Cross-Entropy Is All You Need To Invert the Data Generating Process (2410.21869v3)

Abstract: Supervised learning has become a cornerstone of modern machine learning, yet a comprehensive theory explaining its effectiveness remains elusive. Empirical phenomena, such as neural analogy-making and the linear representation hypothesis, suggest that supervised models can learn interpretable factors of variation in a linear fashion. Recent advances in self-supervised learning, particularly nonlinear Independent Component Analysis, have shown that these methods can recover latent structures by inverting the data generating process. We extend these identifiability results to parametric instance discrimination, then show how insights transfer to the ubiquitous setting of supervised learning with cross-entropy minimization. We prove that even in standard classification tasks, models learn representations of ground-truth factors of variation up to a linear transformation. We corroborate our theoretical contribution with a series of empirical studies. First, using simulated data matching our theoretical assumptions, we demonstrate successful disentanglement of latent factors. Second, we show that on DisLib, a widely-used disentanglement benchmark, simple classification tasks recover latent structures up to linear transformations. Finally, we reveal that models trained on ImageNet encode representations that permit linear decoding of proxy factors of variation. Together, our theoretical findings and experiments offer a compelling explanation for recent observations of linear representations, such as superposition in neural networks. This work takes a significant step toward a cohesive theory that accounts for the unreasonable effectiveness of supervised deep learning.

• The paper demonstrates that minimizing cross-entropy in supervised learning recovers latent factors up to a linear transformation.
• It introduces a cluster-centric data generating model using von Mises-Fisher distributions to establish identifiability in latent representations.
• Empirical evaluations on benchmarks and ImageNet validate that standard classification tasks can yield robust and interpretable disentangled features.

Overview of "Cross-Entropy is All You Need to Invert the Data Generating Process"

The paper "Cross-Entropy is All You Need to Invert the Data Generating Process" explores the foundational aspects of representation learning, especially in the context of supervised learning through cross-entropy minimization. This paper addresses the theoretical gaps in understanding why supervised learning models, typically in classification tasks, can infer meaningful, disentangled latent representations. The authors leverage concepts from nonlinear Independent Component Analysis (ICA) and parametric instance discrimination to demonstrate that supervised learning inherently inverts the data generating process, thus recovering latent variables up to a linear transformation.

Key Contributions

• Framework for Identifiability: The authors propose a novel cluster-centric Data Generating Process (DGP) model, setting the stage for proving identifiability in contexts similar to auxiliary-variable ICA. They extend these insights to parametric instance discrimination and, crucially, show how these theoretical results apply to supervised learning with cross-entropy.
• Empirical Evaluation: Through simulations and tests on disentanglement benchmarks and real-world datasets like ImageNet, the paper empirically substantiates its theoretical claims. The research highlights the ability of simple classification tasks to recover latent structures, providing a robust agreement with their formulated hypotheses.

Theoretical Insights

The paper's theoretical contributions demonstrate that solving supervised classification tasks with cross-entropy minimization can retrieve ground-truth factors of variation in data up to an orthogonal transformation. By utilizing the DIET (Discriminative Image Embedding Through Transformation) framework, the authors show how instance discrimination resembles non-linear feature extraction, offering a bridge between self-supervised learning and classical supervised paradigms.

Cluster-Centric DGP Model: The theoretical model leverages von Mises-Fisher distributions centered around cluster vectors representing semantic classes. This assumption allows the separation of local and global data structures in latent spaces, providing new insights into how neural networks encode feature information.

Cross-Entropy and ICA Parallel: The analysis reveals that the cross-entropy loss, a staple in classification tasks, effectively acts as an ICA mechanism. This manifests significantly as the ability of deep networks to encode interpretable representations linearly, even in tasks traditionally reserved for self-supervised approaches.

Empirical Findings

Synthetic Data and Disentanglement Benchmarks: The paper includes comprehensive evaluations on controlled datasets, such as DisLib and ImageNet-X, demonstrating linear decoding of latent features through learned embeddings. These results show remarkable consistency across architectures, dataset sizes, and latent dimensionalities, reinforcing the theoretical backbone.

Scalability and Real-World Applications: Despite the controlled settings, the findings extend to complex datasets, underscoring the generalizability and scalability of the proposed methods. Models trained on ImageNet revealed robust representations aligned with ground-truth factors of variation, confirming the applicability beyond synthetic benchmarks.

Implications and Future Directions

The work advances the understanding of representation learning, underlining the sufficiency of cross-entropy-based learning in extracting semantic structures from data. This revelation paves the way for reconsidering the relationship between supervised and self-supervised paradigms, suggesting that complex feature engineering or contrastive methods may not be strictly necessary for effective representation learning.

Robustness and Flexibility of Learning Frameworks: By interpreting standard classification tasks through the lens of nonlinear ICA, future research might explore broader applications, including those involving complex modalities or multi-task learning, where disentangled and meaningful representations are crucial.

The Role of Cross-Entropy: The insight that cross-entropy aids in fully capturing latent data structures invites exploration into other loss functions and regularization techniques that might similarly unlock deeper understanding or enhanced performance.

Interdisciplinary Integration: Given the implications for both theoretical and applied machine learning, this research could inspire cross-disciplinary synergies, linking fields such as cognitive neuroscience, where understanding underlying representational mechanisms is pivotal.

In conclusion, this paper takes a significant step toward forming a unified theory that explains the effectiveness of deep learning methodologies, situating supervised classification as a potent framework for inverting data generating processes and highlighting the integral role of cross-entropy in this quest.