Rethinking Softmax with Cross-Entropy: Neural Network Classifier as Mutual Information Estimator

Abstract

Mutual information is widely applied to learn latent representations of observations, whilst its implication in classification neural networks remain to be better explained. We show that optimising the parameters of classification neural networks with softmax cross-entropy is equivalent to maximising the mutual information between inputs and labels under the balanced data assumption. Through experiments on synthetic and real datasets, we show that softmax cross-entropy can estimate mutual information approximately. When applied to image classification, this relation helps approximate the point-wise mutual information between an input image and a label without modifying the network structure. To this end, we propose infoCAM, informative class activation map, which highlights regions of the input image that are the most relevant to a given label based on differences in information. The activation map helps localise the target object in an input image. Through experiments on the semi-supervised object localisation task with two real-world datasets, we evaluate the effectiveness of our information-theoretic approach.

References in corpus (7)

• Adam: A Method for Stochastic Optimization
• Conditional Image Synthesis With Auxiliary Classifier GANs
• Learning deep representations by mutual information estimation and maximization
• InfoVAE: Information Maximizing Variational Autoencoders
• On Variational Bounds of Mutual Information
• Formal Limitations on the Measurement of Mutual Information
• Improved Techniques For Weakly-Supervised Object Localization

Cited by in corpus (12)

• A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges
• FeederGAN: Synthetic Feeder Generation via Deep Graph Adversarial Nets
• X-GOAL: Multiplex Heterogeneous Graph Prototypical Contrastive Learning
• Co-Mixup: Saliency Guided Joint Mixup with Supermodular Diversity
• Frustratingly Easy Transferability Estimation
• Neural Estimators for Conditional Mutual Information Using Nearest Neighbors Sampling
• DICE: Diversity in Deep Ensembles via Conditional Redundancy Adversarial Estimation
• Fusing Higher-order Features in Graph Neural Networks for Skeleton-based Action Recognition
• Generalized Data Weighting via Class-level Gradient Manipulation
• Informative Class Activation Maps
• Are Deep Neural Architectures Losing Information? Invertibility Is Indispensable
• A Thousand Words are Worth More Than One Recording: NLP Based Speaker Change Point Detection