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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.
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 ...
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By Dvoretzky's theorem, every finite-dimensional normed vector space has a high-dimensional subspace on which the norm is approximately Euclidean; the Euclidean norm is the only norm with this property. [24] It can be extended to infinite-dimensional vector spaces as the L 2 norm or L 2 distance. [25]
The 2-norm and ∞-norm are strictly convex, and thus (by convex optimization) the minimizer is unique (if it exists), and exists for bounded distributions. Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any ...
Further, it is easy to show that the Fourier series of any L 2 function converges to it in L 2 norm. After Dirichlet's result, several experts, including Dirichlet, Riemann, Weierstrass and Dedekind, stated their belief that the Fourier series of any continuous function would converge everywhere.
In statistics, k-medians clustering [1] [2] is a cluster analysis algorithm. It is a generalization of the geometric median or 1-median algorithm, defined for a single cluster. k -medians is a variation of k -means clustering where instead of calculating the mean for each cluster to determine its centroid , one instead calculates the median .
The norm (see also Norms) can be used to approximate the optimal norm via convex relaxation. It can be shown that the L 1 {\displaystyle L_{1}} norm induces sparsity. In the case of least squares, this problem is known as LASSO in statistics and basis pursuit in signal processing.