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The Frobenius norm defined by ‖ ‖ = = = | | = = = {,} is self-dual, i.e., its dual norm is ‖ ‖ ′ = ‖ ‖.. The spectral norm, a special case of the induced norm when =, is defined by the maximum singular values of a matrix, that is, ‖ ‖ = (), has the nuclear norm as its dual norm, which is defined by ‖ ‖ ′ = (), for any matrix where () denote the singular values ...
These forms can be useful for the simplification of Boolean functions, which is of great importance in the optimization of Boolean formulas in general and digital circuits in particular. Other canonical forms include the complete sum of prime implicants or Blake canonical form (and its dual), and the algebraic normal form (also called Zhegalkin ...
The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.
ANF is a canonical form, which means that two logically equivalent formulas will convert to the same ANF, easily showing whether two formulas are equivalent for automated theorem proving. Unlike other normal forms, it can be represented as a simple list of lists of variable names— conjunctive and disjunctive normal forms also require ...
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[10] In a linear regression problem modeled as y ∼ G e n e r a l i z e d N o r m a l ( X ⋅ θ , α , p ) {\displaystyle y\sim \mathrm {GeneralizedNormal} (X\cdot \theta ,\alpha ,p)} , the MLE will be the arg min θ ‖ X ⋅ θ − y ‖ p {\displaystyle \arg \min _{\theta }\|X\cdot \theta -y\|_{p}} where the p-norm is used.
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[2] In the social sciences , a result may be considered statistically significant if its confidence level is of the order of a two-sigma effect (95%), while in particle physics and astrophysics , there is a convention of requiring statistical significance of a five-sigma effect (99.99994% confidence) to qualify as a discovery .