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SOM-VAE: Interpretable Discrete Representation Learning on Time Series
Vincent Fortuin, Matthias Hüser, Francesco Locatello, Heiko Strathmann, Gunnar Rätsch
TL;DR
The paper addresses the difficulty of learning interpretable and temporally smooth representations for high-dimensional time series. It proposes a probabilistic discrete representation framework combining deep generative modeling, a differentiable assignment strategy, and a Markov model, and reports improved clustering, interpretability, and medically relevant downstream representations. The authors identify extensions to probabilistic prediction, more principled nondifferentiability solutions, and learned neighborhood structures as future work.
Problem
Existing time-series representation learning methods can be difficult to interpret and may neglect the data’s temporal structure and smooth behavior.
Method
The framework learns topologically interpretable discrete representations with a deep architecture, a differentiable reconstruction path, and a jointly learned Markov model over SOM states.
Results
The model shows improved clustering performance and more informative, interpretable representations across benchmark, chaotic, and real-world medical time series, including downstream medical prediction targets.
Takeaways & Limitations
The learned discrete representations support interpretable visualization, temporal state modeling, and downstream analysis of complex time-series data.
Takeaways & Limitations
Future work is needed to extend the probabilistic component toward prediction, replace empirically motivated nondifferentiability solutions, and learn neighborhood structures rather than fixing a 2D grid.
Abstract
from arXiv · showhide
High-dimensional time series are common in many domains. Since human cognition is not optimized to work well in high-dimensional spaces, these areas could benefit from interpretable low-dimensional representations. However, most representation learning algorithms for time series data are difficult to interpret. This is due to non-intuitive mappings from data features to salient properties of the representation and non-smoothness over time. To address this problem, we propose a new representation learning framework building on ideas from interpretable discrete dimensionality reduction and deep generative modeling. This framework allows us to learn discrete representations of time series, which give rise to smooth and interpretable embeddings with superior clustering performance. We introduce a new way to overcome the non-differentiability in discrete representation learning and present a gradient-based version of the traditional self-organizing map algorithm that is more performant than the original. Furthermore, to allow for a probabilistic interpretation of our method, we integrate a Markov model in the representation space. This model uncovers the temporal transition structure, improves clustering performance even further and provides additional explanatory insights as well as a natural representation of uncertainty. We evaluate our model in terms of clustering performance and interpretability on static (Fashion-)MNIST data, a time series of linearly interpolated (Fashion-)MNIST images, a chaotic Lorenz attractor system with two macro states, as well as on a challenging real world medical time series application on the eICU data set. Our learned representations compare favorably with competitor methods and facilitate downstream tasks on the real world data.
1 INTRODUCTION
The paper addresses the need for interpretable, temporally smooth representations of high-dimensional time series by proposing a probabilistic framework for discrete, topologically structured embeddings. Its contributions include a deep architecture, differentiable discrete assignment, and evidence of improved clustering and interpretability.
- Motivation: Many time-series representation methods are difficult to interpret and neglect temporal smoothness, motivating lower-dimensional representations with topological structure.The paper frames interpretability as important for decisions based on large, complex, and time-evolving data.
- Background: Self-organizing maps provide lower-dimensional representations with predefined, topologically interpretable structure, such as a visualizable two-dimensional grid.The SOM is introduced as a way to equip discrete representation spaces with neighborhood relationships.
- Contributions: The proposed framework learns topologically interpretable discrete representations in a probabilistic fashion using a novel deep architecture.The framework is evaluated on benchmark, synthetic chaotic, and real-world medical time series.
- Contributions: The paper introduces a framework for interpretable discrete representation learning on time series.This is identified as one of the paper’s main contributions.
- Contributions: A latent probabilistic model improves clustering and interpretability of the learned time-series representations.The probabilistic component is presented as an additional contribution beyond the discrete representation framework.
- Contributions: The model achieves superior clustering performance on benchmark and real-world medical data and facilitates downstream tasks on the medical dataset.The claim is stated at the level of benchmark and medical-data evaluations rather than a single metric.
2 PROBABILISTIC SOM-VAE
The model combines a neural encoder-decoder with a topologically organized discrete embedding space and a jointly learned Markov transition model. Its objective addresses discrete-assignment non-differentiability while encouraging reconstruction, SOM organization, temporal transitions, and smoothness.
- Architecture: Inputs are encoded into a lower-dimensional latent space, assigned probabilistically to a discrete embedding dictionary, and decoded back into the original data space.The embedding assignment distribution is flexible; experiments use a categorical distribution to resemble the original SOM algorithm.
- Topological structure: A two-dimensional SOM connects embeddings through immediate-neighbor relationships, providing a topologically interpretable structure that facilitates visualization.Updating an embedding also updates its immediate horizontal and vertical neighbors.
- Objective: The SOM-VAE objective combines reconstruction, commitment, and SOM losses, with separate terms optimizing model components and the reconstruction loss corresponding to the VAE ELBO.The commitment term keeps encodings close to their assigned SOM nodes, compensating for non-differentiable assignment.
- Optimization: The architecture addresses non-differentiable discrete assignment by decoding directly from the continuous encoding, creating a differentiable path to the encoder.This direct reconstruction term also encourages the continuous encoding to remain informative; later, the discrete reconstruction term dominates clustering.
- Temporal model: A Markov model defines system states as assigned SOM nodes and is trained jointly to model observed transitions and discourage high-probability transitions to distant nodes.The transition loss raises probabilities of observed transitions, while the smoothness loss penalizes likely transitions to far-away nodes.
3 RELATED WORK
Related work spans clustering in autoencoder latent spaces, discrete representation learning, self-organizing maps, and probabilistic modeling for temporal or medical data. The paper positions its approach at the intersection of these areas, especially by applying SOM structure within learned representations for dynamic data.
- Clustering and discrete representations: Prior clustering methods include k-means improvements, clustering in autoencoder latent spaces, and mixtures of autoencoders.The VQ-VAE is the closest related method and emphasizes discrete representation as compression rather than clustering.
- Self-organizing maps: SOMs provide interpretable lower-dimensional discrete structures and have been extended or combined with other models, but not, to the authors’ knowledge, used in latent spaces of variational autoencoders.The two-dimensional case is especially easy to visualize.
- Medical time series: Probabilistic and deep generative models have been applied to medical time series, while prior SOM work in healthcare learned interpretable static rather than dynamic patient representations.The distinction is relevant to prediction of future patient states in intensive care settings.
4 EXPERIMENTS
Experiments on benchmark, chaotic, and medical data show that SOM-VAE learns interpretable discrete representations with strong clustering performance. Its probabilistic extension preserves or improves clustering while revealing temporal and clinically meaningful structure.
- Experimental setup: The evaluation covers MNIST, Fashion-MNIST, interpolated image time series, Lorenz trajectories, and real eICU medical time series.The same general architecture was used across experiments, with the latent probabilistic model included in selected settings.
- Benchmark clustering: SOM-VAE outperforms competing methods on benchmark clustering measured by purity and normalized mutual information.The comparisons include k-means, VQ-VAE, a standard SOM, and gradient-based SOM variants.
- Benchmark clustering: 512 embeddings yield an interpretable two-dimensional representation of the MNIST data manifold, despite lower NMI from the cluster-count penalty.The representation is visualized through images generated from a section of the learned latent space.
- Benchmark clustering: The SOM component and VAE both improve clustering, while the proposed reconstruction-based treatment of discrete assignments outperforms gradient copying and classical SOM optimization.Removing both encoder reconstruction guidance and gradient copying causes learning to fail because the encoder receives no gradient information.
- Probabilistic model: The concurrently learned transition matrix reaches negative log likelihood 0.25 versus 0.24 for the maximum-likelihood estimate, while the probabilistic model does not impair NMI and can improve Fashion-MNIST clustering.These results support fitting the temporal model jointly with representation learning.
- Chaotic time series: SOM-VAE representations are closer to Lorenz attractor assignments than k-means representations, assigning low entropy to most trajectories without basin changes and largely ignoring noise.The model slightly overestimates attractor-basin change probabilities, producing a heavier distribution tail.
- Medical time series: On eICU data, the full probabilistic model performs best on the downstream tasks and forms compact clusters enriched for different future physiology scores.Patient trajectories illustrate movement toward low-score regions before healthy discharge versus high-score regions before death.
5 CONCLUSION
The SOM-VAE recovers topologically interpretable state representations and improves clustering, while its probabilistic component remains a future avenue for enabling predictions. On medical data, the learned representations are more informative for medically relevant prediction targets and may help clinicians understand patient states and trajectories.
- SOM-VAE recovers topologically interpretable state representations on time series and static data while improving standard clustering performance.
- On a challenging medical data set, the model learns more informative representations for medically relevant prediction targets than competitor methods.
- The representations can be visualized interpretablely and could help clinicians understand patients’ health states and trajectories more intuitively.
- Extending the probabilistic component to enable predictions remains an open direction for future work.
A SELF-ORGANIZING MAPS
A self-organizing map approximates a high-dimensional data manifold with a lower-dimensional discrete space whose topology supports interpretable neighborhoods. The paper uses gradient-based minibatch updates, while addressing the difficulty of fitting disjoint manifolds through a variational autoencoder.
- A self-organizing map approximates a high-dimensional continuous data manifold with a lower-dimensional discrete representation that can be visualized as a 2D grid.
- Each data point selects the closest node by Euclidean distance, after which the winner and neighboring embeddings are updated toward that point.
- The neighborhood function defines how strongly nodes update relative to the winner, with rectangular, hexagonal, and Gaussian structures available.
- Gradient-based minibatch training uses larger steps for distant nodes and smaller steps for nearby nodes, and seems to outperform the original fixed-step formulation.
- Because a fully connected SOM cannot represent spatial gaps between disjoint manifolds, the VAE maps data into a compact latent manifold first.
B IMPLEMENTATION DETAILS
The implementation evaluates clustering with purity and normalized mutual information, using the latter to penalize overly large clusterings. Model and competitor hyperparameters are selected or reused according to stated optimization and comparison procedures.
- Hyperparameters are optimized with Robust Bayesian Optimization using mean squared reconstruction error, with loss weights tuned for balanced convergence.
- The proposed and related special-case models use the same hyperparameters, while external competitors use published or package-default settings when applicable.
- Purity measures the accuracy of assigning each cluster its most prominent class label, but can reach 1.0 when every data point forms its own cluster.
- NMI balances mutual information against clustering entropy, thereby regularizing the trade-off between low within-cluster variance and few clusters.
- Both purity and NMI are normalized to values in [0, 1].
D.1 CLUSTERING ON MNIST AND FASHION-MNIST
Experiments varying the number of clusters show that purity increases with cluster count, whereas NMI first increases and then decreases because it penalizes excessive cluster numbers. The best NMI appears between 16 and 36 clusters.
- Purity increases monotonically with the number of clusters because it does not penalize larger cluster counts.
- NMI first increases and then decreases as cluster count grows too large, reflecting its penalty for model misspecification.
- The optimal cluster count according to NMI appears to lie between 16 and 36.
D.2 INTERPRETABLE REPRESENTATIONS OF CHAOTIC TIME SERIES
The study uses a chaotic Lorenz system with two attractor basins to assess whether learned representations preserve interpretable macro-state dynamics. Entropy comparisons over trajectory spans evaluate SOM-VAE against k-means and the system’s state representations.
- Experimental system: The Lorenz system is a coupled ordinary differential-equation system used to generate chaotic trajectories.For a = 10, b = 28, and c = 8/3, it forms a strange attractor with two attractor points.
- Experimental system: The experiment simulated 100 trajectories of 10,000 time steps and trained SOM-VAE and k-means with 64 embeddings or clusters.The system switches chaotically between two attractor basins, whose identities are determined from distances to the attractor points.
- Interpretability measure: Interpretability is defined as similarity between the learned representation and the system’s ground-truth macro-state over time.Because single-time-point representations are insufficient for this measure, the study compares their temporal evolution using entropy.
- Interpretability measure: Trajectory spans of 100 time steps are evaluated using entropies from the real system, ground-truth basin, SOM-VAE, and k-means spaces.The analysis also records whether each span contains a switch between attractor basins.
- Medical-data context: The experiments use dynamic eICU data resampled onto a one-hour grid and labeled with a dynamic APACHE-score variant.Selected variables include MAP, temperature, respiratory rate, HCO3, sodium, potassium, and creatinine, with normal and abnormal ranges.
D.4 DETAILED ANALYSIS OF SOMVAEP R O B PATIENT STATES
The detailed eICU analysis examines whether SOMVAEProb organizes patient states into compact, interpretable latent structures. It focuses on future physiology, mortality risk, and specific physiological abnormalities.
- Future physiology: SOMVAEProb uncovers compact and interpretable latent-space structures with respect to future physiology scores.The analysis examines acute physiology scores in greater detail than the main-text presentation.
- Mortality risk: The analysis evaluates enrichment for future mortality risk as a severity indicator in the ICU.It uses dynamic mortality patterns to characterize patient states on the SOMVAEProb map.
- Patient phenotypes: The analysis explores map regions enriched for particular physiological-abnormality phenotypes.The stated examples concern specific abnormalities rather than only aggregate severity scores.
FULL RESULTS FOR FUTURE ACUTE PHYSIOLOGY SCORES
Supplementary analyses examine how SOMVAEProb map locations relate to future physiology, mortality, and laboratory abnormalities. Additional visualizations show interpretable two-dimensional latent organization for MNIST and Fashion-MNIST.
- Future acute physiology: SOMVAEProb distinguishes future acute-physiology risk profiles assigned to an abnormal map cell and a normal middle-map cell.The two distributions are described as largely disjoint for coordinates [2,0] and [4,3].
- Dynamic mortality risk: Map regions enriched for higher acute physiology scores also exhibit elevated mortality rates over baseline.The left-edge region is described as having significantly greater future mortality risk than the right-edge region.
- Patient-state phenotypes: Distinct laboratory-abnormality phenotypes are enriched in different SOMVAEProb map regions.Low sodium is enriched near the left edge, while high potassium is enriched near the right edge.
- Image-manifold representations: MNIST latent-space generations form an interpretable discrete two-dimensional representation of the higher-dimensional data manifold.The visualization uses 512 learned embeddings.
- Image-manifold representations: Fashion-MNIST latent-space generations likewise form an interpretable discrete two-dimensional representation of the data manifold.The visualization uses 512 learned embeddings.