The bottleneck introduced by the low-dimensional latent space is what characterizes the compression and representation learning capabilities ofautoencoders.

Differentially Private-SGD (DP-SGD) of [ACG+16] and its variations are the only known algorithms for private training of large scale neural networks.

Despite their promising performance, the learned knowledge remains implicit in these black-box neural structures, which hinders understanding the importance of input features and how they influencedecisions.

When one models the considered sets of samples as empirical probability distributions, Optimal Transport (OT)frameworkprovides asolution tofind,without supervision, asoft-correspondence mapbetweenthemgivenbyan optimalcoupling.

In this paper, we propose polynomial forms to represent distributions of state variables over time for discrete-time stochastic dynamical systems. This problem arises in a variety of applications in areas ranging from biology to robotics. Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving the state variables. First, we recall ideas from interval arithmetic, and use them to rigorously represent the state variables at time t as a function of the initial state variables and noise symbols that model the random exogenous inputs encountered before time t. Next, we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables. We demonstrate interesting applications that demonstrate how our approach can be useful in some situations to establish mathematically guaranteed bounds that are of a different nature from those obtained through simulations with pseudo-random numbers.