# Meta Learning for Support Recovery in High-dimensional Precision Matrix Estimation

arXiv: https://arxiv.org/abs/2006.12598 · HTML: https://mlsanity.com/paper/2006.12598/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2020.

## Built on by (most cited first, 3 of 3)

- [PI-NLF: A Proportional-Integral Approach for Non-negative Latent Factor Analysis](https://mlsanity.com/paper/2205.02591/) (2022)
- [Meta Sparse Principal Component Analysis](https://mlsanity.com/paper/2208.08938/) (2022)
- [Meta Learning for High-dimensional Ising Model Selection Using ℓ1-regularized Logistic Regression](https://mlsanity.com/paper/2208.09539/) (2022)

## Abstract (arXiv)

In this paper, we study meta learning for support (i.e., the set of non-zero entries) recovery in high-dimensional precision matrix estimation where we reduce the sufficient sample complexity in a novel task with the information learned from other auxiliary tasks. In our setup, each task has a different random true precision matrix, each with a possibly different support. We assume that the union of the supports of all the true precision matrices (i.e., the true support union) is small in size. We propose to pool all the samples from different tasks, and improperly estimate a single precision matrix by minimizing the ℓ1-regularized log-determinant Bregman divergence. We show that with high probability, the support of the improperly estimated single precision matrix is equal to the true support union, provided a sufficient number of samples per task n ∈ O((log N)/K), for N-dimensional vectors and K tasks. That is, one requires less samples per task when more tasks are available. We prove a matching information-theoretic lower bound for the necessary number of samples, which is n ∈ Omega((log N)/K), and thus, our algorithm is minimax optimal. Then for the novel task, we prove that the minimization of the ℓ1-regularized log-determinant Bregman divergence with the additional constraint that the support is a subset of the estimated support union could reduce the sufficient sample complexity of successful support recovery to O(log(|Soff|)) where |Soff| is the number of off-diagonal elements in the support union and is much less than N for sparse matrices. We also prove a matching information-theoretic lower bound of Omega(log(|Soff|)) for the necessary number of samples. Synthetic experiments validate our theory.