# Enhancing State Estimation in Robots: A Data-Driven Approach with
Differentiable Ensemble Kalman Filters

Xiao Liu1, Geoffrey Clark1, Joseph Campbell2, Yifan Zhou1, and Heni Ben Amor1
1X. Liu, G. Clark, Y. Zhou, and H. Ben Amor are with SCAI at Arizona State
University, USA {xliu330, gmclark1, yzhou298<EMAIL_ADDRESS>Campbell is
with the RI at Carnegie Mellon University, USA<EMAIL_ADDRESS>
###### Abstract

This paper introduces a novel state estimation framework for robots using
differentiable ensemble Kalman filters (DEnKF). DEnKF is a reformulation of
the traditional ensemble Kalman filter that employs stochastic neural networks
to model the process noise implicitly. Our work is an extension of previous
research on differentiable filters, which has provided a strong foundation for
our modular and end-to-end differentiable framework. This framework enables
each component of the system to function independently, leading to improved
flexibility and versatility in implementation. Through a series of
experiments, we demonstrate the flexibility of this model across a diverse set
of real-world tracking tasks, including visual odometry and robot
manipulation. Moreover, we show that our model effectively handles noisy
observations, is robust in the absence of observations, and outperforms state-
of-the-art differentiable filters in terms of error metrics. Specifically, we
observe a significant improvement of at least 59% in translational error when
using DEnKF with noisy observations. Our results underscore the potential of
DEnKF in advancing state estimation for robotics. Code for DEnKF is available
at https://github.com/ir-lab/DEnKF

##  I INTRODUCTION

In robotics, Recursive Bayesian filters, especially Kalman filters, play a
crucial role in accurately localizing robots in their surroundings [1],
predicting the future movements of human interaction partners [2], tracking
objects over time [3], and ensuring stability during robot locomotion [4].
Typically, the success of these filters depends on having an accurate model of
the dynamics of the system being observed, as well as a model of the
observation process itself. However, modeling complex systems and their noise
profiles can be a challenging task, often requiring additional steps like
statistical modeling or system identification. Despite advancements in this
field, such as particle filters [5], scalability remains an issue when working
with high-dimensional systems.

The limitations of traditional Bayesian filters have inspired the development
of Deep State-Space Models (DSSM) [6, 7, 8]. DSSM leverages deep learning
techniques to learn approximate, nonlinear models of the underlying states and
measurements from recorded data. This approach aims to overcome the need for
explicit modeling of the processes, which can be challenging for complex
dynamical systems. However, incorporating the nonlinear capabilities of neural
networks into recursive filtering may come with additional linearization steps
or limitations [1], which can impact the quality of the inference.

Figure 1: Differentiable Ensemble Kalman Filter (DEnKF), employs an ensemble
state to represent the probability density and integrates a stochastic neural
network to generate state transitions. In order to project high-dimensional
visual inputs to observation space, a sensor model is utilized. By combining
an ensemble Kalman Filter with a learned observation model, the DEnKF is able
to achieve precise posterior state estimations.

In this paper, we introduce a new approach to robot state estimation by
extending prior research on differentiable recursive filters. Specifically, we
propose the Differentiable Ensemble Kalman Filter (DEnKF), which employs high-
dimensional camera images to estimate and correct the state of a robot arm, as
demonstrated in Fig. 1. Our method builds upon the solid foundation
established by prior works on differentiable filters, including the
contributions made in [7, 8, 9]. Our approach addresses the challenges
encountered by differentiable filtering through both theoretical and practical
innovations. One such innovation involves the sampling of states from the
posterior distribution of a neural network, which eliminates the need to
estimate noise parameters for the recursive filter. Another innovation
involves the ensemble formulation of the filtering process, which eliminates
the need for linearization. Notably, unlike many other Deep State-Space Models
(DSSM) in the literature, our approach avoids the use of Recurrent Neural
Networks (RNNs), which have been shown to limit the accuracy of learned models
and may lead to non-Markovian state-spaces [7].

In this paper, we present an end-to-end learning approach for recursive
filtering that simultaneously learns the observation, dynamics, and noise
characteristics of a robotic system. The key contributions of our work can be
summarized as follows:

  * •

A stochastic state transition model that uses samples from the posterior of a
neural network to implicitly model the process noise, avoiding the need for a
parametric representation of the posterior.

  * •

An ensemble formulation that allows for the efficient inference of both linear
and nonlinear systems, without the need for an explicit covariance matrix,
making it suitable for high-dimensional inputs and noisy observations.

  * •

Empirical evaluations for the autonomous driving task show DEnKF effectively
reduce the translational and rotational errors compared to state-of-the-art
methods, reducing errors by up to 59% and 36% when dealing with noisy
observations, and handling missing observation scenarios with improved error
reductions by 2-fold and 3-fold.

##  II Related work

Kalman filters (KFs) are well-studied and widely-used state-space models with
many applications in robotics [1]. KFs are designed for systems with linear
process and observation models and assume normally distributed noise. To
overcome some of these limitations and extend the inference capabilities to
nonlinear systems, several variants have been proposed, e.g., the Extended
Kalman Filter (EKF) [10] and the Unscented Kalman Filter (UKF) [11]. Still,
even the EKF and UKF face theoretical and computational challenges when
dealing with high-dimensional observations. Among the many reasons for this
limitation is the need for explicitly calculating an error covariance during
the filtering process.

Differentiable filters: Differentiable filters (DFs) aim to adapt the
recursive filtering techniques to handle high-dimensional inputs. For
instance, BackpropKF [12] proposed a way to train Kalman Filters as recurrent
neural networks using backpropagation with the integration of feed-forward
networks and convolutional neural networks. Similarly, Differentiable
algorithm networks [13] introduced neural network components that encode
differentiable robotic algorithms. This methodology is similar to that of
Differentiable Particle Filters (DPFs)[9, 14, 15], which employ algorithmic
priors to increase learning efficiency. Variations of DPFs were explored in
[16] and [17] using adversarial methods for posterior estimation or partial
ground truth particles for semi-supervised learning. The training of DFs and
modeling of uncertainty along with noise profiles were analyzed in [8]. The
authors implemented the components of the DFs as multi-layer perceptrons and
enveloped the sub-modules in an RNN layer. The DFs in [8, 18] were tested on
real-world tasks, indicating that end-to-end learning is crucial for learning
accurate noise models. Similarly, [19] developed a self-supervised visual-
inertial odometry model using the differentiable Extended Kalman Filter (dEKF)
based on the work in [8]. However, as noted in [7], RNNs can often be a
limiting factor in learning accurate models of the system dynamics. The
current lack of DFs that handle missing observations is attributed to the use
of RNN frameworks to model system dynamics.

Ensemble Kalman Filters: A modern variant of KFs that is particularly
successful on high-dimensional, nonlinear tasks is Ensemble Kalman Filters
(EnKFs) [20]. EnKFs have been shown to enable accurate estimation of state-
space dynamics in data assimilation tasks [21] without linearity assumptions.
They have found popularity in modeling and forecasting complex weather
phenomena that may include millions of state dimensions [22]. Rather than
assuming a certain parametric form of the underlying distribution, EnKFs
approximate the posterior distribution through an ensemble (or collection) of
state vectors. They are computationally efficient since they do not require
the explicit calculation (and inversion) of error covariance matrices. In
addition, EnKFs do not require explicit parametric characterizations of the
process and observation noise. Instead, it only requires the ability to
generate samples from the underlying distributions.

In this paper, we leverage this key insight in order to create a theoretical
and practical connection between EnKFs and stochastic Bayesian neural networks
(SNNs) [23, 24]. As stated in [25], SNNs can model two types of uncertainty:
1) aleatoric uncertainty, which arises from inherent stochasticities of a
system, i.e., process noise and observation noise; 2) epistemic uncertainty,
which is caused by a lack of sufficient data to determine the underlying
system uniquely. The integration between EnKFs and SNNs results in a
probabilistic filtering process that leverages the advantages of modern neural
network techniques. First attempts at differentiable EnKFs were proposed in
[26] which searches for optimal parameters utilizing gradient information from
EnKFs. This differs substantially from our approach, which is a fully
differentiable end-to-end framework. Note that EnKF is theoretically related
to Particle Filters (PFs) – both are Monte-Carlo filtering techniques based on
similar principles. However, in contrast to PFs, EnKFs provide equal weight to
each ensemble member thereby eschewing the well-known sample degeneracy
problem. In addition, EnKFs have also been shown to efficiently model complex
phenomena using relatively small ensemble sizes [27]. A study of the
approximation error of these filters [28] also indicates that with increasing
size of state dimensions, EnKFs show a much slower rate of degradation than
the PFs.

##  III Differentiable EnKF

Recursive Bayesian filtering addresses the general challenge of estimating the
state ${\bf x}_{t}$ of a discrete-time dynamical system given a sequence of
noisy observations ${\bf y}_{1:t}$. The posterior distribution of the state
can be represented as:

$\displaystyle p({\bf x}_{t}|{\bf y}_{1:t},{\bf x}_{1:t-1})$
$\displaystyle\propto p({\bf y}_{t}|{\bf x}_{t})\ p({\bf x}_{t}|{\bf
y}_{1:t-1},{\bf x}_{1:t-1}).$ (1)

Let $\text{bel}({\bf x}_{t})=p({\bf x}_{t}|{\bf y}_{1:t},{\bf x}_{1:t-1})$,
applying the Markov property, i.e., the assumption that the next state is
dependent only upon the current state, yields:

$\displaystyle\text{bel}({\bf x}_{t})=\underbrace{p({\bf y}_{t}|{\bf
x}_{t})}_{\text{observation model}}\prod_{t=1}^{t}\overbrace{p({\bf
x}_{t}|{\bf x}_{t-1})}^{\text{state transition model}}\text{bel}({\bf
x}_{t-1}),$ (2)

where $p({\bf y}_{t}|{\bf x}_{t})$ is the observation model and $p({\bf
x}_{t}|{\bf x}_{t-1}$) is the transition model. The transition model describes
the laws that govern the evolution of the system state. By contrast, the
observation model identifies the relationship between the hidden, internal
state of the system and observed, noisy measurements. An alternative approach
of KFs and its variants is to leverage modern deep learning techniques in
order to extract complex, nonlinear transition and observation models.
Starting with the state transition model in linear KFs:

$\displaystyle{\bf x}_{t}={\bf A}{\bf x}_{t-1}+{\bf q}_{t}\quad{\bf
q}_{t}\thicksim\mathcal{N}(0,{\bf Q}_{t}).$ (3)

the work in [8] replaces the transition matrix ${\bf A}$ and the process noise
${\bf Q}_{t}$ with trained neural networks $f_{\boldsymbol{\theta}}$ and
$q_{\boldsymbol{\psi}}$ respectively:

$\displaystyle{\bf x}_{t}=f_{\boldsymbol{\theta}}({\bf
x}_{t-1})+\mathcal{N}\left(0,q_{\boldsymbol{\psi}}({\bf{x}}_{t-1})\right),$
(4)

where ${\boldsymbol{\theta}}$ and ${\boldsymbol{\psi}}$ denote the neural
network weights. Note, that the network $q_{\boldsymbol{\psi}}(\cdot)$
produces the entries of the covariance matrix ${\bf Q}_{t}$ representing a
Gaussian distribution. As shown in Eq. 4, the state estimate ${\bf x}_{t}$ is
calculated by generating a sample from a normal distribution with covariance
${\bf Q}_{t}$ which is then added to the neural network prediction
$f_{\boldsymbol{\theta}}({\bf x}_{t-1})$. Hence, the process model and the
process noise are calculated using two separate neural networks which may not
be producing outputs consistent with each other.

###  III-A Stochastic Neural Models of Dynamics

In this paper, we avoid this separation by using recent insights in stochastic
neural networks (SNNs) [23]. More specifically, the work in [29] has
established a theoretical link between the Dropout training algorithm and
Bayesian inference in deep Gaussian processes. Accordingly, after training a
neural network with Dropout, it is possible to generate empirical samples from
the predictive posterior via _stochastic forward passes_. Hence, for the
purposes of filtering, we can implicitly model the process noise by sampling
state from a neural network trained on the transition dynamics, i.e., ${\bf
x}_{t}\thicksim f_{\boldsymbol{\theta}}({\bf x}_{t-1})$. In contrast to
previous approaches, the transition network $f_{\boldsymbol{\theta}}(\cdot)$
models the system dynamics, as well as the inherent noise model in a
consistent fashion without imposing diagonality.

###  III-B Nonlinear Filtering with Differentiable Ensembles

Introducing non-linearities through neural network realizations of the
transition and observation function invalidates the linearity assumptions that
are the backbone of many recursive Bayesian filters. To overcome this
challenge, we embed our methodology within an EnKF framework. Throughout the
filtering process, each ensemble member is propagated forward in time to yield
a new approximate posterior distribution. EnKF does not require an explicit
representation of the process and observation noise – instead we only need to
be able to sample from the noise distribution. We formulate DEnKF as an
extension of the EnKF while keeping the core algorithmic steps intact. In
particular, we use an initial ensemble of $E$ members to represent the initial
state distribution ${\bf X}_{0}=[{\bf x}^{1}_{0},\dots,{\bf x}^{E}_{0}]$,
$E\in\mathbb{Z}^{+}$.

Prediction Step: We leverage the stochastic forward passes from a trained
state transition model to update each ensemble member:

$\displaystyle{\bf x}^{i}_{t|t-1}$ $\displaystyle\thicksim
f_{\boldsymbol{\theta}}({\bf x}^{i}_{t|t-1}|{\bf x}^{i}_{t-1|t-1}),\ \forall
i\in E.$ (5)

Matrix ${\bf X}_{t|t-1}=[{\bf x}^{1}_{t|t-1},\cdots,{\bf x}^{E}_{t|t-1}]$
holds the updated ensemble members which are propagated one step forward
through the state space. Note that sampling from the transition model
$f_{\boldsymbol{\theta}}(\cdot)$ (using the SNN methodology described above)
implicitly introduces a process noise.

Update Step: Given the updated ensemble members ${\bf X}_{t|t-1}$, a nonlinear
observation model $h_{\boldsymbol{\psi}}(\cdot)$ is applied to transform the
ensemble members from the state space to observation space. Following our main
rationale, the observation model is realized via a neural network with weights
$\boldsymbol{\psi}$. Accordingly, the update equations for the EnKF become:

$\displaystyle{\bf H}_{t}{\bf X}_{t|t-1}$
$\displaystyle=\left[h_{\boldsymbol{\psi}}({\bf
x}^{1}_{t|t-1}),\cdots,h_{\boldsymbol{\psi}}({\bf x}^{E}_{t|t-1})\right],$ (6)
$\displaystyle{\bf H}_{t}{\bf A}_{t}$ $\displaystyle={\bf H}_{t}{\bf
X}_{t|t-1}$ (7)
$\displaystyle-\left[\frac{1}{E}\sum_{i=1}^{E}h_{\boldsymbol{\psi}}({\bf
x}^{i}_{t|t-1}),\cdots,\frac{1}{E}\sum_{i=1}^{E}h_{\boldsymbol{\psi}}({\bf
x}^{i}_{t|t-1})\right].$

${\bf H}_{t}{\bf X}_{t|t-1}$ is the predicted observation, and ${\bf
H}_{t}{\bf A}_{t}$ is the sample mean of the predicted observation at $t$.
EnKF treats observations as random variables. Hence, the ensemble can
incorporate a measurement perturbed by a small stochastic noise thereby
accurately reflecting the error covariance of the best state estimate [20]. In
our differentiable version of the EnKF, we also incorporate a sensor model
which can learn projections between a latent space and higher-dimensional
observations spaces, i.e. images. To this end, we leverage the methodology
from Sec. III-A to train a stochastic sensor model
$s_{\boldsymbol{\xi}}(\cdot)$:

$\displaystyle\tilde{{\bf y}}^{i}_{t}$ $\displaystyle\thicksim
s_{\boldsymbol{\xi}}(\tilde{{\bf y}}^{i}_{t}|{\bf y}_{t}),\ \forall i\in E.$
(8)

where ${\bf y}_{t}$ represents the noisy observation. Sampling yields
observations $\tilde{{\bf Y}}_{t}=[\tilde{{\bf y}}^{1}_{t},\cdots,\tilde{{\bf
y}}^{E}_{t}]$ and sample mean $\tilde{{\bf
y}}_{t}=\frac{1}{E}\sum_{i=1}^{i}\tilde{{\bf y}}^{i}_{t}$. The innovation
covariance ${\bf S}_{t}$ can then be calculated as:

$\displaystyle{\bf S}_{t}$ $\displaystyle=\frac{1}{E-1}({\bf H}_{t}{\bf
A}_{t})({\bf H}_{t}{\bf A}_{t})^{T}+r_{\boldsymbol{\zeta}}(\tilde{{\bf
y}_{t}}).$ (9)

where $r_{\boldsymbol{\zeta}}(\cdot)$ is the measurement noise model
implemented using MLP. We use the same way to model the observation noise as
in [8], $r_{\boldsymbol{\zeta}}(\cdot)$ takes an learned observation
$\tilde{{\bf y}_{t}}$ in time $t$ and provides stochastic noise in the
observation space by constructing the diagonal of the noise covariance matrix.
The final estimate of the ensemble ${\bf X}_{t|t}$ can be obtained by
performing the measurement update step:

$\displaystyle\begin{split}{\bf A}_{t}&={\bf
X}_{t|t-1}-\frac{1}{E}\sum_{i=1}^{E}{\bf x}^{i}_{t|t-1},\end{split}$ (10)
$\displaystyle\begin{split}{\bf K}_{t}&=\frac{1}{E-1}{\bf A}_{t}({\bf
H}_{t}{\bf A}_{t})^{T}{\bf S}_{t}^{-1},\end{split}$ (11)
$\displaystyle\begin{split}{\bf X}_{t|t}&={\bf X}_{t|t-1}+{\bf
K}_{t}(\tilde{{\bf Y}}_{t}-{\bf H}_{t}{\bf X}_{t|t-1}),\end{split}$ (12)

where ${\bf K}_{t}$ is the Kalman gain. In inference, the ensemble mean
${\bf\bar{x}}_{t|t}=\frac{1}{E}\sum_{i=1}^{E}{\bf x}^{i}_{t|t}$ is used as the
updated state. The neural network structures for all learnable modules are
described in Table I. Furthermore, we highlight couple of the theoretical
properties (in Appendix) of EnKF and its relations to DEnKF.

TABLE I: Differentiable EnKF learnable sub-modules.

$f_{\boldsymbol{\theta}}$: | 2$\times$SNN(32, ReLu), 2$\times$SNN(64, ReLu), 1$\times$SNN(S, -)
---|---
$h_{\boldsymbol{\psi}}$: | 2$\times$fc(32, Relu), 2$\times$fc(64, ReLu), 1$\times$ fc(O, -)
$r_{\boldsymbol{\zeta}}$: | 2$\times$fc(16, ReLu), 1$\times$fc(O, -)
$s_{\boldsymbol{\xi}}$: | conv(7$\times$7, 64, stride 2, ReLu), conv(3$\times$3, 32, stride 2, ReLu),
conv(3$\times$3, 16, stride 2, ReLu), flatten(), 2$\times$SNN(64, ReLu),
2$\times$SNN(32, ReLu), 1$\times$SNN(O, -)
fc: fully connected, conv: convolution, S, O: state and observation dimension.

##  IV Experiments

We evaluate the DEnKF framework on two common robotics tasks: a) a visual
odometry task for autonomous driving and b) a robot manipulation task in both
simulation and real-world. We compare our results to a number of state-of-the-
art differential filtering methods [8, 9, 12].

Training: DEnKF contains four sub-modules: a state transition model, an
observation model, an observation noise model, and a sensor model. The entire
framework is trained in an end-to-end manner via a mean squared error (MSE)
loss between the ground truth state $\hat{{\bf x}}_{t|t}$ and the estimated
state ${\bf\bar{x}}_{t|t}$ at every timestep. We also supervise the
intermediate modules via loss gradients
$\mathcal{L}_{f_{\boldsymbol{\theta}}}$ and
$\mathcal{L}_{s_{\boldsymbol{\xi}}}$. Given ground truth at time $t$, we apply
the MSE loss gradient calculated between $\hat{{\bf x}}_{t|t}$ and the output
of the state transition model to $f_{\boldsymbol{\theta}}$ as in Eq. 13. We
apply the intermediate loss gradients computed based on the ground truth
observation $\hat{{\bf y}_{t}}$ and the output of the stochastic sensor model
$\tilde{{\bf y}}_{t}$:

$\displaystyle\mathcal{L}_{f_{\boldsymbol{\theta}}}=\|{\bf\bar{x}}_{t|t-1}-\hat{{\bf
x}}_{t|t}\|_{2}^{2},\ \ \mathcal{L}_{s_{\boldsymbol{\xi}}}=\|\tilde{{\bf
y}_{t}}-\hat{{\bf y}_{t}}\|_{2}^{2}.$ (13)

All models in the experiments were trained for 50 epochs with batch size 64,
and a learning rate of $\eta=10^{-5}$. We chose the model with the best
performance on a validation set for testing. The ensemble size of the DEnKF
was set to 32 ensemble members.

###  IV-A Visual Odometry Task

In this experiment, we investigate performance on the popular KITTI Visual
Odometry dataset [30]. Following the same evaluation procedure as our
baselines [8, 9, 12], we define the state of the moving vehicle as a
5-dimensional vector ${\bf x}=[x,y,\theta,v,\dot{\theta}]^{T}$, including the
position and orientation of the vehicle, and the linear and angular velocity
w.r.t. the current heading direction $\theta$. The raw observation $\bf y$
corresponds to the RGB camera image of the current frame and a difference
image between the current frame and the previous frame, where ${\bf
y}\in\mathbb{R}^{150\times 50\times 6}$. The learned observation $\tilde{{\bf
y}}$ is defined as $\tilde{{\bf y}}=[v,\dot{\theta}]^{T}$, since only the
relative changes of position and orientation can be captured between two
frames.

Data: The KITTI Visual Odometry dataset consists of 11 trajectories with
ground truth pose (translation and rotation matrices) of a vehicle driving in
urban areas with a data collection rate around 10Hz. To facilitate learning,
we standardize the data on every dimension to have a 0 mean and a standard
deviation of 1 during training.

TABLE II: Result evaluations on KITTI Visual Odometry task measured in RMSE
and MAE. m/m and def/m denote the translational error and the rotational
error. Results for dEKF, dPF, and dPF-M-lrn are reproduced for detailed
comparisons.

Method | RMSE | MAE | Test 100 | Test 100/200/400/800 | Wall clock time (s)
---|---|---|---|---|---
m/m | deg/m | m/m | deg/m
dEKF [8] | 2.9366$\pm$0.006 | 1.9738$\pm$0.027 | 0.2646$\pm$0.004 | 0.1386$\pm$0.002 | 0.3159$\pm$0.002 | 0.0923$\pm$0.005 | 0.0463$\pm$0.004
DPF [9] | 2.2575$\pm$0.027 | 1.4671$\pm$0.014 | 0.1344$\pm$0.002 | 0.1203$\pm$0.007 | 0.2255$\pm$0.001 | 0.0716$\pm$0.004 | 0.0486$\pm$0.005
dPF-M-lrn [8] | 2.1001$\pm$0.012 | 1.4016$\pm$0.004 | 0.1720$\pm$0.010 | 0.0974$\pm$0.009 | 0.1848$\pm$0.004 | 0.0611$\pm$0.003 | 0.0693$\pm$0.011
DEnKF | 1.3329$\pm$0.043 | 1.1349$\pm$0.024 | 0.0249$\pm$0.001 | 0.0506$\pm$0.001 | 0.0460$\pm$0.002 | 0.0353$\pm$0.001 | 0.0603$\pm$0.001
Means$\pm$standard errors.

Figure 2: Visual Odometry results with different differentiable filters: the
error rate for LSTM and BKF are reported from [12], dEKF, DPF, and dPF-M are
reproduced.

Results: We assess the performance of state estimation using an 11-fold cross-
validation withholding 1 trajectory at each time. We report the root mean
squared error (RMSE), mean absolute error (MAE), and the standard KITTI
benchmark metrics, the translational error (m/m), and the rotational error
(deg/m) in Table II. The error metrics are computed from the test trajectory
over all subsequences of 100 timesteps, and all subsequences of 100, 200, 400,
and 800 timesteps. Figure 2 shows the performance of DEnKF and other
differentiable filtering techniques. Note that lower error metrics can be
obtained by imposing domain- and data-specific information, i.e., using stereo
images [31], incorporating LiDAR [32, 33], or applying SLAM and loop-closure
related assumptions [31, 34]. However, we opt for the most commonly used setup
when comparing filtering technique in a task-agnostic fashion (as performed in
[8, 9, 12]) to ensure fair and comparable evaluations.

Comparison: Table II presents the outcomes of our proposed method in
comparison with the existing state-of-the-art differentiable filters,
including differentiable Extended Kalman filter (dEKF) [8], differentiable
particle filter (DPF) [9], and modified differentiable particle filter with
learned process and process noise models (dPF-M-lrn) [8]. To provide a fair
comparison, we do not include unstructured LSTM models as baselines since
prior works [12, 8] have shown that LSTM models do not achieve comparable
results. In our comparison, we use the same pre-trained sensor model
$s_{\boldsymbol{\xi}}$ with the same visual inputs and integrate it into all
the DF frameworks evaluated here. In this experiment, the motion model of the
vehicle is known. The only unknown part of the state is the velocities.
Therefore, we use the learnable process model to update those state variables
and use the known motion model to update the ($x,y,\theta$). For dEKF, we
supply the computed Jacobian matrix in training and testing since the motion
model is known. For DPF, we use 100 particles to train and test. DPF contains
a different learnable module called observation likelihood estimation model
$l$, which takes an image embedding and outputs a likelihood for updating each
particle’s weight.

Table II shows that DEnKF achieves a RMSE of 1.33, which is $\thicksim$54%,
$\thicksim$41%, and $\thicksim$37% lower than that of dEKF, DPF, and dPF-M-
lrn, respectively. Specifically, DEnKF reduces the translational error by
$\thicksim$85%, $\thicksim$79%, and $\thicksim$75% for Test 100/200/400/800
compared to dEKF, DPF, and dPF-M-lrn. It also reduces the rotational error by
$\thicksim$62%, $\thicksim$51%, and $\thicksim$42% for each baseline. Notably,
dPF-M-lrn manifests the best performance among all the baselines, it
implements learnable process noise model as described in Eq. 4, and it uses a
Gaussian Mixture Model for computing the likelihood for all particles. While
dPF-M-lrn performs the best among all baselines, DEnKF shows higher tracking
accuracy than dPF-M-lrn and runs 0.009s faster. It is worth noting that the
inference time for dPF-M-lrn is higher than any other DF in Table II.

Figure 3: Different visual inputs during testing. (a) the raw observation; (b)
the visual inputs with 2 types of noise: salt-and-pepper and blurring; (c) the
difference map.

Noisy and missing observation: According to [35], failures of vehicle cameras
may compromise autonomous driving performance – potentially even leading to
injuries and death. Common failures are listed by [35], i.e., brightness,
blurred vision, and brackish. We conducted an evaluation of the performance of
DEnKF and other DFs in the presence of noisy observations during inference. In
Fig.3, we added salt-and-pepper and blurring effects to the test images, and
reported the performance of DFs under these conditions in TableIII (top and
mid). Our findings show that DEnKF with noise performs worse compared to DEnKF
without noise, with a 17% and 29% increase in translational and rotational
error, respectively, compared to the metrics from Table II. However, DEnKF
remains more robust against noise perturbations than dEKF, DPF, and dPF-M-lrn,
achieving $\thicksim$80%, $\thicksim$66%, and $\thicksim$59% improvement on
translational error with salt-and-pepper noise for Test 100/200/400/800. We
also performed an experiment on missing observations by providing no visual
input with a chance of 30% at every timestep. In this case, DEnKF’s modularity
allows the state transition model $f_{\boldsymbol{\theta}}$ to propagate the
state forward through the state space, whereas other RNN-based filters remain
in the same hidden state until an observation is processed. Error metrics for
this scenario are reported in Table III (bottom), where we re-built the
process model for each DFs to account for such problems. DEnKF’s incorporation
of a stochastic neural network in the forward model handling missing
observation scenarios outperforms other DFs, resulting in a reduction of
$\thicksim$54% and $\thicksim$36% for translational and rotational error,
respectively, compared to dPF-M-lrn.

TABLE III: Result evaluations on different types of noisy visual input and
with 30% missing observation.

|  | Test 100 | Test 100/200/400/800
---|---|---|---
|  | m/m | deg/m | m/m | deg/m
S&P Noise | dEKF | 0.267 | 0.094 | 0.364 | 0.061
DPF | 0.249 | 0.159 | 0.296 | 0.052
dPF-M-lrn | 0.207 | 0.090 | 0.246 | 0.048
DEnKF | 0.029 | 0.089 | 0.102 | 0.039
Blurring | dEKF | 0.235 | 0.137 | 0.390 | 0.104
DPF | 0.144 | 0.122 | 0.236 | 0.101
dPF-M-lrn | 0.141 | 0.073 | 0.172 | 0.085
DEnKF | 0.028 | 0.064 | 0.054 | 0.050
Missing | dEKF | 0.254 | 0.173 | 0.298 | 0.134
DPF | 0.245 | 0.165 | 0.273 | 0.131
dPF-M-lrn | 0.204 | 0.116 | 0.230 | 0.093
DEnKF | 0.112 | 0.089 | 0.188 | 0.057

Figure 4: Results on UR5 manipulation in simulation. Top: the robot joint
configuration at varied timestep; Bottom: 3 key joint angle trajectory; The
vertical lines mapping from the top to the bottom align the tracking results
of the exact joint angle in time for each key joint. TABLE IV: Result
evaluations on UR5 manipulation task measured in MAE from 3 different domains
– real-world, simulation, and sim-to-real. Results for dEKF, dPF, and dPF-M-
lrn are reproduced for detailed comparisons.

Method | Real-world | Simulation | Sim-to-real | Wall clock time (s)
---|---|---|---|---
Joint (deg) | EE (cm) | Joint (deg) | EE (cm) | Joint (deg) | EE (cm)
dEKF [8] | 16.0862$\pm$0.063 | 5.6680$\pm$0.060 | 4.9357$\pm$0.224 | 1.9112$\pm$0.148 | 8.3041$\pm$0.525 | 4.3645$\pm$0.072 | 0.0469$\pm$0.003
DPF [9] | 15.9302$\pm$0.080 | 5.0834$\pm$0.301 | 4.4623$\pm$0.220 | 1.5135$\pm$0.191 | 5.9531$\pm$0.031 | 3.9695$\pm$0.006 | 0.0515$\pm$0.002
dPF-M-lrn [8] | 12.8366$\pm$0.086 | 3.9521$\pm$0.436 | 3.8233$\pm$0.230 | 1.2639$\pm$0.081 | 5.4389$\pm$0.011 | 3.9405$\pm$0.014 | 0.0854$\pm$0.001
DEnKF | 11.4222$\pm$0.005 | 3.4260$\pm$0.002 | 2.5587$\pm$0.093 | 0.8241$\pm$0.019 | 3.9531$\pm$0.034 | 2.6368$\pm$0.002 | 0.0712$\pm$0.002
Means$\pm$standard errors.

###  IV-B Robot Manipulation Task

In the second experiment, we assess the efficacy of DEnKF in a challenging
robot manipulation setting. Specifically, we train and employ DEnKF to monitor
the state of a UR5 robot while performing tabletop arrangement tasks. Similar
to behavioral cloning from observation tasks [36], actions are not provided in
this experiment, and the DEnKF is trained to learn to propagate state over
time. The robot state is defined as ${\bf x}=[J_{1},\cdots,J_{7},x,y,z]^{T}$,
where $J_{1}$-$J_{7}$ denote the 7 joint angle of the UR5 robot, and $(x,y,z)$
represents the 3D robot end-effector (EE) position w.r.t. $(0,0,0)$ which is
the center of the manipulation platform. As shown in Fig. 4(top), raw
observations ${\bf y}\in\mathbb{R}^{224\times 224\times 3}$ are images
captured from a camera placed in front of the table. The learned observation
$\tilde{{\bf y}}$ is defined to have the same dimension as the robot state,
where $\tilde{{\bf y}}=[J_{1},\cdots,J_{7},x,y,z]^{T}$.

Data: The data collection process is conducted both in the MuJoCo [37]
simulator and in the real-world. We record the UR5 robot operating on a random
object by performing one of “pick”, “push”, and “put down” actions. We collect
2,000 demonstrations in simulation and 100 on the real robot, changing the
location of each object for each demonstration. We use ABR control and
robosuite [38] in addition to MuJoCo to ensure rigorous dynamics in the
simulator. Each sequence length is around 350 steps with 0.08 sec as the
timestep. We use an 80/20 data split for training and testing.

Result: We conduct a performance evaluation of DEnKF in three different
domains, namely real-world, simulation, and sim-to-real. We train two separate
DEnKFs on simulation and real-world datasets, respectively, and then perform
sim-to-real transfer by fine-tuning the simulation-trained DEnKF on real-world
data. State estimation with uncertainty measurement using distributed ensemble
members in simulation is illustrated in Fig. 4. Following the same comparison
protocol as in Sec. IV-A, we supply all DFs the same pre-trained sensor model
$s_{\boldsymbol{\xi}}$ in each domain, but no known motion model is enabled at
this time. We train all learnable modules except $s_{\boldsymbol{\xi}}$ and
reported the mean absolute error (MAE) in the joint angle space (deg) and end-
effector positions (cm) for DEnKFs and other DF baselines. The experimental
results indicate that DEnKF and the other baseline DFs are capable of
achieving domain adaptation by fine-tuning the simulation framework for real-
world scenarios. Notably, the DEnKF with sim-to-real transfer achieves
accurate state estimation, resulting in a reduction of 29% in MAE for joint
angle space and 33% for end-effector (EE) positions when compared to the dPF-
M-lrn. In Table IV, the DEnKF with sim-to-real transfer exhibits an average of
2.6cm offset (MAE) from the ground truth for EE positions across testing
sequences. We further analyze the state tracking in EE space by visualizing
the EE trajectories in 3D, as depicted in Fig. 5, where the fine-tuned DEnKF
is utilized to estimate the state with two real-robot test examples of action
sequence “pick up" and “put down".

Moreover, an additional experiment is conducted to test the trade-off between
the accuracy and computational performance of proposed DEnKF framework as
shown in Fig. 6. Tellingly, increasing the number of ensemble members can have
a substantial positive effect on performance ($+9.6\%$) while the
computational overhead is only marginally affected (increase from 0.075 sec to
0.134 sec).

Figure 5: EE positions visualization. Top: UR5 executes “pick up” action;
Bottom: UR5 executes “put down” action. Figure 6: Computational time vs. MAE
(EE position) with increasing number of ensemble members.

##  V Conclusions

In this paper, we present the Differentiable Ensemble Kalman Filters (DEnKF)
as an extended version of Ensemble Kalman Filters, and demonstrate their
applications to state estimation tasks. We show that our framework is
applicable in both simulation and real-world, and that it is capable of
performing state estimation with complex tasks, e.g., KITTI visual odometry,
and robot manipulation. We also discuss state tracking in high-dimensional
observation spaces with varied noise conditions and missing observations. In
particular, DEnKF manages to decrease the error metrics on translation and
rotation by at least 59% and 36% with noisy observations versus the state-of-
the-art approaches. These experiments manifest the DEnKF significantly
improves the tracking accuracy and uncertainty estimates, thus, has great
potential in many robotic applications.

The proposed framework is modular, which allows for flexibility in using
individual components separately. However, it should be noted that various
learning tasks may require distinct curricula. For example, challenging visual
tasks may necessitate an extended training period for the sensor model before
it can be incorporated into end-to-end learning. Therefore, a universal
curriculum that guarantees optimal performance of all sub-modules in every
situation does not currently exist. In future research, we plan to explore the
potential of the DEnKF framework in detecting perturbations as a downstream
application. Specifically, leveraging the learned system dynamics from the
state transition model and the mapping from observation to state space learned
by the sensor model and the Kalman update step, we aim to use the distance
between the outputs of the two steps to detect perturbations in the system.
Overall, the results of this study suggest that DEnKF has great potential in
many robotic applications.

## Acknowledgment

The authors gratefully acknowledge support of this work through a grant by
“The Global KAITEKI Center” (TGKC) of the Global Futures Laboratory at Arizona
State University. TGKC is a research alliance between Arizona State University
and The KAITEKI Institute, an affiliate of the Mitsubishi Chemical Group.

## References

  * [1] S. Thrun, W. Burgard, and D. Fox, _Probabilistic robotics_. Cambridge, Mass.: MIT Press, 2005.
  * [2] L. Wang, G. Wang, S. Jia, A. Turner, and S. Ratchev, “Imitation learning for coordinated human–robot collaboration based on hidden state-space models,” _Robotics and Computer-Integrated Manufacturing_ , vol. 76, p. 102310, 2022\.
  * [3] S. Chen, “Kalman filter for robot vision: a survey,” _IEEE Transactions on industrial electronics_ , vol. 59, no. 11, pp. 4409–4420, 2011.
  * [4] J. Reher, W.-L. Ma, and A. D. Ames, “Dynamic walking with compliance on a cassie bipedal robot,” in _2019 18th European Control Conference (ECC)_. IEEE, 2019, pp. 2589–2595.
  * [5] S. Thrun, “Probabilistic robotics,” _Communications of the ACM_ , vol. 45, no. 3, pp. 52–57, 2002.
  * [6] S. S. Rangapuram, M. W. Seeger, J. Gasthaus, L. Stella, Y. Wang, and T. Januschowski, “Deep state space models for time series forecasting,” in _Advances in Neural Information Processing Systems_ , S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, Eds., vol. 31. Curran Associates, Inc., 2018\.
  * [7] A. Klushyn, R. Kurle, M. Soelch, B. Cseke, and P. van der Smagt, “Latent matters: Learning deep state-space models,” _Advances in Neural Information Processing Systems_ , vol. 34, 2021.
  * [8] A. Kloss, G. Martius, and J. Bohg, “How to train your differentiable filter,” _Autonomous Robots_ , pp. 1–18, 2021.
  * [9] R. Jonschkowski, D. Rastogi, and O. Brock, “Differentiable particle filters: End-to-end learning with algorithmic priors,” _arXiv preprint arXiv:1805.11122_ , 2018.
  * [10] H. Sorenson, _Kalman Filtering: Theory and Application_. IEEE Press, Los Alamitos, 1985.
  * [11] R. Van Der Merwe, _Sigma-point Kalman filters for probabilistic inference in dynamic state-space models_. Oregon Health & Science University, 2004.
  * [12] T. Haarnoja, A. Ajay, S. Levine, and P. Abbeel, “Backprop kf: Learning discriminative deterministic state estimators,” in _Advances in neural information processing systems_ , 2016, pp. 4376–4384.
  * [13] P. Karkus, X. Ma, D. Hsu, L. P. Kaelbling, W. S. Lee, and T. Lozano-Pérez, “Differentiable algorithm networks for composable robot learning,” _arXiv preprint arXiv:1905.11602_ , 2019.
  * [14] A. Corenflos, J. Thornton, G. Deligiannidis, and A. Doucet, “Differentiable particle filtering via entropy-regularized optimal transport,” in _International Conference on Machine Learning_. PMLR, 2021, pp. 2100–2111.
  * [15] X. Chen, H. Wen, and Y. Li, “Differentiable particle filters through conditional normalizing flow,” in _2021 IEEE 24th International Conference on Information Fusion (FUSION)_. IEEE, 2021, pp. 1–6.
  * [16] Y. Wang, B. Liu, J. Wu, Y. Zhu, S. S. Du, L. Fei-Fei, and J. B. Tenenbaum, “Dualsmc: Tunneling differentiable filtering and planning under continuous pomdps,” _arXiv preprint arXiv:1909.13003_ , 2019.
  * [17] H. Wen, X. Chen, G. Papagiannis, C. Hu, and Y. Li, “End-to-end semi-supervised learning for differentiable particle filters,” in _2021 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2021, pp. 5825–5831.
  * [18] M. A. Lee, B. Yi, R. Martín-Martín, S. Savarese, and J. Bohg, “Multimodal sensor fusion with differentiable filters,” in _2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 2020, pp. 10 444–10 451.
  * [19] B. Wagstaff, E. Wise, and J. Kelly, “A self-supervised, differentiable kalman filter for uncertainty-aware visual-inertial odometry,” in _2022 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)_. IEEE, 2022, pp. 1388–1395.
  * [20] G. Evensen, “The ensemble kalman filter: Theoretical formulation and practical implementation,” _Ocean dynamics_ , vol. 53, no. 4, pp. 343–367, 2003.
  * [21] M. Roth, G. Hendeby, C. Fritsche, and F. Gustafsson, “The ensemble kalman filter: a signal processing perspective,” _EURASIP Journal on Advances in Signal Processing_ , vol. 2017, no. 1, pp. 1–16, 2017.
  * [22] P. L. Houtekamer and F. Zhang, “Review of the ensemble kalman filter for atmospheric data assimilation,” _Monthly Weather Review_ , vol. 144, no. 12, pp. 4489–4532, 2016.
  * [23] L. V. Jospin, H. Laga, F. Boussaid, W. Buntine, and M. Bennamoun, “Hands-on bayesian neural networks—a tutorial for deep learning users,” _IEEE Computational Intelligence Magazine_ , vol. 17, no. 2, pp. 29–48, 2022.
  * [24] B. Lakshminarayanan, A. Pritzel, and C. Blundell, “Simple and scalable predictive uncertainty estimation using deep ensembles,” _Advances in neural information processing systems_ , vol. 30, 2017.
  * [25] K. Chua, R. Calandra, R. McAllister, and S. Levine, “Deep reinforcement learning in a handful of trials using probabilistic dynamics models,” _Advances in neural information processing systems_ , vol. 31, 2018.
  * [26] Y. Chen, D. Sanz-Alonso, and R. Willett, “Auto-differentiable ensemble kalman filters,” _arXiv preprint arXiv:2107.07687_ , 2021.
  * [27] S. Zhang, M. Harrison, A. Rosati, and A. Wittenberg, “System design and evaluation of coupled ensemble data assimilation for global oceanic climate studies,” _Monthly Weather Review_ , vol. 135, no. 10, pp. 3541–3564, 2007\.
  * [28] P. B. Quang and V. Tran, “High-dimensional simulation experiments with particle filter and ensemble kalman filter,” _Applied Mathematics in Engineering and Reliability_ , p. 1, 2016.
  * [29] Y. Gal and Z. Ghahramani, “Dropout as a bayesian approximation: Representing model uncertainty in deep learning,” in _international conference on machine learning_. PMLR, 2016, pp. 1050–1059.
  * [30] A. Geiger, P. Lenz, and R. Urtasun, “Are we ready for autonomous driving? the kitti vision benchmark suite,” in _2012 IEEE conference on computer vision and pattern recognition_. IEEE, 2012, pp. 3354–3361.
  * [31] K. Lenac, J. Ćesić, I. Marković, and I. Petrović, “Exactly sparse delayed state filter on lie groups for long-term pose graph slam,” _The International Journal of Robotics Research_ , vol. 37, no. 6, pp. 585–610, 2018.
  * [32] J. Zhang and S. Singh, “Visual-lidar odometry and mapping: Low-drift, robust, and fast,” in _2015 IEEE International Conference on Robotics and Automation (ICRA)_. IEEE, 2015, pp. 2174–2181.
  * [33] C.-C. Chou and C.-F. Chou, “Efficient and accurate tightly-coupled visual-lidar slam,” _IEEE Transactions on Intelligent Transportation Systems_ , 2021.
  * [34] I. Cvišić, J. Ćesić, I. Marković, and I. Petrović, “Soft-slam: Computationally efficient stereo visual simultaneous localization and mapping for autonomous unmanned aerial vehicles,” _Journal of field robotics_ , vol. 35, no. 4, pp. 578–595, 2018.
  * [35] A. Ceccarelli and F. Secci, “Rgb cameras failures and their effects in autonomous driving applications,” _IEEE Transactions on Dependable and Secure Computing_ , 2022.
  * [36] F. Torabi, G. Warnell, and P. Stone, “Behavioral cloning from observation,” _arXiv preprint arXiv:1805.01954_ , 2018.
  * [37] E. Todorov, T. Erez, and Y. Tassa, “Mujoco: A physics engine for model-based control,” in _2012 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2012, pp. 5026–5033.
  * [38] Y. Zhu, J. Wong, A. Mandlekar, and R. Martín-Martín, “robosuite: A modular simulation framework and benchmark for robot learning,” in _arXiv preprint arXiv:2009.12293_ , 2020.
  * [39] M. D. Butala, J. Yun, Y. Chen, R. A. Frazin, and F. Kamalabadi, “Asymptotic convergence of the ensemble kalman filter,” in _2008 15th IEEE International Conference on Image Processing_. IEEE, 2008, pp. 825–828.
  * [40] I. Kasanickỳ, “Ensemble kalman filter on high and infinite dimensional spaces,” _Univerzita Karlova, Matematicko-fyzikální fakulta_ , 2017\.
  * [41] J. Mandel, _Efficient implementation of the ensemble Kalman filter_. University of Colorado at Denver and Health Sciences Center, 2006.
  * [42] A. Hommels, A. Murakami, and S.-I. Nishimura, “A comparison of the ensemble kalman filter with the unscented kalman filter: application to the construction of a road embankment,” _Geotechniek_ , vol. 13, no. 1, p. 52, 2009.
  * [43] J. Mandel, L. Cobb, and J. D. Beezley, “On the convergence of the ensemble kalman filter,” _Applications of Mathematics_ , vol. 56, no. 6, pp. 533–541, 2011.
  * [44] G. Cybenko, “Approximation by superpositions of a sigmoidal function,” _Mathematics of control, signals and systems_ , vol. 2, no. 4, pp. 303–314, 1989.
  * [45] Z. Pu and J. Hacker, “Ensemble-based kalman filters in strongly nonlinear dynamics,” _Advances in atmospheric sciences_ , vol. 26, pp. 373–380, 2009\.
  * [46] E. H. Bergou, S. Gratton, and J. Mandel, “On the convergence of a non-linear ensemble kalman smoother,” _Applied Numerical Mathematics_ , vol. 137, pp. 151–168, 2019.

[Theoretical properties of EnKF] This section provides an overview of the
theoretical underpinnings of Ensemble Kalman Filters (EnKF), outlines key
differences between EnKF and Kalman Filters (KF), and highlights the
significance of EnKF in practical applications.

###  -A Asymptotic Convergence to KF

Research in [39] has demonstrated that EnKF asymptotically converges to the
optimal estimates provided by KF in a finite space. It is important to note
that EnKF is a Monte Carlo algorithm that offers computationally tractable
solutions to high-dimensional state estimation problems, which converge to KF
[39].

###  -B Convergence in Hilbert Space

The convergence of EnKF has been proven in [40] for infinite-dimensional state
space models. Specifically, the proof shows that when a sequence of states
${\bf X}_{t}$ represents a system with stochastic dynamics defined on a
Hilbert space and EnKF is applied, the empirical ensemble converges to the
reference ensemble in $L^{p}$ for all $p\in[1,\infty)$. The convergence in the
infinite dimensional Hilbert space is an important insight in machine learning
research, as it offers a theoretical foundation for the use of EnKF in high-
dimensional state estimation problems.

###  -C Computational Performance

EnKF has a computational complexity of around $O(E^{2}n)$, where $E$ is the
number of ensemble members and $n$ is the dimension of the state space,
according to Mandel et al. [41]. In contrast, the computational complexity of
the Extended Kalman Filter (EKF), an alternative filtering approach for non-
linear systems, is $O(n^{3})$ [1]. One of the advantages of EnKF over EKF is
that it does not require explicit construction of the covariance matrix since
this information is implicitly captured by the ensemble. This reduces the
computational and memory burden for large-scale estimation problems.

###  -D Comparison with UKF

The Unscented Kalman Filter (UKF) employs a set of sigma points that are
propagated through the actual non-linear function. These points are selected
to match the mean, covariance, and higher-order moments of a Gaussian random
variable. In some cases, UKF has the tendency to overestimate the state
compared to EnKF, as illustrated in a an example in [42]. Still, both EnKF and
UKF are effective state estimation methods for non-linear systems, and the
choice of method should depend on the specific task at hand, in other words,
the suitability of either method is determined by factors such as the
complexity of the model, the availability of prior information, and the
desired computational efficiency.

###  -E Convergence of DEnKF

Ensemble Kalman Filter (EnKF) are proven to converge in linear systems [43].
In the context of our _differentiable_ Ensemble Kalman Filters (DEnKF), the
prediction step is substituted with a stochastic neural network for the state
transition function, as introduced in Sec. III-A. $\bf H$ in Eq. 6 is learned
through multi-layer perceptrons as part of the observation model. In theory, a
neural network can act as a function approximator capable of modeling exact
linear functions, which has been demonstrated by the universal approximation
theorem [44]. Hence, the proposed state transition function can be
approximated to any degree of accuracy when the dynamics model is linear, and
this holds true for the observation model in DEnKF as it approximates a linear
process. Accordingly, the DEnKF satisfies the criteria necessary for the
linear EnKF convergence proof. With regards to nonlinear systems, there is no
global convergence proof for the EnKF. However, in practice it is often the
preferred filtering approach in this setting since it empirically demonstrates
strong performance in a variety of applications with strong non-linearities
[45]. In addition, recent work has shown that specific variants of the filter
[46], i.e., the ensemble Kalman smoother, are proven to converge even in the
nonlinear settings under certain assumptions.


§ INTRODUCTION

§ ARCHITECTURE

§ BLOCKCHAIN ECONOMICS

In this section we will discuss economical aspects of JaxNet.

§ CONCLUSIONS

Missing 'biblatex' package
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The University of Chicago Press

journaltitleJournal of Political Economy
titleCommodity-reserve currency

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titleA program for monetary stability

journaltitleEconometrica: Journal of the Econometric Society
titleAsset bubbles and overlapping generations

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titleHow to prove yourself: Practical solutions to identification and signature problems

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titleInvestment under uncertainty

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titleImplications of historical trends in the electrical efficiency of computing

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titleTheoretical Bitcoin Attacks with less than Half of the Computational Power (draft)

WW Norton & Company

titleIntermediate microeconomics with calculus: a modern approach

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titleMoore’s Law Might Be Slowing Down, But Not Energy Efficiency

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titleProofs of proofs of work with sublinear complexity

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titleMerged Mining: Analysis of Effects and Implications

journaltitleMathematics in Computer Science
titleOn succinct representations of binary trees

journaltitleIACR Cryptology ePrint Archive
titleNon-Interactive Proofs of Proof-of-Work.

journaltitleAvailable at SSRN 3261063
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Association for Computing Machinery (ACM)

journaltitleCommunications of the ACM
titleMajority is not enough

Wiley Online Library

journaltitleEconomic Inquiry
titleBitcoin 1, bitcoin 2,....: An experiment in privately issued outside monies

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titleSok: Sharding on blockchain

IACR Cryptology ePrint Archive, 2019: 1128

titleSok: Communication across distributed ledgers

titleOn the Profitability of Selfish Mining Against Multiple Difficulty Adjustment Algorithms

journaltitlearXiv preprint arXiv:2004.04605
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titleA survey of distributed consensus protocols for blockchain networks

journaltitlearXiv preprint arXiv:2002.05973
titleAlgebraic Structure of Blockchains: A Group-Theoretical Primer

journaltitleIEEE Access
titleSolutions to Scalability of Blockchain: A Survey

noteAccessed on 20.11.2019
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noteBitcoin white paper
titleBitcoin: A Peer-to-Peer Electronic Cash System

noteAccessed on 20.11.2019
titleOn sharding blockchains

noteAccessed on 10.01.2020

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

The University of Chicago Press

journaltitleJournal of Political Economy
titleCommodity-reserve currency

Ravenio Books

titleA program for monetary stability

journaltitleEconometrica: Journal of the Econometric Society
titleAsset bubbles and overlapping generations

booktitleConference on the Theory and Application of Cryptographic Techniques
titleHow to prove yourself: Practical solutions to identification and signature problems

booktitleConference on the theory and application of cryptographic techniques
titleA digital signature based on a conventional encryption function

booktitleCarnegie-Rochester conference series on public policy
titleDiscretion versus policy rules in practice

Princeton university press

titleInvestment under uncertainty

American Mathematical Soc.

titleLectures on generating functions

journaltitleJournal of Monetary Economics
titleFloat on a Note

journaltitleDatabase trends and applications
noteAccessed on 10.02.2020
titleDatabase Sharding: The Key to Database Scalability

princeton university press

titleThis time is different: Eight centuries of financial folly

journaltitleIEEE Annals of the History of Computing
titleImplications of historical trends in the electrical efficiency of computing

RFC 5905 (Proposed Standard)

titleNetwork time protocol version 4: Protocol and algorithms specification

journaltitleStanford Encyclopedia of Philosophy
titleThe st. petersburg paradox

journaltitlearXiv preprint arXiv:1312.7013
titleTheoretical Bitcoin Attacks with less than Half of the Computational Power (draft)

WW Norton & Company

titleIntermediate microeconomics with calculus: a modern approach

journaltitleIEEE Spectrum
titleMoore’s Law Might Be Slowing Down, But Not Energy Efficiency

titleAn introduction to Catalan numbers

booktitleInternational Conference on Financial Cryptography and Data Security
titleOn scaling decentralized blockchains

Motherboard Tech by Vice

noteAccessed on 10.12.2019
titleBitcoin Could Consume as Much Electricity as Denmark by 2020

booktitleInternational Conference on Financial Cryptography and Data Security
titleProofs of proofs of work with sublinear complexity

Master’s thesis, Vienna University of Technology

titleMerged Mining: Analysis of Effects and Implications

journaltitleMathematics in Computer Science
titleOn succinct representations of binary trees

journaltitleIACR Cryptology ePrint Archive
titleNon-Interactive Proofs of Proof-of-Work.

journaltitleAvailable at SSRN 3261063
titleEquilibrium bitcoin pricing

noteAccessed on 9.01.2020
titleThe Anti-Bitcoin: Inflationary By Design

Association for Computing Machinery (ACM)

journaltitleCommunications of the ACM
titleMajority is not enough

Wiley Online Library

journaltitleEconomic Inquiry
titleBitcoin 1, bitcoin 2,....: An experiment in privately issued outside monies

noteAccessed on 20.11.2019
titleBitcoin Mining Efficiency Set to Double by 2020

noteAccessed on 20.01.2020
titleBitcoin: Disinflating to Death

noteAccessed on 20.12.2019
titleBitcoin Security in One Chart

noteAccessed on 20.12.2019
titleBitcoin Security: a Negative Exponential

CESifo Working Paper Series

titleAn equilibrium model of the market for bitcoin mining

journaltitleMultiCoin Capital
titleModels for scaling trustless computation

noteAccessed on 20.11.2019
titleWhat is Merged Mining? Can You Mine Two Cryptos at the Same Time?

noteAccessed on 20.11.2019
titleThe Scalability Trilemma in Blockchain

noteAccessed on 9.01.2020
titleTime Travelling with Satoshi

journaltitleBitcoin Magazine
noteAccessed on 25.01.2020
titleOp Ed: The Many Faces of Sharding for Blockchain Scalability

noteAccessed on 10.01.2020
titleCase Study: Merged Mining in Dogecoin & Litecoin

journaltitleAvailable at SSRN 3273982
titlePitfalls of Bitcoin’s Proof-of-Work: R&D arms race and mining centralization

journaltitleIACR Cryptology ePrint Archive
titleFlyclient: Super-Light Clients for Cryptocurrencies.
Cambridge Centre for Alternative Finance

noteAccessed on 10.12.2019
titleCambridge Bitcoin Electricity Consumption Index.

booktitleProceedings of the 1st ACM Conference on Advances in Financial Technologies
titleAn axiomatic approach to block rewards

journaltitleAvailable at SSRN 3336367
titleMoney mining and price dynamics

journaltitlearXiv preprint arXiv:1912.06412
titleOn Profitability of Nakamoto double spend

Taylor & Francis

journaltitleApplied Economics Letters
titleBitcoin price and its marginal cost of production: support for a fundamental value

noteAccessed on 20.02.2020
titleThe Longest Blockchain is not the Strongest Blockchain

noteAccessed on 20.11.2019
titleHow Merge Mining and Anchored Blockchain Projects Capitalize on Bitcoin's Security Model

journaltitlearXiv preprint arXiv:1910.00765
titleSharPer: Sharding Permissioned Blockchains Over Network Clusters

booktitleThe World Wide Web Conference
titleRedesigning Bitcoin's fee market

Cowles Foundation Discussion Paper

titleBitcoin: An Impossibility Theorem for Proof-of-Work based Protocols

noteAccessed on 10.01.2020
title'Lost Coins' Study Estimates BTC's True Supply

noteAccessed on 10.01.2020
titleCryptocurrency 'Burning': Can It Manage Inflation?

noteAccessed on 15.01.2020
titleBitcoin’s Block Timestamp Protection Rules

journaltitlearXiv preprint arXiv:1904.03487
titleExploring the attack surface of blockchain: A systematic overview

noteAccessed on 9.01.2020
titleThe Complete Guide to Stablecoins in 2019

journaltitleURL: https://nearprotocol. com/downloads/Nightshade. pdf
titleNightshade: Near Protocol Sharding Design

journaltitlearXiv preprint arXiv:1901.11218
titleReplay attacks and defenses against cross-shard consensus in sharded distributed ledgers

booktitleProceedings of the 1st ACM Conference on Advances in Financial Technologies
titleSok: Sharding on blockchain

IACR Cryptology ePrint Archive, 2019: 1128

titleSok: Communication across distributed ledgers

titleOn the Profitability of Selfish Mining Against Multiple Difficulty Adjustment Algorithms

journaltitlearXiv preprint arXiv:2004.04605
titleThe cost of Bitcoin mining has never really increased

journaltitleIEEE Communications Surveys & Tutorials
titleA survey of distributed consensus protocols for blockchain networks

journaltitlearXiv preprint arXiv:2002.05973
titleAlgebraic Structure of Blockchains: A Group-Theoretical Primer

journaltitleIEEE Access
titleSolutions to Scalability of Blockchain: A Survey

noteAccessed on 20.11.2019
titleMerged Mining

noteBitcoin white paper
titleBitcoin: A Peer-to-Peer Electronic Cash System

noteAccessed on 20.11.2019
titleOn sharding blockchains

noteAccessed on 10.01.2020

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