Which Similarity-Sensitive Entropy?

A canonical step in quantifying a system is to measure its entropy. Shannon entropy and other traditional entropy measures capture only the information encoded in the frequencies of a system's elements. Recently, Leinster, Cobbold, and Reeve (LCR) introduced a method that also captures the rich information encoded in the similarities and differences among elements, yielding similarity-sensitive entropy. More recently, the Vendi score (VS) was introduced as an alternative, raising the question of how LCR and VS compare, and which is preferable. Here we address these questions conceptually, analytically, and experimentally, using 53 machine-learning datasets. We show that LCR and VS can differ by orders of magnitude and can capture complementary information about a system, except in limiting cases. We demonstrate that both LCR and VS depend on how similarities are scaled and introduce the concept of ``half distance'' to parameterize this dependence. We prove that VS provides an upper bound on LCR for several values of the Rényi-Hill order parameter and conjecture that this bound holds for all values. We conclude that VS is preferable only when interpreting elements as linear combinations of a more fundamental set of ``ur-elements'' or when the system or dataset possesses a quantum-mechanical character. In the broader circumstance where one seeks simply to capture the rich information encoded by similarity, LCR is favored; nevertheless, for certain half-distances the two methods can complement each other.

X-Factor: Quality Is a Dataset-Intrinsic Property

In the universal quest to optimize machine-learning classifiers, three factors -- model architecture, dataset size, and class balance -- have been shown to influence test-time performance but do not fully account for it. Previously, evidence was presented for an additional factor that can be referred to as dataset quality, but it was unclear whether this was actually a joint property of the dataset and the model architecture, or an intrinsic property of the dataset itself. If quality is truly dataset-intrinsic and independent of model architecture, dataset size, and class balance, then the same datasets should perform better (or worse) regardless of these other factors. To test this hypothesis, here we create thousands of datasets, each controlled for size and class balance, and use them to train classifiers with a wide range of architectures, from random forests and support-vector machines to deep networks. We find that classifier performance correlates strongly by subset across architectures ($R^2=0.79$), supporting quality as an intrinsic property of datasets independent of dataset size and class balance and of model architecture. Digging deeper, we find that dataset quality appears to be an emergent property of something more fundamental: the quality of datasets' constituent classes. Thus, quality joins size, class balance, and model architecture as an independent correlate of performance and a separate target for optimizing machine-learning-based classification.

Beyond Size and Class Balance: Alpha as a New Dataset Quality Metric for Deep Learning

In deep learning, achieving high performance on image classification tasks requires diverse training sets. However, dataset diversity is incompletely understood. The current best practice is to try to maximize dataset size and class balance. Yet large, class-balanced datasets are not guaranteed to be diverse: images can still be arbitrarily similar. We hypothesized that, for a given model architecture, better model performance can be achieved by maximizing dataset diversity more directly. This could open a path for performance improvement without additional computational resources or architectural advances. To test this hypothesis, we introduce a comprehensive framework of diversity measures, developed in ecology, that generalizes familiar quantities like Shannon entropy by accounting for similarities and differences among images. (Dataset size and class balance emerge from this framework as special cases.) By analyzing thousands of subsets from seven medical datasets representing ultrasound, X-ray, CT, and pathology images, we found that the best correlates of performance were not size or class balance but $A$ -- ``big alpha'' -- a set of generalized entropy measures interpreted as the effective number of image-class pairs in the dataset, after accounting for similarities among images. One of these, $A_0$, explained 67\% of the variance in balanced accuracy across all subsets, vs. 54\% for class balance and just 39\% for size. The best pair was size and $A_1$ (79\%), which outperformed size and class balance (74\%). $A$ performed best for subsets from individual datasets as well as across datasets, supporting the generality of these results. We propose maximizing $A$ as a potential new way to improve the performance of deep learning in medical imaging.