CEDA analyzes large categorical datasets using tree geometry and binary codes.
problem Analyzing large categorical datasets with extreme-K samples. method CEDA uses tree geometry and binary codes to analyze categorical data.
result CEDA discovers patterns and evaluates their reliability in large categorical datasets.
Extends linear representation hypothesis to categorical and hierarchical concepts in LLMs.
problem Representing concepts without natural contrasts in large language models.
method Formalizes linear representation hypothesis for categorical and hierarchical concepts, proving relationships between concept hierarchy and representation geometry.
result Validated theoretical results on large language models, estimating representations for 900+ concepts.
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
We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choo…
A new method for generative modeling of discrete data using geometric latent subspaces.
problem Learning generative models for discrete data with statistical dependencies.
method Geometric latent-subspace framework in exponential parameter space of product manifolds of categorical distributions.
result Low-dimensional latent space encodes statistical dependencies and accurately models high-dimensional discrete data.
Survey on categorifying Jones polynomial.
problem Categorification of Jones polynomial.
method Not explicitly stated, likely involves algebraic and geometric categorification techniques.
result Significance and ramifications in geometry, algebra, and topology.
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…
Develops a new theory of localization in algebraic geometry.
problem Localization of cohomological theories on closed subsets.
method Categorical and algebro-geometric approach, focusing on torsors and translation groupoids.
result Found that localization often results in a torsor of supported refinements rather than a localized class.
We introduce the notions of categorical systoles and categorical volumes of Bridgeland stability conditions on triangulated categories. We prove that for any projective K3 surface, there exists a constant C depending only on the rank and discriminant of its Picard group, such that $$\mathrm{sys}(σ)^2\leq C\cdot\mathrm{…
In this article, we give a survey of Geometric Invariant Theory for Toric Varieties, and present an application to the Einstein-Weyl Geometry. We compute the image of the Minitwistor space of the Honda metrics as a categorical quotient according to the most efficient linearization. The result is the complex weighted pr…
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.
cGAP visualizes high-dimensional categorical data with interpretable geometric structure.
problem Lack of visualization tools for high-dimensional categorical data.
method cGAP uses Homogeneity Analysis (HOMALS) to embed data in a 3D space and maps it to colors.
result cGAP reveals coherent clusters, outliers, and local-to-global structure in categorical data.
cGAP visualizes high-dimensional categorical data with interpretable geometric structure.
problem Lack of visualization tools for high-dimensional categorical data.
method cGAP uses Homogeneity Analysis (HOMALS) to embed data in a 3D space and maps it to colors for visualization.
result cGAP reveals coherent clusters, outliers, and local-to-global structure in categorical data.
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…
SFM matches flows on statistical manifolds for better discrete generation.
problem Discrete generation on statistical manifolds with strong prior assumptions.
method Statistical Flow Matching (SFM) on manifold of categorical distributions using Fisher information metric.
result SFM achieves higher sampling quality and likelihood than other models.
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.
Develops a new theory of localization in algebraic geometry.
problem Understanding localizations in cohomological theories with open-closed structures.
method Categorical and algebro-geometric approach, focusing on torsors and refinements.
result Establishes compatibility with various algebraic operations and recovers classical results.
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.
New framework uses cohomology to analyze probabilistic distortions and arbitrage.
problem Analyzing probabilistic distortions and arbitrage in categorical filtrations.
method Transport cohomological framework, simplicial structure, loop effects, holonomy.
result Nontrivial probabilistic distortions and obstructions generated by loops.
Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categor…
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.
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.
When a complex semisimple group G acts holomorphically on a Kähler manifold (X,ω) such that a maximal compact subgroup K⊂G preserves the symplectic form ω, a basic result of symplectic geometry says that the corresponding categorical quotient X/G can be identified with quotient of the zero-set of the m…
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.
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.
In this article we prove that stratified spaces and other geometric subfamilies satisfy categorical Fraïssé properties, a matter that might be of interest for both geometers and logicians. As a motivation we show a new example of a stratified pseudomanifold that satisfies the finite oscillation property with respect to…
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.
A central extension of the loop group of a Lie group is called transgressive, if it corresponds under transgression to a degree four class in the cohomology of the classifying space of the Lie group. Transgressive loop group extensions are those that can be explored by finite-dimensional, higher-categorical geometry ov…
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.
Unified framework for observables in n-plectic geometry.
problem Quantization of extended objects in higher geometric contexts.
method Develops a semi-simplicial set model for observables, using a Grassmann variable to encode submanifold codimensions.
result Establishes a categorified pre-n-Hilbert space and a quantization scheme matching multisymplectic geometry.
Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sa…
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.
A new model integrates covariates with grade of membership analysis for better latent structure recovery.
problem Improving latent structure recovery in multivariate categorical data analysis.
method Covariate-assisted grade of membership model exploiting shared low-rank simplex geometry.
result Auxiliary covariates can provably improve latent structure recovery, leading to faster convergence rates.
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.
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.
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.
We show that the function sheaf of a Z2n-manifold is a nuclear Fréchet sheaf of Z2n-graded Z2n-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a Z2n-morphism are all continuous. These results are essenti…
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.
This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
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.
A new gradient estimator for categorical distributions reduces bias and variance.
problem Intractability of gradients for categorical distributions in discrete latent variable models.
method CatLog-Derivative trick and IndeCateR gradient estimator.
result IndeCateR reduces bias and variance of gradients for categorical distributions.
Bayesian optimisation tackles high-dimensional categorical and mixed search spaces.
problem Bayesian optimisation on high-dimensional categorical and mixed search spaces is challenging.
method Combining local optimisation with a tailored kernel design.
result Empirically outperforms current baselines in performance and computational costs.
Human categorization is one of the most important and successful targets of cognitive modeling in psychology, yet decades of development and assessment of competing models have been contingent on small sets of simple, artificial experimental stimuli. Here we extend this modeling paradigm to the domain of natural images…
Efficient optimisation of black-box problems that comprise both continuous and categorical inputs is important, yet poses significant challenges. We propose a new approach, Continuous and Categorical Bayesian Optimisation (CoCaBO), which combines the strengths of multi-armed bandits and Bayesian optimisation to select …
The article compares predictor importance in classification problems with categorical outcomes.
problem Comparing predictor importance in classification problems with categorical response variables.
method The approach is based on the categorical Gini correlation (CGC) and tests differences in CGCs across predictor groups.
result The proposed methodology accommodates predictors of arbitrary and unequal dimensions and allows for dependence between predictor groups.
In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic logic of SDG enforces the existence of infinitesimal objects which essentially modi…
Paper introduces new methods for modeling categorical data.
problem Training generative models on categorical data like text and segmentation.
method Argmax Flows and Multinomial Diffusion models.
result Models outperform existing methods in log-likelihood.