Revisits and proves a reparametrization theorem for multi-valued graphs in higher codimension.
problem Analyzing multi-valued sections of vector bundles and proving a reparametrization theorem.
method Develops properties of Q-multisections and provides a geometric proof. result Elementary and purely geometric proof of a reparametrization theorem for multi-valued graphs.
Two new proofs of Gromov's non-squeezing theorem using curve reparametrization and gradient bounds.
problem Gromov's non-squeezing theorem in symplectic geometry.
method Reparametrization of pseudo-holomorphic curves and application of mean value inequality or Gromov-Schwarz lemma.
result Uniform bounds on the gradient of pseudo-holomorphic curves leading to compactness of moduli space.
Study on how reparametrization affects neural nets' parameter spaces from a geometric perspective.
problem Inconsistencies in flatness measures, optimization, and probability densities under reparametrization.
method Riemannian geometry to study invariance of neural nets under reparametrization.
result Invariance of neural nets is an inherent property if the metric is explicitly represented and transformation rules are correct.
A new method for 3D surface registration using dynamic programming.
problem Elastic shape registration of 3D surfaces.
method Optimization over a subset of reparametrizations using dynamic programming.
result Proposes an algorithm that produces a solution closer to optimal than gradient-based methods.
DualVDT improves time-series forecasting with a novel dual reparametrized structure.
problem Time-series forecasting with improved performance and analytical rigor.
method Dual reparametrized variational mechanisms on VAE, latent score based generative model, reverse time stochastic differential equation, variational ancestral sampling, KL divergence reduction.
result Advanced performance in time-series forecasting with reduced KL divergence.
New algorithm combines MCMC and variational methods for flexible implicit distributions.
problem Approximate inference for complex continuous models.
method Combines reparametrization, MCMC, and variational methods to construct flexible implicit distributions.
result Easily applicable to arbitrary continuous models without computing log density ratios.
Proves properness of action on map space for complex reparametrization group.
problem Properness of action of $PSL(n+1, {f C})$ on Lkp-maps. method Proof of properness using v-stability and reparametrization group. result Proved properness of action of $PSL(n+1, {f C})$ on Lkp-maps. Improved VI with Price's gradient estimator for target log-density.
problem Approximating target distributions from unnormalized log-densities.
method Stochastic gradient-based variational inference with Price's gradient estimator.
result Identifies Price's gradient as the key to WVI's superior performance.
Proposes spred for solving L1 penalty with SGD.
problem Solving L1 penalty in optimization problems. method Reparametrization and SGD approach.
result Proves spred as an exact differentiable solver of L1. New findings show a balance between data fit and complexity in kernel hyperparameters.
problem Overcorrelation due to reparametrization of kernel hyperparameters.
method Reparametrization of kernel hyperparameters and analysis of marginal likelihood.
result Data fit term influences all other kernel hyperparameters, not just the complexity penalty.
The square root velocity function (SRVF), introduced by Srivastava et al, has proved to be an effective way to compare absolutely continuous curves in RN modulo reparametrization. Several computational papers have been published based on this method. In this paper, we carefully establish the theoretical foundations …
A new method uses signatures to classify shapes efficiently.
problem Classifying shapes succinctly and invariantly.
method Proposes a method using signatures for shape classification.
result Outperforms current methods like SRV transform and dynamic programming.
Monge SAM improves deep learning by making sharpness-aware minimization invariant to reparametrizations.
problem Non-invariance of sharpness-aware minimization (SAM) to reparametrizations.
method Introduces Monge SAM, a reparametrization-invariant version of SAM using a Riemannian metric.
result Monge SAM enhances robustness and generalization compared to previous methods.
We consider pairs of a non-empty compact connected and locally connected Hausdorff space and a real-valued continuous function. Our aim is to measure the difference between this kind of the pairs. In this notes we introduce new pseudodistances between pairs associated with reparametrization invariant seminorms. We fini…
The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.
problem Understanding conjugate points in Lorentzian geometry.
method Introducing and comparing different definitions of conjugate points in synthetic Lorentzian length spaces.
result All defined notions of conjugate points are compatible with the smooth spacetime setting.
Weight normalization and reparametrized gradient descent adaptively regularize weights and converge to minimum l2 norm solutions.
problem Adapting to non-convex weight normalization for convergence to minimum l2 norm solutions.
method Weight normalization and reparametrized projected gradient descent (rPGD) for overparametrized least-squares regression.
result rPGD converges close to the minimum l2 norm solution, even for far-from-zero initializations.
Paper proves trajectories of Chaplygin systems are reparametrized geodesics.
problem Understanding trajectories of Chaplygin systems.
method Constructive proof using modified Riemannian metrics.
result Reparametrized geodesics of Chaplygin systems.
New method for elastic curve and surface matching.
problem Elastic matching of unparametrized curves and surfaces.
method Combines square root normal fields and varifold fidelity metrics.
result Numerical examples demonstrate the approach's effectiveness.
Study controls volume measure for Lagrangian flows in Calabi-Yau manifolds.
problem Controlling volume measure for Lagrangian flows in Calabi-Yau manifolds.
method Optimal control on time-dependent measure of a measurable set under reparametrized Lagrangian mean curvature flow.
result Classification of Lagrangian translating solitons in Cm that evolve by the reparametrized flow. Completeness of surface metrics established for Sobolev spaces.
problem Ensuring completeness of reparametrization-invariant Sobolev metrics on surface spaces.
method Recasting completeness criteria for infinite-dimensional Riemannian manifolds and applying geometric estimates based on the Michael--Simon--Sobolev inequality.
result Established metric and geodesic completeness for specific Sobolev metrics on immersed surfaces, validating Mumford's conjecture.
A new numerical framework simplifies elastic surface matching and comparison.
problem Challenging problem in surface comparison and matching in computer vision.
method Relaxing the geodesic boundary constraint using a varifold fidelity metric.
result Flexibility to deal with arbitrary topologies and sampling patterns, scalability to large meshes.
Efficiently infers latent SDEs with scalable memory and time costs.
problem Inference of latent SDEs with high time and memory complexity.
method Amortized reparametrization of expectations under linear SDEs, coupled with efficient gradient approximation.
result Achieves similar performance to adjoint sensitivities with fewer model evaluations.
We simplify SVI volatility smile constraints for three sub-SVIs without numerical methods.
problem No arbitrage constraints for SVI volatility smiles.
method Explicit domain derivation for sub-SVIs without numerical procedures.
result Explicit no arbitrage domains for Symmetric SVI, Vanishing Upward/Downward SVI, and SSVI.
Motivated by Demailly's strategy towards the Kobayashi hyperbolicity conjecture, we study the action on the k-jets of germs of holomorphic discs in a complex manifold X of the reparametrization group of k-jets of germs of biholomorphisms of the source. This reparametrization group is a subgroup of the general linear gr…
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
A new variational method improves deep neural network inference.
problem Overparametrized deep neural networks struggle with variational approximations.
method A novel variational family with two independent linear subspaces.
result State-of-the-art performance across various tasks and datasets.
We classify compact Kähler surfaces with nonconstant Killing potentials such that all integral curves of their gradients are reparametrized geodesics.
Paper shows surfaces with same SRNF but different shapes.
problem SRNF degeneracy in shape space.
method Introduced Square Root Normal Field (SRNF) to represent and compare surfaces.
result Examples of surfaces with same SRNF but different shapes.
Quantized Variational Inference improves ELBO optimization with fast convergence.
problem Maximizing Evidence Lower Bound (ELBO) for variational inference.
method Optimal Voronoi Tesselation for variance-free gradients, Richardson extrapolation for asymptotic improvement.
result Quantized Variational Inference leads to fast convergence with comparable computational cost.
This study compares two methods for sampling with transport maps, finding flow-based proposals work better for multimodal distributions.
problem Sampling from distributions with complex geometries.
method Compares two approaches: (i) proposal draws from the flow and (ii) reparametrization.
result Flow-based proposals are more effective for multimodal distributions in high dimensions, while reparametrization methods are more robust in other scenarios.
The paper proves Weyl projective rigidity for sub-Riemannian metrics and shows genericity of such metrics.
problem Investigating the Weyl projective rigidity of sub-Riemannian metrics.
method Analytic and smooth category proofs for specific distributions with minimal order complex abnormal extremals.
result Genericity of Weyl projectively rigid sub-Riemannian metrics and distributions.
Generalizes SRVF to curves in homogeneous spaces for efficient distance computation.
problem Distance on curves in homogeneous spaces.
method Generalizes SRVF to homogeneous spaces and proves existence of optimal reparametrizations.
result Efficient computation of quotient distance on curves in homogeneous spaces.
Study on completeness of Sobolev metrics on manifold-valued curves.
problem Completeness of Sobolev metrics on spaces of manifold-valued curves.
method Analysis of reparametrization invariant Sobolev metrics of order n≥2. result Sobolev immersions are metrically and geodesically complete for several important cases of metrics.
We introduce Natural Neural Networks, a novel family of algorithms that speed up convergence by adapting their internal representation during training to improve conditioning of the Fisher matrix. In particular, we show a specific example that employs a simple and efficient reparametrization of the neural network weigh…
The space of all immersed closed curves of rotation degree 0 in the plane modulo reparametrizations has the same homotopy groups as the circle times the 2-sphere.
New solitons found in curve metric space.
problem Elastic metric on curve spaces for shape analysis.
method Reparametrization-invariant Sobolev metric extension, geodesic equation analysis.
result Geodesics are soliton solutions for elastic metric.
Paper derives CLT for Bayesian neural networks trained with variational inference.
problem Analyzing the fluctuation behavior of Bayesian neural networks trained with different variational inference schemes.
method Rigorous derivation of CLT for three variational inference schemes: idealized, Bayes-by-Backprop, and Minimal VI.
result Minimal VI scheme has larger variances but is more computationally efficient.
Research shows deep generative models' likelihoods are unreliable for anomaly detection.
problem Anomaly detection using deep generative models' likelihoods is unreliable.
method Examined the behavior of distribution densities through reparametrization.
result The likelihoods used for anomaly detection rely on strong and implicit hypotheses.
The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.
problem Comparing and analyzing shapes of curves.
method Construction and theoretical properties of quotient elastic metrics, special case of square root velocity metric, numerical approaches for estimation.
result Simplified expression for the square root velocity metric distance.
The paper extends square root velocity framework to curves in homogeneous spaces.
problem Computing metrics and analyzing curves in homogeneous spaces.
method Generalized square root velocity framework to homogeneous spaces, identifying curves with horizontal lifts in G, computing geodesics, and performing quotient operations. result Geodesics and Karcher means can be computed in quotient spaces of curves in homogeneous spaces.
MetFlow combines MCMC and VI efficiently for better inference.
problem Combining MCMC and VI for efficient inference.
method Introduces MetFlow, a novel MCMC algorithm with Normalizing Flows, and a new method to combine it with VI.
result MetFlow produces expressive variational families with improved computational efficiency.
Study disproves conjecture about metric completion of curve spaces.
problem Completeness properties of spaces of immersed curves with reparametrization-invariant metrics.
method Examined Sobolev-type metrics on real-valued immersed curves, demonstrating multiple distinct limit points.
result Metric completion of spaces of immersed open curves includes multiple distinct limit points, not a single point as previously conjectured.
New connections share geodesics with superintegrable systems.
problem Understanding geodesics in affine connections related to superintegrable systems.
method Analyzing dual-geodesics and comparing them across different connections.
result Certain torsion-free affine connections associated with second order superintegrable systems share the same dual-geodesics.
Develops a lifting theory for exponential maps in semi-Riemannian geometry.
problem Overcoming singularities in exponential maps to prove geodesic connectivity.
method Lifting theory for semi-Riemannian manifolds with path-continuation property.
result General path-lifting theorem extending globally under certain conditions.
New method uses Fisher-Rao metric for non-Gaussian decoders.
problem Existing latent space geometry theory only works for Gaussian decoders.
method Pull back Fisher-Rao metric to latent space for non-Gaussian decoders.
result Achieves meaningful latent geometries for various non-Gaussian decoders.
The paper analyzes Bayesian neural networks trained with VI, proving a law of large numbers for different schemes.
problem Training Bayesian neural networks with variational inference.
method Analyzes three training schemes: exact estimation, Bayes by Backprop, and Minimal VI.
result All training schemes converge to the same mean-field limit.
We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extreme…
Let (Ft) be a smooth flow on a smooth manifold M and h:M→M be a smooth orbit preserving map. The following problem is studied: suppose that for every point z of M there exists a germ of a smooth function fz at z such that near z we have that h(x)=Ffz(x)(x). Can the functions (fz) be glued …