44 research outputs found
Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width
A function is called pseudo-Boolean.
It is well-known that each pseudo-Boolean function can be written as
where ${\cal F}\subseteq \{I:\
I\subseteq [n]\}[n]=\{1,2,...,n\}\chi_I(x)=\prod_{i\in I}x_i\hat{f}(I)f\max \{|I|:\ I\in {\cal
F}\}f\rhoi\in
[n]\rho\cal Fi\in [n]\mathbf{x}_i\mathbf{x}_jj\neq i.\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)pf||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}p\ge 1||f||_q\ge ||f||_pq> p\ge 1ffdq> p>1 ||f||_q\le
(\frac{q-1}{p-1})^{d/2}||f||_p.d\rhoq> p\ge 2 ||f||_q\le
((2r)!\rho^{r-1})^{1/(2r)}||f||_p,r=\lceil q/2\rceilq=4p=2 ||f||_4\le (2\rho+1)^{1/4}||f||_2.
Better Non-Local Games from Hidden Matching
We construct a non-locality game that can be won with certainty by a quantum
strategy using log n shared EPR-pairs, while any classical strategy has winning
probability at most 1/2+O(log n/sqrt{n}). This improves upon a recent result of
Junge et al. in a number of ways.Comment: 11 pages, late
An Exploration of the Role of Principal Inertia Components in Information Theory
The principal inertia components of the joint distribution of two random
variables and are inherently connected to how an observation of is
statistically related to a hidden variable . In this paper, we explore this
connection within an information theoretic framework. We show that, under
certain symmetry conditions, the principal inertia components play an important
role in estimating one-bit functions of , namely , given an
observation of . In particular, the principal inertia components bear an
interpretation as filter coefficients in the linear transformation of
into . This interpretation naturally leads to the
conjecture that the mutual information between and is maximized when
all the principal inertia components have equal value. We also study the role
of the principal inertia components in the Markov chain , where and are binary
random variables. We illustrate our results for the setting where and
are binary strings and is the result of sending through an additive
noise binary channel.Comment: Submitted to the 2014 IEEE Information Theory Workshop (ITW
Systems of Linear Equations over and Problems Parameterized Above Average
In the problem Max Lin, we are given a system of linear equations
with variables over in which each equation is assigned a
positive weight and we wish to find an assignment of values to the variables
that maximizes the excess, which is the total weight of satisfied equations
minus the total weight of falsified equations. Using an algebraic approach, we
obtain a lower bound for the maximum excess.
Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin
introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75,
2009). In Max Lin AA all weights are integral and we are to decide whether the
maximum excess is at least , where is the parameter.
It is not hard to see that we may assume that no two equations in have
the same left-hand side and . Using our maximum excess results,
we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable
for a wide special case: for an arbitrary fixed function
.
Max -Lin AA is a special case of Max Lin AA, where each equation has at
most variables. In Max Exact -SAT AA we are given a multiset of
clauses on variables such that each clause has variables and asked
whether there is a truth assignment to the variables that satisfies at
least clauses. Using our maximum excess results, we
prove that for each fixed , Max -Lin AA and Max Exact -SAT AA can
be solved in time This improves
-time algorithms for the two problems obtained by Gutin et
al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively