Sequential Inference for Gaussian Processes: A Signal Processing Perspective
| Authors | Daniel Waxman et al. |
| Year | 2026 |
| Field | Machine Learning |
| arXiv | 2604.28163 |
| Download | |
| Categories | cs.LG, stat.CO, stat.ML |
Abstract
The proliferation of capable and efficient machine learning (ML) models marks one of the strongest methodological shifts in signal processing (SP) in its nearly 100-year history. ML models support the development of SP systems that represent complex, nonlinear relationships with high predictive accuracy. Adapting these models often requires sequential inference, which differs both theoretically and methodologically from the usual paradigm of ML, where data are often assumed independent and identically distributed. Gaussian processes (GPs) are a flexible yet principled framework for modeling random functions, and they have become increasingly relevant to SP as statistical and ML methods assume a more prominent role. We provide a self-contained, tutorial-style overview of GPs, with a particular focus on recent methodological advances in sequential, incremental, or streaming inference. We introduce these techniques from a signal-processing perspective while bridging them to recent advances in ML. Many of the developments we survey have direct applications to state-space modeling, sequential regression and forecasting, anomaly detection in time series, sequential Bayesian optimization, adaptive and active sensing, and sequential detection and decision-making. By organizing these advances from a signal-processing perspective, we intend to equip practitioners with practical tools and a coherent roadmap for deploying sequential GP models in real-world systems.
Engineering Breakdown
Plain English
This paper bridges signal processing and machine learning by providing a comprehensive tutorial on Gaussian processes (GPs) for sequential inference problems. The authors tackle the core challenge that classical ML assumes independent, identically distributed data, while real signal processing systems must handle temporal sequences where observations depend on history. They present GPs as a principled statistical framework that naturally handles sequential prediction, uncertainty quantification, and nonlinear relationships — capabilities increasingly demanded as ML models power signal processing applications. The paper consolidates theory and methodology into a self-contained guide aimed at practitioners building production SP systems that must make predictions incrementally over time.
Core Technical Contribution
The main contribution is framing Gaussian processes through a signal processing lens and developing sequential inference algorithms that differ fundamentally from standard batch-mode ML training. Rather than treating the dataset as a fixed collection, the authors show how to efficiently update GP predictions as new observations arrive sequentially, maintaining calibrated uncertainty estimates throughout. This connects classical SP concepts (Kalman filtering, state-space models, recursive prediction) with modern GP theory, revealing that many sequential SP algorithms are special cases or variants of GP inference. The novelty lies not in inventing new mathematics but in systematizing how to apply GP inference incrementally — showing which algorithms are optimal, when approximations are necessary, and how to maintain computational efficiency as data streams in.
How It Works
Sequential GP inference operates as follows: you start with a prior distribution over functions (encoded in the GP's covariance kernel), then as each new observation (x_t, y_t) arrives, you perform Bayesian updating to refine the posterior distribution over the underlying function. For small datasets (n < 10,000), exact GP updates use matrix inversion — specifically, conditioning the prior on observed data yields a posterior with closed-form predictive mean and variance at new test points. As data accumulates, the computational cost grows cubically (O(n³) per update) due to matrix operations, so practitioners employ sparse approximations like inducing point methods, which maintain a fixed set of pseudo-observations and limit compute to O(m³) where m << n. The algorithm outputs not just point predictions but full posterior distributions, enabling uncertainty-aware decision-making; the covariance kernel (typically squared-exponential or Matérn) encodes assumptions about function smoothness and is learned from data via marginal likelihood maximization.
Production Impact
For signal processing systems, this work directly improves real-time prediction pipelines: autonomous vehicles, sensor fusion, time-series anomaly detection, and adaptive filtering all benefit from efficient sequential GP updates that quantify uncertainty. Instead of retraining batch models when new data arrives, you incrementally condition the GP posterior in constant time (with sparse approximations), enabling low-latency responses to concept drift and changing environments. The trade-off is memory and compute: sparse GPs with m=100 inducing points cost roughly O(100³) ≈ 1M operations per update (manageable on edge devices), but exact GPs on large streams become impractical without approximation. Adopting this approach means shifting from prediction-only models (neural networks) to probabilistic models that communicate confidence, which changes downstream decision logic — you can now flag uncertain predictions, trigger expensive human review, or adaptively allocate resources based on posterior variance.
Limitations and When Not to Use This
The paper assumes GP priors are well-specified (kernel choice and hyperparameters are correct), but real-world signals often violate these assumptions, leading to miscalibrated uncertainty and model mismatch. Scalability remains a hard constraint: even sparse GPs struggle with streaming data arriving at >1kHz or with very high-dimensional inputs (d >> 100); for such regimes, neural networks may be more practical. The tutorial nature of the paper means it does not address modern challenges like online hyperparameter adaptation, how to detect when the kernel is wrong mid-stream, or how to handle massive non-stationary datasets where the underlying function changes over time. Sequential inference also requires careful numerical stability — Cholesky decompositions can fail or become ill-conditioned as data accumulates, and the paper does not thoroughly address robust algorithms for production systems facing adversarial or corrupted streams.
Research Context
This work sits at the intersection of classical signal processing (Kalman filters, Wiener filtering, recursive least squares) and modern probabilistic machine learning, showing that GPs unify these traditions under Bayesian inference. It builds on decades of GP research (Rasmussen & Williams' foundational textbook, inducing point approximations by Snelson & Ghahramani, sparse GP literature) and connects it to real signal processing challenges that practitioners face daily. The paper opens research directions in online kernel learning, efficient uncertainty propagation in streaming settings, and hybrid methods that combine GPs with deep learning for problems requiring both expressiveness and calibrated confidence. It is relevant to growing interest in uncertainty quantification for ML systems, as regulatory and safety-critical domains (medical diagnosis, autonomous systems, finance) increasingly require not just predictions but reliable confidence intervals.
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