# Sketchy Decisions: Convex Low-Rank Matrix Optimization with Optimal Storage

arXiv: https://arxiv.org/abs/1702.06838 · HTML: https://mlsanity.com/paper/1702.06838/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2017.

## Built on by (most cited first, 11 of 11)

- [Large Scale Private Learning via Low-rank Reparametrization](https://mlsanity.com/paper/2106.09352/) (2021)
- [Training Quantized Neural Networks to Global Optimality via Semidefinite Programming](https://mlsanity.com/paper/2105.01420/) (2021)
- [Sketch-and-Lift: Scalable Subsampled Semidefinite Program for K-means Clustering](https://mlsanity.com/paper/2201.08226/) (2022)
- [Semidefinite Programming versus Burer-Monteiro Factorization for Matrix Sensing](https://mlsanity.com/paper/2208.07469/) (2022)
- [Fast, Scalable, Warm-Start Semidefinite Programming with Spectral Bundling and Sketching](https://mlsanity.com/paper/2312.11801/) (2023)
- [Community Detection with a Subsampled Semidefinite Program](https://mlsanity.com/paper/2102.01419/) (2021)
- [Total Least Squares Phase Retrieval](https://mlsanity.com/paper/2102.00927/) (2021)
- [Screening for a Reweighted Penalized Conditional Gradient Method](https://mlsanity.com/paper/2107.01106/) (2021)
- [Parabolic Relaxation for Quadratically-constrained Quadratic Programming -- Part II: Theoretical & Computational Results](https://mlsanity.com/paper/2208.03625/) (2022)
- [Randomized Approach to Matrix Completion: Applications in Recommendation Systems and Image Inpainting](https://mlsanity.com/paper/2403.01919/) (2024)
- [A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron](https://mlsanity.com/paper/2503.01441/) (2025)

## Abstract (arXiv)

This paper concerns a fundamental class of convex matrix optimization problems. It presents the first algorithm that uses optimal storage and provably computes a low-rank approximation of a solution. In particular, when all solutions have low rank, the algorithm converges to a solution. This algorithm, SketchyCGM, modifies a standard convex optimization scheme, the conditional gradient method, to store only a small randomized sketch of the matrix variable. After the optimization terminates, the algorithm extracts a low-rank approximation of the solution from the sketch. In contrast to nonconvex heuristics, the guarantees for SketchyCGM do not rely on statistical models for the problem data. Numerical work demonstrates the benefits of SketchyCGM over heuristics.