On the smoothness of best 𝐿₂ approximants from nonlinear spline manifolds

Chui, Charles K.; Smith, Philip W.; Ward, Joseph D.

doi:10.1090/s0025-5718-1977-0422955-7

Cited by 4 publications

(1 citation statement)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In one dimension, when f ∈ L p (Ω) for 0 < p ≤ ∞, it was shown (see Rice [14] and Powell [15]) that problem (3.2) has a solution f n ∈ C[0, 1]. Solution of problem (3.2) is not unique in general; but it is unique for sufficiently smooth f and large enough n (see Chui et al [16]).…”

Section: The Best Least-squares Approximation Denote Vectors Of Weigh...mentioning

confidence: 99%

Adaptive Two-Layer ReLU Neural Network: I. Best Least-squares Approximation

Liu¹,

Cai²,

Chen³

2021

Preprint

View full text Add to dashboard Cite

In this paper, we introduce adaptive neuron enhancement (ANE) method for the best least-squares approximation using two-layer ReLU neural networks (NNs). For a given function f (x), the ANE method generates a two-layer ReLU NN and a numerical integration mesh such that the approximation accuracy is within the prescribed tolerance. The ANE method provides a natural process for obtaining a good initialization which is crucial for training nonlinear optimization problems. Numerical results of the ANE method are presented for functions of two variables exhibiting either intersecting interface singularities or sharp interior layers.

show abstract

Section: The Best Least-squares Approximation Denote Vectors Of Weigh...mentioning

confidence: 99%

Adaptive Two-Layer ReLU Neural Network: I. Best Least-squares Approximation

Liu¹,

Cai²,

Chen³

2021

Preprint

View full text Add to dashboard Cite

show abstract

Unicity of best 𝐿₂ approximation by second-order splines with variable knots

Barrow¹,

Chui²,

Smith³

et al. 1977

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Communicated by R. R. Goldberg, February 25, 1977 1. Introduction and results. Let S N C C[0, 1] denote the class of all piecewise linear functions with at most TV + 1 linear segments. In this article we announce some interesting and somewhat surprising approximation properties of S N in the space L 2 [0, 1]. Three main theorems will be stated in this section and the main idea of our proof of the first two theorems will be sketched in §2.Theorem 1 describes a fairly large class of strictly convex functions which have, for each positive integer N, unique best L 2 [0, 1] approximants from the nonlinear (spline) manifold S N . Theorem 2 states that any sufficiently smooth strictly convex function eventually, i.e. for all large N, has a unique best L 2 [0,1] approximant from this manifold. This behavior will be called "eventual uniqueness". Theorem 3 indicates the sharpness of these two results.We emphasize that S N is not only a nonlinear manifold, but also a nonclosed subset of L 2 [0, 1]. Hence, arguments regarding existence, uniqueness, and characterization of best approximants are nontrivial. Since it has been shown in [1] that, for every positive integer TV, any continuous function has at least one best L 2 [0, 1] approximant from S N , we are only concerned with uniqueness and eventual uniqueness of best approximants in this paper.

show abstract

Unicity of best mean approximation by second order splines with variable knots

Barrow¹,

Chui²,

Smith³

et al. 1978

Math. Comp.

View full text Add to dashboard Cite

Let S N 2 S_N^2 denote the nonlinear manifold of second order splines defined on [0, 1] having at most N N interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function f f by elements of S N 2 S_N^2 . Approximation relative to the L 2 [ 0 , 1 ] {L_2}[0,1] norm is treated first, with the results then extended to the best L 1 {L_1} and best one-sided L 1 {L_1} approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function f f satisfying f > 0 f > 0 has a unique best approximant from S N 2 S_N^2 provided either log ⁡ f \log f is concave, or N N is sufficiently large, N ⩾ N 0 ( f ) N \geqslant {N_0}(f) ; for any N N , there is a smooth function f f , with f > 0 f > 0 , having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.

show abstract

On the smoothness of best 𝐿₂ approximants from nonlinear spline manifolds

Cited by 4 publications

References 10 publications

Adaptive Two-Layer ReLU Neural Network: I. Best Least-squares Approximation

Adaptive Two-Layer ReLU Neural Network: I. Best Least-squares Approximation

Unicity of best 𝐿₂ approximation by second-order splines with variable knots

Unicity of best mean approximation by second order splines with variable knots

Contact Info

Product

Resources

About