This paper introduces a new framework for quantifying predictive uncertainty for both data and models that relies on projecting the data into a Gaussian reproducing kernel Hilbert space (RKHS) and transforming the data probability density function (PDF) in a way that quantifies the flow of its gradient as a topological potential field quantified at all points in the sample space. This enables the decomposition of the PDF gradient flow by formulating it as a moment decomposition problem using operators from quantum physics, specifically the Schrodinger's formulation. We experimentally show that the higher order modes systematically cluster the different tail regions of the PDF, thereby providing unprecedented discriminative resolution of data regions having high epistemic uncertainty. In essence, this approach decomposes local realizations of the data PDF in terms of uncertainty moments. We apply this framework as a surrogate tool for predictive uncertainty quantification of point-prediction neural network models, overcoming various limitations of conventional Bayesian based uncertainty quantification methods. Experimental comparisons with some established methods illustrate performance advantages exhibited by our framework.
Reduced modeling in high-dimensional reproducing kernel Hilbert spaces offers the opportunity to approximate efficiently non-linear dynamics. In this work, we devise an algorithm based on low rank constraint optimization and kernel-based computation that generalizes a recent approach called "kernel-based dynamic mode decomposition". This new algorithm is characterized by a gain in approximation accuracy, as evidenced by numerical simulations, and in computational complexity.
Google Research（谷歌研究院）；Harvard University（哈佛大学）；Columbia University（哥伦比亚大学）
Ensemble learning is a standard approach to building machine learning systems that capture complex phenomena in real-world data. An important aspect of these systems is the complete and valid quantification of model uncertainty. We introduce a Bayesian nonparametric ensemble (BNE) approach that augments an existing ensemble model to account for different sources of model uncertainty. BNE augments a model's prediction and distribution functions using Bayesian nonparametric machinery. It has a theoretical guarantee in that it robustly estimates the uncertainty patterns in the data distribution, …
Distribution-Free Uncertainty Quantification for Kernel Methods by Gradient Perturbations
Hungarian Academy of Sciences（匈牙利科学院）；MTA SZTAKI: Institute for Computer Science and Control（MTA SZTAKI：计算机科学与控制研究所）；EPIC Centre of Excellence（EPIC卓越中心）
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron-Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that …
Uncertainty Quantification for Regression: A Unified Framework based on kernel scores
LMU Munich（慕尼黑莱茵恩大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；DLR-German Aerospace Center University of Tromsø（德国航空航天中心-特罗姆瑟大学）；German Research Center for Artificial Intelligence (DFKI, DSA)（德国人工智能研究中心）
Uncertainty Estimates of Predictions via a General Bias-Variance Decomposition
German Cancer Research Center (DKFZ)（德国癌症研究中心（DKFZ））；German Cancer Consortium (DKTK)（德国癌症联盟（DKTK））；Goethe University Frankfurt（法兰克福歌德大学）；Frankfurt Cancer Institute（法兰克福癌症研究所）
本文提出 proper scores 的通用偏差-方差分解，以 Bregman Information 刻画方差项，并将其用于模型集成、置信区域及各程度域漂移下的分布外检测。
Uncertainty Quantification with the Empirical Neural Tangent Kernel
School of Mathematics And Physics University of Queensland（数学与物理学院，昆士兰大学）；Dept. of Electrical and Electronic Engineering University of Melbourne（电子与电气工程系，墨尔本大学）；School of Mathematics and Statistics University of Melbourne（数学与统计学学院，墨尔本大学）；CIRES and School of Mathematics And Physics University of Queensland（CIRES和数学与物理学院，昆士兰大学）
Deep neural networks are increasingly being used for the analysis of medical images. However, most works neglect the uncertainty in the model's prediction. We propose an uncertainty-aware deep kernel learning model which permits the estimation of the uncertainty in the prediction by a pipeline of a Convolutional Neural Network and a sparse Gaussian Process. Furthermore, we adapt different pre-training methods to investigate their impacts on the proposed model. We apply our approach to Bone Age Prediction and Lesion Localization. In most cases, the proposed model shows better performance compar…
The ever-growing size of the datasets renders well-studied learning techniques, such as Kernel Ridge Regression, inapplicable, posing a serious computational challenge. Divide-and-conquer is a common remedy, suggesting to split the dataset into disjoint partitions, obtain the local estimates and average them, it allows to scale-up an otherwise ineffective base approach. In the current study we suggest a fully data-driven approach to quantify uncertainty of the averaged estimator. Namely, we construct simultaneous element-wise confidence bands for the predictions yielded by the averaged estimat…
Traditional Bayesian approaches for model uncertainty quantification rely on notoriously difficult processes of marginalization over each network parameter to estimate its probability density function (PDF). Our hypothesis is that internal layer outputs of a trained neural network contain all of the information related to both its mapping function (quantified by its weights) as well as the input data distribution. We therefore propose a framework for predictive uncertainty quantification of a trained neural network that explicitly estimates the PDF of its raw prediction space (before activatio…
Global sensitivity analysis is the main quantitative technique for identifying the most influential input variables in a numerical simulation model. In particular when the inputs are independent, Sobol' sensitivity indices attribute a portion of the output of interest variance to each input and all possible interactions in the model, thanks to a functional ANOVA decomposition. On the other hand, moment-independent sensitivity indices focus on the impact of input variables on the whole output distribution instead of the variance only, thus providing complementary insight on the inputs / output …
A Posterior-Predictive Variance Decomposition for Epistemic and Aleatoric Uncertainty in Wind Power Forecasting
School of Engineering, Deakin University（德肯大学工程学院）；ARC Training Centre in Energy Technologies for Future Grids, School of Engineering, University of Wollongong（未来电网能源技术培训中心，沃林戈大学工程学院）
Single Image Inpainting and Super-Resolution with Simultaneous Uncertainty Guarantees by Universal Reproducing Kernels
Institute for Computer Science and Control (SZTAKI)（计算机科学与控制研究所）；Institute of Mathematics, Budapest University of Technology and Economics (BME)（布达佩斯技术大学数学研究所）
In this manuscript, we propose a multiclass data description model based on kernel Mahalanobis distance (MDD-KM) with self-adapting hyperparameter setting. MDD-KM provides uncertainty quantification and can be deployed to build classification systems for the realistic scenario where out-of-distribution (OOD) samples are present among the test data. Given a test signal, a quantity related to empirical kernel Mahalanobis distance between the signal and each of the training classes is computed. Since these quantities correspond to the same reproducing kernel Hilbert space, they are commensurable …
This work proposes to learn fair low-rank tensor decompositions by regularizing the Canonical Polyadic Decomposition factorization with the kernel Hilbert-Schmidt independence criterion (KHSIC). It is shown, theoretically and empirically, that a small KHSIC between a latent factor and the sensitive features guarantees approximate statistical parity. The proposed algorithm surpasses the state-of-the-art algorithm, FATR (Zhu et al., 2018), in controlling the trade-off between fairness and residual fit on synthetic and real data sets.
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Microsoft Research New England（微软新英格兰研究院）；New York University（纽约大学）
We introduce a new scalable variational Gaussian process approximation which provides a high fidelity approximation while retaining general applicability. We propose the harmonic kernel decomposition (HKD), which uses Fourier series to decompose a kernel as a sum of orthogonal kernels. Our variational approximation exploits this orthogonality to enable a large number of inducing points at a low computational cost. We demonstrate that, on a range of regression and classification problems, our approach can exploit input space symmetries such as translations and reflections, and it significantly …
UCA - LIMOS UMR 6158 CNRS（克莱蒙奥弗涅大学 - 计算机科学、数学建模与优化实验室（法国国家科学研究中心联合实验室））
The proposed work aims at proposing a alternative kernel decomposition in the context of kernel machines with indefinite kernels. The original paper of KSVM (SVM in Kreǐn spaces) uses the eigen-decomposition, our proposition avoids this decompostion. We explain how it can help in designing an algorithm that won't require to compute the full kernel matrix. Finally we illustrate the good behavior of the proposed method compared to KSVM.
A Bias-Variance-Covariance Decomposition of Kernel Scores for Generative Models
German Cancer Research Center (DKFZ)（德国癌症研究中心）；German Cancer Consortium (DKTK)（德国癌症联盟）；Goethe University Frankfurt（法兰克福大学）；Frankfurt Cancer Institute (FCI)（法兰克福癌症研究所）
Qatar Computing Research Institute, HBKU（卡塔尔计算研究所，哈马德·本·哈利法大学）；University of Padova（帕多瓦大学）
In this paper, we show how the Ordered Decomposition DAGs (ODD) kernel framework, a framework that allows the definition of graph kernels from tree kernels, allows to easily define new state-of-the-art graph kernels. Here we consider a fast graph kernel based on the Subtree kernel (ST), and we propose various enhancements to increase its expressiveness. The proposed DAG kernel has the same worst-case complexity as the one based on ST, but an improved expressivity due to an augmented set of features. Moreover, we propose a novel weighting scheme for the features, which can be applied to other k…
Ludwig Maximilians University Munich（慕尼黑路德维希-马克西米利安大学）；Siemens AG（西门子股份公司）；University Hospital Erlangen（埃尔朗根大学医院）
Recurrent neural network based solutions are increasingly being used in the analysis of longitudinal Electronic Health Record data. However, most works focus on prediction accuracy and neglect prediction uncertainty. We propose Deep Kernel Accelerated Failure Time models for the time-to-event prediction task, enabling uncertainty-awareness of the prediction by a pipeline of a recurrent neural network and a sparse Gaussian Process. Furthermore, a deep metric learning based pre-training step is adapted to enhance the proposed model. Our model shows better point estimate performance than recurren…
University of Oxford（牛津大学）；University College London（伦敦大学学院）
Inducing point Gaussian process approximations are often considered a gold standard in uncertainty estimation since they retain many of the properties of the exact GP and scale to large datasets. A major drawback is that they have difficulty scaling to high dimensional inputs. Deep Kernel Learning (DKL) promises a solution: a deep feature extractor transforms the inputs over which an inducing point Gaussian process is defined. However, DKL has been shown to provide unreliable uncertainty estimates in practice. We study why, and show that with no constraints, the DKL objective pushes "far-away"…
Uncertainty plays a crucial role in the machine learning field. Both model trustworthiness and performance require the understanding of uncertainty, especially for models used in high-stake applications where errors can cause cataclysmic consequences, such as medical diagnosis and autonomous driving. Accordingly, uncertainty decomposition and quantification have attracted more and more attention in recent years. This short report aims to demystify the notion of uncertainty decomposition through an introduction to two types of uncertainty and several decomposition exemplars, including maximum l…
Siemens AG（西门子股份公司）；TU Munich（慕尼黑工业大学）；University of Cambridge（剑桥大学）；Harvard University（哈佛大学）
Bayesian neural networks with latent variables are scalable and flexible probabilistic models: They account for uncertainty in the estimation of the network weights and, by making use of latent variables, can capture complex noise patterns in the data. We show how to extract and decompose uncertainty into epistemic and aleatoric components for decision-making purposes. This allows us to successfully identify informative points for active learning of functions with heteroscedastic and bimodal noise. Using the decomposition we further define a novel risk-sensitive criterion for reinforcement lea…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Freiburg（弗莱堡大学）
This work presents the concept of kernel mean embedding and kernel probabilistic programming in the context of stochastic systems. We propose formulations to represent, compare, and propagate uncertainties for fairly general stochastic dynamics in a distribution-free manner. The new tools enjoy sound theory rooted in functional analysis and wide applicability as demonstrated in distinct numerical examples. The implication of this new concept is a new mode of thinking about the statistical nature of uncertainty in dynamical systems.
KU Leuven（鲁汶大学）；Shanghai Jiao Tong University（上海交通大学）
In this paper, we attempt to solve a long-lasting open question for non-positive definite (non-PD) kernels in machine learning community: can a given non-PD kernel be decomposed into the difference of two PD kernels (termed as positive decomposition)? We cast this question as a distribution view by introducing the \emph{signed measure}, which transforms positive decomposition to measure decomposition: a series of non-PD kernels can be associated with the linear combination of specific finite Borel measures. In this manner, our distribution-based framework provides a sufficient and necessary co…
Siemens Corporate Research and Technology（西门子企业研究与技术部）；Temple University（天普大学）；Vienna University of Technology（维也纳工业大学）
Low-rank matrix decomposition has gained great popularity recently in scaling up kernel methods to large amounts of data. However, some limitations could prevent them from working effectively in certain domains. For example, many existing approaches are intrinsically unsupervised, which does not incorporate side information (e.g., class labels) to produce task specific decompositions; also, they typically work "transductively", i.e., the factorization does not generalize to new samples, so the complete factorization needs to be recomputed when new samples become available. To solve these probl…
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；University of Bath（巴斯大学）；TU Chemnitz（开姆尼茨工业大学）
High-dimensional data in the form of tensors are challenging for kernel classification methods. To both reduce the computational complexity and extract informative features, kernels based on low-rank tensor decompositions have been proposed. However, what decisive features of the tensors are exploited by these kernels is often unclear. In this paper we propose a novel kernel that is based on the Tucker decomposition. For this kernel the Tucker factors are computed based on re-weighting of the Tucker matrices with tuneable powers of singular values from the HOSVD decomposition. This provides a …
Fast Gaussian process inference by exact Matérn kernel decomposition
Guangdong Provincial/Zhuhai Key Laboratory of Interdisciplinary Research and Application for Data Science, Beijing Normal-Hong Kong Baptist University（广东/珠海交叉学科与数据科学应用重点实验室，北京师范大学-香港 Baptist大学）；EDF Lab, FiME (Laboratoire de Finance des Marchés de l’Énergie)（EDF实验室，FiME（能源市场金融实验室））
Luzerne University of Applied Sciences and Arts（卢塞恩应用科学与艺术大学）；Goethe University Frankfurt am Main（法兰克福歌德大学）
A Support Vector Machine (SVM) has become a very popular machine learning method for text classification. One reason for this relates to the range of existing kernels which allow for classifying data that is not linearly separable. The linear, polynomial and RBF (Gaussian Radial Basis Function) kernel are commonly used and serve as a basis of comparison in our study. We show how to derive the primal form of the quadratic Power Kernel (PK) -- also called the Negative Euclidean Distance Kernel (NDK) -- by means of complex numbers. We exemplify the NDK in the framework of text categorization usin…
Institute for Clarity in Documentation（清晰度文献研究所）；Inria Paris-Rocquencourt（法国国家信息与自动化研究所巴黎-罗康库尔中心）；Rajiv Gandhi University（拉吉夫·甘地大学）；Tsinghua University（清华大学）；Palmer Research Laboratories（帕尔默研究实验室）
The emerging edge computing has promoted immense interests in compacting a neural network without sacrificing much accuracy. In this regard, low-rank tensor decomposition constitutes a powerful tool to compress convolutional neural networks (CNNs) by decomposing the 4-way kernel tensor into multi-stage smaller ones. Building on top of Tucker-2 decomposition, we propose a generalized Higher Order Tucker Articulated Kernels (HOTCAKE) scheme comprising four steps: input channel decomposition, guided Tucker rank selection, higher order Tucker decomposition and fine-tuning. By subjecting each CONV …
Marginalising over families of Gaussian Process kernels produces flexible model classes with well-calibrated uncertainty estimates. Existing approaches require likelihood evaluations of many kernels, rendering them prohibitively expensive for larger datasets. We propose a Bayesian Quadrature scheme to make this marginalisation more efficient and thereby more practical. Through use of the maximum mean discrepancies between distributions, we define a kernel over kernels that captures invariances between Spectral Mixture (SM) Kernels. Kernel samples are selected by generalising an information-the…
Cognizant（高知特）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Neural Networks (NNs) have been extensively used for a wide spectrum of real-world regression tasks, where the goal is to predict a numerical outcome such as revenue, effectiveness, or a quantitative result. In many such tasks, the point prediction is not enough: the uncertainty (i.e. risk or confidence) of that prediction must also be estimated. Standard NNs, which are most often used in such tasks, do not provide uncertainty information. Existing approaches address this issue by combining Bayesian models with NNs, but these models are hard to implement, more expensive to train, and usually d…
University of South Florida（南佛罗里达大学）；Farmingdale State College（法明戴尔州立学院）；Changshu Institute of Technology（常熟理工学院）
This manuscript gives a theoretical framework for a new Hilbert space of functions, the so called occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex numbers. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD rout…
Meta-learning to Calibrate Gaussian Processes with Deep Kernels for Regression Uncertainty Estimation
NTT Communication Science Laboratories, NTT Corporation（NTT集团NTT通信科学实验室）；NTT Computer and Data Science Laboratories, NTT Corporation（NTT集团NTT计算机与数据科学实验室）
The deep neural network suffers from many fundamental issues in machine learning. For example, it often gets trapped into a local minimum in training, and its prediction uncertainty is hard to be assessed. To address these issues, we propose the so-called kernel-expanded stochastic neural network (K-StoNet) model, which incorporates support vector regression (SVR) as the first hidden layer and reformulates the neural network as a latent variable model. The former maps the input vector into an infinite dimensional feature space via a radial basis function (RBF) kernel, ensuring absence of local…
This paper presents a method for building a preconditioner for a kernel ridge regression problem, where the preconditioner is not only effective in its ability to reduce the condition number substantially, but also efficient in its application in terms of computational cost and memory consumption. The suggested approach is based on randomized matrix decomposition methods, combined with the fast multipole method to achieve an algorithm that can process large datasets in complexity linear to the number of data points. In addition, a detailed theoretical analysis is provided, including an upper b…
Irish Centre for Maternal and Child Health Research (INFANT)（爱尔兰母婴健康研究中心（INFANT））；University College Cork（科克大学）
Linear classifiers with random convolution kernels are computationally efficient methods that need no design or domain knowledge. Unlike deep neural networks, there is no need to hand-craft a network architecture; the kernels are randomly generated and only the linear classifier needs training. A recently proposed method, RandOm Convolutional KErnel Transforms (ROCKETs), has shown high accuracy across a range of time-series data sets. Here we propose a multi-scale version of this method, using both high- and low-frequency components. We apply our methods to inter-burst detection in a cohort of…
