Computational models of the human head are promising tools for estimating the impact-induced response of brain, and thus play an important role in the prediction of traumatic brain injury. Modern biofidelic head model simulations are associated with very high computational cost, and high-dimensional inputs and outputs, which limits the applicability of traditional uncertainty quantification (UQ) methods on these systems. In this study, a two-stage, data-driven manifold learning-based framework is proposed for UQ of computational head models. This framework is demonstrated on a 2D subject-specific head model, where the goal is to quantify uncertainty in the simulated strain fields (i.e., output), given variability in the material properties of different brain substructures (i.e., input). In the first stage, a data-driven method based on multi-dimensional Gaussian kernel-density estimation and diffusion maps is used to generate realizations of the input random vector directly from the available data. Computational simulations of a small number of realizations provide input-output pairs for training data-driven surrogate models in the second stage. The surrogate models employ nonlinear dimensionality reduction using Grassmannian diffusion maps, Gaussian process regression to create a low-cost mapping between the input random vector and the reduced solution space, and geometric harmonics models for mapping between the reduced space and the Grassmann manifold. It is demonstrated that the surrogate models provide highly accurate approximations of the computational model while significantly reducing the computational cost. Monte Carlo simulations of the surrogate models are used for uncertainty propagation. UQ of strain fields highlight significant spatial variation in model uncertainty, and reveal key differences in uncertainty among commonly used strain-based brain injury predictor variables.
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Reconfigurable intelligent surface (RIS) is very promising for wireless networks to achieve high energy efficiency, extended coverage, improved capacity, massive connectivity, etc. To unleash the full potentials of RIS-aided communications, acquiring accurate channel state information is crucial, which however is very challenging. For RIS-aided multiple-input and multiple-output (MIMO) communications, the existing channel estimation methods have computational complexity growing rapidly with the number of RIS units $N$ (e.g., in the order of $N^2$ or $N^3$) and/or have special requirements on the matrices involved (e.g., the matrices need to be sparse for algorithm convergence to achieve satisfactory performance), which hinder their applications. In this work, instead of using the conventional signal model in the literature, we derive a new signal model obtained through proper vectorization and reduction operations. Then, leveraging the unitary approximate message passing (UAMP), we develop a more efficient channel estimator that has complexity linear with $N$ and does not have special requirements on the relevant matrices, thanks to the robustness of UAMP. These facilitate the applications of the proposed algorithm to a general RIS-aided MIMO system with a larger $N$. Moreover, extensive numerical results show that the proposed estimator delivers much better performance and/or requires significantly less number of training symbols, thereby leading to notable reductions in both training overhead and latency.
Let $X$ be a random variable with unknown mean and finite variance. We present a new estimator of the mean of $X$ that is robust with respect to the possible presence of outliers in the sample, provides tight sub-Gaussian deviation guarantees without any additional assumptions on the shape or tails of the distribution, and moreover is asymptotically efficient. This is the first estimator that provably combines all these qualities in one package. Our construction is inspired by robustness properties possessed by the self-normalized sums. Theoretical findings are supplemented by numerical simulations highlighting strong performance of the proposed estimator in comparison with previously known techniques.
We consider two-stage robust optimization problems, which can be seen as games between a decision maker and an adversary. After the decision maker fixes part of the solution, the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the partial first-stage solution to a full solution. We extend this classic setting by adding another adversary stage after the second decision-maker stage, which results in min-max-min-max problems, thus pushing two-stage settings further towards more general multi-stage problems. We focus on budgeted uncertainty sets and consider both the continuous and discrete case. For the former, we show that a wide range of robust combinatorial optimization problems can be decomposed into polynomially many subproblems, which can be solved in polynomial time for example in the case of (\textsc{representative}) \textsc{selection}. For the latter, we prove NP-hardness for a wide range of problems, but note that the special case where first- and second-stage adversarial costs are equal can remain solvable in polynomial time.
In this paper, we open up new avenues for visual servoing systems built upon the Path Integral (PI) optimal control theory, in which the non-linear partial differential equation (PDE) can be transformed into an expectation over all possible trajectories using the Feynman-Kac (FK) lemma. More precisely, we propose an MPPI-VS control strategy, a real-time and inversion-free control strategy on the basis of sampling-based model predictive control (namely, Model Predictive Path Integral (MPPI) control) algorithm, for both image-based, 3D point, and position-based visual servoing techniques, taking into account the system constraints (such as visibility, 3D, and control constraints) and parametric uncertainties associated with the robot and camera models as well as measurement noise. Contrary to classical visual servoing control schemes, our control strategy directly utilizes the approximation of the interaction matrix, without the need for estimating the interaction matrix inversion or performing the pseudo-inversion. We validate the MPPI-VS control strategy as well as the classical control schemes on a 6-DoF Cartesian robot with an eye-in-hand camera based on the utilization of four points in the image plane as visual features. To better assess and demonstrate the robustness and potential advantages of our proposed control strategy compared to classical schemes, intensive simulations under various operating conditions are carried out and then discussed. The obtained results demonstrate the effectiveness and capability of the proposed scheme in coping easily with the system constraints, as well as its robustness in the presence of large errors in camera parameters and measurements.
We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic Lagrangian (i.e., path-based) predictions which are obtained from dynamical systems generated by uncertain (Eulerian) vector fields. This work is motivated by the desire to improve Lagrangian predictions in complex dynamical systems based either on analytically simplified or data-driven models. We derive a hierarchy of general information bounds on uncertainty in estimates of statistical observables $\mathbb{E}^{\nu}[f]$, evaluated on trajectories of the approximating dynamical system, relative to the "true'' observables $\mathbb{E}^{\mu}[f]$ in terms of certain $\varphi$-divergences, $\mathcal{D}_\varphi(\mu\|\nu)$, which quantify discrepancies between probability measures $\mu$ associated with the original dynamics and their approximations $\nu$. We then derive two distinct bounds on $\mathcal{D}_\varphi(\mu\|\nu)$ itself in terms of the Eulerian fields. This new framework provides a rigorous way for quantifying and mitigating uncertainty in Lagrangian predictions due to Eulerian model error.
This paper deals with a special type of Lyapunov functions, namely the solution of Zubov's equation. Such a function can be used to characterize the domain of attraction for systems of ordinary differential equations. We derive and prove an integral form solution to Zubov's equation. For numerical computation, we develop two data-driven methods. One is based on the integration of an augmented system of differential equations; and the other one is based on deep learning. The former is effective for systems with a relatively low state space dimension and the latter is developed for high dimensional problems. The deep learning method is applied to a New England 10-generator power system model. We prove that a neural network approximation exists for the Lyapunov function of power systems such that the approximation error is a cubic polynomial of the number of generators. The error convergence rate as a function of n, the number of neurons, is proved.
Recovering a dense depth image from sparse LiDAR scans is a challenging task. Despite the popularity of color-guided methods for sparse-to-dense depth completion, they treated pixels equally during optimization, ignoring the uneven distribution characteristics in the sparse depth map and the accumulated outliers in the synthesized ground truth. In this work, we introduce uncertainty-driven loss functions to improve the robustness of depth completion and handle the uncertainty in depth completion. Specifically, we propose an explicit uncertainty formulation for robust depth completion with Jeffrey's prior. A parametric uncertain-driven loss is introduced and translated to new loss functions that are robust to noisy or missing data. Meanwhile, we propose a multiscale joint prediction model that can simultaneously predict depth and uncertainty maps. The estimated uncertainty map is also used to perform adaptive prediction on the pixels with high uncertainty, leading to a residual map for refining the completion results. Our method has been tested on KITTI Depth Completion Benchmark and achieved the state-of-the-art robustness performance in terms of MAE, IMAE, and IRMSE metrics.
We study constrained reinforcement learning (CRL) from a novel perspective by setting constraints directly on state density functions, rather than the value functions considered by previous works. State density has a clear physical and mathematical interpretation, and is able to express a wide variety of constraints such as resource limits and safety requirements. Density constraints can also avoid the time-consuming process of designing and tuning cost functions required by value function-based constraints to encode system specifications. We leverage the duality between density functions and Q functions to develop an effective algorithm to solve the density constrained RL problem optimally and the constrains are guaranteed to be satisfied. We prove that the proposed algorithm converges to a near-optimal solution with a bounded error even when the policy update is imperfect. We use a set of comprehensive experiments to demonstrate the advantages of our approach over state-of-the-art CRL methods, with a wide range of density constrained tasks as well as standard CRL benchmarks such as Safety-Gym.
Despite the state-of-the-art performance for medical image segmentation, deep convolutional neural networks (CNNs) have rarely provided uncertainty estimations regarding their segmentation outputs, e.g., model (epistemic) and image-based (aleatoric) uncertainties. In this work, we analyze these different types of uncertainties for CNN-based 2D and 3D medical image segmentation tasks. We additionally propose a test-time augmentation-based aleatoric uncertainty to analyze the effect of different transformations of the input image on the segmentation output. Test-time augmentation has been previously used to improve segmentation accuracy, yet not been formulated in a consistent mathematical framework. Hence, we also propose a theoretical formulation of test-time augmentation, where a distribution of the prediction is estimated by Monte Carlo simulation with prior distributions of parameters in an image acquisition model that involves image transformations and noise. We compare and combine our proposed aleatoric uncertainty with model uncertainty. Experiments with segmentation of fetal brains and brain tumors from 2D and 3D Magnetic Resonance Images (MRI) showed that 1) the test-time augmentation-based aleatoric uncertainty provides a better uncertainty estimation than calculating the test-time dropout-based model uncertainty alone and helps to reduce overconfident incorrect predictions, and 2) our test-time augmentation outperforms a single-prediction baseline and dropout-based multiple predictions.
Data augmentation has been widely used for training deep learning systems for medical image segmentation and plays an important role in obtaining robust and transformation-invariant predictions. However, it has seldom been used at test time for segmentation and not been formulated in a consistent mathematical framework. In this paper, we first propose a theoretical formulation of test-time augmentation for deep learning in image recognition, where the prediction is obtained through estimating its expectation by Monte Carlo simulation with prior distributions of parameters in an image acquisition model that involves image transformations and noise. We then propose a novel uncertainty estimation method based on the formulated test-time augmentation. Experiments with segmentation of fetal brains and brain tumors from 2D and 3D Magnetic Resonance Images (MRI) showed that 1) our test-time augmentation outperforms a single-prediction baseline and dropout-based multiple predictions, and 2) it provides a better uncertainty estimation than calculating the model-based uncertainty alone and helps to reduce overconfident incorrect predictions.
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