Various algorithms combine deep neural networks (DNNs) and Kalman filters (KFs) to learn from data to track in complex dynamics. Unlike classic KFs, DNN-based systems do not naturally provide the error covariance alongside their estimate, which is of great importance in some applications, e.g., navigation. To bridge this gap, in this work we study error covariance extraction in DNN-aided KFs. We examine three main approaches that are distinguished by the ability to associate internal features with meaningful KF quantities such as the Kalman gain (KG) and prior covariance. We identify the differences between these approaches in their requirements and their effect on the training of the system. Our numerical study demonstrates that the above approaches allow DNN-aided KFs to extract error covariance, with most accurate error prediction provided by model-based/data-driven designs.

Combining the classical Kalman filter (KF) with a deep neural network (DNN) enables tracking in partially known state space (SS) models. A major limitation of current DNN-aided designs stems from the need to train them to filter data originating from a specific distribution and underlying SS model. Consequently, changes in the model parameters may require lengthy retraining. While the KF adapts through parameter tuning, the black-box nature of DNNs makes identifying tunable components difficult. Hence, we propose Adaptive KalmanNet (AKNet), a DNN-aided KF that can adapt to changes in the SS model without retraining. Inspired by recent advances in large language model fine-tuning paradigms, AKNet uses a compact hypernetwork to generate context-dependent modulation weights. Numerical evaluation shows that AKNet provides consistent state estimation performance across a continuous range of noise distributions, even when trained using data from limited noise settings.

People with brain or spinal cord-related paralysis often need to rely on others for basic tasks, limiting their independence. A potential solution is brain-machine interfaces (BMIs), which could allow them to voluntarily control external devices (e.g., robotic arm) by decoding brain activity to movement commands. In the past decade, deep-learning decoders have achieved state-of-the-art results in most BMI applications, ranging from speech production to finger control. However, the 'black-box' nature of deep-learning decoders could lead to unexpected behaviors, resulting in major safety concerns in real-world physical control scenarios. In these applications, explainable but lower-performing decoders, such as the Kalman filter (KF), remain the norm. In this study, we designed a BMI decoder based on KalmanNet, an extension of the KF that augments its operation with recurrent neural networks to compute the Kalman gain. This results in a varying “trust” that shifts between inputs and dynamics. We used this algorithm to predict finger movements from the brain activity of two monkeys. We compared KalmanNet results offline (pre-recorded data, $n=13$ days) and online (real-time predictions, $n=5$ days) with a simple KF and two recent deep-learning algorithms: tcFNN (non-ReFIT version) and LSTM. KalmanNet achieved comparable or better results than other deep learning models in offline and online modes, relying on the dynamical model for stopping while depending more on neural inputs for initiating movements. We further validated this mechanism by implementing a heteroscedastic KF that used the same strategy, and it also approached state-of-the-art performance while remaining in the explainable domain of standard KFs. However, we also see two downsides to KalmanNet. KalmanNet shares the limited generalization ability of existing deep-learning decoders, and its usage of the KF as an inductive bias limits its performance in the presence of unseen noise distributions. Despite this trade-off, our analysis successfully integrates traditional controls and modern deep-learning approaches to motivate high-performing yet still explainable BMI designs.

The Kalman filter (KF) is a widely-used algorithm for tracking dynamical systems that can be faithfully captured by state space (SS) models. The need to fully describe an SS model limits its applicability under complex settings, e.g., when tracking based on visual or graphical data. This challenge can be treated by mapping the measurements into latent features obeying some postulated closed-form SS model, and applying the KF in the latent space. However, the validity of this approximated SS model may constitute a limiting factor. In this work we tackle the challenges associated with tracking from high-dimensional measurements by jointly learning the KF along with the latent space mapping. Our proposed approach combines a learned encoder while tracking in the latent space using the recently proposed data-driven Kalman-Net, and having both modules jointly tuned from data. Our empirical results demonstrate that the proposed approach achieves improved performance over both model-based and data-driven techniques, by learning a surrogate latent representation that most facilitates tracking.

Stochastic control deals with finding an optimal control signal for a dynamical system in a setting with uncertainty, playing a key role in numerous applications. The linear quadratic Gaussian (LQG) is a widely-used setting, where the system dynamics is represented as a linear Gaussian state-space (SS) model, and the objective function is quadratic. For this setting, the optimal controller is obtained in closed form by the separation principle. However, in practice, the underlying system dynamics often cannot be faithfully captured by a fully known linear Gaussian SS model, limiting its performance. Here, we present LQGNet, a stochastic controller that leverages data to operate under partially known dynamics. LQGNet augments the state tracking module of separation-based control with a dedicated trainable algorithm. The resulting system preserves the operation of classic LQG control while learning to cope with partially known SS models without having to fully identify the dynamics. We empirically show that LQGNet outperforms classic stochastic control by overcoming mismatched SS models.

Pairs trading is a family of trading policies based on monitoring the relationships between pairs of assets. A common pairs trading approach relies on state space (SS) modeling, from which financial indicators can be obtained with low complexity and latency using a Kalman filter (KF), and processed using classic policies such as Bollinger bands (BB). However, such SS models are inherently approximated and mismatched, often degrading the revenue. In this work we propose KalmanNet Bollinger Trading (KalmanBOT), a dataaided policy that preserves the advantages of KF-aided BB policies while leveraging data to overcome the approximated nature of the SS model. We adopt the recent KalmanNet architecture, and approximate BB with a differentiable mapping, converting the policy into a trainable model. We empirically demonstrate that KalmanBOT yields improved rewards compared with model-based and data-driven benchmarks.

Electrocardiographic signals (ECG) are used in many healthcare applications, including at-home monitoring of vital signs. These applications often rely on wearable technology and provide low quality ECG signals. Although many methods have been proposed for denoising the ECG to boost its quality and enable clinical interpretation, these methods typically fall short for ECG data obtained with wearable technology, because of either their limited tolerance to noise or their limited flexibility to capture ECG dynamics. This paper presents HKF, a hierarchical Kalman filtering method, that leverages a patient-specific learned structured prior of the ECG signal, and integrates it into a state space model to yield filters that capture both intra- and inter-heartbeat dynamics. HKF is demonstrated to outperform previously proposed methods such as the model-based Kalman filter and data-driven autoencoders, in ECG denoising task in terms of mean-squared error, making it a suitable candidate for application in extramural healthcare settings.

Direction of arrival (DoA) estimation is a fundamental task in array processing. A popular family of DoA estimation algorithms are subspace methods, which operate by dividing the measurements into distinct signal and noise subspaces. Subspace methods, such as Root-MUSIC, require the sources to be non-coherent, and are considerably degraded when this does not hold. In this work we propose Deep Root-MUSIC (DR-MUSIC); a data-driven DoA estimator which augments Root-MUSIC with a deep neural network applied to the empirical autocorrelation of the input. DR-MUSIC learns how to divide the observations into distinguishable subspaces, thus leveraging data to cope with coherent sources, low SNR and limited snapshots, while preserving the interpretability and the suitability of the model-based algorithm.

The Kalman filter (KF) is a widely-used algorithm for tracking the latent state of a dynamical system from noisy observations. For systems that are well-described by linear Gaussian state space models, the KF minimizes the mean-squared error (MSE). However, in practice, observations are corrupted by outliers, severely impairing the KF’s performance. In this work, an outlier-insensitive KF (OIKF) is proposed, where robustness is achieved by modeling a potential outlier as a normally distributed random variable with unknown variance (NUV). The NUV’s variance is estimated online, using both expectation-maximization (EM) and alternating maximization (AM). The former was previously proposed for the task of smoothing with outliers and was adapted here to filtering, while both EM and AM obtained the same performance and outperformed the other algorithms, the AM approach is less complex and thus requires 40% less runtime. Our empirical study demonstrates that the MSE of our proposed outlier-insensitive KF outperforms previously proposed algorithms, and that for data clean of outliers, it reverts to the classic KF, i.e., MSE optimality is preserved.

Providing a metric of uncertainty alongside a state estimate is often crucial when tracking a dynamical system. Classic state estimators, such as the Kalman filter (KF), provide a time-dependent uncertainty measure from knowledge of the underlying statistics; however, deep learning based tracking systems struggle to reliably characterize uncertainty. In this paper, we investigate the ability of KalmanNet, a recently proposed; hybrid; model-based; deep state tracking algorithm, to estimate an uncertainty measure. By exploiting the interpretable nature of KalmanNet, we show that the error covariance matrix can be computed based on its internal features, as an uncertainty measure. We demonstrate that when the system dynamics are known, KalmanNet—which learns its mapping from data without access to the statistics—provides uncertainty similar to that provided by the KF; and while in the presence of evolution model-mismatch, KalmanNet provides a more accurate error estimation.

Direction of arrival (DoA) estimation is a crucial task in sensor array signal processing, giving rise to various successful model-based (MB) algorithms as well as recently developed data-driven (DD) methods. This paper introduces a new hybrid MB/DD DoA estimation architecture, based on the classical multiple signal classification (MUSIC) algorithm. Our approach augments crucial aspects of the original MUSIC structure with specifically designed neural architectures, allowing it to overcome certain limitations of the purely MB method, such as its inability to successfully localize coherent sources. The deep augmented MUSIC algorithm is shown to outperform its unaltered version with a superior resolution.

The smoothing task is the core of many signal processing applications. It deals with the recovery of a sequence of hidden state variables from a sequence of noisy observations in a one-shot manner. In this work we propose RTSNet, a highly efficient model-based and data-driven smoothing algorithm. RTSNet integrates dedicated trainable models into the flow of the classical Rauch-Tung-Striebel (RTS) smoother, and is able to outperform it when operating under model mismatch and non-linearities while retaining its efficiency and interpretability. Our numerical study demonstrates that although RTSNet is based on more compact neural networks, which leads to faster training and inference times, it outperforms the state-of-the-art, data-driven smoother in a non-linear use case.

Anomaly detection among a large number of processes arises in many applications ranging from dynamic spectrum access to cybersecurity. In such problems one can often obtain noisy observations aggregated from a chosen subset of processes that conforms to a tree structure. The distribution of these observations, based on which the presence of anomalies is detected, may be only partially known. This gives rise to the need for a search strategy designed to account for both the sample complexity and the detection accuracy, as well as cope with statistical models that are known only up to some missing parameters. In this work we propose a sequential search strategy using two variations of the Generalized Log Likelihood Ratio statistic. Our proposed Hierarchical Dynamic Search (HDS) strategy is shown to be order-optimal with respect to the size of the search space and asymptotically optimal with respect to the detection accuracy. An explicit upper bound on the error probability of HDS is established for the finite sample regime. Extensive experiments are conducted, demonstrating the performance gains of HDS over existing methods.

The Kalman filter (KF) is a celebrated signal processing algorithm, implementing optimal state estimation of dynamical systems that are well represented by a linear Gaussian state-space model. The KF is model-based, and therefore relies on full and accurate knowledge of the underlying model. We present KalmanNet, a hybrid data-driven/model-based filter that does not require full knowledge of the underlying model parameters. KalmanNet is inspired by the classical KF flow and implemented by integrating a dedicated and compact neural network for the Kalman gain computation. We present an offline training method, and numerically illustrate that KalmanNet can achieve optimal performance without full knowledge of the model parameters. We demonstrate that when facing inaccurate parameters KalmanNet learns to achieve notably improved performance compared to KF.