Geometric recursion constructs measurable functions on moduli spaces.
problem Constructing measurable functions on moduli spaces of bordered Riemann surfaces.
method Inductive construction via excisions of pairs of pants, with convergence conditions.
result Geometric recursion produces functions that can be integrated with respect to the Weil-Petersson measure.
New invariant csm simplifies computing geometric invariants of recursive group orbits.
problem Computing geometric invariants of recursive group orbits is hard.
method Introduced new invariant csm and used it to compute invariants explicitly. result Explicit formulas for local Euler obstructions and sectional Euler characteristics.
The paper explores generalizations of Mirzakhani's recursion and computes volumes for physical gravity models.
problem Computing volumes for physical gravity models.
method Topological recursion and physical two-dimensional gravity models.
result Derivation of Virasoro constraints and cut-and-join equations for generalized Mirzakhani's recursions.
Solves a recursion for Gromov-Witten invariants of the unknot.
problem Determining Gromov-Witten invariants for a specific Lagrangian brane.
method Uses a skein-theoretic recursion and geometric solutions.
result Solves the recursion to find the expected hook-content formula.
This paper proves that lattice point enumeration in moduli spaces satisfies topological recursion.
problem Enumeration of lattice points in moduli spaces of curves.
method Proves topological recursion for lattice point enumeration in moduli spaces.
result The enumeration satisfies local topological recursion.
We calculate volumes of quadratic differentials using topological recursion.
problem Calculating volumes of quadratic differentials on curves.
method Topological recursion and geometric recursion applied to hyperbolic lengths of multicurves.
result Formula for constant terms of polynomials in terms of stable graphs.
This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.
problem Analyzing the relationship between Seiberg-Witten curves and topological recursion.
method Analytical approach using Seiberg-Witten family of curves.
result A generalized formula relating Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.
The paper analyzes and proposes a new stopping criterion for recursive Bayesian classification.
problem Limitations of conventional stopping criteria in recursive Bayesian classification.
method Geometric interpretation of state posterior progression and analysis of conventional criteria.
result Proposes a new stopping criterion to overcome limitations of conventional methods.
LASER compresses recursive model activations by exploiting their low-dimensional structure.
problem Understanding and optimizing the geometric structure of recursive reasoning trajectories.
method Dynamic low-rank basis tracking via matrix-free subspace tracking with a fidelity-triggered reset mechanism.
result Recursive activations occupy a linear, low-dimensional subspace that can be compressed efficiently.
This paper studies recursive ensembles driven by Fibonacci updates, improving learning dynamics.
problem Improving learning dynamics in recursive ensemble learning.
method Develops second-order recursive architectures with Fibonacci-type update flows.
result Establishes global convergence conditions and generalization bounds for recursive ensembles.
Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed, geometric 3-manifolds is algorithmically recognizable. In our terminology, the …
Recursive training of generative models can lead to model collapse, and the recursion converges to a unique limiting distribution.
problem Model collapse in recursive training of generative models
method Recursive training on their own outputs
result Recursive training converges to a unique limiting distribution
Counting lattice points in moduli space of Klein surfaces.
problem Count lattice points in moduli space of Klein surfaces.
method Introduced metric Möbius graphs, counted lattice points weighted by non-orientability measure, deduced recursion for volumes.
result Proved refined version of Norbury's recursion and computed refined Euler characteristic.
Using geometrical approach exposed in arXiv:math/0304245 and arXiv:nlin/0511012, we explore the Camassa-Holm equation (both in its initial scalar form, and in the form of 2x2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local an…
Using methods of math.DG/0304245 and [I.S.Krasil'shchik and P.H.M.Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000], we accomplish an extensive study of the N=1 supersymmetric Korteweg-de Vries equation. The results include: a description of local and non…
Unified framework for observables in n-plectic geometry.
problem Quantization of extended objects in higher geometric contexts.
method Develops a semi-simplicial set model for observables, using a Grassmann variable to encode submanifold codimensions.
result Establishes a categorified pre-n-Hilbert space and a quantization scheme matching multisymplectic geometry.
This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.
problem Understanding the geometry of combinatorial Teichmüller space.
method Developed a parallel between combinatorial Teichmüller space and Weil-Petersson geometry, using measured foliations and Fenchel-Nielsen coordinates.
result Established a geometric recursion and topological recursion for mapping class group invariants.
Paper develops geometry for Kleinian groups using Farey polynomials.
problem Understanding the geometry of Kleinian groups generated by parabolic elements.
method Sakuma-Weeks triangulations and Farey recursive polynomials.
result Simple recursive algorithm to determine link complement geometry.
This research extends topological recursion to hyperbolic surfaces with tight boundaries and conical defects.
problem Calculating volumes of hyperbolic surfaces with special boundaries.
method Generalized topological recursion to handle tight boundaries and conical defects.
result Weil-Petersson volumes are polynomial in boundary lengths for hyperbolic surfaces with tight boundaries and conical defects.
We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based approach and aim to provide a tutorial to the computations.
Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large derivative books. Recursive Marginal Quantization of the Euler scheme has recen…
This paper develops efficient algorithms for multibody dynamics using screw and Lie group theory.
problem Efficient modeling and computation of multibody systems.
method Recursive algorithms and Lie group formulations for multibody dynamics.
result Derivation of efficient Newton-Euler and Lagrange equations for multibody systems.
Classifies homomorphisms between specific braid groups.
problem Classifying homomorphisms between braid groups.
method Complete classification through recursive approach.
result Recursive classification of homomorphisms between braid groups.
This work studies the contraction coefficients of Schrödinger bridge problems in linear systems.
problem Optimally controlling the evolution of a system's state density over time.
method Analyzes and improves the convergence rates of dynamic Schrödinger systems via geometric and control-theoretic interpretations.
result New insights into improving computation of worst-case contraction coefficients by preconditioning.
This paper extends geometric study of neural networks to non-differentiable layers and random walks.
problem Understanding the geometric properties of neural networks, especially those with non-differentiable activation functions.
method Singular Riemannian geometry approach to convolutional, residual, and recursive neural networks.
result Illustrated geometric findings with numerical experiments on image classification and thermodynamic problems.
Adds recursion to deep learning frameworks for better handling of recursive data structures.
problem Lack of support for recursion in existing deep learning frameworks.
method Complements existing frameworks with recursive execution of dataflow graphs and APIs for recursive definitions.
result Recursive implementation reduces training and inference time by more effectively using resources.
Generates infinite-depth hierarchical clusters from few examples.
problem Inadequate finite-sample clustering methods for fine-scale hierarchical structures.
method Classification fields generated by a local refinement rule, approximated by predictors.
result Learned predictors can approximate infinite-depth hierarchical structures.
Geometric framework explains deep learning performance.
problem Understanding why deep learning works well across various tasks.
method Comparing deep learning to quantum computations and diffeomorphic template matching.
result Geometric structures of different deep learning systems.
Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.
problem Understanding HOMFLYPT polynomials and their geometric origins.
method Defining worldsheet skein module and D-module, considering skein valued open curve counts.
result Worldsheet skein D-module for Hopf link conormal is generated by three operator polynomials.
We use categorical skew Howe duality to find recursion rules that compute categorified sl(N) invariants of rational tangles colored by exterior powers of the standard representation. Further, we offer a geometric interpretation of these rules which suggests a connection to Floer theory. Along the way we make progress t…
Paper defines Farey Recursive Functions and explores their properties.
problem Understanding recursive functions on rationals.
method Defined and studied Farey Recursive Functions using Farey graph.
result Farey Recursive Functions naturally connect to 2-bridge knots and links.
This paper contains the technical foundations from stochastic differential geometry for the construction of geometrically intrinsic nonlinear recursive filters. A diffusion X on a manifold N is run for a time interval T, with a random initial condition. There is a single observation consisting of a nonlinear function o…
This paper introduces online algorithms to estimate robust geometric median in large data streams.
problem Detecting outliers in large data sets using robust statistical measures.
method Online stochastic Newton methods for estimating the geometric median.
result Rates of convergence for online estimation of the geometric median.
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof…
Paper proves 1-point recursions for various enumerative problems.
problem Enumerating gluings of polygons to orientable surfaces.
method Proves existence of 1-point recursions for specific classes of problems.
result Recover Harer-Zagier recursion and prove existence for new problems.
Tab-TRM uses recursive model for insurance pricing on tabular data.
problem Insurance pricing on tabular data.
method Adapts recursive latent reasoning to insurance modeling using a compact, parameter-efficient network.
result Improves insurance pricing accuracy using iterative refinement of latent tokens.
New recursion formula for non-orientable surfaces resolves divergences.
problem Computing volumes of moduli spaces for non-orientable surfaces.
method Generalization of Mirzakhani's recursion to non-orientable surfaces, handling divergences with integral kernels.
result Regularized volumes can be computed with a cutoff on crosscap size.
Estimates returns for dollar cost averaging using geometric Brownian motion.
problem Estimating returns for dollar cost averaging investing strategy.
method Uses geometric Brownian motion and log-Normal distribution to construct a lower bound for returns. Computes parameters recursively and in closed form for dollar cost averaging. Compares to lump sum investing for matching wealth distributions.
result Probability of negative returns is less than 2.5% for 40 years of annual dollar cost averaging.
The article improves prediction by aggregating Kalman recursions online.
problem Improving expert aggregation in prediction models.
method Using exponential weights and state-space models to aggregate Kalman recursions.
result New algorithms outperform existing methods in Kalman recursion expert aggregation.
New recursion found for hyperbolic sphere volumes.
problem Volume calculation of hyperbolic sphere moduli spaces.
method Proved a non-linear recursive relation.
result Generalized Zograf's result for conical points and geodesic boundaries.
Character varieties of 2-bridge knots and links explained using Farey recursion.
problem Understanding character varieties of 2-bridge knots and links.
method Using Farey recursion to define polynomials for character varieties.
result Character varieties described in terms of polynomials defined by Farey recursion.
Bijective proof of map enumeration recursion formulae.
problem Counting maps of arbitrary topology.
method Iterating Tutte's algorithm and pair-of-pants decomposition.
result Combinatorial meaning for all terms of topological recursion.
We describe a method for recursively calculating Gromov-Witten invariants of all blowups of the projective plane. This recursive formula is different from the recursive formulas due to Göttsche and Pandharipande in the zero genus case, and Caporaso and Harris in the case of no blowups. We use tropical curves and a recu…
New iteration method for complex and real Monge-Ampere equations converges under certain conditions.
problem Proving convergence of Monge-Ampere iterations for complex and real equations.
method Introduced Monge-Ampere iteration for real equations, established convergence conditions, and provided geometric applications.
result Established sufficient conditions for convergence of Monge-Ampere iteration and provided geometric applications.
Topological recursion recovers a specific partition function for colored knots.
problem Recovering the extended Ooguri-Vafa partition function for colored HOMFLY-PT polynomials of torus knots.
method Applying topological recursion to the spectral curve of colored HOMFLY-PT polynomials of torus knots.
result Topological recursion reproduces the n-point functions of the extended Ooguri-Vafa partition function.
Framework learns asymmetric and local features in multi-dimensional data.
problem Learning features in multi-dimensional data, especially images.
method Bayesian hierarchical modeling with recursive wavelet transforms.
result Framework achieves high computational scalability and adaptivity.
This paper concerns the recursive utility maximization problem under partial information. We first transform our problem under partial information into the one under full information. When the generator of the recursive utility is concave, we adopt the variational formulation of the recursive utility which leads to a s…
We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brezín and Hikami. We express this recursion in special times in which all terms W1(g) of the genus expansion of the one-loop mean are polynomials. We find a generalization of th…