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In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().
In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm () / (+). [16] Here we mean by F-norm some real-valued function ‖ ‖ on an F-space with distance , such that ‖ ‖ = (,).
When learning a linear function , characterized by an unknown vector such that () =, one can add the -norm of the vector to the loss expression in order to prefer solutions with smaller norms. Tikhonov regularization is one of the most common forms.
The FixNorm method divides the output vectors from a transformer by their L2 norms, then multiplies by a learned parameter . The ScaleNorm replaces all LayerNorms inside a transformer by division with L2 norm, then multiplying by a learned parameter g ′ {\displaystyle g'} (shared by all ScaleNorm modules of a transformer).
SVM algorithms categorize binary data, with the goal of fitting the training set data in a way that minimizes the average of the hinge-loss function and L2 norm of the learned weights. This strategy avoids overfitting via Tikhonov regularization and in the L2 norm sense and also corresponds to minimizing the bias and variance of our estimator ...
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Completeness of the space holds provided that whenever a series of elements from l 2 converges absolutely (in norm), then it converges to an element of l 2. The proof is basic in mathematical analysis , and permits mathematical series of elements of the space to be manipulated with the same ease as series of complex numbers (or vectors in a ...
For example, the , norm is used in multi-task learning to group features across tasks, such that all the elements in a given row of the coefficient matrix can be forced to zero as a group. [6] The grouping effect is achieved by taking the ℓ 2 {\displaystyle \ell ^{2}} -norm of each row, and then taking the total penalty to be the sum of these ...