EigenTrajectory Methods Overview

• EigenTrajectory methods are defined as techniques using eigen-decomposition to extract compact, interpretable trajectory descriptors.
• They are applied to pedestrian forecasting, outlier detection on discrete manifolds, and system optimization via Fisher Information Matrix eigenvalue maximization.
• Experiments show significant error reduction, improved anomaly detection accuracy, and enhanced parameter estimation robustness across varied datasets.

EigenTrajectory refers to a family of methods that leverage spectral or low-rank representations for trajectory modeling, embedding, forecasting, optimization, or outlier detection. The term appears prominently in three technical contexts: (1) low-rank descriptor spaces for multi-modal forecasting of pedestrian trajectories (Bae et al., 2023), (2) spectral flow embeddings for interpretable trajectory outlier detection on discrete manifolds via Hodge Laplacian eigenmodes (Frantzen et al., 2021), and (3) continuous-time optimization of system trajectories using Fisher Information Matrix (FIM) eigenvalue criteria for system identification (Wilson et al., 2017). Each instantiation exploits eigen-decomposition to distill trajectory information into compact, interpretable coordinates suitable for downstream analysis, prediction, or optimization.

1. Low-Rank Descriptor Spaces for Trajectory Forecasting

In the context of social and pedestrian trajectory forecasting, "EigenTrajectory" refers to a data-driven, low-rank descriptor construction via spatio-temporal principal components. The ground-truth sets of observed ($A_n$) and future ($B_n$) trajectories for $N$ agents are stacked into matrices $A \in \mathbb{R}^{L_{obs} \times N}$, $B \in \mathbb{R}^{L_{pred} \times N}$ and decomposed using singular value decomposition:

$$A = U_{obs} \Sigma_{obs} V_{obs}^T, \qquad B = U_{pred} \Sigma_{pred} V_{pred}^T$$

The top-$k$ left singular vectors ($U_{obs,k}$ and $U_{pred,k}$) constitute bases for the $\mathbb{ET}$ space (EigenTrajectory space). Trajectories are encoded by coefficients $\mathbf{c}_{obs,n} = U_{obs,k}^T A_n$, and future predictions are similarly represented. This compact $\mathbb{ET}$ space enables sequence models (e.g., RNNs, GCNs, transformers) to operate on low-dimensional, spatio-temporal descriptors instead of raw Euclidean coordinates (Bae et al., 2023).

Key features:

• Spatio-temporal PCA yields dimensional reduction, capturing high-level behavioral modes.
• Social interactions are encoded natively on $\mathbb{ET}$ coefficients, allowing use of tuplewise attention, graph operations, or aggregation.
• Model architectures require only input/output shape adaptations; forecasting is performed on $\mathbb{ET}$ coefficients, and reconstructed via $U_{pred,k}$.
• The anchor-based refinement: future trajectories are clustered in $\mathbb{ET}$ space (using k-means), with anchor-based corrections ensuring multi-modal coverage.

Experiments on ETH, UCY, SDD, and GCS datasets demonstrate that models augmented with EigenTrajectory descriptors achieve superior ADE, FDE, and collision performance, notably outperforming Bézier/B-spline parametric curve methods (Bae et al., 2023).

2. Spectral Flow Embeddings for Outlier Detection on Simplicial Complexes

A distinct EigenTrajectory method is introduced for outlier detection on discrete or discretized manifolds visualized as simplicial complexes (Frantzen et al., 2021). Here, trajectories are conceptualized as edge-flows: vectors $f \in \mathbb{R}^E$ counting algebraically the traversal of each edge. The Hodge 1-Laplacian,

$L_1 = B_1^T B_1 + B_2 B_2^T$

(where $B_1$ and $B_2$ are boundary incidence matrices for vertices/edges and edges/triangles), encodes topological and geometric information. The embedding of $f$ in spectral space uses the eigendecomposition $L_1 = U \Lambda U^T$, projecting $f$ onto the span of the lowest-$k$ eigenvectors:

$\phi_k(f) = [u_1^T f, \ldots, u_k^T f]^T$

The first $b = \dim \ker(L_1)$ components correspond to harmonic modes (homology cycles — holes in the complex); higher modes capture local gradients and curls.

Key features:

• Embedding is geometry-independent; trajectories routed differently around holes are well separated in harmonic coordinates.
• Enables interpretable statistics: e.g., circulation of trajectory $f$ around the $i$-th hole is $u_i^T f$.
• Compact $\phi_k(f)$ representations are amenable to vector-space outlier detectors such as LOF, Isolation Forest, or OC-SVM (Frantzen et al., 2021).
• Pipeline comprises building the simplicial complex, forming boundary matrices, spectral decomposition, trajectory embedding, and anomaly scoring.

Experiments with synthetic Delaunay triangulations and real-world whale migration data illustrate the method’s effectiveness in clustering typical behaviors and detecting topologically anomalous trajectories.

3. Optimization of Trajectories via Fisher Information Eigenvalue Maximization

The EigenTrajectory framework is also deployed as a trajectory synthesis tool aiming at maximization of the information content for parameter estimation in nonlinear dynamic systems (Wilson et al., 2017). The approach uses the Fisher Information Matrix (FIM),

$$I(\theta) = \int_{t_0}^{t_f} \Gamma_\theta(t)^T R^{-1} \Gamma_\theta(t) \, dt$$

where $\Gamma_\theta(t)$ encapsulates measurement sensitivities, and $R$ is measurement noise covariance. The optimization objective is maximizing the smallest eigenvalue $\lambda_{\min}[I(\theta)]$ (E-optimal design), corresponding to improving the worst-case identifiability of the parameter vector $\theta$. The full cost function incorporates tracking regularization and control effort:

$$J(u(\cdot)) = Q_p / \lambda_{\min}[I(\theta)] + \frac{1}{2} \int_{t_0}^{t_f} [(x(t) - x_d(t))^T Q_\tau (x(t) - x_d(t)) + u(t)^T R_\tau u(t)] dt$$

The method efficiently computes gradient information via adjoint sensitivity equations, followed by projected gradient descent on $u(\cdot)$. Techniques include Armijo line search, backward adjoint integration, and explicit handling of ill-conditioning in the FIM.

Experimental validation on double-pendulum carts demonstrates substantial improvements in parameter error (%%%%37$\mathbf{c}_{obs,n} = U_{obs,k}^T A_n$38%%%%–100$\times$), favorable estimator convexity, and efficiency in measurement design (Wilson et al., 2017).

4. Algorithmic Summaries and Pseudocode

Low-Rank Descriptor Generation ($\mathbb{ET}$ space):

1. Assemble observation/future trajectory matrices $A,B$.
2. Compute SVD, retain top-$k$ components $U_{obs,k},U_{pred,k}$.
3. For each sample $n$: compute $\mathbf{c}_{obs,n} = U_{obs,k}^T A_n$, $\mathbf{c}_{pred,n} = U_{pred,k}^T B_n$.
4. Optionally, cluster future coefficients for anchor-based multi-modal prediction.
5. Feed low-rank coefficients into task models; reconstruct trajectories via inverse transform.

Spectral Embedding via Hodge Laplacian:

1. Build simplicial complex $K$ from domain discretization.
2. Assemble boundary matrices $B_1, B_2$.
3. Form $L_1$ and decompose to get $U_k$.
4. Encode trajectories as flow vectors $f$; project to $\phi_k(f)$.
5. Apply standard vector outlier detection.

FIM-based Trajectory Optimization:

1. Integrate state and sensitivity ODEs under initial $u_0$.
2. Compute $I(\theta)$, eigendecomposition, and cost $J$.
3. Backpropagate adjoint equations for $J$ gradient.
4. Update $u$ via projected step, repeat until convergence.
5. Output optimized trajectory.

5. Empirical Performance and Benchmarking

Method Context	Primary Benchmark	Quantitative Gains
$\mathbb{ET}$ Forecasting (Bae et al., 2023)	ETH/UCY/SDD/GCS	FDE reduction (up to 70% over baselines), ADE/FDE = 0.21/0.34 m
Hodge Flow Embedding (Frantzen et al., 2021)	Synthetic, Whale GPS	Outlier detection accuracy $\gtrsim$97% in synthetic, biologically plausible detection in real data
FIM Synthesis (Wilson et al., 2017)	Double-pendulum Cart	$\lambda_{\min}$ up by $10^3$, estimation error reduced by 10$\times$–100$\times$

6. Limitations and Extensions

Limitations are context-dependent:

• In $\mathbb{ET}$-space forecasting, rank-$k$ truncation may omit fine-grained jitter, affecting performance in dense or highly stochastic environments and slightly increasing collision rates.
• In spectral flow embedding, embedding dimension $k$ controls separation capacity between topological and geometric modes; larger $k$ captures richer local behaviors but increases computation.
• FIM-based optimization does not guarantee global optimality and may require periodic re-bootstrapping when parameter estimates drift.

Planned extensions include adaptive rank selection, joint basis learning, multi-scale descriptors, direct uncertainty propagation in descriptor space, and closed-loop bootstrapping for system identification.

7. Connections and Significance

EigenTrajectory methods provide a principled approach to trajectory abstraction grounded in spectral, topological, or information-theoretic rigor. The $\mathbb{ET}$ low-rank PCA descriptor space addresses key challenges in social forecasting by offering improved clustering of feasible futures and robustness to measurement noise relative to classical computer graphics curves. Hodge Laplacian spectral embeddings offer interpretable topological signatures with scalable, geometry-agnostic anomaly detection. FIM eigenvalue-maximizing trajectories have demonstrable impact in experimental system identification.

These approaches share an emphasis on compact, mathematically grounded encodings, facilitating improved generalization, interpretability, and algorithmic efficiency across prediction, optimization, and anomaly detection domains.