Forming a molecular candidate set that contains a wide range of potentially effective compounds is crucial to the success of drug discovery. While most databases and machine-learning-based generation models aim to optimize particular chemical properties, there is limited literature on how to properly measure the coverage of the chemical space by those candidates included or generated. This problem is challenging due to the lack of formal criteria to select good measures of the chemical space. In this paper, we propose a novel evaluation framework for measures of the chemical space based on two analyses: an axiomatic analysis with three intuitive axioms that a good measure should obey, and an empirical analysis on the correlation between a measure and a proxy gold standard. Using this framework, we are able to identify #Circles, a new measure of chemical space coverage, which is superior to existing measures both analytically and empirically. We further evaluate how well the existing databases and generation models cover the chemical space in terms of #Circles. The results suggest that many generation models fail to explore a larger space over existing databases, which leads to new opportunities for improving generation models by encouraging exploration.

GANs for time series data often use sliding windows or self-attention to capture underlying time dependencies. While these techniques have no clear theoretical justification, they are successful in significantly reducing the discriminator size, speeding up the training process, and improving the generation quality. In this paper, we provide both theoretical foundations and a practical framework of GANs for high-dimensional distributions with conditional independence structure captured by a Bayesian network, such as time series data. We prove that several probability divergences satisfy subadditivity properties with respect to the neighborhoods of the Bayes-net graph, providing an upper bound on the distance between two Bayes-nets by the sum of (local) distances between their marginals on every neighborhood of the graph. This leads to our proposed Subadditive GAN framework that uses a set of simple discriminators on the neighborhoods of the Bayes-net, rather than a giant discriminator on the entire network, providing significant statistical and computational benefits. We show that several probability distances including Jensen-Shannon, Total Variation, and Wasserstein, have subadditivity or generalized subadditivity. Moreover, we prove that Integral Probability Metrics (IPMs), which encompass commonly-used loss functions in GANs, also enjoy a notion of subadditivity under some mild conditions. Furthermore, we prove that nearly all f-divergences satisfy local subadditivity in which subadditivity holds when the distributions are relatively close. Our experiments on synthetic as well as real-world datasets verify the proposed theory and the benefits of subadditive GANs.

This paper shows that the fuzzy temporal logic can model figures of thought to describe decision-making behaviors. In order to exemplify, some economic behaviors observed experimentally were modeled from problems of choice containing time, uncertainty and fuzziness. Related to time preference, it is noted that the subadditive discounting is mandatory in positive rewards situations and, consequently, results in the magnitude effect and time effect, where the last has a stronger discounting for earlier delay periods (as in, one hour, one day), but a weaker discounting for longer delay periods (for instance, six months, one year, ten years). In addition, it is possible to explain the preference reversal (change of preference when two rewards proposed on different dates are shifted in the time). Related to the Prospect Theory, it is shown that the risk seeking and the risk aversion are magnitude dependents, where the risk seeking may disappear when the values to be lost are very high.

In this article we describe a cognitive heuristic known as the unpacking effect by using a mathematical model, based on the quantum formalism, already introduced for the conjunction fallacy. We present the basic postulates of such quantum-like model and we show that the presence of interference terms is responsible of the unpacking effect. In particular, the sign of the interference and its functional form are able to describe the experimental results about subadditivity, superadditivity and additivity. A comparison with previous models is presented, as well as new experimental predictions, allowing to conclude that this new formalism and the basic concepts of quantum information processing provide a new promising way to describe and understand human judgement and categorization.