Researchers link tangle invariants for Khovanov and knot Floer homologies.
problem Relating tangle invariants for Khovanov and knot Floer homologies.
method Constructing algebraic DA bimodules for tangles and open braids, showing homotopy equivalence to Ozsvath-Szabo bimodules.
result Homotopy equivalence of DA bimodules for tangles and knot Floer homology.
We investigate a relationship between Ozsváth and Szabó's bordered theory and the algebras and bimodules constructed by Khovanov-Seidel. Specifically, we show that (a variant of) a special case of Ozsváth-Szabó's algebras has a quotient which is isomorphic to the Khovanov-Seidel quiver algebra with coefficients in $\ma…
In this paper we introduce a chain complex C1±1(D) where D is a plat braid diagram for a knot K. This complex is inspired by knot Floer homology, but it the construction is purely algebraic. It is constructed as an oriented cube of resolutions with differential d=d_0+d_1. We show that the E_2 page of the assoc…
We relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of Uq(gl(1∣1)). Specifically, we consider two subalgebras Cr(n,S) and Cl(n,S) of Ozsváth- Szabó's algebra B(n,S), an…
We survey Ozsváth-Szabó's bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts (A∞-modules, type D-structures, box tensor, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and ske…
Foams 2-equivalent to singular Soergel bimodules.
problem Establishing equivalence between foams and Soergel bimodules.
method 2-equivalence between foams and singular Soergel bimodules of type A.
result 2-category of foams 2-equivalent to singular Soergel bimodules.
New algebras and maps defined in knot Floer homology for trivalent vertices.
problem Categorification of knot Floer homology for trivalent vertices.
method Definition of new algebras, local bimodules, and bimodule maps in bordered knot Floer homology.
result Categorification of representations of U_q(gl(1|1)^-).
Paper proposes BSP to find stable bimodules of cross-correlated features.
problem Identify groups of features from two data types with strong cross-correlation.
method Iterative-testing based bimodule search procedure (BSP).
result BSP outperforms existing methods in detecting stable bimodules.
We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.
CAD-DA controls anomaly detection under domain adaptation.
problem Valid statistical inference after domain adaptation.
method Conditional Selective Inference to handle domain adaptation effects.
result Valid statistical inference under domain adaptation achieved.
Study links using Soergel bimodules and Serre duality.
problem Computing the triply-graded homology of the Whitehead link.
method Representability of partial trace functors and diagrammatics for Hochschild cohomology.
result Computed the triply-graded homology of the Whitehead link.
The paper solves a problem in constructing a bicategory of algebra bundles.
problem Defining a well-defined composition law for algebra bundles over a smooth manifold.
method Developed a complete solution for a bicategory of algebra bundles, addressing non-invertible bimodules and non-semisimple algebras.
result A complete solution to the problem of constructing a bicategory of algebra bundles.
Study categorifies link invariants using Soergel bimodules.
problem Categorification of link invariants.
method Explicit computation of derived traces of Soergel bimodules.
result Derived annular Khovanov-Rozansky link invariant.
Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.
problem Decomposing the Goldman-Turaev Lie bialgebra of a surface.
method Algebraic construction of double Lie bimodules and their combination.
result Decomposes the Goldman-Turaev Lie bialgebra along a simple separating curve.
SFS-DA method statistically tests FS reliability under domain adaptation.
problem Feature selection reliability under domain adaptation with limited target data.
method Selective Inference framework to control false positive rate and enhance true positive rate.
result SFS-DA method controls FPR below a pre-specified level α (e.g., 0.05) while maximizing true positive rate. We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule define…
Paper proposes MCC to reduce class confusion for versatile DA.
problem Class confusion in DA methods limits their performance across different scenarios.
method Introduces Minimum Class Confusion (MCC) loss function to handle various DA scenarios.
result MCC significantly improves performance on diverse DA scenarios, including Multi-Source and Multi-Target DA.
Defines new Heegaard Floer invariants with actions of both E and F.
problem Developing a new theory of Heegaard Floer invariants with specific actions.
method Introducing larger spaces and bimodule structures with actions of E and F.
result Shows the new bimodules satisfy necessary gluing properties for a 1+1 open-closed TQFT.
New DA method CIRM outperforms existing methods under structural causal model assumptions.
problem Improving prediction performance in domain adaptation with perturbed source and target data.
method Theoretical framework based on structural causal models to analyze and compare DA methods.
result CIRM method outperforms existing methods when covariates and label distributions are perturbed in target data.
We remark some basic facts on homological aspects of involutive Lie bialgebras and their involutive bimodules, and present some problems on surface topology related to these facts.
Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of bimodule over B(n). We define analogous bimodules for singular crossings…
We trade matrix factorizations and Koszul complexes for Hochschild homology of Soergel bimodules to modify the construction of triply-graded link homology and relate it to Kazhdan-Lusztig theory.
New 4-manifold invariant defined from trisection diagrams.
problem Defining a new 4-manifold invariant from trisection diagrams.
method Algebraic data from bimodule categories and spherical fusion categories, described diagrammatically.
result Includes Hopf algebraic invariants and modular fusion category invariants.
Data augmentation (DA) is commonly used during model training, as it significantly improves test error and model robustness. DA artificially expands the training set by applying random noise, rotations, crops, or even adversarial perturbations to the input data. Although DA is widely used, its capacity to provably impr…
New connections found for quantum flag manifolds modules.
problem Unique connections for relative line modules over quantum flag manifolds.
method Applied general results on quantum principal bundles to Heckenberger-Kolb calculi.
result Found bimodule connections with invertible maps.
New stable homotopy refinement of quantum annular Khovanov homology.
problem Quantum topological Hochschild homology and annular Khovanov spectra.
method Introducing quantum topological Hochschild homology (qTHH) and constructing a new stable homotopy refinement of quantum annular Khovanov homology.
result The new stable homotopy refinement agrees with qTHH of spectral Chen-Khovanov tangle bimodules and recovers earlier work.
Domain adaptation (DA) is an important and emerging field of machine learning that tackles the problem occurring when the distributions of training (source domain) and test (target domain) data are similar but different. Current theoretical results show that the efficiency of DA algorithms depends on their capacity of …
Paper proves Koszul duality for weighted A-infinity algebras.
problem Koszul duality for weighted A-infinity algebras.
method Constructs new box tensor product for weighted A-infinity bimodules and verifies correspondence between maps and bimodules.
result Proves Koszul duality result between weighted A-infinity algebras.
ADDA framework speeds up data augmentation in massive data settings.
problem Slow data augmentation in massive data settings.
method Develops asynchronous and distributed data augmentation (ADDA) framework.
result ADDA significantly speeds up data augmentation compared to parent DA algorithms.
TransCal calibrates DA models with lower bias and variance.
problem Calibrating DA models to estimate accurate predictive uncertainty.
method Transferable Calibration (TransCal) in a unified hyperparameter-free optimization framework.
result TransCal achieves more accurate calibration with lower bias and variance.
OMD and DA perform similarly in static settings but OMD is inferior under dynamic learning rates.
problem Proving and understanding the performance difference between OMD and DA under dynamic learning rates.
method Introducing stabilization to OMD and modifying its convergence analysis.
result OMD with stabilization and DA have the same performance guarantees under dynamic learning rates.
EnFF uses flows to speed up DA in high dimensions.
problem Efficiently assimilating noisy data in high-dimensional systems.
method Flow Matching (FM) for training-free, scalable data assimilation.
result EnFF accelerates DA with improved cost-accuracy tradeoffs and scalability.
For machine learning task, lacking sufficient samples mean the trained model has low confidence to approach the ground truth function. Until recently, after the generative adversarial networks (GAN) had been proposed, we see the hope of small samples data augmentation (DA) with realistic fake data, and many works valid…
GGDA simplifies DA for large models, speeding up attribution by up to 50x.
problem Computational intensity of existing DA methods limits their applicability to large-scale models.
method Generalized Group Data Attribution (GGDA) framework attributing to groups of training points.
result GGDA achieves up to 50x speedups over standard DA methods while maintaining effectiveness.
This review article surveys data augmentation MCMC algorithms.
problem Sampling from intractable probability distributions.
method Comprehensive study of DA MCMC algorithms, their convergence properties, and acceleration strategies.
result Synthesizes recent developments and provides insights for researchers.
STAND-DA improves AD in DA target domains with limited data.
problem Statistical validity of AD after DA with limited data.
method Selective Inference framework for GPU-accelerated p-value computation. result Valid p-values and controlled false positive rate. DA improves solar wind forecasts by updating model boundary conditions.
problem Improving solar wind forecasting accuracy.
method Variational Data Assimilation with solar wind model and in-situ observations.
result DA forecasts are more accurate than non-DA forecasts, especially when STEREO-B's latitude is offset from Earth.
Categorifies a skein relation for links colored by one-column Young diagrams.
problem Categorification of a skein relation for links.
method Homotopy equivalence between two complexes of singular Soergel bimodules.
result Categorification of the colored skein relation.
The paper studies geometric properties of soliton surfaces using an extended Darboux frame field.
problem Geometric analysis of soliton surfaces associated with the Betchov-Da Rios equation.
method Derivative formulas of an extended Darboux frame field, geometric invariants, curvature calculations.
result Construction of curvature ellipse and Wintgen ideal soliton surfaces.
Paper tackles multi-source domain adaptation for regression.
problem Predicting HDL cholesterol levels using gut microbiome data.
method Two-step procedure: 1) Extend a flexible single-source DA algorithm for classification to regression. 2) Augment with ensemble learning for multi-source DA.
result Consistent improvement in HDL cholesterol level prediction performance over existing methods.
Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y te…
In dialogues, an utterance is a chain of consecutive sentences produced by one speaker which ranges from a short sentence to a thousand-word post. When studying dialogues at the utterance level, it is not uncommon that an utterance would serve multiple functions. For instance, "Thank you. It works great." expresses bot…
We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soerg…
There are few papers about the consumption pattern of the Portuguese wine, using econometrics techniques. This work, pretend to analyze the consumers behavior of the wine produced in Portugal, determining the demand equation with panel data methods. There were used statistical data available in the Alentejo Regional Wi…
Defines state sum models with defects in 3-manifolds.
problem Detecting and characterizing defects in 3-manifolds.
method Turaev-Viro-Barrett-Westbury state sum models with defects labeled by bimodule categories and functors.
result State sums are triangulation-independent and can be computed using polygon diagrams.
The paper proves unique Levi-Civita connections on noncommutative forms.
problem Existence and uniqueness of Levi-Civita connections on noncommutative differential forms.
method Combining Hilbert module and algebraic techniques, proving conditions for existence and uniqueness of Hermitian torsion-free connections.
result Existence and uniqueness of Levi-Civita connections on θ-deformations of compact Riemannian manifolds.
Spectral sequence connects knot homologies via algebraic geometry.
problem Connecting algebraic and geometric knot homologies.
method Bigraded spectral sequence from gl(0)-homology to knot Floer homology.
result Constructs a Bockstein-type spectral sequence.
We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the differe…