# Precision Matrix Estimation with Noisy and Missing Data

## TL;DR

This paper introduces an ADMM algorithm for estimating precision matrices from noisy or incomplete data, addressing optimization challenges and comparing its performance with existing methods, with applications to political voting networks.

## Contribution

It develops a novel ADMM-based method for precision matrix estimation with indefinite inputs and nonconvex penalties, filling a gap in high-dimensional noisy data analysis.

## Key findings

- The proposed algorithm effectively estimates precision matrices with noisy and missing data.
- Empirical comparisons show tradeoffs between different estimation methods.
- Application to US senators' voting data reveals meaningful network structures.

## Abstract

Estimating conditional dependence graphs and precision matrices are some of the most common problems in modern statistics and machine learning. When data are fully observed, penalized maximum likelihood-type estimators have become standard tools for estimating graphical models under sparsity conditions. Extensions of these methods to more complex settings where data are contaminated with additive or multiplicative noise have been developed in recent years. In these settings, however, the relative performance of different methods is not well understood and algorithmic gaps still exist. In particular, in high-dimensional settings these methods require using non-positive semidefinite matrices as inputs, presenting novel optimization challenges. We develop an alternating direction method of multipliers (ADMM) algorithm for these problems, providing a feasible algorithm to estimate precision matrices with indefinite input and potentially nonconvex penalties. We compare this method with existing alternative solutions and empirically characterize the tradeoffs between them. Finally, we use this method to explore the networks among US senators estimated from voting records data.