In this paper we study a model of random knots obtained by fixing a space curve in -dimensional Euclidean space with , and orthogonally projecting the space curve on to random dimensional subspaces. By varying the space curve we obtain different models of random parametrized knots, and we will study how the…
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168,742 papers · 148 categories
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48 results for “random curves”
Research examines the distribution of curve components in random multicurves.
problem Distribution of curve components in random multicurves.
method Action of the mapping class group on random multicurves.
result Distribution of curve components analyzed.
Random simple closed curves map Teichmüller space to geodesic currents.
problem Mapping Teichmüller space to geodesic currents.
method Using a formula for intersection numbers of multicurves and Dehn coordinates.
result Proper embedding of Teichmüller space into the space of geodesic currents.
New method calculates geodesic distances in Gaussian random field manifolds.
problem Quantifying similarity between random fields in different regimes.
method Numerical method using geodesic distances in Gaussian random field manifolds.
result Estimation of geodesic distances for various initial conditions.
Fold maps associated to geodesic random walks on curved spaces.
problem Understanding the behavior of geodesic random walks on curved surfaces.
method Analyzing mappings from the unit tangent sphere to a manifold with non-positive curvature.
result For odd powers of the unit tangent sphere, these mappings are fold maps.
Probabilistic model for exhaustion in infinite-genus curve complexes.
problem Action rigidity in infinite-genus curve complexes.
method Costa and Farber's model for random simplicial complexes.
result Probabilistic evidence for exhaustion via rigid expansions.
Random walks on mapping class groups identified with geodesic laminations.
problem Understanding random walks on mapping class groups.
method Electrification of curve graph, identifying Poisson boundary, using geodesic laminations.
result Random walk on mapping class group identified with geodesic laminations.
Random Heegaard splittingsmath.GT
A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic an…
The random graph is an infinite graph with the universal property that any embedding of extends to an embedding of , for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface if and only if has infinite genus, showing that the curve system on an infinite genus s…
Random curves on surfaces have predictable properties as they grow.
problem Characterizing topological properties of random curves on surfaces.
method Analyzing a simple random walk on the Cayley graph of surface groups.
result The properties of random curves are generic and predictable as they grow.
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
Simplified proof for dimension reduction of polygonal curves.
problem Preserving the continuous Fréchet distance of polygonal curves.
method Sparse oblivious subspace embeddings for generalized dissimilarity measures.
result Generalized dimension reduction technique works for various distance measures.
Survey on random walks on mapping class groups and their properties.
problem Understanding random walks on mapping class groups.
method Analyzing actions on Teichmüller spaces and curve complexes.
result Laws of large numbers and central limit theorems for random walks.
Study shows double descent curve in high-dimensional linear regression with random projections.
problem Understanding the generalization performance in high-dimensional settings with random projections.
method Fixed prediction problem, ridge regression estimator, minimum norm least-squares fit, random matrix theory, asymptotic equivalents.
result Exhibit a double descent curve for high-dimensional linear regression with random projections.
Unified analysis of generalization curves in large models using gradient flow.
problem Analyzing generalization error curves in simple learning models.
method Gradient flow in the Gaussian covariate model, using random matrix theory.
result Unified understanding of multiple descent structures in learning curves.
We analyze learning curves of RF models with convex regularization and derive precise asymptotic expressions.
problem Understanding the learning curves of RF models with general convex regularization.
method Novel multi-level application of the convex Gaussian min max theorem (CGMT) to compute precise asymptotic expressions.
result Precise asymptotic expressions for learning curves of RF models with separable strongly convex regularization or regularization.
We consider learning on graphs, guided by kernels that encode similarity between vertices. Our focus is on random walk kernels, the analogues of squared exponential kernels in Euclidean spaces. We show that on large, locally treelike, graphs these have some counter-intuitive properties, specifically in the limit of lar…
Covariance as a commutatormath.ST
Covariance is shown as a commutator in random variable calculus.
problem Expressing covariance as a commutator of operators.
method Demonstrated through commutator identities involving expectations and products of functions.
result Revealed the underlying Lie algebraic structure in efficient influence curve calculus.
We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on . In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time of the random walk to the Lyapunov exponent of …
Enhances Random Forest for imbalanced functional data classification.
problem Challenges in classifying imbalanced functional data.
method Functional Random Forest with Adaptive Cost-Sensitive Splitting (FRF-ACS).
result Significantly improves minority class recall and predictive performance.
Random quotients of mapping class groups have rigid properties.
problem Rigidity of random quotients of mapping class groups.
method Generalization of Ivanov's theorem and use of hierarchically hyperbolic groups.
result Automorphisms and commensurators of random quotients coincide with the groups themselves.
The presence of slipknots in configurations of proteins and DNA has been shown to affect their functionality, or alter it entirely. Historically, polymers are modeled as polygonal chains in space. As an alternative to space curves, we provide a framework for working with subknots inside of knot diagrams via knotoid dia…
We show that a random walk on the mapping class group of an orientable surface of finite type makes linear progress in the relative metric, which is quasi-isometric to the complex of curves.
Study rolling dynamics with random slipping and twisting using large deviation principles.
problem Analyzing the stability of a rolling model with random slipping and twisting.
method Modelled as a stochastic differential equation on the orthonormal frame bundle, examined via large deviations.
result Proved large deviation principles for projection curves and their horizontal lifts on the base manifold.
Reconstructing signature features from randomized vector fields in differential equations.
problem Reconstructing signature features from controlled differential equations with random vector fields.
method Using controlled ordinary differential equations driven by continuous bounded variation curves, the study explores the extent to which signature features can be reconstructed from the non-linear flow of these equations.
result The number of signature features that can be reconstructed from the non-linear flow of controlled ordinary differential equations with random vector fields is exponential in the hidden dimension, under certain conditions.
We prove a sharp estimate on the expected value of the integral of the index of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.
ReLU networks don't exponentially distort curve lengths as previously thought.
problem Understanding how neural networks distort curve lengths with depth.
method Analyzing expected length distortion of ReLU networks with random initialization.
result Expected length distortion does not grow with depth, and shrinks slightly.
Tree ensembles, such as random forests and AdaBoost, are ubiquitous machine learning models known for achieving strong predictive performance across a wide variety of domains. However, this strong performance comes at the cost of interpretability (i.e. users are unable to understand the relationships a trained random f…
We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.
Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…
It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting …
The paper studies multiple descent in multi-component prediction models.
problem Understanding the risk curves in multi-component prediction models.
method Investigates a 'double random feature model' and 'multiple random feature model' in ridge regression.
result Risk curves of multi-component prediction models can exhibit multiple descents.
We show that the probability that a finitely supported random walk on a non-elementary subgroup of the the mapping class group gives a non-pseudo-Anosov element decays exponentially in the length of the random walk. More generally, we show that if R is a set of mapping class group elements with an upper bound on their …
Probabilistic models predict neural network performance across varying hyperparameters.
problem Predicting neural network performance with different hyperparameters.
method Probabilistic models based on random forests and Bayesian recurrent neural networks.
result Models outperform state-of-the-art hyperparameter optimization methods.
Kernel ridgeless regression with random features shows good generalization without explicit regularization.
problem Generalization of kernel ridgeless regression without explicit regularization.
method Investigation of ridgeless regression with random features and stochastic gradient descent, exploring the effect of random features error and spectral density optimization.
result Random features error exhibits the double-descent curve, leading to improved generalization.
Develops ML tool for macroeconomic forecasting with clear interpretations.
problem Forecasting and understanding macroeconomic parameters over time.
method Macroeconomic Random Forest (MRF) algorithm, Generalized Time-Varying Parameters (GTVPs).
result Clear forecasting gains and accurate predictions of unemployment and inflation.
A good classification method should yield more accurate results than simple heuristics. But there are classification problems, especially high-dimensional ones like the ones based on image/video data, for which simple heuristics can work quite accurately; the structure of the data in such problems is easy to uncover wi…
The paper predicts responses on out-of-sample nodes using latent positions on unknown curves.
problem Predicting responses on out-of-sample nodes with latent positions on unknown curves.
method Manifold learning and graph embedding technique using latent positions.
result Convergence guarantees for predicting responses on out-of-sample nodes.
Study benchmarks classical models over quantum in DeFi yield prediction.
problem Accurate yield and performance forecasting for DeFi liquidity allocation.
method Benchmarked six models on Curve Finance pools' historical data.
result Classical models, especially XGBoost, outperform quantum models.
Characterizes RFF regression in large setting, providing precise learning phases and double descent curve.
problem Characterizes RFF regression in large setting.
method Characterizes the exact asymptotics of random Fourier feature (RFF) regression in the realistic setting of large .
result Characterizes two qualitatively different phases of learning and the corresponding double descent test error curve.
Developed a random walk analog of geodesic flow on hyperbolic groups.
problem Geodesic flow on hyperbolic groups due to non-uniqueness of geodesics.
method Introduced a new framework using random walks and bi-infinite trajectories.
result Established ergodicity of the randomized geodesic flow and exponential mixing.
Defines Vassiliev complexity measures for open and closed curves in 3D space.
problem Measuring complexity of curves in 3D space.
method Using enhanced Jones polynomial coefficients and Gauss code diagrams.
result Second Vassiliev measure converges to knot invariants as curve ends coincide.
Random forest model predicts sewer pipe deterioration with high accuracy.
problem Challenges in predicting and scheduling sewer pipe inspections.
method Random forest classification model for sewer pipe condition prediction.
result Model achieved excellent AUC of 0.81 in a case study for City of LA.
We study the -median clustering problem for high-dimensional polygonal curves with finite but unbounded number of vertices. We tackle the computational issue that arises from the high number of dimensions by defining a Johnson-Lindenstrauss projection for polygonal curves. We analyze the resulting error in terms of …
Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order with …
Gradient span algorithms show consistent progress in high dimensions.
problem Understanding consistent training progress in large machine learning models.
method Proving deterministic behavior of gradient span algorithms on Gaussian random functions.
result Gradient span algorithms have asymptotically deterministic behavior in high dimensions.
Bayesian nonparametric method partitions shapes using curves.
problem Capturing complex shapes in multi-dimensional data.
method Proposes a novel spline partitioning approach using curves.
result Demonstrates improved shape modeling compared to existing methods.
Statistical physics approaches can be used to derive accurate predictions for the performance of inference methods learning from potentially noisy data, as quantified by the learning curve defined as the average error versus number of training examples. We analyse a challenging problem in the area of non-parametric inf…