Formal Concepts vs Eigenvectors of Density Operators

In the seventh talk of the ACT@UCR seminar, Tai-Danae Bradley told us about applications of categorical quantum mechanics to formal concept analysis.

She gave her talk on Wednesday May 13th. Afterwards we discussed 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, or download a video here, or watch the video 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.

Her talk is based on her thesis:

• Tai-Danae Bradley, At the Interface of Algebra and Statistics.


6 Responses to Formal Concepts vs Eigenvectors of Density Operators

  1. Stasheff, James says:

    Not to be missed
    Tai is amazing

  2. Bruce Smith says:

    Her lecture will be recorded and put on YouTube.

    I’m depending on this! (Since it sounds very interesting, but I can’t make it to the live version.)

    • John Baez says:

      Great, I’m glad you’ll give it a listen. I tend to put up the YouTube videos by the evening after the talk.

    • John Baez says:

      By the way, one can get a non-technical preview of her talk here:

    • Mark Meckes says:

      I just came across this and was also very happy to hear the video is posted. You might add links to videos on the seminar web site; I had checked for a video of this talk there and was disappointed not to find one. (The slides alone are great, but I couldn’t help feeling that once I got to “the main idea”, it would have been helpful to hear what she said out loud!)

      • John Baez says:

        If you want to see all the talks in one place go here; it’s somewhat more avidly curated than the official seminar website. I can tell the people in charge of that website to add links to everything.

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