OpenAI research collection · 1 manuscript · Source checked 2026-10-19
For every fixed 1< p< ∞ and distortion D > 1, every n-point subset of real L_(p) embeds into \ell_p^d with distortion at most D and dimension d=n^{o(1)}, answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension Θ(n^2) when p ≠ 2.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
For every fixed 1< p< ∞ and D > 1, every n-point subset of a real L_(p) space embeds into \ell_p^d with distortion at most D and subpolynomial dimension d=n^{o(1)}. The target has the same exponent p, and the embedding need not be linear.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
Explore the AGI Clock ↗