Questions tagged [convex-optimization]

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2 votes
0 answers
32 views

constructing a weighted penalty function as function of position for elastic net

I am solving a "special" elastic net like regularized least squares problem $$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_2 + \lambda_1 {\...
  • 616
0 votes
0 answers
21 views

Covariance fitting and Procrustes

I have seen this trick to simplify an optimization problem. I would like to understand the logic behind it. Take two matrices $A$ and $B$ of dimension $N \times N$ and suppose that matrix $B$ is ...
  • 103
1 vote
0 answers
89 views

Implementing optimization part of blind deconvolution in Python

I am trying to implement the paper Qi Shan, Zhaorong Li, Jiaya Jia, Chi-Keung Tang - Fast Image/Video Upsampling and need some help with implementing minimizing an energy function (7) shown below. ...
  • 111
6 votes
1 answer
147 views

Solving a Weighted Basis Pursuit Denoising Problem (BPDN) with MATLAB / CVX

Following up from an answer by @Royi on adding weights to BPDN problem , I would like to use CVX to test this approach. How can we formulate in CVX the regularized LS L1 norm with weights given by a ...
  • 616
4 votes
1 answer
81 views

Adding Variance \ Weights Information When Solving a Basis Pursuit Denoising Problem (BPDN)

Having a "measured" vector $\mathbf{y}$ with its statistics (counts or variance per element), one can use weighted least squares approach to solve the linear system $$\mathbf{A}\mathbf{x} = \...
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5 votes
1 answer
87 views

How Could One Accelerate the Convergence of the Least Mean Squares (LMS) Filter?

How can the convergence of an LMS filter be accelerated? Can we do better than the vanilla algorithm?
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5 votes
1 answer
108 views

Converting Hadamard Product into Matrix Multiplication in Image Deconvolution with Total Variation (TV) Using ADMM

I would like to solve the following Image Deconvolution equation by ADMM. $$\mathbf { \min\frac{1}{2}\Vert{Cx-b}\Vert_2^2+\Vert w\circ (D x)\Vert_1 \tag 1}$$ Where, $x$ is a vector of unknown pixel ...
1 vote
2 answers
112 views

Optimization of square matrix multiplied with another matrix to have the final result a unitary matrix

I have a square matrix $D$ whose size is $m \times m$ multiplied with another $m \times m$ square matrix $C$, I need to optimize the matrix $C$ to have a unitary matrix $DC$. I mean optimize the ...
0 votes
1 answer
70 views

time-domain channel estimation based on two vectors optimization

Let's the input data vector $X = [X_1, X_2, X_3, X_4, X_5,X_6,X_7,X_8];$ where $[X_7,X_8]$ are well known, and the vector $y = h*X$ where $*$ is the convolution operation and $h = [h_1,h_2]$ is the ...
5 votes
1 answer
194 views

Is Sum of Absolute Value / $ {L}_{1} $ Norm of Differences Convex?

I'm not sure how to approach this exercise. One idea is to derive it w.r.t z, show that there is a min-extremum at $z=f_k$ and then show that for each value from the right and the left of the loss ...
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5 votes
2 answers
63 views

Justification for Squared $ {L}_{2} $ Data and Smoothness Term as an Error Bound

Often in variational methods (and not only) we have an energy that is of the form: $$E(u) = \frac{1}{2}\|f-u\|^2_2 + \frac{\alpha}{2}\|\psi(u)\|^2_2,$$ where the first term is referred to as the data ...
7 votes
2 answers
329 views

Quadratic Programming with Linear Equality Constraints

I need to solve an equality constrained minimization problem as give below $$\min_{\textbf{w}} \mathbf{w}^TR\mathbf{w} $$ such that $$X\mathbf{w} = \mathbf{1}$$ where $R\in \mathbb{R}^{n\times n}$ is ...
  • 331
0 votes
1 answer
47 views

Least Squares Filter Design: Deriving the Objective Function

I'm following the derivation in this paper A Comb Filter Design Using Fractional-Sample Delay to obtain the objective function for the least-squares filter design. N-order FIR filter: $H(z) = \sum_{...
  • 113
2 votes
0 answers
30 views

Image Restoration and Standard Forms of Second Order Cone Programming (SOCP)

I'm studying the application of SOCP methods in Image restoration And I want to understand the difference between the two formulas of SOCP and how they are related. Standard form (1) : min $f^{t}x $ ...
6 votes
1 answer
208 views

Solving LASSO (Basis Pursuit Denoising Form) with LARS

I'm now working on using LARS (Least Angle Regression) algorithm to solve a LASSO problem in Basis Pursuit Denoising form like: \begin{align*} \quad && \arg \min_{\beta}{\left\| y - X\beta \...
  • 63
0 votes
1 answer
121 views

On the Use of OMP Algorithm to Estimate Sparse Vector

As known, Orthogonal Matching Pursuit (OMP) Algorithm is to recover the sparse channel after convolution with another vector. But when I implement that in MATLAB, I don't get the sparse vector ...
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5 votes
1 answer
167 views

How to solve ADMM for TV Minimization Problem For Different Sizes $A$ and $x$ in $Ax=b$

I have matrix $A$ that is $(M \times M)$ square matrix, $x$ $(M \times N)$ matrix, $b$ is $(M \times N)$ matrix. Knowing $A$ and $b$ I would like to get the $x$ from the equation $Ax=b$. $N=p \times q$...
2 votes
0 answers
22 views

Solving Sparse Model with given Dictionary Using LASSO

I was trying to solve a problem where the basis matrix contains the components of $\sin(nx)$, $\cos(nx)$, $\sinh(nx)$ and $\cosh(nx)$. Say the $n$ varies from 1 to 100. While solving the lasso linear ...
5 votes
2 answers
300 views

Proximal Gradient Method (PGM) for a Function Model with More than 2 Functions (Sum of Functions)

I am currently working in signal reconstruction. I am trying to develop an algorithm where the user can plug any constraint to the main objective function (let's say chi2, least squares). I was trying ...
2 votes
0 answers
31 views

Rakeness Optimization problem

Rakeness optimization problem demonstrate that increases the rakeness between $a$ , $b$ while leaving $b$ random enough. where $e$ is the energy of the projection waveforms and $r$ is a randomness-...
6 votes
1 answer
127 views

Convex Optimization with $ {L}_{1, 2} $ Regularization Term

I have an optimization problem such as follow: $$\underset{X}{\operatorname{argmin}}\sum _s \left \| T_sX_{:,s} - Y_{:,s} \right \|^2_2 +\lambda\left \| GX \right \|_{2,1} \tag{1}$$ I have introduced ...
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4 votes
1 answer
134 views

Why Does FISTA Algorithm Not Work for Signed Signals?

Using the FISTA Algorithm for compressive sensing from Tiep H. Vu - FISTA, I created the matlab example below. I created 2 sparse signals x_signed and x_pos, where the latter only contains positive ...
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5 votes
1 answer
125 views

Resources on Solving Convex Optimization Problems in the Compressed Sensing Field

When I read papers of compressed sensing, sparse representation and whatever requiring optimization of a cost function, I just find the final results as an iterative equation or so which will converge ...
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5 votes
2 answers
270 views

Constrained LASSO Problem - $ {L}_{1} $ Regularized Least Squares with Linear Equality Constraints

I have an optimization question. I want to solve the following problem: $$ \arg\min_S\frac{1}{2}\|s-c\|_2^2 +\lambda\|\Phi s\|_1 \mbox{ s.t. } As = 0 $$ in which $\Phi$ is the wavelet transform ...
0 votes
1 answer
53 views

Wireless Body Area Networks with Minimum Energy Consumption [closed]

For adaptive compressive sensing(cs),the sensing matrix is related to the input signal. For example, in rakeness-based(cs), the sensing matrix is obtained by solving an optimization problem which ...
1 vote
1 answer
445 views

Fast Optimization for Long FIR Filters

I need FIR filter lengths in the order of 1e4 and above to obtain reasonable accuracy in desired frequency response. The problem is that optimisation in MATLAB (e.g. fircls or the Optimization Toolbox)...
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6 votes
1 answer
761 views

How to Formulate a Constraint Which Ensures All Variables Have the Same Sign

I'm trying to include a constraint in my problem (to be solved by any convex optimization solver). Let {a,b,c,d ...} be a finite set of continuous variables. How to formulate a constraint which ensure ...
5 votes
1 answer
305 views

The Gradient Operator of a Vectorized Image in Matrix Form

I have this optimization problem: $$ \arg \min_{ X \left( i, j \right) } \sum_{i, j} \left\| X \left( i, j \right) - 255 \right\|_{2}^{2} + \lambda \sum_{i, j} \left\| \nabla X \left( i, j \right) - \...
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5 votes
1 answer
712 views

Automatic Image Enhancement of Images of Scanned Documents (Auto Whitening)

Dropbox have make a blog post about there automatic enhancement method for scanned document image - Fast Document Rectification and Enhancement. I followed the post and they mention a formula to make ...
  • 115
6 votes
2 answers
2k views

Solving LASSO ($ {L}_{1} $ Regularized Least Squares) with Gradient Descent

To the best of my knowledge, state of the art methods for optimizing the LASSO objective function include the LARS algorithm and proximal gradient methods. I was wondering however, if the LASSO ...
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0 votes
1 answer
153 views

Difference Between Iteratively Reweighted Least Squares (IRLS) and Sequential Quadratic Programming?

Part of my work is concerned with applications in Sparse Bayesian Learning and therefore I occasionally stumble over interesting papers in the field of compressed sensing. I recently read ...
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0 votes
2 answers
2k views

Why Do Most of The Papers Use the Frobenius Norm for Denoising?

I have an noisy image and I want to remove noise from it; suppose $y$ is noisy image and $A$ is linear mask which makes my image noisy and $x$ is original image, so we have $$ Ax + \eta = y $$ and $\...
11 votes
1 answer
953 views

How Can I Use MATLAB to Solve a Total Variation Denoising / Deblurring Problem?

The Total Variation Denoising Problem is given by: $$ \arg \min_{x} \frac{1}{2} {\left\| A x - y \right\|}_{2}^{2} + \lambda \operatorname{TV} \left( x \right) $$ Where $ \operatorname{TV} \left( \...
user avatar
3 votes
1 answer
3k views

Understanding Soft Thresholding Operator

I want to understand what is soft thresholding operator? Why we use this operator? I came across this term while I was reading the paper - A New Algorithm Based on Linearized Bregman Iteration with ...
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2 votes
1 answer
383 views

How Come RIP Guarantees Unique Restoration of the Sparse Solution by $ {L}_{1} $ Minimization?

We have a sensing matrix $\Phi$, satisfying the restricted isometry property (RIP), and a sparse signal $x$. We want to recover $\hat x$ from the measurement $y=\Phi x$ by using $l_1$-minimization. I ...
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2 votes
1 answer
149 views

Adaptive Filter Gradient Descent

The quadratic performance surface of an adaptive filter is a paraboloid. Its minimum can be found wherever the gradient is zero. However, since there are two types of paraboloids (elliptical and ...
6 votes
1 answer
168 views

Regularized Least Squares by Laplacian Operator - Optimal Value of the Regularization Factor (Lagrangian Multiplier)

Consider the cost function $$f(X,\lambda) = \|AX-b\|_2^2 + \alpha \|LX\|_2^2$$ $A:$Measurement matrix($R_{m\times n}$,$m \ll n$), $b:$observation vector($R_m$), $L:$Laplacian operator($R_{n \times n}...
5 votes
2 answers
884 views

Significance of $ \lambda $ in Basis Pursuit

In basis Pursuit, L1 minimization is done to perform compressed sensing. In the literature there is a $ \lambda $ parameter used as a regularizer. What is its significance?
9 votes
1 answer
319 views

Least Angle Regression (LARS) without Matrix Inversion

Sorry if this is too damned long. I did what I could to abbreviate it. The question is about Least Angle Regression (LARS). I'm new to numerical work with matrices. I believe I have a way to ...
11 votes
1 answer
161 views

Ideas on Matrix Factorization / Transformations for $ {L}_{1} $ Minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where $\...
8 votes
4 answers
802 views

Solving Convex Optimization Problem Used for High Quality Denoising

The highest voted answer to this question suggests that to denoise a signal while preserving sharp transitions one should minimize the objective function: $$ |x-y|^2 + b|f(y)| $$ where $...