Many emerging applications - such as adversarial training, AI alignment, and robust optimization - can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions - spanning initialization, training dynamics, and network width - that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-Łojasiewicz (PŁ) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.

We introduce a novel programming model, Data Spatial Programming, which extends the semantics of Object-Oriented Programming (OOP) by introducing new class-like constructs called archetypes. These archetypes encapsulate spatial relationships between data entities and execution flow in a structured manner, enabling more expressive and semantically rich computations over interconnected data structures. By formalizing the relationships between data elements in space, our approach allows for more intuitive modeling of complex systems where the topology of connections is essential to the underlying computational model. This paradigm addresses limitations in traditional OOP when representing dynamically evolving networks, agent-based systems, and other spatially-oriented computational problems.

A subset of points in a metric space is said to resolve it if each point in the space is uniquely characterized by its distance to each point in the subset. In particular, resolving sets can be used to represent points in abstract metric spaces as Euclidean vectors. Importantly, due to the triangle inequality, points close by in the space are represented as vectors with similar coordinates, which may find applications in classification problems of symbolic objects under suitably chosen metrics. In this manuscript, we address the resolvability of Jaccard spaces, i.e., metric spaces of the form $(2^X,\text{Jac})$, where $2^X$ is the power set of a finite set $X$, and $\text{Jac}$ is the Jaccard distance between subsets of $X$. Specifically, for different $a,b\in 2^X$, $\text{Jac}(a,b)=|a\Delta b|/|a\cup b|$, where $|\cdot|$ denotes size (i.e., cardinality) and $\Delta$ denotes the symmetric difference of sets. We combine probabilistic and linear algebra arguments to construct highly likely but nearly optimal (i.e., of minimal size) resolving sets of $(2^X,\text{Jac})$. In particular, we show that the metric dimension of $(2^X,\text{Jac})$, i.e., the minimum size of a resolving set of this space, is $\Theta(|X|/\ln|X|)$. In addition, we show that a much smaller subset of $2^X$ suffices to resolve, with high probability, all different pairs of subsets of $X$ of cardinality at most $\sqrt{|X|}/\ln|X|$, up to a factor.

Face morphing is one of the seminal problems in computer graphics, with numerous artistic and forensic applications. It is notoriously challenging due to pose, lighting, gender, and ethnicity variations. Generally, this task consists of a warping for feature alignment and a blending for a seamless transition between the warped images. We propose to leverage coordinate-based neural networks to represent such warpings and blendings of face images. During training, we exploit the smoothness and flexibility of such networks, by combining energy functionals employed in classical approaches without discretizations. Additionally, our method is time-dependent, allowing a continuous warping, and blending of the target images. During warping inference, we need both direct and inverse transformations of the time-dependent warping. The first is responsible for morphing the target image into the source image, while the inverse is used for morphing in the opposite direction. Our neural warping stores those maps in a single network due to its inversible property, dismissing the hard task of inverting them. The results of our experiments indicate that our method is competitive with both classical and data-based neural techniques under the lens of face-morphing detection approaches. Aesthetically, the resulting images present a seamless blending of diverse faces not yet usual in the literature.

A common pipeline in learning-based control is to iteratively estimate a model of system dynamics, and apply a trajectory optimization algorithm - e.g.~$\mathtt{iLQR}$ - on the learned model to minimize a target cost. This paper conducts a rigorous analysis of a simplified variant of this strategy for general nonlinear systems. We analyze an algorithm which iterates between estimating local linear models of nonlinear system dynamics and performing $\mathtt{iLQR}$-like policy updates. We demonstrate that this algorithm attains sample complexity polynomial in relevant problem parameters, and, by synthesizing locally stabilizing gains, overcomes exponential dependence in problem horizon. Experimental results validate the performance of our algorithm, and compare to natural deep-learning baselines.

The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank $k$ approximation of a matrix $A$ using matrix-vector products with standard Gaussian vectors. Here, we generalize the theory of randomized SVD to multivariable Gaussian vectors, allowing one to incorporate prior knowledge of $A$ into the algorithm. This enables us to explore the continuous analogue of the randomized SVD for Hilbert--Schmidt (HS) operators using operator-function products with functions drawn from a Gaussian process (GP). We then construct a new covariance kernel for GPs, based on weighted Jacobi polynomials, which allows us to rapidly sample the GP and control the smoothness of the randomly generated functions. Numerical examples on matrices and HS operators demonstrate the applicability of the algorithm.

We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $\ell_2$ norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity. Depending on the scaling limit considered, the mean and variance of the limiting Gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn. We also obtain explicit error bounds on the moments of the norm and the Kolmogorov-Smirnov distance to a Gaussian. Finally, we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with ReLU activations. This provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with ReLU activations and a given architecture.