L0 norm

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# Category: L0 norm

In mathematicsthe L p spaces are function spaces defined using a natural generalization of the p -norm for finite-dimensional vector spaces. L p spaces form an important class of Banach spaces in functional analysisand of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.

In statisticsmeasures of central tendency and statistical dispersionsuch as the meanmedianand standard deviationare defined in terms of L p metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer to penalizing either the L 1 norm of a solution's vector of parameter values i.

Techniques which use an L2 penalty, like ridge regressionencourage solutions where most parameter values are small. Elastic net regularization uses a penalty term that is a combination of the L 1 norm and the L 2 norm of the parameter vector. This is a consequence of the Riesz—Thorin interpolation theoremand is made precise with the Hausdorff—Young inequality.

Hilbert spaces are central to many applications, from quantum mechanics to stochastic calculus. In fact, by choosing a Hilbert basis i. In many situations, the Euclidean distance is insufficient for capturing the actual distances in a given space.

An analogy to this is suggested by taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distancewhich takes into account that streets are either orthogonal or parallel to each other.

The class of p -norms generalizes these two examples and has an abundance of applications in many parts of mathematicsphysicsand computer science. The absolute value bars are unnecessary when p is a rational number and, in reduced form, has an even numerator.

The Euclidean norm from above falls into this class and is the 2 -norm, and the 1 -norm is the norm that corresponds to the rectilinear distance.

It turns out that this limit is equivalent to the following definition:. See L -infinity. Abstractly speaking, this means that R n together with the p -norm is a Banach space.

This Banach space is the L p -space over R n. The grid distance or rectilinear distance sometimes called the " Manhattan distance " between two points is never shorter than the length of the line segment between them the Euclidean or "as the crow flies" distance.

Formally, this means that the Euclidean norm of any vector is bounded by its 1-norm:. This fact generalizes to p -norms in that the p -norm x p of any given vector x does not grow with p :.

For the opposite direction, the following relation between the 1 -norm and the 2 -norm is known:. This inequality depends on the dimension n of the underlying vector space and follows directly from the Cauchy—Schwarz inequality. On the other hand, the formula. It does define an F-normthough, which is homogeneous of degree p. The space of sequences has a complete metric topology provided by the F-norm. Many authors abuse terminology by omitting the quotation marks.

This is not a norm because it is not homogeneous. For example, scaling the vector x by a positive constant does not change the "norm". Despite these defects as a mathematical norm, the non-zero counting "norm" has uses in scientific computinginformation theoryand statistics —notably in compressed sensing in signal processing and computational harmonic analysis.

The associated defective "metric" is known as Hamming distance. This contains as special cases:.Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. Personal Sign In.

For IEEE to continue sending you helpful information on our products and services, please consent to our updated Privacy Policy. Email Address. Sign In. A general approximating approach on l 0 norm-a typical metric of system sparsity, is proposed and integrated into the cost function of the LMS algorithm.

This integration is equivalent to add a zero attractor in the iterations, by which the convergence rate of small coefficients, that dominate the sparse system, can be effectively improved. Moreover, using partial updating method, the computational complexity is reduced.

The simulations demonstrate that the proposed algorithm can effectively improve the performance of LMS-based identification algorithms on sparse system. Article :. Date of Publication: 05 June DOI: Need Help?Low-dose computed tomography CT reconstruction is a challenging problem in medical imaging. To complement the standard filtered back-projection FBP reconstruction, sparse regularization reconstruction gains more and more research attention, as it promises to reduce radiation dose, suppress artifacts, and improve noise properties.

In this work, we present an iterative reconstruction approach using improved smoothed l 0 SL0 norm regularization which is used to approximate l 0 norm by a family of continuous functions to fully exploit the sparseness of the image gradient. Due to the excellent sparse representation of the reconstruction signal, the desired tissue details are preserved in the resulting images. To evaluate the performance of the proposed SL0 regularization method, we reconstruct the simulated dataset acquired from the Shepp-Logan phantom and clinical head slice image.

Additional experimental verification is also performed with two real datasets from scanned animal experiment. Compared to the referenced FBP reconstruction and the total variation TV regularization reconstruction, the results clearly reveal that the presented method has characteristic strengths.

In particular, it improves reconstruction quality via reducing noise while preserving anatomical features. X-ray computed tomography has been widely used clinically for disease diagnosis, surgical guidance, perfusion imaging, and so forth. However, the massive X-ray radiations during CT exams are likely to induce cancer and other diseases in patients [ 12 ].

Therefore, the issue of low-dose computerized tomography reconstruction has been raised and attracted more and more research attention. As far as we know, there are two low-dose strategies widely studied for dose reduction: 1 lowering X-ray tube current values, measured by milliampere mA or milliampere-seconds mAsor lowering X-ray tube voltage, measured by kilovolt KVand 2 lowering the number of sampling views during CT inspection.

The strategy of regulation by mA or KV usually produces high noisy projection data.

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Thus, when the exposure dose is reduced, the images reconstructed using methods such as FBP suffer from increased artifacts and noise [ 3 ]. Diagnostic mistakes may appear in this case. The latter approach may also induce image artifacts due to limited sampling angles. As a result, the diagnostic value of the reconstructed images may be greatly degraded if inappropriate reconstruction approaches are applied. To solve these problems, statistical reconstruction algorithms [ 4 — 9 ] attempt to produce high quality images by better modeling the projection data and the imaging geometry, which have shown superior performance compared to FBP-type reconstructions.

Another path has been recently opened by compressed sensing CS with existing range of applications in medical imaging, for example, magnetic resonance imaging MRIbioluminescence tomography, optical coherence tomography, and low-dose CT reconstruction [ 10 — 24 ]. The CS theory reveals the potential capability of restoring sparse signals even if the Nyquist sampling theorem cannot be satisfied. Although the restricted isometry property RIP condition is not often satisfied in practice, CS-based reconstruction can yield more satisfying results than the traditional FBP algorithms in CT reconstruction [ 25 ].

Among several choices of sparse transforms, the gradient operator is motivated by the assumption that a preferable solution should be of bounded variation. It is known as total variation TV regularization, which favors solutions to be predominantly piecewise constant. TV has been widely used in the CT reconstruction community. However, TV-regularized images may suffer from loss of detail features and contrast, resulting in the staircasing artifacts.

It is well known that l 0 norm regularization can provide a sparser representation than the TV regularization l 1 norm [ 2627 ]. However, the application of l 0 norm in image reconstruction is often a nondeterministic polynomial-time NP hard problem. In addition, l 0 norm is a nonconvex function in discontinuous form.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I am trying to plot different norms with contourf and contour. I have succeeded with all norms except zero-norm and inf-norm.

What is the right way to draw similar graphs for 0-norm and inf-norm? First, let me mention that numpy provides numpy.

In the remainder I will stick to the attempt from the question to calculate the norm manually though. This can easily be calculated using numpy. The L0 "norm" would be defined as the number of non-zero elements. For the 2D case it can hence take the values 0 both zero1 one zeroor 2 both non-zero. Depicting this function as a contour plot is not really succefull because the function essentially deviates from 2 only along two lines in the plot.

Using an imshow plot would show it though. Learn more. Asked 2 years, 8 months ago. Active 2 years, 8 months ago.

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If nothing happens, download Xcode and try again. If nothing happens, download the GitHub extension for Visual Studio and try again. Louizos, M. Welling and D. Not yet strictly measure how sparse the L0 regularized model is, but show histograms of the first convolutional layers' weights. We use optional third-party analytics cookies to understand how you use GitHub.

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Latest commit.The -norm also written " -norm" is a vector norm defined for a complex vector. The -norm is the vector norm that is commonly encountered in vector algebra and vector operations such as the dot productwhere it is commonly denoted. However, if desired, a more explicit but more cumbersome notation can be used to emphasize the distinction between the vector norm and complex modulus together with the fact that the -norm is just one of several possible types of norms.

For real vectorsthe absolute value sign indicating that a complex modulus is being taken on the right of equation 2 may be dropped. So, for example, the -norm of the vector is given by. The -norm is also known as the Euclidean norm. However, this terminology is not recommended since it may cause confusion with the Frobenius norm a matrix norm is also sometimes called the Euclidean norm.

The -norm of a vector is implemented in the Wolfram Language as Norm [ m2], or more simply as Norm [ m ]. The " -norm" denoted with an uppercase is reserved for application with a function. Gradshteyn, I. Tables of Integrals, Series, and Products, 6th ed.

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Least Squares.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. Now, not being properly a norm doesn't make it a pseudonorm. There are different types of "not being properly a norm". But that holds in this case. Moreover, a pseudonorm requires the absolute scalability property, which is the key part that fails here. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Ask Question. Asked 7 years, 4 months ago. Active 5 years, 9 months ago. Viewed 3k times. Thus, we call it "pseudo-norm" because it is not mathematically a norm, but it acts as a norm. Is it correct? Active Oldest Votes. So it's not properly a norm and it's not a pseudonorm. Robert Cardona Robert Cardona 7, 3 3 gold badges 44 44 silver badges bronze badges. I read in two published paper that it's called pseudo-norm!

In the example, scalability fails, so it's not a pseudo-norm under that definition. The pseudonorm definition in Robert's answer is quite standard. The definition of quasinorm seems to be a bit less standard.