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The approach to analysing compositional data has been dominated by the use of logratio transformations, to ensure exact subcompositional coherence and, in some situations, exact isometry as well. A problem with this approach is that data zeros, found in most applications, have to be replaced to allow the logarithmic transformation. An alternative new approach, called the `chiPower' transformation, which allows data zeros, is to combine the standardization inherent in the chi-square distance in correspondence analysis, with the essential elements of the Box-Cox power transformation. The chiPower transformation is justified because it} defines between-sample distances that tend to logratio distances for strictly positive data as the power parameter tends to zero, and are then equivalent to transforming to logratios. For data with zeros, a value of the power can be identified that brings the chiPower transformation as close as possible to a logratio transformation, without having to substitute the zeros. Especially in the area of high-dimensional data, this alternative approach can present such a high level of coherence and isometry as to be a valid approach to the analysis of compositional data. Furthermore, in a supervised learning context, if the compositional variables serve as predictors of a response in a modelling framework, for example generalized linear models, then the power can be used as a tuning parameter in optimizing the accuracy of prediction through cross-validation. The chiPower-transformed variables have a straightforward interpretation, since they are each identified with single compositional parts, not ratios.

Bayesian linear mixed-effects models and Bayesian ANOVA are increasingly being used in the cognitive sciences to perform null hypothesis tests, where a null hypothesis that an effect is zero is compared with an alternative hypothesis that the effect exists and is different from zero. While software tools for Bayes factor null hypothesis tests are easily accessible, how to specify the data and the model correctly is often not clear. In Bayesian approaches, many authors use data aggregation at the by-subject level and estimate Bayes factors on aggregated data. Here, we use simulation-based calibration for model inference applied to several example experimental designs to demonstrate that, as with frequentist analysis, such null hypothesis tests on aggregated data can be problematic in Bayesian analysis. Specifically, when random slope variances differ (i.e., violated sphericity assumption), Bayes factors are too conservative for contrasts where the variance is small and they are too liberal for contrasts where the variance is large. Running Bayesian ANOVA on aggregated data can - if the sphericity assumption is violated - likewise lead to biased Bayes factor results. Moreover, Bayes factors for by-subject aggregated data are biased (too liberal) when random item slope variance is present but ignored in the analysis. These problems can be circumvented or reduced by running Bayesian linear mixed-effects models on non-aggregated data such as on individual trials, and by explicitly modeling the full random effects structure. Reproducible code is available from \url{//osf.io/mjf47/}.

Hopfield networks are an attractive choice for solving many types of computational problems because they provide a biologically plausible mechanism. The Self-Optimization (SO) model adds to the Hopfield network by using a biologically founded Hebbian learning rule, in combination with repeated network resets to arbitrary initial states, for optimizing its own behavior towards some desirable goal state encoded in the network. In order to better understand that process, we demonstrate first that the SO model can solve concrete combinatorial problems in SAT form, using two examples of the Liars problem and the map coloring problem. In addition, we show how under some conditions critical information might get lost forever with the learned network producing seemingly optimal solutions that are in fact inappropriate for the problem it was tasked to solve. What appears to be an undesirable side-effect of the SO model, can provide insight into its process for solving intractable problems.

The work of Kalman and Bucy has established a duality between filtering and optimal estimation in the context of time-continuous linear systems. This duality has recently been extended to time-continuous nonlinear systems in terms of an optimization problem constrained by a backward stochastic partial differential equation. Here we revisit this problem from the perspective of appropriate forward-backward stochastic differential equations. This approach sheds new light on the estimation problem and provides a unifying perspective. It is also demonstrated that certain formulations of the estimation problem lead to deterministic formulations similar to the linear Gaussian case as originally investigated by Kalman and Bucy. Finally, optimal control of partially observed diffusion processes is discussed as an application of the proposed estimators.

Convex PCA, which was introduced by Bigot et al., is a dimension reduction methodology for data with values in a convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this paper is threefold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein geodesic PCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory.

It is a challenge to numerically solve nonlinear partial differential equations whose solution involves discontinuity. In the context of numerical simulators for multi-phase flow in porous media, there exists a long-standing issue known as Grid Orientation Effect (GOE), wherein different numerical solutions can be obtained when considering grids with different orientations under certain unfavorable conditions. Our perspective is that GOE arises due to numerical instability near displacement fronts, where spurious oscillations accompanied by sharp fronts, if not adequately suppressed, lead to GOE. To reduce or even eliminate GOE, we propose augmenting adaptive artificial viscosity when solving the saturation equation. It has been demonstrated that appropriate artificial viscosity can effectively reduce or even eliminate GOE. The proposed numerical method can be easily applied in practical engineering problems.

Nested binomial sums form a particular class of sums that arise in the context of particle physics computations at higher orders in perturbation theory within QCD and QED, but that are also mathematically relevant, e.g., in combinatorics. We present the package RICA (Rule Induced Convolutions for Asymptotics), which aims at calculating Mellin representations and asymptotic expansions at infinity of those objects. These representations are of particular interest to perform analytic continuations of such sums.

Given $n$ independent and identically distributed observations and measuring the value of obtaining an additional observation in terms of Le Cam's notion of deficiency between experiments, we show for certain types of non-parametric experiments that the value of an additional observation decreases at a rate of $1/\sqrt{n}$. This is distinct from the known typical decrease at a rate of $1/n$ for parametric experiments and the non-decreasing value in the case of very large experiments. In particular, the rate of $1/\sqrt{n}$ holds for the experiment given by observing samples from a density about which we know only that it is bounded from below by some fixed constant. Thus there exists an experiment where the value of additional observations tends to zero but for which no estimator that is consistent (in total variation distance) exists.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.