# Core-elements Subsampling for Alternating Least Squares

arXiv: https://arxiv.org/abs/2509.18024 · HTML: https://mlsanity.com/paper/2509.18024/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 57 references that are in the map.
- First posted on arXiv 22 Sep 2025.

## Builds on (most cited first, 26 of 26)

- [Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares](https://mlsanity.com/paper/1410.2596/) (2014)
- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [A Statistical Perspective on Algorithmic Leveraging](https://mlsanity.com/paper/1306.5362/) (2013)
- [Optimal Subsampling for Large Sample Logistic Regression](https://mlsanity.com/paper/1702.01166/) (2017)
- [Optimal Distributed Subsampling for Maximum Quasi-Likelihood Estimators with Massive Data](https://mlsanity.com/paper/2005.10435/) (2020)
- [Information-Based Optimal Subdata Selection for Big Data Linear Regression](https://mlsanity.com/paper/1710.10382/) (2017)
- [Optimal Subsampling Algorithms for Big Data Regressions](https://mlsanity.com/paper/1806.06761/) (2018)
- [Randomly Initialized Alternating Least Squares: Fast Convergence for Matrix Sensing](https://mlsanity.com/paper/2204.11516/) (2022)
- [Efficient Approximation of Gromov-Wasserstein Distance Using Importance Sparsification](https://mlsanity.com/paper/2205.13573/) (2022)
- [Asymptotic Analysis of Sampling Estimators for Randomized Numerical Linear Algebra Algorithms](https://mlsanity.com/paper/2002.10526/) (2020)
- [Optimal subsampling for quantile regression in big data](https://mlsanity.com/paper/2001.10168/) (2020)
- [More efficient approximation of smoothing splines via space-filling basis selection](https://mlsanity.com/paper/2003.10609/) (2020)
- [Orthogonal Subsampling for Big Data Linear Regression](https://mlsanity.com/paper/2105.14647/) (2021)
- [Improved Active Learning via Dependent Leverage Score Sampling](https://mlsanity.com/paper/2310.04966/) (2023)
- [LowCon: A design-based subsampling approach in a misspecified linear modeL](https://mlsanity.com/paper/2010.12178/) (2020)
- [Modern Subsampling Methods for Large-Scale Least Squares Regression](https://mlsanity.com/paper/2105.01552/) (2021)
- [An optimal transport approach for selecting a representative subsample with application in efficient kernel density estimation](https://mlsanity.com/paper/2206.01182/) (2022)
- [Smoothing splines approximation using Hilbert curve basis selection](https://mlsanity.com/paper/2109.11727/) (2021)
- [Core-Elements for Large-Scale Least Squares Estimation](https://mlsanity.com/paper/2206.10240/) (2022)
- [Model-free Subsampling Method Based on Uniform Designs](https://mlsanity.com/paper/2209.03617/) (2022)
- [A Model-free Variable Screening Method Based on Leverage Score](https://mlsanity.com/paper/2109.09936/) (2021)
- [Independence-Encouraging Subsampling for Nonparametric Additive Models](https://mlsanity.com/paper/2302.13441/) (2023)
- [Algorithmic Acceleration of Parallel ALS for Collaborative Filtering: Speeding up Distributed Big Data Recommendation in Spark](https://mlsanity.com/paper/1508.03110/) (2015)
- [Optimal Sampling Designs for Multi-dimensional Streaming Time Series with Application to Power Grid Sensor Data](https://mlsanity.com/paper/2303.08242/) (2023)
- [Sampling-Based Methods for Multi-Block Optimization Problems over Transport Polytopes](https://mlsanity.com/paper/2306.16763/) (2023)
- [Leverage classifier: Another look at support vector machine](https://mlsanity.com/paper/2308.12444/) (2023)

## Abstract (arXiv)

In this paper, we propose a novel element-wise subset selection method for the alternating least squares (ALS) algorithm, focusing on low-rank matrix factorization involving matrices with missing values, as commonly encountered in recommender systems. While ALS is widely used for providing personalized recommendations based on user-item interaction data, its high computational cost, stemming from repeated regression operations, poses significant challenges for large-scale datasets. To enhance the efficiency of ALS, we propose a core-elements subsampling method that selects a representative subset of data and leverages sparse matrix operations to approximate ALS estimations efficiently. We establish theoretical guarantees for the approximation and convergence of the proposed approach, showing that it achieves similar accuracy with significantly reduced computational time compared to full-data ALS. Extensive simulations and real-world applications demonstrate the effectiveness of our method in various scenarios, emphasizing its potential in large-scale recommendation systems.