Nelson–Nguyen Conjecture on Sparse Oblivious Subspace Embeddings

Prove or disprove that a SparseStack embedding with sparsity ζ = O((log r)/ε) and embedding dimension k = O(r/ε^2) is an oblivious subspace embedding for r-dimensional subspaces with multiplicative distortion parameters α = 1−ε and β = 1+ε.

Background

Sparse embeddings are computationally attractive, but fully optimal sparsity and dimension trade-offs remain unproven. The Nelson–Nguyen conjecture posits that near-optimal embedding dimension is attainable with near-optimal sparsity.

Recent advances approach the conjectured bounds up to extra logarithmic or subpolylogarithmic factors, but the exact trade-off remains unresolved.

References

Prove or disprove the Nelson--Nguyen conjecture: A SparseStack embedding with parameters $\zeta = \mathcal{O}((\log r) / \varepsilon)$ and $k = \mathcal{O}(r / \varepsilon2)$ is an oblivious subspace embedding with parameters $\alpha = 1 - \varepsilon$ and $\beta = 1+\varepsilon$.

— Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop  (2602.05394 - Amsel et al., 5 Feb 2026) in Subsection "Sparse dimensionality reduction maps" (Section 5)