February 13,L1, L2 stability. Could some one please tell me wht is meant by L1 and L2 stability? February 14,Re: L1, L2 stability. Hyperbolic system of equations such as Euler equations admit discontinuous solutions. Derivatives of discontinuous functions exist only in generalized distributional sense.
Since L2 norm involves derivatives of function, L1 norm is used in the stability considerations for hyperbolic systems of conservation laws. L2 norm is used for elliptic and parabolic systems. L1 and L2 stability simply measure the growth of instabilities in the solution, hence the notion of L1 and L2 stability.
February 15,Re: L1, L2 stability. February 21,Re: L1, L2 stability. March 6,Re: L1, L2 stability. Hi I could n ot able to understand what does that mean by "Derivatives of discontinuous functions exist only in generalized distributional sense. Thread Tools. BB code is On. Smilies are On. Trackbacks are Off. Pingbacks are On. Refbacks are On. Forum Rules. All times are GMT The time now is Add Thread to del.
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Similar Threads.In mathematicsa norm is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin : it commutes with scaling, obeys a form of the triangle inequalityand is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean normor 2-normwhich may also be defined as the square root of the inner product of a vector with itself. A pseudonorm or seminorm satisfies the first two properties of a norm, but may be zero for other vectors than the origin.
In a similar manner, a vector space with a seminorm is called a seminormed vector space. Suppose that p and q are two norms or seminorms on a vector space V.
L1, L2 stability
The norms p and q are equivalent if and only if they induce the same topology on V. Such notation is also sometimes used if p is only a seminorm. For the length of a vector in Euclidean space which is an example of a norm, as explained belowthe notation v with single vertical lines is also widespread. This is the Euclidean norm, which gives the ordinary distance from the origin to the point X —a consequence of the Pythagorean theorem.
This operation may also be referred to as "SRSS", which is an acronym for the s quare r oot of the s um of s quares. However, all these norms are equivalent in the sense that they all define the same topology. The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis.
Hence, the Euclidean norm can be written in a coordinate-free way as. There are exactly four Euclidean Hurwitz algebras over the real numbers.
In this case, the norm can be expressed as the square root of the inner product of the vector and itself:. This formula is valid for any inner product spaceincluding Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation:. The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x.
The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope of dimension equivalent to that of the norm minus 1. The p -norm is related to the generalized mean or power mean.
These spaces are of great interest in functional analysisprobability theory and harmonic analysis. However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional.
The partial derivative of the p -norm is given by. The derivative with respect to xtherefore, is. The F -norm described above is not a norm in the usual sense because it lacks the required homogeneity property.
In metric geometrythe discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distancewhich is important in coding and information theory.
In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero.
However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.
In signal processing and statisticsDavid Donoho referred to the zero " norm " with quotation marks. Following Donoho's notation, the zero "norm" of x is simply the number of non-zero coordinates of xor the Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of p -norms as p approaches 0. Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous.April 20,How L-norms are used to study stability of a numerical scheme?
Ravindra Shende. Greetings, Can someone tell me in what way L1, L2 or infinity norm is used to study stability analysis of a numerical scheme? I am trying to develop a code to solve compressible-Navier-Stokes equation using compact scheme for spatial discretization and runge-kutta scheme for temporal discretization.
What other methods can be used for stability analysis of this scheme? Thank you. April 20, Filippo Maria Denaro. Originally Posted by Ravindra Shende. Originally Posted by FMDenaro. The general Holder norm is simply an application on the N-dimensional space that give you a "number". That means that you must first define an error for example the exact minus the numerical solution in vector N-dimensional space and apply a norm on it. The norm is studied in general for vanishing time and mesh steps.
Last edited by Ravindra Shende; April 20, at Reason: very large image size. You could use a norm on the error to verify the error slope and the accuracy order. April 21, Now, from this plot, what can I say about the stability of this scheme? This is the link to the plot. Ravindra Shende likes this. Thank you for your replies!!! Sorry for the error. I used this equation just as a simple example to ask my question about the use of L-norms in determining the stability of a numerical scheme.
Now, von-Neumann stability analysis cannot be used for compressible-Navier-Stokes equations. So, can the concept of linear growth of L-norms of error be used as a stability criterion?
What other methods are used to study the stability of a numerical scheme meant for compressible-Navier-Stokes equations? Kind regards.While practicing machine learning, you may have come upon a choice of deciding whether to use the L1-norm or the L2-norm for regularization, or as a loss function, etc. It is basically minimizing the sum of the absolute differences S between the target value Y i and the estimated values f x i :. L2-norm is also known as least squares.
It is basically minimizing the sum of the square of the differences S between the target value Y i and the estimated values f x i :.
The method of least absolute deviations finds applications in many areas, due to its robustness compared to the least squares method.
Least absolute deviations is robust in that it is resistant to outliers in the data. This may be helpful in studies where outliers may be safely and effectively ignored.
If it is important to pay attention to any and all outliers, the method of least squares is a better choice. If this example is an outlier, the model will be adjusted to minimize this single outlier case, at the expense of many other common examples, since the errors of these common examples are small compared to that single outlier case.
The instability property of the method of least absolute deviations means that, for a small horizontal adjustment of a datum, the regression line may jump a large amount. After passing this region of solutions, the least absolute deviations line has a slope that may differ greatly from that of the previous line. In contrast, the least squares solutions is stable in that, for any small adjustment of a data point, the regression line will always move only slightly; that is, the regression parameters are continuous functions of the data.
The top represents L1-norm and the bottom represents L2-norm. The first column represents how a regression line fits these three points using L1-norm and L2-norm respectively.
Suppose we move the green point horizontally slightly towards the right, the L2-norm still maintains the shape of the original regression line but makes a much steeper parabolic curve. However in the L1-norm case, the slope of the regression line is now much more steeper affecting every other predictions even well-beyond the rightmost point.
As such, all future predictions are affected much more seriously than the L2-norm results. Suppose we move the green point even more horizontally further to the right past the first black point third columnthe L2-norm now also changes a bit but not as much as the L1-norm, which the slope has completed turned in direction.
This change of slope will definitely invalidate all previous results. By just a small perturbation of the data points, the regression line changes by a lot. This is what instability of the L1-norm versus the stability of the L2-norm means here.
Solution uniqueness is a simpler case but requires a bit of imagination. First, this picture below:. Generalizing this to n-dimensions. This is why L2-norm has unique solutions while L1-norm does not. This is actually a result of the L1-norm, which tends to produces sparse coefficients explained below. L2-norm produces non-sparse coefficients, so does not have this property. L1-norm has the property of producing many coefficients with zero values or very small values with few large coefficients.
Computational efficiency. L1-norm does not have an analytical solution, but L2-norm does. This allows the L2-norm solutions to be calculated computationally efficiently. However, L1-norm solutions does have the sparsity properties which allows it to be used along with sparse algorithms, which makes the calculation more computationally efficient. Home About. It is basically minimizing the sum of the square of the differences S between the target value Y i and the estimated values f x i : The differences of L1-norm and L2-norm can be promptly summarized as follows: Robustnessper wikipedia, is explained as: The method of least absolute deviations finds applications in many areas, due to its robustness compared to the least squares method.
Stabilityper wikipedia, is explained as: The instability property of the method of least absolute deviations means that, for a small horizontal adjustment of a datum, the regression line may jump a large amount. This is best explained with a picture below mspaint made, sorry for the low quality : The top represents L1-norm and the bottom represents L2-norm.Learn all about your browsing private mode options in this free lesson.
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Navigating gluten-free dining at home and at restaurants can be especially difficult for vegans because gluten-containing proteins and grains are a common focus of a plant-based diet.Great to have the Points of Interest marked as well.
Differences between the L1-norm and the L2-norm (Least Absolute Deviations and Least Squares)
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He also gave us helpful information about trying to book a whale watching tour.The specification of intervals follows the same conventions as those of value. Since confidences are a continuous value, the most common case will be asking for a range, but the service will accept also individual values. It's also possible to specify both a value and a confidence.
Finally, note that it is also possible to specify support, value, and confidence parameters in the same query. Filtering and Paginating Fields from a Model A model might be composed of hundreds or even thousands of fields. Thus when retrieving a model, it's possible to specify that only a subset of fields be retrieved, by using any combination of the following parameters in the query string (unrecognized parameters are ignored): Fields Filter Parameters Parameter TypeDescription fields optional Comma-separated list A comma-separated list of field IDs to retrieve.
To update a model, you need to PUT an object containing the fields that you want to update to the model' s base URL. Once you delete a model, it is permanently deleted. If you try to delete a model a second time, or a model that does not exist, you will receive a "404 not found" response. However, if you try to delete a model that is being used at the moment, then BigML. To list all the models, you can use the model base URL.
By default, only the 20 most recent models will be returned. You can get your list of models directly in your browser using your own username and API key with the following links. You can also paginate, filter, and order your models. This is valid for both regression and classification models.
Objective Weights: submitting a specific weight for each class in classification models. Automatic Balancing: setting the balance argument to true to let BigML automatically balance all the classes evenly.
Let's see each method in more detail. We can use it as an input to create a model that will use to weight each instance accordingly. In this case, fraudulent transactions will weigh 10 times more than valid transactions in the model building computations. You can just use one of the rows and add the corresponding count as a weight field. This will reduce the size of your sources enormously.8. Norms of Vectors and Matrices
Objective Weights The second method for adding weights only applies to classification models. Each instance will be weighted according to its class weight. This means the example below is equivalent to the example above. If every weight does end up zero (this is possible with sampled datasets), then the resulting model will have a single node with a nil output.
Each instance will be weighted individually according to the weight field's value.