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Neural networks excel at discovering statistical patterns in high-dimensional data sets. In practice, higher-order cumulants, which quantify the non-Gaussian correlations between three or more variables, are particularly important for the performance of neural networks. But how efficient are neural networks at extracting features from higher-order cumulants? We study this question in the spiked cumulant model, where the statistician needs to recover a privileged direction or "spike" from the order-$p\ge 4$ cumulants of $d$-dimensional inputs. Existing literature established the presence of a wide statistical-to-computational gap in this problem. We deepen this line of work by finding an exact formula for the likelihood ratio norm which proves that statistical distinguishability requires $n\gtrsim d$ samples, while distinguishing the two distributions in polynomial time requires $n \gtrsim d^2$ samples for a wide class of algorithms, i.e. those covered by the low-degree conjecture. Numerical experiments show that neural networks do indeed learn to distinguish the two distributions with quadratic sample complexity, while "lazy" methods like random features are not better than random guessing in this regime. Our results show that neural networks extract information from higher-ordercorrelations in the spiked cumulant model efficiently, and reveal a large gap in the amount of data required by neural networks and random features to learn from higher-order cumulants.

The accurate representation of multidimensional potential energy surfaces is a necessary requirement for realistic computer simulations of molecular systems. The continued increase in computer power accompanied by advances in correlated electronic structure methods nowadays enable routine calculations of accurate interaction energies for small systems, which can then be used as references for the development of analytical potential energy functions (PEFs) rigorously derived from many-body expansions. Building on the accuracy of the MB-pol many-body PEF, we investigate here the performance of permutationally invariant polynomials, neural networks, and Gaussian approximation potentials in representing water two-body and three-body interaction energies, denoting the resulting potentials PIP-MB-pol, BPNN-MB-pol, and GAP-MB-pol, respectively. Our analysis shows that all three analytical representations exhibit similar levels of accuracy in reproducing both two-body and three-body reference data as well as interaction energies of small water clusters obtained from calculations carried out at the coupled cluster level of theory, the current gold standard for chemical accuracy. These results demonstrate the synergy between interatomic potentials formulated in terms of a many-body expansion, such as MB-pol, that are physically sound and transferable, and machine-learning techniques that provide a flexible framework to approximate the short-range interaction energy terms.

Currently, two main approaches exist to distinguish differential susceptibility from diathesis-stress and vantage sensitivity in genotype x environment interaction (GxE) research: Regions of significance (RoS) and competitive-confirmatory approaches. Each is limited by their single-gene/single-environment foci given that most phenotypes are the product of multiple interacting genetic and environmental factors. We thus addressed these two concerns in a recently developed R package (LEGIT) for constructing GxE interaction models with latent genetic and environmental scores using alternating optimization. Herein we test, by means of computer simulation, diverse GxE models in the context of both single and multiple genes and environments. Results indicate that the RoS and competitive-confirmatory approaches were highly accurate when the sample size was large, whereas the latter performed better in small samples and for small effect sizes. The confirmatory approach generally had good accuracy (a) when effect size was moderate and N >= 500 and (b) when effect size was large and N >= 250, whereas RoS performed poorly. Computational tools to determine the type of GxE of multiple genes and environments are provided as extensions in our LEGIT R package.

Motivated by the goal of expanding currently existing genotype x environment interaction (GxE) models to simultaneously include multiple genetic variants and environmental exposures in a parsimonious way, we developed a novel method to estimate the parameters in a GxE model, where G is a weighted sum of genetic variants (genetic score) and E is a weighted sum of environments (environmental score). The approach uses alternating optimization to estimate the parameters of the GxE model. This is an iterative process where the genetic score weights, the environmental score weights, and the main model parameters are estimated in turn assuming the other parameters to be constant. This technique can be used to construct relatively complex interaction models that are constrained to a particular structure, and hence contain fewer parameters. We present the model as a two-way interaction longitudinal mixed model, for which ordinary linear regression is a special case, but it can easily be extended to be compatible with k-way interaction models and generalized linear mixed models. The model is implemented in R (LEGIT package) and using SAS macros (LEGIT_SAS). Here we present examples from the Maternal Adversity, Vulnerability, and Neurodevelopment (MAVAN) study where we improve significantly upon already existing models using alternating optimization. Furthermore, through simulations, we demonstrate the power and validity of this approach even with small sample sizes.