11 research outputs found
Uniform estimates of nonlinear spectral gaps
By generalizing the path method, we show that nonlinear spectral gaps of a
finite connected graph are uniformly bounded from below by a positive constant
which is independent of the target metric space. We apply our result to an
-ball in the -regular tree, and observe that the asymptotic
behavior of nonlinear spectral gaps of as does not
depend on the target metric space, which is in contrast to the case of a
sequence of expanders. We also apply our result to the -dimensional Hamming
cube and obtain an estimate of its nonlinear spectral gap with respect to
an arbitrary metric space, which is asymptotically sharp as .Comment: to appear in Graphs and Combinatoric
Impossibility of dimension reduction in the nuclear norm
Let (the Schatten--von Neumann trace class) denote the Banach
space of all compact linear operators whose nuclear norm
is finite, where
are the singular values of . We prove that
for arbitrarily large there exists a subset
with that cannot be
embedded with bi-Lipschitz distortion into any -dimensional
linear subspace of . is not even a -Lipschitz
quotient of any subset of any -dimensional linear subspace of
. Thus, does not admit a dimension reduction
result \'a la Johnson and Lindenstrauss (1984), which complements the work of
Harrow, Montanaro and Short (2011) on the limitations of quantum dimension
reduction under the assumption that the embedding into low dimensions is a
quantum channel. Such a statement was previously known with
replaced by the Banach space of absolutely summable sequences via the
work of Brinkman and Charikar (2003). In fact, the above set can
be taken to be the same set as the one that Brinkman and Charikar considered,
viewed as a collection of diagonal matrices in . The challenge is
to demonstrate that cannot be faithfully realized in an arbitrary
low-dimensional subspace of , while Brinkman and Charikar
obtained such an assertion only for subspaces of that consist of
diagonal operators (i.e., subspaces of ). We establish this by proving
that the Markov 2-convexity constant of any finite dimensional linear subspace
of is at most a universal constant multiple of