# Causal Inference (C-inf) -- closed form worst case typical phase transitions

arXiv: https://arxiv.org/abs/2301.00793 · HTML: https://mlsanity.com/paper/2301.00793/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 57 references that are in the map.
- First posted on arXiv 2 Jan 2023.

## Builds on (most cited first, 6 of 6)

- [Causal Matrix Completion](https://mlsanity.com/paper/2109.15154/) (2021)
- [Causal Inference with Noisy and Missing Covariates via Matrix Factorization](https://mlsanity.com/paper/1806.00811/) (2018)
- [Regularly random duality](https://mlsanity.com/paper/1303.7295/) (2013)
- [Design-based Analysis in Difference-In-Differences Settings with Staggered Adoption](https://mlsanity.com/paper/1808.05293/) (2018)
- [Upper-bounding ℓ1-optimization weak thresholds](https://mlsanity.com/paper/1303.7289/) (2013)
- [Randomization Tests in Observational Studies with Staggered Adoption of Treatment](https://mlsanity.com/paper/1912.10610/) (2019)

## Abstract (arXiv)

In this paper we establish a mathematically rigorous connection between Causal inference (C-inf) and the low-rank recovery (LRR). Using Random Duality Theory (RDT) concepts developed in [46,48,50] and novel mathematical strategies related to free probability theory, we obtain the exact explicit typical (and achievable) worst case phase transitions (PT). These PT precisely separate scenarios where causal inference via LRR is possible from those where it is not. We supplement our mathematical analysis with numerical experiments that confirm the theoretical predictions of PT phenomena, and further show that the two closely match for fairly small sample sizes. We obtain simple closed form representations for the resulting PTs, which highlight direct relations between the low rankness of the target C-inf matrix and the time of the treatment. Hence, our results can be used to determine the range of C-inf's typical applicability.