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The Geometry of Manifold Learning: Singular foliations via Deep Learning models
Gabriele Novelli, Eliot Tron, Emanuele Latini, Rita Fioresi
TL;DR
Understanding whether trained networks capture the geometry of their data motivates studying the data-space structure induced by the DIM, especially for non-smooth ReLU networks. The paper proves that the DIM distribution is generically non-smooth and that a suitable smooth modification is integrable, yielding integral manifolds; experimental evidence links these manifolds to training data.
Problem
How the DIM foliation’s non-smoothness and loss of rank behave remains unexplored, limiting understanding of the geometry a trained network induces on data space.
Source: §1 · p. 2Method
The paper studies the DIM distribution for ReLU networks and constructs a suitable smooth distribution whose integrability is checked using Stefan-Sussmann theory.
Source: §1 · p. 3, §3 · p. 8, §4 · p. 10Results
For generic weights, the DIM distribution is non-smooth, while a suitable smooth modification satisfies Stefan-Sussmann integrability conditions and defines integral manifolds.
Source: §3 · p. 8, §4 · p. 10, §5 · p. 11Takeaways & Limitations
The resulting foliation provides a data-space geometric structure, and experimental evidence links its integral manifolds with the dataset used for training.
Source: §5 · pp. 10–11The relation between the integral manifolds and the training dataset is linked by experimental evidence, and the authors leave further investigation of that relation to future work.
Source: §5 · p. 11
Abstract
from arXiv · showhide
Deep learning algorithms have become widely used and successful across a wide range of tasks. A key challenge is determining whether these algorithms truly capture the intrinsic geometry of their training data in order to better understand what they learn from data. In this theoretical work, we explore a new geometric structure defined on the data space by a trained neural network through the Data Information Matrix (DIM), a variation of the Fisher Information Matrix. This new geometric structure is a foliation and the main focus of this work is to explore the non-smooth case of ReLU activation functions, where traditional geometric tools often fail. The foliation is defined through a distribution spanned by the columns of the DIM. We first prove that this distribution is non-smooth in the sense of distributions, rendering the existence of an associated foliation much more complex. To circumvent this issue, we introduce a smoothed version of this distribution that differs from the original only on a set of measure zero. Finally, we prove that this smoothed distribution induces a singular foliation on the data space, which is based on the data the network has been trained with. Our theoretical contributions provide a geometric framework to better understand how neural networks organize their input space in non-smooth settings, bridging differential geometry and deep learning theory.
1 Introduction
The paper studies how a trained network induces geometry on data space through the Data Information Matrix (DIM), focusing on unresolved properties of the resulting foliation. For ReLU networks, it proves the DIM distribution is non-smooth and establishes integrability for a smooth distribution derived from it.
1 Introduction
- The paper uses the DIM to give data space a natural foliation structure, extending geometric study beyond the manifold hypothesis.The DIM is presented as a data-space analogue of the Fisher Information Matrix.Source: §1 · p. 2
- The full geometry of the DIM foliation remains unexplored, particularly the distribution’s non-smoothness and the mechanisms behind its loss of rank.Earlier work links the DIM structure to the amount of noise associated with a data-space point.Source: §1 · p. 2
- ReLU networks’ piecewise-linear structure naturally leads to distributions with varying rank and limited regularity.The introduction notes that ReLU is widely used in deep learning.Source: §1 · p. 3
- For generic ReLU convolutional classifiers, the DIM defines a non-smooth distribution.The introduction points to Theorem 3.2 for this result.Source: §1 · p. 3
- The paper establishes integrability for a smooth distribution obtained from the DIM distribution of ReLU convolutional classifiers.The introduction identifies this as a key application of Frobenius and Stefan-Sussmann results.Source: §1 · p. 3
2 Preliminaries on Information Geometry
This section sets notation for classifier probabilities and information losses, then defines the Fisher Information Matrix (FIM) and the Data Information Matrix (DIM). It interprets the DIM’s image and kernel as directions that change, or leave unchanged, the classifier’s output probabilities.
2 Preliminaries on Information Geometry
- The classifier assigns a probability distribution over C labels to each input x, with p_i denoting the predicted probability of label i.The network weights are w, and the input belongs to the data space.Source: §2 · p. 3
- The loss compares the model distribution with a ground-truth distribution, and for a one-hot target it is the KL divergence up to a zero constant.The section also introduces information loss and the loss function for comparing probability distributions.Source: §2 · pp. 3–4
- A common classifier choice is softmax probabilities derived from scores, with cross-entropy as the corresponding loss.The scores are determined by the network parameters and may use compositions involving ReLUs.Source: §2 · p. 4
- The section defines the FIM from parameter gradients of log probabilities and the DIM from input gradients of log probabilities.Both matrices take an expectation over the model’s label distribution.Source: §2 · p. 4
- The FIM and DIM are positive semidefinite, and each has rank below the number of classes.Their kernels are orthogonal to the spans of the corresponding parameter- or input-space probability gradients.Source: §2 · p. 4
- The DIM distribution assigns each input the span of its class-probability gradients; directions in the kernel leave the predicted distribution unchanged.The kernel directions are orthogonal to the image of the DIM and are called noise directions.Source: §2 · pp. 4–5
3 Singular and non-smooth Distributions
The DIM assigns each data point a distribution spanned by gradients of the network’s output probabilities. For ReLU networks, this distribution is generically non-smooth, but changing it on a closed null set yields a smooth distribution for the subsequent integrability analysis.
Singular and non-smooth distributions
- A distribution is singular when its rank is not constant; with ReLU scores, whether the DIM distribution is smooth is nontrivial.Different vector fields can span the same pointwise distribution, so nonsmooth generators alone do not establish that the distribution is nonsmooth.Source: §3 · pp. 5, 8
DIM distribution in ReLU networks
- The DIM distribution assigns each input the span of gradients of its output probabilities, computed from the network’s score function.The section assumes a ReLU score function and uses a convention for evaluating gradients at ReLU nondifferentiable points.Source: §3 · pp. 6–7
Singular points
- The DIM distribution’s singular points lie in a closed null subset of a finite union of hyperplanes.The stated result assumes softmax probabilities and the specified ReLU score function.Source: §3 · pp. 6–7
Non-smoothness of the DIM distribution
- For generic choices of network weights, the DIM distribution is non-smooth.The claim is not for every weight choice; the paper notes genericity is essential to its scope.Source: §3 · p. 8
Smoothing the distribution
- The paper modifies the DIM distribution on a closed set of measure zero to define a smooth version that agrees with it elsewhere.The modified distribution is set to zero on the union of hyperplanes containing the singular points.Source: §3 · p. 8
- The modified distribution is smooth, with smooth vector fields constructed to span it, setting up the next section’s integrability analysis.The construction spans the modified distribution both on and off the hyperplanes.Source: §3 · pp. 8–9
4 Integrability of the DIM distribution
The section establishes integrability of the DIM distribution away from its singular hyperplanes and of its smooth modification across the full data space. The global result uses the Stefan–Sussmann flow condition, which also handles points on the hyperplanes.
4 Integrability of the DIM distribution
- Integrability asks whether each point lies on a submanifold whose tangent spaces coincide with the distribution along that submanifold.For singular distributions, rank may vary across the ambient manifold, though it is constant along each such submanifold.Source: §4 · p. 9
- Frobenius characterizes integrability for smooth regular distributions through involutivity, while Stefan–Sussmann provides the criterion used for singular distributions.The flow-invariance condition in Stefan–Sussmann is stronger than involutivity.Source: §4 · pp. 9–10
- The DIM distribution is integrable on open regions outside H, where it has constant rank; Frobenius applies there because it is involutive.The section notes that involutivity was proved in.Source: §4 · p. 10
- The modified singular distribution bD is integrable throughout the data space by the Stefan–Sussmann theorem.Away from H, its flows do not reach H in finite time; on H, the flow differential sends zero to zero, satisfying the condition there.Source: §4 · p. 10
5 Conclusions
The conclusions identify the DIM distribution as generally non-smooth and describe a modification that yields an integrable singular smooth distribution. The resulting integral manifolds are linked experimentally to training data, a relation left for future investigation.
5 Conclusions
- For parameters in general position, the DIM distribution spanned by a deep-learning system’s probability gradients is non-smooth outside a measure-zero set.The conclusion describes the exceptional set using non-null sets of polynomial functions.Source: §5 · pp. 10–11
- A suitable modification of the DIM distribution yields a singular smooth distribution satisfying Stefan–Sussmann integrability conditions and therefore a well-defined notion of integral manifold in data space.The distribution arises from the Data Information Matrix, the data-space analogue of the Fisher Information Matrix.Source: §5 · pp. 10–11
- Experimental evidence in links the integral manifolds to the dataset used to train the model, but the conclusion leaves that relation for future investigation.Source: §5 · p. 11
A Smoothness of the DIM distribution
The appendix argues that, under a non-constant DIM distribution, smoothness is exceptional in parameter space. It compares score-function Jacobians across a ReLU hyperplane and uses rank constraints to show that smooth cases lie in a closed measure-zero set.
A Smoothness of the DIM distribution
- Across a ReLU hyperplane H, the appendix compares score-function Jacobians Jx(s)+ and Jx(s)−, which differ by setting a set of weight rows to zero.The two Jacobian forms are specified on the regions on either side of H and on H itself.Source: A Smoothness of the DIM distribution · pp. 14–15
- The proof assumes smoothness near a hyperplane point and derives a contradiction when the adjacent Jacobian images impose incompatible constraints on a smooth vector field.The argument considers points away from intersections of multiple hyperplanes; the appendix says the generic-point argument is the same.Source: A Smoothness of the DIM distribution · pp. 14–15
- If the DIM distribution is non-constant, the parameters for which it is smooth lie in a closed measure-zero subset of parameter space.The theorem assumes non-constancy, distinguishing it from non-regularity; the paper says the constant case is not observed experimentally.Source: A Smoothness of the DIM distribution · p. 15
- Rank constraints across the hyperplane produce polynomial equations in parameter space, placing any smooth cases inside a closed measure-zero set.If the relevant minors vanish identically on one side, the row-zero relation forces them to vanish on the other, contradicting the assumed rank increase.Source: A Smoothness of the DIM distribution · pp. 15–16
B Distributions
The appendix distinguishes pointwise distributions from modules of smooth vector fields, showing that their correspondence loses information. It also explains why Hermann’s integrability theorem has limited applicability to concrete questions involving these distributions.
B Distributions
- The paper distinguishes its pointwise distributions from module distributions, a different notion used in.A module distribution is a C∞(M) submodule of the smooth vector fields X(M).Source: B Distributions · p. 16
- The maps between smooth distributions and module distributions are not inverses: multiple module distributions can yield the same distribution, and some are not global sections of any distribution.Source: B Distributions · p. 16
- In the example on M = R, distinct module distributions Fk yield the same distribution, while the global sections recover only F0.The information carried by xk is lost when evaluating at x.Source: B Distributions · pp. 16–17
- Hermann’s theorem applies to module distributions and is less applicable to concrete questions; its finite-generation condition may fail even when the distribution is integrable.The appendix gives an example whose global sections are not finitely generated, although the distribution is integrable.Source: B Distributions · pp. 17–18