Alexandra Morgan

Phuc Nguyen, Josiah Couch, Rahul Bansal et al.

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.

Miao Li, Phuc Nguyen, Christopher Tam et al.

Tabular data dominates data science but poses challenges for generative models, especially when the data is limited or sensitive. We present a novel approach to generating synthetic tabular data based on the principle of maximum entropy -- MaxEnt -- called GEM-T, for ``generative entropy maximization for tables.'' GEM-T directly captures nth-order interactions -- pairwise, third-order, etc. -- among columns of training data. In extensive testing, GEM-T matches or exceeds deep neural network approaches previously regarded as state-of-the-art in 23 of 34 publicly available datasets representing diverse subject domains (68\%). Notably, GEM-T involves orders-of-magnitude fewer trainable parameters, demonstrating that much of the information in real-world data resides in low-dimensional, potentially human-interpretable correlations, provided that the input data is appropriately transformed first. Furthermore, MaxEnt better handles heterogeneous data types (continuous vs. discrete vs. categorical), lack of local structure, and other features of tabular data. GEM-T represents a promising direction for light-weight high-performance generative models for structured data.

Phuc Nguyen, Miao Li, Alexandra Morgan et al.

Generative models hold great potential, but only if one can trust the evaluation of the data they generate. We show that many commonly used quality scores for comparing two-dimensional distributions of synthetic vs. ground-truth data give better results than they should, a phenomenon we call the "grade inflation problem." We show that the correlation score, Jaccard score, earth-mover's score, and Kullback-Leibler (relative-entropy) score all suffer grade inflation. We propose that any score that values all datapoints equally, as these do, will also exhibit grade inflation; we refer to such scores as "equipoint" scores. We introduce the concept of "equidensity" scores, and present the Eden score, to our knowledge the first example of such a score. We found that Eden avoids grade inflation and agrees better with human perception of goodness-of-fit than the equipoint scores above. We propose that any reasonable equidensity score will avoid grade inflation. We identify a connection between equidensity scores and Rényi entropy of negative order. We conclude that equidensity scores are likely to outperform equipoint scores for generative models, and for comparing low-dimensional distributions more generally.