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Abstract In recommender systems, users rate items, and are subsequently served other product recommendations based on these ratings. Even though users usually rate a tiny percentage of the available items, the system tries to estimate unobserved preferences by finding similarities across users and across items. In this work, we treat the observed ratings data as partially observed, dense, weighted, bipartite networks. For a class of systems without outside information, we adapt an approach developed for dense, weighted networks to account for unobserved edges and the bipartite nature of the problem. The approach begins with clustering both users and items into communities, and locally estimates the patterns of ratings within each subnetwork induced by restricting attention to one community of users and one community of items community. The local fitting procedure relies on estimating local sociability parameters for every user and item, and selecting the function that determines the degree correction contours which best models the underlying data. We compare the performance of our proposed approach to existing methods on a simulated data set, as well as on a data set of joke ratings, examining model performance in both cases at differing levels of sparsity. On the joke ratings data set, our proposed model performs better than existing alternatives in relatively sparse settings, though other approaches achieve better results when more data is available. Collectively, the results indicate that despite struggling to pick up subtler signals, the proposed approach’s recovery of large scale, coarse patterns may still be useful in practical settings where high sparsity is typical. 

We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical window, components merge approximately like the multiplicative coalescent, (ii) asymptotics of the susceptibility functions are the same as that of the Erdős-Rényi random graph, (iii) asymptotic negligibility of the maximal component size and the diameter in the barely subcritical regime, and (iv) macroscopic averaging of distances between vertices in the barely subcritical regime. As an application of the general universality theorem, we establish, under some regularity conditions, the critical percolation scaling limit of graphs that converge, in a suitable topology, to an $L^3$graphon. In particular, we define a notion of the critical window in this setting. The $L^3$assumption ensures that the model is in the Erdős-Rényi universality class and that the scaling limit is Brownian. Our results do not assume any specific functional form for the graphon. As a consequence of our results on graphons, we obtain the metric scaling limit for Aldous-Pittel’s RGIV model inside the critical window (see D.J. Aldous and B. Pittel [Random Structures Algorithms 17 (2000), pp. 79–102]). Our universality principle has applications in a number of other problems including in the study of noise sensitivity of critical random graphs (see E. Lubetzky and Y. Peled [Israel J. Math. 252 (2022), pp. 187–214]). In Bhamidi et al. [Scaling limits and universality II: geometry of maximal components in dynamic random graph models in the critical regime, In preparation], we use our universality theorem to establish the metric scaling limit of critical bounded size rules. Our method should yield the critical metric scaling limit of Ruciński and Wormald’s random graph process with degree restrictions provided an additional technical condition about the barely subcritical behavior of this model can be proved (see A. Ruciński and N. C. Wormald [Combin. Probab. Comput. 1 (1992), pp. 169–180]). 

In this paper, we propose a new framework to detect adversarial examples motivated by the observations that random components can improve the smoothness of predictors and make it easier to simulate the output distribution of a deep neural network. With these observations, we propose a novel Bayesian adversarial example detector, short for BATER, to improve the performance of adversarial example detection. Specifically, we study the distributional difference of hidden layer output between natural and adversarial examples, and propose to use the randomness of the Bayesian neural network to simulate hidden layer output distribution and leverage the distribution dispersion to detect adversarial examples. The advantage of a Bayesian neural network is that the output is stochastic while a deep neural network without random components does not have such characteristics. Empirical results on several benchmark datasets against popular attacks show that the proposed BATER outperforms the state-of-the-art detectors in adversarial example detection.