# Random Subspace Cubic-Regularization Methods, with Applications to Low-Rank Functions

arXiv: https://arxiv.org/abs/2501.09734 · HTML: https://mlsanity.com/paper/2501.09734/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2025.

## Built on by (most cited first, 1 of 1)

- [Blockwise Stabilized Adaptive Cubic Regularization with Subsolvers via Recurrence](https://mlsanity.com/paper/2608.22129/) (2026)

## Abstract (arXiv)

We propose and analyze random subspace variants of the second-order Adaptive Regularization using Cubics (ARC) algorithm. These methods iteratively restrict the search space to some random subspace of the parameters, constructing and minimizing a local model only within this subspace. Thus, our variants only require access to (small-dimensional) projections of first- and second-order problem derivatives and calculate a reduced step inexpensively. Under suitable assumptions, the ensuing methods maintain the optimal first-order, and second-order, global rates of convergence of (full-dimensional) cubic regularization, while showing improved scalability both theoretically and numerically, particularly when applied to low-rank functions. When applied to the latter, our adaptive variant naturally adapts the subspace size to the true rank of the function, without knowing it a priori.