# Randomized Approach to Matrix Completion: Applications in Recommendation Systems and Image Inpainting

arXiv: https://arxiv.org/abs/2403.01919 · HTML: https://mlsanity.com/paper/2403.01919/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 4 Mar 2024.

## Builds on (most cited first, 19 of 19)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Sketching as a Tool for Numerical Linear Algebra](https://mlsanity.com/paper/1411.4357/) (2014)
- [Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent](https://mlsanity.com/paper/2005.08898/) (2020)
- [Understanding Alternating Minimization for Matrix Completion](https://mlsanity.com/paper/1312.0925/) (2013)
- [Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation](https://mlsanity.com/paper/1802.08397/) (2018)
- [Globally Convergent Type-I Anderson Acceleration for Non-Smooth Fixed-Point Iterations](https://mlsanity.com/paper/1808.03971/) (2018)
- [Operator splitting for a homogeneous embedding of the linear complementarity problem](https://mlsanity.com/paper/2004.02177/) (2020)
- [Robust CUR Decomposition: Theory and Imaging Applications](https://mlsanity.com/paper/2101.05231/) (2021)
- [Low-Rank Matrix and Tensor Completion via Adaptive Sampling](https://mlsanity.com/paper/1304.4672/) (2013)
- [Fast Exact Matrix Completion with Finite Samples](https://mlsanity.com/paper/1411.1087/) (2014)
- [Matrix Completion with Cross-Concentrated Sampling: Bridging Uniform Sampling and CUR Sampling](https://mlsanity.com/paper/2208.09723/) (2022)
- [Sketchy Decisions: Convex Low-Rank Matrix Optimization with Optimal Storage](https://mlsanity.com/paper/1702.06838/) (2017)
- [Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion](https://mlsanity.com/paper/2204.03316/) (2022)
- [A determinantal point process for column subset selection](https://mlsanity.com/paper/1812.09771/) (2018)
- [On the Power of Adaptivity in Matrix Completion and Approximation](https://mlsanity.com/paper/1407.3619/) (2014)
- [CUR Algorithm for Partially Observed Matrices](https://mlsanity.com/paper/1411.0860/) (2014)
- [A Preconditioned Riemannian Gradient Descent Algorithm for Low-Rank Matrix Recovery](https://mlsanity.com/paper/2305.02543/) (2023)
- [Provably Correct Algorithms for Matrix Column Subset Selection with Selectively Sampled Data](https://mlsanity.com/paper/1505.04343/) (2015)

## Abstract (arXiv)

We present a novel method for matrix completion, specifically designed for matrices where one dimension is significantly larger than the other. Our Columns Selected Matrix Completion (CSMC) method combines Column Subset Selection with Low-Rank Matrix Completion to efficiently reconstruct incomplete datasets. CSMC substantially reduces computational cost while preserving the solution quality of state-of-the-art convex matrix completion techniques. Each stage of CSMC involves solving a convex optimization problem. We introduce two algorithmic implementations of CSMC, each tailored to problems of different scales. A formal analysis is provided, outlining the necessary assumptions and the probability of exact recovery. To evaluate the impact of matrix size, rank, and the ratio of missing entries on solution quality and runtime, we conducted experiments on synthetic data. The method was also applied to two real-world tasks: recommendation systems and image inpainting. Our results show that CSMC achieves substantial runtime improvements while maintaining competitive accuracy compared to leading convex optimization-based methods.