189 research outputs found
A Matrix Expander Chernoff Bound
We prove a Chernoff-type bound for sums of matrix-valued random variables
sampled via a random walk on an expander, confirming a conjecture due to
Wigderson and Xiao. Our proof is based on a new multi-matrix extension of the
Golden-Thompson inequality which improves in some ways the inequality of
Sutter, Berta, and Tomamichel, and may be of independent interest, as well as
an adaptation of an argument for the scalar case due to Healy. Secondarily, we
also provide a generic reduction showing that any concentration inequality for
vector-valued martingales implies a concentration inequality for the
corresponding expander walk, with a weakening of parameters proportional to the
squared mixing time.Comment: Fixed a minor bug in the proof of Theorem 3.
Towards Stronger Counterexamples to the Log-Approximate-Rank Conjecture
We give improved separations for the query complexity analogue of the
log-approximate-rank conjecture i.e. we show that there are a plethora of total
Boolean functions on input bits, each of which has approximate Fourier
sparsity at most and randomized parity decision tree complexity
. This improves upon the recent work of Chattopadhyay, Mande and
Sherif (JACM '20) both qualitatively (in terms of designing a large number of
examples) and quantitatively (improving the gap from quartic to cubic). We
leave open the problem of proving a randomized communication complexity lower
bound for XOR compositions of our examples. A linear lower bound would lead to
new and improved refutations of the log-approximate-rank conjecture. Moreover,
if any of these compositions had even a sub-linear cost randomized
communication protocol, it would demonstrate that randomized parity decision
tree complexity does not lift to randomized communication complexity in general
(with the XOR gadget)
- …