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Change Surfaces for Expressive Multidimensional Changepoints and Counterfactual Prediction

Identifying changes in model parameters is fundamental in machine learning and statistics. However, standard changepoint models are limited in expressiveness, often addressing unidimensional problems and assuming instantaneous changes. We introduce change surfaces as a multidimensional and highly expressive generalization of changepoints. We provide a model-agnostic formalization of change surfaces, illustrating how they can provide variable, heterogeneous, and non-monotonic rates of change across multiple dimensions. Additionally, we show how change surfaces can be used for counterfactual prediction. As a concrete instantiation of the change surface framework, we develop Gaussian Process Change Surfaces (GPCS). We demonstrate counterfactual prediction with Bayesian posterior mean and credible sets, as well as massive scalability by introducing novel methods for additive non-separable kernels. Using two large spatio-temporal datasets we employ GPCS to discover and characterize complex changes that can provide scientific and policy relevant insights. Specifically, we analyze twentieth century measles incidence across the United States and discover previously unknown heterogeneous changes after the introduction of the measles vaccine. Additionally, we apply the model to requests for lead testing kits in New York City, discovering distinct spatial and demographic patterns.

Experimental Design for Cost-Aware Learning of Causal Graphs

We consider the minimum cost intervention design problem: Given the essential graph of a causal graph and a cost to intervene on a variable, identify the set of interventions with minimum total cost that can learn any causal graph with the given essential graph. We first show that this problem is NP-hard. We then prove that we can achieve a constant factor approximation to this problem with a greedy algorithm. We then constrain the sparsity of each intervention. We develop an algorithm that returns an intervention design that is nearly optimal in terms of size for sparse graphs with sparse interventions and we discuss how to use it when there are costs on the vertices.

Iteratively Learning from the Best

We study a simple generic framework to address the issue of bad training data; both bad labels in supervised problems, and bad samples in unsupervised ones. Our approach starts by fitting a model to the whole training dataset, but then iteratively improves it by alternating between (a) revisiting the training data to select samples with lowest current loss, and (b) re-training the model on only these selected samples. It can be applied to any existing model training setting which provides a loss measure for samples, and a way to refit on new ones. We show the merit of this approach in both theory and practice. We first prove statistical consistency, and linear convergence to the ground truth and global optimum, for two simpler model settings: mixed linear regression, and gaussian mixture models. We then demonstrate its success empirically in (a) saving the accuracy of existing deep image classifiers when there are errors in the labels of training images, and (b) improving the quality of samples generated by existing DC-GAN models, when it is given training data that contains a fraction of the images from a different and unintended dataset. The experimental results show significant improvement over the baseline methods that ignore the existence of bad labels/samples.

Text-Adaptive Generative Adversarial Networks: Manipulating Images with Natural Language

This paper addresses the problem of manipulating images using natural language description. Our task aims to semantically modify visual attributes of an object in an image according to the text describing the new visual appearance. Although existing methods synthesize images having new attributes, they do not fully preserve text-irrelevant contents of the original image. In this paper, we propose the text-adaptive generative adversarial network (TAGAN) to generate semantically manipulated images while preserving text-irrelevant contents. The key to our method is the text-adaptive discriminator that creates word-level local discriminators according to input text to classify fine-grained attributes independently. With this discriminator, the generator learns to generate images where only regions that correspond to the given text are modified. Experimental results show that our method outperforms existing methods on CUB and Oxford-102 datasets, and our results were mostly preferred on a user study. Extensive analysis shows that our method is able to effectively disentangle visual attributes and produce pleasing outputs.

AutoInt: Automatic Feature Interaction Learning via Self-Attentive Neural Networks

Click-through rate (CTR) prediction, which aims to predict the probability of a user clicking an ad or an item, is critical to many online applications such as online advertising and recommender systems. The problem is very challenging since (1) the input features (e.g., the user id, user age, item id, item category) are usually sparse and high-dimensional, and (2) an effective prediction relies on high-order combinatorial features (a.k.a. cross features), which are very time-consuming to hand-craft by domain experts and are impossible to be enumerated. Therefore, there have been efforts in finding low-dimensional representations of the sparse and high-dimensional raw features and their meaningful combinations. In this paper, we propose an effective and efficient algorithm to automatically learn the high-order feature combinations of input features. Our proposed algorithm is very general, which can be applied to both numerical and categorical input features. Specifically, we map both the numerical and categorical features into the same low-dimensional space. Afterward, a multi-head self-attentive neural network with residual connections is proposed to explicitly model the feature interactions in the low-dimensional space. With different layers of the multi-head self-attentive neural networks, different orders of feature combinations of input features can be modeled. The whole model can be efficiently fit on large-scale raw data in an end-to-end fashion. Experimental results on four real-world datasets show that our proposed approach not only outperforms existing state-of-the-art approaches for prediction but also offers good explainability.

An approach to predictively securing critical cloud infrastructures through probabilistic modeling

Cloud infrastructures are being increasingly utilized in critical infrastructures such as banking/finance, transportation and utility management. Sophistication and resources used in recent security breaches including those on critical infrastructures show that attackers are no longer limited by monetary/computational constraints. In fact, they may be aided by entities with large financial and human resources. Hence there is urgent need to develop predictive approaches for cyber defense to strengthen cloud infrastructures specifically utilized by critical infrastructures. Extensive research has been done in the past on applying techniques such as Game Theory, Machine Learning and Bayesian Networks among others for the predictive defense of critical infrastructures. However a major drawback of these approaches is that they do not incorporate probabilistic human behavior which limits their predictive ability. In this paper, a stochastic approach is proposed to predict less secure states in critical cloud systems which might lead to potential security breaches. These less-secure states are deemed as risky’ states in our approach. Markov Decision Process (MDP) is used to accurately incorporate user behavior(s) as well as operational behavior of the cloud infrastructure through a set of features. The developed reward/cost mechanism is then used to select appropriate actions’ to identify risky states at future time steps by learning an optimal policy. Experimental results show that the proposed framework performs well in identifying future `risky’ states. Through this work we demonstrate the effectiveness of using probabilistic modeling (MDP) to predictively secure critical cloud infrastructures.

Failing Loudly: An Empirical Study of Methods for Detecting Dataset Shift

We might hope that when faced with unexpected inputs, well-designed software systems would fire off warnings. Machine learning (ML) systems, however, which depend strongly on properties of their inputs (e.g. the i.i.d. assumption), tend to fail silently. This paper explores the problem of building ML systems that fail loudly, investigating methods for detecting dataset shift and identifying exemplars that most typify the shift. We focus on several datasets and various perturbations to both covariates and label distributions with varying magnitudes and fractions of data affected. Interestingly, we show that while classifier-based methods perform well in high-data settings, they perform poorly in low-data settings. Moreover, across the dataset shifts that we explore, a two-sample-testing-based approach, using pretrained classifiers for dimensionality reduction performs best.

Evolutionary Self-Expressive Models for Subspace Clustering

The problem of organizing data that evolves over time into clusters is encountered in a number of practical settings. We introduce evolutionary subspace clustering, a method whose objective is to cluster a collection of evolving data points that lie on a union of low-dimensional evolving subspaces. To learn the parsimonious representation of the data points at each time step, we propose a non-convex optimization framework that exploits the self-expressiveness property of the evolving data while taking into account representation from the preceding time step. To find an approximate solution to the aforementioned non-convex optimization problem, we develop a scheme based on alternating minimization that both learns the parsimonious representation as well as adaptively tunes and infers a smoothing parameter reflective of the rate of data evolution. The latter addresses a fundamental challenge in evolutionary clustering — determining if and to what extent one should consider previous clustering solutions when analyzing an evolving data collection. Our experiments on both synthetic and real-world datasets demonstrate that the proposed framework outperforms state-of-the-art static subspace clustering algorithms and existing evolutionary clustering schemes in terms of both accuracy and running time, in a range of scenarios.

Deep Reinforcement Learning based Recommendation with Explicit User-Item Interactions Modeling

Recommendation is crucial in both academia and industry, and various techniques are proposed such as content-based collaborative filtering, matrix factorization, logistic regression, factorization machines, neural networks and multi-armed bandits. However, most of the previous studies suffer from two limitations: (1) considering the recommendation as a static procedure and ignoring the dynamic interactive nature between users and the recommender systems, (2) focusing on the immediate feedback of recommended items and neglecting the long-term rewards. To address the two limitations, in this paper we propose a novel recommendation framework based on deep reinforcement learning, called DRR. The DRR framework treats recommendation as a sequential decision making procedure and adopts an ‘Actor-Critic’ reinforcement learning scheme to model the interactions between the users and recommender systems, which can consider both the dynamic adaptation and long-term rewards. Furthermore, a state representation module is incorporated into DRR, which can explicitly capture the interactions between items and users. Three instantiation structures are developed. Extensive experiments on four real-world datasets are conducted under both the offline and online evaluation settings. The experimental results demonstrate the proposed DRR method indeed outperforms the state-of-the-art competitors.

Software Engineering Challenges of Deep Learning

Surprisingly promising results have been achieved by deep learning (DL) systems in recent years. Many of these achievements have been reached in academic settings, or by large technology companies with highly skilled research groups and advanced supporting infrastructure. For companies without large research groups or advanced infrastructure, building high-quality production-ready systems with DL components has proven challenging. There is a clear lack of well-functioning tools and best practices for building DL systems. It is the goal of this research to identify what the main challenges are, by applying an interpretive research approach in close collaboration with companies of varying size and type. A set of seven projects have been selected to describe the potential with this new technology and to identify associated main challenges. A set of 12 main challenges has been identified and categorized into the three areas of development, production, and organizational challenges. Furthermore, a mapping between the challenges and the projects is defined, together with selected motivating descriptions of how and why the challenges apply to specific projects. Compared to other areas such as software engineering or database technologies, it is clear that DL is still rather immature and in need of further work to facilitate development of high-quality systems. The challenges identified in this paper can be used to guide future research by the software engineering and DL communities. Together, we could enable a large number of companies to start taking advantage of the high potential of the DL technology.

Modelling rankings in R: the PlackettLuce package

This paper presents the R package PlackettLuce, which implements a generalization of the Plackett-Luce model for rankings data. The generalization accommodates both ties (of any order) and partial rankings (rankings of only some items). By default, the implementation adds a set of pseudo-comparisons with a hypothetical item, ensuring that the network of wins and losses is always strongly connected, i.e. all items are connected to every other item by both a path of wins and a path of losses. This means that the worth of each item can always be estimated by maximum likelihood, with finite standard error. It also has a regularization effect, shrinking the estimated parameters towards equal item worth. In addition to standard methods for model summary, PlackettLuce provides a method to compute quasi standard errors for the item parameters, so that comparison intervals can be derived even when a reference item is set. Finally the package provides a method for model-based partitioning using ranking-specific covariates, enabling the identification of subgroups where items have been ranked differently. These features are demonstrated through application to classic and novel data sets.

Using Machine Learning to Predict the Evolution of Physics Research

The advancement of science as outlined by Popper and Kuhn is largely qualitative, but with bibliometric data it is possible and desirable to develop a quantitative picture of scientific progress. Furthermore it is also important to allocate finite resources to research topics that have growth potential, to accelerate the process from scientific breakthroughs to technological innovations. In this paper, we address this problem of quantitative knowledge evolution by analysing the APS publication data set from 1981 to 2010. We build the bibliographic coupling and co-citation networks, use the Louvain method to detect topical clusters (TCs) in each year, measure the similarity of TCs in consecutive years, and visualize the results as alluvial diagrams. Having the predictive features describing a given TC and its known evolution in the next year, we can train a machine learning model to predict future changes of TCs, i.e., their continuing, dissolving, merging and splitting. We found the number of papers from certain journals, the degree, closeness, and betweenness to be the most predictive features. Additionally, betweenness increases significantly for merging events, and decreases significantly for splitting events. Our results represent a first step from a descriptive understanding of the Science of Science (SciSci), towards one that is ultimately prescriptive.

Unsupervised Data Selection for Supervised Learning

Recent research put a big effort in the development of deep learning architectures and optimizers obtaining impressive results in areas ranging from vision to language processing. However little attention has been addressed to the need of a methodological process of data collection. In this work we show that high quality data for supervised learning can be selected in an unsupervised manner and that by doing so one can obtain models capable to generalize better than in the case of random training set construction.

Model-Based Active Exploration

Efficient exploration is an unsolved problem in Reinforcement Learning. We introduce Model-Based Active eXploration (MAX), an algorithm that actively explores the environment. It minimizes data required to comprehensively model the environment by planning to observe novel events, instead of merely reacting to novelty encountered by chance. Non-stationarity induced by traditional exploration bonus techniques is avoided by constructing fresh exploration policies only at time of action. In semi-random toy environments where directed exploration is critical to make progress, our algorithm is at least an order of magnitude more efficient than strong baselines.

Causal Inference in Nonverbal Dyadic Communication with Relevant Interval Selection and Granger Causality

Human nonverbal emotional communication in dyadic dialogs is a process of mutual influence and adaptation. Identifying the direction of influence, or cause-effect relation between participants is a challenging task, due to two main obstacles. First, distinct emotions might not be clearly visible. Second, participants cause-effect relation is transient and variant over time. In this paper, we address these difficulties by using facial expressions that can be present even when strong distinct facial emotions are not visible. We also propose to apply a relevant interval selection approach prior to causal inference to identify those transient intervals where adaptation process occurs. To identify the direction of influence, we apply the concept of Granger causality to the time series of facial expressions on the set of relevant intervals. We tested our approach on synthetic data and then applied it to newly, experimentally obtained data. Here, we were able to show that a more sensitive facial expression detection algorithm and a relevant interval detection approach is most promising to reveal the cause-effect pattern for dyadic communication in various instructed interaction conditions.

Learning Comment Generation by Leveraging User-Generated Data

Existing models on open-domain comment generation produce repetitive and uninteresting response. To cope with this issue, we propose a combined approach of retrieval and generation methods. We introduce an attentive scorer to retrieve informative and relevant comments by using user-generated data. Then, we use the retrieved comments to train our sequence-to-sequence model with copy mechanism to copy important keywords from articles. We show the robustness of our model, and it can alleviate the issue. In our experiments, our proposed generative model significantly outperforms the Seq2Seq with attention model and Information Retrieval models by around 27 and 30 BLEU-1 points respectively.

Kalman Gradient Descent: Adaptive Variance Reduction in Stochastic Optimization

We introduce Kalman Gradient Descent, a stochastic optimization algorithm that uses Kalman filtering to adaptively reduce gradient variance in stochastic gradient descent by filtering the gradient estimates. We present both a theoretical analysis of convergence in a non-convex setting and experimental results which demonstrate improved performance on a variety of machine learning areas including neural networks and black box variational inference. We also present a distributed version of our algorithm that enables large-dimensional optimization, and we extend our algorithm to SGD with momentum and RMSProp.

Principled Uncertainty Estimation for Deep Neural Networks

When the cost of misclassifying a sample is high, it is useful to have an accurate estimate of uncertainty in the prediction for that sample. There are also multiple types of uncertainty which are best estimated in different ways, for example, uncertainty that is intrinsic to the training set may be well-handled by a Bayesian approach, while uncertainty introduced by shifts between training and query distributions may be better-addressed by density/support estimation. In this paper, we examine three types of uncertainty: model capacity uncertainty, intrinsic data uncertainty, and open set uncertainty, and review techniques that have been derived to address each one. We then introduce a unified hierarchical model, which combines methods from Bayesian inference, invertible latent density inference, and discriminative classification in a single end-to-end deep neural network topology to yield efficient per-sample uncertainty estimation. Our approach addresses all three uncertainty types and readily accommodates prior/base rates for binary detection.

Three Mechanisms of Weight Decay Regularization

• MaxHedge: Maximising a Maximum Online with Theoretical Performance Guarantees• LPCNet: Improving Neural Speech Synthesis Through Linear Prediction• Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes• Scale Estimation of Monocular SfM for a Multi-modal Stereo Camera• Exploring $k$ out of Top $ρ$ Fraction of Arms in Stochastic Bandits• Consistency of ELBO maximization for model selection• Near-Linear Time Insertion-Deletion Codes and (1+$\varepsilon$)-Approximating Edit Distance via Indexing• Sequential anatomy localization in fetal echocardiography videos• The Chain of Antichains — Box Protocol: the Dual-Blockchain and a Stablecoin• Automatically Evolving CNN Architectures Based on Blocks• Learning Criteria and Evaluation Metrics for Textual Transfer between Non-Parallel Corpora• Bounded Regression with Gaussian Process Projection• FFT, FMM, and Multigrid on the Road to Exascale: performance challenges and opportunities• Pattern formation for colloidal systems• Active Learning of Uniformly Accurate Inter-atomic Potentials for Materials Simulation• An Efficient Implementation of Riemannian Manifold Hamiltonian Monte Carlo for Gaussian Process Models• A unifying picture of generalized thermodynamic uncertainty relations• Language Modeling for Code-Switching: Evaluation, Integration of Monolingual Data, and Discriminative Training• Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons• Accurate, Efficient and Scalable Graph Embedding• Cultural transmission modes of music sampling traditions remain stable despite delocalization in the digital age• Energy-Reliability Aware Link Optimization for Battery-Powered IoT Devices with Non-Ideal Power Amplifiers• Dynamic Thresholding Mechanisms for IR-Based Filtering in Efficient Source Code Plagiarism Detection• Eigenvalue Fluctuations of Symmetric Group Permutation Representations on k-tuples and k-subsets• Sparse Logistic Regression Learns All Discrete Pairwise Graphical Models• Semi-Supervised Translation with MMD Networks• Mean-field theory of graph neural networks in graph partitioning• Learning to Learn without Forgetting By Maximizing Transfer and Minimizing Interference• Probabilistic Multilevel Clustering via Composite Transportation Distance• Rademacher Complexity for Adversarially Robust Generalization• Bumpless Pipedreams, Reduced Word Tableaux and Stanley Symmetric Functions• Location and scale behaviour of the quantiles of a natural exponential family• The Expressive Power of Parameterized Quantum Circuits• Uncertainty Quantification in Three Dimensional Natural Convection using Polynomial Chaos Expansion and Deep Neural Networks• A Distributed Epigenetic Shape Formation and Regeneration Algorithm for a Swarm of Robots• Variational Inference with Tail-adaptive f-Divergence• Optimized Signal Distortion for PAPR Reduction of OFDM Signals with IFFT/FFT Complexity via ADMM Approaches• STFT spectral loss for training a neural speech waveform model• Neural source-filter-based waveform model for statistical parametric speech synthesis• Social Vehicle Swarms: A Novel Perspective on Social-aware Vehicular Communication Architecture• Parallel Entangling Operations on a Universal Ion Trap Quantum Computer• Passivity-Based Analysis of Sampled and Quantized Control Implementations• Characterizing Entities in the Bitcoin Blockchain• An Amalgamation of Classical and Quantum Machine Learning For the Classification of Adenocarcinoma and Squamous Cell Carcinoma Patients• Investigation of enhanced Tacotron text-to-speech synthesis systems with self-attention for pitch accent language• The Universality Class of Nano-Crystal Plasticity: Self-Organization and Localization in Discrete Dislocation Dynamics• R-BBG$_2$: Recursive Bipartition of Bi-connected Graphs• Parameter instability regimes for sparse proximal denoising programs• Complete Weight Distribution and MacWilliams Identities for Asymmetric Quantum Codes• Semi-crowdsourced Clustering with Deep Generative Models• A fast algorithm for globally solving Tikhonov regularized total least squares problem• On Controllable Sparse Alternatives to Softmax• Identification of physical processes via combined data-driven and data-assimilation methods• Altruistic Autonomy: Beating Congestion on Shared Roads• GOT-10k: A Large High-Diversity Benchmark for Generic Object Tracking in the Wild• On chordal phylogeny graphs• The non-linear sewing lemma I: weak formulation• The non-linear sewing lemma II: Lipschitz continuous formulation• Improved multipath time delay estimation using cepstrum subtraction• Distributed Control of Hybrid AC/DC Microgrids: from Synchronous to Asynchronous Algorithms• An Augmented Linear Mixing Model to Address Spectral Variability for Hyperspectral Unmixing• Long-memory Gaussian processes governed by generalized Fokker-Planck equations• Martingale spaces and representations under absolutely continuous changes of probability• The infinitesimal generator of the stochastic Burgers equation• DeepSymmetry : Using 3D convolutional networks for identification of tandem repeats and internal symmetries in protein structures• Simon’s problem for linear functions• Small noise limit and convexity for generalized incompressible flows, Schrödinger problems, and optimal transport• 1-Bit Massive MIMO Downlink Based on Constructive Interference• Logit Pairing Methods Can Fool Gradient-Based Attacks• Patch-based Sparse Representation For Bacterial Detection• Hybrid Analog-Digital Precoding for Interference Exploitation• Simple and Fast BlockQuicksort using Lomuto’s Partitioning Scheme• Refined $q$-Trinomial Coefficients and Two Infinite Hierarcies of $q$-Series Identities• Speaking style adaptation in Text-To-Speech synthesis using Sequence-to-sequence models with attention• Energy-Efficient Soft-Assisted Product Decoders• $\mathbf{2}$-Closure of $\mathbf{\frac{3}{2}}$-transitive group in polynomial time• Traffic Light Queues and the Poisson Clumping Heuristic• Studio e confronto delle strutture di Apache Spark• PartsNet: A Unified Deep Network for Automotive Engine Precision Parts Defect Detection• On the Convergence Rate of Training Recurrent Neural Networks• A tutorial on the synthesis and validation of a closed-loop wind farm controller using a steady-state surrogate model• Marginal Replay vs Conditional Replay for Continual Learning• Asymptotic Upper Capacity Bound for Receive Antenna Selection in Massive MIMO Systems• Learning to Teach with Dynamic Loss Functions• A study on load-balanced variants of the bin packing problem• On the Power of Quantum Queue Automata in Real-time• Burst ranking for blind multi-image deblurring• Efficient Taxonomic Similarity Joins with Adaptive Overlap Constraint• Gradual Machine Learning for Entity Resolution• ActionXPose: A Novel 2D Multi-view Pose-based Algorithm for Real-time Human Action Recognition• Challenges, Designs, and Performances of a 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Feature Selection in Mixtures-of-Experts Models• Real-Time RGB-D Camera Pose Estimation in Novel Scenes using a Relocalisation Cascade• Median activation functions for graph neural networks• Exhaustion approximation for the control problem of the heat or Schrödinger semigroup on unbounded domains• Fast Computation of Genome-Metagenome Interaction Effects• Contextual Speech Recognition with Difficult Negative Training Examples• RF photonic delay lines using space-division multiplexing• Multicore fiber delay line performance against bending and twisting effects• Semi-unsupervised Learning of Human Activity using Deep Generative Models• Variational Calibration of Computer Models• The geometry of the space of branched Rough Paths• An optimal control approach of within day congestion pricing for stochastic transportation networks• Multivariate Analysis and Visualization using R Package muvis• Automatic CNN-based detection of cardiac MR motion artefacts using k-space data augmentation and curriculum learning• DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications• End-to-end music source separation: is it possible in the waveform domain?• Adversarial Attacks on Stochastic Bandits• Novel Near-Optimal Scalar Quantizers with Exponential Decay Rate and Global Convergence• A Coarse-to-fine Pyramidal Model for Person Re-identification via Multi-Loss Dynamic Training• Efficient convex cptimization for optimal PMU placement in large distribution grids• ReviewQA: a relational aspect-based opinion reading dataset• Dreaming neural networks: forgetting spurious memories and reinforcing pure ones• Strong regularization by Brownian noise propagating through a weak Hörmander structure• Sharp Schauder Estimates for some Degenerate Kolmogorov Equations• Leveraging Gaussian Process and Voting-Empowered Many-Objective Evaluation for Fault Identification• Optimal Output Regulation for Square, Over-Actuated and Under-Actuated Linear Systems• Global null-controllability and nonnegative-controllability of slightly superlinear heat equations• Approximate Bayesian Computation via Population Monte Carlo and Classification• Eco-PANDA: A Computationally Economic, Geometrically Converging, Dual Optimization Method on Time-Varying Undirected Graphs• Few-shot 3D Multi-modal Medical Image Segmentation using Generative Adversarial Learning• Seismicity in sheared granular matter• Enabling Factorized Piano Music Modeling and Generation with the MAESTRO Dataset• Efficient Constrained Signal Reconstruction by Randomized Epigraphical Projection• Reflexive polytopes arising from bipartite graphs with $γ$-positivity associated to interior polynomials• Learning Gaussian Processes by Minimizing PAC-Bayesian Generalization Bounds• The problem with probabilistic DAG automata for semantic graphs• Toward Creating Subsurface Camera• Adversarial Risk and Robustness: General Definitions and Implications for the Uniform Distribution• Abelian 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