Recht-Re Noncommutative Arithmetic-Geometric Mean Conjecture is False
Authors: Zehua Lai, Lek-Heng Lim
ICML 2020 | Conference PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | Our approach relies on the noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefnite program and the validity of the conjecture to certain bounds for the optimum values, which we show are false as soon as n = 5. We show how to transform Conjecture 1 into a form where the noncommutative Positivstellensatz applies, which implies in particular that for any specifc values of m and n, the conjecture can be checked via two semidefnite programs. This allows us to show in Section 3 that the conjecture is false as soon as m = n = 5. |
| Researcher Affiliation | Academia | 1Computational and Applied Mathematics Initiative, University of Chicago, Chicago, IL 60637, USA. |
| Pseudocode | No | No structured pseudocode or algorithm blocks are provided in the paper. The methods are described using mathematical formulations and textual explanations. |
| Open Source Code | Yes | The actual numerical entries of the matrices appearing in (7) and (8) are omitted due to space constraints; but they can be found in the output of our program (code in supplement). |
| Open Datasets | No | The paper does not involve traditional dataset-based experiments with training, as it focuses on theoretical proofs and computational verification of mathematical properties. Therefore, no information about publicly available datasets for training is provided. |
| Dataset Splits | No | The paper does not describe experiments involving training, validation, or testing splits of a dataset, as it focuses on theoretical proofs and computational verification of mathematical properties. |
| Hardware Specification | Yes | Using a personal computer with an Intel Core i7-9700k processor and 16GB of RAM, our Se Du Mi (Sturm, 1999) program in Matlab takes 150 seconds. |
| Software Dependencies | Yes | Using a personal computer with an Intel Core i7-9700k processor and 16GB of RAM, our Se Du Mi (Sturm, 1999) program in Matlab takes 150 seconds. |
| Experiment Setup | Yes | For any fxed values of m and n, Conjecture 1C is in a form that can be checked by standard SDP solvers. The dimension of the SDP grows exponentially with m... For m = n = 5, the basis β has 1 + 5 + 52 = 31 monomials; the SDP in (6) has 1 + 5 + 52 + 53 + 54 + 55 = 3906 linear constraints, 312 6 + 1 = 5767 variables, and takes the form: minimize tr(C0Y ) subject to tr(Ci Y ) = bi, i = 1, . . . , 3906, Y = diag(λ, Y1, . . . , Y6) 0. |