The paper proves residual finiteness of certain lattices and constructs surfaces with specific fundamental groups.
problem Residual finiteness of lattices in P U ( 2 , 1 ) ~ \widetilde{\mathrm{PU}(2,1)} PU ( 2 , 1 ) and existence of smooth projective surfaces. method Proved residual finiteness of certain lattices and constructed surfaces using central extensions.
result First examples of residually finite lattices in P U ( 2 , 1 ) ~ \widetilde{\mathrm{PU}(2,1)} PU ( 2 , 1 ) and construction of surfaces with specific fundamental groups. Residually finite groups found in manifold automorphisms.
problem Residual finiteness of automorphism groups of high-dimensional manifolds.
method Embedding calculus, Weiss fibre sequence, convergence of embedding calculus tower, smoothing theory.
result Topological mapping class group of high-dimensional manifolds is residually finite.
ResGCN detects anomalies in attributed networks by capturing sparsity and nonlinearity.
problem Detecting anomalous nodes in attributed networks.
method Attention-based deep residual modeling using Graph Convolutional Networks.
result ResGCN effectively detects anomalies in attributed networks.
Researchers relax the CVF's smoothness requirement to create more flexible flow models.
problem Challenges in constructing flexible density models due to the CVF's smoothness requirement.
method Introduce L \mathcal{L} L -diffeomorphisms as generalized transformations that may violate smoothness on zero Lebesgue-measure sets. result The relaxation allows for the use of non-smooth activation functions like ReLU in residual flows.
The aim of this note is to improve upon our earlier result which translates Weyl's (curvature) formulation of Chern character of a smooth vector bundle into the language of residues. The dualized Chern character is the functional on smooth differential forms on M. In our previous paper, this functional has been express…
In this paper we generalize Leray's calculus of residues in several complex variables, to the situation of an abstract smooth CR manifold M of general type (n,k).
Deep, wide ConvResNets can approximate functions and their smoothness.
problem Function approximation and smoothness in deep networks.
method Analyzing ConvResNets, proving their ability to approximate functions and their smoothness.
result Large ConvResNets can approximate functions and exhibit sufficient first-order smoothness.
ResNets promote smoother interpolations than MLPs, enhancing generalization.
problem Understanding the difference in smoothness between ResNets and MLPs.
method Neural Tangent Kernel (NTK) analysis during gradient descent training.
result ResNet's NTK results in smoother interpolations than MLPs.
In this paper we prove geometric residue theorems for bundle maps over a compact manifold. The theory developed associates residues to the singularity submanifolds of the map for any invariant polynomial. The theory is then applied to a variety of settings: smooth maps between equidimensional manifolds, CR-singularitie…
Adaptive regularization improves deep learning model performance.
problem Improving generalization in deep learning models.
method Adaptive regularization via residual smoothing based on the heat equation.
result Our algorithm outperforms other optimization methods in generalization.
Investigates the relationship between ResNets and Neural ODEs, quantifying their closeness and providing training methods.
problem Quantifying the distance between ResNet dynamics and Neural ODE solutions.
method Bounding the distance between hidden state trajectories and Neural ODE solutions, using gradient descent and Heun's method.
result Gradient descent and Heun's method can implicitly regularize ResNets towards Neural ODEs, especially for smooth residual functions.
Inverse depth scaling found in LLMs due to similar layers averaging error.
problem Understanding how depth affects loss in large language models.
method Analysis of LLMs and toy residual networks.
result Loss scales inversely proportional to depth in LLMs.
Constructs currents representing Baum-Bott residues for foliations.
problem Calculating Baum-Bott residues for complex foliations.
method Explicit construction of currents with support on singular components.
result Currents represent Baum-Bott residues and are independent under certain conditions.
Let M M M be a smooth manifold and G G G a compact connected Lie group acting on M M M by isometries. In this paper, we study the equivariant cohomology of X = T ∗ M {\bf X}=T^\ast M X = T ∗ M , and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case that X \bf X X is a compact symplectic mani…
Proves a Baum--Bott formula for foliations by curves with logarithmic terms.
problem Analyzing singularities and smoothness in foliations by curves.
method Logarithmic Baum--Bott residues for foliated triples ( X , F , D ) (X, \mathcal{F}, D) ( X , F , D ) , relating to Poincaré's Problem and GSV indices. result Logarithmic Baum--Bott residues generalize Aleksandrov logarithmic index for vector fields on hypersurfaces.
A new principle minimizes residual and introduces momentum to improve PDE solution dynamics.
problem Ill-conditioning in Dirac-Frenkel residual minimization leads to non-unique parameter dynamics.
method Introduces a history variable (momentum) to select better-conditioned parameter velocities, preserving residual minimization while promoting smooth parameter evolutions.
result The approach leads to increased robustness in singular and near-singular PDE solution regimes.
Researchers extend a groupoid approach to calculate Wodzicki residue and Kontsevich-Vishik trace.
problem Calculating Wodzicki residue and Kontsevich-Vishik trace for pseudo-differential operators of any order.
method Groupoid approach to pseudo-differential operators.
result Extension of van Erp and Yuncken's work to operators of any order.
Deep GCNII tackles over-smoothing problem in graph convolutional networks.
problem Over-smoothing problem in shallow graph convolutional networks.
method Proposes GCNII with initial residual and identity mapping techniques.
result Deep GCNII outperforms state-of-the-art methods on various tasks.
A beta function for double layers is defined and analyzed.
problem Defining and analyzing a beta function for double layers.
method Holomorphic function definition and analytic continuation.
result Residues of the beta function are integrals of invariants.
Construct algorithms for Frobenius manifolds and residue pairings on Calabi-Yau varieties.
problem Construct algorithms for Frobenius manifolds and residue pairings on Calabi-Yau varieties.
method Analyze a dGBV algebra and introduce weak primitive forms.
result Explicit algorithms for Frobenius manifolds and residue pairings.
The study proves residual finiteness for certain lattice extensions and negatively curved projective varieties.
problem Residual finiteness of central extensions of arithmetic lattices in PU(n,1).
method General theorem on residual finiteness of extensions with characteristic class in span of Poincaré duals to totally geodesic divisors.
result Residual finiteness of central extensions for congruence lattices in PU(n,1) for n ≥ 4.
We show that any smooth bi-Lipschitz h h h can be represented exactly as a composition h m ∘ . . . ∘ h 1 h_m \circ ... \circ h_1 h m ∘ ... ∘ h 1 of functions h 1 , . . . , h m h_1,...,h_m h 1 , ... , h m that are close to the identity in the sense that each ( h i − I d ) \left(h_i-\mathrm{Id}\right) ( h i − Id ) is Lipschitz, and the Lipschitz constant decreases inversely with the number m m m of functions com…
New diagnostic method detects misspecified models in inverse PDE problems.
problem Misleading residual-norm diagnostics in inverse PDE problems.
method Structure-sensitive sequential diagnostic using e-processes.
result Rejects fitted models that produce biased predictions.
Fast nonparametric conditional independence testing via two-stage regression
problem Fast nonparametric conditional independence testing
method BLITZ (Broad-to-Local Independence Testing via residualiZation)
result Better null calibration than fast kernel, random-feature, and regression-based competitors
Spatial Adapter adds structured spatial representation to frozen predictors.
problem Efficiently adding spatial structure to pre-trained models.
method Structured spatial decomposition and closed-form covariance for residual fields.
result Adapter improves spatial prediction and uncertainty quantification.
A conservative drifting method improves generative modeling by using KDE gradients, proving convergence rates.
problem Improving generative modeling by addressing non-conservatism issues.
method Proposes a conservative drifting method using kernel density estimator gradients to address non-conservatism.
result Proves finite-particle convergence rates for the conservative method, providing explicit quadrature constants.
A new linear GCN model improves recommendation performance for large graphs.
problem Training difficulties and over-smoothing in GCN-based CF models.
method Proposes a linear residual graph convolutional network (LRGCCF) to address training difficulties and over-smoothing issues.
result The proposed model yields better efficiency and effectiveness on real datasets.
New method uses observational data to improve trial design efficiency.
problem Scarce randomized controlled trials; inefficiency of using observational data.
method Active Residual Learning, R-Design framework, R-EPIG criterion.
result Efficiently estimating residuals to correct observational bias improves trial design.
Develops a new multivariate regression model for complex outcomes.
problem Flexible, heterogeneous, and residual-dependent multivariate regression problems.
method MultiVCBART framework with Graphical Horseshoe priors.
result Empirically outperforms existing models on sparse, high-dimensional datasets.
Recent work has studied the reasons for the remarkable performance of deep neural networks in image classification. We examine batch normalization on the one hand and the dynamical systems view of residual networks on the other hand. Our goal is in understanding the notions of stability and smoothness of the inter-laye…
Existence of minimizers proven for residual ANNs with ReLU activation.
problem Existence of minimizers in neural network optimization landscapes.
method Proof using closure of search space containing ANNs and additional discontinuous responses.
result Existence of minimizers proven for residual ANNs with ReLU activation.
Gradient descent converges to global minima for ResNets with linearly scaled width.
problem Understanding the convergence of deep residual networks with varying network width and dataset size.
method Analyzing the Jacobian of ResNets and applying gradient descent for quadratic loss.
result Gradient descent converges to global minima for ResNets with linearly scaled width and independent of depth.
Deep ResNets exhibit distinct scaling properties with depth, challenging neural ODE models.
problem Understanding the scaling properties of deep ResNets and their relation to neural ODEs.
method Detailed numerical experiments on weights trained by stochastic gradient descent.
result Deep ResNets can exhibit different scaling regimes, including stochastic differential equations or neither, challenging the neural ODE model.
Improved stochastic approximation method reduces residual error.
problem Reducing residual error in stochastic approximation algorithms.
method Fixed-schedule one-quarter barrier and bias-corrected acceleration.
result Achieves T − 1 / 2 + o ( 1 ) T^{-1/2+o(1)} T − 1/2 + o ( 1 ) residual reduction with O ( 1 ) O(1) O ( 1 ) primitive samples. In this paper, we study Lipschitz-Fredholm vector fields on Bounded-Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if M M M is a connected smooth bounded-Fréchet-Finsler manifold endowed with a strengthened connection K \mathcal{K} K and if ξ ξ ξ is a smooth Lipschitz-Fr…
An analytic approach and description are presented for the moduli cotangent sheaf for suitable stable curve families including noded fibers. For sections of the square of the relative dualizing sheaf, the residue map at a node gives rise to an exact sequence. The residue kernel defines the vanishing residue subsheaf. F…
Paper proposes fast, robust methods for low-rank matrix recovery.
problem Estimating low-rank matrices from incomplete or corrupted data.
method Scaled subgradient methods for nonsmooth, nonconvex formulations.
result Methods converge almost dimension-free and condition-number independent.
Residual neural networks improve collision prediction in planetary simulations.
problem Accurate prediction of planetary collisions in N-body simulations.
method Residual neural networks trained on collision data.
result Residual neural networks outperform existing methods in prediction accuracy and generalization.
AAS optimizes neural network PDE approximations by adaptively sampling.
problem Statistical errors from random samples in neural network PDE approximations.
method Minmax formulation to optimize neural network and training set samples.
result Reduces Monte Carlo approximation error for a given sample size.
Abstract: Non-residually finite hyperbolic groups imply non-residually finite rigid hyperbolic groups.
problem Existence of non-residually finite hyperbolic groups
method Direct implication
result Existence of non-residually finite rigid hyperbolic groups
Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work [2] residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these resul…
In this note, residual finiteness of quandles is defined and investigated. It is proved that free quandles and knot quandles of tame knots are residually finite and Hopfian. Residual finiteness of quandles arising from residually finite groups (conjugation, core and Alexander quandles) is established. Further, residual…
Two methods improve Gaussian process predictive distributions' calibration.
problem Improving the reliability of Gaussian process predictive intervals.
method Introduces two methods: cps-gp and bcr-gp, both adapting conformal predictive systems to GP interpolation.
result Both methods provide finite-sample marginal calibration and smooth predictive distributions.
Every non-trivial knot group is fully residually perfect.
problem Understanding the residual properties of knot groups.
method Analyzing the residual properties of knot groups using group theory.
result Every non-trivial knot group is fully residually perfect.
ConvResNets approximate Besov functions and classify on low-dimensional manifolds.
problem Lack of statistical theories for deep learning on high-dimensional data.
method Exploits low-dimensional geometric structures of real-world data sets using ConvResNets.
result ConvResNets can approximate Besov functions and learn classifiers with optimal excess risk.
Kolmogorov-Arnold Networks improve deep learning adaptivity and can approximate Besov functions optimally.
problem Improving deep learning adaptivity and understanding approximation rates.
method Analyzing Besov norms and using Res-KANs for approximation.
result KANs can optimally approximate Besov functions at the optimal rate.
Residual flows are shown to approximate MMD well.
problem Lack of theoretical understanding of normalizing flows' expressiveness.
method Proved residual flows are universal approximators in MMD.
result Residual flows can approximate MMD with a bounded number of blocks.
We present a primal-dual algorithmic framework to obtain approximate solutions to a prototypical constrained convex optimization problem, and rigorously characterize how common structural assumptions affect the numerical efficiency. Our main analysis technique provides a fresh perspective on Nesterov's excessive gap te…