 Award ID(s):
 1727785
 NSFPAR ID:
 10190501
 Date Published:
 Journal Name:
 Nonlinear Optimal Velocity Car Following Dynamics (II): Rate of Convergence In the Presence of Fast Perturbation
 Page Range / eLocation ID:
 416 to 421
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract We describe a stochastic, dynamical system capable of inference and learning in a probabilistic latent variable model. The most challenging problem in such models—sampling the posterior distribution over latent variables—is proposed to be solved by harnessing natural sources of stochasticity inherent in electronic and neural systems. We demonstrate this idea for a sparse coding model by deriving a continuoustime equation for inferring its latent variables via Langevin dynamics. The model parameters are learned by simultaneously evolving according to another continuoustime equation, thus bypassing the need for digital accumulators or a global clock. Moreover, we show that Langevin dynamics lead to an efficient procedure for sampling from the posterior distribution in the L0 sparse regime, where latent variables are encouraged to be set to zero as opposed to having a small L1 norm. This allows the model to properly incorporate the notion of sparsity rather than having to resort to a relaxed version of sparsity to make optimization tractable. Simulations of the proposed dynamical system on both synthetic and natural image data sets demonstrate that the model is capable of probabilistically correct inference, enabling learning of the dictionary as well as parameters of the prior.more » « less

Tauman_Kalai, Yael (Ed.)Connections between proof complexity and circuit complexity have become major tools for obtaining lower bounds in both areas. These connections  which take the form of interpolation theorems and querytocommunication lifting theorems  translate efficient proofs into small circuits, and vice versa, allowing tools from one area to be applied to the other. Recently, the theory of TFNP has emerged as a unifying framework underlying these connections. For many of the proof systems which admit such a connection there is a TFNP problem which characterizes it: the class of problems which are reducible to this TFNP problem via queryefficient reductions is equivalent to the tautologies that can be efficiently proven in the system. Through this, proof complexity has become a major tool for proving separations in blackbox TFNP. Similarly, for certain monotone circuit models, the class of functions that it can compute efficiently is equivalent to what can be reduced to a certain TFNP problem in a communicationefficient manner. When a TFNP problem has both a proof and circuit characterization, one can prove an interpolation theorem. Conversely, many lifting theorems can be viewed as relating the communication and query reductions to TFNP problems. This is exciting, as it suggests that TFNP provides a roadmap for the development of further interpolation theorems and lifting theorems. In this paper we begin to develop a more systematic understanding of when these connections to TFNP occur. We give exact conditions under which a proof system or circuit model admits a characterization by a TFNP problem. We show:  Every wellbehaved proof system which can prove its own soundness (a reflection principle) is characterized by a TFNP problem. Conversely, every TFNP problem gives rise to a wellbehaved proof system which proves its own soundness.  Every wellbehaved monotone circuit model which admits a universal family of functions is characterized by a TFNP problem. Conversely, every TFNP problem gives rise to a wellbehaved monotone circuit model with a universal problem. As an example, we provide a TFNP characterization of the Polynomial Calculus, answering a question from [Mika Göös et al., 2022], and show that it can prove its own soundness.more » « less

This paper considers the problem of tracking and predicting dynamical processes with model switching. The classical approach to this problem has been to use an interacting multiple model (IMM) which uses multiple Kalman filters and an auxiliary system to estimate the posterior probability of each model given the observations. More recently, datadriven approaches such as recurrent neural networks (RNNs) have been used for tracking and prediction in a variety of settings. An advantage of datadriven approaches like the RNN is that they can be trained to provide good performance even when the underlying dynamic models are unknown. This paper studies the use of temporal convolutional networks (TCNs) in this setting since TCNs are also datadriven but have certain structural advantages over RNNs. Numerical simulations demonstrate that a TCN matches or exceeds the performance of an IMM and other classical tracking methods in two specific settings with model switching: (i) a GilbertElliott burst noise communication channel that switches between two different modes, each modeled as a linear system, and (ii) a maneuvering target tracking scenario where the target switches between a linear constant velocity mode and a nonlinear coordinated turn mode. In particular, the results show that the TCN tends to identify a mode switch as fast or faster than an IMM and that, in some cases, the TCN can perform almost as well as an omniscient Kalman filter with perfect knowledge of the current mode of the dynamical system.more » « less

Many chemical reactions and molecular processes occur on time scales that are significantly longer than those accessible by direct simulations. One successful approach to estimating dynamical statistics for such processes is to use many short time series of observations of the system to construct a Markov state model, which approximates the dynamics of the system as memoryless transitions between a set of discrete states. The dynamical Galerkin approximation (DGA) is a closely related framework for estimating dynamical statistics, such as committors and mean first passage times, by approximating solutions to their equations with a projection onto a basis. Because the projected dynamics are generally not memoryless, the Markov approximation can result in significant systematic errors. Inspired by quasiMarkov state models, which employ the generalized master equation to encode memory resulting from the projection, we reformulate DGA to account for memory and analyze its performance on two systems: a twodimensional triple well and the AIB9 peptide. We demonstrate that our method is robust to the choice of basis and can decrease the time series length required to obtain accurate kinetics by an order of magnitude.

ABSTRACT The ‘spectral age problem’ is our systematic inability to reconcile the maximum cooling time of radiating electrons in the lobes of a radio galaxy with its age as modelled by the dynamical evolution of the lobes. While there are known uncertainties in the models that produce both age estimates, ‘spectral’ ages are commonly underestimated relative to dynamical ages, consequently leading to unreliable estimates of the timeaveraged kinetic feedback of a powerful radio galaxy. In this work, we attempt to solve the spectral age problem by observing two clustercentre powerful radio galaxies; 3C 320 and 3C 444. With highresolution broadband Karl G. Jansky Very Large Array observations of the radio sources and deep XMM–Newton and Chandra observations of their hot intracluster media, coupled with the use of an analytic model, we robustly determine their spectral and dynamical ages. After finding selfconsistent dynamical models that agree with our observational constraints, and accounting for subequipartition magnetic fields, we find that our spectral ages are still underestimated by a factor of two at least. Equipartition magnetic fields will underestimate the spectral age by factors of up to ∼20. The turbulent mixing of electron populations in the radio lobes is likely to be the main remaining factor in the spectral age/dynamical age discrepancy, and must be accounted for in the study of large samples of powerful radio galaxies.