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Your formulation: $$\arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \lambda {\left\| \boldsymbol{x} \right\|}_{1}$$ Has 2 elements: The Fidelity Term This is basically measurements term with the model of AWGN with IID noise. The Regularization Term This is a sparse promoting model by using the Laplace ...

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I will divide my answer into 3 sections. The Distribution of the Derivative of Images Take a real world image, any image. Apply the derivative operator on it (Namely apply the kernel $\left[ 1, -1 \right]$ on it. Display the histogram of the filtered image. I took this image: The histogram I got is this: This distribution is very similar to Laplace ...

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A MATLAB code which implements the problem as defined and solve it using CVX is given by: %% General Parameters close('all'); clear('all'); %% Simulation Parameters numRows = 6; numCols = 10; varianceFctr = 3; paramLambda = 2.75; %% Generate Data mA = randn(numRows, numCols); vX0 = rand(numCols, 1) >= 0.65; vC = varianceFctr * rand(numRows, 1);...

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Let's think about it in a different way - Generate Noise from a Dictionary. Let's create a Dictionary $A \in \mathbb{R}^{m \times n}$ where each of its rows is normalized (Has Euclidean Norm of $1$) and generated by a Gaussian Random Vector. Now, let's create $N$ random vector ${\left\{ {r}_{i} \right\}}_{i = 1}^{N}$ by: $${r}_{i} = A {g}_{i}$$ ...

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There are 2 forms of the Basis Pursuit problem: \begin{align*} \text{The  \lambda  Form:} & \quad && \arg \min_{x} &&\frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1} \\ \text{The  \epsilon Form:} && \quad & \arg \min_{x} && {\left\| x \right\|}_{1} \\ && \quad & \text{... 3 Answer taken from Implementation of Block Orthogonal Matching Pursuit (BOMP) Algorithm. The Block Orthogonal Matching Pursuit (BOMP) Algorithm is basically the Orthogonal Matching Pursuit (OMP) Algorithm with single major difference - Instead of selecting single index which maximizes the correlation we chose a set of indices, sub set of columns of the ... 3 The Block Orthogonal Matching Pursuit (BOMP) Algorithm is basically the Orthogonal Matching Pursuit (OMP) Algorithm with single major difference - Instead of selecting single index which maximizes the correlation we chose a set of indices, sub set of columns of the matrix and the solution vector. A good reference for the algorithm is given in: An Optimal ... 3 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 ... 3 Have a look on the following optimization problem: \arg \min_{x} \frac{1}{2} {\left\| A x - y \right\|} + \lambda {\left\| x \right\|}_{0} $$Where  {\left\| \cdot \right\|}_{0}  is counting the number of non zero elements. It is known that Iterative Hard Thresholding can be utilized to solve this and in some cases guaranteed to find the correct ... 2 If you can't find anything in the literature about a threshold, you can develop your own with the following procedure: Generate$$N=\frac{ln(1-M)}{R}$$random matrices \boldsymbol{A} that you are certain ares sparse using your current method. Here, M is the margin of error and R is the maximum error resolution to be able to detect. Let's assume ... 2 The first equation you have is often called the Quadratic Problem, which through the use of Duality can be shown to be equivalent to the Basis Pursuit De-Noising (BPDN) given as:$$ \arg \min_{\boldsymbol{x}} {\left\| \boldsymbol{x} \right\|}_{1} \mbox{ subject to } {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}< \delta .$$In your 2nd ... 2 This question is typically the subject of a paper like Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information, Emmanuel J. Candès, Justin Romberg, 2006: This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f\in \mathbb{C}^N and ... 2 It depends on the application and on what basis you decide to look at. If there are only a few targets and little to no clutter, then a radar image can be considered sparse in the image domain i.e. it has only a few Fourier components. A possible example of this is ground to air or air to air radars. Synthetic Aperture Radar (SAR) imaging of the ground - ... 2 You can alternatively create a DFT matrix in matlab using this code: exp(-1j*2*pi* ((0:N-1)/N).' * (0:N-1)) And the IDFT matrix thus: 1/N * exp(1j*2*pi* ((0:N-1)/N).' * (0:N-1)) As the only difference betweenm DFT and IDFT is the sign and a scaling factor. You could alternatively just do: ifft(eye(N)) But this doesn't get around needing the full DFT ... 2 How you parameterize your sparsity will depend on your application. The authors of that paper, in a paragraph on page 231 say: which is why they clump the coefficients together in P blocks of start time t_{B_k} of duration n_{B_k}. For one impulse response this is shown in their figure 1(b). The overall sparsity will depend on the reflections and ... 2 The notion of sparsity entails that an object, living for instance in an n-dimensional space, can be described (in the suitable basis/frame) by a number k of meaningful components (each above a threshold, or whose combination is close enough to the signal) "much smaller" than n. When talking about filter identification (adaptive or not), the filter ... 2 My first comment would be why the heck are you using R if you are concerned with processing speed, or are you just prototyping algorithms? Anyway, Without getting into how I derived it, here is a formula that is much much faster: Take the log of your signal (-1 if 0):$$ g[x] = \ln(y[x]) $$Calculate the following value:$$ B = \frac{ \begin{array}{c} ...

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Ha just figured out a faster and better method just using BIC-optimized selection of optimal peak width, using a banded covariate matrix with shifted Gaussian peak shapes of given width & using nonnegative least squares fits (which is solved using an active set method and regularizes the problem a bit, though less of course than with LASSO or L0 norm ...

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The question of the existence of a sparse basis of noise is closely related to the question of the effective dimensionality of the noise subspace. First, it is important to realise that noise is a process, and not a signal. You can think of a full characterisation of any kind of noise by means of a function $p: S \to \mathbb{R}^+_0$ that maps a signal $s$ ...

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