arXiv research

We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We prese…

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomolog…

We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The exi…

Homotopy cardinality counts augmentations of Legendrian knots.

The paper constructs Yang-Baxter solutions using categorical augmented racks.

In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two categorical Legendrian isotopy invariants: the augmentation category, a unital $A_{\infty}$-category, which lifts the set of augmentations of the associated Chekanov-Eliashberg DGA, and a DG category of construct…

In this paper we construct an $\mathcal{A}_\infty$-category associated to a Legendrian submanifold of jet spaces. Objects of the category are augmentations of the Chekanov algebra $\mathcal{A}(Λ)$ and the homology of the morphism spaces forms a new set of invariants of Legendrian submanifolds called the bilinearised Le…

Novel categories are commonly defined as those unobserved during training but present during testing. However, partially labelled training datasets can contain unlabelled training samples that belong to novel categories, meaning these can be present in training and testing. This research is the first to generalise betw…

The abstract introduces a new $A_\infty$ duality via LSFT algebra.

We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a …

Robotic clothing manipulation improved with fashion image analysis techniques.

Functor connects sheaf categories of Legendrian submanifolds.

To model categorical response variables given their covariates, we propose a permuted and augmented stick-breaking (paSB) construction that one-to-one maps the observed categories to randomly permuted latent sticks. This new construction transforms multinomial regression into regression analysis of stick-specific binar…

We describe stable cup-i products on the cochain complex with $F^2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of a…

Enhances graph classification models on small datasets.

This study improves cryptocurrency price forecasting using time series categorization and deep learning.

GeoECG augments ECG data to improve heart disease detection.

We provide an explicit example of a non trivial Legendrian knot $Λ$ such that there exists a Lagrangian concordance from $Λ_0$ to $Λ$ where $Λ_0$ is the trivial Legendrian knot. We then use the map induced in Legendrian contact homology by a concordance and the augmentation category of $Λ$ to show that no Lagrangian co…

Top-down attention allows people to focus on task-relevant visual information. Is the resulting perceptual boost task-dependent in naturalistic settings? We aim to answer this with a large-scale computational experiment. First, we design a collection of visual tasks, each consisting of classifying images from a chosen …

Controller-Augmented Hidden Markov Models (CHMMs) are a framework for constrained sequential inference.

Data availability is a bottleneck during early stages of development of new capabilities for intelligent artificial agents. We investigate the use of text generation techniques to augment the training data of a popular commercial artificial agent across categories of functionality, with the goal of faster development o…

A number of recent approaches to policy learning in 2D game domains have been successful going directly from raw input images to actions. However when employed in complex 3D environments, they typically suffer from challenges related to partial observability, combinatorial exploration spaces, path planning, and a scarc…

OpenViewer tackles multi-view learning challenges with interpretability and generalization.

Proposes robust model through Wasserstein geodesic interpolation of training data.

The paper studies posets from decompositions in symmetric monoidal categories.

G-SimCLR improves unsupervised learning by clustering images into pseudo labels.

Survey of integrating domain knowledge into DL models.

With a sharp rise in fluency and users of "Hinglish" in linguistically diverse country, India, it has increasingly become important to analyze social content written in this language in platforms such as Twitter, Reddit, Facebook. This project focuses on using deep learning techniques to tackle a classification problem…

GANs improve anomaly detection in power plants, achieving nearly perfect classification.

The paper develops spectral networks in symplectic topology and their relation to Lagrangian fillings.

Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various …

Enhances FAVAR models with autoencoder for better economic forecasting and interpretability.

Recent studies have shown promising results in using Deep Learning to detect malignancy in whole slide imaging. However, they were limited to just predicting positive or negative finding for a specific neoplasm. We attempted to use Deep Learning with a convolutional neural network algorithm to build a lymphoma diagnost…

Annotation guidelines used to guide the annotation of training and evaluation datasets can have a considerable impact on the quality of machine learning models. In this study, we explore the effects of annotation guidelines on the quality of app feature extraction models. As a main result, we propose several changes to…

We introduce a generative model to simulate radiation patterns within a jet using the Lund jet plane. We show that using an appropriate neural network architecture with a stochastic generation of images, it is possible to construct a generative model which retrieves the underlying two-dimensional distribution to within…

Temporal information impacts only a fraction of time series datasets, skewing benchmark evaluations.

New operations match Steenrod squares on Khovanov homology.

Semi-supervised learning is a model training method that uses both labeled and unlabeled data. This paper proposes a fully Bayes semi-supervised learning algorithm that can be applied to any multi-category classification problem. We assume the labels are missing at random when using unlabeled data in a semi-supervised …

Given two distinct datasets, an important question is if they have arisen from the the same data generating function or alternatively how their data generating functions diverge from one another. In this paper, we introduce an approach for measuring the distance between two datasets with high dimensionality using varia…

Studies amenable category's monotonicity and its relation to topological complexity.

To each oriented surface S, we associate a differential graded category Ko(S). The homotopy category Ho(Ko(S)) is a triangulated category which satisfies properties akin to those of the contact categories studied by K. Honda. These categories are also related to the algebraic contact categories of Y. Tian and to the bo…

Formulates a new connection between topological and geometric categories.

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free c…

We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.

The paper defines new TQFTs from non-semisimple categories and proves spherical categories are chromatic.

Spider category comparison proves equivalence to Sikora's quotient category.

We generalize the notion of monetary value measures developed with category theory in [Adachi, 2014] by extending their base category from the category \c{hi} to the category of probability spaces Prob introduced in [Adachi and Ryu, 2016].

ETQFTs created from non-semisimple modular categories.