Close the polynomial space gap for higher Gaussian density moments

Close the remaining polynomial gap between the one-pass streaming upper and lower bounds for estimating higher Gaussian density moments M_p(P) for fixed integer p>1 under Gaussian kernel similarity.

Background

For a stream P of n points in fixed-dimensional Euclidean space, the paper gives a one-pass (1+ε)-approximation algorithm for the p-th Gaussian density moment M_p(P) using O_{p,d}(ε{-2} n{1-1/(p+1)} log n * log(1/δ)) words. It also proves a lower bound of Ω_p(ε{-2} n{1-2/(p+1)}/log n) bits in one dimension over the stated accuracy range.

The exponents of n in these bounds differ, leaving a polynomial separation between the known upper and lower space complexities. The authors explicitly identify eliminating this separation as an unresolved issue.

References

For higher density moments, a small polynomial gap remains between the upper and lower bounds.

— Streaming Algorithms for Gaussian Kernel Density Statistics  (2609.09622 - Zhang, 9 Sep 2026) in Section Conclusion

For the diversity index, our lower bound shows that a dimension-dependent polylogarithmic amount of bit space is necessary; closing the gap between the exponents in the upper and lower bounds remains open.

— Streaming Algorithms for Gaussian Kernel Density Statistics  (2609.09622 - Zhang, 9 Sep 2026) in Section Conclusion