# Decomposed Linear Dynamical Systems (dLDS) for learning the latent components of neural dynamics

arXiv: https://arxiv.org/abs/2206.02972 · HTML: https://mlsanity.com/paper/2206.02972/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 75 references that are in the map.
- First posted on arXiv 7 Jun 2022.

## Builds on (most cited first, 15 of 15)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Methods for Interpreting and Understanding Deep Neural Networks](https://mlsanity.com/paper/1706.07979/) (2017)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [Variational Autoencoder with Learned Latent Structure](https://mlsanity.com/paper/2006.10597/) (2020)
- [Recurrent switching linear dynamical systems](https://mlsanity.com/paper/1610.08466/) (2016)
- [Tree-Structured Recurrent Switching Linear Dynamical Systems for Multi-Scale Modeling](https://mlsanity.com/paper/1811.12386/) (2018)
- [Enabling hyperparameter optimization in sequential autoencoders for spiking neural data](https://mlsanity.com/paper/1908.07896/) (2019)
- [Representing Closed Transformation Paths in Encoded Network Latent Space](https://mlsanity.com/paper/1912.02644/) (2019)
- [Hierarchical Coupled Geometry Analysis for Neuronal Structure and Activity Pattern Discovery](https://mlsanity.com/paper/1511.02086/) (2015)
- [Dynamic Filtering of Time-Varying Sparse Signals via l1 Minimization](https://mlsanity.com/paper/1507.06145/) (2015)
- [Time-varying Autoregression with Low Rank Tensors](https://mlsanity.com/paper/1905.08389/) (2019)
- [Variational Sparse Coding with Learned Thresholding](https://mlsanity.com/paper/2205.03665/) (2022)
- [Learning Identity-Preserving Transformations on Data Manifolds](https://mlsanity.com/paper/2106.12096/) (2021)
- [Linear State-Space Model with Time-Varying Dynamics](https://mlsanity.com/paper/1410.0555/) (2014)

## Built on by (most cited first, 16 of 16)

- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data](https://mlsanity.com/paper/2410.04814/) (2024)
- [Modeling Latent Neural Dynamics with Gaussian Process Switching Linear Dynamical Systems](https://mlsanity.com/paper/2408.03330/) (2024)
- [Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systems](https://mlsanity.com/paper/2307.06538/) (2023)
- [Probabilistic Decomposed Linear Dynamical Systems for Robust Discovery of Latent Neural Dynamics](https://mlsanity.com/paper/2408.16862/) (2024)
- [SiBBlInGS: Similarity-driven Building-Block Inference using Graphs across States](https://mlsanity.com/paper/2306.04817/) (2023)
- [Self-Supervised Dynamical System Representations for Physiological Time-Series](https://mlsanity.com/paper/2512.00239/) (2025)
- [LINOCS: Lookahead Inference of Networked Operators for Continuous Stability](https://mlsanity.com/paper/2404.18267/) (2024)
- [Unsupervised discovery of the shared and private geometry in multi-view data](https://mlsanity.com/paper/2408.12091/) (2024)
- [Modeling Neural Activity with Conditionally Linear Dynamical Systems](https://mlsanity.com/paper/2502.18347/) (2025)
- [Accelerating Learned Image Compression Through Modeling Neural Training Dynamics](https://mlsanity.com/paper/2505.18107/) (2025)
- [Spectral learning of Bernoulli linear dynamical systems models](https://mlsanity.com/paper/2303.02060/) (2023)
- [MANGO: Learning Disentangled Image Transformation Manifolds with Grouped Operators](https://mlsanity.com/paper/2409.09542/) (2024)
- [Interpretable time series analysis with Gumbel dynamics](https://mlsanity.com/paper/2509.21578/) (2025)
- [Learning Mixtures of Linear Dynamical Systems via Hybrid Tensor-EM Method](https://mlsanity.com/paper/2510.06091/) (2025)
- [Behavior-dLDS: A decomposed linear dynamical systems model for neural activity partially constrained by behavior](https://mlsanity.com/paper/2603.05612/) (2026)

## Abstract (arXiv)

Learning interpretable representations of neural dynamics at a population level is a crucial first step to understanding how observed neural activity relates to perception and behavior. Models of neural dynamics often focus on either low-dimensional projections of neural activity, or on learning dynamical systems that explicitly relate to the neural state over time. We discuss how these two approaches are interrelated by considering dynamical systems as representative of flows on a low-dimensional manifold. Building on this concept, we propose a new decomposed dynamical system model that represents complex non-stationary and nonlinear dynamics of time series data as a sparse combination of simpler, more interpretable components. Our model is trained through a dictionary learning procedure, where we leverage recent results in tracking sparse vectors over time. The decomposed nature of the dynamics is more expressive than previous switched approaches for a given number of parameters and enables modeling of overlapping and non-stationary dynamics. In both continuous-time and discrete-time instructional examples we demonstrate that our model can well approximate the original system, learn efficient representations, and capture smooth transitions between dynamical modes, focusing on intuitive low-dimensional non-stationary linear and nonlinear systems. Furthermore, we highlight our model's ability to efficiently capture and demix population dynamics generated from multiple independent subnetworks, a task that is computationally impractical for switched models. Finally, we apply our model to neural "full brain" recordings of C. elegans data, illustrating a diversity of dynamics that is obscured when classified into discrete states.