Impossibility of dimension reduction in the nuclear norm

Abstract

Abstractvet S 1 @the htten!von xeumnn tre lssA denote the fnh spe of ll ompt liner opertors T : 2 → 2 whose nuler norm T S 1 = ∞ j=1 σ j (T ) is (niteD where {σ j (T )} ∞ j=1 re the singulr vlues of T F e prove tht for ritrrily lrge n ∈ N there exists suset C ⊆ S 1 with |C| = n tht nnot e emedded with iEvipshitz distortion O(1) into ny n o(1) Edimensionl liner suspe of S 1 F C is not even O(1)Evipshitz quotient of ny suset of ny n o(1) Edimensionl liner suspe of S 1 F husD S 1 does not dmit dimension redution result á l tohnson nd vindenstruss @IWVRAD whih omplements the work of rrrowD wontnro nd hort @PHIIA on the limittions of quntum dimension redution under the ssumption tht the emedding into low dimensions is quntum hnnelF uh sttement ws previously known with S 1 repled y the fnh spe 1 of solutely summle sequenes vi the work of frinkmn nd ghrikr @PHHQAF sn ftD the ove set C n e tken to e the sme set s the one tht frinkmn nd ghrikr onsideredD viewed s olletion of digonl mtries in S 1 F he hllenge is to demonstrte tht C nnot e fithfully relized in n ritrry lowEdimensionl suspe of S 1 D while frinkmn nd ghrikr otined suh n ssertion only for suspes of S 1 tht onsist of digonl opertors @iFeFD suspes of 1 AF e estlish this y proving tht the wrkov PEonvexity onstnt of ny (nite dimensionl liner suspe X of S 1 is t most universl onstnt multiple of log dim(X)F

Metric Dimension Reduction: A Snapshot of the Ribe Program

“…The similarity of the lower bounds in (13) and (15) is not coincidental. One can view the Brinkman-Charikar example C BC ⊆ 1 also as a collection of diagonal matrices in S 1 , and [193] treats this very same subset by strengthening the assertion of [58] that C BC does not well-embed into low-dimensional subspaces of S 1 which consist entirely of diagonal matrices, to the same assertion for low-dimensional subspaces of S 1 which are now allowed to consist of any matrices whatsoever. Using our notation for relative dimension reduction moduli, this gives the stronger assertion k α n ( 1 , S 1 ) n c/α 2 .…”

“…By [176,91], the Brinkman-Charikar subset C BC ⊆ 1 (as well as a variant of it due to Laakso [140] which has [151] the same non-embeddability property into low-dimensional subspaces of 1 ) satisfies Π q (C BC ) (log n) 1/q for every q 2 (recall that in our notation |C BC | = n). At the same time, it is proved in [193] that Π 2 (F ) log dim(F ) for every finite dimensional subset of S 1 . It remains to contrast these asymptotic behaviors (for q = 2) to deduce that if c F (C BC ) α, then necessarily dim(F ) n c/α 2 .…”

“…The above difficulty was overcome for S 1 in [193], where (15) was proven. The similarity of the lower bounds in (13) and (15) is not coincidental.…”

“…Very recently, a further geometric approach was obtained in [193], where it was used to derive a stronger statement that, as shown in [193], cannot follow from the method of [152] (the statement is that the n-point subset C BC ⊆ 1 is not even an α-Lipschitz quotient of any subset of a low dimensional subspace of S 1 ; see [193] for the relevant definition an a complete discussion). The approach of [193] relies on an invariant that arose in the Ribe program and is called Markov convexity. Fix q > 0.…”

“…Failure of average metric dimension reduction is not known for any non-universal (finite cotype) Banach space, and it would be very interesting to provide such an example. By [58,193] we know that 1 and S 1 fail to admit metric dimension reduction, but this is not known for average distortion, thus leading to the following question.…”