Questions tagged [convex-optimization]

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3
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1answer
553 views

How Can I Use MATLAB to Solve a Total Variation Denoising 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( \...
8
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4answers
557 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 $...
0
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1answer
50 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 ...
1
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0answers
23 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 ...
1
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0answers
13 views

Proximal operation evaluate at different point

I have a question: Consider we have the following procedure: $$ z = \text{prox}_{\lambda r_1}\left(x - \nabla f(y)\right) $$ In which $r_1$ is just a simple convex function, for example, $\|\cdot\|_1$....
1
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1answer
70 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
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1answer
40 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
1answer
76 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
2answers
109 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 ...
2
votes
1answer
67 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 ...
2
votes
1answer
325 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 ...
0
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1answer
50 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
196 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
146 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 ...
0
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1answer
59 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
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1answer
124 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 $\...
0
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2answers
316 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?
2
votes
1answer
113 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 ...
0
votes
1answer
52 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) - \...
0
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2answers
525 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 $\...
3
votes
1answer
204 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 ...
0
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2answers
146 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 ...