本文批判性地考察了在单输出高斯过程（GP）模型中使用 Matérn 核的加性混合这一普遍做法，并探讨了 Matérn 核的乘性混合在多输出 GP 模型中的性质。对于单输出情形，我们推导出一系列理论结果，表明 Matérn 核混合的光滑性由最不光滑的分量决定，并且具有这种核的 GP 实际上等价于最不光滑的核分量。此外，我们证明单个核分量内的混合权重或参数均不可识别。随后，我们将注意力转向多输出 GP 模型，分析乘性核 $K(x,y) = AK_0(x,y)$ 中协方差矩阵 $A$ 的可识别性，其中 $K_0$ 是诸如 Matérn 之类的标准单输出核。我们证明 $A$ 在相差一个乘法常数的意义下是可识别的，这表明乘性混合非常适合多输出任务。我们的发现得到了单输出和多输出情形下大量模拟和实际应用的支持。这项工作为 GP 模型的核选择与解释提供了见解，强调了针对不同任务选择合适核结构的重要性。
英文摘要：
In this paper, we critically examine the prevalent practice of using additive mixtures of Matérn kernels in single-output Gaussian process (GP) models and explore the properties of multiplicative mixtures of Matérn kernels for multi-output GP models. For the single-output case, we derive a series of theoretical results showing that the smoothness of a mixture of Matérn kernels is determined by the least smooth component and that a GP with such a kernel is effectively equivalent to the least smooth kernel component. Furthermore, we demonstrate that none of the mixing weights or parameters within individual kernel components are identifiable. We then turn our attention to multi-output GP models and analyze the identifiability of the covariance matrix $A$ in the multiplicative kernel $K(x,y) = AK_0(x,y)$, where $K_0$ is a standard single output kernel such as Matérn. We show that $A$ is identifiable up to a multiplicative constant, suggesting that multiplicative mixtures are well suited for multi-output tasks. Our findings are supported by extensive simulations and real applications for both single- and multi-output settings. This work provides insight into kernel selection and interpretation for GP models, emphasizing the importance of choosing appropriate kernel structures for different tasks.
In this paper, we prove that some Gaussian structural equation models with dependent errors having equal variances are identifiable from their corresponding Gaussian distributions. Specifically, we prove identifiability for the Gaussian structural equation models that can be represented as Andersson-Madigan-Perlman chain graphs (Andersson et al., 2001). These chain graphs were originally developed to represent independence models. However, they are also suitable for representing causal models with additive noise (Peña, 2016. Our result implies then that these causal models can be identified fr…
Multi-View Causal Discovery without Non-Gaussianity: Identifiability and Algorithms
Inria（法国国家信息与自动化技术研究院）；CEA（法国国家科学研究中心）；University of Paris-Saclay（巴黎-萨克勒大学）；University of Carnegie Mellon（卡内基梅隆大学）；Apple（苹果公司）；University of Helsinki（赫尔辛基大学）
University of Minnesota（明尼苏达大学）；University of Virginia（弗吉尼亚大学）
We study the problem of learning a mixture model of non-parametric product distributions. The problem of learning a mixture model is that of finding the component distributions along with the mixing weights using observed samples generated from the mixture. The problem is well-studied in the parametric setting, i.e., when the component distributions are members of a parametric family -- such as Gaussian distributions. In this work, we focus on multivariate mixtures of non-parametric product distributions and propose a two-stage approach which recovers the component distributions of the mixture…
The vast majority of work in self-supervised learning, both theoretical and empirical (though mostly the latter), have largely focused on recovering good features for downstream tasks, with the definition of "good" often being intricately tied to the downstream task itself. This lens is undoubtedly very interesting, but suffers from the problem that there isn't a "canonical" set of downstream tasks to focus on -- in practice, this problem is usually resolved by competing on the benchmark dataset du jour. In this paper, we present an alternative lens: one of parameter identifiability. More prec…
University of Chicago（芝加哥大学）；Carnegie Mellon University（卡内基梅隆大学）
We prove identifiability of a broad class of deep latent variable models that (a) have universal approximation capabilities and (b) are the decoders of variational autoencoders that are commonly used in practice. Unlike existing work, our analysis does not require weak supervision, auxiliary information, or conditioning in the latent space. Specifically, we show that for a broad class of generative (i.e. unsupervised) models with universal approximation capabilities, the side information $u$ is not necessary: We prove identifiability of the entire generative model where we do not observe $u$ a…
Responsible AI Research Centre（负责任人工智能研究中心）；Australian Institute for Machine Learning（澳大利亚机器学习研究所）；School of Computer Science and Engineering（计算机科学与工程学院）；School of Mathematics and Statistics（数学与统计学学院）；Halicioğlu Data Science Institute（Halicioğlu数据科学研究所）；Department of Philosophy（哲学系）
Gaussian Process (GP) models are a class of flexible non-parametric models that have rich representational power. By using a Gaussian process with additive structure, complex responses can be modelled whilst retaining interpretability. Previous work showed that additive Gaussian process models require high-dimensional interaction terms. We propose the orthogonal additive kernel (OAK), which imposes an orthogonality constraint on the additive functions, enabling an identifiable, low-dimensional representation of the functional relationship. We connect the OAK kernel to functional ANOVA decompos…
This paper studies identifiability and convergence behaviors for parameters of multiple types in finite mixtures, and the effects of model fitting with extra mixing components. First, we present a general theory for strong identifiability, which extends from the previous work of Nguyen [2013] and Chen [1995] to address a broad range of mixture models and to handle matrix-variate parameters. These models are shown to share the same Wasserstein distance based optimal rates of convergence for the space of mixing distributions --- $n^{-1/2}$ under $W_1$ for the exact-fitted and $n^{-1/4}$ under $W…
In this work, we consider the identifiability assumption of Gaussian linear structural equation models (SEMs) in which each variable is determined by a linear function of its parents plus normally distributed error. It has been shown that linear Gaussian structural equation models are fully identifiable if all error variances are the same or known. Hence, this work proves the identifiability of Gaussian SEMs with both homogeneous and heterogeneous unknown error variances. Our new identifiability assumption exploits not only error variances, but edge weights; hence, it is strictly milder than p…
Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable, by introducing a novel framework involving clustering overfitted \emph{parametric} (i.e. misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature, and are flexible enough to include for example mixtures of Gaussian mixtures. In contrast to the recent literature on estimating nonparametric mixtures, we allow for general nonparametric mixture components, and instead impose regularity assumptions on …
Synthetic Potential Outcomes and Causal Mixture Identifiability
Eric and Wendy Schmidt Center（埃里克和温迪·施密特中心）；Broad Institute of MIT and Harvard（麻省理工学院与哈佛大学布罗德研究所）；Laboratory for Information and Decision Systems（信息与决策系统实验室）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
Bayesian network (BN) structure learning from complete data has been extensively studied in the literature. However, fewer theoretical results are available for incomplete data, and most are related to the Expectation-Maximisation (EM) algorithm. Balov (2013) proposed an alternative approach called Node-Average Likelihood (NAL) that is competitive with EM but computationally more efficient; and he proved its consistency and model identifiability for discrete BNs. In this paper, we give general sufficient conditions for the consistency of NAL; and we prove consistency and identifiability for co…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Adobe Systems, Inc.（奥多比系统公司）；Analog Devices（亚德诺半导体）
In this paper, we develop a parameter estimation method for factorially parametrized models such as Factorial Gaussian Mixture Model and Factorial Hidden Markov Model. Our contributions are two-fold. First, we show that the emission matrix of the standard Factorial Model is unidentifiable even if the true assignment matrix is known. Secondly, we address the issue of identifiability by making a one component sharing assumption and derive a parameter learning algorithm for this case. Our approach is based on a dictionary learning problem of the form $X = O R$, where the goal is to learn the dict…
Discrepancy Modeling with Intermediate Variables: A New Framework for Robust Gaussian Process Calibration
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Argonne National Laboratory（阿贡国家实验室）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Gaussian processes models are widely adopted for nonparameteric/semi-parametric modeling. Identifiability issues occur when the mean model contains polynomials with unknown coefficients. Though resulting prediction is unaffected, this leads to poor estimation of the coefficients in the mean model, and thus the estimated mean model loses interpretability. This paper introduces a new Gaussian process model whose stochastic part is orthogonal to the mean part to address this issue. This paper also discusses applications to multi-fidelity simulations using data examples.
Department of Engineering Science, University of Oxford（牛津大学工程科学系）；Man AHL（Man AHL（可能是一个机构或部门，但根据现有信息无法确定具体中文名称，此处按原文保留））
We propose a graph spectrum-based Gaussian process for prediction of signals defined on nodes of the graph. The model is designed to capture various graph signal structures through a highly adaptive kernel that incorporates a flexible polynomial function in the graph spectral domain. Unlike most existing approaches, we propose to learn such a spectral kernel, where the polynomial setup enables learning without the need for eigen-decomposition of the graph Laplacian. In addition, this kernel has the interpretability of graph filtering achieved by a bespoke maximum likelihood learning algorithm …
Equilibrium Causal Games: Separation, Identification, and the Identifiability of Cyclic Latent States
Department of Mechanical & Industrial Engineering, Montana State University（机械与工业工程系，蒙大拿州立大学）；Gianforte School of Computing, Montana State University（计算机学院，蒙大拿州立大学）
School of Electrical Engineering and Computer Science, Oregon State University（俄勒冈州立大学电气工程与计算机科学学院）；Department of Computer and Information Science and Engineering, University of Florida（佛罗里达大学计算机与信息科学与工程系）；Department of Electrical and Computer Engineering, University of Virginia（弗吉尼亚大学电气与计算机工程系）；Department of Electronic Engineering, The Chinese University of Hong Kong（香港中文大学电子工程系）
Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communiti…
Centuries of development in natural sciences and mathematical modeling provide valuable domain expert knowledge that has yet to be explored for the development of machine learning models. When modeling complex physical systems, both domain knowledge and data provide necessary information about the system. In this paper, we present a data-driven model that takes advantage of partial domain knowledge in order to improve generalization and interpretability. The presented approach, which we call EVGP (Explicit Variational GaussianProcess), has the following advantages: 1) using available domain kn…
Identifiable Latent Polynomial Causal Models Through the Lens of Change
Australian Institute for Machine Learning, The University of Adelaide（阿德莱德大学澳大利亚机器学习研究所）；School of Computer Science and Engineering, The University of New South Wales（新南威尔士大学计算机科学与工程学院）；School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；Halicioğlu Data Science Institute (HDSI), University of California San Diego（加州大学圣地亚哥分校哈利西奥格鲁数据科学研究所）；Carnegie Mellon University（卡内基梅隆大学）
Towards Flexibility and Interpretability of Gaussian Process State-Space Model
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Future Network of Intelligence Institute（未来网络智能研究院）；Universidad Autónoma de Madrid（马德里自治大学）；Cognizant AI（高知特人工智能）
Learning of low dimensional structure in multidimensional data is a canonical problem in machine learning. One common approach is to suppose that the observed data are close to a lower-dimensional smooth manifold. There are a rich variety of manifold learning methods available, which allow mapping of data points to the manifold. However, there is a clear lack of probabilistic methods that allow learning of the manifold along with the generative distribution of the observed data. The best attempt is the Gaussian process latent variable model (GP-LVM), but identifiability issues lead to poor per…
For many survey-based spatial modelling problems, responses are observed as spatially aggregated over survey regions due to limited resources. Covariates, from weather models and satellite imageries, can be observed at many different spatial resolutions, making the pre-processing of covariates a key challenge for any spatial modelling task. We propose a Gaussian process regression model to flexibly handle multiresolution covariates by employing an additive kernel that can efficiently aggregate features across resolutions. Compared to existing approaches that rely on resolution matching, our ap…
We introduce Gaussian orthogonal latent factor processes for modeling and predicting large correlated data. To handle the computational challenge, we first decompose the likelihood function of the Gaussian random field with a multi-dimensional input domain into a product of densities at the orthogonal components with lower-dimensional inputs. The continuous-time Kalman filter is implemented to compute the likelihood function efficiently without making approximations. We also show that the posterior distribution of the factor processes is independent, as a consequence of prior independence of f…
The relative value of interventional and observational samples in Bayesian Causal Linear Gaussian Models
Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge（剑桥大学数学科学中心统计实验室）；Imperial College London（帝国理工学院）；University of Oxford（牛津大学）
We consider the problem of learning a conditional Gaussian graphical model in the presence of latent variables. Building on recent advances in this field, we suggest a method that decomposes the parameters of a conditional Markov random field into the sum of a sparse and a low-rank matrix. We derive convergence bounds for this estimator and show that it is well-behaved in the high-dimensional regime as well as "sparsistent" (i.e. capable of recovering the graph structure). We then show how proximal gradient algorithms and semi-definite programming techniques can be employed to fit the model to…
Identification of parameters in ordinary differential equations (ODEs) is an important and challenging task when modeling dynamic systems in biomedical research and other scientific areas, especially with the presence of time-varying parameters. This article proposes a fast and accurate method, TVMAGI (Time-Varying MAnifold-constrained Gaussian process Inference), to estimate both time-constant and time-varying parameters in the ODE using noisy and sparse observation data. TVMAGI imposes a Gaussian process model over the time series of system components as well as time-varying parameters, and …
We study identifiability of Andersson-Madigan-Perlman (AMP) chain graph models, which are a common generalization of linear structural equation models and Gaussian graphical models. AMP models are described by DAGs on chain components which themselves are undirected graphs. For a known chain component decomposition, we show that the DAG on the chain components is identifiable if the determinants of the residual covariance matrices of the chain components are monotone non-decreasing in topological order. This condition extends the equal variance identifiability criterion for Bayes nets, and it …
LiD-GLM: Lipschitz-constrained Deep Generalized Linear Models
Competence Center for Clinical Trials Bremen（不来梅临床研究能力中心）；University of Bremen（不来梅大学）；Leibniz Institute for Prevention Research and Epidemiology – BIPS（莱布尼茨预防研究与流行病学研究所——BIPS）
