James Fairbanks

James P. Fairbanks, Geoffrey D. Sanders, David A. Bader

Many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy, which leads to the first meaningful stopping criterion for two way spectral partitioning. More generally, we provide pointwise convergence guarantees so that blends (linear combinations) of eigenvectors can be employed to solve data analysis problems with confidence in their accuracy. We demonstrate this theory on an accessible model problem, the Ring of Cliques, by deriving the relevant eigenpairs and comparing the predicted results to numerical solutions. These results bridge the gap between linear algebra based data analysis methods and the convergence theory of iterative approximation methods.

Mason Lary, Richard Samuelson, Alexander Wilentz et al.

Motivated by deep learning regimes with multiple interacting yet distinct model components, we introduce learning diagrams, graphical depictions of training setups that capture parameterized learning as data rather than code. A learning diagram compiles to a unique loss function on which component models are trained. The result of training on this loss is a collection of models whose predictions ``agree" with one another. We show that a number of popular learning setups such as few-shot multi-task learning, knowledge distillation, and multi-modal learning can be depicted as learning diagrams. We further implement learning diagrams in a library that allows users to build diagrams of PyTorch and Flux.jl models. By implementing some classic machine learning use cases, we demonstrate how learning diagrams allow practitioners to build complicated models as compositions of smaller components, identify relationships between workflows, and manipulate models during or after training. Leveraging a category theoretic framework, we introduce a rigorous semantics for learning diagrams that puts such operations on a firm mathematical foundation.

Tyler Hanks, James Fairbanks, Matthew Klawonn

Cartesian reverse derivative categories (CRDCs) provide an axiomatic generalization of the reverse derivative, which allows generalized analogues of classic optimization algorithms such as gradient descent to be applied to a broad class of problems. In this paper, we show that generalized gradient descent with respect to a given CRDC induces a hypergraph functor from a hypergraph category of optimization problems to a hypergraph category of dynamical systems. The domain of this functor consists of objective functions that are 1) general in the sense that they are defined with respect to an arbitrary CRDC, and 2) open in that they are decorated spans that can be composed with other such objective functions via variable sharing. The codomain is specified analogously as a category of general and open dynamical systems for the underlying CRDC. We describe how the hypergraph functor induces a distributed optimization algorithm for arbitrary composite problems specified in the domain. To illustrate the kinds of problems our framework can model, we show that parameter sharing models in multitask learning, a prevalent machine learning paradigm, yield a composite optimization problem for a given choice of CRDC. We then apply the gradient descent functor to this composite problem and describe the resulting distributed gradient descent algorithm for training parameter sharing models.

Angeline Aguinaldo, Evan Patterson, James Fairbanks et al.

Classical planning representation languages based on first-order logic have preliminarily been used to model and solve robotic task planning problems. Wider adoption of these representation languages, however, is hindered by the limitations present when managing implicit world changes with concise action models. To address this problem, we propose an alternative approach to representing and managing updates to world states during planning. Based on the category-theoretic concepts of $\mathsf{C}$-sets and double-pushout rewriting (DPO), our proposed representation can effectively handle structured knowledge about world states that support domain abstractions at all levels. It formalizes the semantics of predicates according to a user-provided ontology and preserves the semantics when transitioning between world states. This method provides a formal semantics for using knowledge graphs and relational databases to model world states and updates in planning. In this paper, we conceptually compare our category-theoretic representation with the classical planning representation. We show that our proposed representation has advantages over the classical representation in terms of handling implicit preconditions and effects, and provides a more structured framework in which to model and solve planning problems.

Kun Cao, James Fairbanks

The scientific literature is a rich source of information for data mining with conceptual knowledge graphs; the open science movement has enriched this literature with complementary source code that implements scientific models. To exploit this new resource, we construct a knowledge graph using unsupervised learning methods to identify conceptual entities. We associate source code entities to these natural language concepts using word embedding and clustering techniques. Practical naming conventions for methods and functions tend to reflect the concept(s) they implement. We take advantage of this specificity by presenting a novel process for joint clustering text concepts that combines word-embeddings, nonlinear dimensionality reduction, and clustering techniques to assist in understanding, organizing, and comparing software in the open science ecosystem. With our pipeline, we aim to assist scientists in building on existing models in their discipline when making novel models for new phenomena. By combining source code and conceptual information, our knowledge graph enhances corpus-wide understanding of scientific literature.

Micah Halter, Christine Herlihy, James Fairbanks

Scientists construct and analyze computational models to understand the world. That understanding comes from efforts to augment, combine, and compare models of related phenomena. We propose SemanticModels.jl, a system that leverages techniques from static and dynamic program analysis to process executable versions of scientific models to perform such metamodeling tasks. By framing these metamodeling tasks as metaprogramming problems, SemanticModels.jl enables writing programs that generate and expand models. To this end, we present a category theory-based framework for defining metamodeling tasks, and extracting semantic information from model implementations, and show how this framework can be used to enhance scientific workflows in a working case study.