Analytical Softmax Temperature Setting from Feature Dimensions for Model- and Domain-Robust Classification

In deep learning-based classification tasks, the softmax function's temperature parameter $T$ critically influences the output distribution and overall performance. This study presents a novel theoretical insight that the optimal temperature $T^*$ is uniquely determined by the dimensionality of the feature representations, thereby enabling training-free determination of $T^*$. Despite this theoretical grounding, empirical evidence reveals that $T^*$ fluctuates under practical conditions owing to variations in models, datasets, and other confounding factors. To address these influences, we propose and optimize a set of temperature determination coefficients that specify how $T^*$ should be adjusted based on the theoretical relationship to feature dimensionality. Additionally, we insert a batch normalization layer immediately before the output layer, effectively stabilizing the feature space. Building on these coefficients and a suite of large-scale experiments, we develop an empirical formula to estimate $T^*$ without additional training while also introducing a corrective scheme to refine $T^*$ based on the number of classes and task complexity. Our findings confirm that the derived temperature not only aligns with the proposed theoretical perspective but also generalizes effectively across diverse tasks, consistently enhancing classification performance and offering a practical, training-free solution for determining $T^*$.