# Recommendation on a Budget: Column Space Recovery from Partially Observed Entries with Random or Active Sampling

arXiv: https://arxiv.org/abs/2002.11589 · HTML: https://mlsanity.com/paper/2002.11589/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Feb 2020.

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

- [The Retriever Should Remember: Experience-Amortized Reranking for Long-Term Agent Memory](https://mlsanity.com/paper/2608.22767/) (2026)

## Abstract (arXiv)

We analyze alternating minimization for column space recovery of a partially observed, approximately low rank matrix with a growing number of columns and a fixed budget of observations per column. In this work, we prove that if the budget is greater than the rank of the matrix, column space recovery succeeds -- as the number of columns grows, the estimate from alternating minimization converges to the true column space with probability tending to one. From our proof techniques, we naturally formulate an active sampling strategy for choosing entries of a column that is theoretically and empirically (on synthetic and real data) better than the commonly studied uniformly random sampling strategy.