# Distributed Estimation of Gaussian Correlations

arXiv: https://arxiv.org/abs/1805.12472 · HTML: https://mlsanity.com/paper/1805.12472/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2018.

## Built on by (most cited first, 4 of 4)

- [Statistical Inference with Limited Memory: A Survey](https://mlsanity.com/paper/2312.15225/) (2023)
- [Batches Stabilize the Minimum Norm Risk in High Dimensional Overparameterized Linear Regression](https://mlsanity.com/paper/2306.08432/) (2023)
- [Information-Theoretic Bounds for Sparse Covariance Estimation in the Vertical-Split Distributed Model](https://mlsanity.com/paper/2606.07124/) (2026)
- [Joint Extremum Compression and Detection of a Time-Delayed Signal for Distributed Sensing](https://mlsanity.com/paper/2607.10874/) (2026)

## Abstract (arXiv)

We study a distributed estimation problem in which two remotely located parties, Alice and Bob, observe an unlimited number of i.i.d. samples corresponding to two different parts of a random vector. Alice can send k bits on average to Bob, who in turn wants to estimate the cross-correlation matrix between the two parts of the vector. In the case where the parties observe jointly Gaussian scalar random variables with an unknown correlation ρ, we obtain two constructive and simple unbiased estimators attaining a variance of (1-ρ2)/(2kln 2), which coincides with a known but non-constructive random coding result of Zhang and Berger. We extend our approach to the vector Gaussian case, which has not been treated before, and construct an estimator that is uniformly better than the scalar estimator applied separately to each of the correlations. We then show that the Gaussian performance can essentially be attained even when the distribution is completely unknown. This in particular implies that in the general problem of distributed correlation estimation, the variance can decay at least as O(1/k) with the number of transmitted bits. This behavior, however, is not tight: we give an example of a rich family of distributions for which local samples reveal essentially nothing about the correlations, and where a slightly modified estimator attains a variance of 2-Omega(k).