Dimension reduction algorithms are essential in data science for tasks such as data exploration, feature selection, and denoising. However, many non-linear dimension reduction algorithms are poorly understood from a theoretical perspective. This work considers a generalized version of multidimensional scaling, which seeks to construct a map from high to low dimension which best preserves pairwise inner products or norms. We investigate the variational properties of this problem, leading to the following insights: 1) Particle-wise descent methods implemented in standard libraries can produce non-deterministic embeddings, 2) A probabilistic formulation leads to solutions with interpretable necessary conditions, and 3) The globally optimal solutions to the relaxed, probabilistic problem is only minimized by deterministic embeddings. This progression of results mirrors the classical development of optimal transportation, and in a case relating to the Gromov-Wasserstein distance actually gives explicit insight into the structure of the optimal embeddings, which are parametrically determined and discontinuous on smooth surfaces. Our results also imply that a standard computational implementation for this problem learns sub-optimal mappings, and we discuss how the embeddings learned in that context have highly misleading clustering structure, underscoring the delicate nature of solving this problem computationally.

Gradient-descent (GD) based algorithms are an indispensable tool for optimizing modern machine learning models. The paper considers distributed stochastic GD (D-SGD)--a network-based variant of GD. Distributed algorithms play an important role in large-scale machine learning problems as well as the Internet of Things (IoT) and related applications. The paper considers two main issues. First, we study convergence of D-SGD to critical points when the loss function is nonconvex and nonsmooth. We consider a broad range of nonsmooth loss functions including those of practical interest in modern deep learning. It is shown that, for each fixed initialization, D-SGD converges to critical points of the loss with probability one. Next, we consider the problem of avoiding saddle points. It is well known that classical GD avoids saddle points; however, analogous results have been absent for distributed variants of GD. For this problem, we again assume that loss functions may be nonconvex and nonsmooth, but are smooth in a neighborhood of a saddle point. It is shown that, for any fixed initialization, D-SGD avoids such saddle points with probability one. Results are proved by studying the underlying (distributed) gradient flow, using the ordinary differential equation (ODE) method of stochastic approximation.

We study a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance $\varepsilon$, using tools from variational analysis. In particular, we describe necessary conditions associated with the optimal classifier subject to such an adversary. Using the necessary conditions, we derive a geometric evolution equation which can be used to track the change in classification boundaries as $\varepsilon$ varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension, and under mild assumptions on the data distribution, we rigorously prove that one can use the initial value problem starting from $\varepsilon=0$, which is simply the Bayes classifier, in order to solve for the global minimizer of the adversarial problem for small values of $\varepsilon$. In higher dimensions we provide a similar result, albeit conditional to the existence of regular solutions of the initial value problem. In the process of proving our main results we obtain a result of independent interest connecting the original adversarial problem with an optimal transport problem under no assumptions on whether classes are balanced or not. Numerical examples illustrating these ideas are also presented.