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### What is an exact measure of sparsity?

"Is there any exact, i.e. numerical, definition for sparsity?" And by numerical, I understand both computable, and practically "usable". My take is that: not yet, as least, there ...
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### Best Metric to Compare Sparsity of Vectors

I am sorry I cannot comment your answer due to my low reputation. Gini and your suggested sparsity ratio ($l_1(x)/l_2(x)$) both give me the same value for $\lambda$. But The problem I still see is ...
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### Compressive Sensing vs. Sparse Coding

As you correctly noted compressed sensing, compressive sampling, sparse sampling all mean the same thing. Some authors also call it sparse sensing. The idea behind compressed sensing is that a sparse ...
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Accepted

### Compressive Sensing vs. Sparse Coding

A couple of reference works offer an exaplanation: A neurological interpretation described in Scholarpedia Stanford's Unsupervised Feature Learning and Deep Learning tutorial If we look at the ...
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### Real world application of signal sparsity?

Sparsity concept is extensively being used in computer vision and image processing. The Idea is that natural image can be pretty sparse when it is transformed to different bases. this bases can be ...
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### Terminologies - sparse channel, sparse input, compressed sensing

The term sparse, as you mention, refers to the fact that some "signal", usually represented by a vector $x$ contains mostly zero or negligible values and only a few non-zero or significant ...
• 1,322
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### What exactly is "sparse representation"?

Take a sine-like signal $s$. In the appropriate Fourier $\mathcal{F}$ domain, it is represented by two "peaks", the other coefficients being zero. Fourier is a sparse representation for sines or close-...
• 31.9k
Accepted

### Estimating Convolution Input Under the Assumption of Sparsity and Constant Non Zero Values Using Compressive Sensing Approach

Basically your problem is called Blind Deconvolution. It means we want to estimate both the operator and the input given the output. You model is Linear Time Invariant Operator so we have LTI Blind ...
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• 19.7k

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

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 ...
• 19.7k

### Best Metric to Compare Sparsity of Vectors

Norms like $\ell_p$, $p \ge 1$, or quasi-norms ($0<p< 1$) are all $1$-homogeneous: $\ell_p(\lambda x) = |\lambda|\ell_p( x)$. Which is not the case for the $\ell_0$ count measure, which is scale ...
• 31.9k
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• 19.7k

### Coherence Calculation in Sparse Sensing

Update I would like to rephrase the original problem with slightly different notation and claim that there is no need to measure the coherence of the product of the circulant matrix A and the DCT ...

### Is There a Sparse Representation for Noise?

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 ...
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You can approach this problem as a special case of the "$k$-simple bounded signal" class described in (Donoho & Tanner, 2010 - Precise Undersampling Theorems ), see page 2, Example 3. Particularly,...
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 $1$...