Generalized Cauchy-Schwarz Divergence and Its Deep Learning Applications

TL;DR

The paper tackles the challenge of quantifying divergence among more than two distributions in deep learning. It introduces the Generalized Cauchy-Schwarz Divergence (GCSD) and a kernel-based, closed-form empirical estimator, enabling efficient integration into loss functions and regularizers. The authors prove key properties (non-negativity, symmetry, projective invariance) and validate GCSD through extensive experiments in deep clustering and multi-source domain adaptation, where GCSD-based methods outperform state-of-the-art baselines. The work offers a scalable, principled tool for learning from and aligning multiple distributions with practical impact across clustering, domain adaptation, and related multi-distribution learning tasks.

Abstract

Divergence measures play a central role and become increasingly essential in deep learning, yet efficient measures for multiple (more than two) distributions are rarely explored. This becomes particularly crucial in areas where the simultaneous management of multiple distributions is both inevitable and essential. Examples include clustering, multi-source domain adaptation or generalization, and multi-view learning, among others. While computing the mean of pairwise distances between any two distributions is a prevalent method to quantify the total divergence among multiple distributions, it is imperative to acknowledge that this approach is not straightforward and necessitates significant computational resources. In this study, we introduce a new divergence measure tailored for multiple distributions named the generalized Cauchy-Schwarz divergence (GCSD). Additionally, we furnish a kernel-based closed-form sample estimator, making it convenient and straightforward to use in various machine-learning applications. Finally, we explore its profound implications in the realm of deep learning by applying it to tackle two thoughtfully chosen machine-learning tasks: deep clustering and multi-source domain adaptation. Our extensive experimental investigations confirm the robustness and effectiveness of GCSD in both scenarios. The findings also underscore the innovative potential of GCSD and its capability to significantly propel machine learning methodologies that necessitate the quantification of multiple distributions.

Figures (8)

• Figure 1: Complexity statistics
• Figure 2: Run time on synthetic data
• Figure 3: Comparison of divergence measures. (a) The evaluation is conducted on univariate random data samples from 10 distributions. To ensure a fair comparison, measurements of all the metrics are normalized by dividing them by their respective minimum values. (b) The evaluation is conducted on multivariate samples from 3 distributions (${\cal{N}}_1,{\cal{N}}_2$, and ${\cal{U}}_1$ as illustrated in \ref{['sec:sata_preparation']} and Figure \ref{['fig:syn_distributions']}) with their dimension varying from $10^1$ to $10^4$. To provide a clear visual representation, we use logarithmic scaling for all measured values.
• Figure 4: Deep divergence-based clustering framework. The Encoder can be implemented using various neural network architectures, such as a simple multilayer perceptron (MLP) for flattened data, a convolutional neural network (CNN) for image-like two-dimensional data, or recurrent neural networks (RNNs) like GRU or LSTM for sequential data. The Cluster can be implemented using a fully connected layer followed by softmax activation, allowing for the creation of a cluster assignment matrix. The loss function $\rm{Loss}$ integrates a generalized divergence measure $D_*^A(\cdot)$ and two regularization terms on assignment matrix $A$ to preserve its simplex property.
• Figure 5: Visualization of the clustered examples. A mini-batch of 100 samples clustered into 10 groups, each of which presented in a line. Groups with fewer clustered samples are filled with random noises.

Theorems & Definitions (4)

• Definition 1
• Remark 1
• Proposition 1
• Proposition 1