Numerically Robust Fixed-Point Smoothing Without State Augmentation
Authors: Nicholas Krämer
TMLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Experimental | The experiments serve two purposes. To start with, they investigate whether the proposed Cholesky-based implementation (Algorithm 3) of the fixed-point smoother recursion (Algorithm 1) holds its promises about memory, runtime, and numerical robustness. |
| Researcher Affiliation | Academia | Nicholas Krämer EMAIL Technical University of Denmark Kongens Lyngby, Denmark |
| Pseudocode | Yes | Algorithm 1 (Fixed-point smoother). To compute the solution to the fixed-point smoothing problem, assemble p(x K | y1:K) with a Kalman filter and evaluate p(x0 | x K, y1:K) as follows. (To simplify the index-related notation in this algorithm, read y1: 1 = y1:0 = .) ... Algorithm 2 (Covariance-based implementation of Equation 12). ... Algorithm 3 (Cholesky-based implementation of Equation 12). |
| Open Source Code | Yes | Code: https://github.com/pnkraemer/code-numerically-robust-fixedpoint-smoother |
| Open Datasets | Yes | Problem setup We solve a boundary value problem based on an ordinary differential equation. More specifically, we solve the 15th in the collection of test problems by Mazzia and Cash (2015) (Figure 3) |
| Dataset Splits | No | The paper describes generating synthetic data for experiments (e.g., "randomly populate all system matrices in the state-space model with independent samples from N(0, 1/K2)" or "sample an initial condition"). It does not use or specify traditional training/test/validation splits for any pre-existing dataset. |
| Hardware Specification | No | All experiments run on the CPU of a consumer-grade laptop and finish within a few minutes. |
| Software Dependencies | No | Our JAX implementation (Bradbury et al., 2018) of Kalman filters, Rauch Tung Striebel smoothers, and fixed-point smoothers is at https://github.com/pnkraemer/code-numerically-robust-fixedpoint-smoother. The paper mentions JAX but does not specify its version number, nor does it list versions for any other key software libraries or dependencies. |
| Experiment Setup | Yes | For Experiment I, we choose K = 1,000, vary d, set the size of the hidden state to D = 2d... For Experiment II, We choose a twice-integrated Wiener process prior and discretise it on K equispaced points in [ 1, 1]... We choose the initial mean m0|0 = (1, 0, 0) and initial covariance C0|0 = diag(0, 1, 1)... For the case study, Define the Wiener velocity model on K = 10 equispaced points ( t = 1/10)... We implement the fixed-point smoother recursion in Cholesky-based arithmetic and run expectation maximisation for three iterations. We initialise the mean guess by sampling all entries independently from a centred normal distribution with a variance of 100. |