Trajectory EM: Principles and Applications

• Trajectory EM is a framework that extends the Expectation-Maximization algorithm along temporal and spatial trajectories to enable robust parameter inference and error mitigation.
• It integrates geometric and probabilistic methods to address challenges in quantum error mitigation, multi-object tracking, and mixed linear regression with provable convergence properties.
• Applications range from quantum NISQ devices and particle tracking to adaptive prediction and control, demonstrating both scalability and interpretable convergence dynamics.

Trajectory Expectation Maximization (Trajectory EM) encompasses a class of statistical and quantum algorithms in which the Expectation-Maximization (EM) principle is leveraged specifically in the context of parameter inference or error mitigation on systems evolving along temporal or measurement-driven trajectories. These algorithms are characterized by either operating directly on sequential or spatial trajectories (e.g., for track fitting or control), or by utilizing trajectory-level statistics to enable robust parameter estimation and inference. Prominent examples span multi-object tracking and curve fitting in particle physics (Magniette, 2018), trajectory optimization in stochastic control (Mallick et al., 2020), quantum error mitigation on NISQ devices (Donvil et al., 2023), and predictive modeling of object trajectories in dynamic environments (Jin, 2022). A distinct, technically rigorous usage refers to the trajectory-based analysis of EM dynamics in mixture models—most notably in mixed linear regression, where the sequence of iterates can be geometrically characterized as a cycloid (Luo et al., 2024, Luo et al., 7 Nov 2025). This article synthesizes main methodologies, theoretical results, and applications, with particular attention to the formal and empirical properties of trajectory EM across these domains.

1. Trajectory EM in Quantum Error Mitigation

In quantum information, "Trajectory EM" refers to a quantum error mitigation framework developed for noisy intermediate scale quantum (NISQ) systems where full quantum error correction is infeasible. Traditional quantum EM approaches aim to undo the effect of a known noise channel $\mathcal N$ by applying its formal inverse, typically constructed as a quasi-probability-weighted sum of completely positive trace-preserving (CPTP) maps. The Trajectory EM approach realizes this inversion by exploiting the structure of quantum jump trajectories—namely, by unraveling the Lindblad evolution into stochastic pure-state trajectories, then reweighting each trajectory with an "influence martingale" quasi-probability weight to achieve, in aggregate, the effect of the inverse noise map (Donvil et al., 2023).

Given a Lindblad master equation for the system's density matrix $\rho(t)$ under Hamiltonian $H(t)$ and noise rates $\gamma_k(t)$,

$$\frac{d}{dt}\rho(t) = -i[H(t),\rho(t)] + \sum_k \gamma_k(t)\left(L_k \rho(t) L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho(t)\} \right),$$

the inversion is operationalized by engineering auxiliary jump rates $\gamma_k(t)$ such that $\gamma_k(t)+\Gamma_k(t)=m(t) \ge 0$, where $\Gamma_k(t)$ are the physical error rates. The influence martingale along a trajectory with jumps at times $\{t_j\}$ in channels $\{k_j\}$ is

$$\mu(t) = \exp\left(\int_0^t m(s) ds\right) \prod_{j: t_j \le t} \frac{-\Gamma_{k_j}(t_j)}{\gamma_{k_j}(t_j)}.$$

Reweighting trajectory outcomes by $\mu(t)$ and averaging recovers evolution under the formal negative-rate "inverse" Lindbladian. The method is realizable using a single engineered reservoir or via discrete simulation with an ancilla qubit in the style of Lloyd–Viola schemes, thus bypassing the need for explicit decomposition into a full CPTP operator basis for the inverse map. Resource considerations include a sampling overhead scaling as $O(\exp(2 \int_0^T \max(0,\Gamma_{\min}(s)) ds))$ for achieving additive error $\varepsilon$ (Donvil et al., 2023).

2. Trajectory EM in Multi-Track Fitting and Shape Mixtures

In high-dimensional particle tracking applications, the "GEM" algorithm is a Trajectory EM technique that generalizes the EM paradigm to cluster point clouds into parameterized track classes, such as straight lines or circles, each subject to orthogonal Gaussian measurement noise (Magniette, 2018). Such tasks inherently involve inference over latent trajectory assignments.

Let $P = \{P_i\}_{i=1}^n$ be observed points in $\mathbb{R}^Q$, modeled as a mixture of $J$ latent tracks $L_j$ (with mixture weights $\pi_j$ and noise scales $\sigma_j$). The log-likelihood is

$$\mathcal{L}(\Theta) = \sum_{i=1}^n \log\left[\sum_{j=1}^J \pi_j \lambda(d_{ij};\sigma_j)\right], \quad \lambda(d;\sigma) = \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{d^2}{2\sigma^2}\right),$$

where $d_{ij}$ is the orthogonal distance from $P_i$ to track $L_j$. The EM recursion alternates between responsibility assignment (E-step) and updating shape, noise, and mixture weight parameters (M-step), employing specialized geometric regression steps for the shape update. GEM achieves $O(nJ)$ complexity per iteration through the "Refsplit" heuristic, and extends to both circular tracks and mixtures thereof (UFit), enabling joint inference for charged particle and multi-modality tracking scenarios. The accompanying libgem library offers scalable implementation and plug-gable track shape modules (Magniette, 2018).

3. Trajectory EM in Stochastic Optimal Control and Trajectory Optimization

Trajectory-based EM methods have been instrumental in stochastic control, specifically for trajectory optimization in systems modeled as linear–Gaussian POMDPs (Mallick et al., 2020). In this paradigm, the SOC-EM (Stochastic Optimal Control-EM) algorithm recasts optimal control as an inference problem over trajectories of states, controls, and outputs, parameterized via a stochastic (e.g., Gaussian) control policy: $u_t \sim \mathcal{N}(K_t x_t + d_t, \Sigma_t).$ The E-step computes expectations over latent states $x_{0:T}$ given noisy measurements $y_{0:T-1}$ and current policy parameters, typically using Kalman smoothing. The M-step updates $K_t, d_t, \Sigma_t$ via closed-form maximizing the expected complete-data log-likelihood: $$K_t^{(k+1)} = \mathbb{E}[u_t x_t^\top] \big(\mathbb{E}[x_t x_t^\top]\big)^{-1}, \quad d_t^{(k+1)} = \mathbb{E}[u_t] - K_t^{(k+1)} \mathbb{E}[x_t], \quad \Sigma_t^{(k+1)} = \mathbb{E}\left[(u_t - K_t x_t - d_t)(\cdot)^\top\right].$$ Each iteration provably decreases the expected cumulative cost, and the covariance $\Sigma_t$ manages the exploitation–exploration trade-off, shrinking as uncertainty resolves. Empirically, SOC-EM outperforms several policy optimization methods in cost and robustness (Mallick et al., 2020).

4. Trajectory-Based EM Analysis in Mixed Linear Regression

In the context of mixture models—especially two-component mixed linear regression (2MLR)—the "trajectory EM" nomenclature refers to the explicit study of the sequence $\{\theta^t\}_{t=0}^\infty$ of EM iterates in the parameter space, both at the population and sample levels (Luo et al., 2024, Luo et al., 7 Nov 2025). Recent results provide closed-form EM updates for $(\theta_t, \nu_t)$ (regression vector and log odds of mixing weights) in terms of expectations involving correlated Gaussians, or using integral representations over Bessel functions: $$\theta^{t+1} = \mathbb{E}[\tanh((y \langle x, \theta^t \rangle)/\sigma^2 + \nu^t) y x],$$

$$\nu^{t+1} = \mathbb{E}[\tanh((y \langle x, \theta^t \rangle)/\sigma^2 + \nu^t)].$$

In the noiseless (infinite SNR) limit, the EM update trajectory can be described by a cycloid in a two-dimensional invariant subspace: $$\tfrac{\theta^{t+1}}{\|\theta^*\|} = \frac{2}{\pi}[\operatorname{sgn}(\rho^t) \varphi^t \hat{e}_1 + \cos\varphi^t \hat{e}_2],$$ where $\varphi^t$ is a suboptimality angle, and the parametric recurrence

$$\tan \varphi^{t+1} = \tan \varphi^t + \varphi^t (1 + \tan^2 \varphi^t)$$

traces a classical cycloid generated by a rolling circle of radius $R = \|\theta^*\|/\pi$. Global convergence is linear in the early phases and super-linear (quadratic) once the iterate is sufficiently aligned. Non-asymptotic error bounds demonstrate finite-sample EM inherits this geometric structure with error rates scaling as $O(\sqrt{d/n})$ or tighter, subject to concentration properties (Luo et al., 2024, Luo et al., 7 Nov 2025).

5. Bayesian and Variational Trajectory EM for Adaptive Prediction

Trajectory EM is also fundamental in time-series modeling and prediction for moving objects under uncertainty. In vehicular networks and dynamic object prediction, environment-adaptive models such as ESATP employ variational Bayesian EM (VBEM) for fitting Variational Gaussian Mixture Models (VGMMs) to segmented, denoised trajectories (Jin, 2022). Here, the EM principle is extended via mean-field variational inference to jointly infer cluster assignments, mixture weights, means, and precision matrices with fully Bayesian treatment. Iterative E- and M-steps coordinate to maximize an Evidence Lower Bound (ELBO), with the E-step assigning soft responsibilities

$$r_{nk} \propto \exp\{\mathbb{E}_q[\log \pi_k] + \frac{1}{2} \mathbb{E}_q[\log|\Lambda_k|] - \frac{D}{2} \log 2\pi - \frac{1}{2} \mathbb{E}_q[(x_n-\mu_k)^\top \Lambda_k (x_n-\mu_k)]\},$$

and the M-step updating Dirichlet, normal-Wishart, and Wishart parameters. A parameter-adaptive selection layer tunes the number of components and hyperparameters based on validation performance, enabling auto-adaptation to variation in environmental statistics. This approach demonstrably outperforms standard EM-GMM, HMM, and GP predictors on vehicle trajectory data in both RMSE and negative log-likelihood metrics (Jin, 2022).

6. Synthesis: Scope and Theoretical/Practical Consequences

Trajectory EM, in its various incarnations, exploits sequential or geometric structure to achieve efficient, principled parameter inference, error mitigation, or prediction in domains where latent structure is organized along observed or simulated trajectories. Rigorous trajectory-based analyses—especially in the context of mixture models—reveal geometric invariants (e.g., cycloid paths for EM in 2MLR), sharp phase transitions between linear and super-linear convergence regimes, and refined non-asymptotic bounds connecting population- and finite-sample error (Luo et al., 2024, Luo et al., 7 Nov 2025). In quantum error mitigation, Trajectory EM realizes quasi-probabilistic inversion using quantum trajectories and influence martingale reweighting, which is both operationally parsimonious and compatible with NISQ-era constraints (Donvil et al., 2023). In stochastic control and tracking, trajectory-level EM combines classical inference and modern optimization, resulting in robust, scalable solutions with provable monotonic cost reduction, unique optima, and interpretable convergence dynamics (Mallick et al., 2020, Magniette, 2018).

In summary, Trajectory EM designates a mathematically principled, algorithmically versatile toolkit wherein the EM paradigm is applied or analyzed on the level of system trajectories—yielding both practical algorithms for NISQ devices, multi-object tracking, control, and adaptive prediction, and a deeper geometric and probabilistic understanding of EM's convergence and error propagation.