Consider supervised learning from i.i.d. samples {(y_i, x_i )}_{i≤n} where x_i ∈ R_p are feature vectors and y_i ∈ R are labels. We study empirical risk minimization over a class of functions that are parameterized by k = O(1) vectors θ_1 , . . . , θ_k ∈ R_p, and prove universality results both for the training and test error. Namely, under the proportional asymptotics n, p → ∞ , with n/p = Θ(1), we prove that the training error depends on the random features distribution only through its covariance structure. Further, we prove that the minimum test error over near-empirical risk minimizers enjoys similar universality properties. In particular, the asymptotics of these quantities can be computed —to leading order— under a simpler model in which the feature vectors x_i are replaced by Gaussian vectors g_i with the same covariance. Earlier universality results were limited to strongly convex learning procedures, or to feature vectors x_i with independent entries. Our results do not make any of these assumptions. Our assumptions are general enough to include feature vectors x_i that are produced by randomized featurization maps. In particular we explicitly check the assumptions for certain random features models (computing the output of a one-layer neural network with random weights) and neural tangent models (first-order Taylor approximation of two-layer networks).

We consider the task of learning a causal graph in the presence of latent confounders given i.i.d.samples from the model. While current algorithms for causal structure discovery in the presence of latent confounders are constraint-based, we here propose a hybrid approach. We prove that under assumptions weaker than faithfulness, any sparsest independence map (IMAP) of the distribution belongs to the Markov equivalence class of the true model. This motivates the Sparsest Poset formulation - that posets can be mapped to minimal IMAPs of the true model such that the sparsest of these IMAPs is Markov equivalent to the true model. Motivated by this result, we propose a greedy algorithm over the space of posets for causal structure discovery in the presence of latent confounders and compare its performance to the current state-of-the-art algorithms FCI and FCI+ on synthetic data.

We consider distributions arising from a mixture of causal models, where each model is represented by a directed acyclic graph (DAG). We provide a graphical representation of such mixture distributions and prove that this representation encodes the conditional independence relations of the mixture distribution. We then consider the problem of structure learning based on samples from such distributions. Since the mixing variable is latent, we consider causal structure discovery algorithms such as FCI that can deal with latent variables. We show that such algorithms recover a “union” of the component DAGs and can identify variables whose conditional distribution across the component DAGs vary. We demonstrate our results on synthetic and real data showing that the inferred graph identifies nodes that vary between the different mixture components. As an immediate application, we demonstrate how retrieval of this causal information can be used to cluster samples according to each mixture component.

We consider the problem of learning a causal graph in the presence of measurement error.This setting is for example common in genomics, where gene expression is corrupted through the measurement process. We develop a provably consistent procedure for estimating the causal structure in a linear Gaussian structural equation model from corrupted observations on its nodes, under a variety of measurement error models. We provide an estimator based on the method-of-moments, which can be used in conjunction with constraint-based causal structure discovery algorithms. We prove asymptotic consistency of the procedure and also discuss finite-sample considerations. We demonstrate our method’s performance through simulations and on real data, where we recover the underlying gene regulatory network from zero-inflated single-cell RNA-seq data.

Network Coding is a relatively new forwarding paradigm where intermediate nodes perform a store, code, and forward operation on incoming packets. Traditional forwarding approaches, which employed a store and forward operation, suffered from the limitations of the max-flow min-cut theorem wherein sources transmitting information over bottleneck links had to compete for access to these links. With Network Coding, multiple sources are now able to transmit packets over bottleneck links simultaneously, increasing network capacity. While the majority of the contemporary literature has focused on the performance of Network Coding from a capacity perspective, the aim of this research has taken a new direction focusing on two Quality of Service metrics, Packet Delivery Ratio (PDR) and latency, in conjunction with Network Coding protocols in Mobile Ad-Hoc Networks (MANETs). Initial simulations will be performed on static environments to determine a Quality of Service baseline comparison between Network Coding protocols and traditional ad-hoc routing protocols. Additional simulations will then be performed for mobile scenarios to determine how the Network Coding protocols will compare to that of the standard ad-hoc routing protocols in the presence of mobility.

Physical-Layer Network Coding (PLNC) has recently emerged as a promising new communications paradigm that has the ability to greatly increase the capacity of wireless networks. Current research has also demonstrated that PLNC can decrease the eavesdropping region of an external entity. In this paper we show how it is possible to mitigate passive eavesdropping in the presence of co-located nodes in a PLNC environment.