This study aims to solve the over-reliance on the rank estimation strategy in the standard tensor factorization-based tensor recovery and the problem of a large computational cost in the standard t-SVD-based tensor recovery. To this end, we proposes a new tensor norm with a dual low-rank constraint, which utilizes the low-rank prior and rank information at the same time. In the proposed tensor norm, a series of surrogate functions of the tensor tubal rank can be used to achieve better performance in harness low-rankness within tensor data. It is proven theoretically that the resulting tensor completion model can effectively avoid performance degradation caused by inaccurate rank estimation. Meanwhile, attributed to the proposed dual low-rank constraint, the t-SVD of a smaller tensor instead of the original big one is computed by using a sample trick. Based on this, the total cost at each iteration of the optimization algorithm is reduced to $\mathcal{O}(n^3\log n +kn^3)$ from $\mathcal{O}(n^4)$ achieved with standard methods, where $k$ is the estimation of the true tensor rank and far less than $n$. Our method was evaluated on synthetic and real-world data, and it demonstrated superior performance and efficiency over several existing state-of-the-art tensor completion methods.
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The aim of this paper is to describe a novel non-parametric noise reduction technique from the point of view of Bayesian inference that may automatically improve the signal-to-noise ratio of one- and two-dimensional data, such as e.g. astronomical images and spectra. The algorithm iteratively evaluates possible smoothed versions of the data, the smooth models, obtaining an estimation of the underlying signal that is statistically compatible with the noisy measurements. Iterations stop based on the evidence and the $\chi^2$ statistic of the last smooth model, and we compute the expected value of the signal as a weighted average of the whole set of smooth models. In this paper, we explain the mathematical formalism and numerical implementation of the algorithm, and we evaluate its performance in terms of the peak signal to noise ratio, the structural similarity index, and the time payload, using a battery of real astronomical observations. Our Fully Adaptive Bayesian Algorithm for Data Analysis (FABADA) yields results that, without any parameter tuning, are comparable to standard image processing algorithms whose parameters have been optimized based on the true signal to be recovered, something that is impossible in a real application. State-of-the-art non-parametric methods, such as BM3D, offer slightly better performance at high signal-to-noise ratio, while our algorithm is significantly more accurate for extremely noisy data (higher than $20-40\%$ relative errors, a situation of particular interest in the field of astronomy). In this range, the standard deviation of the residuals obtained by our reconstruction may become more than an order of magnitude lower than that of the original measurements. The source code needed to reproduce all the results presented in this report, including the implementation of the method, is publicly available at //github.com/PabloMSanAla/fabada
We consider the problem of unfair discrimination between two groups and propose a pre-processing method to achieve fairness. Corrective methods like statistical parity usually lead to bad accuracy and do not really achieve fairness in situations where there is a correlation between the sensitive attribute S and the legitimate attribute E (explanatory variable) that should determine the decision. To overcome these drawbacks, other notions of fairness have been proposed, in particular, conditional statistical parity and equal opportunity. However, E is often not directly observable in the data, i.e., it is a latent variable. We may observe some other variable Z representing E, but the problem is that Z may also be affected by S, hence Z itself can be biased. To deal with this problem, we propose BaBE (Bayesian Bias Elimination), an approach based on a combination of Bayes inference and the Expectation-Maximization method, to estimate the most likely value of E for a given Z for each group. The decision can then be based directly on the estimated E. We show, by experiments on synthetic and real data sets, that our approach provides a good level of fairness as well as high accuracy.
We propose a data-driven, closure model for Reynolds-averaged Navier-Stokes (RANS) simulations that incorporates aleatoric, model uncertainty. The proposed closure consists of two parts. A parametric one, which utilizes previously proposed, neural-network-based tensor basis functions dependent on the rate of strain and rotation tensor invariants. This is complemented by latent, random variables which account for aleatoric model errors. A fully Bayesian formulation is proposed, combined with a sparsity-inducing prior in order to identify regions in the problem domain where the parametric closure is insufficient and where stochastic corrections to the Reynolds stress tensor are needed. Training is performed using sparse, indirect data, such as mean velocities and pressures, in contrast to the majority of alternatives that require direct Reynolds stress data. For inference and learning, a Stochastic Variational Inference scheme is employed, which is based on Monte Carlo estimates of the pertinent objective in conjunction with the reparametrization trick. This necessitates derivatives of the output of the RANS solver, for which we developed an adjoint-based formulation. In this manner, the parametric sensitivities from the differentiable solver can be combined with the built-in, automatic differentiation capability of the neural network library in order to enable an end-to-end differentiable framework. We demonstrate the capability of the proposed model to produce accurate, probabilistic, predictive estimates for all flow quantities, even in regions where model errors are present, on a separated flow in the backward-facing step benchmark problem.
Observational studies are frequently used to estimate the effect of an exposure or treatment on an outcome. To obtain an unbiased estimate of the treatment effect, it is crucial to measure the exposure accurately. A common type of exposure misclassification is recall bias, which occurs in retrospective cohort studies when study subjects may inaccurately recall their past exposure. Specifically, differential recall bias can be problematic when examining the effect of a self-reported binary exposure since the magnitude of recall bias can differ between groups. In this paper, we provide the following contributions: 1) we derive bounds for the average treatment effect (ATE) in the presence of recall bias; 2) we develop several estimation approaches under different identification strategies; 3) we conduct simulation studies to evaluate their performance under several scenarios of model misspecification; 4) we propose a sensitivity analysis method that can examine the robustness of our results with respect to different assumptions; and 5) we apply the proposed framework to an observational study, estimating the effect of childhood physical abuse on adulthood mental health.
Probability density estimation is a core problem of statistics and signal processing. Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely affects the performance. In this paper, we propose a non-classical parametrization for density estimation using sample moments, which does not require the choice of such functions. The parametrization is induced by the squared Hellinger distance, and the solution of it, which is proved to exist and be unique subject to a simple prior that does not depend on data, and can be obtained by convex optimization. Statistical properties of the density estimator, together with an asymptotic error upper bound are proposed for the estimator by power moments. Applications of the proposed density estimator in signal processing tasks are given. Simulation results validate the performance of the estimator by a comparison to several prevailing methods. To the best of our knowledge, the proposed estimator is the first one in the literature for which the power moments up to an arbitrary even order exactly match the sample moments, while the true density is not assumed to fall within specific function classes.
Throughout the life sciences we routinely seek to interpret measurements and observations using parameterised mechanistic mathematical models. A fundamental and often overlooked choice in this approach involves relating the solution of a mathematical model with noisy and incomplete measurement data. This is often achieved by assuming that the data are noisy measurements of the solution of a deterministic mathematical model, and that measurement errors are additive and normally distributed. While this assumption of additive Gaussian noise is extremely common and simple to implement and interpret, it is often unjustified and can lead to poor parameter estimates and non-physical predictions. One way to overcome this challenge is to implement a different measurement error model. In this review, we demonstrate how to implement a range of measurement error models in a likelihood-based framework for estimation, identifiability analysis, and prediction. We focus our implementation within a frequentist profile likelihood-based framework, but our approach is directly relevant to other approaches including sampling-based Bayesian methods. Case studies, motivated by simple caricature models routinely used in the systems biology and mathematical biology literature, illustrate how the same ideas apply to different types of mathematical models. Open-source Julia code to reproduce results is available on GitHub.
Over-the-air computation (AirComp), as a data aggregation method that can improve network efficiency by exploiting the superposition characteristics of wireless channels, has received much attention recently. Meanwhile, the orthogonal time frequency space (OTFS) modulation can provide a strong Doppler resilience and facilitates reliable transmission for high-mobility communications. Hence, in this work, we investigate an OTFS-based AirComp system in the presence of time-frequency dual-selective channels. In particular, we commence from the development of a novel transmission framework for the considered system, where the pilot signal is sent together with data and the channel estimation is implemented according to the echo from the access point to the sensor, thereby reducing the overhead of channel state information (CSI) feedback. Hereafter, based on the CSI estimated from the previous frame, a robust precoding matrix aiming at minimizing mean square error in the current frame is designed, which takes into account the estimation error from the receiver noise and the outdated CSI. The simulation results demonstrate the effectiveness of the proposed robust precoding scheme by comparing it with the non-robust precoding. The performance gain is more obvious in high signal-to-noise ratio in case of large channel estimation errors.
We study the problems of offline and online contextual optimization with feedback information, where instead of observing the loss, we observe, after-the-fact, the optimal action an oracle with full knowledge of the objective function would have taken. We aim to minimize regret, which is defined as the difference between our losses and the ones incurred by an all-knowing oracle. In the offline setting, the decision-maker has information available from past periods and needs to make one decision, while in the online setting, the decision-maker optimizes decisions dynamically over time based a new set of feasible actions and contextual functions in each period. For the offline setting, we characterize the optimal minimax policy, establishing the performance that can be achieved as a function of the underlying geometry of the information induced by the data. In the online setting, we leverage this geometric characterization to optimize the cumulative regret. We develop an algorithm that yields the first regret bound for this problem that is logarithmic in the time horizon. Finally, we show via simulation that our proposed algorithms outperform previous methods from the literature.
Structural data well exists in Web applications, such as social networks in social media, citation networks in academic websites, and threads data in online forums. Due to the complex topology, it is difficult to process and make use of the rich information within such data. Graph Neural Networks (GNNs) have shown great advantages on learning representations for structural data. However, the non-transparency of the deep learning models makes it non-trivial to explain and interpret the predictions made by GNNs. Meanwhile, it is also a big challenge to evaluate the GNN explanations, since in many cases, the ground-truth explanations are unavailable. In this paper, we take insights of Counterfactual and Factual (CF^2) reasoning from causal inference theory, to solve both the learning and evaluation problems in explainable GNNs. For generating explanations, we propose a model-agnostic framework by formulating an optimization problem based on both of the two casual perspectives. This distinguishes CF^2 from previous explainable GNNs that only consider one of them. Another contribution of the work is the evaluation of GNN explanations. For quantitatively evaluating the generated explanations without the requirement of ground-truth, we design metrics based on Counterfactual and Factual reasoning to evaluate the necessity and sufficiency of the explanations. Experiments show that no matter ground-truth explanations are available or not, CF^2 generates better explanations than previous state-of-the-art methods on real-world datasets. Moreover, the statistic analysis justifies the correlation between the performance on ground-truth evaluation and our proposed metrics.
Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.
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