Neural network residuals isolate and locate unknown faults.
problem Locating unknown faults in industrial systems.
method Neural network-based residuals combining physical insights and machine learning.
result Neural network residuals can isolate and locate unknown faults.
The paper proves localization formulas for contact manifolds.
problem Localization of integrals on contact manifolds.
method Equivariant basic cohomology and contact moment map.
result Analogue of localization formulas for contact manifolds.
A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a proje…
Formulae for divergent integrals with singularities, useful in string theory.
problem Handling divergent integrals with singularities.
method Formulae for finite part of divergent integrals, using local residue map.
result Formulae for finite part of divergent integrals expressed in terms of local residue map.
Study of pure mapping class groups on infinite graphs.
problem Classifying graphs with specific mapping class groups.
method Completely classified graphs with pure mapping class groups.
result Established semidirect product decomposition and computed first integral cohomology.
Formula calculates residues for maps near holomorphic distributions.
problem Calculating residues for maps near holomorphic distributions.
method Residues formula for maps generically transversal to regular holomorphic distributions.
result Established a residues formula for maps near holomorphic distributions.
The paper introduces a diagnostic method to detect grokking transitions in models before test accuracy improves.
problem Detecting the transition from training to generalization in machine learning models.
method Summarize task-dependent observables as empirical distributions, map them to Wasserstein/quantile coordinates, and analyze using Hankel dynamic mode decomposition.
result The diagnostic method achieves AUROC \(\approx\) 0.93 for grokking-vs-non-grokking discrimination at the run level.
This paper generalizes monodromy maps for projective structures with poles.
problem Understanding the monodromy of projective structures with poles.
method Generalizing the monodromy map from compact curves to those with poles.
result The monodromy map for projective structures with poles is a local biholomorphism.
Holomorphic residue formula for complex supermanifolds.
problem Residue localization on complex supermanifolds.
method Holomorphic residue localization formula for odd vector fields.
result Explicit local residue formula under isolated non-degeneracy hypotheses.
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
Proves Singer conjecture for specific geometric varieties.
problem Singer conjecture for certain geometric varieties.
method Semismall Albanese map and residually finite fundamental group.
result Proves Singer conjecture for specified varieties.
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…
Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.
problem Residual finiteness of hyperbolic groups and congruence subgroup property for mapping class groups.
method Assumption of residual finiteness of hyperbolic groups leads to proof of congruence subgroup property.
result Congruence subgroup property for mapping class groups of hyperbolic surfaces.
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.
Using the notion of equivariant Kirwan map, as defined by Goldin, we prove that -- in the case of Hamiltonian torus actions with isolated fixed points -- Tolman and Weitsman's description of the kernel of the Kirwan map can be deduced directly from the residue theorem of Jeffrey and Kirwan. A characterization of the ke…
This paper studies the regularity of constrained Willmore immersions into Rm≥3 locally around both "regular" points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion, its first, and its second derivatives, given in terms of resi…
The study shows how quotients of mapping class groups are hierarchically hyperbolic.
problem Understanding the hierarchical hyperbolicity of mapping class groups and their quotients.
method A combinatorial criterion for hierarchical hyperbolicity applied to mapping class groups.
result Quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic.
New findings show mapping class groups of certain high-dimensional manifolds are not residually finite.
problem Understanding the mapping class groups of simply connected high-dimensional manifolds.
method Provided a counterexample showing mapping class groups are not residually finite.
result Mapping class groups of some high-dimensional manifolds are not residually finite.
The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators associated to the Guillemin-Wodzicki residue trace. The resulting functional is multiplicative, a local invariant, and not defined by a regularization procedure. The residue determinan…
Classifies mapping tori of specific groups, generalizing known results.
problem Classifying mapping tori of specific groups.
method Using Hopf-type properties and Poincaré Duality groups.
result Generalizes and provides new proofs for fibered 3-manifolds.
Study excess logarithmic residues for foliations to bound invariant hypersurfaces and test log canonicity.
problem Bounding invariant hypersurfaces and testing log canonicity of singularities.
method Introduce excess logarithmic residues, prove residue formula, derive Poincaré-type bound, and use them to recover log discrepancies.
result Componentwise logarithmic residues of a lifted foliation along the exceptional divisor recover log discrepancies of singularities.
We prove the existence of "half-plane differentials" with prescribed local data on any Riemann surface. These are meromorphic quadratic differentials with higher-order poles which have an associated singular flat metric isometric to a collection of euclidean half-planes glued by an interval-exchange map on their bounda…
Localizes Wodzicki residue for logarithm of differential operators.
problem Localizing Wodzicki residue for logarithm of differential operators.
method Localisation formula using rescaled differential operators and spinor bundles.
result Expresses index of Dirac operator in terms of local density involving logarithm.
Let p be a prime. In this paper, we classify the geometric 3-manifolds whose fundamental groups are virtually residually p. Let M=M3 be a virtually fibered 3-manifold. It is well-known that G=π1(M) is residually solvable and even residually finite solvable. We prove that G is always virtually residually p…
Homology growth of specific mapping tori vanishes for certain groups.
problem Homology growth of polynomially growing mapping tori in various groups.
method Proof of the cheap rebuilding property for specific groups.
result Torsion homology growth vanishes for Farber sequences in every degree.
Study meromorphic k-differentials with prescribed singularities on Riemann surfaces.
problem Understanding local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeros and poles, and k-residues at poles.
result For a given pattern of zeros, there exists a primitive holomorphic k-differential with these zeros.
CAM-GAN improves GANs for continual learning with efficient feature map transformations.
problem Efficient continual learning for GANs with reduced parameter growth.
method Designing and leveraging parameter-efficient feature map transformations, including global and task-specific parameters, residual bias, and Fisher information matrix.
result Significantly improved model performance and high-quality samples with fewer parameters.
The paper shows dense and residual sets of continuous maps with positive metric mean dimension.
problem Understanding the genericity of continuous maps with positive metric mean dimension.
method Analyzing continuous maps on compact Riemannian manifolds and Cantor sets.
result The set of continuous maps with metric mean dimension equal to a given value is dense and, for the dimension, residual in the space of continuous maps.
The class of metrizable spaces M with the following approximation property is introduced and investigated: M∈AP(n,0) if for every $\e>0$ and a map $g\colon\I^n\to M$ there exists a 0-dimensional map $g'\colon\I^n\to M$ which is $\e$-homotopic to g. It is shown that this class has very nice properties. For exam…
Deep learning networks are approximated using dynamical systems theory.
problem Understanding the approximation capabilities of deep learning networks.
method Modeling deep residual networks as continuous-time dynamical systems and using approximation theories in Lp. result Established general sufficient conditions for universal approximation of deep residual networks.
Framework calculates positional influence in causal residual Transformers.
problem Understanding positional influence in causal residual Transformers.
method Adjoint-sensitivity framework for positional influence in causal residual Transformers.
result Exact evolution of adjoint-energy influence density and decomposition into residual transmission, nonlocal Volterra, and local channels.
The paper develops a theory of Bott-Chern cohomology and proves a residue theorem.
problem Localization of Bott-Chern classes and Hermitian residues.
method Developed a unified theory of Cech-Bott-Chern cohomology, defined cup product and integration, and introduced local and global duality homomorphisms.
result Proved a residue theorem for vector bundles with compatible Hermitian connections.
Jeffrey and Kirwan suggested expressions for intersection pairings on the reduced space of a Hamiltonian G-space in terms of multiple residues. In this paper we prove a residue formula for symplectic volumes of reduced spaces of a quasi-Hamiltonian SU(2)-space. The definition of quasi-Hamiltonian G-spaces was recently …
A new model improves CT image quality from low-dose scans.
problem Improving CT image quality from low-dose scans.
method Multi-layer Residual Sparsifying Transform (MRST) learning model for low-dose CT reconstruction.
result The MRST model outperforms conventional methods in maintaining subtle details.
The paper studies residues of manifolds and their applications in geometry.
problem Understanding the residues of manifolds and their geometric implications.
method Analytic continuation and Möbius invariance of residues, introduction of relative and weighted residues.
result Scalar curvature, mean curvature, and Euler characteristic can be expressed in terms of residues.
Proposes a method to make deep networks' derivatives more stable.
problem Making deep networks' derivatives more stable over larger regions.
method A learning problem to encourage stable derivatives, with an inference step and optimization step.
result Proposes a novel relaxation to scale the algorithm to realistic models.
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…
A new method improves maximum inner product search by locally decomposing residual vectors.
problem Maximum inner product search efficiency and accuracy.
method Local Orthogonal Decomposition (LOD) combined with multiscale quantization.
result LOD consistently achieves higher recall than previous methods under the same bitrates.
Proposes a new regression method using Lp-norms for non-Gaussian noise.
problem Non-Gaussian noise in residuals affects the performance of local least squares regression.
method Introduces local polynomial Lp-norm regression, replacing weighted least squares with weighted Lp-norm estimation. result Demonstrates superior performance over local least squares in one-dimensional data and higher dimensions.
We count meromorphic differentials with fixed residues and poles of fixed orders.
problem Counting meromorphic differentials with fixed residues and poles of fixed orders.
method Intersection theory on compactified moduli spaces of differentials.
result Complete solution to the problem with interesting combinatorial properties.
Dynamic steerable blocks improve deep networks by learning filter invariances.
problem Pixel-based filters ignore image properties, leading to suboptimal performance.
method Developed frame-based ResNets and Densenets, which are steerable under predefined transformations.
result Dynamic steerable blocks outperform other approaches on contour detection datasets.
New results on homology torsion growth for various groups.
problem Understanding the growth of higher torsion homologies for arithmetic lattices and other groups.
method Quantitative homotopical method called effective rebuilding, constructing small classifying spaces of finite index subgroups.
result Strong asymptotic bounds for the torsion growth in principal congruence subgroups.
Non-coherence proven for certain groups with specific mapping tori.
problem Proving non-coherence for groups with specific mapping tori.
method Analyzing the structure of groups as extensions and properties of mapping tori.
result Groups with certain mapping tori are not coherent.
Develops a novel method to estimate non-Gaussian hydraulic conductivities efficiently.
problem Estimation of non-Gaussian hydraulic conductivity fields in subsurface flow models.
method Integrates adversarial autoencoders with residual dense convolutional networks for parameterization and surrogate modeling.
result Significantly reduces computation time for accurate inversion results.
Study meromorphic k-differentials on Riemann surfaces, focusing on their singularities.
problem Characterize the local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeroes and poles, and k-residues at poles, for different genera.
result For genus g ≥ 2, every expected tuple of k-residues appears as actual residues.
Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this paper, we study the equivariant cohomology of X=T∗M, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case that X is a compact symplectic mani…
Researchers compute a residue cocycle for Dirac-type operators using modified Getzler calculus.
problem Computing the residue cocycle for Dirac-type operators.
method Modified Getzler calculus for computation.
result Computed residue cocycle for a class of Dirac-type operators.
The paper computes the mapping class group of certain 6-manifolds.
problem Computing the mapping class group of specific 6-manifolds.
method Algebraic properties and homology groups of the mapping class group were determined.
result The mapping class group is residually finite and virtually torsion-free.