Scalable Gaussian Processes for Integrated and Overlapping Measurements via Augmented State-space Models

Abstract Astronomical measurements are often integrated over finite exposures, which can obscure latent variability on comparable timescales. Correctly accounting for exposure integration with Gaussian processes (GPs) in such scenarios is essential but computationally challenging: once exposure times vary or overlap across measurements, the covariance matrix forfeits any quasi-separability, forcing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> memory and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> runtime costs. Linear Gaussian state-space models (SSMs) are equivalent to GPs and have well-known <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> solutions via the Kalman filter and Rauch–Tung–Striebel smoother. In this work, we extend the GP–SSM equivalence to handle integrated measurements while maintaining scalability by augmenting the SSM with an integral state that resets at exposure start times and is observed at exposure end times. This construction yields exactly the same posterior as a fully integrated GP but in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> time on a CPU and is parallelizable down to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>/</mml:mo> <mml:mi>T</mml:mi> <mml:mo>+</mml:mo> <mml:mi>log</mml:mi> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> on a GPU with T parallel workers. We present State-space Models for O(Linear/log) GPs ( smolgp ), an open-source Python/JAX package offering drop-in compatibility with tinygp while supporting both standard and exposure-aware GP modeling. Since SSMs provide a framework for representing general GP kernels via their series expansion, smolgp also brings scalable performance to many commonly used covariance kernels in astronomy that lack quasi-separability, such as the quasiperiodic kernel. The substantial performance boosts at large N will enable massive multi-instrument cross-comparisons where exposure overlap is ubiquitous and unlocks the potential for analyses with more complex models and/or higher dimensional datasets.