List-Decodable Regression via Expander Sketching

TL;DR

This paper presents a new expander-sketching approach for list-decodable linear regression that improves sample complexity and runtime, enabling robust and efficient estimation even with high contamination.

Contribution

It introduces an expander-sketching framework that achieves optimal sample complexity and near input-sparsity runtime for list-decodable regression without relying on sum-of-squares methods.

Findings

Achieves sample complexity $ ilde{O}((d+ ext{log}(1/ ext{delta}))/ ext{alpha})$

List size $O(1/ ext{alpha})$

Near input-sparsity runtime $ ilde{O}( ext{nnz}(X)+d^{3}/ ext{alpha})$

Abstract

We introduce an expander-sketching framework for list-decodable linear regression that achieves sample complexity $\tilde{O}((d+\log(1/\delta))/\alpha)$, list size $O(1/\alpha)$, and near input-sparsity running time $\tilde{O}(\mathrm{nnz}(X)+d^{3}/\alpha)$ under standard sub-Gaussian assumptions. Our method uses lossless expanders to synthesize lightly contaminated batches, enabling robust aggregation and a short spectral filtering stage that matches the best known efficient guarantees while avoiding SoS machinery and explicit batch structure.