Convex Analysis at Infinity: An Introduction to Astral Space and Fireside Chat
summary
Constructs a compact extension of R^n called astral space that adds directional limits so every linear map extends continuously, proves any convex f on R^n has a minimizer in the extension, and represents points uniquely as a finite part plus a leftward-ordered sum of astron directions (astral rank k). Uses one-dimensional projections (inner products) to define sequence equivalence, a nonmetric nonvector topology (Stone–Čech-like compactification), and applies the framework to show convergence for descent methods on problems like logistic regression and boosting.
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chapters (9)
- 0:00Introduction to speaker
- 2:20Minimizing convex functions
- 8:15Motivation for astral space
- 12:24Defining minimizers at infinity
- 19:48Constructing astral space
- 29:01Properties of astral points
- 34:09Representing astral points
- 40:07Extending functions
- 48:32Convergence and conclusion
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