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Abstract
We study twolayer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of $\mathbb R^d$valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with MonteCarlo rates for a certain class of functions, extending existing results for the finitedimensional case. In the second part of the paper, we study, from the regularisation theory viewpoint, the problem of finding optimal representations of such functions via signed measures on a latent space from a finite number of noisy observations. We discuss regularity conditions known as source conditions and obtain convergence rates in a Bregman distance for the representing measure in the regime when both the noise level goes to zero and the number of samples goes to infinity at appropriate rates.
Original language  English 

Pages (fromto)  63586389 
Number of pages  32 
Journal  SIAM Journal on Mathematical Analysis 
Volume  54 
Issue number  6 
Early online date  8 Dec 2022 
DOIs  
Publication status  Published  31 Dec 2022 
Bibliographical note
Funding:The work of the author was supported by EPSRC Fellowship grant EP/V003615/1, the Cantab Capital Institute for the Mathematics of Information at the University of Cambridge, and the National Physical Laboratory
Keywords
 cs.LG
 cs.NA
 math.FA
 math.NA
 math.PR
 68Q32, 68T07, 46E40, 41A65, 65J22
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 1 Finished

Regularisation theory in the data driven setting
Korolev, Y. (PI)
Engineering and Physical Sciences Research Council
1/09/22 → 31/10/24
Project: Research council