Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning with Hidden Factorial Structure

Charles Arnal, Clement Berenfeld, Simon Rosenberg, Vivien Cabannes

TL;DR

The paper addresses learning with high-dimensional discrete data by proposing hidden factorial structure as a key prior and introducing a factorization hypothesis for both inputs and outputs. It develops two embedding schemes—learned and factorization-compatible—and provides theoretical results on approximation and statistical complexities showing potential exponential gains when structure is present. Through controlled MLP experiments, the authors show that neural networks can leverage the factorized structure to learn discrete conditional distributions more efficiently and generalize under structured data regimes. The work highlights design principles for representations of discrete data and suggests potential connections to transfer learning, offering a bridge between classical nonparametric insights and modern neural architectures.

Abstract

Statistical learning in high-dimensional spaces is challenging without a strong underlying data structure. Recent advances with foundational models suggest that text and image data contain such hidden structures, which help mitigate the curse of dimensionality. Inspired by results from nonparametric statistics, we hypothesize that this phenomenon can be partially explained in terms of decomposition of complex tasks into simpler subtasks. In this paper, we present a controlled experimental framework to test whether neural networks can indeed exploit such "hidden factorial structures". We find that they do leverage these latent patterns to learn discrete distributions more efficiently. We also study the interplay between our structural assumptions and the models' capacity for generalization.

Learning with Hidden Factorial Structure

TL;DR

The paper addresses learning with high-dimensional discrete data by proposing hidden factorial structure as a key prior and introducing a factorization hypothesis for both inputs and outputs. It develops two embedding schemes—learned and factorization-compatible—and provides theoretical results on approximation and statistical complexities showing potential exponential gains when structure is present. Through controlled MLP experiments, the authors show that neural networks can leverage the factorized structure to learn discrete conditional distributions more efficiently and generalize under structured data regimes. The work highlights design principles for representations of discrete data and suggests potential connections to transfer learning, offering a bridge between classical nonparametric insights and modern neural architectures.

Abstract

Statistical learning in high-dimensional spaces is challenging without a strong underlying data structure. Recent advances with foundational models suggest that text and image data contain such hidden structures, which help mitigate the curse of dimensionality. Inspired by results from nonparametric statistics, we hypothesize that this phenomenon can be partially explained in terms of decomposition of complex tasks into simpler subtasks. In this paper, we present a controlled experimental framework to test whether neural networks can indeed exploit such "hidden factorial structures". We find that they do leverage these latent patterns to learn discrete distributions more efficiently. We also study the interplay between our structural assumptions and the models' capacity for generalization.

Paper Structure

This paper contains 30 sections, 3 theorems, 54 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Context and motivations
  3. Contributions
  4. Related works
  5. Setting
  6. The Factorization Hypothesis
  7. Input Embeddings
  8. Learned embedding
  9. Factorization-compatible embedding
  10. Theoretical Analysis
  11. Approximation Complexity
  12. Statistical Complexity
  13. Experiments
  14. Experimental Design
  15. Data specification
  16. ...and 15 more sections

Key Result

Theorem 1

The log-likelihood matrix $\Lambda$ of the conditional probability distribution $p(y|x)$ can be factorized as $\Lambda = U^\top G$, where $U \in {\mathbb{R}}^{\bar{\chi}(p) \times M}$, $G \in {\mathbb{R}}^{\bar{\chi}(p) \times N}$ with and

Figures (19)

  • Figure 1: A graphical representation of a model for $k = 4$ and $\ell = 3$. In this example, $\mathrm{pa}_1 = (x_1,x_2)$, $\mathrm{pa}_2 = (x_4)$, $\mathrm{pa}_3 = (x_1,x_2,x_4)$, and the third hidden variable $x_3$ has no impact on the output $y$.
  • Figure 2: A diagram of a factorization-compatible embedding $x \mapsto \tilde{e}_x$.
  • Figure 3: Value of the population loss ${\cal L}$\ref{['eq:loss']} in the single pass setting as a function of the number of epochs for various values of the statistical complexity parameter $\chi$\ref{['eq:SC']}. We obtain various values of $\chi$ by modifying the data model parameter (P1).
  • Figure 4: Value of the population loss ${\cal L}$\ref{['eq:loss']} in the compression setting after $T=10^6$ epochs of training as a function of the embedding dimension $d$ for $|I_j| \in \{1, 2, 3, 4\}$ (P3). The dashed lines indicate $d = \bar{\chi}$.
  • Figure 5: Value of the population loss ${\cal L}$\ref{['eq:loss']} in the compression setting as a function of the compute for various values of the embedding dimension $d$ (P5). Note that $\bar{\chi} = 16$\ref{['eq:AC']} with our default experimental parameters.
  • ...and 14 more figures

Theorems & Definitions (11)

  • Remark 1: Compression and representation
  • Example 1: Text data
  • Example 2: Recommender systems
  • Remark 2: Link to mixture estimation
  • Remark 3: Link to transformers
  • Remark 4: Link to other statistical problems
  • Theorem 1
  • Theorem 2: see e.g. canonne2020short
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:ac']}
  • ...and 1 more