Permutation invariance is among the most common symmetry that can be exploited to simplify complex problems in machine learning (ML). There has been a tremendous surge of research activities in building permutation invariant ML architectures. However, less attention is given to: (1) how to statistically test for permutation invariance of coordinates in a random vector where the dimension is allowed to grow with the sample size; (2) how to leverage permutation invariance in estimation problems and how does it help reduce dimensions. In this paper, we take a step back and examine these questions in several fundamental problems: (i) testing the assumption of permutation invariance of multivariate distributions; (ii) estimating permutation invariant densities; (iii) analyzing the metric entropy of permutation invariant function classes and compare them with their counterparts without imposing permutation invariance; (iv) deriving an embedding of permutation invariant reproducing kernel Hilbert spaces for efficient computation. In particular, our methods for (i) and (iv) are based on a sorting trick and (ii) is based on an averaging trick. These tricks substantially simplify the exploitation of permutation invariance.

We consider the problem of nonlinear stochastic optimal control. This problem is thought to be fundamentally intractable owing to Bellman's infamous "curse of dimensionality". We present a result that shows that repeatedly solving an open-loop deterministic problem from the current state, similar to Model Predictive Control (MPC), results in a feedback policy that is $O(\epsilon^4)$ near to the true global stochastic optimal policy. Furthermore, empirical results show that solving the Stochastic Dynamic Programming (DP) problem is highly susceptible to noise, even when tractable, and in practice, the MPC-type feedback law offers superior performance even for stochastic systems.

We make inroads into understanding the robustness of Variational Autoencoders (VAEs) to adversarial attacks and other input perturbations. While previous work has developed algorithmic approaches to attacking and defending VAEs, there remains a lack of formalization for what it means for a VAE to be robust. To address this, we develop a novel criterion for robustness in probabilistic models: $r$-robustness. We then use this to construct the first theoretical results for the robustness of VAEs, deriving margins in the input space for which we can provide guarantees about the resulting reconstruction. Informally, we are able to define a region within which any perturbation will produce a reconstruction that is similar to the original reconstruction. To support our analysis, we show that VAEs trained using disentangling methods not only score well under our robustness metrics, but that the reasons for this can be interpreted through our theoretical results.