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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.
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.
This regularization function, while attractive for the sparsity that it guarantees, is very difficult to solve because doing so requires optimization of a function that is not even weakly convex. Lasso regression is the minimal possible relaxation of ℓ 0 {\displaystyle \ell _{0}} penalization that yields a weakly convex optimization problem.
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 ...
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).
However, there are RKHSs in which the norm is an L 2-norm, such as the space of band-limited functions (see the example below). An RKHS is associated with a kernel that reproduces every function in the space in the sense that for every x {\displaystyle x} in the set on which the functions are defined, "evaluation at x {\displaystyle x} " can be ...
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