Tryphon T. Georgiou

Accurate characterization of brain tissue microstructure using diffusion MRI (dMRI) depends on data acquisition protocols. While optimal design has been explored through Fisher information, empirical subsampling of large datasets and machine learning strategies, a unified approach of optimization for diverse white and gray matter biophysical models remains necessary. To this end, we propose a generalized protocol design framework based on the Cramér-Rao Lower Bound (CRLB), which accommodates user-defined constraints while comprehensively optimizing across parameter spaces. By implementing biophysical models in a differentiable environment to compute gradients using automatic differentiation, we avoid the approximation errors of finite-difference methods and the scalability limitations of derivative-free optimization. This enables stable, reproducible convergence for high-dimensional, nonlinear models. Moreover, this approach produces data acquisition protocols that improve the estimation accuracy of microstructural metrics such as axonal diameter indices, diffusivities, and compartment water-exchange rates, thereby enhancing the fidelity of biophysical modeling for research and clinical applications.

Discrete notions of curvature have yielded important insights into the fragility of networks, including financial, gene regulatory, and social networks. These quantitative measures help identify critical nodes and pathways whose failure may cause disruption to the network's overall functionality. One such measure, the Ollivier-Ricci (OR) curvature, which is the focus of this paper, extends this inherently geometric concept into the setting of graphs by evaluation via the cost of transporting node distributions. However, a previously unstudied and salient feature of graphs is that links between nodes may reflect more than a spatial separation - links may be inhibitory or antagonistic, a quality that is not captured in the geometry of continuous spaces. To this end, we present the notions of a balanced graph and of graph frustration, to capture antagonistic effects of signed edge weights modeling promotion (+) or inhibition (-); a balanced graph is one where every cycle has an even number of negative edge weights, and frustration quantifies the degree of deviation from a balanced graph. Based on these concepts, we introduce modified Ollivier-Ricci-inspired fragility indices that point to pathways that magnify frustration in unbalanced graphs. We study two types of networks, gene regulatory and social networks, to demonstrate the utility of the fragility indices to impede or enhance functionality with respect to graph frustration. Our results demonstrate that, indeed, these new indices better identify critical edges, as quantified by several global measures, than other commonly used indices.

This paper highlights an apparent, yet relatively unknown link between algorithm design in optimization theory and controller synthesis in robust control. Specifically, quadratic optimization can be recast as a regulation problem within the framework of $\mathcal {H}_\infty$ control. From this vantage point, the optimality of Polyak's fastest heavy-ball algorithm can be ascertained as a solution to a gain margin optimization problem. The approach is independent of Polyak's original and brilliant argument, and relies on foundational work by Tannenbaum, who introduced and solved gain margin optimization via Nevanlinna–Pick interpolation theory. The link between first-order optimization methods and robust control sheds new light on the limits of algorithmic performance of such methods, and suggests a framework where similar computational tasks can be systematically studied and algorithms optimized. In particular, it raises the question as to whether periodically scheduled algorithms can achieve faster rates for quadratic optimization, in a manner analogous to periodic control that extends the gain margin beyond that of time-invariant control. This turns out not to be the case, due to the analytic obstruction of a transmission zero that is inherent in causal schemes. Interestingly, this obstruction can be removed with implicit algorithms, cast as feedback regulation problems with causal, but not strictly causal dynamics, thereby devoid of the transmission zero at infinity and able to achieve superior convergence rates.

We revisit the problem of finding the shortest path between two selected vertices of a graph and formulate this as an $\ell_1$-regularized regression -- Least Absolute Shrinkage and Selection Operator (lasso). We draw connections between a numerical implementation of this lasso-formulation, using the so-called LARS algorithm, and a more established algorithm known as the bi-directional Dijkstra. Appealing features of our formulation include the applicability of the Alternating Direction of Multiplier Method (ADMM) to the problem to identify short paths, and a relatively efficient update to topological changes.

We show that the continuous-time gradient descent in Rn can be viewed as an optimal controlled evolution for a suitable action functional; a similar result holds for stochastic gradient descent. We then provide an analogous characterization for the Wasserstein gradient flow of the (relative) entropy, with an action that mirrors the classical case where the Euclidean gradient is replaced by the Wasserstein gradient of the relative entropy. In the small-step limit, these continuous-time actions align with the Jordan Kinderlehrer Otto scheme. Next, we consider gradient flows for the relative entropy over a Wasserstein product space-a study motivated by the stochastic-control formulation of Schrodinger bridges. We characterize the product-space steepest descent as the solution to a variational problem with two control velocities and a product-space Wasserstein gradient, and we show that the induced fluxes in the two components are equal and opposite. This framework suggests applications to the optimal control evolution of microrobotic swarms that can communicate their present distribution to the other swarm.

Synthesis of optimization algorithms typically follows a {\em design-then-analyze\/} approach, which can obscure fundamental performance limits and hinder the systematic development of algorithms that operate near these limits. Recently, a framework grounded in robust control theory has emerged as a powerful tool for automating algorithm synthesis. By integrating design and analysis stages, fundamental performance bounds are revealed and synthesis of algorithms that achieve them is enabled. In this paper, we apply this framework to design algorithms for solving strongly convex optimization problems with linear equality constraints. Our approach yields a single-loop, gradient-based algorithm whose convergence rate is independent of the condition number of the constraint matrix. This improves upon the best known rate within the same algorithm class, which depends on the product of the condition numbers of the objective function and the constraint matrix.

The theory of Monge–Kantorovich optimal mass transport has in recent years spurred a fast developing phase of research in stochastic control, control of ensemble systems, thermodynamics, data science, and several other fields in engineering and science. We herein introduce a new type of transportation problems. The salient feature of these problems is that particles/agents in the ensemble are labeled and their relative position along their journey is of interest. Of particular importance in our program are control laws that steer ensembles along cycles ensuring that individual particles return to their original position. This feature is in contrast with the classical theory of optimal transport where the primary object of study is the path of probability densities, without any concern about particle labels. In the theory that we present, we focus on the case of Gaussian distributions and linear dynamics, and explore a hitherto unstudied sub-Riemannian structure of Monge–Kantorovich transport where the relative position of particles along their journey is modeled by the holonomy of the transportation schedule. From this vantage point, we discuss several other problems of independent interest.