Relating Implicit Bias and Adversarial Attacks through Intrinsic Dimension

Despite their impressive performance in classification, neural networks are known to be vulnerable to adversarial attacks. These attacks are small perturbations of the input data designed to fool the model. Naturally, a question arises regarding the potential connection between the architecture, settings, or properties of the model and the nature of the attack. In this work, we aim to shed light on this problem by focusing on the implicit bias of the neural network, which refers to its inherent inclination to favor specific patterns or outcomes. Specifically, we investigate one aspect of the implicit bias, which involves the essential Fourier frequencies required for accurate image classification. We conduct tests to assess the statistical relationship between these frequencies and those necessary for a successful attack. To delve into this relationship, we propose a new method that can uncover non-linear correlations between sets of coordinates, which, in our case, are the aforementioned frequencies. By exploiting the entanglement between intrinsic dimension and correlation, we provide empirical evidence that the network bias in Fourier space and the target frequencies of adversarial attacks are closely tied.

DADApy: Distance-based Analysis of DAta-manifolds in Python

DADApy is a python software package for analysing and characterising high-dimensional data manifolds. It provides methods for estimating the intrinsic dimension and the probability density, for performing density-based clustering and for comparing different distance metrics. We review the main functionalities of the package and exemplify its usage in toy cases and in a real-world application. The package is freely available under the open-source Apache 2.0 license and can be downloaded from the Github page https://github.com/sissa-data-science/DADApy.

Estimating the intrinsic dimension of datasets by a minimal neighborhood information

Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number of coordinates. Such manifold is generally twisted and curved, in addition points on it will be non-uniformly distributed: two factors that make the identification of the ID and its exploitation really hard. Here we propose a new ID estimator using only the distance of the first and the second nearest neighbor of each point in the sample. This extreme minimality enables us to reduce the effects of curvature, of density variation, and the resulting computational cost. The ID estimator is theoretically exact in uniformly distributed datasets, and provides consistent measures in general. When used in combination with block analysis, it allows discriminating the relevant dimensions as a function of the block size. This allows estimating the ID even when the data lie on a manifold perturbed by a high-dimensional noise, a situation often encountered in real world data sets. We demonstrate the usefulness of the approach on molecular simulations and image analysis.

Automatic topography of high-dimensional data sets by non-parametric Density Peak clustering

Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach for charting data spaces, providing a topography of the probability distribution from which the data are harvested. This topography includes information on the number and the height of the probability peaks, the depth of the "valleys" separating them, the relative location of the peaks and their hierarchical organization. The topography is reconstructed by using an unsupervised variant of Density Peak clustering exploiting a non-parametric density estimator, which automatically measures the density in the manifold containing the data. Importantly, the density estimator provides an estimate of the error. This is a key feature, which allows distinguishing genuine probability peaks from density fluctuations due to finite sampling.