Computed tomography has propelled scientific advances in fields from biology to materials science. This technology allows for the elucidation of 3-dimensional internal structure by the attenuation of x-rays through an object at different rotations relative to the beam. By imaging 2-dimensional projections, a 3-dimensional object can be reconstructed through a computational algorithm. Imaging at a greater number of rotation angles allows for improved reconstruction. However, taking more measurements increases the x-ray dose and may cause sample damage. Deep neural networks have been used to transform sparse 2-D projection measurements to a 3-D reconstruction by training on a dataset of known similar objects. However, obtaining high-quality object reconstructions for the training dataset requires high x-ray dose measurements that can destroy or alter the specimen before imaging is complete. This becomes a chicken-and-egg problem: high-quality reconstructions cannot be generated without deep learning, and the deep neural network cannot be learned without the reconstructions. This work develops and validates a self-supervised probabilistic deep learning technique, the physics-informed variational autoencoder, to solve this problem. A dataset consisting solely of sparse projection measurements from each object is used to jointly reconstruct all objects of the set. This approach has the potential to allow visualization of fragile samples with x-ray computed tomography. We release our code for reproducing our results at: https://github.com/vganapati/CT_PVAE .
The versatility to learn from a handful of samples is the hallmark of human intelligence. Few-shot learning is an endeavour to transcend this capability down to machines. Inspired by the promise and power of probabilistic deep learning, we propose a novel variational inference network for few-shot classification (coined as TRIDENT) to decouple the representation of an image into semantic and label latent variables, and simultaneously infer them in an intertwined fashion. To induce task-awareness, as part of the inference mechanics of TRIDENT, we exploit information across both query and support images of a few-shot task using a novel built-in attention-based transductive feature extraction module (we call AttFEX). Our extensive experimental results corroborate the efficacy of TRIDENT and demonstrate that, using the simplest of backbones, it sets a new state-of-the-art in the most commonly adopted datasets miniImageNet and tieredImageNet (offering up to 4% and 5% improvements, respectively), as well as for the recent challenging cross-domain miniImagenet --> CUB scenario offering a significant margin (up to 20% improvement) beyond the best existing cross-domain baselines. Code and experimentation can be found in our GitHub repository: https://github.com/anujinho/trident
The ability to estimate epistemic uncertainty is often crucial when deploying machine learning in the real world, but modern methods often produce overconfident, uncalibrated uncertainty predictions. A common approach to quantify epistemic uncertainty, usable across a wide class of prediction models, is to train a model ensemble. In a naive implementation, the ensemble approach has high computational cost and high memory demand. This challenges in particular modern deep learning, where even a single deep network is already demanding in terms of compute and memory, and has given rise to a number of attempts to emulate the model ensemble without actually instantiating separate ensemble members. We introduce FiLM-Ensemble, a deep, implicit ensemble method based on the concept of Feature-wise Linear Modulation (FiLM). That technique was originally developed for multi-task learning, with the aim of decoupling different tasks. We show that the idea can be extended to uncertainty quantification: by modulating the network activations of a single deep network with FiLM, one obtains a model ensemble with high diversity, and consequently well-calibrated estimates of epistemic uncertainty, with low computational overhead in comparison. Empirically, FiLM-Ensemble outperforms other implicit ensemble methods, and it and comes very close to the upper bound of an explicit ensemble of networks (sometimes even beating it), at a fraction of the memory cost.
Accurate uncertainty quantification is a major challenge in deep learning, as neural networks can make overconfident errors and assign high confidence predictions to out-of-distribution (OOD) inputs. The most popular approaches to estimate predictive uncertainty in deep learning are methods that combine predictions from multiple neural networks, such as Bayesian neural networks (BNNs) and deep ensembles. However their practicality in real-time, industrial-scale applications are limited due to the high memory and computational cost. Furthermore, ensembles and BNNs do not necessarily fix all the issues with the underlying member networks. In this work, we study principled approaches to improve uncertainty property of a single network, based on a single, deterministic representation. By formalizing the uncertainty quantification as a minimax learning problem, we first identify distance awareness, i.e., the model's ability to quantify the distance of a testing example from the training data, as a necessary condition for a DNN to achieve high-quality (i.e., minimax optimal) uncertainty estimation. We then propose Spectral-normalized Neural Gaussian Process (SNGP), a simple method that improves the distance-awareness ability of modern DNNs with two simple changes: (1) applying spectral normalization to hidden weights to enforce bi-Lipschitz smoothness in representations and (2) replacing the last output layer with a Gaussian process layer. On a suite of vision and language understanding benchmarks, SNGP outperforms other single-model approaches in prediction, calibration and out-of-domain detection. Furthermore, SNGP provides complementary benefits to popular techniques such as deep ensembles and data augmentation, making it a simple and scalable building block for probabilistic deep learning. Code is open-sourced at https://github.com/google/uncertainty-baselines
The transition to a fully renewable energy grid requires better forecasting of demand at the low-voltage level to increase efficiency and ensure reliable control. However, high fluctuations and increasing electrification cause huge forecast variability, not reflected in traditional point estimates. Probabilistic load forecasts take future uncertainties into account and thus allow more informed decision-making for the planning and operation of low-carbon energy systems. We propose an approach for flexible conditional density forecasting of short-term load based on Bernstein polynomial normalizing flows, where a neural network controls the parameters of the flow. In an empirical study with 363 smart meter customers, our density predictions compare favorably against Gaussian and Gaussian mixture densities. Also, they outperform a non-parametric approach based on the pinball loss for 24h-ahead load forecasting for two different neural network architectures.
The worldwide variation in vegetation height is fundamental to the global carbon cycle and central to the functioning of ecosystems and their biodiversity. Geospatially explicit and, ideally, highly resolved information is required to manage terrestrial ecosystems, mitigate climate change, and prevent biodiversity loss. Here, we present the first global, wall-to-wall canopy height map at 10 m ground sampling distance for the year 2020. No single data source meets these requirements: dedicated space missions like GEDI deliver sparse height data, with unprecedented coverage, whereas optical satellite images like Sentinel-2 offer dense observations globally, but cannot directly measure vertical structures. By fusing GEDI with Sentinel-2, we have developed a probabilistic deep learning model to retrieve canopy height from Sentinel-2 images anywhere on Earth, and to quantify the uncertainty in these estimates. The presented approach reduces the saturation effect commonly encountered when estimating canopy height from satellite images, allowing to resolve tall canopies with likely high carbon stocks. According to our map, only 5% of the global landmass is covered by trees taller than 30 m. Such data play an important role for conservation, e.g., we find that only 34% of these tall canopies are located within protected areas. Our model enables consistent, uncertainty-informed worldwide mapping and supports an ongoing monitoring to detect change and inform decision making. The approach can serve ongoing efforts in forest conservation, and has the potential to foster advances in climate, carbon, and biodiversity modelling.
The COVID-19 pandemic has significantly impacted the tourism and hospitality sector. Public policies such as travel restrictions and stay-at-home orders had significantly affected tourist activities and service businesses' operations and profitability. To this end, it is essential to develop an interpretable forecast model that supports managerial and organizational decision-making. We developed DemandNet, a novel deep learning framework for predicting time series data under the influence of the COVID-19 pandemic. The framework starts by selecting the top static and dynamic features embedded in the time series data. Then, it includes a nonlinear model which can provide interpretable insight into the previously seen data. Lastly, a prediction model is developed to leverage the above characteristics to make robust long-term forecasts. We evaluated the framework using daily hotel demand and revenue data from eight cities in the US. Our findings reveal that DemandNet outperforms the state-of-art models and can accurately predict the impact of the COVID-19 pandemic on hotel demand and revenues.
Neural density estimators have proven remarkably powerful in performing efficient simulation-based Bayesian inference in various research domains. In particular, the BayesFlow framework uses a two-step approach to enable amortized parameter estimation in settings where the likelihood function is implicitly defined by a simulation program. But how faithful is such inference when simulations are poor representations of reality? In this paper, we conceptualize the types of model misspecification arising in simulation-based inference and systematically investigate the performance of the BayesFlow framework under these misspecifications. We propose an augmented optimization objective which imposes a probabilistic structure on the latent data space and utilize maximum mean discrepancy (MMD) to detect potentially catastrophic misspecifications during inference undermining the validity of the obtained results. We verify our detection criterion on a number of artificial and realistic misspecifications, ranging from toy conjugate models to complex models of decision making and disease outbreak dynamics applied to real data. Further, we show that posterior inference errors increase as a function of the distance between the true data-generating distribution and the typical set of simulations in the latent summary space. Thus, we demonstrate the dual utility of MMD as a method for detecting model misspecification and as a proxy for verifying the faithfulness of amortized Bayesian inference.
Data-driven forecasts of air quality have recently achieved more accurate short-term predictions. Despite their success, most of the current data-driven solutions lack proper quantifications of model uncertainty that communicate how much to trust the forecasts. Recently, several practical tools to estimate uncertainty have been developed in probabilistic deep learning. However, there have not been empirical applications and extensive comparisons of these tools in the domain of air quality forecasts. Therefore, this work applies state-of-the-art techniques of uncertainty quantification in a real-world setting of air quality forecasts. Through extensive experiments, we describe training probabilistic models and evaluate their predictive uncertainties based on empirical performance, reliability of confidence estimate, and practical applicability. We also propose improving these models using "free" adversarial training and exploiting temporal and spatial correlation inherent in air quality data. Our experiments demonstrate that the proposed models perform better than previous works in quantifying uncertainty in data-driven air quality forecasts. Overall, Bayesian neural networks provide a more reliable uncertainty estimate but can be challenging to implement and scale. Other scalable methods, such as deep ensemble, Monte Carlo (MC) dropout, and stochastic weight averaging-Gaussian (SWAG), can perform well if applied correctly but with different tradeoffs and slight variations in performance metrics. Finally, our results show the practical impact of uncertainty estimation and demonstrate that, indeed, probabilistic models are more suitable for making informed decisions. Code and dataset are available at \url{https://github.com/Abdulmajid-Murad/deep_probabilistic_forecast}
For many novel applications, such as patient-specific computer-aided surgery, conventional solution techniques of the underlying nonlinear problems are usually computationally too expensive and are lacking information about how certain can we be about their predictions. In the present work, we propose a highly efficient deep-learning surrogate framework that is able to accurately predict the response of bodies undergoing large deformations in real-time. The surrogate model has a convolutional neural network architecture, called U-Net, which is trained with force-displacement data obtained with the finite element method. We propose deterministic and probabilistic versions of the framework. The probabilistic framework utilizes the Variational Bayes Inference approach and is able to capture all the uncertainties present in the data as well as in the deep-learning model. Based on several benchmark examples, we show the predictive capabilities of the framework and discuss its possible limitations
The continued spread of electric vehicles raises new challenges for the supporting digital infrastructure. For example, long-distance route planning for such vehicles relies on the prediction of both the expected travel time as well as energy use. We envision a two-tier architecture to produce such predictions. First, a routing and travel-time-prediction subsystem generates a suggested route and predicts how the speed will vary along the route. Next, the expected energy use is predicted from the speed profile and other contextual characteristics, such as weather information and slope. To this end, the paper proposes deep-learning models that are built from EV tracking data. First, as the speed profile of a route is one of the main predictors for energy use, different simple ways to build speed profiles are explored. Next, eight different deep-learning models for energy-use prediction are proposed. Four of the models are probabilistic in that they predict not a single-point estimate but parameters of a probability distribution of energy use on the route. This is particularly relevant when predicting EV energy use, which is highly sensitive to many input characteristics and, thus, can hardly be predicted precisely. Extensive experiments with two real-world EV tracking datasets validate the proposed methods. The code for this research has been made available on GitHub.
Causal discovery, i.e., inferring underlying causal relationships from observational data, has been shown to be highly challenging for AI systems. In time series modeling context, traditional causal discovery methods mainly consider constrained scenarios with fully observed variables and/or data from stationary time-series. We develop a causal discovery approach to handle a wide class of non-stationary time-series that are conditionally stationary, where the non-stationary behaviour is modeled as stationarity conditioned on a set of (possibly hidden) state variables. Named State-Dependent Causal Inference (SDCI), our approach is able to recover the underlying causal dependencies, provably with fully-observed states and empirically with hidden states. The latter is confirmed by experiments on synthetic linear system and nonlinear particle interaction data, where SDCI achieves superior performance over baseline causal discovery methods. Improved results over non-causal RNNs on modeling NBA player movements demonstrate the potential of our method and motivate the use of causality-driven methods for forecasting.
To validate the safety of automated vehicles (AV), scenario-based testing aims to systematically describe driving scenarios an AV might encounter. In this process, continuous inputs such as velocities result in an infinite number of possible variations of a scenario. Thus, metamodels are used to perform analyses or to select specific variations for examination. However, despite the safety criticality of AV testing, metamodels are usually seen as a part of an overall approach, and their predictions are not questioned. This paper analyzes the predictive performance of Gaussian processes (GP), deep Gaussian processes, extra-trees, and Bayesian neural networks (BNN), considering four scenarios with 5 to 20 inputs. Building on this, an iterative approach is introduced and evaluated, which allows to efficiently select test cases for common analysis tasks. The results show that regarding predictive performance, the appropriate selection of test cases is more important than the choice of metamodels. However, the choice of metamodels remains crucial: Their great flexibility allows BNNs to benefit from large amounts of data and to model even the most complex scenarios. In contrast, less flexible models like GPs convince with higher reliability. Hence, relevant test cases are best explored using scalable virtual test setups and flexible models. Subsequently, more realistic test setups and more reliable models can be used for targeted testing and validation.
This is a tutorial and survey paper on Boltzmann Machine (BM), Restricted Boltzmann Machine (RBM), and Deep Belief Network (DBN). We start with the required background on probabilistic graphical models, Markov random field, Gibbs sampling, statistical physics, Ising model, and the Hopfield network. Then, we introduce the structures of BM and RBM. The conditional distributions of visible and hidden variables, Gibbs sampling in RBM for generating variables, training BM and RBM by maximum likelihood estimation, and contrastive divergence are explained. Then, we discuss different possible discrete and continuous distributions for the variables. We introduce conditional RBM and how it is trained. Finally, we explain deep belief network as a stack of RBM models. This paper on Boltzmann machines can be useful in various fields including data science, statistics, neural computation, and statistical physics.
Bayesian neural networks provide a direct and natural way to extend standard deep neural networks to support probabilistic deep learning through the use of probabilistic layers that, traditionally, encode weight (and bias) uncertainty. In particular, hybrid Bayesian neural networks utilize standard deterministic layers together with few probabilistic layers judicially positioned in the networks for uncertainty estimation. A major aspect and benefit of Bayesian inference is that priors, in principle, provide the means to encode prior knowledge for use in inference and prediction. However, it is difficult to specify priors on weights since the weights have no intuitive interpretation. Further, the relationships of priors on weights to the functions computed by networks are difficult to characterize. In contrast, functions are intuitive to interpret and are direct since they map inputs to outputs. Therefore, it is natural to specify priors on functions to encode prior knowledge, and to use them in inference and prediction based on functions. To support this, we propose hybrid Bayesian neural networks with functional probabilistic layers that encode function (and activation) uncertainty. We discuss their foundations in functional Bayesian inference, functional variational inference, sparse Gaussian processes, and sparse variational Gaussian processes. We further perform few proof-of-concept experiments using GPflus, a new library that provides Gaussian process layers and supports their use with deterministic Keras layers to form hybrid neural network and Gaussian process models.
Partition of unity networks (POU-Nets) have been shown capable of realizing algebraic convergence rates for regression and solution of PDEs, but require empirical tuning of training parameters. We enrich POU-Nets with a Gaussian noise model to obtain a probabilistic generalization amenable to gradient-based minimization of a maximum likelihood loss. The resulting architecture provides spatial representations of both noiseless and noisy data as Gaussian mixtures with closed form expressions for variance which provides an estimator of local error. The training process yields remarkably sharp partitions of input space based upon correlation of function values. This classification of training points is amenable to a hierarchical refinement strategy that significantly improves the localization of the regression, allowing for higher-order polynomial approximation to be utilized. The framework scales more favorably to large data sets as compared to Gaussian process regression and allows for spatially varying uncertainty, leveraging the expressive power of deep neural networks while bypassing expensive training associated with other probabilistic deep learning methods. Compared to standard deep neural networks, the framework demonstrates hp-convergence without the use of regularizers to tune the localization of partitions. We provide benchmarks quantifying performance in high/low-dimensions, demonstrating that convergence rates depend only on the latent dimension of data within high-dimensional space. Finally, we introduce a new open-source data set of PDE-based simulations of a semiconductor device and perform unsupervised extraction of a physically interpretable reduced-order basis.
Bayesian neural networks utilize probabilistic layers that capture uncertainty over weights and activations, and are trained using Bayesian inference. Since these probabilistic layers are designed to be drop-in replacement of their deterministic counter parts, Bayesian neural networks provide a direct and natural way to extend conventional deep neural networks to support probabilistic deep learning. However, it is nontrivial to understand, design and train Bayesian neural networks due to their complexities. We discuss the essentials of Bayesian neural networks including duality (deep neural networks, probabilistic models), approximate Bayesian inference, Bayesian priors, Bayesian posteriors, and deep variational learning. We use TensorFlow Probability APIs and code examples for illustration. The main problem with Bayesian neural networks is that the architecture of deep neural networks makes it quite redundant, and costly, to account for uncertainty for a large number of successive layers. Hybrid Bayesian neural networks, which use few probabilistic layers judicially positioned in the networks, provide a practical solution.
Probabilistic deep learning is deep learning that accounts for uncertainty, both model uncertainty and data uncertainty. It is based on the use of probabilistic models and deep neural networks. We distinguish two approaches to probabilistic deep learning: probabilistic neural networks and deep probabilistic models. The former employs deep neural networks that utilize probabilistic layers which can represent and process uncertainty; the latter uses probabilistic models that incorporate deep neural network components which capture complex non-linear stochastic relationships between the random variables. We discuss some major examples of each approach including Bayesian neural networks and mixture density networks (for probabilistic neural networks), and variational autoencoders, deep Gaussian processes and deep mixed effects models (for deep probabilistic models). TensorFlow Probability is a library for probabilistic modeling and inference which can be used for both approaches of probabilistic deep learning. We include its code examples for illustration.
Quantized neural networks (NN) are the common standard to efficiently deploy deep learning models on tiny hardware platforms. However, we notice that quantized NNs are as vulnerable to adversarial attacks as the full-precision models. With the proliferation of neural networks on small devices that we carry or surround us, there is a need for efficient models without sacrificing trust in the prediction in presence of malign perturbations. Current mitigation approaches often need adversarial training or are bypassed when the strength of adversarial examples is increased. In this work, we investigate how a probabilistic framework would assist in overcoming the aforementioned limitations for quantized deep learning models. We explore Stochastic-Shield: a flexible defense mechanism that leverages input filtering and a probabilistic deep learning approach materialized via Monte Carlo Dropout. We show that it is possible to jointly achieve efficiency and robustness by accurately enabling each module without the burden of re-retraining or ad hoc fine-tuning.
Simultaneous Localization and Mapping (SLAM) system typically employ vision-based sensors to observe the surrounding environment. However, the performance of such systems highly depends on the ambient illumination conditions. In scenarios with adverse visibility or in the presence of airborne particulates (e.g. smoke, dust, etc.), alternative modalities such as those based on thermal imaging and inertial sensors are more promising. In this paper, we propose the first complete thermal-inertial SLAM system which combines neural abstraction in the SLAM front end with robust pose graph optimization in the SLAM back end. We model the sensor abstraction in the front end by employing probabilistic deep learning parameterized by Mixture Density Networks (MDN). Our key strategies to successfully model this encoding from thermal imagery are the usage of normalized 14-bit radiometric data, the incorporation of hallucinated visual (RGB) features, and the inclusion of feature selection to estimate the MDN parameters. To enable a full SLAM system, we also design an efficient global image descriptor which is able to detect loop closures from thermal embedding vectors. We performed extensive experiments and analysis using three datasets, namely self-collected ground robot and handheld data taken in indoor environment, and one public dataset (SubT-tunnel) collected in underground tunnel. Finally, we demonstrate that an accurate thermal-inertial SLAM system can be realized in conditions of both benign and adverse visibility.