| Abstract: |
| In the past decade, there has been a significant shift in the types of mathematical objects under investigation, moving from vectors and matrices in the Euclidean spaces to functions residing in Hilbert spaces, and ultimately extending to probability measures within the probability measure space. Many questions that were originally posed in the context of linear function spaces are now being revisited in the realm of probability measures. One such question is to the efficiently find a probability measure that minimizes a given objective functional.
In Euclidean space, we devised optimization techniques such as gradient descent and introduced momentum-based methods to accelerate its convergence. Now, the question arises: Can we employ analogous strategies to expedite convergence within the probability measure space?
In this presentation, we provide an affirmative answer to this question. Specifically, we present a series of momentum-inspired acceleration method under the framework of Hamiltonian flow, and we prove the new class of method can achieve arbitrary high-order of convergence. This opens the door of developing methods beyond standard gradient flow. |
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