arXiv subjects

Anthony Bloch

Publications and source records attributed to Anthony Bloch.

At least 55 records · Page 3Linked to original sources

arXiv2021

Nonparametric Continuous Sensor Registration

This paper develops a new mathematical framework that enables nonparametric joint semantic and geometric representation of continuous functions using data. The joint embedding is modeled by representing the processes in a reproducing kernel Hilbert space. The functions can be defined on arbitrary smooth manifolds where the action of a Lie group aligns them. The continuous functions allow the registration to be independent of a specific signal resolution. The framework is fully analytical with a closed-form derivation of the Riemannian gradient and Hessian. We study a more specialized but widely used case where the Lie group acts on functions isometrically. We solve the problem by maximizing the inner product between two functions defined over data, while the continuous action of the rigid body motion Lie group is captured through the integration of the flow in the corresponding Lie algebra. Low-dimensional cases are derived with numerical examples to show the generality of the proposed framework. The high-dimensional derivation for the special Euclidean group acting on the Euclidean space showcases the point cloud registration and bird's-eye view map registration abilities. An implementation of this framework for RGB-D cameras outperforms the state-of-the-art robust visual odometry and performs well in texture and structure-scarce environments.

math.OC↗

arXiv2021

Controllability of Hypergraphs

In this paper, we develop a notion of controllability for hypergraphs via tensor algebra and polynomial control theory. Inspired by uniform hypergraphs, we propose a new tensor-based multilinear dynamical system representation, and derive a Kalman-rank-like condition to determine the minimum number of control nodes (MCN) needed to achieve controllability of even uniform hypergraphs. We present an efficient heuristic to obtain the MCN. MCN can be used as a measure of robustness, and we show that it is related to the hypergraph degree distribution in simulated examples. Finally, we use MCN to examine robustness in real biological networks.

math.OC↗

arXiv2020

Multilinear Control Systems Theory

In this paper, we provide a system theoretic treatment of a new class of multilinear time-invariant (MLTI) systems in which the states, inputs and outputs are tensors, and the system evolution is governed by multilinear operators. The MLTI system representation is based on the Einstein product and even-order paired tensors. There is a particular tensor unfolding which gives rise to an isomorphism from this tensor space to the general linear group, i.e. the group of invertible matrices. By leveraging this unfolding operation, one can extend classical linear time-invariant (LTI) system notions including stability, reachability and observability to MLTI systems. While the unfolding based formulation is a powerful theoretical construct, the computational advantages of MLTI systems can only be fully realized while working with the tensor form, where hidden patterns/structures can be exploited for efficient representations and computations. Along these lines, we establish new results which enable one to express tensor unfolding based stability, reachability and observability criteria in terms of more standard notions of tensor ranks/decompositions. In addition, we develop a generalized CANDECOMP/PARAFAC decomposition and tensor train decomposition based model reduction framework, which can significantly reduce the number of MLTI system parameters. We demonstrate our framework with numerical examples.

math.OC↗

arXiv2020

Data-Driven Model Reduction for Multilinear Control Systems via Tensor Trains

In this paper, we explore the role of tensor algebra in balanced truncation (BT) based model reduction/identification for high-dimensional multilinear/linear time invariant systems. In particular, we employ tensor train decomposition (TTD), which provides a good compromise between numerical stability and level of compression, and has an associated algebra that facilitates computations. Using TTD, we propose a new BT approach which we refer to as higher-order balanced truncation, and consider different data-driven variations including higher-order empirical gramians, higher-order balanced proper orthogonal decomposition and a higher-order eigensystem realization algorithm. We perform computational and memory complexity analysis for these different flavors of TTD based BT methods, and compare with the corresponding standard BT methods in order to develop insights into where the proposed framework may be beneficial. We provide numerical results on simulated and experimental datasets showing the efficacy of the proposed framework.

eess.SY↗

arXiv2019

Variational collision and obstacle avoidance of multi-agent systems on Riemannian manifolds

In this paper we study a path planning problem from a variational approach to collision and obstacle avoidance for multi-agent systems evolving on a Riemannian manifold. The problem consists of finding non-intersecting trajectories between the agent and prescribed obstacles on the workspace, among a set of admissible curves, to reach a specified configuration, based on minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to prevent collision with the obstacles and among the agents. We apply the results to examples of a planar rigid body, and collision and obstacle avoidance for agents evolving on a sphere.

eess.SY↗

arXiv2019

Adaptive Continuous Visual Odometry from RGB-D Images

In this paper, we extend the recently developed continuous visual odometry framework for RGB-D cameras to an adaptive framework via online hyperparameter learning. We focus on the case of isotropic kernels with a scalar as the length-scale. In practice and as expected, the length-scale has remarkable impacts on the performance of the original framework. Previously it was handled using a fixed set of conditions within the solver to reduce the length-scale as the algorithm reaches a local minimum. We automate this process by a greedy gradient descent step at each iteration to find the next-best length-scale. Furthermore, to handle failure cases in the gradient descent step where the gradient is not well-behaved, such as the absence of structure or texture in the scene, we use a search interval for the length-scale and guide it gradually toward the smaller values. This latter strategy reverts the adaptive framework to the original setup. The experimental evaluations using publicly available RGB-D benchmarks show the proposed adaptive continuous visual odometry outperforms the original framework and the current state-of-the-art. We also make the software for the developed algorithm publicly available.

cs.RO↗

arXiv2019

Variational point-obstacle avoidance on Riemannian manifolds

In this letter we study variational obstacle avoidance problems on complete Riemannian manifolds. The problem consists of minimizing an energy functional depending on the velocity, covariant acceleration and a repulsive potential function used to avoid a static obstacle on the manifold, among a set of admissible curves. We derive the dynamical equations for extrema of the variational problem, in particular on compact connected Lie groups and Riemannian symmetric spaces. Numerical examples are presented to illustrate the proposed method.

eess.SY↗

arXiv2019

The Bouncing Penny and Nonholonomic Impacts

The evolution of a Lagrangian mechanical system is variational. Likewise, when dealing with a hybrid Lagrangian system (a system with discontinuous impacts), the impacts can also be described by variations. These variational impacts are given by the so-called Weierstrass-Erdmann corner conditions. Therefore, hybrid Lagrangian systems can be completely understood by variational principles. Unlike typical (unconstrained / holonomic) Lagrangian systems, nonholonomically constrained Lagrangian systems are not variational. However, by using the Lagrange-d'Alembert principle, nonholonomic systems can be described as projections of variational systems. This paper works out the analogous version of the Weierstrass-Erdmann corner conditions for nonholonomic systems and examines the billiard problem with a rolling disk.

math.DS↗

arXiv2019

Continuous Direct Sparse Visual Odometry from RGB-D Images

This paper reports on a novel formulation and evaluation of visual odometry from RGB-D images. Assuming a static scene, the developed theoretical framework generalizes the widely used direct energy formulation (photometric error minimization) technique for obtaining a rigid body transformation that aligns two overlapping RGB-D images to a continuous formulation. The continuity is achieved through functional treatment of the problem and representing the process models over RGB-D images in a reproducing kernel Hilbert space; consequently, the registration is not limited to the specific image resolution and the framework is fully analytical with a closed-form derivation of the gradient. We solve the problem by maximizing the inner product between two functions defined over RGB-D images, while the continuous action of the rigid body motion Lie group is captured through the integration of the flow in the corresponding Lie algebra. Energy-based approaches have been extremely successful and the developed framework in this paper shares many of their desired properties such as the parallel structure on both CPUs and GPUs, sparsity, semi-dense tracking, avoiding explicit data association which is computationally expensive, and possible extensions to the simultaneous localization and mapping frameworks. The evaluations on experimental data and comparison with the equivalent energy-based formulation of the problem confirm the effectiveness of the proposed technique, especially, when the lack of structure and texture in the environment is evident.

cs.RO↗

arXiv2019

Multilinear Time Invariant System Theory

In biological and engineering systems, structure, function and dynamics are highly coupled. Such interactions can be naturally and compactly captured via tensor based state space dynamic representations. However, such representations are not amenable to the standard system and controls framework which requires the state to be in the form of a vector. In order to address this limitation, recently a new class of multiway dynamical systems has been introduced in which the states, inputs and outputs are tensors. We propose a new form of multilinear time invariant (MLTI) systems based on the Einstein product and even-order paired tensors. We extend classical linear time invariant (LTI) system notions including stability, reachability and observability for the new MLTI system representation by leveraging recent advances in tensor algebra.

math.OC↗

arXiv2019

Time-minimum control of quantum purity for 2-level Lindblad equations

We study time-minimum optimal control for a class of quantum two-dimensional dissipative systems whose dynamics are governed by the Lindblad equation and where control inputs acts only in the Hamiltonian. The dynamics of the control system are analyzed as a bi-linear control system on the Bloch ball after a decoupling of such dynamics into intra- and inter-unitary orbits. The (singular) control problem consists of finding a trajectory of the state variables solving a radial equation in the minimum amount of time, starting at the completely mixed state and ending at the state with the maximum achievable purity. The boundary value problem determined by the time-minimum singular optimal control problem is studied numerically. If controls are unbounded, simulations show that multiple local minimal solutions might exist. To find the unique globally minimal solution, we must repeat the algorithm for various initial conditions and find the best solution out of all of the candidates. If controls are bounded, optimal controls are given by bang-bang controls using the Pontryagin minimum principle. Using a switching map we construct optimal solutions consisting of singular arcs. If controls are bounded, the analysis of our model also implies classical analysis done previously for this problem.

math.OC↗

arXiv2018

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimizing a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem. We illustrate the results with examples of rigid bodies, both planar and spatial, and underactuated vehicles including a unicycle and an underactuated unmanned vehicle.

math.OC↗

arXiv2018

The variational discretization of the constrained higher-order Lagrange-Poincaré equations

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.

math.DS↗

arXiv2018

Variational collision avoidance problems on Riemannian manifolds

In this article we introduce a variational approach to collision avoidance of multiple agents evolving on a Riemannian manifold and derive necessary conditions for extremals. The problem consists of finding non-intersecting trajectories of a given number of agents, among a set of admissible curves, to reach a specified configuration, based on minimizing an energy functional that depends on the velocity, covariant acceleration and an artificial potential function used to prevent collision among the agents. The results are validated through numerical experiments on the manifolds $\mathbb{R}^{2}$ and $S^2$.

eess.SY↗

arXiv2018

Poincaré-Bendixson Theorem for Hybrid Systems

The Poincaré-Bendixson theorem plays an important role in the study of the qualitative behavior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincaré-Bendixson Theorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincaré return map, a useful object for the stability analysis of hybrid systems. We also prove a Poincaré-Bendixson Theorem for a class of one dimensional hybrid dynamical systems.

math.DS↗

arXiv2017

An Algorithm for Cellular Reprogramming

The day we understand the time evolution of subcellular elements at a level of detail comparable to physical systems governed by Newton's laws of motion seems far away. Even so, quantitative approaches to cellular dynamics add to our understanding of cell biology, providing data-guided frameworks that allow us to develop better predictions about and methods for control over specific biological processes and system-wide cell behavior. In this paper we describe an approach to optimizing the use of transcription factors in the context of cellular reprogramming. We construct an approximate model for the natural evolution of a synchronized population of fibroblasts, based on data obtained by sampling the expression of some 22,083 genes at several times along the cell cycle. (These data are based on a colony of cells that have been cell cycle synchronized) In order to arrive at a model of moderate complexity, we cluster gene expression based on the division of the genome into topologically associating domains (TADs) and then model the dynamics of the expression levels of the TADs. Based on this dynamical model and known bioinformatics, we develop a methodology for identifying the transcription factors that are the most likely to be effective toward a specific cellular reprogramming task. The approach used is based on a device commonly used in optimal control. From this data-guided methodology, we identify a number of validated transcription factors used in reprogramming and/or natural differentiation. Our findings highlight the immense potential of dynamical models models, mathematics, and data guided methodologies for improving methods for control over biological processes.

q-bio.GN↗

arXiv2017

Variational obstacle avoidance problem on Riemannian manifolds

We introduce variational obstacle avoidance problems on Riemannian manifolds and derive necessary conditions for the existence of their normal extremals. The problem consists of minimizing an energy functional depending on the velocity and covariant acceleration, among a set of admissible curves, and also depending on a navigation function used to avoid an obstacle on the workspace, a Riemannian manifold. We study two different scenarios, a general one on a Riemannian manifold and, a sub-Riemannian problem. By introducing a left-invariant metric on a Lie group, we also study the variational obstacle avoidance problem on a Lie group. We apply the results to the obstacle avoidance problem of a planar rigid body and an unicycle.

math.OC↗

arXiv2017

Optimal Control of Quantum Purity for $n=2$ Systems

The objective of this work is to study time-minimum and energy-minimum global optimal control for dissipative open quantum systems whose dynamics is governed by the Lindblad equation. The controls appear only in the Hamiltonian. Using recent results regarding the decoupling of such dissipative dynamics into intra- and inter-unitary orbits, we transform the control system into a bi-linear control system on the Bloch ball (the unitary sphere together with its interior). We then design a numerical algorithm to construct an optimal path to achieve a desired point given initial states close to the origin (the singular point) of the Bloch ball. This is done both for the minimum-time and minimum -energy control problems.

math.OC↗