Paper describes how to extend multiple conjugation quandles using maps.
problem Understanding affine extensions of multiple conjugation quandles.
method Introduces augmented MCQ Alexander pairs for affine extensions.
result Affine extensions of multiple conjugation quandles can be described by quadruples of maps.
Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
problem Creating integrable discrete analogs of surface nets and conjugate nets.
method Interpreting classical differential geometry results through Bäcklund transformations and applying permutability properties.
result Integrable discrete analogs of asymptotic and conjugate nets are constructed.
A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair, …
Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
The paper studies conjugating complex representations into real ones.
problem Understanding representations of surface groups into complex Lie groups.
method Analyzes representations of finitely generated groups into PGL(k,C) and determines conjugacy conditions. result Identifies representations in the larger variety that are conjugate in PGL(k,C) to a representation in PGL(k,R). The paper defines Fenchel conjugate and biconjugate on Hadamard manifolds.
problem Defining Fenchel conjugate and biconjugate on curved spaces.
method Introduced a new definition of Fenchel conjugate and biconjugate on Hadamard manifolds based on the tangent bundle.
result Developed a Fenchel-Moreau Theorem for geodesically convex functions on Hadamard manifolds.
The paper introduces a new framework to understand and optimize deep neural networks.
problem Understanding and optimizing the trainability and generalization of deep neural networks.
method Developed a conjugate learning theoretical framework based on convex conjugate duality.
result Demonstrated that training deep neural networks with SGD achieves global optima of empirical risk.
We consider minimal immersions in MxR. We study existence and uniqueness of associate and conjugate isometric immersions to a given minimal surface. We use the theory of univalent harmonic map between surfaces. Then we study the geometry of associate minimal vertical graphs. We prove that an associate surface of a vert…
We show that the quantum field theoretical formulation of the τ-function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that i) the partial charge transformations preserving the neutral sector are Laplace transformations, ii) the basic ver…
Develops quantum character theory for complex reductive groups.
problem Quantum analogue of conjugation equivariant D-modules. method Schur-Weyl functor and double affine Hecke algebra.
result Computes endomorphism algebras of quantum Hotta-Kashiwara modules.
Unified Kantorovich duality for multimarginal optimal transport on Polish spaces.
problem Optimal transport of multiple probability distributions.
method Unified Kantorovich duality theory for multimarginal optimal transport on general Polish product spaces.
result Unified duality theory for multimarginal optimal transport, extending classical two-marginal conjugacy.
The paper tackles efficient computation of optimal transport by approximating conjugates with amortized optimization.
problem Efficient computation of convex conjugates in optimal transport is challenging and limits the quality of transport maps.
method The approach combines amortized approximations of conjugates with a fine-tuning solver to improve transport map quality.
result The method significantly improves the quality of transport maps for the Wasserstein-2 benchmark and models many 2D couplings and flows.
Develops theory of homogeneous statistical manifolds and classifies Lie groups.
problem Understanding statistical manifolds and Lie groups.
method Constructs examples and classifies Lie groups using information geometry.
result Explicit examples of homogeneous statistical manifolds of low dimension constructed.
Conjugate pairs of distributions over infinite dimensional spaces are prominent in statistical learning theory, particularly due to the widespread adoption of Bayesian nonparametric methodologies for a host of models and applications. Much of the existing literature in the learning community focuses on processes posses…
Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which a…
Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.
problem Understanding symmetries and anomalies in 2D Yang-Mills theory.
method Combining continuum methods, topological defects, and higher gauge theory.
result Unified description of higher and lower form gauge fields, identifying spontaneous symmetry breaking.
We address the challenge of effective exploration while maintaining good performance in policy gradient methods. As a solution, we propose diverse exploration (DE) via conjugate policies. DE learns and deploys a set of conjugate policies which can be conveniently generated as a byproduct of conjugate gradient descent. …
NeuralIF uses neural networks to improve preconditioning for faster CG convergence.
problem Improving convergence of conjugate gradient method for large-scale sparse systems.
method Data-driven approach using graph neural networks to generate incomplete factorization.
result Data-driven preconditioners accelerate convergence of conjugate gradient method.
The paper studies p-harmonic functions and their conjugates, showing they converge to calibrations of laminations.
problem Behavior of q-harmonic functions and their conjugates in the limit as qo1. method Analysis of p-harmonic conjugates and their convergence to calibrations of laminations. result The laminations calibrated by the limiting p-harmonic conjugates are exactly those arising from the 1-Laplacian. We consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in S3, which are elements of the braid groups Bg,n. We prove a geometric version of the Markov theorem for braid equivalence in the handlebody, which is based on the L-moves. Using this we then prove tw…
We compute the equivariant K-theory KG∗(G) for a simply connected Lie group G (acting on itself by conjugation). We prove that KG∗(G) is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group G, namely PSU(3),…
A quandle is a set that has a binary operation satisfying three conditions corresponding to the Reidemeister moves. Homology theories of quandles have been developed in a way similar to group homology, and have been applied to knots and knotted surfaces. In this paper, a homology theory is defined that unifies group an…
For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: Our prices will be expressed using Fenche…
This paper tackles global Nash equilibrium in non-convex multi-player games.
problem Challenges in finding global Nash equilibrium due to non-convexity.
method Conjugate transformation and variational inequality formulation to prove existence and design algorithms.
result Designs an ODE-based algorithm with exponential convergence rate and proves its effectiveness in practical scenarios.
Sturm theory applied to symplectic geometry and mechanics.
problem Detecting geometric properties of solutions in symplectic geometry and mechanics.
method Generalization of symplectic Sturm theory to Hamiltonians and application to semi-Riemannian manifolds and singular Lagrangian systems.
result Detection of conjugate and focal points on semi-Riemannian manifolds and geometrical properties of solutions space.
The paper defines subdifferentials on Hadamard manifolds and identifies conditions for Fenchel conjugate equality.
problem Understanding convex analysis on Riemannian manifolds.
method Using Busemann functions to define subdifferentials and investigate Fenchel conjugate equality.
result Identifies conditions for equality in the Fenchel-Young inequality on Hadamard manifolds.
A new method speeds up deep neural network training.
problem Nonconvex optimization in deep neural networks.
method Scaled conjugate gradient method for nonconvex optimization.
result The method converges faster and achieves lower scores in practical applications.
Classifies surfaces supporting alignable nets with geodesic and conjugate properties.
problem Classifying surfaces with specific geometric properties.
method Cartan's theory of moving frames, coordinate-free classification, explicit immersion formulas.
result Two classes of alignable Voss surfaces, each with two two-parameter families, including one with an isothermal-conjugate geodesic net.
Paper provides criteria to detect non-admissible quandles via coloring.
problem Determining non-admissibility of quandles.
method Using colorings of (1, 1)-tangles to detect non-admissibility.
result Constructed numerous non-admissible quandles.
Conjugate gradient methods improve efficiency for high-dimensional GLMMs.
problem Efficiency bottleneck in computing high-dimensional GLMM precision matrices.
method Combining spectral analysis and random graph theory with conjugate gradient methods.
result CG-based methods achieve linear scaling in cost with model parameters and observations.
The natural gradient method has been used effectively in conjugate Gaussian process models, but the non-conjugate case has been largely unexplored. We examine how natural gradients can be used in non-conjugate stochastic settings, together with hyperparameter learning. We conclude that the natural gradient can signific…
The paper provides guarantees for a tangent transform algorithm in logistic regression models.
problem Finding theoretical guarantees for statistical optimality and algorithmic convergence in non-conjugate models.
method Exploiting convex duality and minorizing the marginal likelihood, the paper derives non-asymptotic upper bounds and convergence guarantees for a tangent transform algorithm in logistic regression models.
result The tangent transform algorithm is shown to be locally asymptotically stable without assumptions on the data-generating process.
We construct non-trapping asymptotically hyperbolic manifolds with boundary conjugate points but no interior conjugate points.
Develops multi-modal neural network models for improved prediction and uncertainty quantification.
problem Improving prediction accuracy and uncertainty quantification for multi-modal data.
method Multi-modal Bayesian neural network models with conjugate last-layer estimation using SVI.
result Improved prediction accuracy and uncertainty quantification compared to uni-modal models.
Improved Gaussian process regression with tighter log marginal likelihood bounds.
problem Improving predictive performance in Gaussian process regression models.
method Lower bound on log marginal likelihood using conjugate gradients.
result Improved predictive performance compared to other conjugate gradient based approaches.
A condition for a statistical manifold to have an equiaffine structure is studied. The facts that dual flatness and conjugate symmetry of a statistical manifold are sufficient conditions for a statistical manifold to have an equiaffine structure were obtained in [2] and [3]. In this paper, a fact that a statistical man…
The paper studies quandles of hyperbolic 3-space isometries.
problem Investigating quandles of hyperbolic 3-space isometries.
method Introducing a new quandle Q(Γ,γ) and constructing a canonical map to the conjugate quandle. result The canonical map from Q(Γ,γ) to the conjugate quandle is injective and has a discrete image. Estimates geodesics on surfaces without conjugate points.
problem Counting geodesics on surfaces without conjugate points.
method Margulis-type asymptotic estimates.
result Asymptotic estimates for geodesics on surfaces.
Simplified proof for Tsallis-INF algorithm without conjugate functions.
problem Deriving a best-of-both-worlds guarantee for Tsallis-INF.
method Modern tools from online convex optimization, avoiding conjugate functions.
result A slimmer proof with simplified constants.
New method speeds up inference for non-conjugate Gaussian processes.
problem Inference for non-conjugate Gaussian processes is slow and unreliable.
method Automated augmented conjugate inference method that constructs auxiliary variables to make the model conditionally conjugate.
result Our method is up to two orders of magnitude faster and more robust than existing methods.
We introduce a multiple conjugation biquandle, and show that it is the universal algebra to define a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of n-parallel biquandle operations for any integer n, an…
This paper classifies reversible and strongly reversible elements in affine groups.
problem Classifying reversible and strongly reversible elements in affine groups.
method Identifying affine transformations and using conjugacy by involutions.
result Classification of reversible and strongly reversible elements in affine groups.
Develops a new sampling method for gauge theories.
problem Sampling from SU(N) gauge theories. method Gauge-equivariant flows for SU(N) variables. result Constructs a class of flows respecting matrix conjugation symmetry.
The paper studies Jacobi fields and conjugate points in projective sprays.
problem Investigating Jacobi fields and conjugate points in projective sprays.
method Proved that conjugate points are preserved under projective changes and established conditions for the existence of conjugate points.
result Conditions for the existence of conjugate points in projectively deformed sprays.
We present both, theory and an algorithm for solving time-harmonic wave problems in a general setting. The time-harmonic solutions will be achieved by computing time-periodic solutions of the original wave equations. Thus, an exact controllability technique is proposed to solve the time-dependent wave equations. We dis…
Given a Lorentzian manifold (M,g), a geodesic γ in M and a timelike Jacobi field Y along γ, we introduce a special class of instants along γ that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate insta…
For a surface group, a new bound is given for conjugator length function.
problem Finding an explicit bound for the conjugator length function of a surface group.
method Detailed analysis of conjugation reductions.
result An explicit bound n−1≤CL(2n)≤n+8g−1 for the conjugator length function of a surface group. The study examines Hopfian properties of conjugation quandles and their underlying groups.
problem Understanding the relationship between Hopfian properties of conjugation quandles and their underlying groups.
method Examined Hopfian and residual finiteness properties of conjugation quandles of specific groups.
result Conjugation quandles of Baumslag-Solitar groups are infinitely generated and not necessarily Hopfian.