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

Philipp Becker, Harit Pandya, Gregor Gebhardt, Cheng Zhao, James Taylor, Gerhard Neumann

arXiv:1905.07357v1cs.LGstat.ML

TL;DR

Deep time-series models for partial, high-dimensional observations often lack explicit uncertainty estimates, while Kalman-based deep approaches rely on approximations and costly matrix inversions. The RKN uses a learned, factorized latent representation with locally linear transitions for direct end-to-end Kalman inference. It reports improved uncertainty estimates and state-estimation performance against LSTMs and GRUs, and better image-imputation results than several generative models.

  • Problem

    Deep recurrent models lack uncertainty estimates, while Kalman-based deep approaches use inference approximations and matrix inversions that complicate end-to-end learning and scaling.

    Source: §1 · pp. 1–2

  • Method

    The RKN learns high-dimensional latent features with locally linear transitions and factorized beliefs, enabling end-to-end Kalman inference without matrix inversions.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    The RKN provides more accurate uncertainty estimates and slightly improves state-estimation errors over LSTMs and GRUs, and outperforms several generative models on image imputation.

    Source: §1 · p. 2, §4 · p. 8

  • Takeaways & Limitations

    The RKN provides an explicit uncertainty representation for time-series state estimation and can also be applied to generative tasks such as image imputation.

    Source: §1 · p. 2, §4 · p. 8

    The transition noise is assumed diagonal and state-independent, and the factorization neglects other covariances, which the authors note may be crude for many systems.

    Source: §2.2 · p. 4, §2.3 · p. 4

Abstract

from arXiv · show

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.

1. Introduction

The RKN combines learned high-dimensional features, locally linear transitions, and factorized inference to model uncertainty without costly matrix inversions. It addresses limitations of prior deep recurrent and Kalman-based approaches and reports improved state-estimation and image-imputation results.

1. Introduction

  • High-dimensional, partial, noisy observations make state estimation difficult, while uncertainty estimates matter for decisions such as acting now or waiting for more information.Prior deep recurrent models lack uncertainty estimates, while Kalman-based deep approaches rely on approximations and computationally demanding matrix inversions.Source: §1 · pp. 1–2
  • The RKN learns a high-dimensional latent representation with locally linear transitions and factorized beliefs, simplifying Kalman inference and avoiding expensive matrix inversions.It is designed for end-to-end learning by direct optimization rather than a variational lower bound.Source: §1 · p. 2, §1.1 · p. 3
  • The RKN provides more accurate uncertainty estimates than LSTMs and GRUs, slightly improves state-estimation errors, and outperforms several generative models on image imputation.The paper also reports applicability to real-world scenarios.Source: §1 · p. 2

1.1. Related Work

  • Unlike BackpropKF, which estimates observable state features before filtering, the RKN learns the feature space used for inference so factorization can simplify Kalman computations.The feature-space idea is related to kernel methods, but the RKN learns its features through a deep encoder end to end.Source: §1.1 · p. 2, §1 · p. 2
  • Prior approaches differ in support for scalable latent states, state estimation, uncertainty, noisy or missing data, and direct optimization versus lower-bound objectives.The cited comparison identifies matrix inversions as a constraint on latent-state size for several Kalman-based methods; Table 1 summarizes qualitative properties.Source: §1.1 · p. 3, Table 1 · p. 3

2. Factorized Inference in Deep Feature Spaces

The RKN learns a high-dimensional deep feature space and factorized Gaussian belief state so Kalman inference can use locally linear transitions and efficient updates.

2. Factorized Inference in Deep Feature Spaces

  • The RKN learns a high-dimensional deep feature space with independent Gaussian belief factors, enabling efficient Kalman inference for complex systems.The representation is learned end-to-end rather than fixed as in kernel feature spaces.Source: §2 · p. 3

2.1. Latent Observation and State Spaces

  • Each latent state pairs observation-derived units with memory units that can retain information inferred over time, such as velocities.The paper assumes n = 2m, providing one memory unit for each observation unit.Source: §2.1 · p. 3

2.2. The Transition Model

  • Locally linear transitions combine learned constant matrices using state-dependent coefficients, with diagonal state-independent transition noise and stability-conscious initialization.The transition is initialized to copy encoder output so the encoder can learn useful features when observations are available.Source: §2.2 · pp. 3–4

2.3. Factorized Covariance Representation

  • The covariance approximation retains each observation unit’s correlation with its corresponding memory unit while neglecting other covariances.The authors call this potentially crude, but argue the learned state representation can make the factorization work well in practice.Source: §2.3 · p. 4

2.4. Factorized Inference in the Latent Space

  • Covariance factorization reduces storage from n^2 to 3m values per matrix, avoids Kalman-gain matrix inversion, and simplifies covariance propagation.Representing covariance components as vectors also helps preserve symmetry and positive definiteness despite numerical issues.Source: §2.4 · pp. 4–5

2.5. Loss and Training

  • Decoders use Gaussian distributions for low-dimensional observations and Bernoulli distributions for images, with training via truncated backpropagation through time and Adam.The paper defines separate log-likelihood objectives for state estimation and image prediction.Source: §2.5 · p. 5

2.6. The Recurrent Kalman Network

  • The Kalman gain gates observation influence using uncertainty; parameters scale linearly with state size versus quadratically for LSTMs and GRUs, and missing inputs skip updates.For absent inputs, the posterior is set to the prior.Source: §2.6 · p. 5

3. Evaluation and Experiments

Across simulated and real-world evaluations, RKN outperforms recurrent baselines in state-estimation settings and alternatives on image imputation, while performing comparably to DeepVO on KITTI.

3.1. Pendulum

  • RKN outperforms LSTM and GRU across pendulum filtering settings; sparse transitions and factorized covariances do not affect performance.The factorized representation uses fewer parameters and more efficient computations; full covariances were infeasible in higher-dimensional experiments because of memory and computation costs.Source: §3.1 · p. 6, Table 2 · p. 6
  • RKN outperforms all models on pendulum image imputation; only the informed KVAE is comparable, and it is still slightly worse.E2C and SIN fail to capture the dynamics, while the uninformed KVAE fails to identify invalid images.Source: Table 3 · p. 6, §3.1 · p. 6

3.2. Multiple Pendulums

  • RKN clearly outperforms competing methods on multiple pendulums, and its normalized-error histogram fits a unit-variance Gaussian while LSTM errors are multimodal.The experiment varies noise independently across image regions, so different pendulums can be occluded while others remain visible.Source: §3.2 · pp. 6–7

3.3. Quad Link

  • RKN scored −44.470 ± 0.208 (informed) and −44.584 ± 0.236 (uninformed) on quad-link imputation; KVAE scored −52.608 ± 0.602 (smoothing) and −59.0218 ± 0.580 (filtering).The comparison uses the informed KVAE, the only model reported as competitive on the pendulum imputation task.Source: §3.3 · p. 7
  • On quad-link filtering, RKN performs significantly better than LSTM and GRU, which fail to perform well.The evaluation uses a 48 × 48 image of a four-link pendulum, with differing occlusion and both noisy and noise-free settings.Source: Table 5 · p. 7, §3.3 · p. 7

3.5. Pneumatic Brook Robot Arm

  • Figure 3 shows predicted link sine values and uncertainty; during extended observation gaps, predictions become less precise and variance increases.Ground truth is blue, predicted values with two standard deviations are green, and crosses encode visibility from yellow (fully visible) to red (fully occluded).Source: Figure 3 · p. 8

3.4. KITTI Dataset for Visual Odometry

  • RKN performs better than sfmlearner-s, LSTM, and GRU, and comparably to the visual-odometry-specific DeepVO approach.The KITTI comparison uses the common evaluation scheme.Source: §3.4 · p. 7, Table 6 · p. 8

3.5. Pneumatic Brook Robot Arm

  • 4.930 log likelihood for RKN exceeds the LSTM (0.952) and GRU (1.186) baselines on pneumatic-joint prediction.The task predicts joint position two seconds ahead using future control inputs after an initial burn-in phase.Source: §3.5 · p. 8

4. Conclusion

The conclusion presents the RKN as an end-to-end model with high-dimensional latent representations, locally linear transitions, and factorized covariances. The accompanying appendix explains the classical filter assumptions and how factorization simplifies prediction and update computations.

4. Conclusion

  • The RKN outperforms generic LSTMs and GRUs on varied state-estimation tasks while providing reasonable uncertainty estimates, and outperforms generative models on image imputation.The paper also reports applicability to real-world scenarios and end-to-end training.Source: §4 · p. 8

Appendix

  • Classical Kalman filtering assumes zero-mean additive Gaussian noise on transitions and observations, with given covariances; its update weights state and observation estimates according to their uncertainty.If no observation is present, or future states are being predicted, the update is omitted.Source: Appendix · p. 10

A. Simplified Kalman Filter Formulas

A.1. NOTATION

  • The notation subsection defines diagonal-matrix and diagonal-vector shorthand, vector indexing, and elementwise product and division.Source: §A.1 · p. 10

A.2. PREDICTION STEP

  • Prediction leaves the mean computation unsimplified, while covariance propagation computes only diagonal terms and can use elementwise multiplication and sum reduction.Sparse matrix multiplications may be exploited for the mean, depending on state size and bandwidth.Source: §A.2 · pp. 10–11

A.3. UPDATE STEP

  • With the factorized structure, the Kalman gain uses diagonal matrices, and the update simplifies the mean and covariance computations.Source: §A.3 · p. 11

B. Root Mean Square Error Results

The appendix reports prediction-performance experiments using RMSE and identifies Table 7 as the results table.

B. Root Mean Square Error Results

C. Visualization of Imputation Results

The section points to Figure 5 for exemplary imputation results from the Pendulum and Quad Link experiments.

C. Visualization of Imputation Results

D. Network Architectures and Hyper Parameters

The appendix lists the optimizer, gradient-computation, convolutional-layer, and variance-output settings used in the experiments.

D. Network Architectures and Hyper Parameters

D.1. PENDULUM AND MULTIPLE PENDULUM EXPERIMENTS

The pendulum experiments use short image sequences with task-specific observation formats and randomly removed frames for imputation. Their encoders and transition models use different latent dimensions for the single- and multiple-pendulum tasks.

D.1. PENDULUM AND MULTIPLE PENDULUM EXPERIMENTS

  • Single-pendulum observations are 24 × 24 grayscale images, while multiple-pendulum observations are 24 × 24 RGB images.Source: §D.1 · p. 12
  • 1,000 training and 500 test sequences of length 150 were used; imputation randomly removed 50% of images.Filtering experiments added noise according to section E.Source: §D.1 · p. 12
  • Both tasks use convolutional encoders; pendulum has latent observation/state dimensions 15/30 and multiple pendulum 45/90, while both transition models use bandwidth 3 and 15 basis functions.The encoder has two convolutional layers, one fully connected layer and a linear output; its fully connected layer uses 30 units for pendulum and 90 for multiple pendulum.Source: §D.1 · p. 12
  • The image decoders use a 144-unit fully connected layer followed by two transposed convolutions, with one output channel for pendulum and three for multiple pendulum.The listed convolution filters and strides differ between the decoder layers.Source: §D.1 · p. 12

D.2. QUAD LINK

The quad-link experiment uses 48 × 48 grayscale image sequences, with randomly removed frames for imputation. Its encoder and transition model use a 100-dimensional latent observation and a 200-dimensional latent state.

D.2. QUAD LINK

  • Observations are 48 × 48 grayscale images.Source: §D.2 · p. 13
  • 4,000 training and 1,000 test sequences of length 150 were used; imputation randomly removed 50% of images.Filtering with additional noise used noise specified in section D.Source: §D.2 · p. 13
  • The encoder uses two convolutional layers and a 200-unit fully connected layer; the transition model has 100-dimensional latent observations and 200-dimensional latent states.The transition model uses bandwidth 3 and 15 basis functions.Source: §D.2 · p. 13
  • The image decoder starts with a 144-unit fully connected layer and uses two transposed convolutions.The listed decoder layers use 16 and 12 filters, with 5 × 5 and 3 × 3 filters, respectively.Source: §D.2 · p. 13

D.3. KITTI

The KITTI and pneumatic robot arm experiments use separate data splits and feature or observation inputs. Each has a task-specific transition model with a latent state twice the size of its latent observation.

D.3. KITTI

  • KITTI features come from the conv6 layer of the pose network, with sequences 00, 01, 02, 08, and 09 for training and 03–07 and 10 for testing.The pose network is based on Zhou et al. (2017).Source: §D.3 · p. 13
  • The KITTI encoder adds a convolution after the pose network; its transition model uses 50-dimensional latent observations and 100-dimensional latent states.The transition model uses bandwidth 1 and 16 basis functions.Source: §D.3 · p. 13

D.4. PNEUMATIC BROOK ROBOT ARM

  • The pneumatic robot arm data comprise six sequences of 30,000 current and joint-position samples at 100 Hz, with five sequences for training and one for testing.Its encoder uses a 30-unit fully connected layer, and its transition model uses 30-dimensional latent observations and 60-dimensional latent states, bandwidth 3, and 32 basis functions.Source: §D.4 · p. 13

E. Observation Noise generation process

The pendulum noise process generates temporally correlated noise factors for each sequence, then blends each image with uniform random noise according to its factor. Thresholding makes some observations fully noise-free or fully noise.

E. Observation Noise generation process

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