# Astral Space: Convex Analysis at Infinity

arXiv: https://arxiv.org/abs/2205.03260 · HTML: https://mlsanity.com/paper/2205.03260/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2022.

## Built on by (most cited first, 6 of 6)

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- [The equivalence of dynamic and strategic stability under regularized learning in games](https://mlsanity.com/paper/2311.02407/) (2023)
- [How Transformers Learn In-Context Recall Tasks? Optimality, Training Dynamics and Generalization](https://mlsanity.com/paper/2505.15009/) (2025)
- [Faster Margin Maximization Rates for Generic and Adversarially Robust Optimization Methods](https://mlsanity.com/paper/2305.17544/) (2023)

## Abstract (arXiv)

Not all convex functions on ℝn have finite minimizers; some can only be minimized by a sequence as it heads to infinity. In this work, we aim to develop a theory for understanding such minimizers at infinity. We study astral space, a compact extension of ℝn to which such points at infinity have been added. Astral space is constructed to be as small as possible while still ensuring that all linear functions can be continuously extended to the new space. Although astral space includes all of ℝn, it is not a vector space, nor even a metric space. However, it is sufficiently well-structured to allow useful and meaningful extensions of concepts of convexity, conjugacy, and subdifferentials. We develop these concepts and analyze various properties of convex functions on astral space, including the detailed structure of their minimizers, exact characterizations of continuity, and convergence of descent algorithms.