Using sums-of-squares to prove Gaussian product inequalities
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Bibliographic record
Abstract
Abstract The long-standing Gaussian product inequality (GPI) conjecture states that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mrow> <m:mo>∏</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:msub> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:msubsup> <m:mrow> <m:mrow> <m:mo>∏</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:mi>E</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:msub> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msup> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> E\left[{\prod }_{j=1}^{n}{| {X}_{j}| }^{{y}_{j}}]\ge {\prod }_{j=1}^{n}E\left[{| {X}_{j}| }^{{y}_{j}}] for any centered Gaussian random vector <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left({X}_{1},\ldots ,{X}_{n}) and any non-negative real numbers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>y</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {y}_{j} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>n</m:mi> </m:math> j=1,\ldots ,n . In this study, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of the novel method, we apply it to prove new four- and five-dimensional GPIs: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:mrow> </m:msubsup> <m:msubsup> <m:mrow> <m:mi>X</m:mi>
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it