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Math-2641 · mathematics · Geometry

Subpolynomial nonlinear dimension reduction in Lp

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed $1<p<\infty $, $p\neq 2$, and $D>1$, does a constant $C_{p,D}$ exist such that every n-point subset of any real Lp space embeds, with freely chosen positive scale and distortion at most D, into $\ell p^d$ for $d\leq \exp (C_{p,D}(\log $ $n)^\gamma )$, where $\gamma =2-p$ for $p<2$ and $\gamma =1-2/p$ for $p>2$? Maps need not be linear, and $n\geq 2$.

Short English statement

For each fixed $1<p<\infty $, $p\neq 2$, and $D>1$, does a constant $C_{p,D}$ exist such that every n-point subset of any real Lp space embeds, with freely chosen positive scale and distortion at most D, into $\ell p^d$ for $d\leq \exp (C_{p,D}(\log $ $n)^\gamma )$, where $\gamma =2-p$ for $p<2$ and $\gamma =1-2/p$ for $p>2$? Maps need not be linear, and $n\geq 2$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2641-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2641

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 094; 2026-09-23).

Subpolynomial nonlinear dimension reduction in Lp

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 094; 2026-09-23), the asserted result is that fixed-distortion embeddings of finite real $L_p$ subsets into the same-exponent finite-dimensional space require only $n^{o(1)}$ dimensions for $1<p<\infty$, with nonlinear maps allowed.

Mathematical statement of the claimed result

Manuscript 1.
Fix $1<p<\infty$, $p\ne2$. Let $d_p(n,D)$ be the minimum dimension sufficient for embedding every $n$-point subset of an $L^p$ space into a finite-dimensional $L^p$ subspace with distortion at most $D$. Put $\gamma(p)=2-p$ for $p<2$ and $\gamma(p)=1-2/p$ for $p>2$. For each $D>1$, the claim gives $C_{p,D}<\infty$ with \[ \frac{\log n}{\log(1+2D)}\le d_p(n,D) \le\exp(C_{p,D}(\log n)^{\gamma(p)})\quad(n\ge2). \] For exact embeddings it instead asserts \[ \lfloor(n-1)/4\rfloor^2\le d_p(n,1)\le\binom n2\quad(n\ge9). \] Consequently the limit of $\log d_p(n,D)/\log n$ is zero for $D>1$ and two for $D=1$. The embeddings need not be linear on the ambient space.

Outline of the argument

For manuscript 1 (family 094; 2026-09-23), homogeneous increments are represented by signed Poisson samples with a shared heavy-tailed marginal. A ramp of clipping thresholds and a controlled projection bound the error before bounded-gradient sampling reduces the dimension; packing supplies the lower comparison.

Formal verification

For manuscript 1 (family 094; 2026-09-23), the documented Lean scope is finite-dimensional approximation bounds for $p>1$ with $p\ne2$, including the fixed-distortion subpolynomial upper estimate and the logarithmic lower estimate; a quadratic comparison is listed for distortion one.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are SubpolynomialLp.lean.

Their named declarations include OAI.SubpolynomialLp.source_main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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