Many recent advances in machine learning are driven by a challenging trifecta: large data size N, high dimensions, and expensive algorithms. In this setting, cross-validation (CV) serves as an important tool for model assessment. Recent advances in approximate cross validation (ACV) provide accurate approximations to CV with only a single model fit, avoiding traditional CV's requirement for repeated runs of expensive algorithms. Unfortunately, these ACV methods can lose both speed and accuracy in high dimensions --- unless sparsity structure is present in the data. Fortunately, there is an alternative type of simplifying structure that is present in most data: approximate low rank (ALR).

Many modern data analyses benefit from explicitly modeling dependence structure in data -- such as measurements across time or space, ordered words in a sentence, or genes in a genome. A gold standard evaluation technique is structured cross-validation (CV), which leaves out some data subset (such as data within a time interval or data in a geographic region) in each fold. But CV here can be prohibitively slow due to the need to re-run already-expensive learning algorithms many times. Previous work has shown approximate cross-validation (ACV) methods provide a fast and provably accurate alternative in the setting of empirical risk minimization. But this existing ACV work is restricted to simpler models by the assumptions that (i) data across CV folds are independent and (ii) an exact initial model fit is available. In structured data analyses, both these assumptions are often untrue.

The evolving field of mobile robotics has indeed increased the demand for simultaneous localization and mapping (SLAM) systems. To augment the localization accuracy and mapping efficacy of SLAM, we refined the core module of the SLAM system. Within the feature matching phase, we introduced cross-validation matching to filter out mismatches. In the keyframe selection strategy, an exponential threshold function is constructed to quantify the keyframe selection process. Compared with a single robot, the multi-robot collaborative SLAM (CSLAM) system substantially improves task execution efficiency and robustness. By employing a centralized structure, we formulate a multi-robot SLAM system and design a coarse-to-fine matching approach for multi-map point cloud registration. Our system, built upon ORB-SLAM3, underwent extensive evaluation utilizing the TUM RGB-D, EuRoC MAV, and TUM_VI datasets. The experimental results demonstrate a significant improvement in the positioning accuracy and mapping quality of our enhanced algorithm compared to those of ORB-SLAM3, with a 12.90% reduction in the absolute trajectory error.

Recent years have seen substantial advances in our understanding of high-dimensional ridge regression, but existing theories assume that training examples are independent. By leveraging recent techniques from random matrix theory and free probability, we provide sharp asymptotics for the in- and out-of-sample risks of ridge regression when the data points have arbitrary correlations. We demonstrate that in this setting, the generalized cross validation estimator (GCV) fails to correctly predict the out-of-sample risk. However, in the case where the noise residuals have the same correlations as the data points, one can modify the GCV to yield an efficiently-computable unbiased estimator that concentrates in the high-dimensional limit, which we dub CorrGCV. We further extend our asymptotic analysis to the case where the test point has nontrivial correlations with the training set, a setting often encountered in time series forecasting. Assuming knowledge of the correlation structure of the time series, this again yields an extension of the GCV estimator, and sharply characterizes the degree to which such test points yield an overly optimistic prediction of long-time risk. We validate the predictions of our theory across a variety of high dimensional data.

It is crucial to assess the predictive performance of a model in order to establish its practicality and relevance in real-world scenarios, particularly for high-dimensional data analysis. Among data splitting or resampling methods, cross-validation (CV) is extensively used for several tasks such as estimating the prediction error, tuning the regularization parameter, and selecting the most suitable predictive model among competing alternatives. The K-fold cross-validation is a popular CV method but its limitation is that the risk estimates are highly dependent on the partitioning of the data (for training and testing). Here, the issues regarding the reproducibility of the K-fold CV estimator is demonstrated in hypothesis testing wherein different partitions lead to notably disparate conclusions. This study presents an alternative novel predictive performance test and valid confidence intervals based on exhaustive nested cross-validation for determining the difference in prediction error between two model-fitting algorithms. A naive implementation of the exhaustive nested cross-validation is computationally costly. Here, we address concerns regarding computational complexity by devising a computationally tractable closed-form expression for the proposed cross-validation estimator using ridge regularization. Our study also investigates strategies aimed at enhancing statistical power within high-dimensional scenarios while controlling the Type I error rate. To illustrate the practical utility of our method, we apply it to an RNA sequencing study and demonstrate its effectiveness in the context of biological data analysis.

Emulating the mapping between quantities of interest and their control parameters using surrogate models finds widespread application in engineering design, including in numerical optimization and uncertainty quantification. Gaussian process models can serve as a probabilistic surrogate model of unknown functions, thereby making them highly suitable for engineering design and decision-making in the presence of uncertainty. In this work, we are interested in emulating quantities of interest observed from models of a system at multiple fidelities, which trade accuracy for computational efficiency. Using multifidelity Gaussian process models, to efficiently fuse models at multiple fidelities, we propose a novel method to actively learn the surrogate model via leave-one-out cross-validation (LOO-CV). Our proposed multifidelity cross-validation (\texttt{MFCV}) approach develops an adaptive approach to reduce the LOO-CV error at the target (highest) fidelity, by learning the correlations between the LOO-CV at all fidelities. \texttt{MFCV} develops a two-step lookahead policy to select optimal input-fidelity pairs, both in sequence and in batches, both for continuous and discrete fidelity spaces. We demonstrate the utility of our method on several synthetic test problems as well as on the thermal stress analysis of a gas turbine blade.

Two key tasks in high-dimensional regularized regression are tuning the regularization strength for good predictions and estimating the out-of-sample risk. It is known that the standard approach -- $k$-fold cross-validation -- is inconsistent in modern high-dimensional settings. While leave-one-out and generalized cross-validation remain consistent in some high-dimensional cases, they become inconsistent when samples are dependent or contain heavy-tailed covariates. To model structured sample dependence and heavy tails, we use right-rotationally invariant covariate distributions - a crucial concept from compressed sensing. In the common modern proportional asymptotics regime where the number of features and samples grow comparably, we introduce a new framework, ROTI-GCV, for reliably performing cross-validation. Along the way, we propose new estimators for the signal-to-noise ratio and noise variance under these challenging conditions. We conduct extensive experiments that demonstrate the power of our approach and its superiority over existing methods.

This paper presents a comparative study of sampling methods within the FedHome framework, designed for personalized in-home health monitoring. FedHome leverages federated learning (FL) and generative convolutional autoencoders (GCAE) to train models on decentralized edge devices while prioritizing data privacy. A notable challenge in this domain is the class imbalance in health data, where critical events such as falls are underrepresented, adversely affecting model performance. To address this, the research evaluates six oversampling techniques using Stratified K-fold cross-validation: SMOTE, Borderline-SMOTE, Random OverSampler, SMOTE-Tomek, SVM-SMOTE, and SMOTE-ENN. These methods are tested on FedHome's public implementation over 200 training rounds with and without stratified K-fold cross-validation. The findings indicate that SMOTE-ENN achieves the most consistent test accuracy, with a standard deviation range of 0.0167-0.0176, demonstrating stable performance compared to other samplers. In contrast, SMOTE and SVM-SMOTE exhibit higher variability in performance, as reflected by their wider standard deviation ranges of 0.0157-0.0180 and 0.0155-0.0180, respectively. Similarly, the Random OverSampler method shows a significant deviation range of 0.0155-0.0176. SMOTE-Tomek, with a deviation range of 0.0160-0.0175, also shows greater stability but not as much as SMOTE-ENN. This finding highlights the potential of SMOTE-ENN to enhance the reliability and accuracy of personalized health monitoring systems within the FedHome framework.

Cross-validation is a common method for estimating the predictive performance of machine learning models. In a data-scarce regime, where one typically wishes to maximize the number of instances used for training the model, an approach called "leave-one-out cross-validation" is often used. In this design, a separate model is built for predicting each data instance after training on all other instances. Since this results in a single test data point available per model trained, predictions are aggregated across the entire dataset to calculate common rank-based performance metrics such as the area under the receiver operating characteristic or precision-recall curves. In this work, we demonstrate that this approach creates a negative correlation between the average label of each training fold and the label of its corresponding test instance, a phenomenon that we term distributional bias. As machine learning models tend to regress to the mean of their training data, this distributional bias tends to negatively impact performance evaluation and hyperparameter optimization. We show that this effect generalizes to leave-P-out cross-validation and persists across a wide range of modeling and evaluation approaches, and that it can lead to a bias against stronger regularization. To address this, we propose a generalizable rebalanced cross-validation approach that corrects for distributional bias. We demonstrate that our approach improves cross-validation performance evaluation in synthetic simulations and in several published leave-one-out analyses.

This paper introduced a clustered estimator of the Network Information Criterion (NICc) to approximate leave-one-cluster-out cross-validated deviance, which can be used as an alternative to cluster-based cross-validation when modeling clustered data. Stone proved that Akaike Information Criterion (AIC) is an asymptotic equivalence to leave-one-observation-out cross-validation if the parametric model is true. Ripley pointed out that the Network Information Criterion (NIC) derived in Stone's proof, is a better approximation to leave-one-observation-out cross-validation when the model is not true. For clustered data, we derived a clustered estimator of NIC, referred to as NICc, by substituting the Fisher information matrix in NIC with its estimator that adjusts for clustering. This adjustment imposes a larger penalty in NICc than the unclustered estimator of NIC when modeling clustered data, thereby preventing overfitting more effectively. In a simulation study and an empirical example, we used linear and logistic regression to model clustered data with Gaussian or binomial response, respectively. We showed that NICc is a better approximation to leave-one-cluster-out deviance and prevents overfitting more effectively than AIC and Bayesian Information Criterion (BIC). NICc leads to more accurate model selection, as determined by cluster-based cross-validation, compared to AIC and BIC.