arXiv subjects

Patrick Finke

Publications and source records attributed to Patrick Finke.

3 recordsLinked to original sources

arXiv2026

Memorization Capacity of Invariant Convolutional Neural Nets

Neural networks are powerful memorizers which are often able to interpolate their training data even when the labels have been randomized. While there exists an extensive literature on the memorization capacity of fully connected networks, in practice more specific architectures with problem-dependent properties are preferred, such as convolutional networks for image classification. In this paper we consider convolutional networks which are invariant with respect to a given arbitrary finite group of orthogonal transformations, which may include translations, reflections, and/or rotations of images and invariances of higher order tensors. We prove upper bounds on the size of an invariant convolutional neural network with threshold activations that is required to interpolate a given dataset with two classes. Our bounds are formulated in terms of a complexity measure that takes the invariant structure into account and is instance-specific, i.e., it is related to the mutual arrangement of the classes in the given dataset and their geometric properties rather than the number of samples. Our proof is constructive: for given input data, instances of an interpolating network can be computed via a randomized algorithm with high probability. In numerical experiments we investigate the practical effectiveness of this algorithm.

stat.ML↗

arXiv2026

Optimal estimates for the $\ell^p$-Lipschitz constants of deep random ReLU neural networks

This paper studies the $\ell^p$-Lipschitz constants of ReLU neural networks $Φ: \mathbb{R}^d \to \mathbb{R}$ with random parameters for $p \in [1,\infty]$. The distribution of the weights follows a variant of the He initialization. In the case of zero-bias networks, we derive matching high-probability upper and lower bounds for wide networks that differ at most by an absolute constant. The bounds reveal two remarkable phenomena. First, the bounds grow root-logarithmically in terms of the depth of the net. Second, the behavior of the $\ell^p$-Lipschitz constants varies significantly between the regimes $p \in [1,2)$ and $p \in [2,\infty]$. For $p \in [2,\infty]$, the $\ell^p$-Lipschitz constant behaves similarly to $\|g\|_{p'}$, where $g \in \mathbb{R}^d$ is a standard Gaussian vector and $1/p + 1/p' = 1$. In contrast, for $p \in [1,2)$, the $\ell^p$-Lipschitz constant aligns more closely with $\|g\|_2$. For networks with possibly nonzero biases, we obtain the same lower bounds without assumptions on the bias distributions. For symmetric bias distributions with bounded densities, we also derive high-probability upper bounds for the $\ell^p$-Lipschitz constants. For fixed sub-exponential bias distributions, these upper bounds differ from the lower bounds by a factor that has square-root dependence on the depth and logarithmic dependence on the width.

stat.ML↗

arXiv2024

Memorization With Neural Nets: Going Beyond the Worst Case

In practice, deep neural networks are often able to easily interpolate their training data. To understand this phenomenon, many works have aimed to quantify the memorization capacity of a neural network architecture: the largest number of points such that the architecture can interpolate any placement of these points with any assignment of labels. For real-world data, however, one intuitively expects the presence of a benign structure so that interpolation already occurs at a smaller network size than suggested by memorization capacity. In this paper, we investigate interpolation by adopting an instance-specific viewpoint. We introduce a simple randomized algorithm that, given a fixed finite data set with two classes, with high probability constructs an interpolating three-layer neural network in polynomial time. The required number of parameters is linked to geometric properties of the two classes and their mutual arrangement. As a result, we obtain guarantees that are independent of the number of samples and hence move beyond worst-case memorization capacity bounds. We verify our theoretical result with numerical experiments and additionally investigate the effectiveness of the algorithm on MNIST and CIFAR-10.

stat.ML↗