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Hypocoercivity in finite dimension (2/2)

math

This is the continuation of my previous blog post about hypocoercivity. If you haven’t already, I would recommend starting with the first part. Recall that:

Hypocoercivity in finite dimension (1/2)

math

For a linear dynamical system \(\dot{x}_t = -M x_t\), coercivity is when \(\frac{d}{dt} \|x_t\| \leq -\alpha \|x_t\|\) for some \(\alpha>0\), and hypocoercivity is when \(\|x_t\| = O(e^{-\alpha t})\) even though there’s no coercivity. Coercivity is relatively simple to understand, but hypocoercivity can be pretty weird, even in finite dimension. In optimization and machine learning, coercivity covers gradient descent-type situations, and hypocoercivity arises when studying min-max or accelerated gradient algorithms.