We consider estimation of generalized additive models using basis expansions with Bayesian model selection. Although Bayesian model selection is an intuitively appealing tool for regression splines by virtue of the flexible knot placement and model-averaged function estimates, its use has traditionally been limited to Gaussian additive regression, as posterior search of the model space requires a tractable form of the marginal model likelihood. We introduce an extension of the method to the exponential family of distributions using the Laplace approximation to the likelihood. Although the Laplace approximation is successful with all Gaussian-type prior distributions in providing a closed-form expression of the marginal likelihood, there is no broad consensus on the best prior distribution to be used for nonparametric regression via model selection. We observe that the classical unit information prior distribution for variable selection may not be suitable for nonparametric regression using basis expansions. Instead, our study reveals that mixtures of g-priors are more suitable. A large family of mixtures of g-priors is considered for a detailed examination of how various mixture priors perform in estimating generalized additive models. Furthermore, we compare several priors of knots for model selection-based spline approaches to determine the most practically effective scheme. The model selection-based estimation methods are also compared with other Bayesian approaches to function estimation. Extensive simulation studies demonstrate the validity of the model selection-based approaches. We provide an R package for the proposed method.
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ACM/IEEE第23屆模型驅動工程語言和系統國際會議，是模型驅動軟件和系統工程的首要會議系列，由ACM-SIGSOFT和IEEE-TCSE支持組織。自1998年以來，模型涵蓋了建模的各個方面，從語言和方法到工具和應用程序。模特的參加者來自不同的背景，包括研究人員、學者、工程師和工業專業人士。MODELS 2019是一個論壇，參與者可以圍繞建模和模型驅動的軟件和系統交流前沿研究成果和創新實踐經驗。今年的版本將為建模社區提供進一步推進建模基礎的機會，并在網絡物理系統、嵌入式系統、社會技術系統、云計算、大數據、機器學習、安全、開源等新興領域提出建模的創新應用以及可持續性。
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We propose two robust methods for testing hypotheses on unknown parameters of predictive regression models under heterogeneous and persistent volatility as well as endogenous, persistent and/or fat-tailed regressors and errors. The proposed robust testing approaches are applicable both in the case of discrete and continuous time models. Both of the methods use the Cauchy estimator to effectively handle the problems of endogeneity, persistence and/or fat-tailedness in regressors and errors. The difference between our two methods is how the heterogeneous volatility is controlled. The first method relies on robust t-statistic inference using group estimators of a regression parameter of interest proposed in Ibragimov and Muller, 2010. It is simple to implement, but requires the exogenous volatility assumption. To relax the exogenous volatility assumption, we propose another method which relies on the nonparametric correction of volatility. The proposed methods perform well compared with widely used alternative inference procedures in terms of their finite sample properties.
Conditional Neural Processes~(CNPs) formulate distributions over functions and generate function observations with exact conditional likelihoods. CNPs, however, have limited expressivity for high-dimensional observations, since their predictive distribution is factorized into a product of unconstrained (typically) Gaussian outputs. Previously, this could be handled using latent variables or autoregressive likelihood, but at the expense of intractable training and quadratically increased complexity. Instead, we propose calibrating CNPs with an adversarial training scheme besides regular maximum likelihood estimates. Specifically, we train an energy-based model (EBM) with noise contrastive estimation, which enforces EBM to identify true observations from the generations of CNP. In this way, CNP must generate predictions closer to the ground-truth to fool EBM, instead of merely optimizing with respect to the fixed-form likelihood. From generative function reconstruction to downstream regression and classification tasks, we demonstrate that our method fits mainstream CNP members, showing effectiveness when unconstrained Gaussian likelihood is defined, requiring minimal computation overhead while preserving foundation properties of CNPs.
Locating 3D objects from a single RGB image via Perspective-n-Point (PnP) is a long-standing problem in computer vision. Driven by end-to-end deep learning, recent studies suggest interpreting PnP as a differentiable layer, allowing for partial learning of 2D-3D point correspondences by backpropagating the gradients of pose loss. Yet, learning the entire correspondences from scratch is highly challenging, particularly for ambiguous pose solutions, where the globally optimal pose is theoretically non-differentiable w.r.t. the points. In this paper, we propose the EPro-PnP, a probabilistic PnP layer for general end-to-end pose estimation, which outputs a distribution of pose with differentiable probability density on the SE(3) manifold. The 2D-3D coordinates and corresponding weights are treated as intermediate variables learned by minimizing the KL divergence between the predicted and target pose distribution. The underlying principle generalizes previous approaches, and resembles the attention mechanism. EPro-PnP can enhance existing correspondence networks, closing the gap between PnP-based method and the task-specific leaders on the LineMOD 6DoF pose estimation benchmark. Furthermore, EPro-PnP helps to explore new possibilities of network design, as we demonstrate a novel deformable correspondence network with the state-of-the-art pose accuracy on the nuScenes 3D object detection benchmark. Our code is available at //github.com/tjiiv-cprg/EPro-PnP-v2.
Efficient and accurate estimation of multivariate empirical probability distributions is fundamental to the calculation of information-theoretic measures such as mutual information and transfer entropy. Common techniques include variations on histogram estimation which, whilst computationally efficient, are often unable to precisely capture the probability density of samples with high correlation, kurtosis or fine substructure, especially when sample sizes are small. Adaptive partitions, which adjust heuristically to the sample, can reduce the bias imparted from the geometry of the histogram itself, but these have commonly focused on the location, scale and granularity of the partition, the effects of which are limited for highly correlated distributions. In this paper, I reformulate the differential entropy estimator for the special case of an equiprobable histogram, using a k-d tree to partition the sample space into bins of equal probability mass. By doing so, I expose an implicit rotational orientation parameter, which is conjectured to be suboptimally specified in the typical marginal alignment. I propose that the optimal orientation minimises the variance of the bin volumes, and demonstrate that improved entropy estimates can be obtained by rotationally aligning the partition to the sample distribution accordingly. Such optimal partitions are observed to be more accurate than existing techniques in estimating entropies of correlated bivariate Gaussian distributions with known theoretical values, across varying sample sizes (99% CI).
Digital image inpainting is an interpolation problem, inferring the content in the missing (unknown) region to agree with the known region data such that the interpolated result fulfills some prior knowledge. Low-rank and nonlocal self-similarity are two important priors for image inpainting. Based on the nonlocal self-similarity assumption, an image is divided into overlapped square target patches (submatrices) and the similar patches of any target patch are reshaped as vectors and stacked into a patch matrix. Such a patch matrix usually enjoys a property of low rank or approximately low rank, and its missing entries are recoveried by low-rank matrix approximation (LRMA) algorithms. Traditionally, $n$ nearest neighbor similar patches are searched within a local window centered at a target patch. However, for an image with missing lines, the generated patch matrix is prone to having entirely-missing rows such that the downstream low-rank model fails to reconstruct it well. To address this problem, we propose a region-wise matching (RwM) algorithm by dividing the neighborhood of a target patch into multiple subregions and then search the most similar one within each subregion. A non-convex weighted low-rank decomposition (NC-WLRD) model for LRMA is also proposed to reconstruct all degraded patch matrices grouped by the proposed RwM algorithm. We solve the proposed NC-WLRD model by the alternating direction method of multipliers (ADMM) and analyze the convergence in detail. Numerous experiments on line inpainting (entire-row/column missing) demonstrate the superiority of our method over other competitive inpainting algorithms. Unlike other low-rank-based matrix completion methods and inpainting algorithms, the proposed model NC-WLRD is also effective for removing random-valued impulse noise and structural noise (stripes).
The success of a football team depends on various individual skills and performances of the selected players as well as how cohesively they perform. This work proposes a two-stage process for selecting optimal playing eleven of a football team from its pool of available players. In the first stage, for the reference team, a LASSO-induced modified trinomial logistic regression model is derived to analyze the probabilities of the three possible outcomes. The model takes into account strengths of the players in the team as well as those of the opponent, home advantage, and also the effects of individual players and player combinations beyond the recorded performances of these players. Careful use of the LASSO technique acts as an appropriate enabler of the player selection exercise while keeping the number of variables at a reasonable level. Then, in the second stage, a GRASP-type meta-heuristic is implemented for the team selection which maximizes the probability of win for the team. The work is illustrated with English Premier League data from 2008/09 to 2015/16. The application demonstrates that the model in the first stage furnishes valuable insights about the deciding factors for different teams whereas the optimization steps can be effectively used to determine the best possible starting lineup under various circumstances. Based on the adopted model and methodology, we propose a measure of efficiency in team selection by the team management and analyze the performance of EPL teams on this front.
We review Quasi Maximum Likelihood estimation of factor models for high-dimensional panels of time series. We consider two cases: (1) estimation when no dynamic model for the factors is specified (Bai and Li, 2016); (2) estimation based on the Kalman smoother and the Expectation Maximization algorithm thus allowing to model explicitly the factor dynamics (Doz et al., 2012). Our interest is in approximate factor models, i.e., when we allow for the idiosyncratic components to be mildly cross-sectionally, as well as serially, correlated. Although such setting apparently makes estimation harder, we show, in fact, that factor models do not suffer of the curse of dimensionality problem, but instead they enjoy a blessing of dimensionality property. In particular, we show that if the cross-sectional dimension of the data, $N$, grows to infinity, then: (i) identification of the model is still possible, (ii) the mis-specification error due to the use of an exact factor model log-likelihood vanishes. Moreover, if we let also the sample size, $T$, grow to infinity, we can also consistently estimate all parameters of the model and make inference. The same is true for estimation of the latent factors which can be carried out by weighted least-squares, linear projection, or Kalman filtering/smoothing. We also compare the approaches presented with: Principal Component analysis and the classical, fixed $N$, exact Maximum Likelihood approach. We conclude with a discussion on efficiency of the considered estimators.
We present a method for selecting valuable projections in computed tomography (CT) scans to enhance image reconstruction and diagnosis. The approach integrates two important factors, projection-based detectability and data completeness, into a single feed-forward neural network. The network evaluates the value of projections, processes them through a differentiable ranking function and makes the final selection using a straight-through estimator. Data completeness is ensured through the label provided during training. The approach eliminates the need for heuristically enforcing data completeness, which may exclude valuable projections. The method is evaluated on simulated data in a non-destructive testing scenario, where the aim is to maximize the reconstruction quality within a specified region of interest. We achieve comparable results to previous methods, laying the foundation for using reconstruction-based loss functions to learn the selection of projections.
We introduce a new regression framework designed to deal with large-scale, complex data that lies around a low-dimensional manifold. Our approach first constructs a graph representation, referred to as the skeleton, to capture the underlying geometric structure. We then define metrics on the skeleton graph and apply nonparametric regression techniques, along with feature transformations based on the graph, to estimate the regression function. In addition to the included nonparametric methods, we also discuss the limitations of some nonparametric regressors with respect to the general metric space such as the skeleton graph. The proposed regression framework allows us to bypass the curse of dimensionality and provides additional advantages that it can handle the union of multiple manifolds and is robust to additive noise and noisy observations. We provide statistical guarantees for the proposed method and demonstrate its effectiveness through simulations and real data examples.
Classic machine learning methods are built on the $i.i.d.$ assumption that training and testing data are independent and identically distributed. However, in real scenarios, the $i.i.d.$ assumption can hardly be satisfied, rendering the sharp drop of classic machine learning algorithms' performances under distributional shifts, which indicates the significance of investigating the Out-of-Distribution generalization problem. Out-of-Distribution (OOD) generalization problem addresses the challenging setting where the testing distribution is unknown and different from the training. This paper serves as the first effort to systematically and comprehensively discuss the OOD generalization problem, from the definition, methodology, evaluation to the implications and future directions. Firstly, we provide the formal definition of the OOD generalization problem. Secondly, existing methods are categorized into three parts based on their positions in the whole learning pipeline, namely unsupervised representation learning, supervised model learning and optimization, and typical methods for each category are discussed in detail. We then demonstrate the theoretical connections of different categories, and introduce the commonly used datasets and evaluation metrics. Finally, we summarize the whole literature and raise some future directions for OOD generalization problem. The summary of OOD generalization methods reviewed in this survey can be found at //out-of-distribution-generalization.com.
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