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In estimation theory, the Kushner equation provides the evolution of the probability density of the state of a dynamical system given continuous-time observations. Building upon our recent work, we propose a new way to approximate the solution of the Kushner equation through tractable variational Gaussian approximations of two proximal losses associated with the propagation and Bayesian update of the probability density. The first is a proximal loss based on the Wasserstein metric and the second is a proximal loss based on the Fisher metric. The solution to this last proximal loss is given by implicit updates on the mean and covariance that we proposed earlier. These two variational updates can be fused and shown to satisfy a set of stochastic differential equations on the Gaussian's mean and covariance matrix. This Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the linear case and generalize them in the nonlinear one., Comment: Lecture Notes in Computer Science, 2023

Along with Markov chain Monte Carlo (MCMC) methods, variational inference (VI) has emerged as a central computational approach to large-scale Bayesian inference. Rather than sampling from the true posterior $\pi$, VI aims at producing a simple but effective approximation $\hat \pi$ to $\pi$ for which summary statistics are easy to compute. However, unlike the well-studied MCMC methodology, algorithmic guarantees for VI are still relatively less well-understood. In this work, we propose principled methods for VI, in which $\hat \pi$ is taken to be a Gaussian or a mixture of Gaussians, which rest upon the theory of gradient flows on the Bures--Wasserstein space of Gaussian measures. Akin to MCMC, it comes with strong theoretical guarantees when $\pi$ is log-concave., Comment: 52 pages, 15 figures

This paper proposes a learning method for denoising gyroscopes of Inertial Measurement Units (IMUs) using ground truth data, and estimating in real time the orientation (attitude) of a robot in dead reckoning. The obtained algorithm outperforms the state-of-the-art on the (unseen) test sequences. The obtained performances are achieved thanks to a well-chosen model, a proper loss function for orientation increments, and through the identification of key points when training with high-frequency inertial data. Our approach builds upon a neural network based on dilated convolutions, without requiring any recurrent neural network. We demonstrate how efficient our strategy is for 3D attitude estimation on the EuRoC and TUM-VI datasets. Interestingly, we observe our dead reckoning algorithm manages to beat top-ranked visual-inertial odometry systems in terms of attitude estimation although it does not use vision sensors. We believe this paper offers new perspectives for visual-inertial localization and constitutes a step toward more efficient learning methods involving IMUs. Our open-source implementation is available at https://github.com/mbrossar/denoise-imu-gyro., Comment: IEEE Robotics and Automation Letters, IEEE In press

In this paper we propose a novel accurate method for dead-reckoning of wheeled vehicles based only on an Inertial Measurement Unit (IMU). In the context of intelligent vehicles, robust and accurate dead-reckoning based on the IMU may prove useful to correlate feeds from imaging sensors, to safely navigate through obstructions, or for safe emergency stops in the extreme case of exteroceptive sensors failure. The key components of the method are the Kalman filter and the use of deep neural networks to dynamically adapt the noise parameters of the filter. The method is tested on the KITTI odometry dataset, and our dead-reckoning inertial method based only on the IMU accurately estimates 3D position, velocity, orientation of the vehicle and self-calibrates the IMU biases. We achieve on average a 1.10% translational error and the algorithm competes with top-ranked methods which, by contrast, use LiDAR or stereo vision. We make our implementation open-source at: https://github.com/mbrossar/ai-imu-dr

Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically., Comment: A slightly shorter version has been published in IEEE Transactions Automatic Control