Study normal bundle and deformation to get new pushforward maps.
problem Construct pushforward maps in various homology theories.
method Use deformation Lie groupoids to construct pushforward maps.
result Functoriality of pushforward maps recovers and generalizes previous cases.
This study improves GANs by learning latent distributions and pushforward maps.
problem Improving the performance of GANs with optimal transport metrics.
method Focuses on the interplay between latent distribution and generator complexity.
result Learning latent distributions and pushforward maps can significantly reduce sample complexity.
This paper learns prior models from indirect data efficiently.
problem Learning prior models from indirect data in Bayesian inversion.
method Generative model of prior as pushforward of Gaussian in latent space, learned by minimizing loss function.
result Efficient residual-based neural operator approximation for forward model learning.
Following Hopkins and Singer, we give a definition for the differential equivariant K-theory of a smooth manifold acted upon by a finite group. The ring structure for differential equivariant K-theory is developed explicitly. We also construct a pushforward map which parallels the topological pushforward in equivariant…
Study X-ray transform on manifolds, desingularize, and improve mapping properties.
problem Characterize mapping properties of X-ray transform and its adjoint on manifolds with strictly convex boundary.
method Use b-fibrations, desingularize, and apply Melrose's Pushforward Theorem to analyze polyhomogeneous functions.
result Improved mapping properties of X-ray transform and its adjoint, recovering sharp results.
Paper introduces a novel map learning algorithm for domain translation and adaptation.
problem Learning a map between related data spaces that can be applied to out-of-sample data and satisfies application-specific constraints.
method Utilizes normalizing flows to parameterize a map that minimizes a probability distance and application-specific regularizers, solving a modified optimal transport problem.
result The proposed method (parOT) outperforms existing optimal transport approaches in domain adaptation and translation tasks.
New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
Transforms classical connections using pushforwards and gauge transformations.
problem Transforming classical connections in categorical settings.
method Constructing pushforwards and applying gauge transformations to decorated path spaces.
result Combines traditional gauge transformation with affine translation.
New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.
problem Unique recovery of transport maps and vector fields from finite measure-valued data.
method Use of Whitney and Takens embedding theorems to establish conditions for unique identification.
result New metric for comparing diffeomorphisms and analogous results in infinitesimal settings.
For L↪X a Lagrangian embedding associated with a real homogeneous space, we construct the moduli space of stable holomorphic discs mapping to (X,L) as an orbifold with corners equipped with a group action. Some essential constructions involving orbifolds with corners are also discussed, including th…
New operations defined on moduli spaces for bundles with orientations.
problem Pushforward operations for principal bundles with orientations.
method Developed a general theory of pushforward operations for principal G-bundles, constructing specific operations for G=BU(1). result Classified all stable pushforward operations and showed they are generated by the projective Euler and rank operations.
The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a …
The paper examines convergence of currents and forms under smooth diffeomorphisms.
problem Analyzing convergence of currents and forms under C0-limits of diffeomorphisms. method Geometric analysis, measure theory, homotopy theory.
result Pushforwards of rectifiable currents converge in the flat norm.
We define disentanglement in generative models and prove it's related to identifiable factors.
problem Understanding disentanglement in generative models like VAEs and GANs.
method Characterized disentanglement in smooth generative pushforward models using the SVD of the Jacobian.
result Disentanglement is identifiable under certain conditions on the generator, promoting separable factors.
Generalized Stacey-Roberts lemma for Banach manifolds.
problem Constructing Lie groupoids of smooth mappings in infinite-dimensional geometry.
method Generalization of the Stacey-Roberts lemma to Banach manifolds with smooth partitions of unity.
result Remedied an error in the original proof for finite-dimensional setting.
We propose a general framework to learn deep generative models via \textbf{V}ariational \textbf{Gr}adient Fl\textbf{ow} (VGrow) on probability spaces. The evolving distribution that asymptotically converges to the target distribution is governed by a vector field, which is the negative gradient of the first variation o…
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
Paper develops methods for analyzing forms with synchronized singularities.
problem Analyzing forms with synchronized singularities.
method Exact reduction, analytic transfer, and geometric recomposition.
result Transfer of sparse domination principle to synchronized singular forms.
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
This work proposes a new neural implicit manifold model for more accurate density estimation on manifolds.
problem Current generative models struggle with representing manifolds accurately and learning densities within them.
method Proposes a neural implicit manifold model and a constrained energy-based model to learn manifold-supported distributions.
result The proposed model can learn manifold-supported distributions with complex topologies more accurately than pushforward models.
In this article we investigate a monoid of smooth mappings on the space of arrows of a Lie groupoid and its group of units. The group of units turns out to be an infinite-dimensional Lie group which is regular in the sense of Milnor. Furthermore, this group is closely connected to the group of bisections of the Lie gro…
Estimates conditional Brenier maps using entropic optimal transport.
problem Non-parametric estimation of conditional Brenier maps.
method Entropic optimal transport for scalable non-parametric estimation.
result Entropic optimal transport maps asymptotically converge to conditional Brenier maps.
Novel construction of Bauer--Furuta invariant using sheaves of spectra.
problem Constructing the Bauer--Furuta invariant without finite-dimensional approximations.
method Using sheaves of spectra and Borel--Moore homology, avoiding approximations.
result Defines the shriek functors and Thom spectra for index calculations.
Paper organizes sampling methods for generative modeling.
problem Challenges in sampling with diffusion models.
method Reviews and organizes existing sampling methods.
result Reveals links between methods to overcome challenges.
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant PU(H)−principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of …
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
Efficiently computes optimal transport maps and Wasserstein barycenters using conditional normalizing flows.
problem Computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces.
method Uses conditional normalizing flows to approximate distributions and solve the primal problem.
result Shows computational feasibility for hundreds of input distributions and yields accurate results.
The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.
problem Analyzing the behavior of harmonic map heat flow to moduli space of flat tori.
method Investigates stability and ergodic behavior of harmonic map heat flow using hyperbolic structure and relative entropy.
result The flow converges weak--∗ to the normalized hyperbolic measure on the moduli space. For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. Fo…
The paper develops a theory for one-step Wasserstein-guided models for PDE-induced measures.
problem Theoretical understanding of generative models' accuracy in scientific computing.
method Regularity theory for optimal transport between doubling measures, excess-risk bounds.
result One-step Wasserstein-guided generative models can approximate PDE-induced measures with Hölder continuity.
We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This construct…
We prove an index theorem concerning the pushforward of flat B-vector bundles, where B is an appropriate algebra. We construct the associated analytic torsion form T. If Z is a smooth closed aspherical manifold, we show that T gives invariants of the homotopy groups of Diff(Z).
Generative models help make decisions under changing data distributions.
problem Making decisions based on historical data when the actual data distribution changes.
method Flow- and score-based generative models to represent and transform distributions.
result Generative models can learn nominal uncertainty, create stressed distributions, and produce conditional distributions.
Study nonparametric density estimation via measure transport, achieving optimal rates.
problem Nonparametric density estimation with optimal rates.
method Measure transport, penalized maximum likelihood, and sieved wavelet estimators.
result Achieve minimax optimal convergence rates over Hölder classes of densities.
Paper compares Bergman kernel and Masur-Veech measure on Teichmüller space.
problem Comparing Bergman kernel and Masur-Veech measure on Teichmüller space.
method Comparison between Bergman kernel form and pushforward measure of Masur-Veech measure.
result Obtained a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech measure.
QFIL improves offline RL by filtering data to reduce bias and variance.
problem Improving offline reinforcement learning policies with limited data.
method QFIL uses a filtered dataset to improve policies, trading off bias and variance through quantile selection.
result QFIL provides a safe policy improvement step with function approximation and effectively balances bias and variance.
We study the algebraic properties of the generalized Futaki invariant of an almost Fano variety and prove that it is in fact a pushforward to a point of an appropriate equivariant Chow cohomology class of the variety. This allows us to use Bott-type formulae for calculating the invariant. We show this use on some examp…
Unified methodology for estimating optimal transport maps in various function spaces.
problem Estimating the function T given samples from P and T♯P. method Unified methodology based on Poincaré inequality and smooth convex function gradient.
result Nearly sharp results in various settings, including normal distribution and neural networks.
Framework for worst-case generation using Wasserstein space optimization.
problem Evaluating robustness and stress-testing systems under distribution shifts.
method Min-max optimization over continuous probability distributions in Wasserstein space.
result Global convergence guarantees for the proposed Gradient Descent Ascent scheme.
Constructs a model for differential KO-theory using Clifford modules.
problem Refining Atiyah and Singer's families index with differential structure.
method Builds a model using families of Clifford modules with superconnection.
result Affords a differential refinement of Atiyah and Singer's families index.
Algorithm samples polygons of fixed edge lengths in any dimension.
problem Sampling random closed polygons with fixed edge lengths in any dimension.
method Weighted edge vectors on unit sphere, Möbius transformation, reweighting factors.
result Algorithm samples polygons according to standard probability measures efficiently.
New Hilbert bundles with ends defined from indexed bases.
problem Defining new structures in Hilbert bundles.
method Indexed bases and unitary operators of finite propagation.
result Characteristic classes of Hilbert bundles with ends.
The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.
problem Understanding the behavior of geodesics and measures on fibered hyperbolic 3-manifolds.
method Properties of geodesics and measures on the circle and sphere are analyzed to prove singularity.
result Natural measures on the circle become singular with respect to measures on the sphere.
Measures time-delay embedding for noisy, sparse data.
problem Applying Takens' embedding theorem to real-world, noisy data.
method Formulated a measure-theoretic generalization of the embedding theorem, using optimal transport.
result Reconstructed full state of dynamical systems from time-lagged partial observations robust to noise and sparsity.
The Cannon-Thurston map's measures become singular with respect to sphere measures.
problem Characterizing the behavior of measures under the Cannon-Thurston map.
method Analyzing the geometric properties of hyperbolic geodesics and quasi-geodesics.
result Natural measures on the circle become singular with respect to measures on the sphere.
We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups U(n). Every word w in the free…
Toeplitz operators linked to submultiplicative filtrations and weighted Bergman kernels.
problem Analyzing the asymptotics of weighted Bergman kernels for submultiplicative filtrations.
method Demonstrated that weight operator is a Toeplitz operator; analyzed asymptotics of weighted Bergman kernels.
result Local refinement of convergence of jumping measures towards geodesic ray pushforward measure.
The purpose of this study is to analyse two related topics: the Rumin cohomology and the H-orientability in the Heisenberg group Hn. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator D, giving examples in th…