# Tag Info

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Boyd has A Matlab Solver for Large-Scale ℓ1-Regularized Least Squares Problems. The problem formulation in there is slightly different, but the method can be applied for the problem. Classical majorization-minimization approach also works well. This corresponds to iteratively perform soft-thresholding (for TV, clipping). The solutions can be seen from the ...

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The soft-thresholding function finds the minimizer of an objective function that involves data fitting in an $\ell_2$ sense as well as minimization of the $\ell_1$ norm (i.e. absolute value). The Lecture Notes - Penalty and Shrinkage Functions for Sparse Signal Processing gives a good discussion of how the soft threshold function is derived.

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The Frobenius Norm has multiple equivalent definitions – the useful for error measure is probably this one: $$\left\|M\right\|_\mathrm F = \sqrt{\sum_{p\in M}\left\lvert p\right\rvert^2}$$ That's a root square over all pixels. Root mean squares are very useful cost functions, as they describe the power of a signal.

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Let's solve a more general problem (Least Squares with Linear Equality Constraints): \begin{alignat*}{3} \arg \min_{x} & \quad & \frac{1}{2} \left\| A x - b \right\|_{2}^{2} \\ \text{subject to} & \quad & C x = d \end{alignat*} The Lagrangian is given by: $$L \left( x, \nu \right) = \frac{1}{2} \left\| A x - b \right\|_{2}^{2} + {\... 4 To solve optimization problems with TV penalty, we use a recently proposed algorithm called Fast Gradient Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems (FISTA), which has better convergence rate than conventional iterative methods, such as ASD-POCS. 4 The problem is given by:$$$$\arg \min_{X} \frac{1}{2} \sum_{k} {\left\| {T}_{k} {X}_{:, k} - {Y}_{:, k} \right\|}_{2}^{2} + \lambda {\left\| G X \right\|}_{2, 1} \\ = \arg \min_{X} \frac{1}{2} \sum_{k} {\left\| {T}_{k} {X}_{:, k} - {Y}_{:, k} \right\|}_{2}^{2} + \lambda \sum_{l} {\left\| G {X}_{:, l} \right\|}_{2}$$$$In the ... 3 Well, in your example, the channel isn't exactly sparse. It has been shown that \ell_0 minimization can recover any K-sparse vector x from observations \Phi x as long as 2K < {\rm spark}(\Phi) \leq M+1, when \Phi is M \times N (so that x is N\times 1), i.e., K<M/2 more or less. This is a necessary condition which means that if K ... 3 There are few options: Stephen Boyd, Lieven Vandenberghe - Convex Optimization. This is the classic in this field. Very well written book. Also have a look on other papers of Boyd on similar subjects such as the The Alternating Direction Method of Multipliers (ADMM). They also have a great MOOC Course Stanford Online CVX 101 - Convex Optimization. Amir Beck ... 3 SQP is a method for solving smooth (objective and constraint functions are at least twice differentiable) constrained nonlinear optimization problems. It solves a series of quadratic programming problems to converge to a solution to the Karush-Kuhn-Tucker conditions for the constrained optimization problem. IRLS is a method for solving unconstrained ... 3 Our goal is to obtain proximal operator of the following function$$ g \left( x \right) = {\left\| x \right\|}_{1} + \operatorname{TV}(x). $$The involved optimization problem for any z \in \mathbb{R}^d is the following$$\text{argmin}_{x}\left\{g(x) + \frac{1}{2}\|x-z\|^2_2\right\}$$Denote the following$$g_1(x) := {\left\| x \right\|}_{1} + \frac{1}...

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Indeed the model for the Proximal Gradient Method (Also see Proximal Gradient Methods for Learning) is in the form of: $$F \left( x \right) = f \left( x \right) + g \left( x \right)$$ Where usually $f \left( x \right)$ is convex smooth function and $g \left( x \right)$ is convex non smooth function. Yet the model is quite flexible and you may define ...

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The solution from the blog you linked goes as following (Coordinating Variable Signs by Paul Rubin, Web Archive): Someone asked me today (or yesterday, depending on whose time zone you go by) how to force a group of variables in an optimization model to take the same sign (all nonpositive or all nonnegative). Assuming that all the variables are ...

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It is pretty simple to create those Matrices. The real issue with them is their size which is enormous for real world images. For small kernels they are sparse which saves the day. Indeed for the Derivative Operator, which has only 2 elements, they are highly sparse. I built them in MATLAB using: mI = im2double(imread(imageFileName)); mI = mI(11:410, 201:...

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Their intention is to solve: $$\arg \min_{ J \left( x, y \right) } \sum_{x, y} \left\| J \left( x, y \right) - 255 \right\|_{2}^{2} + \lambda \sum_{x, y} \left\| \nabla J \left( x, y \right) - \nabla I \left( x, y \right) \right\|_{2}^{2}$$ Where $I \left( x, y \right)$ is the input image and $J \left( x, y \right)$ is the output image. They state ...

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It can easily solved by the Gradient Descent Framework with one adjustment in order to take care of the ${L}_{1}$ norm term. Since the ${L}_{1}$ norm isn't smooth you need to use the concept of Sub Gradient / Sub Derivative. When you integrate Sub Gradient instead of Gradient into the Gradient Descent Method it becomes the Sub Gradient Method. In the ...

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The quadratic surface is determined by the autocorrelation matrix of the data, which is always positive definite or positive semi-definite. This means that any stationary point is always a minimum. In the worst case, this minimum is not unique if the matrix is singular, but it can never be a saddle point.

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In the particular case where $f(y)=\|y\|_1$, the objective function can be written as $$\|x-y\|^2 + b\|y\|_1 = \sum_i(x_i - y_i)^2 + b\sum_i |y_i|,$$ minimizing it requires to minimize each entry of the sum: $$\hat{y_i} = argmin \{(x_i-y_i)^2 + b|y_i|\}$$ Using subdifferentials it is possible to show that the minimizer is the soft-thresholding ...

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If you want to solve for single value of $\lambda$ in the model: $$\arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1}$$ Then you can use Coordinate Descent method which is the fastest and simplest and doesn't require any matrix inversion. I have a MATLAB code for in my ${L}_{1}$ Regularized Least Squares ...

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Since $\epsilon$ is a parameter you need to set why not trade it with another parameter you need to set to create an easily solvable problem (Relaxation of the Problem)? You can transform the problem into the following form (${L}_{1}$ Regularized Least Squares): $$\arg \min_{x} \frac{1}{2} \left\| A x - z \right\|^{2} + \lambda \left\| x \right\|_{1} ... 2 Added: if f(x) = \ell_1(x) = \sum |x_i|, the terms are all independent — as @Alejandro points out, you can just minimize each term by itself. It's more interesting to minimize \qquad\qquad \| Ax - b \|_2^2 + \lambda \|x\|_1  where \|x\|_1 instead of \|x\|_2 is intended to push many x_i to 0. The following notes are for this case. (I ... 2 This parameter control sparsity of the basis. Look for articles on L1 norm "oracle property". In essence, by minimizing the L1 norm you're driving your representation to have few non-zero components. The lambda tunes how much of your squared error vs. L1 norm you pay most attention to. Large lambda means you care a lot about sparsity, but not exact matching. ... 2 I will try explaining why would someone use such an option as defining the solution as positive. The Proximal Gradient Method which is the hurt of the FISTA algorithm is basically a generalization of Projected (Sub) Gradient Descent Method. As such, if one knows that the solution lies in a Convex Set and one knows the projection onto that set one could ... 2 There are many types of matrix norms. Three are quite standard: element-wise norms: unfolding the matrix into a long vector, and compute a norm for that vector. Schatten norms: (power) vector noms over singular values of the matrix. induced norm: maxima over vector norms with uni-norm vectors. The Frobenius norm is the most simple: the square root of the ... 2 Sampling will be at about 1 GHz, output should be about 10 kHz So, you're decimating by a factor of 10^5=2^5\cdot5^5. This very much says that you'd normally go ahead and successively decimate. In an FPGA, halfband filters are especially cheap and efficient to implement; better yet: they're extremely easy to design, and you don't have to change the ... 2 One of the motivations to use the  {L}_{2}  norm comes from the Maximum a Posteriori Estimation (MAP) framework. If you model  \psi \left( u \right) \sim \mathcal{N} \left( 0, \alpha \right)  then if you derive the MAP Estimator in case the added noise is Gaussian you'd get the exact model you posted above. An example of the derivation of MAP model to ... 1 The Error in the Model The problem is in the dimensions of the Linear Operator  A  in your model compared to the Data Matrix  X . The number of columns of the matrix  A  must match the number of pixels in each column of  X  (Each image). While in your case it matches the number of images. Matrix Form of 2D Linear Operator Let's try to build the ... 1 Indeed you can not solve the problem ignoring the equality constraints and then project the solution onto the set of solution for the constraint. It is easy to build real world example which shows that. Yet, it might be that in most cases it will work reasonably well. You didn't mention how you solve the LASSO Problem but one of the easiest ways to solve ... 1 There are two optimisation problems here: To get the sensing matrix \mathcal{A} To recover the signal by y = \mathcal{A} x Out of these two problems, the first one is the one that is solved off-line to obtain a sensing matrix \mathcal{A} that appears to be more "suitable" to a particular family of signals. The second optimisation problem is the ... 1 Due to the non-smoothness of the l_1 norm, the algorithm is called subgradient descent. Because the you are looking for a solution that has a lot of zeros in it, you are still going to have to evaluate sub-gradients around points where elements of \mathbf{x} are zero. In fact most of the algorithms effectively treat elements below a certain threshold as ... 1 I will solve this for 1D but it could easily generalize into 2D. The nice thing about the TV Norm that it can be re formulated by the  {L}_{1}  of the Derivative Operator:$$ \operatorname{TV} \left( x \right) = \sum_{i = 1}^{N - 1} \left| {x}_{i + 1} - {x}_{i} \right| = {\left\| D x \right\|}_{1}  Where $D$ is the matrix form of the Derivative ...

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