Questions tagged [convex-optimization]
The convex-optimization tag has no usage guidance.
28
questions
1
vote
0answers
16 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 $
...
1
vote
1answer
26 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 \...
0
votes
1answer
38 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 ...
1
vote
1answer
105 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 $Ax=b$. $N=p \times q$...
1
vote
0answers
17 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 ...
3
votes
2answers
87 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 ...
1
vote
0answers
24 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-enforcing ...
2
votes
1answer
79 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 ...
1
vote
1answer
76 views
Why Does FISTA Algorithm Not Work for Signed Signals?
Using the FISTA Algorithm for compressive sensing from https://github.com/tiepvupsu/FISTA, I created the matlab example below.
I created 2 sparse signals x_signed and x_pos, where the latter only ...
0
votes
1answer
44 views
Resources on Solving Convex Optimization Problems in the Compress 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 ...
2
votes
2answers
137 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
1answer
52 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
1answer
294 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)...
2
votes
1answer
111 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 ...
1
vote
1answer
66 views
The Gradient Operator of a Vectorized Image in Matrix Form
I have this optimization problem:
$$ \arg \min_{ X \left( i, j \right) } \sum_{i, j} \left\| X \left( i, j \right) - 255 \right\|_{2}^{2} + \lambda \sum_{i, j} \left\| \nabla X \left( i, j \right) - \...
2
votes
1answer
270 views
Automatic Image Enhancement of Images of Scanned Documents (Auto Whitening)
Dropbox have make a blog post about there automatic enhancement method for scanned document image - Fast Document Rectification and Enhancement.
I followed the post and they mention a formula to make ...
1
vote
2answers
296 views
Solving LASSO ($ {L}_{1} $ Regularized Least Squares) with Gradient Descent
To the best of my knowledge, state of the art methods for optimizing the LASSO objective function include the LARS algorithm and proximal gradient methods.
I was wondering however, if the LASSO ...
0
votes
1answer
92 views
Difference Between Iteratively Reweighted Least Squares (IRLS) and Sequential Quadratic Programming?
Part of my work is concerned with applications in Sparse Bayesian Learning and therefore I occasionally stumble over interesting papers in the field of compressed sensing.
I recently read ...
0
votes
2answers
698 views
Why Do Most of The Papers Use the Frobenius Norm for Denoising?
I have an noisy image and I want to remove noise from it; suppose $y$ is noisy image and $A$ is linear mask which makes my image noisy and $x$ is original image, so we have
$$
Ax + \eta = y
$$
and $\...
6
votes
1answer
656 views
How Can I Use MATLAB to Solve a Total Variation Denoising / Deblurring Problem?
The Total Variation Denoising Problem is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| A x - y \right\|}_{2}^{2} + \lambda \operatorname{TV} \left( x \right) $$
Where $ \operatorname{TV} \left( \...
3
votes
1answer
604 views
Understanding Soft Thresholding Operator
I want to understand what is soft thresholding operator? Why we use this operator? I came across this term while I was reading the paper - A New Algorithm Based on Linearized Bregman Iteration with ...
2
votes
1answer
338 views
How Come RIP Guarantees Unique Restoration of the Sparse Solution by $ {L}_{1} $ Minimization?
We have a sensing matrix $\Phi$, satisfying the restricted isometry property (RIP), and a sparse signal $x$. We want to recover $\hat x$ from the measurement $y=\Phi x$ by using $l_1$-minimization.
I ...
2
votes
1answer
123 views
Adaptive Filter Gradient Descent
The quadratic performance surface of an adaptive filter is a paraboloid. Its minimum can be found wherever the gradient is zero. However, since there are two types of paraboloids (elliptical and ...
2
votes
1answer
72 views
Regularized Least Squares by Laplacian Operator - Optimal Value of the Regularization Factor (Lagrangian Multiplier)
Consider the cost function
$$f(X,\lambda) = \|AX-b\|_2^2 + \alpha \|LX\|_2^2$$
$A:$Measurement matrix($R_{m\times n}$,$m \ll n$), $b:$observation vector($R_m$), $L:$Laplacian operator($R_{n \times n}...
1
vote
2answers
372 views
Significance of $ \lambda $ in Basis Pursuit
In basis Pursuit, L1 minimization is done to perform compressed sensing. In the literature there is a $ \lambda $ parameter used as a regularizer.
What is its significance?
5
votes
1answer
227 views
Least Angle Regression (LARS) without Matrix Inversion
Sorry if this is too damned long. I did what I could to abbreviate it.
The question is about Least Angle Regression (LARS).
I'm new to numerical work with matrices.
I believe I have a way to ...
7
votes
1answer
136 views
Ideas on Matrix Factorization / Transformations for $ {L}_{1} $ Minimization
I am starting with a typical $\ell_1$ basis pursuit problem:
$$
\min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon,
$$
where $\...
8
votes
4answers
597 views
Solving Convex Optimization Problem Used for High Quality Denoising
The highest voted answer to this question suggests that to denoise a signal while preserving sharp transitions one should
minimize the objective function:
$$ |x-y|^2 + b|f(y)| $$
where $...
