1,485 research outputs found
Alternating direction algorithms for regularization in compressed sensing
In this paper we propose three iterative greedy algorithms for compressed
sensing, called \emph{iterative alternating direction} (IAD), \emph{normalized
iterative alternating direction} (NIAD) and \emph{alternating direction
pursuit} (ADP), which stem from the iteration steps of alternating direction
method of multiplier (ADMM) for -regularized least squares
(-LS) and can be considered as the alternating direction versions of
the well-known iterative hard thresholding (IHT), normalized iterative hard
thresholding (NIHT) and hard thresholding pursuit (HTP) respectively. Firstly,
relative to the general iteration steps of ADMM, the proposed algorithms have
no splitting or dual variables in iterations and thus the dependence of the
current approximation on past iterations is direct. Secondly, provable
theoretical guarantees are provided in terms of restricted isometry property,
which is the first theoretical guarantee of ADMM for -LS to the best of
our knowledge. Finally, they outperform the corresponding IHT, NIHT and HTP
greatly when reconstructing both constant amplitude signals with random signs
(CARS signals) and Gaussian signals.Comment: 16 pages, 1 figur
Nonextensive information theoretical machine
In this paper, we propose a new discriminative model named \emph{nonextensive
information theoretical machine (NITM)} based on nonextensive generalization of
Shannon information theory. In NITM, weight parameters are treated as random
variables. Tsallis divergence is used to regularize the distribution of weight
parameters and maximum unnormalized Tsallis entropy distribution is used to
evaluate fitting effect. On the one hand, it is showed that some well-known
margin-based loss functions such as loss, hinge loss, squared
hinge loss and exponential loss can be unified by unnormalized Tsallis entropy.
On the other hand, Gaussian prior regularization is generalized to Student-t
prior regularization with similar computational complexity. The model can be
solved efficiently by gradient-based convex optimization and its performance is
illustrated on standard datasets
Bayesian linear regression with Student-t assumptions
As an automatic method of determining model complexity using the training
data alone, Bayesian linear regression provides us a principled way to select
hyperparameters. But one often needs approximation inference if distribution
assumption is beyond Gaussian distribution. In this paper, we propose a
Bayesian linear regression model with Student-t assumptions (BLRS), which can
be inferred exactly. In this framework, both conjugate prior and expectation
maximization (EM) algorithm are generalized. Meanwhile, we prove that the
maximum likelihood solution is equivalent to the standard Bayesian linear
regression with Gaussian assumptions (BLRG). The -EM algorithm for BLRS is
nearly identical to the EM algorithm for BLRG. It is showed that -EM for
BLRS can converge faster than EM for BLRG for the task of predicting online
news popularity
Deterministic Construction of Binary Measurement Matrices with Various Sizes
We introduce a general framework to deterministically construct binary
measurement matrices for compressed sensing. The proposed matrices are composed
of (circulant) permutation submatrix blocks and zero submatrix blocks, thus
making their hardware realization convenient and easy. Firstly, using the
famous Johnson bound for binary constant weight codes, we derive a new lower
bound for the coherence of binary matrices with uniform column weights.
Afterwards, a large class of binary base matrices with coherence asymptotically
achieving this new bound are presented. Finally, by choosing proper rows and
columns from these base matrices, we construct the desired measurement matrices
with various sizes and they show empirically comparable performance to that of
the corresponding Gaussian matricesComment: 5 pages, 3 figure
Johnson Type Bounds on Constant Dimension Codes
Very recently, an operator channel was defined by Koetter and Kschischang
when they studied random network coding. They also introduced constant
dimension codes and demonstrated that these codes can be employed to correct
errors and/or erasures over the operator channel. Constant dimension codes are
equivalent to the so-called linear authentication codes introduced by Wang,
Xing and Safavi-Naini when constructing distributed authentication systems in
2003. In this paper, we study constant dimension codes. It is shown that
Steiner structures are optimal constant dimension codes achieving the
Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension
codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain
Steiner structures. Then, we derive two Johnson type upper bounds, say I and
II, on constant dimension codes. The Johnson type bound II slightly improves on
the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known
Steiner structures is actually a family of optimal constant dimension codes
achieving both the Johnson type bounds I and II.Comment: 12 pages, submitted to Designs, Codes and Cryptograph
Minimum Pseudo-Weight and Minimum Pseudo-Codewords of LDPC Codes
In this correspondence, we study the minimum pseudo-weight and minimum
pseudo-codewords of low-density parity-check (LDPC) codes under linear
programming (LP) decoding. First, we show that the lower bound of Kelly,
Sridhara, Xu and Rosenthal on the pseudo-weight of a pseudo-codeword of an LDPC
code with girth greater than 4 is tight if and only if this pseudo-codeword is
a real multiple of a codeword. Then, we show that the lower bound of Kashyap
and Vardy on the stopping distance of an LDPC code is also a lower bound on the
pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this
lower bound is tight if and only if this pseudo-codeword is a real multiple of
a codeword. Using these results we further show that for some LDPC codes, there
are no other minimum pseudo-codewords except the real multiples of minimum
codewords. This means that the LP decoding for these LDPC codes is
asymptotically optimal in the sense that the ratio of the probabilities of
decoding errors of LP decoding and maximum-likelihood decoding approaches to 1
as the signal-to-noise ratio leads to infinity. Finally, some LDPC codes are
listed to illustrate these results.Comment: 17 pages, 1 figur
Sparse signal recovery by minimization under restricted isometry property
In the context of compressed sensing, the nonconvex minimization
with has been studied in recent years. In this paper, by generalizing
the sharp bound for minimization of Cai and Zhang, we show that the
condition in terms of
\emph{restricted isometry constant (RIC)} can guarantee the exact recovery of
-sparse signals in noiseless case and the stable recovery of approximately
-sparse signals in noisy case by minimization. This result is more
general than the sharp bound for minimization when the order of RIC is
greater than and illustrates the fact that a better approximation to
minimization is provided by minimization than that provided
by minimization
Reconstruction Guarantee Analysis of Binary Measurement Matrices Based on Girth
Binary 0-1 measurement matrices, especially those from coding theory, were
introduced to compressed sensing (CS) recently. Good measurement matrices with
preferred properties, e.g., the restricted isometry property (RIP) and
nullspace property (NSP), have no known general ways to be efficiently checked.
Khajehnejad \emph{et al.} made use of \emph{girth} to certify the good
performances of sparse binary measurement matrices. In this paper, we examine
the performance of binary measurement matrices with uniform column weight and
arbitrary girth under basis pursuit. Explicit sufficient conditions of exact
reconstruction %only including and are obtained, which improve the
previous results derived from RIP for any girth and results from NSP when
is odd. Moreover, we derive explicit , and
sparse approximation guarantees. These results further show that
large girth has positive impacts on the performance of binary measurement
matrices under basis pursuit, and the binary parity-check matrices of good LDPC
codes are important candidates of measurement matrices.Comment: accepted by IEEE ISIT 201
Unifying Decision Trees Split Criteria Using Tsallis Entropy
The construction of efficient and effective decision trees remains a key
topic in machine learning because of their simplicity and flexibility. A lot of
heuristic algorithms have been proposed to construct near-optimal decision
trees. ID3, C4.5 and CART are classical decision tree algorithms and the split
criteria they used are Shannon entropy, Gain Ratio and Gini index respectively.
All the split criteria seem to be independent, actually, they can be unified in
a Tsallis entropy framework. Tsallis entropy is a generalization of Shannon
entropy and provides a new approach to enhance decision trees' performance with
an adjustable parameter . In this paper, a Tsallis Entropy Criterion (TEC)
algorithm is proposed to unify Shannon entropy, Gain Ratio and Gini index,
which generalizes the split criteria of decision trees. More importantly, we
reveal the relations between Tsallis entropy with different and other split
criteria. Experimental results on UCI data sets indicate that the TEC algorithm
achieves statistically significant improvement over the classical algorithms.Comment: 6 pages, 2 figure
Maximum Multiflow in Wireless Network Coding
In a multihop wireless network, wireless interference is crucial to the
maximum multiflow (MMF) problem, which studies the maximum throughput between
multiple pairs of sources and sinks. In this paper, we observe that network
coding could help to decrease the impacts of wireless interference, and propose
a framework to study the MMF problem for multihop wireless networks with
network coding. Firstly, a network model is set up to describe the new conflict
relations modified by network coding. Then, we formulate a linear programming
problem to compute the maximum throughput and show its superiority over one in
networks without coding. Finally, the MMF problem in wireless network coding is
shown to be NP-hard and a polynomial approximation algorithm is proposed.Comment: 5 pages, 3 figures, submitted to ISIT 201
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