251 research outputs found
Separating NOF communication complexity classes RP and NP
We provide a non-explicit separation of the number-on-forehead communication
complexity classes RP and NP when the number of players is up to \delta log(n)
for any \delta<1. Recent lower bounds on Set-Disjointness [LS08,CA08] provide
an explicit separation between these classes when the number of players is only
up to o(loglog(n))
Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity
We develop a new method for estimating the discrepancy
of tensors associated with multiparty communication problems
in the ``Number on the Forehead\u27\u27 model of Chandra, Furst and Lipton.
We define an analogue of the Hadamard property of matrices
for tensors in multiple dimensions and show that any -party communication
problem represented by a Hadamard tensor must have
multiparty communication complexity.
We also exhibit constructions of Hadamard tensors,
giving lower bounds
on multiparty communication complexity
for a new class of explicitly defined Boolean functions
Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness
We show that the deterministic number-on-forehead communication complexity of set disjointness for k parties on a universe of size n is Omega(n/4^k). This gives the first lower bound that is linear in n, nearly matching Grolmusz\u27s upper bound of O(log^2(n) + k^2n/2^k). We also simplify the proof of Sherstov\u27s Omega(sqrt(n)/(k2^k)) lower bound for the randomized communication complexity of set disjointness
Disentangling causal webs in the brain using functional Magnetic Resonance Imaging: A review of current approaches
In the past two decades, functional Magnetic Resonance Imaging has been used
to relate neuronal network activity to cognitive processing and behaviour.
Recently this approach has been augmented by algorithms that allow us to infer
causal links between component populations of neuronal networks. Multiple
inference procedures have been proposed to approach this research question but
so far, each method has limitations when it comes to establishing whole-brain
connectivity patterns. In this work, we discuss eight ways to infer causality
in fMRI research: Bayesian Nets, Dynamical Causal Modelling, Granger Causality,
Likelihood Ratios, LiNGAM, Patel's Tau, Structural Equation Modelling, and
Transfer Entropy. We finish with formulating some recommendations for the
future directions in this area
Simultaneous Multiparty Communication Protocols for Composed Functions
In the Number On the Forehead (NOF) multiparty communication model,
players want to evaluate a function on some input by broadcasting bits according to a
predetermined protocol. The input is distributed in such a way that each player
sees all of it except . In the simultaneous setting, the players
cannot speak to each other but instead send information to a referee. The
referee does not know the players' input, and cannot give any information back.
At the end, the referee must be able to recover from what
she obtained.
A central open question, called the barrier, is to find a function
which is hard to compute for or more players (where the 's
have size ) in the simultaneous NOF model. This has important
applications in circuit complexity, as it could help to separate from
other complexity classes. One of the candidates belongs to the family of
composed functions. The input to these functions is represented by a boolean matrix , whose row is the input and is a
block-width parameter. A symmetric composed function acting on is specified
by two symmetric - and -variate functions and , that output
where is the -th block of width
of . As the majority function is conjectured to be outside of
, Babai et. al. suggested to study , with large
enough.
So far, it was only known that is not enough for to
break the barrier in the simultaneous deterministic NOF model. In this
paper, we extend this result to any constant block-width , by giving a
protocol of cost for any symmetric composed
function when there are players.Comment: 17 pages, 1 figure; v2: improved introduction, better cost analysis
for the 2nd protoco
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