Computation-Aware Kalman Filtering and Smoothing (2405.08971v1)

Abstract: Kalman filtering and smoothing are the foundational mechanisms for efficient inference in Gauss-Markov models. However, their time and memory complexities scale prohibitively with the size of the state space. This is particularly problematic in spatiotemporal regression problems, where the state dimension scales with the number of spatial observations. Existing approximate frameworks leverage low-rank approximations of the covariance matrix. Since they do not model the error introduced by the computational approximation, their predictive uncertainty estimates can be overly optimistic. In this work, we propose a probabilistic numerical method for inference in high-dimensional Gauss-Markov models which mitigates these scaling issues. Our matrix-free iterative algorithm leverages GPU acceleration and crucially enables a tunable trade-off between computational cost and predictive uncertainty. Finally, we demonstrate the scalability of our method on a large-scale climate dataset.

• The paper introduces Computation-Aware Kalman Filters and Smoothers that lower computational costs while preserving accurate uncertainty estimates.
• It employs low-dimensional projection and covariance truncation to mitigate expensive matrix operations and reduce memory requirements.
• Empirical results on large state spaces, including a 230k-dimension dataset, validate the scalability and precision of the proposed methods.

Computation-Aware Kalman Filters for Temporal Data

What is This Research About?

This research introduces new algorithms called Computation-Aware Kalman Filters (CAKFs) and Computation-Aware Kalman Smoothers (CAKSs). These algorithms are designed to handle high-dimensional data in applications where temporal correlations play a critical role, such as climate science and robotics. The primary aim is to reduce computational costs while maintaining accuracy in uncertainty estimates.

Motivation Behind the Study

When dealing with temporal data in machine learning, one of the common approaches is to use State Space Models (SSMs). These models allow us to perform efficient Bayesian inference via filtering and smoothing techniques. The well-known Kalman filter is a prime example. However, as the state dimension grows, the computational cost becomes prohibitive due to:

1. Memory Requirements: Need to store large covariance matrices in memory, requiring quadratic to cubic memory and computational resources.
2. Matrix Inversions: These are computationally expensive and can become a bottleneck.

Key Innovations

Computation-Aware Filtering and Smoothing

The paper proposes two main innovations to address these challenges:

1. Low-Dimensional Projection: The data is projected onto a lower-dimensional subspace, thus reducing the computational cost of matrix operations.
2. Covariance Truncation: The state covariance matrices are truncated to a manageable size, reducing memory requirements while still accounting for approximation errors in uncertainty estimates.

Strong Numerical Results

• The algorithms scale to larger state space dimensions more efficiently than existing methods. For example, the proposed algorithms were applied to a climate dataset with a state dimension of up to 230k, requiring significantly less memory than traditional methods.
• On empirical tests, the algorithms demonstrated a remarkable ability to resolve finer details in spatiotemporal Gaussian process regression tasks.

How It Works

1. Projection-Based Updates: The CAKFs use low-dimensional projections to reduce the cost of matrix multiplications and inversions. This ensures that each update step requires less computational power without compromising accuracy.
2. Matrix-Free Implementation: Instead of storing large matrices, the algorithms use iterative, matrix-free methods that leverage modern parallel hardware like GPUs.
3. Downdate Truncation: By retaining only the most informative parts of the covariance matrices, the algorithm manages to keep the memory footprint small while quantifying the approximation error effectively.

Implications of the Research

Practical Implications

1. Scalable Data Processing: The proposed CAKFs and CAKSs make it feasible to handle high-dimensional temporal data efficiently, impacting fields like climate science, finance, and robotics.
2. Improved Performance on GPUs: These algorithms are designed to exploit the parallelism offered by GPUs, making them suitable for large-scale data processing tasks.

Theoretical Insights

1. Combined Uncertainty Estimates: One of the notable theoretical guarantees is that the uncertainty estimates provided by these algorithms account for both epistemic uncertainty and approximation errors, making them robust for real-world applications.
2. Pointwise Error Bounds: The paper provides rigorous bounds on the prediction errors, ensuring that these approximations do not compromise the integrity of the results.

Future Directions

While the paper presents a significant advancement in handling high-dimensional temporal data, several future directions can be explored:

1. Extension to Non-Linear Models: The current focus is on linear Gaussian models. Extending these techniques to non-linear models could widen their applicability.
2. Real-Time Applications: Further refinement can make these algorithms more suitable for real-time applications in robotics and autonomous systems.
3. Hybrid Methods: Combining CAKFs with other approximate inference techniques could lead to even more efficient algorithms.

Conclusion

This research introduces Computation-Aware Kalman Filters and Smoothers, providing efficient methods to handle high-dimensional temporal data with lower computational costs and accurate uncertainty estimates. The practical and theoretical implications of these algorithms promise significant advancements in machine learning applications involving temporal dynamics.