…
Study the exponential map on surfaces using fluid dynamics.
problem Exponential map of volume-preserving diffeomorphisms on closed surfaces.
method Fluid dynamical proof of Ebin--Misiołek--Preston theorem and extension of Shnirelman's rigidity result.
result Exponential map is a nonlinear Fredholm mapping of index zero and Fredholm quasiregular.
Logarithmic regret for continuous-time reinforcement learning.
problem Continuous-time Markov decision processes with unknown transition probabilities and holding times.
method Upper confidence reinforcement learning, mean holding time estimation, stochastic comparison of point processes.
result Logarithmic regret bound achieved in finite time.
Study on stability of harmonic maps with sub-Riemannian geometry.
problem Stability of exponentially subelliptic harmonic maps.
method Derived first and second variation formulas, applied to prove stability under certain conditions.
result Exponentially subelliptic harmonic maps are stable if the target manifold has nonpositive curvature.
Paper develops Riemannian geometry for SPSD matrices with DA applications.
problem Riemannian geometry of SPSD matrices for DA.
method Closed-form expressions, approximations of geodesic path, PT, canonical representation.
result Proposes an algorithm for DA with improved performance.
Improved GNN simulation of WL test with exponentially lower complexity.
problem Improving the complexity of simulating the Weisfeiler-Lehman test with GNNs.
method Exponentially lower complexity simulation of WL test using GNNs with polylogarithmic parameters and O(log n) bits feature vectors.
result Near-optimal construction with logarithmic lower bounds for feature vector length and neural network size.
Let S be a nonexceptional oriented surface of finite type. We construct an uncountable family of probability measures on the space of area on holomorphic quadratic differentials over the moduli space for S containing the usual Lebesgue measure. These measures are invariant under the Teichmueller geodesic flow, and they…
New exponential map for Lie groups connects to sub-Riemannian geometry.
problem Developing a new exponential map for Lie groups.
method Introducing a new exponential map related to sub-Riemannian geometry.
result New exponential map connects to sub-Riemannian geometry.
Study shows exponential growth of Laplacian determinant on random hyperbolic surfaces.
problem Understanding the behavior of Laplacian determinants on random hyperbolic surfaces.
method Investigated various models of random hyperbolic surfaces and their Laplacian determinants as genus increases.
result For all popular models, the determinant grows exponentially with a universal exponent as the genus goes to infinity.
We generalize Cartan's logarithmic derivative of a smooth map from a manifold into a Lie group G to smooth maps into a homogeneous space M=G/H, and determine the global monodromy obstruction to reconstructing such maps from infinitesimal data. The logarithmic derivative of the embedding of a submanifold $Σ\subset M…
Study logarithmic flat connections on principal bundles using Lie groupoids.
problem Classify flat connections on principal bundles with logarithmic singularities.
method Use tools from Lie groupoid theory to classify representations and establish van Kampen theorems.
result Obtain a functorial Riemann-Hilbert correspondence for logarithmic connections.
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
Proves properties of sub-Riemannian exponential map, showing it's not injective.
problem Regularity and continuity of sub-Riemannian exponential map.
method Used sub-Riemannian Jacobi fields and Maslov index of Jacobi curves.
result Exponential map of 3D Heisenberg group is not injective near conjugate vectors.
The paper triangulates Heisenberg groups with horizontal and straight simplexes.
problem Triangulating Heisenberg groups with specific regularity properties.
method Constructing triangulations with horizontal and straight simplexes on a polyhedral structure and extending to the whole Heisenberg group.
result Explicit examples of grid and triangulations provided.
Paper generalizes VB-FTRL for online learning of quantum states with logarithmic loss.
problem Online learning of quantum states with logarithmic loss.
method Generalizes VB-FTRL algorithm for LL-OLQS with polynomial-time implementation.
result Achieves a regret rate of O(d2log(d+T)) for LL-OLQS. Logarithmic regret achieved in continuous-time linear-quadratic reinforcement learning.
problem Optimizing control actions in unknown continuous-time systems over a finite time horizon.
method Least-squares algorithm based on continuous-time observations and controls, with perturbation analysis and parameter estimation error analysis.
result Logarithmic regret bound of order O((lnM)(lnlnM)). Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.
problem Investigate non-geodesic connections in warped Segre-Veronese manifolds.
method Investigate a one-parameter family of warped geometries, presenting closed expressions for maps and distance.
result Segre-Veronese manifolds are not geodesically connected in Euclidean geometry but can be for some warping parameters.
Develops a new exponential map for time-varying vector fields.
problem Lack of global flows for general time-varying vector fields.
method Categorical development of spaces of vector fields and flows, allowing for systematic localisation.
result Derives the homeomorphism of the exponential map for vector fields with measurable time-dependence.
Honest traders can outperform insiders in a Black-Scholes market with positive probability.
problem Comparing the performance of honest and insider traders in a financial market.
method Using anticipating stochastic calculus and forward integral analysis of the Doléans-Dade exponential process.
result The honest trader can achieve higher logarithmic utility and wealth than the insider with positive probability.
This paper analyzes two Lie group momentum optimization algorithms and their convergence rates.
problem Optimizing functions on Lie groups using momentum-based dynamics.
method Investigates Lie Heavy-Ball and Lie NAG-SC algorithms, quantifying their convergence rates under smoothness and convexity assumptions.
result Lie NAG-SC accelerates optimization over the momentumless case, while Lie Heavy-Ball does not.
Analogous exponential map defined for Hopf algebras.
problem Defining an exponential map for Hopf algebras.
method Analogy with Lie groups, interpretation as states, Hilbert C* bimodules, dual Hopf algebra elements.
result Multiple interpretations of exponential map values in Hopf algebras.
Exponential rate of convergence for harmonic heat flow maps.
problem Analyzing the convergence rate of harmonic heat flow maps.
method Proving exponential convergence rate for harmonic heat flow maps.
result Exponential convergence rate of the harmonic heat flow.
We create a new online reduction of multiclass classification to binary classification for which training and prediction time scale logarithmically with the number of classes. Compared to previous approaches, we obtain substantially better statistical performance for two reasons: First, we prove a tighter and more comp…