Research resolves sign conventions in Floer theory for Morse-Bott case.
problem Sign conventions in filtered A∞-operations for Lagrangian Floer theory. method Defined filtered A∞-operations and verified formulae using de Rham model. result Resolved sign issues in Bott-Morse setting.
This paper introduces a new distance metric for filtered A-infinity categories, focusing on Lagrangian submanifolds.
problem Measuring the distance between filtered A-infinity categories associated with Lagrangian submanifolds.
method Developed a Gromov-Hausdorff distance to measure the difference between these categories.
result Established that the sequence of filtered A-infinity categories forms a Cauchy sequence in Gromov-Hausdorff distance.
Extends quantization theory to mixed polarizations using transverse differential operators.
problem Quantization in mixed polarization.
method Developed a theory of transverse differential operators associated to non-singular polarizations.
result Obtained a geometric interpretation of deformation quantization and sheaf of subalgebras acting on polarized sections.
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 L∞-model. result Recover classical obstruction classes as leading terms of Maurer-Cartan representatives and proves the existence of higher obstructions.
We study the geometry and topology of (filtered) algebra-bundles ΨZ over a smooth manifold X with typical fibre ΨZ(Z;V), the algebra of classical pseudodifferential operators of integral order on the compact manifold Z acting on smooth sections of a vector bundle V. First a theorem…
The optimal predictor for a linear dynamical system (with hidden state and Gaussian noise) takes the form of an autoregressive linear filter, namely the Kalman filter. However, a fundamental problem in reinforcement learning and control theory is to make optimal predictions in an unknown dynamical system. To this end, …
Constructs a cyclic, filtered, strictly unital curved A∞ category for Lagrangian submanifolds and develops Floer theory.
problem Proving that any Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by a given collection of Lagrangian submanifolds.
method Develops a cyclic, filtered, strictly unital curved A∞ category and uses it to prove the above statement. result Any Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by a given collection of Lagrangian submanifolds.
Defines and characterizes operators on Lie ∞-algebras with respect to actions.
problem Characterizing operators on Lie ∞-algebras with respect to actions.
method Using higher derived brackets construction and Maurer-Cartan elements.
result Determines the Lie ∞-algebra controlling the deformation of operators.
A new filter adapts to heavy-tailed data without tuning, improving performance in challenging conditions.
problem Degraded performance of Kalman and EnKF in heavy-tailed distributions.
method Generalizes EnKF using t-distributions, estimating parameters via EM algorithm.
result Improves performance on challenging filtering problems with heavy-tailed noise.
Researchers extend pseudodifferential calculus on filtered manifolds using fixed point algebras.
problem Defining operators with varying orders on filtered manifolds.
method Using generalized fixed point algebras and nilpotent Lie groups, they construct a new calculus.
result They establish a new calculus that reflects the behavior of differential operators on filtered manifolds.
A new surgery formula for knot lattice homology.
problem Developing a new surgery formula for knot lattice homology.
method Provided an iterable version of the surgery formula using doubly-filtered spaces and involutive data.
result Computed knot lattice spaces for specific knots and three-manifolds.
Paper presents efficient algorithms for convolutional neural networks using Winograd minimal filtering.
problem Resource-efficient implementation of convolutional neural networks.
method Winograd minimal filtering trick applied to M-tap filters (M=3,5,7,9,11) for parallel hardware implementation.
result Approximately 30% reduction in multipliers for fully parallel hardware implementation.
We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Z≥0∪{∞}. It is zero for overtwisted contact structures, ∞ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable …
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …
Kernel-based Bayesian filter for nonlinear systems using infinite-dimensional operators.
problem Modeling and predicting nonlinear dynamical systems.
method Functional Bayesian perspective, reproducing kernel Hilbert space, Gaussian kernel.
result Effective approximation and accurate results for nonlinear systems.
We discuss the problem of adaptive discrete-time signal denoising in the situation where the signal to be recovered admits a "linear oracle" -- an unknown linear estimate that takes the form of convolution of observations with a time-invariant filter. It was shown by Juditsky and Nemirovski (2009) that when the $\ell_2…
Let f:A-->B be a covering map. We say A has e filtered ends with respect to f (or B) if for some filtration {K_n} of B by compact subsets, A - f^{-1}(K_n) "eventually" has e components. The main theorem states that if Y is a (suitable) free H-space, if K < H has infinite index, and if Y has a positive finite number of …
In this paper, we 'construct' a 2-functor from the unobstructed immersed Weinstein category to the category of all filtered A∞ categories. We consider arbitrary (compact) symplectic manifolds and its arbitrary (relatively spin) immersed Lagrangian submanifolds. The filtered A∞ category associated to…
Paper revisits graph-CNNs using Laplace-Beltrami spectral filters and polynomials.
problem Improving spectral graph convolutional neural networks (graph-CNNs).
method Developed Laplace-Beltrami CNN (LB-CNN) by replacing graph Laplacian with LB operator and approximating spectral filters using Chebyshev, Laguerre, and Hermite polynomials.
result Classification accuracy of LB-CNN is not dependent on the type of polynomials or operators.
The classical Rankin-Cohen brackets are bi-differential operators from C∞(R)×C∞(R) into C∞(R). They are covariant for the (diagonal) action of SL(2,R) through principal series representations. We construct generalizations of these operators, replacing…
For a scalar evolution equation ut=K(t,x,u,ux,…,un),n≥2 the cohomology spaces H1,s(R∞) vanishes for s≥3 while the space H1,2(R∞) is isomorphic to the space of variational operators. The cohomology space H1,2(R∞) is also shown to be …
This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving L∞-algebra structure. result Construction of an intrinsic L∞-algebra on the Khovanov-Sano complex. In the co-sparse analysis model a set of filters is applied to a signal out of the signal class of interest yielding sparse filter responses. As such, it may serve as a prior in inverse problems, or for structural analysis of signals that are known to belong to the signal class. The more the model is adapted to the cla…
New invariants show stronger virtual knot sets.
problem Classifying virtual knots using invariants.
method Defining F-order invariants using forbidden moves. result Set of F-order invariants is strictly stronger. Isomorphism found between filtered calculus and crossed products.
problem Tackles isomorphism in filtered calculus and crossed products.
method Uses natural R-action and structure result for C*-algebra of graded nilpotent Lie groups.
result Found isomorphism between kernel of tangent groupoid and crossed product.
SC-Net learns interpretable filters for inverse problems, achieving optimal convergence and super-resolution.
problem Solving ill-posed inverse problems with effective regularization and interpretability.
method SC-Net operates in the spectral domain, learning a pointwise adaptive filter function based on signal-to-noise ratio.
result SC-Net achieves optimal convergence rate and zero-shot super-resolution, matching theoretical bounds.
We study the behavior of the spectrum of the Dirac operator together with a symmetric W1,∞-potential on spin manifolds under a collapse of codimension one with bounded sectional curvature and diameter. If there is an induced spin structure on the limit space N then there are convergent eigenvalues which co…
A method makes particle filters differentiable without altering their forward pass.
problem Compatibility issues between particle filters and automatic differentiation.
method Introduces a correction to particle weights using the stop-gradient operator.
result Automatic differentiation produces good estimators for gradients and second-order derivatives.
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
The paper characterizes vector bundles and differential operators using Lie algebras and their symbols.
problem Characterizing vector bundles and differential operators using algebraic methods.
method Lie-algebraic characterization of vector bundles and differential operators.
result The Lie algebras P(E,M) and S(P(E,M)) characterize vector bundles and their smooth sections. We establish higher-order weighted Sobolev and Holder regularity for solutions to variational equations defined by the elliptic Heston operator, a linear second-order degenerate-elliptic operator arising in mathematical finance. Furthermore, given C∞-smooth data, we prove C∞-regularity of solutions up t…
Defines Wodzicki residue using groupoids and fibered distributions.
problem Defining and understanding the Wodzicki residue in noncommutative geometry.
method Using groupoid language and filtered manifolds, defining the residue and showing its properties.
result The groupoidal residue is a trace on pseudodifferential operators and matches the usual residue in certain cases.
The paper classifies symbols of differential operators on vector bundles.
problem Classifying symbols of linear differential operators on vector bundles.
method Associated tuples of linear operators to non-degenerate symbols and used C. Procesi's results to find rational invariants and equivalence criteria.
result Generators for rational invariants and a criterion for symbol equivalence.
We prove R-bisectoriality and boundedness of the H∞-functional calculus in Lp for all 1<p<∞ for the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on k-forms.
BankGCN improves graph convolution networks by handling multi-channel signals with adaptive filter banks.
problem Handling multi-channel graph signals with limited architectures.
method BankGCN decomposes multi-channel signals into subspaces and uses adapted filters for each subspace.
result BankGCN achieves excellent performance in graph classification on benchmark datasets.
Identifies directed graphs from node measurements using polynomial filters.
problem Inferring directed network topology from nodal measurements.
method System identification of graph convolutional filter followed by topology inference.
result Effective recovery of directed graphs from measurements.
The paper analyzes MACD using operator theory.
problem Understanding the mathematical foundation of MACD.
method Developed a functional-analytic framework interpreting MACD as a phase-corrected, smoothed derivative operator.
result MACD is structurally equivalent to a band-pass filter and can be expressed as a finite difference of delayed and doubly averaged signals.
New equivariant filters improve graph classification.
problem Designing deep learning models for graph symmetries.
method Nonlinear spectral filters (NLSFs) that are equivariant to graph functional shifts.
result NLSFs outperform existing spectral GNNs in graph classification.
The paper quantizes Kähler manifolds using differential operators.
problem Quantizing classical observables on Kähler manifolds as differential operators.
method Constructing higher-order differential operators using Fedosov-type constructions and proving asymptotic equivalence to Berezin-Toeplitz operators.
result Holomorphic differential operators are precisely those that arise as Berezin-Toeplitz operators for quantizable functions.
It is well known that Convolutional Neural Networks (CNNs) have significant redundancy in their filter weights. Various methods have been proposed in the literature to compress trained CNNs. These include techniques like pruning weights, filter quantization and representing filters in terms of a basis functions. Our ap…
A new SOHP filter improves trend estimation in economic time series.
problem Improving trend estimation in nonlinear economic time series.
method Recursive application of one-sided HP filter on updated cyclical components, combined with an incremental HP filtering algorithm.
result Better performance of SOHP filter compared to other HP-type filters on real economic data.
We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern character of the (equivariant family) index, the result being given by an Atiyah-Singe…
Paper proposes a DNN-driven AF framework for improved generalization.
problem Generalization challenge in adaptive filtering.
method Structural embedding of DNN into AF system, using maximum likelihood as implicit cost function.
result Demonstrates improved generalization capability through extensive experiments.
Augments GNNs with diversification to preserve node identity.
problem Current GNNs filter node information, potentially losing node identity.
method Integrates diversification operators with aggregation to enrich node representations.
result Significant performance boost on 9 node classification tasks.
Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.
problem Quantizing the relationship between Koszul and Schouten brackets in Poisson geometry.
method Employing Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of L∞-algebroids. result Quantizes the L∞-morphism into a single linear operator, a formal Fourier integral operator. An important problem in fiber-optic communications is to invert the nonlinear Schrödinger equation in real time to reverse the deterministic effects of the channel. Interestingly, the popular split-step Fourier method (SSFM) leads to a computation graph that is reminiscent of a deep neural network. This observation all…
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
Study Morse theory on loop spaces and Hecke algebras.
problem Morse theory applied to loop spaces and Hecke algebras.
method Defined a Morse-type A∞-algebra and showed equivalence to Heegaard Floer algebras. result Equivalence of based multiloop A∞-algebra to wrapped higher-dimensional Heegaard Floer algebras.