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In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
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 ().
The L 2 space of square-integrable functions; L 2 norm; The ℓ 2 space of square-summable sequences; L 2 cohomology, a cohomology theory for smooth non-compact manifolds with Riemannian metric; L 2 (n), the family of 2-dimensional projective special linear groups on finite fields. Ridge regression, regression and regularization method also ...
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.
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 ...
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L p-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete , i.e. a Banach space .
In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform.Specifically, given a domain Ω in n-dimensional Euclidean space R n, then the square-integrable function k : Ω × Ω → C belonging to L 2 (Ω×Ω) such that
Notice that ‖ ‖ is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), ‖ ‖ is the trace class norm (see trace class), and ‖ ‖ is the operator norm (see operator norm). Note that the matrix p-norm is often also written as ‖ ⋅ ‖ p {\displaystyle \|\cdot \|_{p}} , but it is not the same as Schatten norm.