Implicit Hypersurface Approximation Capacity in Deep ReLU Networks
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arXiv
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| Format: | Preprint |
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2024
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| author | Vallin, Jonatan Larsson, Karl Larson, Mats G. |
| author_facet | Vallin, Jonatan Larsson, Karl Larson, Mats G. |
| contents | We develop a geometric approximation theory for deep feed-forward neural networks with ReLU activations. Given a $d$-dimensional hypersurface in $\mathbb{R}^{d+1}$ represented as the graph of a $C^2$-function $ϕ$, we show that a deep fully-connected ReLU network of width $d+1$ can implicitly construct an approximation as its zero contour with a precision bound depending on the number of layers. This result is directly applicable to the binary classification setting where the sign of the network is trained as a classifier, with the network's zero contour as a decision boundary. Our proof is constructive and relies on the geometrical structure of ReLU layers provided in [doi:10.48550/arXiv.2310.03482]. Inspired by this geometrical description, we define a new equivalent network architecture that is easier to interpret geometrically, where the action of each hidden layer is a projection onto a polyhedral cone derived from the layer's parameters. By repeatedly adding such layers, with parameters chosen such that we project small parts of the graph of $ϕ$ from the outside in, we, in a controlled way, construct a network that implicitly approximates the graph over a ball of radius $R$. The accuracy of this construction is controlled by a discretization parameter $δ$ and we show that the tolerance in the resulting error bound scales as $(d-1)R^{3/2}δ^{1/2}$ and the required number of layers is of order $d\big(\frac{32R}δ\big)^{\frac{d+1}{2}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Implicit Hypersurface Approximation Capacity in Deep ReLU Networks Vallin, Jonatan Larsson, Karl Larson, Mats G. Machine Learning We develop a geometric approximation theory for deep feed-forward neural networks with ReLU activations. Given a $d$-dimensional hypersurface in $\mathbb{R}^{d+1}$ represented as the graph of a $C^2$-function $ϕ$, we show that a deep fully-connected ReLU network of width $d+1$ can implicitly construct an approximation as its zero contour with a precision bound depending on the number of layers. This result is directly applicable to the binary classification setting where the sign of the network is trained as a classifier, with the network's zero contour as a decision boundary. Our proof is constructive and relies on the geometrical structure of ReLU layers provided in [doi:10.48550/arXiv.2310.03482]. Inspired by this geometrical description, we define a new equivalent network architecture that is easier to interpret geometrically, where the action of each hidden layer is a projection onto a polyhedral cone derived from the layer's parameters. By repeatedly adding such layers, with parameters chosen such that we project small parts of the graph of $ϕ$ from the outside in, we, in a controlled way, construct a network that implicitly approximates the graph over a ball of radius $R$. The accuracy of this construction is controlled by a discretization parameter $δ$ and we show that the tolerance in the resulting error bound scales as $(d-1)R^{3/2}δ^{1/2}$ and the required number of layers is of order $d\big(\frac{32R}δ\big)^{\frac{d+1}{2}}$. |
| title | Implicit Hypersurface Approximation Capacity in Deep ReLU Networks |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2407.03851 |