Geometry of Singular Foliations and Learning Manifolds in ReLU Networks via the Data Information Matrix

Understanding how real data is distributed in high dimensional spaces is the key to many tasks in machine learning. We want to provide a natural geometric structure on the space of data employing a ReLU neural network trained as a classifier. Through the Data Information Matrix (DIM), a variation of the Fisher information matrix, the model will discern a singular foliation structure on the space of data. We show that the singular points of such foliation are contained in a measure zero set, and that a local regular foliation exists almost everywhere. Experiments show that the data is correlated with leaves of such foliation. Moreover we show the potential of our approach for knowledge transfer by analyzing the spectrum of the DIM to measure distances between datasets.

💡 Research Summary

The paper proposes a geometric framework for high‑dimensional data based on the internal structure of a ReLU‑based classifier. By defining the Data Information Matrix (DIM) – a data‑space analogue of the Fisher information matrix – the authors extract a distribution D(x) on the input space, where D(x) is the span of the gradients ∇ₓ log p(yᵢ|x,w) for all classes. This distribution can be interpreted as a tangent subspace at each point and, under suitable conditions, integrates to a foliation L_D that partitions the data space into immersed submanifolds (leaves).

Theoretical contributions are centered on two results. Lemma 3.4 characterises singular points of D as locations where the rank of D(x) drops relative to a neighbourhood, typically occurring at ReLU’s nondifferentiable hyperplanes or at parameter‑induced boundaries. Theorem 3.6 proves that the set of such singular points has Lebesgue measure zero, guaranteeing that almost everywhere the distribution has constant rank and the foliation is regular.

A crucial observation is that the involutivity of D – required by Frobenius’ theorem for integrability – holds automatically for piecewise‑linear activations such as ReLU and MaxPool. The authors verify this by showing that the Lie brackets of the generating vector fields vanish, leading to a well‑defined foliation. In contrast, smooth activations like GeLU or sigmoid produce non‑vanishing brackets; Table 1 demonstrates that the dimension of the Lie‑closed space V_D(x) exceeds that of D(x), indicating a failure of involutivity and thus the absence of a foliation.

Experimentally, the authors train shallow ReLU networks on the XOR problem, MNIST, and Fashion‑MNIST. Visualizations (Fig. 1‑2) illustrate that moving along a leaf of L_D smoothly morphs an image while preserving class probabilities, whereas moving orthogonally changes the input without affecting the predicted label. The DIM’s eigenvalue spectrum is then used to define a distance between datasets: datasets with more similar spectra (e.g., MNIST vs. a rotated MNIST) are closer, while those with divergent spectra (MNIST vs. Fashion‑MNIST) are farther apart. This spectral distance predicts the difficulty of knowledge transfer; fine‑tuning a model trained on one dataset to another with a large spectral gap incurs higher error.

The paper also discusses practical implications: (1) the foliation provides a natural way to identify data points that lie on the same intrinsic submanifold, which can aid in data augmentation or semi‑supervised learning; (2) the measure‑zero nature of singularities justifies ignoring pathological regions during training; (3) the DIM offers a computationally cheap tool (via automatic differentiation) to assess dataset similarity without requiring costly optimal transport calculations.

Limitations are acknowledged. The framework relies on piecewise‑linear activations; extending it to smooth activations would require sub‑Riemannian geometry or alternative integrability conditions. The authors do not provide a formal proof that the DIM‑based distance satisfies metric properties (symmetry, triangle inequality). Moreover, the impact of network depth and width on the rank and stability of D is left for future work.

In summary, the work bridges information geometry, differential geometry, and deep learning by introducing the Data Information Matrix as a bridge between classifier gradients and geometric structure. It establishes that ReLU networks endow the data space with an almost‑everywhere regular singular foliation, whose leaves align closely with the training data. The spectral analysis of DIM opens a promising avenue for dataset comparison and knowledge transfer, suggesting that the geometry induced by a trained network can be harnessed for a variety of downstream tasks.