arXiv subjects
Publications and source records attributed to Tryphon T. Georgiou.
We show that on a compact, connected, Riemannian manifold without boundary, every isotopic to the identity diffeomorphism can be written as a finite composition of optimal mass transport maps. This result represents a non-linear infinite dimensional generalization of Ballantine's factorization results.
We prove that, on a smooth manifold $M$ equipped with a constant rank horizontal distribution $H$ and a smooth inner product $g_H$ on $H$, any horizontal vector field can be written as a linear combination of, at most, $3N$ Lie brackets of horizontal gradient fields of depth-$(1)$, where $N$ is the minimal immersion dimension of $M$, recovering the Riemannian case trivially when $H$ is the entire tangent bundle. When $H$ is also bracket-generating with a uniformly bounded step, we show that any vector field can be written as a linear combination of a uniformly bounded number of iterated Lie brackets of horizontal gradient fields with uniformly bounded depth, both bounds depending solely on the immersion dimension $N$ and the uniform upper bound on the step of $H$. Utilizing this, in conjunction with the Trotter property, we show that, if the manifold is compact, connected, and without boundary, the group generated by flows of horizontal gradient fields is dense in the identity component of the diffeomorphism group. When the manifold is also Riemannian, we show that the same density statement holds for the group generated by diffeomorphic optimal mass transport maps.
We study the controllability of the differential Lyapunov equation under isospectral rotation of a linear gradient field. Specifically, control is effected by a symmetric time-varying gain-matrix constrained to have fixed eigenvalues; that is, by exclusively modulating the eigen-vectors of the state matrix and not its eigenvalues. Motivation for this problem stems from a certain type of control objectives (minimum shear/attention) aimed to reduce anisotropic deformation when ensembles are steered by a common law--optimality necessitates constancy of eigenvalues. In the paper we introduce and motivate this type of isospectral steering, and describe the reachable set of covariances for any specified terminal time and eigenvalues of the gain matrix. The theory we develop is intimately linked to multilinear algebra as well as to positive linear algebra and the Birkoff-von Neumann theorem for doubly stochastic matrices.
We revisit Brockett's attention in the context of bilinear gradient flow of an ensemble, and explore an alternative formalism that aims to reduce shear by minimizing the conditioning number of the dynamics; equivalently, we minimize the range of the eigenvalues of the dynamics. Remarkably, the evolution is isospectral, and this property is inherited by the coupled nonlinear dynamics of the control problem from a Lax isospectral flow.
Positive-definite matrices materialize as state transition matrices of linear time-invariant gradient flows, and the composition of such materializes as the state transition after successive steps where the driving potential is suitably adjusted. Thus, factoring an arbitrary matrix (with positive determinant) into a product of positive-definite ones provides the needed schedule for a time-varying potential to have a desired effect. The present work provides a detailed analysis of this factorization problem by lifting it into a sequence of Monge-Kantorovich transportation steps on Gaussian distributions and studying the induced holonomy of the optimal transportation problem. From this vantage point we determine the minimal number of positive-definite factors that have a desired effect on the spectrum of the product, e.g., ensure specified eigenvalues or being a rotation matrix. Our approach is computational and allows to identify the needed number of factors as well as trade off their conditioning number with their actual number.
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 concept of `attention' as a technical term to quantify the effort in calibrating control action based on available data. While Wiener, in his work on Cybernetics, anticipated key principles on prioritizing task-relevant signals, it was not until the late 1990's when Brockett first formulated pertinent optimization problems that have inspired subsequent as well as the present work. `Attention,' as a technical term, is defined so as to quantify the dependence of the control law on the time and space/state coordinate; a control law that is independent of time and space, assuming it meets specifications, requires vanishing attention. In the present work we focus on Linear-Markovian dynamics with Gaussian state uncertainty so as to analyze the structure of minimal-attention control schemes that steer the dynamics between terminal states with Gaussian uncertainty profile.
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.
The synthesis of optimization algorithms typically follows a design-first-analyze-later approach, which often obscures fundamental performance limitations and hinders the systematic design of algorithms operating at the achievable theoretical boundaries. Recently, a framework based on frequency-domain techniques from robust control theory has emerged as a powerful tool for automating algorithm synthesis. By integrating the design and analysis stages, this framework enables the identification of fundamental performance limits. In this paper, we build on this framework and extend it to address algorithms for solving strongly convex problems with equality constraints. As a result, we obtain a new class of algorithms that offers sharp trade-off between number of matrix multiplication per iteration and convergence rate.
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.
We consider control and inference problems where control protocols and internal dynamics are informed by two types of constraints. Our data consist of i) statistics on the ensemble and ii) trajectories or final disposition of selected tracer particles embedded in the flow. Our aim is i') to specify a control protocol to realize a flow that meets such constraints or ii') to recover the internal dynamics that are consistent with such a data set. We analyze these problems in the setting of linear flows and Gaussian distributions. The control cost is taken to be a suitable action integral constrained by either the trajectories of tracer particles or their terminal placements.
The theory of Monge-Kantorovich Optimal Mass Transport (OMT) 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 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.
In this Letter, we interpret the Husimi function as the conditional probability density of continuously measuring a stream of constant position and momentum outcomes, indefinitely. This gives rise to an alternative definition that naturally extends to an arbitrary collection of self-adjoint operators without reference to coherent states. This definition recovers the Husimi distribution for a spin-half particle when monitoring the three Pauli matrices, as well as Born's rule for quantum measurement when monitoring commuting quantum observables. Ultimately, the proposed paradigm generates positive representations of quantum states as conditional densities, on both finite and infinite time classical experiments, as expectations of a fundamental operator, the Gaussian semigroup.
We study the control problem of regulating the purity of a quantum harmonic oscillator in a Gaussian state via weak measurements. Specifically, we assume time-invariant Hamiltonian dynamics and that control is exerted via the back-action induced from monitoring the oscillator's position and momentum observables; the manipulation of the detector measurement strengths regulates the purity of the target Gaussian quantum state. After briefly drawing connections between Gaussian quantum dynamics and stochastic control, we focus on the effect of inefficient detectors and derive closed-form expressions for the transient and steady-state dynamics of the state covariance. We highlight the degradation of attainable purity that is due to inefficient detectors, as compared to that dictated by the Robertson-Schrödinger uncertainty relation. Our results suggest that quantum correlations can enhance the purity at steady-state. The quantum harmonic oscillator represents a basic system where analytic formulae may provide insights into the role of inefficient measurements in quantum control; the gained insights are pertinent to measurement-based quantum engines and cooling experiments.
Quantum counterparts of Schrodinger's classical bridge problem have been around for the better part of half a century. During that time, several quantum approaches to this multifaceted classical problem have been introduced. In the present work, we unify, extend, and interpret several such approaches through a classical large deviations perspective. To this end, we consider time-symmetric ensembles that are pre- and post-selected before and after a Markovian experiment is performed. The Schrodinger bridge problem is that of finding the most likely joint distribution of initial and final outcomes that is consistent with obtained endpoint results. The derived distribution provides quantum Markovian dynamics that bridge the observed endpoint states in the form of density matrices. The solution retains its classical structure in that density matrices can be expressed as the product of forward-evolving and backward-evolving matrices. In addition, the quantum Schrodinger bridge allows inference of the most likely distribution of outcomes of an intervening measurement with unknown results. This distribution may be written as a product of forward- and backward-evolving expressions, in close analogy to the classical setting, and in a time-symmetric way. The derived results are illustrated through a two-level amplitude damping example.
The problem of reconciling a prior probability law on paths with data was introduced by E. Schrödinger in 1931/32. It represents an early formulation of a maximum likelihood problem. This specific formulation can also be seen as the control problem to modify the law of a diffusion process so as to match specifications on marginal distributions at given times. Thereby, in recent years, this so-called Schrödinger's bridge problem has been at the center of the uncertainty control development. However, an understudied facet of this program has been to address uncertainty in space (state) and time, modeling the effect of tasks being completed contingent on meeting a certain condition at some random time instead of imposing specifications at fixed times. The present work is a study to extend Schrödinger's paradigm on such an issue, and herein, it is tackled in the context of random walks on directed graphs. Specifically, we study the case where one marginal is the initial probability distribution on a Markov chain, while others are marginals of stopping (first-arrival) times at absorbing states, signifying completion of tasks. We show when the prior law on paths is Markov, a Markov policy is once again optimal to satisfy those marginal constraints with respect to a likelihood cost following Schrödinger's dictum. Based on this, we present the mathematical formulation involving a Sinkhorn-type iteration to construct the optimal probability law on paths matching the spatio-temporal marginals.
We consider the problem of steering a collection of n particles that obey identical n-dimensional linear dynamics via a common state feedback law towards a rearrangement of their positions, cast as a controllability problem for a dynamical system evolving on the space of matrices with positive determinant. We show that such a task is always feasible and, moreover, that it can be achieved arbitrarily fast. We also show that an optimal feedback control policy to achieve a similar feat, may not exist. Furthermore, we show that there is no universal formula for a linear feedback control law to achieve a rearrangement, optimal or not, that is everywhere continuous with respect to the specifications. We conclude with partial results on the broader question of controllability of dynamics on orientation-preserving diffeomorphisms.
Reaction-diffusion equations are commonly used to model a diverse array of complex systems, including biological, chemical, and physical processes. Typically, these models are phenomenological, requiring the fitting of parameters to experimental data. In the present work, we introduce a novel formalism to construct reaction-diffusion models that is grounded in the principle of maximum entropy. This new formalism aims to incorporate various types of experimental data, including ensemble currents, distributions at different points in time, or moments of such. To this end, we expand the framework of Schrödinger bridges and Maximum Caliber problems to nonlinear interacting systems. We illustrate the usefulness of the proposed approach by modeling the evolution of (i) a morphogen across the fin of a zebrafish and (ii) the population of two varieties of toads in Poland, so as to match the experimental data.