Study on categorical bundles for gauge theories with new product concept.
problem Developing a new framework for gauge theories involving multiple gauge groups.
method Investigate product bundles in the categorical sense, construct cocycles, and introduce twisted-product bundles.
result Established a new concept of local triviality for categorical bundles.
Transforms classical connections using pushforwards and gauge transformations.
problem Transforming classical connections in categorical settings.
method Constructing pushforwards and applying gauge transformations to decorated path spaces.
result Combines traditional gauge transformation with affine translation.
For a principal bundle P→M equipped with a connection Aˉ, we study an infinite dimensional bundle PAˉdecP over the space of paths on M, with the points of PAˉdecP being horizontal paths on P decorated with elements of a second structure group. We co…
We study a type of connection forms, given by Chen integrals, over pathspaces by placing such forms within a category-theoretic framework of principal bundles and connections. We introduce a notion of 'decorated' principal bundles, develop parallel transport on such bundles, and explore specific examples in the context…
In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals.
This work defines a categorical notion of principal bundles.
problem Different definitions of principal bundles in various categories.
method Formulated in join-restriction categories, which generalize partial maps.
result Shows the tangent bundle as the product of tangent space and group object.
This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.
problem Characterizing vector bundles in smooth manifolds.
method Introducing differential bundles in a tangent category and proving equivalence with vector bundles in smooth manifolds.
result Differential bundles in smooth manifolds are equivalent to vector bundles.
The paper constructs a free abelian group from Anosov representations on bundles.
problem Constructing a free abelian group from Anosov representations on bundles.
method Extending the Γ-action via ρ to the space of connections on pullbacks of tangent bundles. result A free abelian group Fab is constructed and acts properly discontinuously on it. We define the thin fundamental categorical group P2(M,∗) of a based smooth manifold (M,∗) as the categorical group whose objects are rank-1 homotopy classes of based loops on M, and whose morphisms are rank-2 homotopy classes of homotopies between based loops on M. Here two maps are rank-n homotop…
Ehresmann connections in tangent categories
problem Generalizing Ehresmann connections to a categorical setting
method Introducing tangent categories and Ehresmann connections
result Defining and proving properties of Ehresmann connections in tangent categories
This article reviews ∞-bundles and their applications in geometry and physics.
problem Understanding higher bundles in geometry and physics.
method An ∞-categorical formulation of higher bundles. result Identification of higher bundles in various contexts.
Develops 2-Hilbert spaces for bundle gerbes in geometric quantization.
problem Higher geometric quantization of bundle gerbes.
method Introduces 2-Hilbert spaces, rig-categories, and duals for bundle gerbes.
result Constructs a 2-Hilbert space of sections for bundle gerbes.
Proves natural operations on holomorphic forms are limited.
problem Identifying natural differential operations on holomorphic forms.
method Developed theory of natural holomorphic bundles, proving a categorical equivalence.
result Only linear combinations, exterior product, and exterior differential are natural operations.
Khovanov homology extended to 3-manifolds, linking tangles.
problem Extending Khovanov homology to 3-manifolds.
method Cutting 3-manifolds into intervals, constructing type D and A structures, gluing back.
result Recovering Khovanov homology of links in 3-manifolds.
The paper explores grids and warps in triple vector bundles, proving a zero-sum property and applying it to manifold and vector bundle contexts.
problem Understanding the structure and commutativity of triple vector bundles.
method Intrinsic proof of the sum of warps being zero, applied to specific cases like tangent bundles and vector bundles.
result The sum of warps in a triple vector bundle is zero, with applications to manifold and vector bundle structures.
The main result of this paper is a new classification theorem for links (smooth embeddings in codimension 2). The classifying space is the rack space (defined in [Trunks and classifying spaces, Applied Categorical Structures, 3 (1995) 321--356]) and the classifying bundle is the first James bundle (defined in "James bu…
Quantizes Kähler manifolds using sheaves and differential operators.
problem Quantizing Kähler manifolds with sheaves and differential operators.
method Constructing a category enriched over sheaves of modules, defining quantizable morphisms, and showing equivalence to differential operator categories.
result Equivalence of quantized categories under certain conditions.
Constructs a fusion product on spinor bundle over loop space.
problem Describes a fusion product on the spinor bundle over loop space.
method Uses Connes fusion of von Neumann bimodules and string structures.
result Establishes a novel relation between string structures, loop fusion, and Connes fusion of Fock spaces.
We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. …
Unit-free approach to Jacobi geometry and Hamiltonian mechanics.
problem Lack of a preferred unit in differential geometry.
method Unit-free categorical language for line bundle geometry and Jacobi manifolds.
result Unit-free Jacobi geometry as a direct generalization of Poisson geometry.
The paper defines supermanifolds via multilinear bundles and shows their structure.
problem Defining and understanding supermanifolds in a clear, accessible way.
method Categorical approach, using multilinear bundles and projective limits.
result Supermanifolds can be seen as infinite-dimensional fiber bundles.
We show that general relativity can be viewed as a higher gauge theory involving a categorical group, or 2-group, called the teleparallel 2-group. On any semi-Riemannian manifold M, we first construct a principal 2-bundle with the Poincare 2-group as its structure 2-group. Any flat metric-preserving connection on M giv…
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
problem Developing a new homology theory for orbifolds with corners.
method Using stratification and triangulation theories of Lie groupoids and their orbit spaces, extending to Lie groupoids with corners.
result Proposes and proves the geometric homology theory (GHT), a flexible generalization of singular homology.
StructureBoost improves gradient boosting for complex categorical variables efficiently.
problem Efficiently handling complex categorical variables with known structure.
method Two methods to overcome computational obstacles in SCDT enumeration for structured categorical variables.
result StructureBoost outperforms existing packages on complex categorical problems.
Bayesian model improves categorization of explosions from sparse data.
problem Challenges in categorizing explosions from limited data.
method Bayesian update to Event Categorization Matrix model with Bayesian Decision Theory.
result Consistent gains in overall accuracy and lower false negative rates.
We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their …
A new method for backpropagating through categorical distributions.
problem Difficulty in backpropagating through categorical latent variables in neural networks.
method Introducing Gumbel-Softmax distribution for differentiable sampling.
result Gumbel-Softmax estimator outperforms existing methods on tasks with categorical latent variables.
In our previous paper (arXiv:1306.5449) we have given a sufficient and necessary condition when the coupling between Lie algebra bundle (LAB) and the tangent bundle exists in the sense of Mackenzie (\cite{Mck-2005}, Definition 7.2.2) for the theory of transitive Lie algebroids. Namely we have defined a new topology on …
UNTIE learns representations of coupled categorical data.
problem Challenges in learning from unlabeled categorical data with complex couplings.
method UNTIE approach for unsupervised representation learning of heterogeneous couplings.
result UNTIE significantly improves categorical data representations on 25 diverse datasets.
The paper studies graded manifolds and their functorial relationship.
problem Understanding the functor between two categories of graded manifolds.
method Examines polynomial filtrations and homogeneity structures, applying the Batchelor-Gawedzki theorem and Borel-Whitney theorem.
result The functor is full and surjective on objects between the categories of graded vector bundles and manifolds.
Paper introduces Categorical Normalizing Flows for better handling of categorical data.
problem Limited application of normalizing flows on categorical data due to lack of intrinsic order.
method Categorical Normalizing Flows use continuous transformations to model latent relations in categorical data, optimizing both continuous representation and model likelihood.
result GraphCNF, a permutation-invariant generative model, outperforms state-of-the-art on molecule generation.
Authors conjecture categorically diagonalizable complex for full twists.
problem Categorically diagonalizing the complex of Soergel bimodules for full twists.
method Utilizes categorical diagonalization theory.
result Proves conjecture in type A, categorifies Young idempotents.
This paper proposes a method to reduce complexity in GLMs with categorical predictors.
problem Wasteful, hard-to-interpret, and prone to overfitting of traditional one-hot encoding for high-cardinality categorical predictors.
method Clustering categories of categorical predictors through a numerical method that preserves or improves accuracy while reducing the number of coefficients.
result Clustering categories of categorical predictors reduces complexity substantially without harming accuracy.
Gaussian-Dirichlet posterior dominance proven for sequential categorical data.
problem Sequential learning from categorical observations bounded in [0,1]
method Establishing an ordering between Dirichlet and Gaussian posteriors under N(0,1) noise
result Posterior mean of categorical distribution stochastically dominates Gaussian distribution
A new method optimises problems with both continuous and categorical inputs.
problem Optimising black-box problems with mixed continuous and categorical inputs.
method Continuous and Categorical Bayesian Optimisation (CoCaBO) combining multi-armed bandits and Bayesian optimisation.
result CoCaBO outperforms existing methods on synthetic and real-world tasks.
The paper shows how integrating categorical semantics can enhance unsupervised domain translation.
problem Improving unsupervised domain translation between perceptually different domains.
method Learning invariant categorical semantic features in an unsupervised manner and conditioning them on the style encoder.
result Conditioning the style encoder on learned categorical semantics improves translation and stylization.
New algorithm recovers labels from noisy categorical data.
problem Recovering latent labels from noisy observations in structured instances.
method Approximate algorithm for graphs with categorical variables.
result Logarithmic dependency of Hamming error to the number of categories.
Method trains GANs on categorical data, outperforming existing models.
problem GANs struggle with discrete data, especially categorical values.
method Proposes architectures with multiple Gumbel softmax output layers.
result Proposed architecture outperforms existing models on various datasets.
Study categorizes mutual funds using natural language processing from unstructured data.
problem Categorizing mutual funds using unstructured data for financial analysis.
method Used natural language processing models to classify mutual funds from their investment strategy descriptions.
result High accuracy in categorizing mutual funds using NLP from unstructured data.
Paper improves anomaly detection and categorization in multi-cloud environments.
problem Differentiating among different types of attacks for better defense.
method Used supervised machine learning techniques (LR and RF) on a public dataset.
result More than 99% detection accuracy and 93.6% categorization accuracy.
Develops 2-categorical methods for multi-parameter persistence.
problem Fundamental limitations of traditional persistence modules.
method 2-categorical structures to capture hierarchical interactions.
result New invariants effectively characterize multidimensional topological features.
The paper explores efficient ways to represent categorical data.
problem Wasteful one-hot encoding of categorical variables.
method Investigates alternative, lower-dimensional real-valued representations.
result Proposed methods retain all predictive information without one-hot encoding.
CADM proposes a cluster-specific distance metric for categorical data clustering.
problem Inadequate distance metrics for categorical data, especially varying within clusters.
method Cluster-customized adaptive distance metric for categorical data.
result Achieved competitive performance in categorical data clustering.
Develops a new method for decision trees using categorical variable structure.
problem Lack of structure in treating categorical variables as predictors.
method Introduces a mathematical framework to represent categorical structure and generalizes decision trees to utilize this structure.
result Improves prediction accuracy on weather data using the new method.
Categorical d-separation criterion simplifies probability graph analysis.
problem Detecting causal relationships in probability distributions.
method Introducing categorical definitions for causal models and d-separation.
result Abstract version of d-separation criterion applies to various probability theories.
A technique for clustering categorical data using ensembled dissimilarity matrices.
problem Clustering categorical data efficiently and accurately.
method Generate many dissimilarity matrices, average them, and extend to high dimensions using alignment techniques.
result Our method provides better clustering results, especially for genome sequences.
nTreeClus clusters categorical sequences using tree-based learners and k-mers.
problem Challenges in clustering categorical and sequential data.
method nTreeClus uses Tree-based Learners, k-mers, and autoregressive models for categorical time series.
result nTreeClus outperformed baseline methods in various validation metrics.
Study extends cognitive modeling to natural images, revealing the importance of image representation.
problem Extending cognitive modeling to natural images and understanding human categorization.
method Conducted a large-scale study with over 500,000 human judgments. Used deep and shallow machine learning methods to represent images. Applied psychological models of categorization to natural images.
result Simple models with abstract prototypes outperform complex exemplar accounts when using expressive, data-driven image representations.