Boltzmann Classifier: A Thermodynamic-Inspired Approach to Supervised Learning

The Boltzmann Classifier introduces a thermodynamically inspired supervised learning approach that uses an energy-based model derived from the Boltzmann distribution to estimate class probabilities, achieving competitive accuracy on benchmark datasets while offering interpretability and computational efficiency.

Supervised Learning, Classification, Energy-Based Model, Interpretability, Efficiency

Muhamed Amin, Bernard R. Brooks

National Heart, Lung and Blood Institute, National Institutes of Health, Bethesda, Maryland, USA, University College Groningen, University of Groningen, Groningen, Netherlands

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Background Problem

Classification is a cornerstone of machine learning with applications in diverse fields like biomedicine and industry, yet traditional methods such as logistic regression and support vector machines often lack physical interpretability. Inspired by statistical physics, the Boltzmann Classifier addresses this gap by introducing a novel energy-based approach to supervised classification, using the Boltzmann distribution to estimate class probabilities based on feature deviations from class centroids. The key problem it aims to solve is providing a physically motivated, interpretable, and computationally efficient alternative to existing classifiers, while maintaining competitive performance and offering insights into decision uncertainty through probabilistic outputs.

Method

The Boltzmann Classifier computes class probabilities for an input vector by defining an energy function based on the L1-norm deviation between the input and class-specific mean feature vectors (centroids) derived from training data, formulated as $E_c(\mathbf{x}) = \sum_{i=1}^{n} |\mathbf{x}_i - \mu_{ic}|$. These energies are transformed into probabilities using the Boltzmann distribution: $P(c|x) = \frac{e^{-E_c(\mathbf{x})/kT}}{\sum_f e^{-E_f(\mathbf{x})/kT}}$, where $kT$ is a tunable scaling factor controlling the softness of the distribution. The class with the highest probability is selected as the predicted label. Features are preprocessed with MinMax scaling to ensure uniform contribution to the energy calculation. Implemented in Python within the scikit-learn framework, the method avoids iterative optimization or backpropagation, prioritizing simplicity and efficiency. The energy function can be extended to other distance metrics beyond L1-norm, offering potential flexibility.

Experiment

The Boltzmann Classifier was evaluated on two datasets: the Breast Cancer Wisconsin dataset (569 samples, 30 features, binary classification of malignant/benign) and the Cobalt Oxidation States dataset from the Cambridge Crystallographic Data Center (9396 structures, 6 features based on Co-Ligand bond distances). Experimental setup involved MinMax scaling of features and comparison with logistic regression and support vector machines (SVM). On the Breast Cancer dataset, it achieved 95% accuracy, slightly below logistic regression and SVM (both 98%), with stable performance across cross-validation folds; misclassified samples showed close probability differences (average 0.21 vs. 0.83 for correct predictions), indicating useful uncertainty insights. On the Cobalt dataset, it scored 87% accuracy, marginally outperforming logistic regression and SVM (both 86%). The setup is reasonable for initial validation but limited, as it lacks testing on complex or multimodal datasets and comparisons with modern deep learning models. The effect of the $kT$ parameter on probability distribution was explored, showing expected softening of decisions at higher values, though its practical impact on robustness remains unclear. Overall, results match the expectation of competitive performance for simple datasets but highlight untested generalizability.

Further Thoughts

The Boltzmann Classifier’s reliance on thermodynamic principles opens an interesting dialogue between physics and machine learning, particularly in the context of energy-based models (EBMs). Its simplicity and lack of iterative optimization are reminiscent of early machine learning approaches, but this raises questions about scalability to high-dimensional, non-linear data—could integrating it with feature learning mechanisms (e.g., autoencoders) enhance its applicability? Additionally, the $kT$ parameter’s role in modulating decision softness could be explored in safety-critical applications, such as medical diagnostics, where controlling false positives is paramount; this ties into broader discussions on AI safety and alignment, where tunable uncertainty might mitigate overconfidence. Comparing this approach to other EBMs, like restricted Boltzmann machines, could reveal whether the thermodynamic analogy offers unique advantages or merely repackages existing concepts. Finally, testing on datasets with known multimodal distributions (e.g., in image classification) could validate or challenge the centroid-based assumption, potentially guiding hybrid models that combine this method with clustering techniques for more robust class representations.