From low to high-dimensional moments without magic

TL;DR

This paper develops methods to reconstruct high-dimensional moments from low-dimensional projections, providing algebraic conditions, optimization techniques, and randomized approaches for approximate recovery.

Contribution

It introduces algebraic conditions and a computational framework for reconstructing high-dimensional moments from low-dimensional projections, including randomized methods for approximation.

Findings

Explicit reconstruction formulas derived from algebraic conditions

Optimization-based method for selecting suitable projections

Randomized projections enable approximate recovery

Abstract

We aim to compute the first few moments of a high-dimensional random vector from the first few moments of a number of its low-dimensional projections. To this end, we identify algebraic conditions on the set of low-dimensional projectors that yield explicit reconstruction formulas. We also provide a computational framework, with which suitable projectors can be derived by solving an optimization problem. Finally, we show that randomized projections permit approximate recovery.