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Does the Data Processing Inequality Reflect Practice? On the Utility of Low-Level Tasks

Roy Turgeman, Tom Tirer

TL;DR

The paper questions the practical reach of the data processing inequality by showing that, in a high-dimensional binary classification setting, there exists a linear pre-processing step that improves accuracy for any finite training size, even when the classifier approaches Bayes optimality with more data. It provides a theoretical framework using a Gaussian Mixture Model with separation parameter $\mathcal{S}$ and derives approximations for the error probabilities before and after processing, including balanced and imbalanced training scenarios. The authors also validate the theory with empirical simulations and extend the insight to practical deep-learning pipelines, demonstrating that denoising and encoding of inputs can yield meaningful gains in finite-data regimes on benchmarks like CIFAR-10 and Mini-ImageNet. The findings highlight the potential benefit of carefully designed low-level tasks and unlabeled-data–driven learning for improving high-level classification, with implications for out-of-distribution robustness and future work on non-linear preprocessing and broader downstream tasks.

Abstract

The data processing inequality is an information-theoretic principle stating that the information content of a signal cannot be increased by processing the observations. In particular, it suggests that there is no benefit in enhancing the signal or encoding it before addressing a classification problem. This assertion can be proven to be true for the case of the optimal Bayes classifier. However, in practice, it is common to perform "low-level" tasks before "high-level" downstream tasks despite the overwhelming capabilities of modern deep neural networks. In this paper, we aim to understand when and why low-level processing can be beneficial for classification. We present a comprehensive theoretical study of a binary classification setup, where we consider a classifier that is tightly connected to the optimal Bayes classifier and converges to it as the number of training samples increases. We prove that for any finite number of training samples, there exists a pre-classification processing that improves the classification accuracy. We also explore the effect of class separation, training set size, and class balance on the relative gain from this procedure. We support our theory with an empirical investigation of the theoretical setup. Finally, we conduct an empirical study where we investigate the effect of denoising and encoding on the performance of practical deep classifiers on benchmark datasets. Specifically, we vary the size and class distribution of the training set, and the noise level, and demonstrate trends that are consistent with our theoretical results.

Does the Data Processing Inequality Reflect Practice? On the Utility of Low-Level Tasks

TL;DR

The paper questions the practical reach of the data processing inequality by showing that, in a high-dimensional binary classification setting, there exists a linear pre-processing step that improves accuracy for any finite training size, even when the classifier approaches Bayes optimality with more data. It provides a theoretical framework using a Gaussian Mixture Model with separation parameter and derives approximations for the error probabilities before and after processing, including balanced and imbalanced training scenarios. The authors also validate the theory with empirical simulations and extend the insight to practical deep-learning pipelines, demonstrating that denoising and encoding of inputs can yield meaningful gains in finite-data regimes on benchmarks like CIFAR-10 and Mini-ImageNet. The findings highlight the potential benefit of carefully designed low-level tasks and unlabeled-data–driven learning for improving high-level classification, with implications for out-of-distribution robustness and future work on non-linear preprocessing and broader downstream tasks.

Abstract

The data processing inequality is an information-theoretic principle stating that the information content of a signal cannot be increased by processing the observations. In particular, it suggests that there is no benefit in enhancing the signal or encoding it before addressing a classification problem. This assertion can be proven to be true for the case of the optimal Bayes classifier. However, in practice, it is common to perform "low-level" tasks before "high-level" downstream tasks despite the overwhelming capabilities of modern deep neural networks. In this paper, we aim to understand when and why low-level processing can be beneficial for classification. We present a comprehensive theoretical study of a binary classification setup, where we consider a classifier that is tightly connected to the optimal Bayes classifier and converges to it as the number of training samples increases. We prove that for any finite number of training samples, there exists a pre-classification processing that improves the classification accuracy. We also explore the effect of class separation, training set size, and class balance on the relative gain from this procedure. We support our theory with an empirical investigation of the theoretical setup. Finally, we conduct an empirical study where we investigate the effect of denoising and encoding on the performance of practical deep classifiers on benchmark datasets. Specifically, we vary the size and class distribution of the training set, and the noise level, and demonstrate trends that are consistent with our theoretical results.
Paper Structure (37 sections, 11 theorems, 258 equations, 15 figures, 4 tables)

This paper contains 37 sections, 11 theorems, 258 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background and related work
  3. Theory
  4. Problem setup: data model, classifier, and data processing
  5. Theoretical results
  6. Performance gain due to data processing
  7. Factors that affect the performance gain
  8. Empirical verification
  9. Experiments in Practical Settings
  10. Noisy CIFAR-10 and pre-classification denoising
  11. Noisy Mini-ImageNet and pre-classification encoding
  12. Noisy CIFAR-10 and pre-classification encoding
  13. Conclusion
  14. Proofs
  15. Existing results
  16. ...and 22 more sections

Key Result

Theorem 1

Let $y \to x \to z$ be a Markov chain where $y \in \{1,2\}$ denotes the sample class. We have where $c_{opt}$ and $\tilde{c}_{opt}$ denote optimal Bayes classifiers.

Figures (15)

  • Figure 1: The theoretical setup. Efficiency of the data processing procedure versus the number of training samples $N_{\text{train}}$, for various values of the training imbalance factor, $\gamma$, and the SNR, $\mathcal{S}$.
  • Figure 2: Noisy CIFAR-10 and pre-classification denoising. Efficiency versus $N_{\text{train}}$.
  • Figure 3: Noisy Mini-ImageNet and pre-classification encoding. Efficiency versus $N_{\text{train}}$.
  • Figure 4: Training and testing error as a function of epochs for (a) noisy data and (b) denoised data. The noise level is $\sigma = 0.25$, $\gamma=1$, and $N_{\text{train}} = 35{,}000$. The denoiser is trained using MSE loss.
  • Figure 5: Practical deep learning setup with noisy CIFAR-10 and SURE-based denoiser. Efficiency of the data processing procedure versus the number of training samples for various values of the training imbalance factor, $\gamma$, and the standard deviation of the noise, $\sigma$.
  • ...and 10 more figures

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2: The probability of error before the processing
  • Theorem 3: The existence and learnability of the processing
  • Theorem 4: The probability of error on the processed data
  • Theorem 5: Performance gain under balanced training data
  • Theorem 6: Performance gain under imbalanced training data
  • Definition 1
  • Theorem 7: Analysis of the asymptotic efficiency
  • Theorem 8: Analysis of the maximal efficiency
  • proof
  • ...and 11 more