196 research outputs found
Non-convex Fraction Function Penalty: Sparse Signals Recovered from Quasi-linear Systems
The goal of compressed sensing is to reconstruct a sparse signal under a few
linear measurements far less than the dimension of the ambient space of the
signal. However, many real-life applications in physics and biomedical sciences
carry some strongly nonlinear structures, and the linear model is no longer
suitable. Compared with the compressed sensing under the linear circumstance,
this nonlinear compressed sensing is much more difficult, in fact also NP-hard,
combinatorial problem, because of the discrete and discontinuous nature of the
-norm and the nonlinearity. In order to get a convenience for sparse
signal recovery, we set most of the nonlinear models have a smooth quasi-linear
nature in this paper, and study a non-convex fraction function in
this quasi-linear compressed sensing. We propose an iterative fraction
thresholding algorithm to solve the regularization problem
for all . With the change of parameter , our algorithm could get a
promising result, which is one of the advantages for our algorithm compared
with other algorithms. Numerical experiments show that our method performs much
better compared with some state-of-art methods
Modified lp-norm regularization minimization for sparse signal recovery
In numerous substitution models for the \l_{0}-norm minimization problem
, the \l_{p}-norm minimization with have been
considered as the most natural choice. However, the non-convex optimization
problem are much more computational challenges, and are also NP-hard.
Meanwhile, the algorithms corresponding to the proximal mapping of the
regularization \l_{p}-norm minimization are limited to
few specific values of parameter . In this paper, we replace the
-norm with a modified function
. With change the
parameter , this modified function would like to interpolate the
\l_{p}-norm . By this transformation, we translated the
\l_{p}-norm regularization minimization into a modified
\l_{p}-norm regularization minimization . Then,
we develop the thresholding representation theory of the problem
, and based on it, the IT algorithm is proposed to
solve the problem for all . Indeed, we
could get some much better results by choosing proper , which is one of the
advantages for our algorithm compared with other methods. Numerical results
also show that, for some proper , our algorithm performs the best in some
sparse signal recovery problems compared with some state-of-art methods
Minimization of fraction function penalty in compressed sensing
In the paper, we study the minimization problem of a non-convex sparsity
promoting penalty function
in compressed sensing, which is called fraction function. Firstly, we discuss
the equivalence of minimization and fraction function minimization.
It is proved that there corresponds a constant such that, whenever
, every solution to also solves , that the
uniqueness of global minimizer of and its equivalence to
if the sensing matrix satisfies a restricted isometry property (RIP) and,
last but the most important, that the optimal solution to the regularization
problem also solves if the certain condition is
satisfied, which is similar to the regularization problem in convex optimal
theory. Secondly, we study the properties of the optimal solution to the
regularization problem including the first-order and the
second optimality condition and the lower and upper bound of the absolute value
for its nonzero entries. Finally, we derive the closed form representation of
the optimal solution to the regularization problem () for all
positive values of parameter , and propose an iterative thresholding
algorithm to solve the regularization problem . We also
provide a series of experiments to assess performance of the algorithm,
and the experiment results show that, compared with soft thresholding algorithm
and half thresholding algorithms, the algorithm performs the best in
sparse signal recovery with and without measurement noise.Comment: 12 page
Generalized singular value thresholding operator to affine matrix rank minimization problem
It is well known that the affine matrix rank minimization problem is NP-hard
and all known algorithms for exactly solving it are doubly exponential in
theory and in practice due to the combinational nature of the rank function. In
this paper, a generalized singular value thresholding operator is generated to
solve the affine matrix rank minimization problem. Numerical experiments show
that our algorithm performs effectively in finding a low-rank matrix compared
with some state-of-art methods
Nonconvex fraction function recovery sparse signal by convex optimization algorithm
In this paper, we will generate a convex iterative FP thresholding algorithm
to solve the problem . Two schemes of convex iterative FP
thresholding algorithms are generated. One is convex iterative FP thresholding
algorithm-Scheme 1 and the other is convex iterative FP thresholding
algorithm-Scheme 2. A global convergence theorem is proved for the convex
iterative FP thresholding algorithm-Scheme 1. Under an adaptive rule, the
convex iterative FP thresholding algorithm-Scheme 2 will be adaptive both for
the choice of the regularized parameter and parameter . These are
the advantages for our two schemes of convex iterative FP thresholding
algorithm compared with our previous proposed two schemes of iterative FP
thresholding algorithm. At last, we provide a series of numerical simulations
to test the performance of the convex iterative FP thresholding
algorithm-Scheme 2, and the simulation results show that our convex iterative
FP thresholding algorithm-Scheme 2 performs very well in recovering a sparse
signal
A New Nonconvex Strategy to Affine Matrix Rank Minimization Problem
The affine matrix rank minimization (AMRM) problem is to find a matrix of
minimum rank that satisfies a given linear system constraint. It has many
applications in some important areas such as control, recommender systems,
matrix completion and network localization. However, the problem (AMRM) is
NP-hard in general due to the combinational nature of the matrix rank function.
There are many alternative functions have been proposed to substitute the
matrix rank function, which lead to many corresponding alternative minimization
problems solved efficiently by some popular convex or nonconvex optimization
algorithms. In this paper, we propose a new nonconvex function, namely,
function (with ), to
approximate the rank function, and translate the NP-hard problem (AMRM) into
the function affine matrix rank minimization (TLAMRM)
problem. Firstly, we study the equivalence of problem (AMRM) and (TLAMRM), and
proved that the uniqueness of global minimizer of the problem (TLAMRM) also
solves the NP-hard problem (AMRM) if the linear map satisfies a
restricted isometry property (RIP). Secondly, an iterative thresholding
algorithm is proposed to solve the regularization problem (RTLAMRM) for all
. At last, some numerical results on low-rank
matrix completion problems illustrated that our algorithm is able to recover a
low-rank matrix, and the extensive numerical on image inpainting problems shown
that our algorithm performs the best in finding a low-rank image compared with
some state-of-art methods
Sparse Portfolio Selection via Non-convex Fraction Function
In this paper, a continuous and non-convex promoting sparsity fraction
function is studied in two sparse portfolio selection models with and without
short-selling constraints. Firstly, we study the properties of the optimal
solution to the problem including the first-order and
the second optimality condition and the lower and upper bound of the absolute
value for its nonzero entries. Secondly, we develop the thresholding
representation theory of the problem . Based on it, we
prove the existence of the resolvent operator of gradient of ,
calculate its analytic expression, and propose an iterative fraction penalty
thresholding (IFPT) algorithm to solve the problem .
Moreover, we also prove that the value of the regularization parameter
can not be chosen too large. Indeed, there exists
such that the optimal solution to the problem is equal
to zero for any . At last, inspired by the thresholding
representation theory of the problem , we propose an
iterative nonnegative fraction penalty thresholding (INFPT) algorithm to solve
the problem . Empirical results show that our
methods, for some proper , perform effective in finding the sparse
portfolio weights with and without short-selling constraints
Twin-Load: Building a Scalable Memory System over the Non-Scalable Interface
Commodity memory interfaces have difficulty in scaling memory capacity to
meet the needs of modern multicore and big data systems. DRAM device density
and maximum device count are constrained by technology, package, and signal in-
tegrity issues that limit total memory capacity. Synchronous DRAM protocols
require data to be returned within a fixed latency, and thus memory extension
methods over commodity DDRx interfaces fail to support scalable topologies.
Current extension approaches either use slow PCIe interfaces, or require
expensive changes to the memory interface, which limits commercial
adoptability. Here we propose twin-load, a lightweight asynchronous memory
access mechanism over the synchronous DDRx interface. Twin-load uses two
special loads to accomplish one access request to extended memory, the first
serves as a prefetch command to the DRAM system, and the second asynchronously
gets the required data. Twin-load requires no hardware changes on the processor
side and only slight soft- ware modifications. We emulate this system on a
prototype to demonstrate the feasibility of our approach. Twin-load has
comparable performance to NUMA extended memory and outperforms a page-swapping
PCIe-based system by several orders of magnitude. Twin-load thus enables
instant capacity increases on commodity platforms, but more importantly, our
architecture opens opportunities for the design of novel, efficient, scalable,
cost-effective memory subsystems.Comment: submitted to PACT1
Adaptive iterative singular value thresholding algorithm to low-rank matrix recovery
The problem of recovering a low-rank matrix from the linear constraints,
known as affine matrix rank minimization problem, has been attracting extensive
attention in recent years. In general, affine matrix rank minimization problem
is a NP-hard. In our latest work, a non-convex fraction function is studied to
approximate the rank function in affine matrix rank minimization problem and
translate the NP-hard affine matrix rank minimization problem into a
transformed affine matrix rank minimization problem. A scheme of iterative
singular value thresholding algorithm is generated to solve the regularized
transformed affine matrix rank minimization problem. However, one of the
drawbacks for our iterative singular value thresholding algorithm is that the
parameter , which influences the behaviour of non-convex fraction function
in the regularized transformed affine matrix rank minimization problem, needs
to be determined manually in every simulation. In fact, how to determine the
optimal parameter is not an easy problem. Here instead, in this paper, we
will generate an adaptive iterative singular value thresholding algorithm to
solve the regularized transformed affine matrix rank minimization problem. When
doing so, our new algorithm will be intelligent both for the choice of the
regularized parameter and the parameter
Iterative thresholding algorithm based on non-convex method for modified lp-norm regularization minimization
Recently, the \l_{p}-norm regularization minimization problem
has attracted great attention in compressed sensing.
However, the \l_{p}-norm in problem is
nonconvex and non-Lipschitz for all , and there are not many
optimization theories and methods are proposed to solve this problem. In fact,
it is NP-hard for all and . In this paper, we study two
modified \l_{p} regularization minimization problems to approximate the
NP-hard problem . Inspired by the good performance of Half
algorithm and algorithm in some sparse signal recovery problems, two
iterative thresholding algorithms are proposed to solve the problems
and
respectively. Numerical results show that our algorithms perform effectively in
finding the sparse signal in some sparse signal recovery problems for some
proper
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