Bounds on Dimension Reduction in the Nuclear Norm

For all n ≥ 1, we give an explicit construction of m × m matrices A1, …, An with m = 2⌊n∕2⌋ such that for any d and d × d matrices \(A^{\prime }_1,\ldots ,A^{\prime }_n\) that satisfy $$\displaystyle \|A^{\prime }_i-A^{\prime }_j\|{ }_{\scriptstyle {\mathsf {S}}_1} \,\leq \, \|A_i-A_j\|{ }_{\scriptstyle {\mathsf {S}}_1}\,\leq \, (1+\delta ) \|A^{\prime }_i-A^{\prime }_j\|{ }_{\scriptstyle {\mathsf {S}}_1} $$ for all i, j ∈{1, …, n} and small enough δ = O(n−c), where c > 0 is a universal constant, it must be the case that d ≥ 2⌊n∕2⌋−1. This stands in contrast to the metric theory of commutative lp spaces, as it is known that for any p ≥ 1, any n points in lp embed exactly in \(\ell _p^d\) for d = n(n − 1)∕2.