Classification logit two-sample testing by neural networks for differentiating near manifold densities.

Area of Science:

• Machine Learning
• Statistical Inference
• Data Science

Background:

• The classical two-sample problem aims to distinguish between two probability densities using finite data samples.
• Recent advancements in generative adversarial networks and variational learning suggest potential for classification networks in solving this problem.
• Existing network-based methods offer scalability to large datasets but require further analysis for accuracy and error bounds.

Purpose of the Study:

• To develop and analyze a novel network-based statistic for the two-sample problem using the classification logit function.
• To investigate the approximation and estimation errors of the logit function for differentiating near-manifold densities.
• To establish theoretical guarantees for the logit function's ability to differentiate densities, considering network parametrization and data dimensionality.

Main Methods:

• Utilizing the classification logit function from a trained neural network as a two-sample statistic.
• Introducing a new theoretical result on near-manifold integral approximation by neural networks to analyze logit function errors.
• Proving the logit function's ability to differentiate sub-exponential densities with sufficient network parametrization and analyzing complexity for manifold data.

Main Results:

• The network logit test demonstrates superior performance over previous network-based classification accuracy tests.
• Experimental results show favorable comparisons with kernel maximum mean discrepancy tests on synthetic and real-world datasets.
• Theoretical analysis confirms provable differentiation of densities and reduced network complexity for manifold data.

Conclusions:

• The proposed network logit test is an effective and computationally efficient method for the two-sample problem.
• The method scales well to large datasets and offers improved performance over existing techniques.
• Theoretical insights provide a foundation for understanding the capabilities and limitations of network-based density differentiation.