A novel GPUM constructs Gaussian Processes for unknown manifolds with probabilistic metrics.
problem High-dimensional data on unknown manifolds with non-Euclidean geometry.
method Bayesian Gaussian Processes latent variable models (BGPLVM), Riemannian geometry, probabilistic metric tensor, Brownian Motion.
result GPUM provides more accurate predictions on unknown manifolds compared to traditional methods.
The paper projects unknown manifolds onto hyperspheres for efficient function approximation.
problem Function approximation from data on unknown manifolds with added errors.
method Projects unknown manifold onto hypersphere and uses localized spherical polynomial kernels.
result Optimal rates of approximation for rough functions are given.
Simple geodesic kNN achieves optimal regression on unknown manifolds.
problem Semi-supervised regression on unknown manifolds.
method Estimate manifold geodesic distances and apply k nearest neighbor regression.
result Geodesic kNN achieves optimal mean squared error bound.
Researchers reconstruct simple Riemannian manifolds from boundary wave arrival times.
problem Reconstructing Riemannian manifolds from unknown interior sources and arrival times.
method Discrete metric approximation using labeled Gromov--Hausdorff distance.
result Finite-time approximations converge to the true Riemannian manifold.
Directly approximates functions on unknown data manifolds without complex computations.
problem Function approximation on unknown data-defined manifolds with conservative results from traditional methods.
method Direct approach using graph Laplacian and local approximation techniques without eigen-decomposition or atlas.
result Universal estimates for smooth functions without prior knowledge of the target function.
ILDM combines diffusion and latent learning for generative modeling on unknown manifolds.
problem Diffusion models struggle with high-dimensional data and lack geometric structure.
method ILDM integrates probabilistic dimensionality reduction with geometry-aware diffusion on unknown manifolds.
result ILDM significantly improves generation quality compared to standard models.
Kernel smoothing on unknown manifolds with bounds and asymptotic normality.
problem Data on unknown manifolds without boundaries.
method Finite sample bounds and asymptotic normality for kernel smoothing and its derivatives.
result Established finite sample bounds and asymptotic normality for kernel smoothing.
Paper introduces method to estimate animal motion on unknown submanifolds using Koopman operator.
problem Estimating animal motion on unknown submanifolds in high-dimensional space.
method Data-dependent approximation of Koopman operator in RKHS over ambient space.
result Strong rates of convergence derived for estimates in terms of fill distance.
Estimates functions on unknown manifolds using multiscale regression.
problem Regression on unknown low-dimensional manifolds embedded in high-dimensional spaces.
method Low-dimensional coordinates at multiple scales, local polynomial fitting, data-driven wavelet thresholding.
result Optimal learning rates for estimating functions with nonuniform regularity.
MAGI-X learns unknown dynamics from data without numerical integration.
problem Difficult to propose ODEs in closed-form for complex systems.
method MAGI-X uses neural networks within a manifold-constrained Gaussian process framework.
result MAGI-X achieves competitive accuracy in fitting and forecasting with reduced computational time.
Guaranteed reachable set for unknown nonlinear systems on manifolds.
problem Determining reachable set for unknown nonlinear systems on manifolds.
method Underapproximations of reachable set using local dynamics and bounds on dynamics rate of change.
result Guaranteed set of reachable states for systems on complete Riemannian manifolds.
RGI improves robustness of GAN-inversion for image restoration and anomaly detection.
problem Robustness of GAN-inversion to unknown gross corruptions.
method Proposes RGI and R-RGI methods with provable robustness guarantees.
result Restored images and corrupted region masks converge to ground truth under mild assumptions.
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.
New method estimates geodesic distances using spherelets.
problem Accurately estimating geodesic distances on unknown manifolds.
method Uses spherelets to locally approximate unknown subspaces and estimate geodesic distances.
result Lower error for many manifolds, validated through simulations and real data.
We prove the existence of Sasaki-Einstein metrics on certain simply connected 5-manifolds where until now existence was unknown. All of these manifolds have non-trivial torsion classes. On several of these we show that there are a countable infinity of deformation classes of Sasaki-Einstein structures.
Normalizing flows can now estimate densities on unknown manifolds.
problem Normalizing flows struggle with data on unknown low-dimensional manifolds.
method Conformal Embedding Flows, which combine standard flows with trainable conformal embeddings.
result Tractable density estimation on manifold-supported data is possible.
Measuring wave sources uniquely identifies manifold properties.
problem Determining Riemannian manifold structure from wave observations.
method Semilinear wave equation measurements at a single point.
result Topological, differential, and geometric structure can be inferred.
New contact manifolds with many fillings found.
problem Contact manifolds with infinite fillings in odd dimensions.
method Spinal open books to construct contact manifolds.
result Contact manifolds with infinitely many different Weinstein fillings constructed.
DBSCAN estimates density level sets on manifolds with i.i.d. samples.
problem Estimating connected components of density level sets on manifolds.
method DBSCAN algorithm applied to i.i.d. samples.
result Rates of estimation error for different data settings.
Small Nijenhuis tensor found on compact manifolds.
problem Finding compact manifolds with small Nijenhuis tensor.
method Provided explicit examples of manifolds with small Nijenhuis tensor.
result Examples of manifolds with small Nijenhuis tensor in various dimensions.
New method uses manifold learning to infer latent positions of 1D submanifolds in random dot product graphs.
problem Inference on latent positions of unknown 1D submanifolds in RDPGs.
method Apply Isomap for manifold learning to estimate arc lengths on the unknown submanifold.
result Test statistics based on Isomap converge to known submanifold power as auxiliary vertices increase.
We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as…
In its most general form, the recognition problem in Riemannian geometry asks for the identification of an unknown Riemannian manifold via measurements of metric invariants on the manifold. We introduce a new infinite sequence of invariants, the first term of which is the usual diameter, and illustrate the role of thes…
This paper addresses the problem of identifying a lower dimensional space where observed data can be sparsely represented. This under-complete dictionary learning task can be formulated as a blind separation problem of sparse sources linearly mixed with an unknown orthogonal mixing matrix. This issue is formulated in a…
Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped virtually fibred non-fibred census 3-manifolds, incl…
Paper solves complex signal processing problem efficiently.
problem Learning an unknown filter from multiple sparse convolutions.
method Nonconvex optimization over the sphere manifold using manifold gradient descent.
result Manifold gradient descent provably recovers the filter under random data model.
RVGP learns vector fields over unknown manifolds, preserving singularities.
problem Learning vector fields over unknown non-Euclidean manifolds.
method RVGP uses positional encoding with eigenfunctions of the connection Laplacian.
result RVGP preserves singularities in vector fields over unknown manifolds.
We survey known (and unknown) results about the behavior of Heegaard genus of 3-manifolds constructed via various gluings. The constructions we consider are (1) gluing together two 3-manifolds with incompressible boundary, (2) gluing together the boundary components of surface times I, and (3) gluing a handlebody to th…
AI agent learns to handle unknown unknown states in reinforcement learning.
problem Handling unexpected, previously unseen states in reinforcement learning.
method Proposes EMDP-GA model with NIVE approach to expand value functions.
result Asymptotically consistent regret and comparable computational complexity.
This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely man…
By considering non-orientable surfaces in the surgered manifolds, we show that the 10/3- and -10/3-Dehn surgeries on the 2-bridge knot 927=S(49,19) are not cosmetic, i.e., they give mutually non-homeomorphic manifolds. The knot is unknown to have no cosmetic surgeries by previously known results; in particular, …
We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold M such that the fundamental group π1(M) coarsely co…
New method tackles unknown unknowns in machine learning.
problem Unknown classes in training data misperceived as other labels.
method Exploratory machine learning with rejection model, feature exploration, and model cascade.
result The method discovers potentially hidden classes and improves model performance.
Researchers found the minimum volume of a 3-cusped hyperbolic 3-manifold.
problem Finding the minimum volume of a 3-cusped orientable hyperbolic 3-manifold.
method Using guts in sutured and pared manifolds.
result The volume of a 3-cusped orientable hyperbolic 3-manifold is at least 5.49... = 6 × Catalan's constant.
New method improves Gaussian process regression on complex, sparse point clouds.
problem Traditional Gaussian processes struggle with restricted domains and point clouds.
method Atlas Gaussian Processes (RC-AGPs) combining heat kernel and RBF kernels.
result RC-AGPs outperform existing methods in regression accuracy.
Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows and general relativity. Recent work by the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existen…
Constructs odd Euler characteristic 4-manifolds.
problem Finding 4-manifolds with odd Euler characteristics.
method Explicit construction of aspherical 4-manifolds with odd Euler characteristics.
result Explicit examples of aspherical 4-manifolds with odd Euler characteristics greater than 12.
New algorithms estimate unknown training examples to improve ML model generalization.
problem Distribution shift between training and testing data leads to poor generalization.
method Combining species-estimation techniques with data-driven methods.
result Correcting the training set with unknown examples improves model robustness.
Paper reveals a surprising connection between curl and Dirac operators on 3-manifolds.
problem Understanding the spectrum and properties of the curl operator on 3-manifolds.
method Relating the curl operator to the Dirac operator on spinc structures and proving ellipticity. result Eigenvalues of curl are always lower bounded by those of the Dirac operator, with equality on the round 3-sphere.
A graph manifold rational homology 3-sphere W with a left-orderable fundamental group admits a co-oriented taut foliation, though it is unknown whether it admits a smooth co-oriented taut foliation. In this paper we extend the gluing theorem of arXiv:1401.7726 to graph manifold rational homology solid tori and use …
New method for predicting neuron activity with unknown stimuli.
problem Statistical inference of neuron activity with missing data and unknown sources.
method Maximum likelihood estimation with fixed-point iteration.
result Model increases system likelihood and reveals neural connections.
The paper shows how to create manifolds with no geodesics converging to a sphere.
problem Creating manifolds with specific geometric properties.
method Constructing a sequence of Riemannian manifolds converging to a unit sphere in intrinsic flat sense.
result The resulting limit space has no geodesics achieving distances between points.
Method locates equilibria on unknown Riemannian manifolds using iterative sampling and parallel transport.
problem Locating equilibria on unknown Riemannian manifolds defined by point-clouds.
method Iterative sampling, parallel transport, and generalized isoclines.
result Algorithm reliably locates equilibria of dynamical systems on unknown manifolds.
New method models unknown systems with hidden parameters using neural networks.
problem Modeling unknown dynamical systems with hidden parameters.
method Training a deep neural network (DNN) model using trajectory data of the unknown system.
result DNN model accurately predicts unknown dynamical systems with new initial conditions.
We consider some infinitesmal and global deformations of G_2 structures on 7-manifolds. We discover a canonical way to deform a G_2 structure by a vector field in which the associated metric gets "twisted" in some way by the vector cross product. We present a system of partial differential equations for an unknown vect…
The paper uses graph Laplacians and maximum principles to study learning problems on unknown manifolds.
problem Learning problems on unknown manifolds with noise.
method Maximum principle arguments and techniques from partial differential equations and the Calculus of variations.
result Asymptotic consistency guarantees for noise-corrupted, non-parametric regression.
A method for learning distributions on complex manifolds using normalizing flows.
problem Learning distributions on non-Euclidean manifolds with high efficiency and accuracy.
method Learning a distribution on a manifold by combining local models that form an open cover.
result The method achieves better sample efficiency and competitive performance on manifolds of unknown topology.
We use surgery along 2-tori embedded in a union of two copies of a product of punctured 2-tori to produce a new collection of homotopy 4-spheres (4-manifolds homotopy equivalent to S4 and hence homeomorphic to S4 but possibly not diffeomorphic to S4). It is still unknown if these new examples are in fact exoti…