Deep Self-Organization: Interpretable Discrete Representation Learning on Time Series

Deep Self-Organization: Interpretable Discrete Representation Learning on Time Series

Vincent Fortuin Institute of Machine Learning, ETH Zürich, Universitätsstrasse 6, 8092 Zürich, Switzerland Matthias Hüser Institute of Machine Learning, ETH Zürich, Universitätsstrasse 6, 8092 Zürich, Switzerland Francesco Locatello Heiko Strathmann Gunnar Rätsch Institute of Machine Learning, ETH Zürich, Universitätsstrasse 6, 8092 Zürich, Switzerland
Abstract

Human professionals are often required to make decisions based on complex multivariate time series measurements in an online setting, e.g. in health care. Since human cognition is not optimized to work well in high-dimensional spaces, these decisions benefit from interpretable low-dimensional representations. However, many 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 to couple a variational autoencoder to a discrete latent space and introduce a topological structure through the use of self-organizing maps. This allows us to learn discrete representations of time series, which give rise to smooth and interpretable embeddings with superior clustering performance. Furthermore, to allow for a probabilistic interpretation of our method, we integrate a Markov model in the latent 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 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. In the latter experiment, our representation uncovers meaningful structure in the acute physiological state of a patient.

1 Introduction

Interpretable representation learning of time series is a seminal problem for uncovering the latent structure in complex physical systems, such as chaotic dynamical systems or medical time series from the intensive care unit. Clustering is a classical method for unsupervised representation learning, and is a natural solution for this problem at first sight. However, clustering used in the naïve sense makes misleading i.i.d. assumptions about the data, neglecting its rich temporal structure and smooth behaviour over time. Another concern for interpretability is the lack of topological structure in the discrete embedding space.

Deep neural networks have a very successful tradition in representation learning [5]. In recent years, they have increasingly been combined with generative modeling through the advent of generative adversarial networks (GANs) [12] and variational autoencoders (VAEs) [19]. However, the representations learned by these models can sometimes be cryptic and do not offer the necessary interpretability. This is why a lot of work has been done to improve them in this regard, in GANs [6] as well as VAEs [15, 9].

In areas where humans have to make decisions based on large amounts of data, interpretability is fundamental to ease the human task. When decisions have to be made in a timely manner and rely on observing some external process over time, such as finance or medicine, the need for intuitive interpretations is even stronger. Clinicians, for instance, might need to visualize and manually process the trajectory of a patient in order to decide on their treatment [17]. They would benefit heavily from a representation learning model, in which those patient trajectories become intuitively understandable and relevant health features are salient.

From this point of view, an interesting development in the field of VAEs is the discretization of the latent space, as implemented in the vector quantized VAE (VQ-VAE) [35]. The VQ-VAE encodes a data point into a continuous latent space and then maps it to the closest of prototype latent vectors, from which the decoder has to reconstruct the original input. The prototype vector is then pulled in the direction of the latent encoding. This algorithm leads to a vector quantization of the latent space and yields an interpretable clustering of latent representations.

Clinicians in the intensive care unit (ICU) are already used to classifying patients into a finite set of discrete categories (e.g. using diagnosis codes). Learned discrete health states may thus fit their intuition of a patient’s development. This advantage may be even more pronounced, if one introduces a neighborhood relationship between the latent states, such as in a self-organizing map (SOM) [22]. The discrete latent space of the SOM can then be used to visualize dynamic health trajectories of the patients, since it provides more structure [33]. Lastly, modeling uncertainty is equally important in this scenario, as it yields a distribution over the likely future states of the patient. It is therefore potentially fruitful to combine the ideas of probabilistic modeling and representation learning in an end-to-end model for health trajectories.

In order to demonstrate this idea, we propose a new VQ-VAE-like architecture that is coupled to a SOM in the latent space, train it on synthetic and real world time series and examine the arising representations. To test the basic idea of modeling an evolving discrete latent space, we use benchmark data sets and a synthetic dynamical system with chaotic behavior, before we evaluate the model on real world medical data.

Our main contributions are to

  • Devise a novel framework for discrete representation learning based on the vector quantized autoencoder and self-organizing maps that achieves superior clustering performance on benchmark tasks.

  • Include a latent probabilistic model into the representation learning architecture and show that it further improves clustering and interpretability of the representations.

  • Show superior performance of the full model on benchmark tasks and a challenging real world medical data set.

2 Related Work

From the early inception of the k-means algorithm for clustering [25], there has been much methodological improvement, including methods that perform clustering in the latent space of (variational) autoencoders [2] or use a mixture of autoencoders for the clustering [38, 26]. The method most related to our work is the VQ-VAE [35] which also forms the basis for our architecture. Its authors have put a stronger focus on the discrete representation as a form of compression instead of clustering. Hence, we have made a few changes in adapting their model (see Sec. 3.1). All these methods have in common that they only yield a single number as a cluster assignment and provide no interpretable structure of relationships between clusters.

An algorithm that provides such an interpretable structure is the self-organizing map (SOM) [22]. It maps the data manifold to a lower-dimensional discrete space, which can be easily visualized in the 2D case. It has been extended to model dynamical systems [4] and combined with probabilistic models for time series [31], although without using learned representations. While there are approaches to turn the SOM into a “deeper” model [8], combine it with multi-layer perceptrons [10] or with metric learning [30], it has (to the best of our knowledge) not been proposed to use SOMs in the latent space of (variational) autoencoders or any other form of unsupervised deep learning model.

Interpretable models for clustering and temporal predictions are especially crucial in fields where humans have to take responsibility for the model’s predictions, such as in health care. The prediction of a patient’s future state is an important problem, particularly on the intensive care unit (ICU) [14, 3]. Probabilistic models, such as Gaussian processes, are a popular choice in this domain [7, 32]. Recently, deep generative models have been proposed [16], sometimes even in combination with probabilistic modeling [24]. To the best of our knowledge, SOMs have only been used to learn interpretable static representations of patients [33], but not dynamic ones.

3 Probabilistic SOM-VAE

Our proposed model is a combination of a self-organizing map [22], a vector quantized variational autoencoder [35] and a Markov model. In the following, we review the SOM and VQ-VAE in order to introduce the general concepts and the notation.

A self-organizing map consists of nodes , where every node corresponds to an embedding in the data space and a representation in a lower-dimensional discrete space , where usually . During training on a data set , a winner node is chosen for every point according to . The embedding vector for every node is then updated according to , where is the learning rate and is a neighborhood function between the nodes defined on the representation space . There can be different design choices for .

The VQ-VAE is a variational autoencoder which discretizes its latent space with an objective function that induces a vector quantization. The encoder neural network maps an input to a latent encoding (usually ) which is then assigned to an embedding in the set of embeddings according to . The decoder neural network reconstructs the input from , which can therefore be seen as a discrete representation of the input. The objective function is defined as

(1)

where is the reconstruction of from decoding , is an operator stopping the gradient flow, and is a tuning parameter. The first term in this loss function is the reconstruction error, while the second and third term train the embeddings and encoder, respectively. The authors call the third term “commitment loss” and claim that the optimization is robust to a large range of -values, including , such that the last two terms could potentially be written as one.

The main challenge in optimizing the VQ-VAE and related architectures is the non-differentiability of the discrete cluster assignment step. Due to this, the gradients from the reconstruction loss cannot flow back into the encoder. In order to mitigate this, the authors of the VQ-VAE propose to copy the gradients from to . They acknowledge that this is an ad hoc approximation, but observed that it works well in their experiments [35].

3.1 Self-organizing map variational autoencoder

Figure 1: Schematic overview of our model architecture. Time series from the data space [green] are encoded by a neural network [black] time-point-wise into the latent space. The latent data manifold is approximated with a self-organizing map (SOM) [red]. In order to achieve a discrete representation, every latent data point () is mapped to its closest node in the SOM (). A Markov transition model [blue] is learned to predict the next discrete representation () given the current one (). The discrete representations can then be decoded by another neural network back into the original data space.

By connecting the embeddings in the VQ-VAE model in the form of a SOM, we can induce an interpretable neighborhood relation on them. A schematic overview of our proposed model is depicted in Figure 1. We choose to use a two-dimensional map because it facilitates visualization similar to [33]. We implement it in a way such that any time an embedding at position in the map gets updated, it also updates all the embeddings in its immediate neighborhood, which is defined as for a two-dimensional map.

The loss function for a single input looks like

(2)

where is the reconstruction of from decoding , is the reconstruction of from decoding and and are weighting hyperparameters.

Every term in this function is specifically designed to optimize a different model component. The first term is the reconstruction loss . The first subterm of this is the discrete reconstruction loss as used in [35] (see Eq. 1), which encourages the assigned SOM node to be an informative representation of the input.

Due to our smaller number of embeddings compared to the original VQ-VAE paper [35], the average distance between an encoding and its closest embedding is much larger in our case. The gradient copying (see above) thus ceases to be a feasible approximation, because the true gradients at points in the latent space which are farther apart will likely be very different. In order to still overcome the non-differentiability issue, we propose to add the second reconstruction subterm to , where the reconstruction is decoded directly from the encoding . This adds a fully differentiable credit assignment path from the loss to the encoder and encourages to also be an informative representation of the input, which is a desirable model feature. Most importantly, it works well in practice (see Sec. 4.1). Note that since is continuous and therefore much less constrained than , this term is optimized easily and becomes small early in training. After that, mostly the -term contributes to . One could therefore view the -term as an initial encouragement to place the data encodings at sensible positions in the latent space, after which the actual clustering task dominates the training objective.

The term encourages the encodings and assigned SOM nodes to be close to each other and is defined as . Closeness of encodings and embeddings should be expected to already follow from the term in a fully differentiable architecture. However, due to the non-differentiability in our model, the term has to be explicitly added to the objective in order for the encoder to get gradient information about . The term is very similar to the commitment loss in the VQ-VAE (see Eq. 1), except that we do not split it into two terms with gradient stopping. We observe that in our model it works just as well this way.

The SOM loss encourages the neighbors of the assigned SOM node to also be close to , thus enabling the embeddings to exhibit a self-organizing map property, while stopping the gradients on such that the encoding is not pulled in the direction of the neighbors. It is defined as , where is the set of neighbors in the discrete space as defined above. This term enforces a neighborhood relation between the discrete codes and encourages all SOM nodes to ultimately receive gradient information from the data. The gradient stopping in this term is motivated similarly to the use in the VQ-VAE loss (see Eq. 1 and [35]). We want to optimize the embeddings based on their neighbors, but not the respective encodings, since any single encoding should be as close as possible to its assigned embedding and not receive gradient information from any other embeddings that it is not assigned to.

3.2 Latent Markov Transition Model

Our ultimate goal is to predict the development of time series in an interpretable way. This means that not only the state representations should be interpretable, but so should be the prediction as well. To this end, we use a temporal probabilistic model.

Learning a probabilistic model in a high-dimensional continuous space can be challenging. Thus, we exploit the low-dimensional discrete space induced by our SOM to learn a temporal model. For that, we define a system state as the assigned node in the SOM and then learn a Markov model for the transitions between those states. The model is learned jointly with the SOM-VAE, where the loss function becomes

(3)

with weighting hyperparameters and .

The term encourages the probabilities of actually observed transitions to be high. It is defined as , with being the probability of a transition from state to state in the Markov model.

The term encourages the probabilities for transitions to nodes that are far away from the current data point to be low or respectively the nodes with high transition probabilities to be proximal. It achieves this by taking large values only for transitions to far away nodes that have a high probability under the model. It is defined as . The probabilistic model can inform the evolution of the SOM through this term which encodes our prior belief that transitions in natural data happen smoothly and that future time points will therefore mostly be found in the neighborhood of previous ones. In a setting where the data measurements are noisy, this improves the clustering by acting as a temporal smoother.

4 Experiments

We performed experiments on MNIST handwritten digits [23], Fashion-MNIST images of clothing [37], synthetic time series of linear interpolations of those images, time series from a chaotic dynamical system and real world medical data from the eICU Collaborative Research Database [11]. For model implementation details, we refer to the appendix (Sec. A).

We found that our method achieves a superior clustering performance compared to other methods. We also show that we can learn a temporal probabilistic model concurrently with the clustering, which is on par with the maximum likelihood solution, while improving the clustering performance. Moreover, we can learn interpretable state representations of a chaotic dynamical system and discover meaningful patterns in real medical data.

4.1 Clustering on benchmark data sets

In order to test the clustering component of the SOM-VAE, we performed experiments on MNIST and Fashion-MNIST. We compare our model (including different adjustments to the loss function) against k-means [25] (sklearn-package [29]), the VQ-VAE [35], a standard implementation of a SOM (minisom-package [36]) and our version of a TF-SOM (TensorFlow-SOM), which is basically a SOM-VAE where the encoder and decoder are set to be identity functions.

The results of the experiment in terms of purity and normalized mutual information are shown in Table 1. The SOM-VAE outperforms the other methods w.r.t. the clustering performance measures. It should be noted here that while k-means is a strong baseline, it is not density matching, i.e. the density of cluster centers is not proportional to the density of data points. Hence, the representation of data in a space induced by the k-means clusters can be misleading.

As argued in the appendix (Sec. B), NMI is a more balanced measure for clustering performance than purity. If one uses 512 embeddings in the SOM, one gets a lower NMI due to the penalty term for the number of clusters, but it yields an interpretable two-dimensional representation of the manifolds of MNIST (Fig. 2, Supp. Fig. S1) and Fashion-MNIST (Supp. Fig. S2).

Figure 2: Images generated from a section of the SOM-VAE’s latent space with 512 embeddings trained on MNIST. It yields a discrete two-dimensional representation of the data manifold in the higher-dimensional latent space.

The experiment shows that the SOM in our architecture improves the clustering (SOM-VAE vs. VQ-VAE) and that the VAE does so as well (SOM-VAE vs. TF-SOM). Both parts of the model therefore seem to be beneficial for our task. It also becomes apparent that our reconstruction loss term on works better in practice than the gradient copying trick from the VQ-VAE (SOM-VAE vs. gradcopy), due to the reasons described in Section 3.1. If one removes the reconstruction loss and does not copy the gradients, the encoder network does not receive any gradient information any more and the learning fails completely (no_grads). Another interesting observation is that stochastically optimizing a SOM using Adam [18] seems to discover a more performant solution than the classical SOM algorithm (TF-SOM vs. minisom). Since k-means seems to be the strongest competitor, we are including it as a reference baseline in the following experiments as well.

MNIST Fashion-MNIST
Method Purity NMI Purity NMI
k-means 0.690 0.000 0.541 0.001 0.654 0.001 0.545 0.000
minisom 0.406 0.006 0.342 0.012 0.413 0.006 0.475 0.002
TF-SOM 0.653 0.007 0.519 0.005 0.606 0.006 0.514 0.004
VQ-VAE 0.538 0.067 0.409 0.065 0.611 0.006 0.517 0.002
no_grads* 0.114 0.000 0.001 0.000 0.110 0.009 0.018 0.016
gradcopy* 0.583 0.004 0.436 0.004 0.556 0.008 0.444 0.005
SOM-VAE* 0.731 0.004 0.594 0.004 0.678 0.005 0.590 0.003
Table 1: Performance comparison of our method and some baselines in terms of purity and normalized mutual information on different benchmark data sets. The methods marked with an asterisk are variants of our proposed method. The values are the means of 10 runs and the respective standard errors. Each method was used to fit 16 embeddings/clusters.

4.2 Markov transition model on the discrete representations

In order to test the probabilistic model in our architecture and its effect on the clustering, we generated synthetic time series data sets of (Fashion-)MNIST images being linearly interpolated into each other. Each time series consists of 64 frames, starting with one image from (Fashion-)MNIST and smoothly changing sequentially into four other images over the length of the time course.

After training the model on these data, we constructed the maximum likelihood estimate (MLE) for the Markov model’s transition matrix by fixing all the weights in the SOM-VAE and making another pass over the training set, counting all the observed transitions. This MLE transition matrix reaches a negative log likelihood of , while our transition matrix, which is learned concurrently with the architecture, yields . Our model is therefore on par with the MLE solution.

Comparing these results with the clustering performance on the standard MNIST and Fashion-MNIST test sets, we observe that the performance in terms of NMI is not impaired by the inclusion of the probabilistic model into the architecture (Tab. 2). On the contrary, the probabilistic model even slightly increases the performance on Fashion-MNIST. Note that we are using 64 embeddings in this experiment instead of 16, leading to a higher clustering performance in terms of purity, but a slightly lower performance in terms of NMI compared to Table 1. This shows again that the measure of purity has to be interpreted with care when comparing different experimental set-ups and that therefore the normalized mutual information should be preferred to make quantitative arguments.

This experiment shows that we can indeed fit a valid probabilistic transition model concurrently with the SOM-VAE training, while at the same time not hurting the clustering performance. It also shows that for certain types of data the clustering performance can even be improved by the probabilistic model (see Sec. 3.2).

MNIST Fashion-MNIST
Method Purity NMI Purity NMI
k-means 0.791 0.005 0.537 0.001 0.703 0.002 0.492 0.001
SOM-VAE 0.868 0.003 0.595 0.002 0.739 0.002 0.520 0.002
SOM-VAE-prob 0.858 0.004 0.596 0.001 0.724 0.003 0.525 0.002
Table 2: Performance comparison of the SOM-VAE with and without latent Markov model (SOM-VAE-prob) against k-means in terms of purity and normalized mutual information on different benchmark data sets. The values are the means of 10 runs and the respective standard errors. Each method is used to fit 64 embeddings/clusters.

4.3 Interpretable representations of chaotic time series

In order to assess whether our model can learn an interpretable representation of more realistic chaotic time series, we train it on synthetic trajectories simulated from the famous Lorenz system [27]. The Lorenz system is a good example for this assessment, since it offers two well defined macro-states (given by the attractor basins) which are occluded by some chaotic noise in the form of periodic fluctuations around the attractors. A good interpretable representation should therefore learn to largely ignore the noise and model the changes between attractor basins. For a review of the Lorenz system and details about the simulations and the performance measure, we refer to the appendix (Sec. C.1).

In order to compare the interpretability of the learned representations, we computed entropy distributions over simulated subtrajectories in the real system space, the attractor assignment space and the representation spaces for k-means and our model. The computed entropy distributions over all subtrajectories in the test set are depicted in Figure 3.

\thesubsubfigure Lorenz attractor
\thesubsubfigure Real space
\thesubsubfigure Attractor assigment
\thesubsubfigure SOM-VAE
\thesubsubfigure k-means
\thesubsubfigure Simulated trajectories
Figure 3: Histograms of entropy distributions (entropy on the x-axes) over all Lorenz attractor subtrajectories [a] of 100 time steps length in our test set. Subtrajectories without a change in attractor basin are colored in blue, the ones where a change has taken place in green.

The experiment shows that the SOM-VAE representations (Fig. 3) are much closer in entropy to the ground-truth attractor basin assignments (Fig. 3) than the k-means representations (Fig. 3). For most of the subtrajectories without attractor basin change they assign a very low entropy, effectively ignoring the noise, while the k-means representations partially assign very high entropies to those trajectories. In total, the k-means representations’ entropy distribution is similar to the entropy distribution in the noisy system space (Fig. 3). The representations learned by the SOM-VAE are therefore more interpretable than the k-means representations with regard to this interpretability measure. As could be expected from these figures, the SOM-VAE representation is also superior to the k-means one in terms of purity with respect to the attractor assignment ( vs. ) as well as NMI ( vs. ).

Finally, we use the learned probabilistic model on our SOM-VAE representations to sample new latent system trajectories and compute their entropies. The distribution looks qualitatively similar to the one over real trajectories (Fig. 3), but our model slightly overestimates the attractor basin change probabilities, leading to a heavier tail of the distribution.

4.4 Learning representations of acute physiological states in the ICU

In order to demonstrate interpretable representation learning on a complex real world task, we trained our model on time series measurements of intensive care unit (ICU) patients. We analyze the performance of the resulting clustering w.r.t. the acute patient physiology state in Table 3. For details regarding the data selection and processing, we refer to the appendix (Sec. C.2).

Method score_6 score_12 score_24
k-means 0.0411 0.0007 0.0384 0.0006 0.0366 0.0005
SOM-VAE 0.0407 0.0005 0.0376 0.0004 0.0354 0.0004
SOM-VAE-prob 0.0474 0.0006 0.0444 0.0006 0.0421 0.0005
Table 3: Performance comparison of our method with and without probabilistic model (SOM-VAE-prob and SOM-VAE) against k-means in terms of normalized mutual information on a challenging temporal clustering task on real eICU data. The dynamic endpoints are the maximum of the physiology score within the next 6, 12 or 24 hours (score_6, score_12, score_24). The values are the means of 10 runs and the respective standard errors. Each method is used to fit 64 embeddings/clusters.

Our full model (including the latent Markov model) performs best on the given tasks, i.e. better than k-means and also better than the SOM-VAE without probabilistic model. This could be due to the noisiness of the medical data and the probabilistic model’s smoothing tendency (see Sec. 3.2).

In order to qualitatively assess the interpretability of the probabilistic SOM-VAE, we analyzed the average physiology score per cluster (Fig. 4). Our model exhibits clusters where higher scores are enriched compared to the background level. Moreover, these clusters form compact structures, facilitating interpretability. We do not observe such interpretable structures in the other methods.

As an illustrative example, we show the trajectories of two patients that start in the same cluster. One patient stays in the regions of the map with low average physiology score and eventually gets discharged from the hospital healthily. The other one moves into map regions with high average physiology score and ultimately dies. Such knowledge could be helpful for doctors, who could determine the risk of a patient for certain endpoints from a glance at their trajectory in the SOM-VAE representation.

\thesubsubfigure k-means
\thesubsubfigure VQ-VAE
\thesubsubfigure SOM-VAE-prob
\thesubsubfigure Patient trajectories
Figure 4: Average of a dynamic variant of the APACHE physiology score of all patient time points assigned to a cluster for different methods. A higher score indicates a higher degree of patient abnormality. White squares correspond to unused clusters, i.e. clusters that contain less than 0.1 percent of the data points. Subfigure (d) shows two patient trajectories in the SOM-VAE-prob representation over their respective whole stays in the ICU. The dots mark the ICU admission, the stars the discharge from the ICU (cured or dead).

5 Conclusion

The SOM-VAE provides an improvement to the standard VQ-VAE framework in terms of clustering performance and offers a way to learn discrete two-dimensional representations of the data manifold in concurrence with the reconstruction task. The learned map offers interpretability of states in different data sets and outperforms baseline methods with respect to different clustering performance measures. The probabilistic component of our model can be learned end-to-end and concurrently with the rest of the architecture, while offering predictive performance close to the maximum likelihood solution and further improving the clustering on noisy data. The model can recover interpretable state representations on time series of chaotic dynamical systems.

On a challenging real world medical data set, our model learns more informative representations with respect to medically relevant dynamical endpoints than competitor methods. It uses the probabilistic model component in this setting to improve the clustering. The learned representations can be visualized in an interpretable way and could be helpful for clinicians to understand patients’ health states and trajectories more intuitively. It will be interesting to see in future work whether the probabilistic model can be extended to not just improve the clustering and interpretability of the whole model, but also enable us to make predictions. Promising avenues in that direction could be to increase the complexity by applying a higher order Markov Model, a Hidden Markov Model or a Gaussian Process.

Acknowledgments

FL is supported by the Max Planck/ETH Center for Learning Systems. MH is supported by the Grant No. 205321_176005 “Novel Machine Learning Approaches for Data from the Intensive Care Unit” of the Swiss National Science Foundation (to GR). VF, FL, MH and HS are partially supported by ETH core funding (to GR). We thank Natalia Marciniak for her administrative efforts, Marc Zimmermann for technical support, Gideon Dresdner, Stephanie Hyland, Viktor Gal and Maja Rudolph for helpful discussions; and Ron Swanson for his inspirational attitude.

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Appendix

Appendix A Implementation details

The hyperparameters of our model were optimized using Robust Bayesian Optimization with the packages sacred and labwatch [13] for the parameter handling and RoBo [20] for the optimization, using the mean squared reconstruction error as the optimization criterion. Since our model defines a general framework, some competitor models can be seen as special cases of our model, where certain parts of the loss function are set to zero or parts of the architecture are omitted. We used the same hyperparameters for those models. For external competitor methods, we used the hyperparameters from the respective publications where applicable and otherwise the default parameters from their packages. The models were implemented in TensorFlow [1] and optimized using Adam [18].

Appendix B Clustering performance measures

Given that one of our most interesting tasks at hand is the clustering of data, we need some performance measures to objectively compare the quality of this clustering with other methods. The measures that we decided to use and that have been used extensively in the literature are purity and normalized mutual information (NMI) [28]. We briefly review them in the following.

Let the set of ground truth classes in the data be and the set of clusters that result from the algorithm . The purity is then defined as where is the total number of data points. Intuitively, the purity is the accuracy of the classifier that assigns the most prominent class label in each cluster to all of its respective data points.

While the purity has a very simple interpretation, it also has some shortcomings. One can for instance easily observe that a clustering with , i.e. one cluster for every single data point, will yield a purity of but still probably not be very informative for most tasks. It would therefore be more sensible to have another measure that penalizes the number of clusters. The normalized mutual information is one such measure.

The NMI is defined as where is the mutual information between and and is the Shannon information entropy. While the entropy of the classes is a data-dependent constant, the entropy of the clustering increases with the number of clusters. It can therefore be seen as a penalty term to regularize the trade-off between low intra-cluster variance and a small number of clusters. Both NMI and purity are normalized, i.e. take values in .

Appendix C Experimental methods

c.1 Interpretable representations of chaotic time series

The Lorenz system is the system of coupled ordinary differential equations defined by

with tuning parameters , and . For parameter choices , and , the system shows chaotic behavior by forming a strange attractor [34] with the two attractor points being given by .

We simulated 100 trajectories of 10,000 time steps each from the chaotic system and trained the SOM-VAE as well as k-means on it with 64 clusters/embeddings respectively. The system chaotically switches back and forth between the two attractor basins. By computing the Euclidian distance between the current system state and each of the attractor points , we can identify the current attractor basin at each time point.

In order to assess the interpretability of the learned representations, we have to define an objective measure of interpretability. We define interpretability as the similarity between the representation and the system’s ground truth macro-state. Since representations at single time points are meaningless with respect to this measure, we compare the evolution of representations and system state over time in terms of their entropy.

We divided the simulated trajectories from our test set into spans of 100 time steps each. For every subtrajectory, we computed the entropies of those subtrajectories in the real system space (macro-state and noise), the assigned attractor basin space (noise-free ground-truth macro-state), the SOM-VAE representation and the k-means representation. We also observed for every subtrajectory whether or not a change between attractor basins has taken place. Note that the attractor assignments and representations are discrete, while the real system space is continuous. In order to make the entropies comparable, we discretize the system space into unit hypercubes for the entropy computation. For a representation with assignments at time and starting time of the subtrajectory, the entropies are defined as

(4)

with being the Shannon information entropy of a discrete set.

c.2 Learning representations of acute physiological states in the ICU

All experiments were performed on dynamic data extracted from the eICU Collaborative Research Database [11]. Irregularly sampled time series data were extracted from the raw tables and then resampled to a regular time grid using a combination of forward filling and missing value imputation using global population statistics. We chose a grid interval of one hour to capture the rapid dynamics of patients in the ICU.

Each sample in the time-grid was then labeled using a dynamic variant of the APACHE score [21], which is a proxy for the instantaneous physiological state of a patient in the ICU. Specifically, the variables MAP, Temperature, Respiratory rate, HCO3, Sodium, Potassium, and Creatinine were selected from the score definition, because they could be easily defined for each sample in the eICU time series. The value range of each variable was binned into ranges of normal and abnormal values, in line with the definition of the APACHE score, where a higher score for a variable is obtained for abnormally high or low values. The scores were then summed up, and we define the predictive score as the worst (highest) score in the next hours, for . Patients are thus stratified by their expected pathology in the near future, which corresponds closely to how a physician would perceive the state of a patient. The training set consisted of 7000 unique patient stays, while the test set contained 3600 unique stays.

Figure S1: Images generated from the SOM-VAE’s latent space with 512 embeddings trained on MNIST. It yields an interpretable discrete two-dimensional representation of the data manifold in the higher-dimensional latent space.
Figure S2: Images generated from the SOM-VAE’s latent space with 512 embeddings trained on MNIST. It yields an interpretable discrete two-dimensional representation of the data manifold in the higher-dimensional latent space.
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