Recurrent Kalman Networks: Factorized Inference in High-Dimensional Deep Feature Spaces

Abstract

In order to integrate uncertainty estimates into deep time-series modelling, Kalman Filters (KFs) (Kalman et al., 1960) have been integrated with deep learning models, however, such approaches typically rely on approximate inference techniques such as variational inference which makes learning more complex and often less scalable due to approximation errors. We propose a new deep approach to Kalman filtering which can be learned directly in an end-to-end manner using backpropagation without additional approximations. Our approach uses a high-dimensional factorized latent state representation for which the Kalman updates simplify to scalar operations and thus avoids hard to backpropagate, computationally heavy and potentially unstable matrix inversions. Moreover, we use locally linear dynamic models to efficiently propagate the latent state to the next time step. The resulting network architecture, which we call Recurrent Kalman Network (RKN), can be used for any time-series data, similar to a LSTM (Hochreiter & Schmidhuber, 1997) but uses an explicit representation of uncertainty. As shown by our experiments, the RKN obtains much more accurate uncertainty estimates than an LSTM or Gated Recurrent Units (GRUs) (Cho et al., 2014) while also showing a slightly improved prediction performance and outperforms various recent generative models on an image imputation task.

Learning Dynamical Systems from Noisy Sensor Measurements using Multiple Shooting

“…Conceptually, training a RNN is not ideal as the recurrent units are to some extent learning again the dynamics of the system. A more elegant solution would be to use a CNN to extract the relevant features and then a probabilistic filter similar to [20]. However, this is beyond the scope of the paper.…”

“…This is counter intuitive as during prediction, there are no measurement available. To circumvent this issue, we follow [20] and use boolean values to distinguish between filtering and prediction phases and the measurement are artificially set to zero during the prediction phase. It is important to note that the RNN that we use for filtering in our method does not need this additional boolean input as the RNN is not used for prediction.…”

“…Another line of work is to replace the latent dynamics learning with a RNN mapping sequences to sequences such as LSTM [18] or GRU [19] and train in a supervised way. [20] introduced Recurrent Kalman networks (RKN), a RNN with gates handling the measurement noise as a Kalman filter. A large state is necessary in order to match the linearity assumptions.…”

Deep Variational Luenberger-type Observer for Stochastic Video Prediction

“…The main contribution of our addressed approach is to increase the robustness of the hidden initial state value, and according to a specific network structure, we can disentangle the representation into two components: static feature and dynamics. Compared with the state-of-the-art video prediction methods such as DDPAE [17], MCnet [40] and DeepRNN [33] as baselines, we evaluate the effectiveness of our model using two datasets, namely the Bouncing Balls dataset [12] and the Pendulum dataset [5]. The specific prediction process and structure can be seen in Figs.…”

Learning needle insertion from sample task executions

“…Becker et al [14] proposed a Kalman filter network (KFN) in the latent space which takes auto-encoded high dimensional input data whereas the filtering process is simplified by avoiding matrix inversion calculation by assuming diagonal covariance matrices. In contrast to KFN, D-LfD benefits from CNN, inherently more powerful feature extractor without any simplification assumption whereas the robot kinematic and calibration data are also fed to the network in latent space.…”