• DOI:10.1093/IMAIAI/IAAA036
• Corpus ID: 4601550

Identification of Shallow Neural Networks by Fewest Samples

@article{Fornasier2018IdentificationOS,
  title={Identification of Shallow Neural Networks by Fewest Samples},
  author={Massimo Fornasier and Jan Vyb{\'i}ral and Ingrid Daubechies},
  journal={ArXiv},
  year={2018},
  volume={abs/1804.01592}
}

• M. Fornasier, J. Vybíral, I. Daubechies
• Published 4 April 2018
• Computer Science, Mathematics
• ArXiv

We address the uniform approximation of sums of ridge functions $\sum_{i=1}^m g_i(a_i\cdot x)$ on ${\mathbb R}^d$, representing the shallowest form of feed-forward neural network, from a small number of query samples, under mild smoothness assumptions on the functions $g_i$'s and near-orthogonality of the ridge directions $a_i$'s. The sample points are randomly generated and are universal, in the sense that the sampled queries on those points will allow the proposed recovery algorithms to perform a uniform approximation of any sum of ridge functions with high-probability. Our general approximation strategy is developed as a sequence of algorithms to perform individual sub-tasks. We first approximate the span of the ridge directions. Then we use a straightforward substitution, which reduces the dimensionality of the problem from $d$ to $m$. The core of the construction is then the approximation of ridge directions expressed in terms of rank-$1$ matrices $a_i \otimes a_i$, realized by formulating their individual identification as a suitable nonlinear program, maximizing the spectral norm of certain competitors constrained over the unit Frobenius sphere. The final step is then to approximate the functions $g_1,\dots,g_m$ by $\hat g_1,\dots,\hat g_m$. Higher order differentiation, as used in our construction, of sums of ridge functions or of their compositions, as in deeper neural network, yields a natural connection between neural network weight identification and tensor product decomposition identification. In the case of the shallowest feed-forward neural network, we show that second order differentiation and tensors of order two (i.e., matrices) suffice.