3,521 research outputs found
The sign rule and beyond: Boundary effects, flexibility, and noise correlations in neural population codes
Over repeat presentations of the same stimulus, sensory neurons show variable
responses. This "noise" is typically correlated between pairs of cells, and a
question with rich history in neuroscience is how these noise correlations
impact the population's ability to encode the stimulus. Here, we consider a
very general setting for population coding, investigating how information
varies as a function of noise correlations, with all other aspects of the
problem - neural tuning curves, etc. - held fixed. This work yields unifying
insights into the role of noise correlations. These are summarized in the form
of theorems, and illustrated with numerical examples involving neurons with
diverse tuning curves. Our main contributions are as follows.
(1) We generalize previous results to prove a sign rule (SR) - if noise
correlations between pairs of neurons have opposite signs vs. their signal
correlations, then coding performance will improve compared to the independent
case. This holds for three different metrics of coding performance, and for
arbitrary tuning curves and levels of heterogeneity. This generality is true
for our other results as well.
(2) As also pointed out in the literature, the SR does not provide a
necessary condition for good coding. We show that a diverse set of correlation
structures can improve coding. Many of these violate the SR, as do
experimentally observed correlations. There is structure to this diversity: we
prove that the optimal correlation structures must lie on boundaries of the
possible set of noise correlations.
(3) We provide a novel set of necessary and sufficient conditions, under
which the coding performance (in the presence of noise) will be as good as it
would be if there were no noise present at all.Comment: 41 pages, 5 figure
Hessian and concavity of mutual information, differential entropy, and entropy power in linear vector Gaussian channels
Within the framework of linear vector Gaussian channels with arbitrary
signaling, closed-form expressions for the Jacobian of the minimum mean square
error and Fisher information matrices with respect to arbitrary parameters of
the system are calculated in this paper. Capitalizing on prior research where
the minimum mean square error and Fisher information matrices were linked to
information-theoretic quantities through differentiation, closed-form
expressions for the Hessian of the mutual information and the differential
entropy are derived. These expressions are then used to assess the concavity
properties of mutual information and differential entropy under different
channel conditions and also to derive a multivariate version of the entropy
power inequality due to Costa.Comment: 33 pages, 2 figures. A shorter version of this paper is to appear in
IEEE Transactions on Information Theor
One-way quantum key distribution: Simple upper bound on the secret key rate
We present a simple method to obtain an upper bound on the achievable secret
key rate in quantum key distribution (QKD) protocols that use only
unidirectional classical communication during the public-discussion phase. This
method is based on a necessary precondition for one-way secret key
distillation; the legitimate users need to prove that there exists no quantum
state having a symmetric extension that is compatible with the available
measurements results. The main advantage of the obtained upper bound is that it
can be formulated as a semidefinite program, which can be efficiently solved.
We illustrate our results by analysing two well-known qubit-based QKD
protocols: the four-state protocol and the six-state protocol. Recent results
by Renner et al., Phys. Rev. A 72, 012332 (2005), also show that the given
precondition is only necessary but not sufficient for unidirectional secret key
distillation.Comment: 11 pages, 1 figur
Some upper and lower bounds on PSD-rank
Positive semidefinite rank (PSD-rank) is a relatively new quantity with
applications to combinatorial optimization and communication complexity. We
first study several basic properties of PSD-rank, and then develop new
techniques for showing lower bounds on the PSD-rank. All of these bounds are
based on viewing a positive semidefinite factorization of a matrix as a
quantum communication protocol. These lower bounds depend on the entries of the
matrix and not only on its support (the zero/nonzero pattern), overcoming a
limitation of some previous techniques. We compare these new lower bounds with
known bounds, and give examples where the new ones are better. As an
application we determine the PSD-rank of (approximations of) some common
matrices.Comment: 21 page
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