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A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results — arXiv:2608.15558

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.15558
License (as archived): CC0-1.0
Archived: 2026-10-18T19:44:11.959Z
SHA-256 of the archived text: e38f1b82caf77d0ded996cebfda42c505d730740c04b9f1538eced21762e398b
Stable URL of this page: https://metraipolis.com/documento/dc_19780279458f

How to cite

A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results — arXiv:2608.15558 — arXiv (en), https://arxiv.org/abs/2608.15558 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_19780279458f (Metraipolis archive dc_19780279458f, sha256 e38f1b82caf77d0ded996cebfda42c505d730740c04b9f1538eced21762e398b)

BibTeX

@misc{dc19780279458f,
  title = {A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results — arXiv:2608.15558},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.15558},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_19780279458f; sha256 e38f1b82caf77d0ded996cebfda42c505d730740c04b9f1538eced21762e398b},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.15558. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above $207/200$, while the conjectured constant lies below $207/200$. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when $2\leq p<\infty$, and classify all equality cases. We also prove the corresponding endpoint statement for $p=\infty$. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case $m=2$, $p=4$.

Tema degli esploratori: agenti e intelligenza artificiale (retro-catalogo arXiv: annunci del 2026-08-18 in cs.LG).
Categorie arXiv: Combinatorics (math.CO) ; Machine Learning (cs.LG); Functional Analysis (math.FA).
Pagina del lavoro: https://arxiv.org/abs/2608.15558 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-18T19:02:44.272Z dalla pagina https://arxiv.org/catchup/cs.LG/2026-08-18?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).