Organizations: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. · School of Mathematics, Northwest University, Xi’an 710127, China.
Abstract
We consider establishing the interpretability theory of deep learning through constructing a corresponding relationship between the renormalization group (RG) method in statistical physics and the training process of deep neural networks (DNNs). We have proved the constructed relationship using the one-dimensional Ising model as the input data. In this paper we generalize our results to the case of continuous input data, which is a necessary preparation for applying the corresponding framework to real-world data. To be representative, we consider a class of data distribution in the exponential family. We prove that when the parameters of fully connected (FC) DNNs achieve their optimal value after training, the characteristic parameters of the feature layer output of DNNs are equal to the fixed points of the characteristic parameters of input data under RG method for continuous fields. This conclusion shows that the training process of DNNs is equivalent to RG calculation on this kind of data and therefore the network can extract main features from the input data just like RG. Also, the equivalence further validates the correspondence framework we have established, providing an explanation for the outstanding performance of DNNs on real-world data.
A scientific theory of deep learning, comprising learning dynamics and statistical properties of learned models, is rapidly gaining attention. One of the corner stones of this development are analytically solvable toy models, allowing for the fully tractable analysis of the learning dynamics. Here we analytically investigate such a toy model using the regularization strength as a tunable external parameter - akin to external fields in statistical physics. In previous studies, (i) an onset of learning transition was predicted analytically and (ii) it was phenomenologically/numerically established that tuning the regularization strength can result in a cascade of phase transitions. The number of those transitions was linked to the geometry of the loss landscape determined by the model complexity. Setting up a rigorous framework underpinning the previous numerical observations, our investigation reveals a precise connection between those cascades of phase transitions, learnable features and the underlying geometry. We provide analytic predictions of these phase transitions as well as tractable order parameters related to learned features. At the level of the minimal model, we connect this macroscopic perspective (that can be condensed into an effective description) to the microscopic perspective in terms of the geometry of the loss landscape characterized by the Hessian spectrum. Thus, the presented model provides a platform to explore and sharpen advances made in the scientific theory of deep learning rooted in statistical physics concepts.
Björn Ladewig, Ibrahim Talha Ersoy, Karoline Wiesner
The analogy between deep neural network forward passes and renormalization group (RG) flows has been repeatedly noted in the literature, but existing treatments remain qualitative: depth is described as a coarse-graining scale, attention is likened to a partition function, and representations are said to flow toward fixed points. No existing work has defined a measurable RG order parameter, tested it under controlled variation of the input distribution, or made quantitative predictions that are empirically verified. We study the simplest architecture for which the analogy is tractable: a pure MLP residual stack trained on masked token prediction over synthetic Markov chain sequences with known spectral properties. We report three findings. (i) The effective rank of the residual stream decreases monotonically with depth after training, consistent with progressive integration of irrelevant degrees of freedom. (ii) This rank collapse is selective: it occurs for chains with short correlation length approximately 1 but is absent for chains with long correlation length approximately 7, measured at the position level to control for mean-pooling artifacts. The network preserves exactly the degrees of freedom relevant to the prediction task, the content of the RG relevance criterion. (iii) Inter-layer kernel drift is concentrated at one or two specific transitions, with the remainder of the network near a fixed point, consistent with a discrete fixed-point plateau. Together these findings constitute the first quantitative, position-level evidence that MLP residual networks implement a selective coarse-graining procedure governed by the spectral structure of the input distribution.
We address the fundamental question of why deep neural networks generalize by establishing a pointwise generalization theory for fully connected networks. This framework resolves long-standing barriers to characterizing the rich nonlinear feature-learning regime and builds a new statistical foundation for representation learning. For each trained model, we characterize the hypothesis via a pointwise Riemannian Dimension, derived from the eigenvalues of the learned feature representations across layers. This establishes a principled framework for deriving hypothesis-dependent, representation-aware generalization bounds. These bounds offer a systematic upgrade over approaches based on model size, products of norms, and infinite-width linearizations, yielding guarantees that are orders of magnitude tighter in both theory and experiment. Analytically, we identify the structural properties and mathematical principles that explain the tractability of deep networks. Empirically, the pointwise Riemannian Dimension exhibits substantial feature compression, decreases with increased over-parameterization, and captures the implicit bias of optimizers. Taken together, our results indicate that deep networks are mathematically tractable in practical regimes and that their generalization is sharply explained by pointwise, feature-spectrum-aware complexity.