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Domain generalization approaches aim to learn a domain invariant prediction model for unknown target domains from multiple training source domains with different distributions. Significant efforts have recently been committed to broad domain generalization, which is a challenging and topical problem in machine learning and computer vision communities. Most previous domain generalization approaches assume that the conditional distribution across the domains remain the same across the source domains and learn a domain invariant model by minimizing the marginal distributions. However, the assumption of a stable conditional distribution of the training source domains does not really hold in practice. The hyperplane learned from the source domains will easily misclassify samples scattered at the boundary of clusters or far from their corresponding class centres. To address the above two drawbacks, we propose a discriminative domain-invariant adversarial network (DDIAN) for domain generalization. The discriminativeness of the features are guaranteed through a discriminative feature module and domain-invariant features are guaranteed through the global domain and local sub-domain alignment modules. Extensive experiments on several benchmarks show that DDIAN achieves better prediction on unseen target data during training compared to state-of-the-art domain generalization approaches.

Variational approximation, such as mean-field (MF) and tree-reweighted (TRW), provide a computationally efficient approximation of the log-partition function for a generic graphical model. TRW provably provides an upper bound, but the approximation ratio is generally not quantified. As the primary contribution of this work, we provide an approach to quantify the approximation ratio through the property of the underlying graph structure. Specifically, we argue that (a variant of) TRW produces an estimate that is within factor $\frac{1}{\sqrt{\kappa(G)}}$ of the true log-partition function for any discrete pairwise graphical model over graph $G$, where $\kappa(G) \in (0,1]$ captures how far $G$ is from tree structure with $\kappa(G) = 1$ for trees and $2/N$ for the complete graph over $N$ vertices. As a consequence, the approximation ratio is $1$ for trees, $\sqrt{(d+1)/2}$ for any graph with maximum average degree $d$, and $\stackrel{\beta\to\infty}{\approx} 1+1/(2\beta)$ for graphs with girth (shortest cycle) at least $\beta \log N$. In general, $\kappa(G)$ is the solution of a max-min problem associated with $G$ that can be evaluated in polynomial time for any graph. Using samples from the uniform distribution over the spanning trees of G, we provide a near linear-time variant that achieves an approximation ratio equal to the inverse of square-root of minimal (across edges) effective resistance of the graph. We connect our results to the graph partition-based approximation method and thus provide a unified perspective. Keywords: variational inference, log-partition function, spanning tree polytope, minimum effective resistance, min-max spanning tree, local inference

Existing calibration algorithms address the problem of covariate shift via unsupervised domain adaptation. However, these methods suffer from the following limitations: 1) they require unlabeled data from the target domain, which may not be available at the stage of calibration in real-world applications and 2) their performance depends heavily on the disparity between the distributions of the source and target domains. To address these two limitations, we present novel calibration solutions via domain generalization. Our core idea is to leverage multiple calibration domains to reduce the effective distribution disparity between the target and calibration domains for improved calibration transfer without needing any data from the target domain. We provide theoretical justification and empirical experimental results to demonstrate the effectiveness of our proposed algorithms. Compared against state-of-the-art calibration methods designed for domain adaptation, we observe a decrease of 8.86 percentage points in expected calibration error or, equivalently, an increase of 35 percentage points in improvement ratio for multi-class classification on the Office-Home dataset.

The predict-then-optimize framework is fundamental in practical stochastic decision-making problems: first predict unknown parameters of an optimization model, then solve the problem using the predicted values. A natural loss function in this setting is defined by measuring the decision error induced by the predicted parameters, which was named the Smart Predict-then-Optimize (SPO) loss by Elmachtoub and Grigas [arXiv:1710.08005]. Since the SPO loss is typically nonconvex and possibly discontinuous, Elmachtoub and Grigas [arXiv:1710.08005] introduced a convex surrogate, called the SPO+ loss, that importantly accounts for the underlying structure of the optimization model. In this paper, we greatly expand upon the consistency results for the SPO+ loss provided by Elmachtoub and Grigas [arXiv:1710.08005]. We develop risk bounds and uniform calibration results for the SPO+ loss relative to the SPO loss, which provide a quantitative way to transfer the excess surrogate risk to excess true risk. By combining our risk bounds with generalization bounds, we show that the empirical minimizer of the SPO+ loss achieves low excess true risk with high probability. We first demonstrate these results in the case when the feasible region of the underlying optimization problem is a polyhedron, and then we show that the results can be strengthened substantially when the feasible region is a level set of a strongly convex function. We perform experiments to empirically demonstrate the strength of the SPO+ surrogate, as compared to standard $\ell_1$ and squared $\ell_2$ prediction error losses, on portfolio allocation and cost-sensitive multi-class classification problems.

We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the samples sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.

Time-to-event endpoints show an increasing popularity in phase II cancer trials. The standard statistical tool for such endpoints in one-armed trials is the one-sample log-rank test. It is widely known, that the asymptotic providing the correctness of this test does not come into effect to full extent for small sample sizes. There have already been some attempts to solve this problem. While some do not allow easy power and sample size calculations, others lack a clear theoretical motivation and require further considerations. The problem itself can partly be attributed to the dependence of the compensated counting process and its variance estimator. We provide a framework in which the variance estimator can be flexibly adopted to the present situation while maintaining its asymptotical properties. We exemplarily suggest a variance estimator which is uncorrelated to the compensated counting process. Furthermore, we provide sample size and power calculations for any approach fitting into our framework. Finally, we compare several methods via simulation studies and the hypothetical setup of a Phase II trial based on real world data.

We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts.

In the domain generalization literature, a common objective is to learn representations independent of the domain after conditioning on the class label. We show that this objective is not sufficient: there exist counter-examples where a model fails to generalize to unseen domains even after satisfying class-conditional domain invariance. We formalize this observation through a structural causal model and show the importance of modeling within-class variations for generalization. Specifically, classes contain objects that characterize specific causal features, and domains can be interpreted as interventions on these objects that change non-causal features. We highlight an alternative condition: inputs across domains should have the same representation if they are derived from the same object. Based on this objective, we propose matching-based algorithms when base objects are observed (e.g., through data augmentation) and approximate the objective when objects are not observed (MatchDG). Our simple matching-based algorithms are competitive to prior work on out-of-domain accuracy for rotated MNIST, Fashion-MNIST, PACS, and Chest-Xray datasets. Our method MatchDG also recovers ground-truth object matches: on MNIST and Fashion-MNIST, top-10 matches from MatchDG have over 50% overlap with ground-truth matches.

Importance sampling is one of the most widely used variance reduction strategies in Monte Carlo rendering. In this paper, we propose a novel importance sampling technique that uses a neural network to learn how to sample from a desired density represented by a set of samples. Our approach considers an existing Monte Carlo rendering algorithm as a black box. During a scene-dependent training phase, we learn to generate samples with a desired density in the primary sample space of the rendering algorithm using maximum likelihood estimation. We leverage a recent neural network architecture that was designed to represent real-valued non-volume preserving ('Real NVP') transformations in high dimensional spaces. We use Real NVP to non-linearly warp primary sample space and obtain desired densities. In addition, Real NVP efficiently computes the determinant of the Jacobian of the warp, which is required to implement the change of integration variables implied by the warp. A main advantage of our approach is that it is agnostic of underlying light transport effects, and can be combined with many existing rendering techniques by treating them as a black box. We show that our approach leads to effective variance reduction in several practical scenarios.

In this paper we introduce a covariance framework for the analysis of EEG and MEG data that takes into account observed temporal stationarity on small time scales and trial-to-trial variations. We formulate a model for the covariance matrix, which is a Kronecker product of three components that correspond to space, time and epochs/trials, and consider maximum likelihood estimation of the unknown parameter values. An iterative algorithm that finds approximations of the maximum likelihood estimates is proposed. We perform a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components. Apart from that, we illustrate our method on real EEG and MEG data sets. The proposed covariance model is applicable in a variety of cases where spontaneous EEG or MEG acts as source of noise and realistic noise covariance estimates are needed for accurate dipole localization, such as in evoked activity studies, or where the properties of spontaneous EEG or MEG are themselves the topic of interest, such as in combined EEG/fMRI experiments in which the correlation between EEG and fMRI signals is investigated.