# Dimensionality Collapse: Optimal Measurement Selection for Low-Error Infinite-Horizon Forecasting

arXiv: https://arxiv.org/abs/2303.15407 · HTML: https://mlsanity.com/paper/2303.15407/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 54 references that are in the map.
- First posted on arXiv 27 Mar 2023.

## Builds on (most cited first, 14 of 14)

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- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [Bayesian Neural Ordinary Differential Equations](https://mlsanity.com/paper/2012.07244/) (2020)
- [Sequential Bayesian optimal experimental design via approximate dynamic programming](https://mlsanity.com/paper/1604.08320/) (2016)
- [Neural ODEs as the Deep Limit of ResNets with constant weights](https://mlsanity.com/paper/1906.12183/) (2019)

## Abstract (arXiv)

This work introduces a method to select linear functional measurements of a vector-valued time series optimized for forecasting distant time-horizons. By formulating and solving the problem of sequential linear measurement design as an infinite-horizon problem with the time-averaged trace of the Cram'er-Rao lower bound (CRLB) for forecasting as the cost, the most informative data can be collected irrespective of the eventual forecasting algorithm. By introducing theoretical results regarding measurements under additive noise from natural exponential families, we construct an equivalent problem from which a local dimensionality reduction can be derived. This alternative formulation is based on the future collapse of dimensionality inherent in the limiting behavior of many differential equations and can be directly observed in the low-rank structure of the CRLB for forecasting. Implementations of both an approximate dynamic programming formulation and the proposed alternative are illustrated using an extended Kalman filter for state estimation, with results on simulated systems with limit cycles and chaotic behavior demonstrating a linear improvement in the CRLB as a function of the number of collapsing dimensions of the system.