Prashant G. Mehta

Tao Yang, Prashant G. Mehta, Sean P. Meyn

A new formulation of the particle filter for nonlinear filtering is presented, based on concepts from optimal control, and from the mean-field game theory. The optimal control is chosen so that the posterior distribution of a particle matches as closely as possible the posterior distribution of the true state given the observations. This is achieved by introducing a cost function, defined by the Kullback-Leibler (K-L) divergence between the actual posterior, and the posterior of any particle. The optimal control input is characterized by a certain Euler-Lagrange (E-L) equation, and is shown to admit an innovation error-based feedback structure. For diffusions with continuous observations, the value of the optimal control solution is ideal. The two posteriors match exactly, provided they are initialized with identical priors. The feedback particle filter is defined by a family of stochastic systems, each evolving under this optimal control law. A numerical algorithm is introduced and implemented in two general examples, and a neuroscience application involving coupled oscillators. Some preliminary numerical comparisons between the feed- back particle filter and the bootstrap particle filter are described.

Tao Yang, Richard S. Laugesen, Prashant G. Mehta et al.

In recent work it is shown that importance sampling can be avoided in the particle filter through an innovation structure inspired by traditional nonlinear filtering combined with Mean-Field Game formalisms. The resulting feedback particle filter (FPF) offers significant variance improvements; in particular, the algorithm can be applied to systems that are not stable. The filter comes with an up-front computational cost to obtain the filter gain. This paper describes new representations and algorithms to compute the gain in the general multivariable setting. The main contributions are, (i) Theory surrounding the FPF is improved: Consistency is established in the multivariate setting, as well as well-posedness of the associated PDE to obtain the filter gain. (ii) The gain can be expressed as the gradient of a function, which is precisely the solution to Poisson's equation for a related MCMC diffusion (the Smoluchowski equation). This provides a bridge to MCMC as well as to approximate optimal filtering approaches such as TD-learning, which can in turn be used to approximate the gain. (iii) Motivated by a weak formulation of Poisson's equation, a Galerkin finite-element algorithm is proposed for approximation of the gain. Its performance is illustrated in numerical experiments.

Amirhossein Taghvaei, Jana de Wiljes, Prashant G. Mehta et al.

This paper is concerned with the filtering problem in continuous-time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter which provides an exact solution for the linear Gaussian problem, (ii) the ensemble Kalman-Bucy filter (EnKBF) which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems, and (iii) the feedback particle filter (FPF) which represents an extension of the EnKBF and furthermore provides for an consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of non-uniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

Amirhossein Taghvaei, Prashant G. Mehta, Sean P. Meyn

Feedback particle filter (FPF) is a numerical algorithm to approximate the solution of the nonlinear filtering problem in continuous-time settings. In any numerical implementation of the FPF algorithm, the main challenge is to numerically approximate the so-called gain function. A numerical algorithm for gain function approximation is the subject of this paper. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian $Δ_ρ$. The numerical problem is to approximate this solution using {\em only} finitely many particles sampled from the probability distribution $ρ$. A diffusion map-based algorithm was proposed by the authors in a prior work to solve this problem. The algorithm is named as such because it involves, as an intermediate step, a diffusion map approximation of the exact semigroup $e^{Δ_ρ}$. The original contribution of this paper is to carry out a rigorous error analysis of the diffusion map-based algorithm. The error is shown to include two components: bias and variance. The bias results from the diffusion map approximation of the exact semigroup. The variance arises because of finite sample size. Scalings and upper bounds are derived for bias and variance. These bounds are then illustrated with numerical experiments that serve to emphasize the effects of problem dimension and sample size. The proposed algorithm is applied to two filtering examples and comparisons provided with the sequential importance resampling (SIR) particle filter.

Amirhossein Taghvaei, Prashant G. Mehta, Sean P. Meyn

This paper is concerned with the analysis of the kernel-based algorithm for gain function approximation in the feedback particle filter. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian. The kernel-based method -- introduced in our prior work -- allows one to approximate this solution using {\em only} particles sampled from the probability distribution. This paper describes new representations and algorithms based on the kernel-based method. Theory surrounding the approximation is improved and a novel formula for the gain function approximation is derived. A procedure for carrying out error analysis of the approximation is introduced. Certain asymptotic estimates for bias and variance are derived for the general nonlinear non-Gaussian case. Comparison with the constant gain function approximation is provided. The results are illustrated with the aid of some numerical experiments.

Tao Yang, Geng Huang, Prashant G. Mehta

This paper introduces a novel feedback-control based particle filter for the solution of the filtering problem with data association uncertainty. The particle filter is referred to as the joint probabilistic data association-feedback particle filter (JPDA-FPF). The JPDA-FPF is based on the feedback particle filter introduced in our earlier papers. The remarkable conclusion of our paper is that the JPDA-FPF algorithm retains the innovation error-based feedback structure of the feedback particle filter, even with data association uncertainty in the general nonlinear case. The theoretical results are illustrated with the aid of two numerical example problems drawn from multiple target tracking applications.

Xiaotian Zhang, Ali Albazroun, Tixian Wang et al.

Limbless terrestrial animals exhibit exceptional locomotor versatility and control, currently unmatched by engineered counterparts. Here, we introduce a computational framework that enables soft synthetic snakes to navigate unstructured, heterogeneous 3D terrains. Our approach is grounded in bio-inspired actuation and sensing models that reduce the control complexity inherent to high-degree-of-freedom, continuum bodies. These models are integrated into a reinforcement learning architecture to derive environment-traversing policies. Training first occurs in simplified, homogeneous terrains to learn locomotion primitives. These are then composed into adaptive strategies for complex landscapes. We demonstrate robustness by deploying a snake in high-fidelity 3D environments reconstructed from real-world imaging, achieving reliable navigation. Overall, this work provides a physically-realistic simulation platform and practical insights for the control of continuum systems in natural terrains.

Aditya Kudre, Heng-Sheng Chang, Prashant G. Mehta

This work develops a duality theory for partially observed linear Gaussian models in discrete time. The state process evolves according to a causal but non-Markovian (or higher-order Gauss-Markov) structure, captured by a lower-triangular transition operator, which is related to transformer, with $T$ as the context length. The main contributions are: (i) a dual control system for the linear Gaussian model, formulated as a backward difference equation (B $Δ$ E); (ii) a duality principle establishing that a specific linear-quadratic optimal control problem for the B $Δ$ E is dual to the filtering problem for the partially observed model; and (iii) an explicit optimal control formula yielding a novel (transformer-like) linear predictor, referred to as the dual filter, whose computational complexity scales linearly in the time horizon $T$, in contrast to the $O(T^3)$ cost of classical smoothing and Wiener-Hopf approaches.

Aditya Kudre, Heng-Sheng Chang, Prashant G. Mehta

Decoder-only transformers compute the conditional probability of the next token from a sequence of past observations. This paper derives, from first principles, inference architectures that solve the same prediction problem - and in doing so, recovers transformer-like layer operations as a consequence of optimal control theory. The framework is developed for two model classes: a nonlinear model of discrete-valued processes, directly motivated by the transformer, and a linear Gaussian model as a tractable baseline. For both model classes, the prediction objective is reformulated as an optimal control problem whose solution yields an explicit inference algorithm, the dual filter, with a layer structure that mirrors the layer structure of a decoder-only transformer. Numerical experiments provide a comparison of the optimal control to attention weights from a trained transformer. These experiments reveal that when the embedding dimension is insufficient, the transformer implicitly exploits non-Markovian structure.

Reginald Zhiyan Chen, Heng-Sheng Chang, Prashant G. Mehta

Hidden Markov Models (HMMs) are fundamental for modeling sequential data, yet learning their parameters from observations remains challenging. Classical methods like the Baum-Welch (EM) algorithm are computationally intensive and prone to local optima, while modern spectral algorithms offer provable guarantees but may produce probability outputs outside valid ranges. This work introduces Belief Net, a novel framework that learns HMM parameters through gradient-based optimization by formulating the HMM's forward filter as a structured neural network. Unlike black-box Transformer models, Belief Net's learnable weights are explicitly the logits of the initial distribution, transition matrix, and emission matrix, ensuring full interpretability. The model processes observation sequences using a decoder-only architecture and is trained end-to-end with standard autoregressive next-observation prediction loss. On synthetic HMM data, Belief Net achieves superior convergence speed compared to Baum-Welch, successfully recovering parameters in both undercomplete and overcomplete settings where spectral methods fail. Comparisons with Transformer-based models are also presented on real-world language data.

Heng-Sheng Chang, Prashant G. Mehta

This paper presents a mathematical framework for causal nonlinear prediction in settings where observations are generated from an underlying hidden Markov model (HMM). Both the problem formulation and the proposed solution are motivated by the decoder-only transformer architecture, in which a finite sequence of observations (tokens) is mapped to the conditional probability of the next token. Our objective is not to construct a mathematical model of a transformer. Rather, our interest lies in deriving, from first principles, transformer-like architectures that solve the prediction problem for which the transformer is designed. The proposed framework is based on an original optimal control approach, where the prediction objective (MMSE) is reformulated as an optimal control problem. An analysis of the optimal control problem is presented leading to a fixed-point equation on the space of probability measures. To solve the fixed-point equation, we introduce the dual filter, an iterative algorithm that closely parallels the architecture of decoder-only transformers. These parallels are discussed in detail along with the relationship to prior work on mathematical modeling of transformers as transport on the space of probability measures. Numerical experiments are provided to illustrate the performance of the algorithm using parameter values used in researchscale transformer models.

Heng-Sheng Chang, Prashant G. Mehta

In the 1940s, Wiener introduced a linear predictor, where the future prediction is computed by linearly combining the past data. A transformer generalizes this idea: it is a nonlinear predictor where the next-token prediction is computed by nonlinearly combining the past tokens. In this essay, we present a probabilistic model that interprets transformer signals as surrogates of conditional measures, and layer operations as fixed-point updates. An explicit form of the fixed-point update is described for the special case when the probabilistic model is a hidden Markov model (HMM). In part, this paper is in an attempt to bridge the classical nonlinear filtering theory with modern inference architectures.

Seung Hyun Kim, Heng-Sheng Chang, Chia-Hsien Shih et al.

This paper is concerned with the problem of estimating (interpolating and smoothing) the shape (pose and the six modes of deformation) of a slender flexible body from multiple camera measurements. This problem is important in both biology, where slender, soft, and elastic structures are ubiquitously encountered across species, and in engineering, particularly in the area of soft robotics. The proposed mathematical formulation for shape estimation is physics-informed, based on the use of the special Cosserat rod theory whose equations encode slender body mechanics in the presence of bending, shearing, twisting and stretching. The approach is used to derive numerical algorithms which are experimentally demonstrated for fiber reinforced and cable-driven soft robot arms. These experimental demonstrations show that the methodology is accurate (<5 mm error, three times less than the arm diameter) and robust to noise and uncertainties.

Tixian Wang, Udit Halder, Heng-Sheng Chang et al.

In this paper, we use the optimal control methodology to control a flexible, elastic Cosserat rod. An inspiration comes from stereotypical movement patterns in octopus arms, which are observed in a variety of manipulation tasks, such as reaching or fetching. To help uncover the mechanisms underlying these observed morphologies, we outline an optimal control-based framework. A single octopus arm is modeled as a Hamiltonian control system, where the continuum mechanics of the arm is modeled after the Cosserat rod theory, and internal, distributed muscle forces and couples are considered as controls. First order necessary optimality conditions are derived for an optimal control problem formulated for this infinite dimensional system. Solutions to this problem are obtained numerically by an iterative forward-backward algorithm. The state and adjoint equations are solved in a dynamic simulation environment, setting the stage for studying a broader class of optimal control problems. Trajectories that minimize control effort are demonstrated and qualitatively compared with observed behaviors.

Heng-Sheng Chang, Udit Halder, Ekaterina Gribkova et al.

This paper presents an application of the energy shaping methodology to control a flexible, elastic Cosserat rod model of a single octopus arm. The novel contributions of this work are two-fold: (i) a control-oriented modeling of the anatomically realistic internal muscular architecture of an octopus arm; and (ii) the integration of these muscle models into the energy shaping control methodology. The control-oriented modeling takes inspiration in equal parts from theories of nonlinear elasticity and energy shaping control. By introducing a stored energy function for muscles, the difficulties associated with explicitly solving the matching conditions of the energy shaping methodology are avoided. The overall control design problem is posed as a bilevel optimization problem. Its solution is obtained through iterative algorithms. The methodology is numerically implemented and demonstrated in a full-scale dynamic simulation environment Elastica. Two bio-inspired numerical experiments involving the control of octopus arms are reported.

S. Yagiz Olmez, Amirhossein Taghvaei, Prashant G. Mehta

In this paper, we present a novel approach to approximate the gain function of the feedback particle filter (FPF). The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian. The numerical problem is to approximate the exact gain function using only finitely many particles sampled from the probability distribution. Inspired by the recent success of the deep learning methods, we represent the gain function as a gradient of the output of a neural network. Thereupon considering a certain variational formulation of the Poisson equation, an optimization problem is posed for learning the weights of the neural network. A stochastic gradient algorithm is described for this purpose. The proposed approach has two significant properties/advantages: (i) The stochastic optimization algorithm allows one to process, in parallel, only a batch of samples (particles) ensuring good scaling properties with the number of particles; (ii) The remarkable representation power of neural networks means that the algorithm is potentially applicable and useful to solve high-dimensional problems. We numerically establish these two properties and provide extensive comparison to the existing approaches.

Prashant G. Mehta, Sean P. Meyn

It is well known that the extension of Watkins' algorithm to general function approximation settings is challenging: does the projected Bellman equation have a solution? If so, is the solution useful in the sense of generating a good policy? And, if the preceding questions are answered in the affirmative, is the algorithm consistent? These questions are unanswered even in the special case of Q-function approximations that are linear in the parameter. The challenge seems paradoxical, given the long history of convex analytic approaches to dynamic programming. The paper begins with a brief survey of linear programming approaches to optimal control, leading to a particular over parameterization that lends itself to applications in reinforcement learning. The main conclusions are summarized as follows: (i) The new class of convex Q-learning algorithms is introduced based on the convex relaxation of the Bellman equation. Convergence is established under general conditions, including a linear function approximation for the Q-function. (ii) A batch implementation appears similar to the famed DQN algorithm (one engine behind AlphaZero). It is shown that in fact the algorithms are very different: while convex Q-learning solves a convex program that approximates the Bellman equation, theory for DQN is no stronger than for Watkins' algorithm with function approximation: (a) it is shown that both seek solutions to the same fixed point equation, and (b) the ODE approximations for the two algorithms coincide, and little is known about the stability of this ODE. These results are obtained for deterministic nonlinear systems with total cost criterion. Many extensions are proposed, including kernel implementation, and extension to MDP models.

Heng-Sheng Chang, Udit Halder, Chia-Hsien Shih et al.

This paper entails application of the energy shaping methodology to control a flexible, elastic Cosserat rod model. Recent interest in such continuum models stems from applications in soft robotics, and from the growing recognition of the role of mechanics and embodiment in biological control strategies: octopuses are often regarded as iconic examples of this interplay. Here, the dynamics of the Cosserat rod, modeling a single octopus arm, are treated as a Hamiltonian system and the internal muscle actuators are modeled as distributed forces and couples. The proposed energy shaping control design procedure involves two steps: (1) a potential energy is designed such that its minimizer is the desired equilibrium configuration; (2) an energy shaping control law is implemented to reach the desired equilibrium. By interpreting the controlled Hamiltonian as a Lyapunov function, asymptotic stability of the equilibrium configuration is deduced. The energy shaping control law is shown to require only the deformations of the equilibrium configuration. A forward-backward algorithm is proposed to compute these deformations in an online iterative manner. The overall control design methodology is implemented and demonstrated in a dynamic simulation environment. Results of several bio-inspired numerical experiments involving the control of octopus arms are reported.

Amirhossein Taghvaei, Prashant G. Mehta

Controlled interacting particle systems such as the ensemble Kalman filter (EnKF) and the feedback particle filter (FPF) are numerical algorithms to approximate the solution of the nonlinear filtering problem in continuous time. The distinguishing feature of these algorithms is that the Bayesian update step is implemented using a feedback control law. It has been noted in the literature that the control law is not unique. This is the main problem addressed in this paper. To obtain a unique control law, the filtering problem is formulated here as an optimal transportation problem. An explicit formula for the (mean-field type) optimal control law is derived in the linear Gaussian setting. Comparisons are made with the control laws for different types of EnKF algorithms described in the literature. Via empirical approximation of the mean-field control law, a finite-$N$ controlled interacting particle algorithm is obtained. For this algorithm, the equations for empirical mean and covariance are derived and shown to be identical to the Kalman filter. This allows strong conclusions on convergence and error properties based on the classical filter stability theory for the Kalman filter. It is shown that, under certain technical conditions, the mean squared error (m.s.e.) converges to zero even with a finite number of particles. A detailed propagation of chaos analysis is carried out for the finite-$N$ algorithm. The analysis is used to prove weak convergence of the empirical distribution as $N\rightarrow\infty$. For a certain simplified filtering problem, analytical comparison of the m.s.e. with the importance sampling-based algorithms is described. The analysis helps explain the favorable scaling properties of the control-based algorithms reported in several numerical studies in recent literature.

Amirhossein Taghvaei, Prashant G. Mehta

This paper presents a methodology and numerical algorithms for constructing accelerated gradient flows on the space of probability distributions. In particular, we extend the recent variational formulation of accelerated gradient methods in (wibisono, et. al. 2016) from vector valued variables to probability distributions. The variational problem is modeled as a mean-field optimal control problem. The maximum principle of optimal control theory is used to derive Hamilton's equations for the optimal gradient flow. The Hamilton's equation are shown to achieve the accelerated form of density transport from any initial probability distribution to a target probability distribution. A quantitative estimate on the asymptotic convergence rate is provided based on a Lyapunov function construction, when the objective functional is displacement convex. Two numerical approximations are presented to implement the Hamilton's equations as a system of $N$ interacting particles. The continuous limit of the Nesterov's algorithm is shown to be a special case with $N=1$. The algorithm is illustrated with numerical examples.

Amirhossein Taghvaei, Prashant G. Mehta

This paper is concerned with the convergence and long-term stability analysis of the feedback particle filter (FPF) algorithm. The FPF is an interacting system of $N$ particles where the interaction is designed such that the empirical distribution of the particles approximates the posterior distribution. It is known that in the mean-field limit ($N=\infty$), the distribution of the particles is equal to the posterior distribution. However little is known about the convergence to the mean-field limit. In this paper, we consider the FPF algorithm for the linear Gaussian setting. In this setting, the algorithm is similar to the ensemble Kalman-Bucy filter algorithm. Although these algorithms have been numerically evaluated and widely used in applications, their convergence and long-term stability analysis remains an active area of research. In this paper, we show that, (i) the mean-field limit is well-defined with a unique strong solution; (ii) the mean-field process is stable with respect to the initial condition; (iii) we provide conditions such that the finite-$N$ system is long term stable and we obtain some mean-squared error estimates that are uniform in time.

Amirhossein Taghvaei, Jin W. Kim, Prashant G. Mehta

This paper is concerned with the problem of representing and learning a linear transformation using a linear neural network. In recent years, there has been a growing interest in the study of such networks in part due to the successes of deep learning. The main question of this body of research and also of this paper pertains to the existence and optimality properties of the critical points of the mean-squared loss function. The primary concern here is the robustness of the critical points with regularization of the loss function. An optimal control model is introduced for this purpose and a learning algorithm (regularized form of backprop) derived for the same using the Hamilton's formulation of optimal control. The formulation is used to provide a complete characterization of the critical points in terms of the solutions of a nonlinear matrix-valued equation, referred to as the characteristic equation. Analytical and numerical tools from bifurcation theory are used to compute the critical points via the solutions of the characteristic equation. The main conclusion is that the critical point diagram can be fundamentally different even with arbitrary small amounts of regularization.