Jared Culbertson

Jared Culbertson
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Jared Culbertson
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Mathematics - Category Theory (2)
 
Computer Science - Learning (2)
 
Statistics - Machine Learning (2)
 
Mathematics - Probability (1)
 
Mathematics - Metric Geometry (1)

Publications Authored By Jared Culbertson

This work draws its inspiration from three important sources of research on dissimilarity-based clustering and intertwines those three threads into a consistent principled functorial theory of clustering. Those three are the overlapping clustering of Jardine and Sibson, the functorial approach of Carlsson and Memoli to partition-based clustering, and the Isbell/Dress school's study of injective envelopes. Carlsson and Memoli introduce the idea of viewing clustering methods as functors from a category of metric spaces to a category of clusters, with functoriality subsuming many desirable properties. Read More

We examine overlapping clustering schemes with functorial constraints, in the spirit of Carlsson--Memoli. This avoids issues arising from the chaining required by partition-based methods. Our principal result shows that any clustering functor is naturally constrained to refine single-linkage clusters and be refined by maximal-linkage clusters. Read More

In his pioneering work on injective metric spaces Isbell attempted a characterization of cellular complexes admitting the structure of an injective metric space, following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure. Considerable advances in the understanding, classification and applications of injective envelopes have been made by Dress, Huber, Sturmfels and collaborators (producing, among other results, many specific examples of injective polyhedra), and most recently by Lang, yet a combination theory explaining how to glue injective polyhedra together to produce large families of injective spaces is still unavailable. In this paper we apply the duality theory of cubings -- simply connected non-positively curved cubical complexes -- to provide a more principled and accessible proof of a result of Mai and Tang on the injective metrizability of collapsible simplicial complexes. Read More

From the Bayesian perspective, the category of conditional probabilities (a variant of the Kleisli category of the Giry monad, whose objects are measurable spaces and arrows are Markov kernels) gives a nice framework for conceptualization and analysis of many aspects of machine learning. Using categorical methods, we construct models for parametric and nonparametric Bayesian reasoning on function spaces, thus providing a basis for the supervised learning problem. In particular, stochastic processes are arrows to these function spaces which serve as prior probabilities. Read More

Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow H$ which is unique up to a set of measure zero. Thus, given a data measurement $\mu: 1 \rightarrow D$, a posterior probability $\widehat{P_H}= I \circ \mu$ can be computed. This procedure is iterative: with each updated probability $P_H$, we obtain a new joint distribution which in turn yields a new inference map $I$ and the process repeats with each additional measurement. Read More