scholarly journals Computational complexity versus statistical performance on sparse recovery problems

2019 ◽  
Vol 9 (1) ◽  
pp. 1-32
Author(s):  
Vincent Roulet ◽  
Nicolas Boumal ◽  
Alexandre d’Aspremont
Keyword(s):  
Computational Complexity ◽  
Compressed Sensing ◽  
Condition Number ◽  
Sparse Recovery ◽  
Algorithmic Complexity ◽  
Complexity Measure ◽  
Condition Numbers ◽  
Worst Case ◽  
Observation Matrix ◽  
Recovery Performance

Abstract We show that several classical quantities controlling compressed-sensing performance directly match classical parameters controlling algorithmic complexity. We first describe linearly convergent restart schemes on first-order methods solving a broad range of compressed-sensing problems, where sharpness at the optimum controls convergence speed. We show that for sparse recovery problems, this sharpness can be written as a condition number, given by the ratio between true signal sparsity and the largest signal size that can be recovered by the observation matrix. In a similar vein, Renegar’s condition number is a data-driven complexity measure for convex programmes, generalizing classical condition numbers for linear systems. We show that for a broad class of compressed-sensing problems, the worst case value of this algorithmic complexity measure taken over all signals matches the restricted singular value of the observation matrix which controls robust recovery performance. Overall, this means in both cases that, in compressed-sensing problems, a single parameter directly controls both computational complexity and recovery performance. Numerical experiments illustrate these points using several classical algorithms.

scholarly journals Wilkinson’s Bus: Weak Condition Numbers, with an Application to Singular Polynomial Eigenproblems

2020 ◽  
Vol 20 (6) ◽  
pp. 1439-1473
Author(s):  
Martin Lotz ◽  
Vanni Noferini
Keyword(s):  
Condition Number ◽  
Stochastic Perturbation ◽  
Weak Condition ◽  
Condition Numbers ◽  
Difficult Case ◽  
Singular Matrix ◽  
Machine Precision ◽  
Worst Case ◽  
New Approach ◽  
Matrix Polynomials

AbstractWe propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theories fail to predict the observed accuracy of computed solutions. To motivate our ideas, we present examples of problems that are discontinuous at a given input and even have infinite stochastic condition number, but where the solution is still computed to machine precision without relying on structured algorithms. Stimulated by the failure of classical and stochastic perturbation theory in capturing such phenomena, we define and analyse a weak worst-case and a weak stochastic condition number. This new theory is a more powerful predictor of the accuracy of computations than existing tools, especially when the worst-case and the expected sensitivity of a problem to perturbations of the input is not finite. We apply our analysis to the computation of simple eigenvalues of matrix polynomials, including the more difficult case of singular matrix polynomials. In addition, we show how the weak condition numbers can be estimated in practice.

scholarly journals Block Compressed Sensing of Images Using Adaptive Granular Reconstruction

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Ran Li ◽  
Hongbing Liu ◽  
Yu Zeng ◽  
Yanling Li
Keyword(s):  
Computational Complexity ◽  
Compressed Sensing ◽  
Structural Characteristic ◽  
Rate Distortion ◽  
Reconstructed Image ◽  
Hard Thresholding ◽  
Low Computational Complexity ◽  
Principle Components Analysis ◽  
Reconstruction Performance ◽  
Block Compressed Sensing

In the framework of block Compressed Sensing (CS), the reconstruction algorithm based on the Smoothed Projected Landweber (SPL) iteration can achieve the better rate-distortion performance with a low computational complexity, especially for using the Principle Components Analysis (PCA) to perform the adaptive hard-thresholding shrinkage. However, during learning the PCA matrix, it affects the reconstruction performance of Landweber iteration to neglect the stationary local structural characteristic of image. To solve the above problem, this paper firstly uses the Granular Computing (GrC) to decompose an image into several granules depending on the structural features of patches. Then, we perform the PCA to learn the sparse representation basis corresponding to each granule. Finally, the hard-thresholding shrinkage is employed to remove the noises in patches. The patches in granule have the stationary local structural characteristic, so that our method can effectively improve the performance of hard-thresholding shrinkage. Experimental results indicate that the reconstructed image by the proposed algorithm has better objective quality when compared with several traditional ones. The edge and texture details in the reconstructed image are better preserved, which guarantees the better visual quality. Besides, our method has still a low computational complexity of reconstruction.

WORST-CASE SCENARIOS FOR AMERICAN OPTIONS

2000 ◽  
Vol 03 (01) ◽  
pp. 25-58
Author(s):  
ROBERT BUFF
Keyword(s):  
Computational Complexity ◽  
Parameter Space ◽  
American Options ◽  
Worst Case ◽  
Knock Out ◽  
Early Exercise ◽  
Pricing Options ◽  
Diffusion Parameters ◽  
Volatility Models ◽  
Compute Time

One approach to cope with uncertain diffusion parameters when pricing options portfolios is to identify the parameters [Formula: see text] in a subset [Formula: see text] of the parameter space which form the worst-case for a particular portfolio. For the sell-side, this leads to a nonlinear algorithm that maximizes the expected liability under the risk-neutral measure. [Formula: see text] depends on the portfolio under consideration. Moreover, the algorithm must take into account that the exposure to [Formula: see text]-risk changes when non-vanilla components such as barrier or American options knock out or are exercised early. In this paper, we describe techniques to price portfolios with American options under worst-case scenarios based on uncertain volatility models. We also present heuristics which reduce the computational complexity that arises from the necessity to consider many early exercise combinations at a time. These heuristics reduce the compute time by almost one half.

scholarly journals Sparse Recovery Algorithm for Compressed Sensing Using Smoothed l0 Norm and Randomized Coordinate Descent

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 834
Author(s):  
Jin ◽  
Yang ◽  
Li ◽  
Liu
Keyword(s):  
Compressed Sensing ◽  
Image Denoising ◽  
Signal To Noise Ratio ◽  
Sparse Recovery ◽  
Coordinate Descent ◽  
Sparse Signal ◽  
Sparse Signal Recovery ◽  
Signal Recovery ◽  
Recovery Algorithm ◽  
Core Concepts

Compressed sensing theory is widely used in the field of fault signal diagnosis and image processing. Sparse recovery is one of the core concepts of this theory. In this paper, we proposed a sparse recovery algorithm using a smoothed l0 norm and a randomized coordinate descent (RCD), then applied it to sparse signal recovery and image denoising. We adopted a new strategy to express the (P0) problem approximately and put forward a sparse recovery algorithm using RCD. In the computer simulation experiments, we compared the performance of this algorithm to other typical methods. The results show that our algorithm possesses higher precision in sparse signal recovery. Moreover, it achieves higher signal to noise ratio (SNR) and faster convergence speed in image denoising.

Array DOA Estimation by the Model of Four Elements in Quare Array Based on the Theory of Compressed Sensing

2014 ◽  
Vol 635-637 ◽  
pp. 971-977 ◽  
Author(s):  
Rui Yuan ◽  
Jun Yue ◽  
Hong Xiu Gao
Keyword(s):  
Computer Simulation ◽  
Computational Complexity ◽  
Compressed Sensing ◽  
Matching Pursuit ◽  
Doa Estimation ◽  
Low Computational Complexity ◽  
Matching Pursuit Algorithm ◽  
Square Array

The DOA estimation by the model of four elements in the square array has studied based on the theory of compressed sensing. Using matching pursuit algorithm and orthogonality matching pursuit algorithm, the computer simulation was presented. The results show the method of DOA estimation by compressed sensing theory is simple, practical and low computational complexity.

scholarly journals A Fast Sparse Recovery Algorithm for Compressed Sensing Using Approximate l0 Norm and Modified Newton Method

Materials ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 1227 ◽  
Author(s):  
Dingfei Jin ◽  
Yue Yang ◽  
Tao Ge ◽  
Daole Wu
Keyword(s):  
Compressed Sensing ◽  
Newton Method ◽  
Simulation Experiment ◽  
Sparse Recovery ◽  
Signal Recovery ◽  
Recovery Efficiency ◽  
Recovery Algorithm ◽  
Modified Newton Method ◽  
Computer Simulation Experiment ◽  
Recovery Algorithms

In this paper, we propose a fast sparse recovery algorithm based on the approximate l0 norm (FAL0), which is helpful in improving the practicability of the compressed sensing theory. We adopt a simple function that is continuous and differentiable to approximate the l0 norm. With the aim of minimizing the l0 norm, we derive a sparse recovery algorithm using the modified Newton method. In addition, we neglect the zero elements in the process of computing, which greatly reduces the amount of computation. In a computer simulation experiment, we test the image denoising and signal recovery performance of the different sparse recovery algorithms. The results show that the convergence rate of this method is faster, and it achieves nearly the same accuracy as other algorithms, improving the signal recovery efficiency under the same conditions.

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].

A Grayscale Image Vulnerability Authentication System Based on Compressed Sensing

2015 ◽  
Vol 738-739 ◽  
pp. 533-537
Author(s):  
Xu Zhan ◽  
Yue Rong Lei ◽  
Hui Ming Zeng ◽  
Jian Ling Chen
Keyword(s):  
Compressed Sensing ◽  
Image Authentication ◽  
Singular Value ◽  
Grayscale Image ◽  
Arnold Transform ◽  
Edge Information ◽  
Observation Matrix ◽  
Image Edge ◽  
Authentication System ◽  
Value Decomposition

In this paper, we study compressed sensing algorithm and image authentication algorithm, present a grayscale image vulnerability authentication system based on compressed sensing. The system extracts the original grayscale image edge information by prewitt algorithm and observes the edge information by compressed sensing algorithm of OMP to generate the observation matrix . Then, the system scrambles the observation matrix by arnold transform algorithm and embeds it into the original grayscale image by singular value decomposition algorithm. We make experiment in order to test the system. The result is shown that the algorithm has good imperceptibility and can resist copying attack.

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