Dependence Induced Representations (2411.15328v1)

Abstract: We study the problem of learning feature representations from a pair of random variables, where we focus on the representations that are induced by their dependence. We provide sufficient and necessary conditions for such dependence induced representations, and illustrate their connections to Hirschfeld--Gebelein--R\'{e}nyi (HGR) maximal correlation functions and minimal sufficient statistics. We characterize a large family of loss functions that can learn dependence induced representations, including cross entropy, hinge loss, and their regularized variants. In particular, we show that the features learned from this family can be expressed as the composition of a loss-dependent function and the maximal correlation function, which reveals a key connection between representations learned from different losses. Our development also gives a statistical interpretation of the neural collapse phenomenon observed in deep classifiers. Finally, we present the learning design based on the feature separation, which allows hyperparameter tuning during inference.

• The paper introduces dependence induced representations by establishing necessary and sufficient conditions that link feature learning to the dependence structure between variables.
• It characterizes a family of D-loss functions, including common losses like cross-entropy, to enable practical feature adaptation and efficient hyperparameter tuning.
• The study reveals deep connections between maximal correlation, minimal sufficiency, and phenomena in deep learning, offering theoretical insights for improved representation learning.

An Overview of "Dependence Induced Representations"

In "Dependence Induced Representations," Xiangxiang Xu and Lizhong Zheng present a comprehensive paper of learning feature representations derived from the dependencies between two random variables. They focus on the representations that are directly influenced by the dependence between the variables $X$ and $Y$, referred to as dependence induced representations. This work presents novel theoretical foundations, detailing sufficient and necessary conditions for these types of representations and their connections to Hirschfeld–Gebelein–Rényi (HGR) maximal correlation functions and minimal sufficient statistics.

Key Contributions

1. Dependence Induced Representations: The authors introduce and explore representations as a function dependent solely on the relationship between variables, characterized as invariant to transformations preserving their dependence structure.
2. Characterization of Loss Functions: The paper characterizes a broad family of loss functions, termed D-losses, capable of learning dependence induced representations. This encompasses well-known functions like cross-entropy and hinge loss, revealing that the learned features are a composition of a loss-dependent function and the maximal correlation function.
3. Deep Connections with Maximal Correlation and Sufficiency: A significant contribution is illustrating that dependence induced representations are deeply connected to HGR maximal correlation functions and minimal sufficiency. This affiliation offers a statistical lens to interpret phenomena observed in deep learning, such as neural collapse in deep classifiers.
4. Feature Adaptation for Practical Learning: The authors propose a learning design strategy influenced by feature separation, enabling hyperparameter tuning during inference. This adaptation mechanism leverages maximal correlation features, facilitating efficient representation learning without requiring retraining.

Theoretical and Practical Implications

The paper analytically connects the theoretical underpinnings of dependence induced representations to widely used deep learning practices. By framing such representations through the lens of maximal correlation and minimal sufficiency, the work presents significant theoretical insights, enhancing our understanding of feature learning mechanisms underlying deep learning models.

The practical implications are also noteworthy; the results suggest that robust representations amenable to various tasks can be achieved by learning from the dependence structure, potentially leading to more efficient algorithms that are less reliant on large labeled datasets. The introduction of feature adapters further enhances this practicality, offering a mechanism for improving adaptability and performance across diverse tasks with minimal retraining.

Future Directions

While this work lays a robust foundation for understanding dependence induced representations, it also opens several avenues for future exploration. Potential directions include extending these principles to more complex network architectures, exploring their integration with unsupervised learning frameworks, and thoroughly evaluating the effect of specific loss function choices on broader generalization capabilities. Additionally, given the ongoing advancements in AI, exploring how these representations can adapt to or leverage real-time data streaming effectively presents an exciting research frontier.

Overall, "Dependence Induced Representations" offers a unique perspective on representation learning, combining rigorous statistical fundamentals with applicable insights for enhancing deep learning methodologies. As the field continues to evolve, understanding and implementing such dependence-structured learning could drive significant advancements in artificial intelligence and machine learning.