1,256 research outputs found
On U-Statistics and Compressed Sensing II: Non-Asymptotic Worst-Case Analysis
In another related work, U-statistics were used for non-asymptotic
"average-case" analysis of random compressed sensing matrices. In this
companion paper the same analytical tool is adopted differently - here we
perform non-asymptotic "worst-case" analysis.
Simple union bounds are a natural choice for "worst-case" analyses, however
their tightness is an issue (and questioned in previous works). Here we focus
on a theoretical U-statistical result, which potentially allows us to prove
that these union bounds are tight. To our knowledge, this kind of (powerful)
result is completely new in the context of CS. This general result applies to a
wide variety of parameters, and is related to (Stein-Chen) Poisson
approximation. In this paper, we consider i) restricted isometries, and ii)
mutual coherence. For the bounded case, we show that k-th order restricted
isometry constants have tight union bounds, when the measurements m =
\mathcal{O}(k (1 + \log(n/k))). Here we require the restricted isometries to
grow linearly in k, however we conjecture that this result can be improved to
allow them to be fixed. Also, we show that mutual coherence (with the standard
estimate \sqrt{(4\log n)/m}) have very tight union bounds.
For coherence, the normalization complicates general discussion, and we
consider only Gaussian and Bernoulli cases here.Comment: 12 pages. Submitted to IEEE Transactions on Signal Processin
The Single-Uniprior Index-Coding Problem: The Single-Sender Case and The Multi-Sender Extension
Index coding studies multiterminal source-coding problems where a set of
receivers are required to decode multiple (possibly different) messages from a
common broadcast, and they each know some messages a priori. In this paper, at
the receiver end, we consider a special setting where each receiver knows only
one message a priori, and each message is known to only one receiver. At the
broadcasting end, we consider a generalized setting where there could be
multiple senders, and each sender knows a subset of the messages. The senders
collaborate to transmit an index code. This work looks at minimizing the number
of total coded bits the senders are required to transmit. When there is only
one sender, we propose a pruning algorithm to find a lower bound on the optimal
(i.e., the shortest) index codelength, and show that it is achievable by linear
index codes. When there are two or more senders, we propose an appending
technique to be used in conjunction with the pruning technique to give a lower
bound on the optimal index codelength; we also derive an upper bound based on
cyclic codes. While the two bounds do not match in general, for the special
case where no two distinct senders know any message in common, the bounds
match, giving the optimal index codelength. The results are expressed in terms
of strongly connected components in directed graphs that represent the
index-coding problems.Comment: Author final manuscrip
The Multi-Sender Multicast Index Coding
We focus on the following instance of an index coding problem, where a set of
receivers are required to decode multiple messages, whilst each knows one of
the messages a priori. In particular, here we consider a generalized setting
where they are multiple senders, each sender only knows a subset of messages,
and all senders are required to collectively transmit the index code. For a
single sender, Ong and Ho (ICC, 2012) have established the optimal index
codelength, where the lower bound was obtained using a pruning algorithm. In
this paper, the pruning algorithm is simplified, and used in conjunction with
an appending technique to give a lower bound to the multi-sender case. An upper
bound is derived based on network coding. While the two bounds do not match in
general, for the special case where no two senders know any message bit in
common, the bounds match, giving the optimal index codelength. The results are
derived based on graph theory, and are expressed in terms of strongly connected
components.Comment: This is an extended version of the same-titled paper accepted and to
be presented at the IEEE International Symposium on Information Theory
(ISIT), Istanbul, in July 201
Potential and Technological Advancement of Biofuels
This scientific paper examines the feasibility of biofuels as a solution to the world‟s energy crisis. It studies the development of the four different generations of biofuel that have been discerned over the years, determining the pros and cons of each. The paper further investigates the issues concerning each generation, and determines how their successors have solved and improved on those problems. In order to give the reader an unbiased perspective, the paper studies both general advantages and disadvantages that encompasses social, economic and environmental impacts. Research and development on the first two generations of biofuels have matured, and case studies have been used to allude to their current applications. The challenge of making third and fourth generation biofuels economically viable has also been highlighted due to their significant environmental and production benefits over the first two generations. The prospects of third and fourth generation biofuels have also been looked into to determine its outlook in the near future. If these next generation biofuels can garner enough support and become cost-competitive, mankind‟s quest for an alternative, renewable source of energy may finally be completed
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