PCA+KMeans Trajectory Selection Strategy

• The paper introduces the (ω, k)-means algorithm that generalizes PCA and k-means by modeling each trajectory cluster as an affine subspace.
• It employs a weighted distance metric and iterative subspace fitting to capture both global structure and local dynamics within trajectory data.
• The strategy enhances data summarization and compression, with applications spanning motion analysis, robotics, financial time series, and image processing.

A PCA+KMeans trajectory selection strategy is an integrated framework for partitioning high-dimensional trajectory or time-series datasets into representative groups, enabling succinct summarization, compression, and selection of prototypical trajectory behaviors. This approach originates from conceptual and algorithmic generalizations of Principal Component Analysis (PCA, Karhunen-Loève transform) and k–means clustering, most systematically formulated in the (ω, k)-means algorithm, where each trajectory cluster is modeled by an affine subspace, not just a single point. The strategy has been widely applied in domains such as motion analysis, robotics, financial time series, image compression, airspace defense, and citation analysis.

1. Algorithmic Foundations: Simultaneous Generalization of PCA and k–means

The core innovation is the (ω, k)-means algorithm, which generalizes both PCA and k-means by representing each cluster as an affine subspace of given dimension $n$, where classical k-means is recovered for $n=0$ and PCA for $k=1$ (Misztal et al., 2011). For a dataset $X \subset \mathbb{R}^N$, each affine subspace is expressed as $$aff(v_0, v_1, \ldots, v_n) = v_0 + lin(v_1, \ldots, v_n)$$, with $v_0$ the center and $(v_1, \ldots, v_n)$ an orthonormal basis.

The weighted distance from a data point $x$ to affine subspace $v$ is:

$$DIST_{(\omega)}(x; v) = \left( \sum_{i=0}^n \omega_i \cdot dist(x; aff(v_0, \ldots, v_i))^2 \right)^{1/2}$$

where weights $\omega$ modulate the contribution of each subspace dimension ($\omega_i \in [0,1]$, $\sum \omega_i = 1$).

The energy of cluster $S$ with subspace $v$ is:

$$E_{(\omega)}(S, v) = \sum_{x \in S} DIST_{(\omega)}(x; v)^2$$

Cluster assignment and subspace fitting are alternated to minimize total energy across $k$ clusters, with each iteration updating subspaces via PCA.

2. Trajectory Selection and Local Linear Approximation

In trajectory selection, this strategy excels by partitioning nonlinear, time-evolving data into clusters that are locally well-approximated by affine subspaces. Each trajectory segment is assigned to a cluster whose subspace captures its inherent directionality and variance. This nuanced grouping enables extraction of representative trajectory segments that summarize dynamics, accommodate intermittently linear motions, and compress information efficiently.

Empirically, for navigation or surveillance scenarios, the global model from PCA may fail to resolve multimodal or locally oriented trajectory patterns, while k-means does not preserve local geometry. (ω, k)-means allows selecting a small set of key trajectory modes by clustering and then describing each segment via its low-dimensional subspace approximation. As a result, segments with different dynamic regimes (e.g., turns, straight motion, or complex maneuvers) are separated and succinctly characterized.

3. Comparisons and Trade-Offs with Classical PCA and k-means

A direct comparison reveals:

Method	Cluster Representation	Captures Directionality	Typical Use Case
k-means	Point (center)	No	Grouping by proximity
PCA	Single affine subspace	Yes (global only)	Global mode extraction
(ω, k)-means	k affine subspaces	Yes (local & global)	Locally linear decomposition

• Effectiveness: (ω, k)-means improves approximation error in settings where data is a union of locally linear structures, surpassing both k-means (which ignores subspace structure) and PCA (which misses multimodality).
• Efficiency: Classical methods are computationally simpler, as (ω, k)-means requires iterative PCA per cluster, but practical convergence is rapid due to monotonic energy decrease.
• Sensitivity: Initialization affects convergence; multiple restarts or advanced seeding (e.g., k-means++) help mitigate local minima.

4. Implementation, Computational Complexity, and Scaling

Implementation entails alternating subspace assignment and fitting. Each iteration:

1. Assigns each trajectory to the best-fitting cluster (using $DIST_{(\omega)}$).
2. Updates cluster subspaces via PCA (eigen-decomposition of the within-cluster covariance matrix).
3. Terminates when overall energy $E_{(\omega)}$ ceases to decrease.

Computational complexity increases with both $k$ and the subspace dimension $n$, as each cluster requires PCA over its assigned data. Nonetheless, dimensionality reduction typically yields sufficient compression, and practical convergence requires modest iterations.

High-dimensional data may require prior preprocessing, e.g., scaling or outlier removal. For large-scale problems, distributed or stochastic variants are applicable.

5. Representative Applications and Generalizations

PCA+KMeans trajectory selection has direct utility across domains:

• Motion Analysis and Robotics: Segmentation of movement into interpretable local linear modes, aiding in motion prediction, anomaly detection, and navigation (Misztal et al., 2011).
• Financial Time Series: Cluster-based summarization of asset trajectories for portfolio selection, with periodic updating to track changes in correlations (Zhan et al., 2021).
• Image and Video Compression: Local subspace modeling improves reconstruction accuracy, outperforming global PCA (Misztal et al., 2011).
• Surveillance and Airspace Protection: Segmenting flight paths into critical regions for risk assessment and region modeling, potentially in concert with Gaussian Process smoothing (Eerland et al., 2016).
• Citation Analysis and Knowledge Diffusion: Feature extraction and clustering of citation trajectories, supporting robust paper impact recommendation schemes (Chakraborty et al., 2023).

Further generalizations include ensemble clustering, vector field or policy-centric grouping, integration with autoencoders, or submodular selection mechanisms to enhance diversity and balance.

6. Advantages, Limitations, and Theoretical Guarantees

Advantages:

• Flexibility to interpolate between k-means and PCA; able to tune local vs global summary.
• Improved data approximation, compression, and segmentation for datasets with local structure.
• Adaptability to non-stationary, multimodal, or heterogeneous regimes.

Limitations:

• Sensitivity to initialization; may require multiple runs or sophisticated seeding strategies.
• Increased computational burden with higher $k$ or subspace dimension—even though each PCA step is efficient, the sheer volume may be significant for very high dimensionality or massive datasets.
• Convergence to local minima inherent to energy landscape complexity.

Theoretical guarantees include monotonic energy reduction per iteration; finite step convergence, especially in energy-based formulations (Misztal et al., 2011).

7. Outlook and Integration with Related Methods

Recent developments integrate PCA+KMeans strategies with:

• Learning-augmented predictors, improving robustness under label noise (Jabari et al., 6 Jan 2024).
• Submodular optimization for coreset selection in autonomous driving (Yang et al., 25 Sep 2024).
• Policy-centric clustering in offline RL, inspiring latent subspace regularization and assignment likelihood objectives (Hu et al., 10 Jun 2025).
• Graph-based representations and GNNs, where clusters define graph edges for relational learning, improving downstream tasks such as dropout prediction (Almeida et al., 9 Aug 2025).

A plausible implication is that the PCA+KMeans paradigm continues to furnish a unifying architecture for trajectory selection and summarization in high-dimensional dynamic datasets, both as a stand-alone tool and as a modular component within more elaborate learning frameworks.