The paper reconstructs Lorentzian spacetimes from causal sets.
problem Reconstructing Lorentzian spacetimes from causal sets.
method Introduced a concept of isomorphy and three types of convergence.
result Established Gromov's reconstruction theorem in Lorentzian geometry.
The paper describes fitting submanifolds to data using Sussmann's orbit theorem.
problem Fitting an immersed submanifold to random samples.
method Uses Sussmann's orbit theorem to ensure submanifold fitting. Reconstruction involves encoding times and decoding via flows of vector fields.
result A high-probability bound on excess risk for the reconstruction error.
Finite approximations help reconstruct countable metric and ultrametric spaces.
problem Reconstructing countable metric and ultrametric spaces.
method Topological reconstruction using inverse limits of finite T0 spaces. result Countable metric and ultrametric spaces can be reconstructed as finite approximations.
Defines precise general manifolds from gluing data.
problem Formalizing the concept of general manifolds.
method Formalized definition using gluing data, equivalence-partially ordered set (e-pos), and natural relations.
result Reconstruction theorems allow reconstructing manifolds and their morphisms from gluing data.
Reconstruct spacetime from order and number of points.
problem Reconstruct spacetime from chronological relations and i.i.d. samples.
method Relaxing hypotheses of Gromov reconstruction theorem, using random adjacency matrices and chronological relations.
result Spacetime can be recovered by only knowing 'order' and 'number' of its points.
The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.
problem Formulating and proving a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for tetrahedra.
method Formulated and proved a Lorentzian analogue of Minkowski theorem for tetrahedra in dS3 and AdS3.
result A unique strictly convex tetrahedron can be reconstructed from four non-trivial based SO+(1,2) holonomies.
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.
This work reconstructs knot invariants from Alexander polynomials, proving consistency with known theorems.
problem Reconstructing knot invariants from Alexander polynomials.
method Quantization, deformation, and rewriting of Alexander polynomials.
result Derives new formulae for colored superpolynomials and proves consistency with Melvin-Morton-Rozansky theorem.
Paper develops a differentiable approach for 3D imaging models using Fourier slice theorem.
problem Uncertainty in 3D structure modeling and pose estimation in scientific imaging.
method Differentiable probabilistic models in Fourier space with backpropagation through projection.
result Validates approach on 3D protein reconstruction and extends to probabilistic models.
A hybrid network improves MRI reconstruction from undersampled data.
problem Reducing MRI acquisition time while maintaining image quality.
method Hybrid architecture combining k-space and image domains using complex-valued and real-valued U-nets.
result Hybrid approach outperformed image-only deep learning methods in hard-to-reconstruct regions.
The paper explores holographic structures on exotic spheres and their implications.
problem Understanding holographic structures on exotic spheres.
method Introducing and studying holographic structures on closed manifolds Y. result Generalizing the Holography Theorem to fillable holographic structures on Y. Paper generalizes regularization methods for Banach spaces.
problem Ill-posedness in machine learning and signal reconstruction.
method Generalizes regularization to Banach spaces and presents a representer theorem.
result Retrieves and extends known results in optimization and machine learning.
The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.
problem Understanding the structure of hyperbolic log-orbi curves.
method Formulates hyperbolic uniformization as a Tannakian reconstruction theorem and constructs a canonical maximal parahoric PSL2-Higgs object.
result Reconstructs the absolute Galois group of a one-variable complex function field as the inverse limit of etale fundamental groups of orbifold models.
The study explores highly supersymmetric backgrounds in 11D supergravity.
problem Understanding and constructing highly supersymmetric backgrounds in 11D supergravity.
method Definition of abstract symbols and a strong version of the Reconstruction Theorem, proposing a strategy to construct backgrounds, and providing an example with detailed computation.
result Bijective correspondence between highly supersymmetric backgrounds and abstract symbols, and a classical supersymmetry gap result.
New method simplifies tomographic reconstruction using RKHS.
problem Tomographic reconstruction challenges.
method RKHS framework for X-ray transform.
result Sharp stability results without Fourier transform.
In the previous paper, the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c_1(M) of the tangent bundle is nef. In this paper, even when c_1(M) is not nef, w…
The paper recovers contact forms from boundary data using vector fields and Lyapunov functions.
problem Recovering contact forms from boundary data.
method Using vector fields and Lyapunov functions, the paper describes boundary data and proves reconstruction of (X,β) up to diffeomorphism. result Boundary data allow for the reconstruction of (X,β) up to a diffeomorphism of X. A classical theorem of Riemannian geometry, due in its original form to Cartan, states that the Taylor expansion of the metric in geodesic normal coordinates is a universal formal power series involving only the symmetrizations of the iterated covariant derivatives of the curvature tensor; this is known as the jet isom…
Defines discrete channel surfaces in Lie sphere geometry.
problem Defining discrete channel surfaces in Lie sphere geometry.
method Definition and associated data sets for reconstruction.
result Proof of a discrete version of Vessiot's Theorem for isothermic discrete channel surfaces.
The Rips complex at scale r is homotopy equivalent to the nerve of a cover of diameter r.
problem Reconstructing spaces using Rips complexes and covers.
method Functorial Dowker-Nerve Diagram, homotopy equivalence, cover of diameter r.
result General framework for reconstructing spaces by Rips complexes.
Bayesian image reconstruction using pre-trained generative models.
problem Distribution shifts and latent variable changes in image data.
method Combining SOTA generative models with Bayes' theorem for image restoration tasks.
result Competitive performance on super-resolution and in-painting tasks without training.
Designs CNNs for better image reconstruction.
problem Image reconstruction from limited data.
method Parseval convolution operators and chaining of elementary modules.
result CNN-based algorithm yields better results than sparsity-based methods.
What do auto-encoders learn about the underlying data generating distribution? Recent work suggests that some auto-encoder variants do a good job of capturing the local manifold structure of data. This paper clarifies some of these previous observations by showing that minimizing a particular form of regularized recons…
Proposes a new category of bundles for Lagrangian reduction in field theory.
problem Lagrangian reduction in field theory.
method Introduces a category of bundles to perform Lagrangian reduction by stages in covariant Field Theory.
result Formulates the Noether theorem in this new theoretical framework.
This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to a…
New insights into knot homology growth, extending previous results.
problem Understanding the growth of homology torsion in knot exteriors.
method Analyzing meromorphic continuation of generating functions for knot homology.
result New proofs of Silver-Williams asymptotic, Fried's theorem, and Gordon's theorem.
Kernel regression improves graph signal estimation.
problem Estimating graph signals from noisy observations.
method Kernel regression in reproducing kernel Hilbert spaces.
result Kernel methods offer richer prior information and simpler estimators.
A locally conformally Kaehler (l.c.K.) manifold is a complex manifold admitting a Kaehler covering M~, with each deck transformation acting by Kaehler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M~. We prove a structure theorem f…
The paper explores polysymplectic structures and their reductions in field theories.
problem Invariance of Lagrangian and Hamiltonian field theories under symmetry groups.
method Application of polysymplectic reduction theorem for both Lagrangian and Hamiltonian field equations.
result Identification and relation of polysymplectic structures through Routhian function and Legendre transformation.
We prove that the Euler form of a metric connection on real oriented vector bundle E over a compact oriented manifold M can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilisticall…
New coding theorem shows achievable rate matches theoretical limit.
problem Unknown existence of encoders and decoders for RDPF.
method Used stochastic, variable-length codes to prove RDPF achievable.
result Achievable rate matches theoretical rate-distortion-perception function.
This paper belongs to a series devoted to the study of the cohomology of classifying spaces. Generalizing the Weil algebra of a Lie algebra and Kalkman's BRST model, here we introduce the Weil algebra W(A) associated to any Lie algebroid A. We then show that this Weil algebra is related to the Bott-Shulman-Stasheff…
A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given space-time is discussed. This result can be seen as part of Penrose and Low's programme intending to describe the causal structure of a space-time M in terms of the topological and geometrical…
Paper develops a new method for solving IBVPs on star-shaped domains.
problem Solving Inverse Boundary Value Problems (IBVP) for parallel transport equations.
method Covariant tomography, integrating geometric decomposition with specific interior extensions.
result Formal solvability criterion for higher-order IBVPs, validated through examples.
New results on inferring hidden states in trackable weak models.
problem Inferring hidden states in trackable weak models.
method Analyzing strongly-connected trackable weak models and reconstructing branch choices.
result The number of hypotheses in strongly-connected trackable models is bounded by a constant.
New methods for signal reconstruction using guiding sets and frame-less pathways.
problem Signal reconstruction in Hilbert spaces with specified properties.
method Axiomatic approach involving sample consistent and guiding sets, with reconstruction set defined as a shortest pathway.
result Existence and uniqueness of reconstruction set in Hilbert space, with derived stability and error bounds.
This article is a follow-up of ``Holonomy and Path Structures in General Relativity and Yang-Mills Theory" by Barrett, J. W. (Int.J.Theor.Phys., vol.30, No.9, 1991). Its main goal is to provide an alternative proof of this part of the reconstruction theorem which concerns the existence of a connection. A construction o…
Develops a reduction theory for covariant field theories with gauge symmetries.
problem Handling gauge symmetries in covariant field theories.
method Utilizes generalized principal connections and fiberwise action of Lie groups.
result Relates vertical reduced equations to the Noether theorem.
Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Some of its tools (e.g., flat norm distance and decomposition in generalized surface space) hold interest from a theoretical pe…
Paper reconstructs training samples from neural network loss functions.
problem Revealing the inputs of training samples from loss functions.
method Introducing virtual polynomials and semi-algebraic sets to analyze loss surfaces.
result Loss functions can reconstruct training samples up to scalar multiplication.
Often noisy point clouds are given as an approximation of a particular compact set of interest. A finite point cloud is a compact set. This paper proves a reconstruction theorem which gives a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets …
Generative adversarial network improves signal reconstruction from magnitude spectrograms.
problem Reconstructing a time-domain signal from a magnitude spectrogram.
method Deep neural network and generative adversarial network approach.
result Our method reconstructs signals faster with higher quality than the Griffin-Lim method.
XPDNet wins MRI reconstruction challenge with neural network.
problem MRI reconstruction from under-sampled data.
method Inspired by MRI and computer vision best practices, XPDNet uses a neural network.
result XPDNet achieved state-of-the-art results in the 2020 fastMRI challenge.
CoRAS adapts image acquisition rates for accurate reconstruction.
problem Determining when enough measurements are collected for accurate image reconstruction.
method Adaptive acquisition rate selection based on reconstruction error probability.
result CoRAS achieves target stopping-time coverage with fewer measurements.
The paper analyzes the excess risk of PCA and provides a precise characterization.
problem Understanding the excess risk of principal component analysis (PCA).
method Established a central limit theorem for PCA error and derived the excess risk distribution.
result Obtained a non-asymptotic upper bound on the excess risk of PCA.
PTOPOFL uses topological descriptors to protect privacy in federated learning.
problem Privacy and data reconstruction attacks in federated learning.
method PTOPOFL replaces gradient communication with persistent homology feature vectors for privacy and topology-guided aggregation.
result PTOPOFL achieves higher AUC and reduces reconstruction risk compared to gradient sharing.
We focus on an interpolation method referred to Bayesian reconstruction in this paper. Whereas in standard interpolation methods missing data are interpolated deterministically, in Bayesian reconstruction, missing data are interpolated probabilistically using a Bayesian treatment. In this paper, we address the framewor…
New method reconstructs moving parts of proteins in cryo-EM.
problem Reconstructing non-rigid molecules with moving parts in cryo-EM.
method Graph Laplacian construction from multiple projection images, followed by spectral volume expansion.
result High-resolution visualization of molecular dynamics using spectral volumes.