105 research outputs found
A Channel Coding Perspective of Collaborative Filtering
We consider the problem of collaborative filtering from a channel coding
perspective. We model the underlying rating matrix as a finite alphabet matrix
with block constant structure. The observations are obtained from this
underlying matrix through a discrete memoryless channel with a noisy part
representing noisy user behavior and an erasure part representing missing data.
Moreover, the clusters over which the underlying matrix is constant are {\it
unknown}. We establish a sharp threshold result for this model: if the largest
cluster size is smaller than (where the rating matrix is of size
), then the underlying matrix cannot be recovered with any
estimator, but if the smallest cluster size is larger than , then
we show a polynomial time estimator with diminishing probability of error. In
the case of uniform cluster size, not only the order of the threshold, but also
the constant is identified.Comment: 32 pages, 1 figure, Submitted to IEEE Transactions on Information
Theor
On network coding for sum-networks
A directed acyclic network is considered where all the terminals need to
recover the sum of the symbols generated at all the sources. We call such a
network a sum-network. It is shown that there exists a solvably (and linear
solvably) equivalent sum-network for any multiple-unicast network, and thus for
any directed acyclic communication network. It is also shown that there exists
a linear solvably equivalent multiple-unicast network for every sum-network. It
is shown that for any set of polynomials having integer coefficients, there
exists a sum-network which is scalar linear solvable over a finite field F if
and only if the polynomials have a common root in F. For any finite or cofinite
set of prime numbers, a network is constructed which has a vector linear
solution of any length if and only if the characteristic of the alphabet field
is in the given set. The insufficiency of linear network coding and
unachievability of the network coding capacity are proved for sum-networks by
using similar known results for communication networks. Under fractional vector
linear network coding, a sum-network and its reverse network are shown to be
equivalent. However, under non-linear coding, it is shown that there exists a
solvable sum-network whose reverse network is not solvable.Comment: Accepted to IEEE Transactions on Information Theor
Dirty Paper Arbitrarily Varying Channel with a State-Aware Adversary
In this paper, we take an arbitrarily varying channel (AVC) approach to
examine the problem of writing on a dirty paper in the presence of an
adversary. We consider an additive white Gaussian noise (AWGN) channel with an
additive white Gaussian state, where the state is known non-causally to the
encoder and the adversary, but not the decoder. We determine the randomized
coding capacity of this AVC under the maximal probability of error criterion.
Interestingly, it is shown that the jamming adversary disregards the state
knowledge to choose a white Gaussian channel input which is independent of the
state
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