scholarly journals Block-Sparse Recovery with Optimal Block Partition

2021 ◽  
Author(s):  
Hiroki Kuroda ◽  
Daichi Kitahara
Keyword(s):  
Penalty Function ◽  
A Priori ◽  
Convex Relaxation ◽  
Optimal Solution ◽  
Sparse Recovery ◽  
Sparse Signals ◽  
L1 Norm ◽  
Recovery Method ◽  
Block Partition ◽  
L2 Norm

This paper presents a convex recovery method for block-sparse signals whose block partitions are unknown a priori. We first introduce a nonconvex penalty function, where the block partition is adapted for the signal of interest by minimizing the mixed l2/l1 norm over all possible block partitions. Then, by exploiting a variational representation of the l2 norm, we derive the proposed penalty function as a suitable convex relaxation of the nonconvex one. For a block-sparse recovery model designed with the proposed penalty, we develop an iterative algorithm which is guaranteed to converge to a globally optimal solution. Numerical experiments demonstrate the effectiveness of the proposed method.

scholarly journals Block-Sparse Recovery with Optimal Block Partition

2021 ◽  
Author(s):  
Hiroki Kuroda ◽  
Daichi Kitahara
Keyword(s):  
Penalty Function ◽  
A Priori ◽  
Convex Relaxation ◽  
Optimal Solution ◽  
Sparse Recovery ◽  
Sparse Signals ◽  
L1 Norm ◽  
Recovery Method ◽  
Block Partition ◽  
L2 Norm

This paper presents a convex recovery method for block-sparse signals whose block partitions are unknown a priori. We first introduce a nonconvex penalty function, where the block partition is adapted for the signal of interest by minimizing the mixed l2/l1 norm over all possible block partitions. Then, by exploiting a variational representation of the l2 norm, we derive the proposed penalty function as a suitable convex relaxation of the nonconvex one. For a block-sparse recovery model designed with the proposed penalty, we develop an iterative algorithm which is guaranteed to converge to a globally optimal solution. Numerical experiments demonstrate the effectiveness of the proposed method.

scholarly journals Block-Sparse Recovery with Optimal Block Partition

2021 ◽  
Author(s):  
Hiroki Kuroda ◽  
Daichi Kitahara
Keyword(s):  
Penalty Function ◽  
A Priori ◽  
Convex Relaxation ◽  
Optimal Solution ◽  
Sparse Recovery ◽  
Sparse Signals ◽  
L1 Norm ◽  
Recovery Method ◽  
Block Partition ◽  
L2 Norm

This paper presents a convex recovery method for block-sparse signals whose block partitions are unknown a priori. We first introduce a nonconvex penalty function, where the block partition is adapted for the signal of interest by minimizing the mixed l2/l1 norm over all possible block partitions. Then, by exploiting a variational representation of the l2 norm, we derive the proposed penalty function as a suitable convex relaxation of the nonconvex one. For a block-sparse recovery model designed with the proposed penalty, we develop an iterative algorithm which is guaranteed to converge to a globally optimal solution. Numerical experiments demonstrate the effectiveness of the proposed method.

Nonuniform recovery of fusion frame structured sparse signals

2017 ◽  
Vol 15 (03) ◽  
pp. 333-352
Author(s):  
Yu Xia ◽  
Song Li
Keyword(s):  
Probability Distribution ◽  
Compressed Sensing ◽  
Fourier Coefficients ◽  
A Priori ◽  
Sparse Recovery ◽  
A Priori Knowledge ◽  
Sparse Signals ◽  
Fusion Frame ◽  
Redundant Representation ◽  
Vector Valued

This paper considers the nonuniform sparse recovery of block signals in a fusion frame, which is a collection of subspaces that provides redundant representation of signal spaces. Combined with specific fusion frame, the sensing mechanism selects block-vector-valued measurements independently at random from a probability distribution [Formula: see text]. If the probability distribution [Formula: see text] obeys a simple incoherence property and an isotropy property, we can faithfully recover approximately block sparse signals via mixed [Formula: see text]-minimization in ways similar to Compressed Sensing. The number of measurements is significantly reduced by a priori knowledge of a certain incoherence parameter [Formula: see text] associated with the angles between the fusion frame subspaces. As an example, the paper shows that an [Formula: see text]-sparse block signal can be exactly recovered from about [Formula: see text] Fourier coefficients combined with fusion frame [Formula: see text], where [Formula: see text].

scholarly journals Kernel Sparse Representation with Hybrid Regularization for On-Road Traffic Sensor Data Imputation

Sensors ◽  
2018 ◽  
Vol 18 (9) ◽  
pp. 2884 ◽  
Author(s):  
Xiaobo Chen ◽  
Cheng Chen ◽  
Yingfeng Cai ◽  
Hai Wang ◽  
Qiaolin Ye
Keyword(s):  
Data Analysis ◽  
Sparse Representation ◽  
Intelligent Transportation Systems ◽  
Missing Values ◽  
Road Traffic ◽  
Linear Representation ◽  
Transportation Systems ◽  
Sensor Data ◽  
L1 Norm ◽  
L2 Norm

The problem of missing values (MVs) in traffic sensor data analysis is universal in current intelligent transportation systems because of various reasons, such as sensor malfunction, transmission failure, etc. Accurate imputation of MVs is the foundation of subsequent data analysis tasks since most analysis algorithms need complete data as input. In this work, a novel MVs imputation approach termed as kernel sparse representation with elastic net regularization (KSR-EN) is developed for reconstructing MVs to facilitate analysis with traffic sensor data. The idea is to represent each sample as a linear combination of other samples due to inherent spatiotemporal correlation, as well as periodicity of daily traffic flow. To discover few yet correlated samples and make full use of the valuable information, a combination of l1-norm and l2-norm is employed to penalize the combination coefficients. Moreover, the linear representation among samples is extended to nonlinear representation by mapping input data space into high-dimensional feature space, which further enhances the recovery performance of our proposed approach. An efficient iterative algorithm is developed for solving KSR-EN model. The proposed method is verified on both an artificially simulated dataset and a public road network traffic sensor data. The results demonstrate the effectiveness of the proposed approach in terms of MVs imputation.

scholarly journals Gradient Projection with Approximate L0 Norm Minimization for Sparse Reconstruction in Compressed Sensing

Sensors ◽  
2018 ◽  
Vol 18 (10) ◽  
pp. 3373 ◽  
Author(s):  
Ziran Wei ◽  
Jianlin Zhang ◽  
Zhiyong Xu ◽  
Yongmei Huang ◽  
Yong Liu ◽  
...  
Keyword(s):  
Image Reconstruction ◽  
Compressed Sensing ◽  
Objective Function ◽  
Signal Reconstruction ◽  
Reconstruction Error ◽  
Sparse Signal ◽  
Reconstruction Accuracy ◽  
L1 Norm ◽  
Function Model ◽  
L2 Norm

In the reconstruction of sparse signals in compressed sensing, the reconstruction algorithm is required to reconstruct the sparsest form of signal. In order to minimize the objective function, minimal norm algorithm and greedy pursuit algorithm are most commonly used. The minimum L1 norm algorithm has very high reconstruction accuracy, but this convex optimization algorithm cannot get the sparsest signal like the minimum L0 norm algorithm. However, because the L0 norm method is a non-convex problem, it is difficult to get the global optimal solution and the amount of calculation required is huge. In this paper, a new algorithm is proposed to approximate the smooth L0 norm from the approximate L2 norm. First we set up an approximation function model of the sparse term, then the minimum value of the objective function is solved by the gradient projection, and the weight of the function model of the sparse term in the objective function is adjusted adaptively by the reconstruction error value to reconstruct the sparse signal more accurately. Compared with the pseudo inverse of L2 norm and the L1 norm algorithm, this new algorithm has a lower reconstruction error in one-dimensional sparse signal reconstruction. In simulation experiments of two-dimensional image signal reconstruction, the new algorithm has shorter image reconstruction time and higher image reconstruction accuracy compared with the usually used greedy algorithm and the minimum norm algorithm.

scholarly journals Smoothing Approximation to the Square-Root Exact Penalty Function

2016 ◽  
Vol 4 (1) ◽  
pp. 87-96
Author(s):  
Yaqiong Duan ◽  
Shujun Lian
Keyword(s):  
Penalty Function ◽  
Original Problem ◽  
Optimal Solution ◽  
Penalty Functions ◽  
Exact Penalty Functions ◽  
Exact Penalty ◽  
Square Root ◽  
Inequality Constrained Optimization ◽  
Numerical Examples ◽  
Mild Conditions

AbstractIn this paper, smoothing approximation to the square-root exact penalty functions is devised for inequality constrained optimization. It is shown that an approximately optimal solution of the smoothed penalty problem is an approximately optimal solution of the original problem. An algorithm based on the new smoothed penalty functions is proposed and shown to be convergent under mild conditions. Three numerical examples show that the algorithm is efficient.

MISSION-ADAPTABLE ROUTE PLANNING IN UNCERTAIN AND ADVERSARIAL ENVIRONMENT

2006 ◽  
Vol 15 (05) ◽  
pp. 803-821 ◽  
Author(s):  
PING YAN ◽  
MINGYUE DING ◽  
CHANGWEN ZHENG
Keyword(s):  
Real Time ◽  
A Priori ◽  
Route Planning ◽  
Optimal Solution ◽  
Dijkstra Algorithm ◽  
Solution Quality ◽  
Model Complex ◽  
Local Methods ◽  
Domain Specific Knowledge ◽  
Planning Algorithm

In this paper, the route-planning problems of Unmanned Aerial Vehicle (UAV) in uncertain and adversarial environment are addressed, including not only single-mission route planning in known a priori environment, but also the route replanning in partially known and mission-changeable environments. A mission-adaptable hybrid route-planning algorithm based on flight roadmap is proposed, which combines existing global and local methods (Dijkstra algorithm, SAS and D*) into a two-level framework. The environment information and constraints for UAV are integrated into the procedure of building flight roadmap and searching for routes. The route-planning algorithm utilizes domain-specific knowledge and operates in real time with near-optimal solution quality, which is important to uncertain and adversarial environment. Other planners do not provide all of the functionality, namely real-time planning and replanning, near-optimal solution quality, and the ability to model complex 3D constraints.

Smoothing Approximation to the New Exact Penalty Function with Two Parameters

2021 ◽  
pp. 2140010
Author(s):  
Jing Qiu ◽  
Jiguo Yu ◽  
Shujun Lian
Keyword(s):  
Global Solution ◽  
Penalty Function ◽  
Original Problem ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Exact Penalty ◽  
Constrained Optimization Problems ◽  
Inequality Constrained Optimization ◽  
Nonlinear Inequality ◽  
Two Parameters

In this paper, we propose a new non-smooth penalty function with two parameters for nonlinear inequality constrained optimization problems. And we propose a twice continuously differentiable function which is smoothing approximation to the non-smooth penalty function and define the corresponding smoothed penalty problem. A global solution of the smoothed penalty problem is proved to be an approximation global solution of the non-smooth penalty problem. Based on the smoothed penalty function, we develop an algorithm and prove that the sequence generated by the algorithm can converge to the optimal solution of the original problem.

Velocity obstacle–based conflict resolution and recovery method

2021 ◽  
pp. 1-23
Author(s):  
F. Sun ◽  
Y. Chen ◽  
X. Xu ◽  
Y. Mu ◽  
Z. Wang
Keyword(s):  
Conflict Resolution ◽  
Optimal Solution ◽  
Mitigation Strategy ◽  
Resolution Method ◽  
Recovery Method ◽  
Aircraft Flight ◽  
Resolution Model ◽  
Group Welfare ◽  
Velocity Obstacle ◽  
Conflict Mitigation

ABSTRACT Considering the shortcomings of current methods for real-time resolution of two-aircraft flight conflicts, a geometric optimal conflict resolution and recovery method based on the velocity obstacle method for two aircraft and a cooperative conflict resolution method for multiple aircraft are proposed. The conflict type was determined according to the relative position and velocity of the aircraft, and a corresponding conflict mitigation strategy was selected. A resolution manoeuvre and a recovery manoeuvre were performed. On the basis of a two-aircraft conflict resolution model, a multi-aircraft cooperative conflict resolution game was constructed to identify an optimal solution for maximising group welfare. The solution and recovery method is simple and effective, and no new flight conflicts are introduced during track recovery. For multi-aircraft conflict resolution, an equilibrium point that maximises the welfare function of the group was identified, and thus, an optimal strategy for multi-aircraft conflict resolution was obtained.

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