The “fast iterative shrinkage-thresholding algorithm,” a.k.a. FISTA, is one of the most well known first-order optimization scheme in literature, as it achieves worst-case $O(1/k^2)$ optimal convergence rate for objective function value. However, despite such an theoretical rate, practice (local) oscillatory behavior FISTA often damps its efficiency. Over past years, various efforts have been m...

Finding a sparse representation of a possibly noisy signal can be modeled as a variational minimization with `q-sparsity constraints for q less than one. Especially for real-time and on-line applications, one requires fast computations of these minimizers. However, there are no sufficiently fast algorithms, and to circumvent this limitation, we consider minimization up to a constant factor. We ...

This paper discusses a class of thresholding-based iterative selection procedures (TISP) for model selection and shrinkage. People have long before noticed the weakness of the convex l1-constraint (or the softthresholding) in wavelets and have designed many different forms of nonconvex penalties to increase model sparsity and accuracy. But for a nonorthogonal regression matrix, there is great d...

Signal denoising can not only enhance the signal to noise ratio (SNR) but also reduce the effect of noise. In order to satisfy the requirements of real-time signal denoising, an improved semisoft shrinkage real-time denoising method based on lifting wavelet transform was proposed. The moving data window technology realizes the real-time wavelet denoising, which employs wavelet transform based o...

We propose an accelerated path-following iterative shrinkage thresholding algorithm (APISTA) for solving high dimensional sparse nonconvex learning problems. The main difference between APISTA and the path-following iterative shrinkage thresholding algorithm (PISTA) is that APISTA exploits an additional coordinate descent subroutine to boost the computational performance. Such a modification, t...

— Image denoising is an important step in image compression and other image processing algorithms. Hard and soft thresholding algorithms are often used to denoise the images. Recently wavelet transform has been used as a tool to denoise the images. However, there are problems associated with the thresholding algorithms. There is no subjective way to determine the threshold. In this work, we imp...

The procedure of reducing noise or reduction before analyzing the time series is important to get accurate and reliable outcomes when building models. Wavelet Shrinkage consisting wavelets with thresholding a powerful mathematical method used reduce that can be exposed. It has time-series observations selects cut-off threshold level suitable for removing most noise. In this paper, natural gas p...

We consider recovery of low-rank matrices from noisy data by hard thresholding of singular values, in which empirical singular values below a threshold λ are set to 0. We study the asymptotic MSE (AMSE) in a framework where the matrix size is large compared to the rank of the matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. The AMSE-optimal choice of h...

Soft-thresholding is a sparse modeling method typically applied to wavelet denoising in statistical signal processing. It is also important in machine learning since it is an essential nature of the well-known LASSO (Least Absolute Shrinkage and Selection Operator). It is known that soft-thresholding, thus, LASSO suffers from a problem of dilemma between sparsity and generalization. This is cau...