We propose a regularization method for ensemble Kalman filtering (EnKF) with elliptic observation operators. Commonly used EnKF regularization methods suppress state correlations at long distances. For observations described by elliptic partial differential equations, such as the pressure Poisson equation (PPE) in incompressible fluid flows, distance localization cannot be applied, as we cannot disentangle slowly decaying physical interactions from spurious long-range correlations. This is particularly true for the PPE, in which distant vortex elements couple nonlinearly to induce pressure. Instead, these inverse problems have a low effective dimension: low-dimensional projections of the observations strongly inform a low-dimensional subspace of the state space. We derive a low-rank factorization of the Kalman gain based on the spectrum of the Jacobian of the observation operator. The identified eigenvectors generalize the source and target modes of the multipole expansion, independently of the underlying spatial distribution of the problem. Given rapid spectral decay, inference can be performed in the low-dimensional subspace spanned by the dominant eigenvectors. This low-rank EnKF is assessed on dynamical systems with Poisson observation operators, where we seek to estimate the positions and strengths of point singularities over time from potential or pressure observations. We also comment on the broader applicability of this approach to elliptic inverse problems outside the context of filtering.

There has been an increasing interest in utilizing machine learning methods in inverse problems and imaging. Most of the work has, however, concentrated on image reconstruction problems, and the number of studies regarding the full solution of the inverse problem is limited. In this work, we study a machine learning based approach for the Bayesian inverse problem of photoacoustic tomography. We develop an approach for estimating the posterior distribution in photoacoustic tomography using an approach based on the variational autoencoder. The approach is evaluated with numerical simulations and compared to the solution of the inverse problem using a Bayesian approach.

Score matching (SM) is a convenient method for training flexible probabilistic models, which is often preferred over the traditional maximum-likelihood (ML) approach. However, these models are less interpretable than normalized models; as such, training robustness is in general difficult to assess. We present a critical study of existing variational SM objectives, showing catastrophic failure on a wide range of datasets and network architectures. Our theoretical insights on the objectives emerge directly from their equivalent autoencoding losses when optimizing variational autoencoder (VAE) models. First, we show that in the Fisher autoencoder, SM produces far worse models than maximum-likelihood, and approximate inference by Fisher divergence can lead to low-density local optima. However, with important modifications, this objective reduces to a regularized autoencoding loss that resembles the evidence lower bound (ELBO). This analysis predicts that the modified SM algorithm should behave very similarly to ELBO on Gaussian VAEs. We then review two other FD-based objectives from the literature and show that they reduce to uninterpretable autoencoding losses, likely leading to poor performance. The experiments verify our theoretical predictions and suggest that only ELBO and the baseline objective robustly produce expected results, while previously proposed SM methods do not.

Conditional inference on arbitrary subsets of variables is a core problem in probabilistic inference with important applications such as masked language modeling and image inpainting. In recent years, the family of Any-Order Autoregressive Models (AO-ARMs) -- closely related to popular models such as BERT and XLNet -- has shown breakthrough performance in arbitrary conditional tasks across a sweeping range of domains. But, in spite of their success, in this paper we identify significant improvements to be made to previous formulations of AO-ARMs. First, we show that AO-ARMs suffer from redundancy in their probabilistic model, i.e., they define the same distribution in multiple different ways. We alleviate this redundancy by training on a smaller set of univariate conditionals that still maintains support for efficient arbitrary conditional inference. Second, we upweight the training loss for univariate conditionals that are evaluated more frequently during inference. Our method leads to improved performance with no compromises on tractability, giving state-of-the-art likelihoods in arbitrary conditional modeling on text (Text8), image (CIFAR10, ImageNet32), and continuous tabular data domains.

Causal learning has long concerned itself with the accurate recovery of underlying causal mechanisms. Such causal modelling enables better explanations of out-of-distribution data. Prior works on causal learning assume that the high-level causal variables are given. However, in machine learning tasks, one often operates on low-level data like image pixels or high-dimensional vectors. In such settings, the entire Structural Causal Model (SCM) -- structure, parameters, \textit{and} high-level causal variables -- is unobserved and needs to be learnt from low-level data. We treat this problem as Bayesian inference of the latent SCM, given low-level data. For linear Gaussian additive noise SCMs, we present a tractable approximate inference method which performs joint inference over the causal variables, structure and parameters of the latent SCM from random, known interventions. Experiments are performed on synthetic datasets and a causally generated image dataset to demonstrate the efficacy of our approach. We also perform image generation from unseen interventions, thereby verifying out of distribution generalization for the proposed causal model.

The satisfiability problem is one of the most famous problems in computer science. Its NP-completeness has been used to argue that SAT is intractable. However, there have been tremendous advances that allow SAT solvers to solve instances with millions of variables. A particularly successful paradigm is stochastic local search. In most cases, there are different ways of formulating the underlying problem. While it is known that this has an impact on the runtime of solvers, finding a helpful formulation is generally non-trivial. The recently introduced GapSAT solver [Lorenz and W\"orz 2020] demonstrated a successful way to improve the performance of an SLS solver on average by learning additional information which logically entails from the original problem. Still, there were cases in which the performance slightly deteriorated. This justifies in-depth investigations into how learning logical implications affects runtimes for SLS. In this work, we propose a method for generating logically equivalent problem formulations, generalizing the ideas of GapSAT. This allows a rigorous mathematical study of the effect on the runtime of SLS solvers. If the modification process is treated as random, Johnson SB distributions provide a perfect characterization of the hardness. Since the observed Johnson SB distributions approach lognormal distributions, our analysis also suggests that the hardness is long-tailed. As a second contribution, we theoretically prove that restarts are useful for long-tailed distributions. This implies that additional restarts can further refine all algorithms employing above mentioned modification technique. Since the empirical studies compellingly suggest that the runtime distributions follow Johnson SB distributions, we investigate this property theoretically. We succeed in proving that the runtimes for Sch\"oning's random walk algorithm are approximately Johnson SB.

A critical element in the development and deployment of AVs is the design of car-following (CF) controllers capable of producing desirable performance in real-world settings. Ideally, a CF control system would effectively and safely handle the longitudinal maneuvers of the vehicle at every encounter it faces. However, designing and training such a controller requires enormous data, testing, and experimentation that covers all possible driving scenarios/encounters. In other words, it requires us to have a perfect understanding of the environment these AVs would be operating under. Clearly, this is very challenging and, possibly, unattainable. AVs are likely to encounter unseen scenarios and be exposed to exogenous and endogenous uncertainties in the physical world. The sources of exogenous and endogenous uncertainties are vast and roughly classified into (Macfarlane and Stroila, 2016; Yao et al., 2020; Katrakazas et al., 2015): (i) vehicular and system dynamics (e.g., vehicle condition, road gradient, aerodynamic drag force, external loads, transmission, brake, the performance of the engine, etc.), (ii) environmental conditions (snow, dust, wind, wet conditions, etc.), and (iii) situational detection (e.g., sensor/measurement errors, radar errors, vehicle speed fluctuations, vehicle localization, communication latency, etc.). All these types of uncertainties can hinder desirable performance (e.g., stability). Yet, a major challenge lies in the complexity of integrating these uncertainties into the control system and the design of the AV.

The problem of sequentially maximizing the expectation of a function seeks to maximize the expected value of a function of interest without having direct control on its features. Instead, the distribution of such features depends on a given context and an action taken by an agent. In contrast to Bayesian optimization, the arguments of the function are not under agent's control, but are indirectly determined by the agent's action based on a given context. If the information of the features is to be included in the maximization problem, the full conditional distribution of such features, rather than its expectation only, needs to be accounted for. Furthermore, the function is itself unknown, only counting with noisy observations of such function, and potentially requiring the use of unmatched data sets. We propose a novel algorithm for the aforementioned problem which takes into consideration the uncertainty derived from the estimation of both the conditional distribution of the features and the unknown function, by modeling the former as a Bayesian conditional mean embedding and the latter as a Gaussian process. Our algorithm empirically outperforms the current state-of-the-art algorithm in the experiments conducted.

Estimating causal queries, such as changes in protein abundance in response to a perturbation, is a fundamental task in the analysis of biomolecular pathways. The estimation requires experimental measurements on the pathway components. However, in practice many pathway components are left unobserved (latent) because they are either unknown, or difficult to measure. Latent variable models (LVMs) are well-suited for such estimation. Unfortunately, LVM-based estimation of causal queries can be inaccurate when parameters of the latent variables are not uniquely identified, or when the number of latent variables is misspecified. This has limited the use of LVMs for causal inference in biomolecular pathways. In this manuscript, we propose a general and practical approach for LVM-based estimation of causal queries. We prove that, despite the challenges above, LVM-based estimators of causal queries are accurate if the queries are identifiable according to Pearl's do-calculus, and describe an algorithm for its estimation. We illustrate the breadth and the practical utility of this approach for estimating causal queries in four synthetic and two experimental case studies, where structures of biomolecular pathways challenge the existing methods for causal query estimation. The code and the data documenting all the case studies are available at \url{https://github.com/srtaheri/LVMwithDoCalculus}

When sampling for Bayesian inference, one popular approach is to use Hamiltonian Monte Carlo (HMC) and specifically the No-U-Turn Sampler (NUTS) which automatically decides the end time of the Hamiltonian trajectory. However, HMC and NUTS can require numerous numerical gradients of the target density, and can prove slow in practice. We propose Hamiltonian neural networks (HNNs) with HMC and NUTS for solving Bayesian inference problems. Once trained, HNNs do not require numerical gradients of the target density during sampling. Moreover, they satisfy important properties such as perfect time reversibility and Hamiltonian conservation, making them well-suited for use within HMC and NUTS because stationarity can be shown. We also propose an HNN extension called latent HNNs (L-HNNs), which are capable of predicting latent variable outputs. Compared to HNNs, L-HNNs offer improved expressivity and reduced integration errors. Finally, we employ L-HNNs in NUTS with an online error monitoring scheme to prevent sample degeneracy in regions of low probability density. We demonstrate L-HNNs in NUTS with online error monitoring on several examples involving complex, heavy-tailed, and high-local-curvature probability densities. Overall, L-HNNs in NUTS with online error monitoring satisfactorily inferred these probability densities. Compared to traditional NUTS, L-HNNs in NUTS with online error monitoring required 1--2 orders of magnitude fewer numerical gradients of the target density and improved the effective sample size (ESS) per gradient by an order of magnitude.