38 research outputs found
Group representations that resist random sampling
We show that there exists a family of groups and nontrivial irreducible
representations such that, for any constant , the average of
over uniformly random elements has
operator norm with probability approaching 1 as .
More quantitatively, we show that there exist families of finite groups for
which random elements are required to bound the norm of
a typical representation below . This settles a conjecture of A. Wigderson
Small-Bias Sets for Nonabelian Groups: Derandomizing the Alon-Roichman Theorem
In analogy with epsilon-biased sets over Z_2^n, we construct explicit
epsilon-biased sets over nonabelian finite groups G. That is, we find sets S
subset G such that | Exp_{x in S} rho(x)| <= epsilon for any nontrivial
irreducible representation rho. Equivalently, such sets make G's Cayley graph
an expander with eigenvalue |lambda| <= epsilon. The Alon-Roichman theorem
shows that random sets of size O(log |G| / epsilon^2) suffice. For groups of
the form G = G_1 x ... x G_n, our construction has size poly(max_i |G_i|, n,
epsilon^{-1}), and we show that a set S \subset G^n considered by Meka and
Zuckerman that fools read-once branching programs over G is also epsilon-biased
in this sense. For solvable groups whose abelian quotients have constant
exponent, we obtain epsilon-biased sets of size (log |G|)^{1+o(1)}
poly(epsilon^{-1}). Our techniques include derandomized squaring (in both the
matrix product and tensor product senses) and a Chernoff-like bound on the
expected norm of the product of independently random operators that may be of
independent interest.Comment: Our results on solvable groups have been significantly improved,
giving eps-biased sets of polynomial (as opposed to quasipolynomial) siz
Quantum Hashing for Finite Abelian Groups
We propose a generalization of the quantum hashing technique based on the
notion of the small-bias sets. These sets have proved useful in different areas
of computer science, and here their properties give an optimal construction for
succinct quantum presentation of elements of any finite abelian group, which
can be used in various computational and cryptographic scenarios. The known
quantum fingerprinting schemas turn out to be the special cases of the proposed
quantum hashing for the corresponding abelian group
A Matrix Hyperbolic Cosine Algorithm and Applications
In this paper, we generalize Spencer's hyperbolic cosine algorithm to the
matrix-valued setting. We apply the proposed algorithm to several problems by
analyzing its computational efficiency under two special cases of matrices; one
in which the matrices have a group structure and an other in which they have
rank-one. As an application of the former case, we present a deterministic
algorithm that, given the multiplication table of a finite group of size ,
it constructs an expanding Cayley graph of logarithmic degree in near-optimal
O(n^2 log^3 n) time. For the latter case, we present a fast deterministic
algorithm for spectral sparsification of positive semi-definite matrices, which
implies an improved deterministic algorithm for spectral graph sparsification
of dense graphs. In addition, we give an elementary connection between spectral
sparsification of positive semi-definite matrices and element-wise matrix
sparsification. As a consequence, we obtain improved element-wise
sparsification algorithms for diagonally dominant-like matrices.Comment: 16 pages, simplified proof and corrected acknowledging of prior work
in (current) Section