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Machine Learning Model Homotopy

By John Mount on September 17, 2024 * ( 2 Comments )

For a long time I have been thinking on how can one apply the ideal
of homotopy to statistics and machine learning. The idea is: for many
modeling situations, a single model doesn't tell the story. One wants
to use trajectory of fits. Examples of this include the Lasso showing
regularization trade-offs, and also fitting a classification model
with different data prevalences. All of these individual models are
different, but they also are a continuously parameterized family.


                            HomotopySmall

                 Homotopy image: Wikimedia, Jim Belk

I have a really neat new example called "Surely That Can't Change
Sign Back and Forth". In this example I show in even a linear
regression we can expect a fit coefficient to change signs as many
times as the number of variables minus 1! This contradicts a
plausible intuition that when you interpolate continuously (by
weights) that the fit coefficients also somewhat interpolate.

B 3 path

So please, check it out.

(Note: I post these announcement notes so I can leave the original
full article in an easily maintainable form.)

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Categories: Tutorials

Tagged as: Machine Learning Probability Model Homotopy symPy

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John Mount

ChatGPT is Getting Scary Better at Math

2 replies >

 1. [5ca1d5] Tomas Petit says:
    September 22, 2024 at 6:30 am

    Have you heard about Topological Data Analysis (TDA)? I feel like
    this could be something that may interest you, given the topic
    that you're looking at here.

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     1. [92a734] jmount says:
        September 22, 2024 at 12:35 pm

        Thanks for the idea and note. I had seen TDA, it reminds me
        most of measuring compactness or fractal dimension (watching
        how the size of a covering grows as you shrink the elements)
        and "topology" in the discrete graph senes (what connects to
        what). But the parameterized dependence is in fact an
        interesting point.

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