Formal Concepts vs Eigenvectors of Density Operators
Azimuth 2020-05-07
In the seventh talk of the ACT@UCR seminar, Tai-Danae Bradley will tell us about applications of categorical quantum mechanics to formal concept analysis.
She will give her talk on Wednesday May 13th at 5 pm UTC, which is 10 am in California, or 1 pm on the east coast of the United States, or 6 pm in England. It will be held online via Zoom, here:
https://ucr.zoom.us/j/607160601
Afterwards we will discuss her talk at the Category Theory Community Server. You can see those discussions here if you become a member:
You can see her slides here.
• Tai-Danae Bradley: Formal concepts vs. eigenvectors of density operators.
Abstract. In this talk, I’ll show how any probability distribution on a product of finite sets gives rise to a pair of linear maps called density operators, whose eigenvectors capture “concepts” inherent in the original probability distribution. In some cases, the eigenvectors coincide with a simple construction from lattice theory known as a formal concept. In general, the operators recover marginal probabilities on their diagonals, and the information stored in their eigenvectors is akin to conditional probability. This is useful in an applied setting, where the eigenvectors and eigenvalues can be glued together to reconstruct joint probabilities. This naturally leads to a tensor network model of the original distribution. I’ll explain these ideas from the ground up, starting with an introduction to formal concepts. Time permitting, I’ll also share how the same ideas lead to a simple framework for modeling hierarchy in natural language. As an aside, it’s known that formal concepts arise as an enriched version of a generalization of the Isbell completion of a category. Oftentimes, the construction is motivated by drawing an analogy with elementary linear algebra. I like to think of this talk as an application of the linear algebraic side of that analogy.
