L1 Optimization Problem Research Articles

Channel Estimation for Underwater Acoustic Communications in Impulsive Noise Environments: A Sparse, Robust, and Efficient Alternating Direction Method of Multipliers-Based Approach

Channel estimation in Underwater Acoustic Communication (UAC) faces significant challenges due to the non-Gaussian, impulsive noise in ocean environments and the inherent high dimensionality of the estimation task. This paper introduces a robust channel estimation algorithm by solving an l1−l1 optimization problem via the Alternating Direction Method of Multipliers (ADMM), effectively exploiting channel sparsity and addressing impulsive noise outliers. A non-monotone backtracking line search strategy is also developed to improve the convergence behavior. The proposed algorithm is low in complexity and has robust performance. Simulation results show that it exhibits a small performance deterioration of less than 1 dB for Channel Impulse Response (CIR) estimation in impulsive noise environments, nearly matching its performance under Additive White Gaussian Noise (AWGN) conditions. For Delay-Doppler (DD) doubly spread channel estimation, it maintains Bit Error Rate (BER) performance comparable to using ground truth channel information in both AWGN and impulsive noise environments. At-sea experimental validations for channel estimation in Orthogonal Frequency Division Multiplexing (OFDM) systems further underscore the fast convergence speed and high estimation accuracy of the proposed method.

On-the-fly compressive single-pixel foveation using the STOne transform

Compressive imaging allows one to sample an image below the Nyquist rate yet still accurately recover it from the measurements by solving an L1 optimization problem. The L1 solvers, however, are iterative and can require significant time to reconstruct the original signal. Intuitively, the reconstruction time can be reduced by reconstructing fewer total pixels. The human eye reduces the total amount of data it processes by having a spatially varying resolution, a method called foveation. In this work, we use foveation to achieve a 4x improvement in L1 compressive sensing reconstruction speed for hyperspectral images and video. Unlike previous works, the presented technique allows the high-resolution region to be placed anywhere in the scene after the subsampled measurements have been acquired, has no moving parts, and is entirely non-adaptive.

Fast interval estimation for discrete-time linear systems: An [formula omitted] optimization method

This paper studies interval estimation for discrete-time linear systems with unknown but bounded disturbances. Inspired by the parity space approach, we propose a point estimator with fixed-time convergence property. The estimator is combined with the zonotope-based interval analysis to achieve fast interval estimation. The parameter matrix in the estimator is optimized by minimizing the length of the edges of the outer box of the error zonotope. It is formulated as L1 optimization problem and can be efficiently solved by linear programming. Comparison studies illustrate the superiority of the proposed method over existing techniques.

Formula omitted] model-driven recursive multi-scale denoising network for image super-resolution

Most existing deep learning based single-image super-resolution (SISR) methods mainly improve the reconstruction performances from the perspective of data-driven, i.e., widening or deepening the networks according to the huge scale of the training data. However, it will bring a huge amount of weights and biases, and cost the expensive computations. Recently, some people have proposed a new frame for designing the deep networks according to the algorithms deduced from the ℓ2-optimization problem. But they did not consider the case with outliers. Since ℓ1-norm can describe the sparsity of the outliers better than ℓ2-norm, we propose an effective deep network designed according to the new algorithm deduced from the ℓ1-optimization problem. In our proposed method, an effective iterative algorithm for the ℓ1 reconstructed optimization problem is deduced based on the split Bregman algorithm, majorization–minimization algorithm, and soft thresholding operator. Then according to the deduced iterative algorithm, an effective deep network, named ℓ1 Model-Driven Recursive Multi-Scale Denoising Network (ℓ1-MRMDN), is designed. Due to the iteration form of the deduced algorithm, the proposed ℓ1-MRMDN contains an inner recursion and an outer recursion. Therefore, our proposed method can not only relieve its sensitiveness to the outliers because of the ℓ1 data fidelity term, but also avoid designing the deep network blindly via the guidance of prior knowledge. Extensive experimental results illustrate that our proposed method is superior to some related popular SISR methods.

L1 optimal controller synthesis for sampled‐data systems via piecewise linear kernel approximation

AbstractThis article provides a new framework for the so‐called L1 optimal control problem of sampled‐data systems, that is, the synthesis problem of a discrete‐time optimal controller minimizing the continuous‐time L∞‐induced norm of the closed‐loop system for a given continuous‐time plant. The main idea is to develop the approximation method called the piecewise linear kernel approximation (PLKA) method, by which the kernel function of the input operator together with the hold function of the output operator in the lifted representation of sampled‐data systems are approximated by piecewise linear functions. By defining adequate preadjoint operators, the PLKA method is shown to have the associated convergence rate of 1/N2 for the deterioration of the attainable L∞‐induced norm performance when the optimal controller synthesis is conducted approximately under the approximation parameter N. Compared with another existing procedure for controller synthesis through different approximation treatment called the piecewise linear input approximation method, we further show that the proposed PLKA method has a quantitatively improved bound on the performance deterioration. Finally, numerical examples are studied to verify the effectiveness of the proposed PLKA method.

Sparse optimal stochastic control

In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the L0 cost functional, in general, the value function is not differentiable in the domain. Then, we characterize the value function as a viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation. Based on the result, we derive a necessary and sufficient condition for the L0 optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with L1 optimal control problem and show an equivalence theorem.

Recovery guarantees for polynomial coefficients from weakly dependent data with outliers

Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; however, additional information about the data’s structure or on the behavior of the unknown function can make the task well-posed. In this work, we study the problem of learning nonlinear functions from corrupted and weakly dependent data. The learning problem is recast as a sparse robust linear regression problem where we incorporate both the unknown coefficients and the corruptions in a basis pursuit framework. The main contribution of our paper is to provide a reconstruction guarantee for the associated ℓ1-optimization problem where the sampling matrix is formed from weakly dependent data. Specifically, we prove that the sampling matrix satisfies the null space property and the stable null space property, provided that the data is compact and satisfies a suitable concentration inequality. We show that our recovery results are applicable to various types of weakly dependent data such as exponentially strongly α-mixing data, geometrically C-mixing data, and uniformly ergodic Markov chain. Our theoretical results are verified via several numerical simulations.

Direct Least Absolute Deviation Fitting of Ellipses

Scattered data from edge detection usually involve undesired noise which seriously affects the accuracy of ellipse fitting. In order to alleviate this kind of degradation, a method of direct least absolute deviation ellipse fitting by minimizing the ℓ1 algebraic distance is presented. Unlike the conventional ℓ2 estimators which tend to produce a satisfied performance on ideal and Gaussian noise data, while do a poor job for non-Gaussian outliers, the proposed method shows very competitive results for non-Gaussian noise. In addition, an efficient numerical algorithm based on the split Bregman iteration is developed to solve the resulting ℓ1 optimization problem, according to which the computational burden is significantly reduced. Furthermore, two classes of ℓ2 solutions are introduced as the initial guess, and the selection of algorithm parameters is studied in detail; thus, it does not suffer from the convergence issues due to poor initialization which is a common drawback existing in iterative-based approaches. Numerical experiments reveal that the proposed method is superior to its ℓ2 counterpart and outperforms some of the state-of-the-art algorithms for both Gaussian and non-Gaussian artifacts.

Real Time Trajectory Optimization of L1 Optimal Control Problem using Gauss-Pseudo-spectral Method

Real time solution to a L1 optimal control problem is presented in this paper. The solution is obtained by solving the states, control and co-states in real time, satisfying the necessary conditions of Pontryagin’s minimum principle. In space-craft trajectory optimization, L1 norm is used as the objective function for minimizing fuel consumption. In this paper, a real-time solution to the orbit transfer problem is presented with the objective to correct any mid-course deviations of the spacecraft by generating optimal trajectories in every guidance cycle. Generally, for a real-time solution, a quadratic cost function is preferred for faster convergence. But, L2 norm does not minimize the fuel consumption when applied to spacecraft trajectory optimization problems. Therefore, an optimal control formulation with L1 norm is considered in this work. Solution to such problems are challenging because of the non-smooth nature of control profile. Hence, closed form solutions for L1 problems are very difficult and sometimes not feasible. Numerical solution to such problems is feasible, with penalty of high computation time. Additionally, the solution obtained is very sensitive to the initial guess values and convergence is not ensured always. In this paper, a simple Newton-Raphson based procedure is used for obtaining the solution to L1 optimal control problem. The proposed method is least sensitive to initial guess values which is an advantage over general gradient based algorithms. A comparison of L1 optimal control problem over L2 problem is discussed with the help of a double integrator system to present the advantages of using L1 norm for fuel minimization. The proposed method is able to converge within a guidance computation cycle making it suitable for real time implementation.

Spline-like Chebyshev polynomial model for compressive imaging

This paper introduces a novel spline-like parametric model for an image representation obtained directly from compressive imaging (CI) measurements. As a representation basis we use Chebyshev polynomials. To avoid common problem of blocking artifacts in block-based reconstruction algorithms, a desired number of derivatives are equated on the block boundaries in a spline-like fashion. This introduces a new set of constraints that fits into CI setup. Unlike splines, the proposed system of equations is underdetermined to provide a necessary degree of freedom for achieving sparsity by solving an ℓ1 optimization problem. Recovered coefficients of the parametric model can be further used for image processing where operations can be elegantly defined and calculated. This offers a new framework for acquisition and processing of analog signals without converting them into samples. Experiments on real measurements show that our model achieves sparse representation without visible blocking artifacts from a reduced set of CI measurements.

Application of a Priori Hyper-Reduction to the Nonlinear Dynamic Finite Element Simulation of a Rolling Car Tire

Abstract The a priori multi-expansion modal (MEM) hyper-reduction method for nonlinear structural dynamics finite element (FE) problems is described, extended, and applied to the dynamic nonlinear FE simulation of a car tire rolling over a rough road surface, including geometrical, material, follower force, and contact nonlinearities. Rather than using time-domain training simulation data, a priori calculated static nonlinear contact configurations and eigenmodes are used as a reduction basis and to perform the hyper-reduction element sampling. The hyper-reduction element sampling is performed by solving an L1 optimization problem subject to a set of equality constraints. This yields a reduced set of elements with an a priori known cardinality, which depends on the amount of constraints taken into consideration and the reduction basis dimension. It is shown that care has to be taken during the hyper-reduction process when considering distributed contact constraints, as is the case for, e.g., a tire rolling over a rough road surface. Large speedup factors can be obtained while still retaining a relatively high accuracy, making application of the MEM method suitable to, for instance, industrial design optimization cases.

External prior learning and internal mean sparse coding for image denoising

Image prior and sparse coding learning methods have important uses in image denoising. Many denoising methods learn priors either from the noisy image itself or an external clean image dataset. But using only these as priors does not always reconstruct the image effectively. In addition, when the image is corrupted by noise, the local sparse coding coefficient obtained from a noisy image patch is inaccurate, restricting denoising performance. We present a noise removal framework based on external prior learning and an internal mean sparse coding method, making use of the innate sparsity and nonlocal self-similarity (NSS) of natural images. Specifically, we first obtain external priors from a clean natural image dataset by Gaussian mixture model. The external priors are applied to guide the subspace clustering of internal noisy image patches, and a compact dictionary is generated for each internal noisy patch cluster. Then an internal mean sparse coding strategy based on NSS is introduced into the sparse representation model, whose regularization parameters then are deduced through a Bayesian framework. An iterative shrinkage method is employed to solve the l1-optimization problem in the sparse representation model. Application of the noise removal model to 16 test images demonstrates denoising performance exceeding other competing methods.

Kernel Approximation Approach to the L1 Optimal Sampled-Data Controller Synthesis Problem

This paper is concerned with a new framework called the kernel approximation approach to the L1 optimal controller synthesis problem of sampled-data systems. On the basis of the lifted representation of sampled-data systems, which contains an input operator and an output operator, this paper introduces a method for approximating the kernel function of the input operator and the hold function of the output operator by piecewise constant functions. Through such a method, the L1 optimal sampled-data controller synthesis problem could be (almost) equivalently converted into the discrete-time l1 optimal controller synthesis problem. This paper further establishes an important inequality that forms the theoretical validity of the kernel approximation approach for tackling the L1 optimal sampled-data controller synthesis problem.

Fast algorithm for the lasso based L 1 -graph construction

The lasso-based L 1 -graph is used in many applications since it can effectively model a set of data points as a graph. The lasso is a popular regression approach and the L 1 -graph represents data points as nodes by using the regression result. More specifically, by solving the L 1 -optimization problem of the lasso, the sparse regression coefficients are used to obtain the weights of the edges in the graph. Conventional graph structures such as k -NN graph use two steps, adjacency searching and weight selection, for constructing the graph whereas the lasso-based L 1 -graph derives the adjacency structure as well as the edge weights simultaneously by using a coordinate descent. However, the construction cost of the lasso-based L 1 -graph is impractical for large data sets since the coordinate descent iteratively updates the weights of all edges until convergence. Our proposal, Castnet, can efficiently construct the lasso-based L 1 -graph. In order to avoid updating the weights of all edges, we prune edges that cannot have nonzero weights before entering the iterations. In addition, we update edge weights only if they are nonzero in the iterations. Experiments show that Castnet is significantly faster than existing approaches.

Large‐Volume Data Compression Using Compressed Sensing for Meteorological Radar

SUMMARYIn Japan, severe weather phenomena such as heavy rains and tornados sometimes cause meteorological disasters. In many cases, these are micro scale phenomena in the sense of spatial and temporal resolutions, which make it difficult to detect them with conventional meteorological radars due to their insufficient spatial and temporal resolutions. Therefore, we have been developing meteorological radars with high resolution and accuracy such as phased array radar (PAR) and Ku‐band broad band radar (BBR), and radar network systems consisting of multiple PARs and BBRs to realize further enhancement of the radar performance in terms of efficiency and accuracy. These high‐resolution radars, however, definitely produce large‐volume data, which is unacceptable in a current backbone information network. In order to solve this problem, in this paper, we tackle the compression of the large‐volume radar data by using Compressed sensing (CS), which can realize highly efficient data compression for sparse signals. When using CS, the radar data are compressed by projecting it onto a randomly generated subspace, and the compressed data are reconstructed by solving a simple ℓ1 optimization problem. We apply the CS‐based data compression scheme to measured radar reflectivity factor, and evaluate the relation between compression ratio and reconstruction accuracy. For the compression ratio of 0.3, rainfall rate calculated from the reconstructed radar reflectivity factor has a mean error of ‐0.89 mm/h with more than 30 dBZ precipitation.

Sparse RBF surface representations

In this paper, we propose a sparse surface representation for arbitrary surface models (point clouds, mesh models, continuous surface models, etc.). We approximate the input surface model with radial basis functions (RBF) whose centers are located on the medial axis of the input object surface. The sparsity of the RBF representation is achieved by solving an L1 optimization problem. Experimental results demonstrate that our method needs much less number of parameters to represent the input surface model with good accuracy. The sparse representation is useful in various applications, such as saving memory space in storing the surface models, and saving time in transmission of the surface models on the internet.

A discriminant sparse representation graph-based semi-supervised learning for hyperspectral image classification

The classification of hyperspectral image with a paucity of labeled samples is a challenging task. In this paper, we present a discriminant sparse representation (DSR) graph for semi-supervised learning (SSL) to address this problem. For graph-based methods, how to construct a graph among the pixels is the key to a successful classification. Our graph construction method contains two steps. Sparse representation (SR) method is first employed to estimate the probability matrix of the pairwise pixels belonging to the same class, and then this probability matrix is integrated into the SR graph, which can be obtained by solving an l1 optimization problem, to form a DSR graph. Experiments on Hyperion and AVIRIS hyperspectral data show that our proposed method outperforms state of the art.

스크린 콘텐츠 비디오의 압축을 위한 인접 화소의 가중 합을 이용한 적응적 Residual DPCM 기법

lx'`yz$ -ze`{('IJ|l$g. p01G}=> !~€E ‚3.1% o' k%g. AbstractIn this paper, we propose a novel residual differential pulse-code modulation (RDPCM) coding technique to improve coding efficiency of screen content videos. The proposed method uses a weighted combination of adjacent residues to provide an accurate estimate in RDPCM. The weights are trained in previously coded samples by using an L1 optimization problem with the least absolute shrinkage and selection operation (LASSO). The proposed method achieves BD-rate saving about 3.1% in all-intra coding. Keyword : HEVC, screen content video coding, RDPCM

Inverse sparse tracker with a locally weighted distance metric.

Sparse representation has been recently extensively studied for visual tracking and generally facilitates more accurate tracking results than classic methods. In this paper, we propose a sparsity-based tracking algorithm that is featured with two components: 1) an inverse sparse representation formulation and 2) a locally weighted distance metric. In the inverse sparse representation formulation, the target template is reconstructed with particles, which enables the tracker to compute the weights of all particles by solving only one l1 optimization problem and thereby provides a quite efficient model. This is in direct contrast to most previous sparse trackers that entail solving one optimization problem for each particle. However, we notice that this formulation with normal Euclidean distance metric is sensitive to partial noise like occlusion and illumination changes. To this end, we design a locally weighted distance metric to replace the Euclidean one. Similar ideas of using local features appear in other works, but only being supported by popular assumptions like local models could handle partial noise better than holistic models, without any solid theoretical analysis. In this paper, we attempt to explicitly explain it from a mathematical view. On that basis, we further propose a method to assign local weights by exploiting the temporal and spatial continuity. In the proposed method, appearance changes caused by partial occlusion and shape deformation are carefully considered, thereby facilitating accurate similarity measurement and model update. The experimental validation is conducted from two aspects: 1) self validation on key components and 2) comparison with other state-of-the-art algorithms. Results over 15 challenging sequences show that the proposed tracking algorithm performs favorably against the existing sparsity-based trackers and the other state-of-the-art methods.

圧縮センシングを用いた気象用レーダの大容量観測データの圧縮

In Japan, severe weather phenomena such as heavy rains and tornados sometimes cause meteorological disasters. In many cases, these are micro scale phenomena in the sense of spatial and temporal resolutions, which make it difficult to detect them with conventional meteorological radars due to their insufficient spatial and temporal resolutions. Therefore, we have been developing meteorological radars with high resolution and accuracy such as phased array radar (PAR) and Ku-band broadband radar (BBR), and radar network systems consisting of multiple PARs and BBRs to realize further enhancement of the radar performance in terms of efficiency and accuracy. These high-resolution radars, however, definitely produce large-volume data, which is unacceptable in a current backbone information network. In order to solve this problem, in this paper, we tackle the compression of the large-volume radar data by using Compressed sensing (CS), which can realize highly efficient data compression for sparse signals. When using CS, the radar data is compressed by projecting it onto a randomly generated subspace, and the compressed data is reconstructed by solving a simple ℓ1 optimization problem. We apply the CS-based data compression scheme to measured radar reflectivity factor, and evaluate the relation between compression ratio and reconstruction accuracy. For the compression ratio of 0.3, rainfall rate calculated from the reconstructed radar reflectivity factor has a mean error of -0.89 mm/h with more than 30 dBZ precipitation.