# On the Expected Value of the Determinant of Random Sum of Rank-One Matrices

## TL;DR

This paper derives a simple formula for the expected determinant of a sum of random rank-one matrices, enabling efficient computation in applications like D-optimal estimation and random graphs.

## Contribution

It introduces a novel, efficient method to compute the expected determinant of sums of random rank-one matrices, simplifying complex calculations.

## Key findings

- Expected determinant can be computed via a single determinant calculation.
- Applicable to D-optimal estimation and random graph analysis.
- Reduces computational complexity from exponential to polynomial.

## Abstract

We present a simple, yet useful result about the expected value of the determinant of random sum of rank-one matrices. Computing such expectations in general may involve a sum over exponentially many terms. Nevertheless, we show that an interesting and useful class of such expectations that arise in, e.g., D-optimal estimation and random graphs can be computed efficiently via computing a single determinant.