A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results

arXiv:2608.15558v1 Announce Type: cross Abstract: For mgeq 2, let c_p(m) be the all-dimensional best constant in $ left|sum_{k=1}^m A_kright|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 2imes 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 2leq 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$.

Source: arXiv cs.LG | 2026-08-18