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In designing generative models, it is commonly believed that in order to learn useful latent structure, we face a fundamental tension between expressivity and structure. In this paper we challenge this view by proposing a new approach to…

We introduce a new algebra called the shifted $q=0$ affine algebra, which arises naturally from the study of coherent sheaves on Grassmannians and n-step partial flag varieties via a natural correspondence. It has similar presentation as…

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a…

We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $\Sigma_\beta$ hierarchy. We focus on linear orderings. We show that at the $\Sigma_1$ level all linear…

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of…

Examples of SL(2, Z) actions on differential graded categories are defined and explored.

We study the circumstances under which one can reconstruct a stack from its associated functor of isomorphism classes. This is possible surprisingly often: we show that many of the standard examples of moduli stacks are determined by their…

This paper describes a computational framework for a grammar architecture in which different linguistic domains such as morphology, syntax, and semantics are treated not as separate components but compositional domains. Word and phrase…

Identifying latent representations or causal structures is important for good generalization and downstream task performance. However, both fields have been developed rather independently. We observe that several methods in both…

We examine the use of classes to formulate several categorical notions. This leads to two proposals: an explicit structure for working with subobjects, and a hierarchy of $k$-classes. We apply the latter to both ordinary and higher…

Gaussian Processes (GPs) provide a powerful framework for making predictions and understanding uncertainty for classification with kernels and Bayesian non-parametric learning. Building such models typically requires strong prior knowledge…

Basic principles of set theory have been applied in the context of probability and binary computation. Applying the same principles on inequalities is less common but can be extremely beneficial in a variety of fields. This paper formulates…

Many recent machine learning approaches to floorplanning represent placement decisions using discrete canvas coordinates, which creates scalability bottlenecks as the action space grows. In this work, we study the effect of learning a…

We provide a new perspective on the problem how high-level state machine models with abstract actions can be related to low-level models in which these actions are refined by sequences of concrete actions. We describe the connection between…

Category theory provides a compact method of encoding mathematical structures in a uniform way, thereby enabling the use of general theorems on, for example, equivalence and universal constructions. In this article we develop the method of…

We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out…

Generative pretraining (the "GPT" in ChatGPT) enables language models to learn from vast amounts of internet text without human supervision. This approach has driven breakthroughs across AI by allowing deep neural networks to learn from…

In arXiv:0805.0157v5, the authors define a class of derived stacks, called "perfect stacks" and show that for this class the categories of quasi-coherent sheaves satisfy a categorical K\"unneth formula. Motivated to extend their results to…

Symbolic planning is a powerful technique to solve complex tasks that require long sequences of actions and can equip an intelligent agent with complex behavior. The downside of this approach is the necessity for suitable symbolic…

We construct a cofibrantly generated Quillen model structure on the category of small differential graded categories. ----- Nous construisons une structure de categorie de modeles de Quillen a engendrement cofibrant sur la categorie des…