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.
We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show th…
Alternative definition of cobordism map in ECH theory.
problem Defining and proving equivalence of cobordism map in ECH theory.
method Reformulating and proving independence of filtered ECH cobordism map using Seiberg-Witten theory.
result The new definition of cobordism map is equivalent to existing definitions.
Generalizes graded bundles with more flexible transformation laws.
problem No specific problem stated; focuses on generalization of graded bundles.
method Introduces filtered bundles with more general polynomial transformation laws.
result Linearisation of filtered bundles is well-defined.
We find a splitting in a special cohomology theory for complex manifolds.
problem Finding a splitting in a specific cohomology theory.
method Construct Hodge filtered function spaces and show they satisfy an unstable splitting.
result Obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology.
Adaptive Heston model calibration using PCRLB and switching filters.
problem Estimating volatility in stochastic volatility models like Heston.
method Bayesian filtering (EKF, UKF, PF) with PCRLB for parameter estimation.
result Adaptive estimation of Heston model parameters improves volatility estimation.
We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E_2-pages can be described in terms of groups arising from the act…
New invariants help study satellite knots and their concordance.
problem Understanding the concordance of satellite knots.
method Filtered instanton homology and classical knot theory techniques.
result Criteria for satellite knots to be linearly independent.
Developed a framework for designing filters in spectral GCNNs with improved performance.
problem Designing effective filters for spectral GCNNs with regularization properties.
method Exploring regularization properties of graph Laplacian and proposing a generalized framework for filter design.
result New filters derived from the framework outperform state-of-the-art techniques in semi-supervised node classification.
Develops inverse EKF for non-linear systems with stability guarantees and learning unknown dynamics.
problem Estimating adversary's Kalman-filtered estimates in highly non-linear systems.
method Proposes inverse extended Kalman filter (I-EKF) for second-order, Gaussian sum, and dithered forward models. Uses reproducing kernel Hilbert space for learning unknown dynamics.
result Derives theoretical stability guarantees for inverse second-order EKF.
We bound the symmetry algebra of a vector distribution, possibly equipped with an additional structure, by the corresponding Tanaka algebra. The main tool is the theory of weighted jets.
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.
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.
Paper learns to rotate filters for group convolutions.
problem Difficult to rotate 3x3 filters on pixel grids.
method Learn filter basis and rotation-invariant coefficients; switch basis for rotation.
result Produces feature maps insensitive to input rotations.
Novel method uses Bayesian filters and PCRLB for state estimation of option prices.
problem Estimating unobserved latent variables from option prices.
method Posterior Cramer-Rao Lower Bound (PCRLB) based adaptive state estimation using various Bayesian filters.
result Proposed method outperforms individual filters and improves forecasting.
New method filters large networks from financial data to reveal key subnetworks.
problem Filtering large dimensional networks to isolate key constituents.
method Exploits spectral properties of high-dimensional data networks, tuning for sparsity and consistency.
result Shows method can interpolate between zero and maximal filtering, preserving spectral properties.
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.
We introduce a family of adaptive estimators on graphs, based on penalizing the ℓ1 norm of discrete graph differences. This generalizes the idea of trend filtering [Kim et al. (2009), Tibshirani (2014)], used for univariate nonparametric regression, to graphs. Analogous to the univariate case, graph trend filteri…
This work preserves linear invariants in ensemble filters for non-Gaussian data assimilation.
problem Maintaining critical invariants like mass, stoichiometric balance, and charge in non-Gaussian data assimilation.
method Introducing a novel class of nonlinear ensemble filters using measure transport theory.
result Recovery of a constrained Kalman filter for Gaussian settings and combination with regularization techniques.
This work ranks CNN filters based on their importance.
problem Unclear role of CNN neurons in producing output.
method Two methods: Shapley value game theory and Importance switch variational inference.
result Filters with higher importance are more crucial for output.
This paper proves long-time accuracy of ensemble Kalman filters for chaotic and machine-learned systems.
problem Ensuring long-term accuracy of ensemble Kalman filters for complex dynamical systems.
method Established conditions for long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems.
result Ensemble Kalman filters maintain small estimation error over long time horizons for chaotic and machine-learned systems.
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.
Transformers can approximate Kalman Filtering in linear systems with small error.
problem Approximating Kalman Filtering using Transformers for linear dynamical systems.
method Two-step reduction: 1) Softmax self-attention block approximates Nadaraya-Watson kernel smoothing, 2) This estimator approximates Kalman Filter.
result Constructs a Transformer that implements the Kalman Filter with small additive error, uniformly bounded in time.
Deep learning explained through spectral filtering of hierarchical features.
problem Understanding how deep neural networks learn useful representations from data.
method Neural Low-Degree Filtering (Neural LoFi) as a stylized limit of gradient-based training.
result Predicts how representations are selected layer by layer and explains emergence of concepts.
The problem of filtering information from large correlation matrices is of great importance in many applications. We have recently proposed the use of the Kullback-Leibler distance to measure the performance of filtering algorithms in recovering the underlying correlation matrix when the variables are described by a mu…
We use GANs and signatures to approximate conditional laws in filtering and prediction of diffusion processes.
problem Approximating conditional laws for diffusion processes with noisy observations.
method Conditional GANs combined with signatures for approximation.
result Efficient approximation of conditional laws for diffusion processes.
Following the approach of standard filtering theory, we analyse investor-valuation of firms, when these are modelled as geometric-Brownian state processes that are privately and partially observed, at random (Poisson) times, by agents. Tasked with disclosing forecast values, agents are able purposefully to withhold the…
The paper learns an autoregressive filter for unknown dynamical systems with robust guarantees.
problem Learning optimal predictions in an unknown dynamical system.
method Directly learns an autoregressive filter using an L∞-based objective, regressing on both inputs and outputs. result The algorithm has optimal sample complexity in terms of the rollout length.
Three situations in which filtering theory is used in mathematical finance are illustrated at different levels of detail. The three problems originate from the following different works: 1) On estimating the stochastic volatility model from observed bilateral exchange rate news, by R. Mahieu, and P. Schotman; 2) A stat…
The paper introduces a quantum state system to count perfect matchings in graphs.
problem Counting perfect matchings in graphs using quantum state systems.
method Topological quantum field theory (TQFT) and spectral sequences.
result The filtered n-color vertex homology for n=2 is generated by perfect matchings. Develops inverse extended Kalman filter for predicting adversarial steps.
problem Predicting adversarial Kalman filter estimates from limited information.
method Proposes inverse extended Kalman filter (I-EKF) for non-linear systems with unknown inputs.
result Derives I-EKF with theoretical stability guarantees and consistency proofs.
We introduce filtered cohomologies of differential forms on symplectic manifolds. They generalize and include the cohomologies discussed in Paper I and II as a subset. The filtered cohomologies are finite-dimensional and can be associated with differential elliptic complexes. Algebraically, we show that the filtered co…
Paper revisits Kalman filter, connects it to HMM, and applies it to financial markets.
problem Understanding and applying Kalman filter in financial markets.
method Intuition, graphical models, Hidden Markov Models, CMA-ES optimization.
result New algorithms and parameter estimation methods for Kalman filters.
OBF optimally filters features under independent Gaussian models.
problem Biomarker discovery from complex data.
method Optimal Bayesian feature selection under independent Gaussian models.
result OBF is consistent and optimal under mild conditions.
Discrete knot theory models use lattice-filtered graphs to detect merging knot components.
problem Detecting merging knot components in discrete models.
method Lattice-filtered move graphs to model knot types, identifying connected components and merge scales.
result Merge scale defined by connected components of lattice-filtered move graphs, with specific examples for the figure-eight knot.
The paper analyzes how gradient descent learns convolutional filters for non-Gaussian inputs.
problem Learning convolutional filters with ReLU for non-Gaussian input distributions.
method Analysis of gradient descent convergence for ReLU activation with polynomial time complexity.
result Gradient descent can learn convolutional filters in polynomial time, with convergence rate dependent on input distribution smoothness and patch similarity.
Computes knot filtered ECH for torus knots on tight 3-sphere.
problem Computing knot filtered ECH for torus knots.
method Generalized knot filtered ECH definition, Morse-Bott methods, energy filtered Seiberg-Witten theory.
result Computed knot filtered ECH for T(2,q) knots (q odd, positive).
The paper presents a machine learning approach to multidimensional item response theory.
problem Modeling and predicting student performance from assessment data.
method Inspired by collaborative filtering, the paper defines a general class of models using penalized joint maximum likelihood (JML) for estimation and cross-validation for model selection.
result The high-dimensional model fit to large and sparse data does not lend itself well to traditional factor interpretation.
Improved price forecasting using Kalman filters reduces trading hiccups.
problem Sudden price whipsaws causing trading losses.
method Kalman filter theory and pseudo code implementation.
result Kalman filter models outperform traditional moving averages in price forecasting.
The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.
problem Understanding knot types in bounded ropelength sublevel spaces.
method Study thick representatives in bounded ropelength sublevel spaces through lifted Reidemeister graphs.
result Define characteristic Reidemeister patterns and finite recognition length.
In this paper we use continuous family of multisections of the moduli space of pseudo holomorphic discs to partially improve, in the case of real coefficient, the construction of Lagrangian Floer cohomology of which the author developed jointly with Oh-Ohta-Ono. Namely we associate cyclically symmetric filtered A infin…
Paper uses optimal transport for Bayesian filtering, deriving new EnKF and FPF formulations.
problem Bayesian filtering for nonlinear systems with non-Gaussian observations.
method Optimal transport theory applied to Bayes' law, constructing Brenier maps.
result New variational formulations of EnKF and FPF for non-Gaussian settings.
Boosted HP filter improves trend estimation in economic data.
problem Inadequate HP filter settings for trend estimation in economic data.
method Iterating the HP smoother to improve trend estimation and eliminate trends.
result Asymptotic recovery of trend mechanisms involving unit root processes and polynomial drifts.
We introduce a covariance matrix estimator that both takes into account the heteroskedasticity of financial returns (by using an exponentially weighted moving average) and reduces the effective dimensionality of the estimation (and hence measurement noise) via techniques borrowed from random matrix theory. We calculate…
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous…
New graph metafeatures improve algorithm selection in collaborative filtering.
problem Selecting the best algorithm for collaborative filtering problems.
method Proposes new graph metafeatures and multicriteria metatargets to evaluate algorithms.
result Graph metafeatures are a good alternative to existing metafeatures.
Study of BGG sequences on foliated manifolds with transverse parabolic geometry.
problem Analysis of BGG sequences on foliated manifolds with transverse parabolic structures.
method Filtered calculus and transversal index theory for filtered manifolds.
result Derived curved BGG sequences for foliated manifolds with transverse parabolic geometry.
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.