# Ultralow-dimensionality reduction for identifying critical transitions by spatial-temporal PCA

arXiv: https://arxiv.org/abs/2501.12582 · HTML: https://mlsanity.com/paper/2501.12582/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 90 references that are in the map.
- First posted on arXiv 22 Jan 2025.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Temporal Convolutional Networks for Action Segmentation and Detection](https://mlsanity.com/paper/1611.05267/) (2016)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Med-BERT: pre-trained contextualized embeddings on large-scale structured electronic health records for disease prediction](https://mlsanity.com/paper/2005.12833/) (2020)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Learnable latent embeddings for joint behavioral and neural analysis](https://mlsanity.com/paper/2204.00673/) (2022)
- [Linking connectivity, dynamics and computations in low-rank recurrent neural networks](https://mlsanity.com/paper/1711.09672/) (2017)
- [Data-driven discovery of intrinsic dynamics](https://mlsanity.com/paper/2108.05928/) (2021)
- [Graph similarity learning for change-point detection in dynamic networks](https://mlsanity.com/paper/2203.15470/) (2022)
- [Spatiotemporal information conversion machine for time-series prediction](https://mlsanity.com/paper/2107.01353/) (2021)
- [Locally embedded presages of global network bursts](https://mlsanity.com/paper/1703.04176/) (2017)
- [Online Change-Point Detection in High-Dimensional Covariance Structure with Application to Dynamic Networks](https://mlsanity.com/paper/1911.07762/) (2019)

## Built on by (most cited first, 2 of 2)

- [Delayformer: spatiotemporal transformation for predicting high-dimensional dynamics](https://mlsanity.com/paper/2506.11528/) (2025)
- [Noise-robust Contrastive Learning for Critical Transition Detection in Dynamical Systems](https://mlsanity.com/paper/2512.12523/) (2025)

## Abstract (arXiv)

Discovering dominant patterns and exploring dynamic behaviors especially critical state transitions and tipping points in high-dimensional time-series data are challenging tasks in study of real-world complex systems, which demand interpretable data representations to facilitate comprehension of both spatial and temporal information within the original data space. Here, we proposed a general and analytical ultralow-dimensionality reduction method for dynamical systems named spatial-temporal principal component analysis (stPCA) to fully represent the dynamics of a high-dimensional time-series by only a single latent variable without distortion, which transforms high-dimensional spatial information into one-dimensional temporal information based on nonlinear delay-embedding theory. The dynamics of this single variable is analytically solved and theoretically preserves the temporal property of original high-dimensional time-series, thereby accurately and reliably identifying the tipping point before an upcoming critical transition. Its applications to real-world datasets such as individual-specific heterogeneous ICU records demonstrated the effectiveness of stPCA, which quantitatively and robustly provides the early-warning signals of the critical/tipping state on each patient.