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We consider the supermarket model in the usual Markovian setting where jobs arrive at rate nλn for some λn > 0, with n parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among dn ≤ n randomly selected service queues. We show that when dn → ∞ and λn → λ ∈ (0, ∞), under natural conditions on the initial queues, the state occupancy process converges in probability, in a suitable path space, to the unique solution of an infinite system of constrained ordinary differential equations parametrized by λ. Our main interest is in the study of fluctuations of the state process about its near equilibrium state in the critical regime, namely when λn → 1. Previous papers e.g. [25] have considered the regime dn √ n log n → ∞ while the objective of the current work is to develop diffusion approximations for the state occupancy process that allow for all possible rates of growth of dn. In particular we consider the three canonical regimes (a)In all three regimes we show, by establishing suitable functional limit theorems, that (under conditions on λn) fluctuations of the state process about its near equilibrium are of order n −1/2 and are governed asymptotically by a one dimensional Brownian motion. The forms of the limit processes in the three regimes are quite different; in the first case we get a linear diffusion; in the second case we get a diffusion with an exponential drift; and in the third case we obtain a reflected diffusion in a half space. In the special case dn/( √ n log n) → ∞ our work gives alternative proofs for the universality results established in [25].

Multi-view data, in which data of different types are obtained from a common set of samples, is now common in many applied scientific problems. An important problem in the analysis of multi-view data is identifying interactions between groups of features from different data types. A bimodule is a pair (A, B) of feature sets from two different data types such that the aggregate cross-correlation between the features in A and those in B is large. A bimodule (A, B) is stable if A coincides with the set of features having significant aggregate correlation with the features in B, and vice-versa. At the population level, stable bimodules correspond to connected components of the cross-correlation network, which is the bipartite graph whose edges are pairs of features with non-zero cross-correlations.We develop and investigate an iterative, testing-based procedure, called BSP, to identify stable bimodules in two moderate-to highdimensional data sets. BSP relies on permutation-based p-values for test statistics equal to sums of squared cross-correlations. These pvalues are approximated using tail probabilities of gamma distributions that are fit using estimates of the permutation moments of the test statistic. Our moment estimates depend on the eigenvalues of the intra-correlation matrices of A and B, and as a result the significance of observed cross-correlations accounts for the correlations within each data type.We carry out a thorough simulation study to assess the performance of BSP, and present an extended application of BSP to the problem of expression quantitative trait loci (eQTL) analysis using recent data from the GTEx project. In addition, we apply BSP to climatology data in order to identify regions in North America where annual temperature variation affects precipitation.The method is available as an R package at https://github.com/ miheerdew/cbce.

We propose a new technique for performing state space exploration of closed loop control systems with neural network feedback controllers. Our approach involves approximating the sensitivity of the trajectories of the closed loop dynamics. Using such an approximator and the system simulator, we present a guided state space exploration method that can generate trajectories visiting the neighborhood of a target state at a specified time. We present a theoretical framework which establishes that our method will produce a sequence of trajectories that will reach a suitable neighborhood of the target state. We provide thorough evaluation of our approach on various systems with neural network feedback controllers of different configurations. We outperform earlier state space exploration techniques and achieve significant improvement in both the quality (explainability) and performance (convergence rate). Finally, we adopt our algorithm for the falsification of a class of temporal logic specification, assess its performance against a state-of-the-art falsification tool, and show its potential in supplementing existing falsification algorithms.

Likelihood-based inferences have been remarkably successful in wide-spanning application areas. However, even after due diligence in selecting a good model for the data at hand, there is inevitably some amount of model misspecification: outliers, data contamination or inappropriate parametric assumptions such as Gaussianity mean that most models are at best rough approximations of reality. A significant practical concern is that for certain inferences, even small amounts of model misspecification may have a substantial impact; a problem we refer to as brittleness. This article attempts to address the brittleness problem in likelihood-based inferences by choosing the most model friendly data generating process in a discrepancybased neighbourhood of the empirical measure. This leads to a new Optimistically Weighted Likelihood (OWL), which robustifies the original likelihood by formally accounting for a small amount of model misspecification. Focusing on total variation (TV) neighborhoods, we study theoretical properties, develop inference algorithms and illustrate the methodology in applications to mixture models and regression.