Nonlinear approximation problems in vector norms

1978 ◽  
pp. 117-132 ◽  
Author(s):  
M R Osborne ◽  
G A Watson
Keyword(s):  
Nonlinear Approximation ◽  
Approximation Problems ◽  
Vector Norms

Optimal Approximation Methods

1995 ◽  
Author(s):  
Marek A. Kowalski ◽  
Krzystof A. Sikorski ◽  
Frank Stenger
Keyword(s):  
Computational Methods ◽  
Nonlinear Approximation ◽  
Optimal Approximation ◽  
Approximation Methods ◽  
Optimal Algorithms ◽  
Optimal Methods ◽  
Sequential Methods ◽  
Linear Problems ◽  
Approximation Problems

In this chapter we outline a theory of optimal computational methods for general, nonlinear approximation problems. We define the notions of optimal algorithms and information, and we analyze the classes of parallel and sequential methods. We put special emphasis on linear problems as well as linear and spline algorithms. Several relationships between optimal methods and n-widths and s-numbers are also exhibited.

Discrete, nonlinear approximation problems in polyhedral norms

1977 ◽  
Vol 28 (2) ◽  
pp. 157-170 ◽  
Author(s):  
D. H. Anderson ◽  
M. R. Osborne
Keyword(s):  
Nonlinear Approximation ◽  
Polyhedral Norms ◽  
Approximation Problems

Discrete, nonlinear approximation problems in polyhedral norms

1977 ◽  
Vol 28 (2) ◽  
pp. 143-156 ◽  
Author(s):  
D. H. Anderson ◽  
M. R. Osborne
Keyword(s):  
Nonlinear Approximation ◽  
Polyhedral Norms ◽  
Approximation Problems

Nonlinear Approximation and (Deep) $$\mathrm {ReLU}$$ Networks

2021 ◽  
Author(s):  
I. Daubechies ◽  
R. DeVore ◽  
S. Foucart ◽  
B. Hanin ◽  
G. Petrova
Keyword(s):  
Nonlinear Approximation

scholarly journals The Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

2021 ◽  
Vol 88 (1) ◽  
Author(s):  
Antoine Gautier ◽  
Matthias Hein ◽  
Francesco Tudisco
Keyword(s):  
Global Convergence ◽  
Convergence Theorem ◽  
Matrix Norm ◽  
Nonnegative Matrices ◽  
Np Hard ◽  
Power Method ◽  
Contraction Ratio ◽  
Matrix Norms ◽  
Vector Norms

AbstractWe analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

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