The article consist of two main parts: an analog of the Leray Theory for Singular Varieties and its application to the Theory of Parshin's Residues. The first part is independent from the second. It uses the theory of Whitney stratifications. The second part is an application of the first. In particular, a geometric an…
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48 results for “residue theory”
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…
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…
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.
Extends residue theory to flags of holomorphic distributions.
problem Calculating the residue class of flags of holomorphic distributions.
method Developed an effective method to calculate the class in certain cases.
result Established a relation between degrees, tangency order, Euler characteristic, and curve degree.
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…
Deep residual networks can approximate any continuous function using control theory.
problem Universal approximation capabilities of deep residual neural networks.
method Relating residual networks to control systems and using Lie algebraic techniques.
result Deep residual networks with adequately deep layers can approximate any continuous function on a compact set.
Residual networks with depthwise hyperparameter scaling transfer optimal hyperparameters across width and depth.
problem The challenge of hyperparameter tuning in deep learning, especially for large models.
method Combining P parameterization with residual networks having a residual branch scale of .
result Optimal hyperparameters transfer across width and depth in residual networks trained with this parameterization.
GT-DDP optimizer trains residual networks using game theory.
problem Applying DDP to residual networks.
method Game-theoretic DDP optimizer for residual networks.
result Improved training convergence and variance reduction.
RFRBoost uses random features to boost deep residual neural networks, improving performance and computational efficiency.
problem Improving performance of deep residual neural networks (RFNNs) while preserving convex optimization benefits.
method Random Feature Representation Boosting (RFRBoost) using boosting theory and random features at each layer.
result RFRBoost significantly outperforms RFNNs and end-to-end trained MLP ResNets in small- to medium-scale tabular datasets.
In this paper, we build tests for the presence of residual noise in a model where the market microstructure noise is a known parametric function of some variables from the limit order book. The tests compare two distinct quasi-maximum likelihood estimators of volatility, where the related model includes a residual nois…
We study conformal -subgeometry of submanifolds in a semi-Riemannian -manifold, focusing on conformal -manifolds and their Poincaré-Einstein metrics . Our approach is based on the spectral theory of Dirac operator in the ambient -manifold, and associated spinor valued meromorp…
Identifying critical residues of a protein using meaningfully-thresholded Random Geometric Graphsq-bio.BM
Researchers identify critical protein residues using advanced graph theory.
problem Identifying essential residues in proteins for function.
method Learning Random Geometric Graphs (RGG) with Cramer's V correlation and organic thresholding.
result Advanced RGG methods accurately identify critical residues compared to existing techniques.
The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.
problem Analyzing Poisson-flat connections with logarithmic poles.
method Defining an Euler-Poisson principal part and residue theory, establishing a Poisson Poincaré-Dulac theorem.
result Any logarithmic Poisson-flat connection is holomorphically gauge equivalent to a pure Euler-Poisson normal form.
Field theory for optimal signal propagation in ResNetscond-mat.dis-nn
Field theory explains optimal scaling in ResNets for signal propagation.
problem Understanding optimal scaling parameter for ResNet performance.
method Finite-size field theory for ResNets to study signal propagation and scaling.
result Analytical expressions for optimal scaling parameter, independent of other hyperparameters.
We add size factor to CAPM and normalize residuals by Volatility Index.
problem Capturing the size effect in CAPM and making residuals Gaussian.
method Insert size effect, normalize residuals by Volatility Index, and fit model to real-world data.
result The new model shows long-term stability and connects to Stochastic Portfolio Theory.
Global analysis of Dixmier traces and Wodzicki residues on compact Lie groups.
problem Computing Dixmier traces and Wodzicki residues on compact Lie groups.
method Global quantisation approach, using global symbols and representation theory.
result Explicit formulae for Dixmier traces and Wodzicki residues on compact Lie groups.
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.
The paper develops residue currents for cohesive modules and proves a generalized Poincaré-Lelong formula.
problem Analyzing coherent sheaves on complex manifolds using global analytic methods.
method Developing residue currents for cohesive modules and proving their properties.
result Proves a generalized Poincaré-Lelong formula for cohesive modules.
3D BF theory on certain 3-manifolds evaluated via residues and large k limits.
problem Singular and ill-defined partition function of 3D BF theory.
method Direct evaluation of path integral for specific 3-manifolds, using residues and large k limits of Chern-Simons matrix integrals.
result 3 definitions of the integral offer insights into the sum/integral over all flat connections.
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.
Information bottleneck theory of high-dimensional regression: relevancy, efficiency and optimalitycs.IT
New approach quantifies overfitting in high-dimensional regression.
problem Quantifying and avoiding overfitting in large neural networks.
method Information bottleneck theory to minimize residual information while maximizing relevant bits.
result Characterized the relative information efficiency of randomized regression compared to optimal algorithms.
New groups with special properties found.
problem Finding new groups with specific geometric properties.
method Proved actions on CAT(0) cubical complexes under certain conditions.
result Many groups admit cocompact actions on CAT(0) cubical complexes.
In this article we develop the theory of residually finite rationally (RFR) groups, where is a prime. We first prove a series of results about the structure of finitely generated RFR groups (either for a single prime , or for infinitely many primes), including torsion-freeness, a Tits alternative, and …
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.
We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems, from the approximation perspective. In particular, we establish general sufficient conditions for universal approximation using continuous-time deep residual networks, which can …
We show that Residual Networks (ResNet) is equivalent to boosting feature representation, without any modification to the underlying ResNet training algorithm. A regret bound based on Online Gradient Boosting theory is proved and suggests that ResNet could achieve Online Gradient Boosting regret bounds through neural n…
Stacking improves deep neural network training efficiency.
problem Improving the efficiency of training deep neural networks.
method Proposes stacking as a form of accelerated gradient descent.
result Proves stacking provides accelerated training for certain deep linear residual networks.
Enhances machine learning interpretability using category theory.
problem Improving machine learning interpretability and social implementation.
method Develops a categorical framework for structured understanding of supervised learning.
result Introduces the Gauss-Markov Adjunction for clarifying residuals and parameters.
An emerging design principle in deep learning is that each layer of a deep artificial neural network should be able to easily express the identity transformation. This idea not only motivated various normalization techniques, such as \emph{batch normalization}, but was also key to the immense success of \emph{residual …
A new method boosts exploration in bandit algorithms, reducing regret.
problem Improving exploration in bandit algorithms with bounded or unbounded rewards.
method Residual Bootstrap Exploration (ReBoot) method that injects data-driven randomness.
result Proves logarithmic regret in Gaussian multi-armed bandits with appropriate variance inflation.
Deep heteroskedastic models overfit, showing a phase transition with regularization strength.
problem Overfitting in deep heteroskedastic regression models.
method Theoretical framework based on statistical field theory, empirical verification, and hyperparameter simplification.
result A phase transition in model behavior with varying regularization strength.
In this paper, we consider a framework adapting the notion of cointegration when two asset prices are generated by a driftless Itô-semimartingale featuring jumps with infinite activity, observed regularly and synchronously at high frequency. We develop a regression based estimation of the cointegrated relations method …
Effective theory for Transformer initialization improves model performance.
problem Improving performance of Transformers at initialization.
method Effective-theory analysis of signal propagation in wide and deep Transformers.
result Particular width scalings of initialization and training hyperparameters.
Study the intersection of positive closed currents using tangent currents and King's residue formula.
problem Investigate the intersection of positive closed currents in complex manifolds.
method Employ tangent currents and King's residue formula to establish a natural condition for intersection.
result Derive an integral representation of the intersection of positive closed currents.
Sharpe Ratio Analysis in High Dimensions: Residual-Based Nodewise Regression in Factor Modelsq-fin.PM
New method for Sharpe ratio analysis in high dimensions using residual-based nodewise regression.
problem Consistency of Sharpe ratio estimators in high-dimensional portfolios.
method Residual-based nodewise regression for estimating precision matrix of errors and returns.
result Consistent Sharpe ratio estimators in various portfolio settings.
In this paper we give formulae for the Dixmier trace and the noncommutative residue (also called Wodzicki's residue) of pseudo-differential operators by using the notion of global symbol. We consider both cases, compact manifolds with or without boundary. Our analysis on the Dixmier trace of invariant pseudo-differenti…
Improved regression analysis using Padé approximants with new residuals and regularization.
problem Improving regression analysis with Padé approximants for accuracy and avoiding overfitting.
method New residuals in least squares method, system of linear equations for rational functions, Tikhonov regularization.
result Demonstrated efficiency in practical cases from physics and reliability theory.
We present the theory of pseudodifferential operators acting on a vector orbibundle over an orbifold, construct the zeta function of an elliptic pseudodifferential operator and show the existence of a meromorphic extension to the complex plane with at most simple poles. We give formulas for generalized densities on the…
Study minimizers in large volume isoperimetric problems with a new flatness criterion.
problem Minimizers in isoperimetric problems with a compact obstacle.
method Study Plateau-type problem with free boundary, develop mesoscale flatness criterion.
result Identify isoperimetric residue in energy expansion for large volume.
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
Develops formal moduli theory for splitting complex supermanifolds.
problem Tackles the splitting problem of complex supermanifolds.
method Constructs a filtered dg Lie algebra to control splittings and transfers the theory to a minimal filtered -model.
result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.
Proposes a robust method for high-dimensional linear models.
problem Inference in high-dimensional settings with heavy-tailed errors and clustered data.
method Residual randomization procedure for Lasso-based inference.
result Outperforms state-of-the-art methods in challenging settings.
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…
D2SRM solves complex PDEs using deep learning.
problem High-dimensional, Hessian-dependent fully nonlinear parabolic PDEs.
method Single scalar space-time network generating derivative-consistent approximations trained through residuals and penalties.
result Well-posedness and convergence theory established for globally Lipschitz equations.
The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.
problem Proving the Singer conjecture and refining Gromov's inequality for aspherical complex surfaces.
method Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group, and refinement of Gromov's inequality for non-residually finite surfaces.
result Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group.
Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework o…
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.