Regression evaluation has been performed for decades. Some metrics have been identified to be robust against shifting and scaling of the data but considering the different distributions of data is much more difficult to address (imbalance problem) even though it largely impacts the comparability between evaluations on different datasets. In classification, it has been stated repeatedly that performance metrics like the F-Measure and Accuracy are highly dependent on the class distribution and that comparisons between different datasets with different distributions are impossible. We show that the same problem exists in regression. The distribution of odometry parameters in robotic applications can for example largely vary between different recording sessions. Here, we need regression algorithms that either perform equally well for all function values, or that focus on certain boundary regions like high speed. This has to be reflected in the evaluation metric. We propose the modification of established regression metrics by weighting with the inverse distribution of function values $Y$ or the samples $X$ using an automatically tuned Gaussian kernel density estimator. We show on synthetic and robotic data in reproducible experiments that classical metrics behave wrongly, whereas our new metrics are less sensitive to changing distributions, especially when correcting by the marginal distribution in $X$. Our new evaluation concept enables the comparison of results between different datasets with different distributions. Furthermore, it can reveal overfitting of a regression algorithm to overrepresented target values. As an outcome, non-overfitting regression algorithms will be more likely chosen due to our corrected metrics.

Multi-view data, in which data of different types are obtained from a common set of samples, is now common in many scientific problems. An important problem in the analysis of multi-view data is identifying interactions between groups of features from different data types. A bimodule is a pair $(A,B)$ of feature sets from two different data types such that the aggregate cross-correlation between the features in $A$ and those in $B$ is large. A bimodule $(A,B)$ is stable if $A$ coincides with the set of features having significant aggregate correlation with the features in $B$, and vice-versa. At the population level, stable bimodules correspond to connected components of the cross-correlation network, which is the bipartite graph whose edges are pairs of features with non-zero cross-correlations. We develop an iterative, testing-based procedure, called BSP, to identify stable bimodules in two moderate- to high-dimensional data sets. BSP relies on permutation-based p-values for sums of squared cross-correlations. We efficiently approximate the p-values using tail probabilities of gamma distributions that are fit using analytical estimates of the permutation moments of the test statistic. Our moment estimates depend on the eigenvalues of the intra-correlation matrices of $A$ and $B$ and as a result, the significance of observed cross-correlations accounts for the correlations within each data type. We carry out a thorough simulation study to assess the performance of BSP, and present an extended application of BSP to the problem of expression quantitative trait loci (eQTL) analysis using recent data from the GTEx project. In addition, we apply BSP to climatology data in order to identify regions in North America where annual temperature variation affects precipitation.

In supervised learning, we fit a single statistical model to a given data set, assuming that the data is associated with a singular task, which yields well-tuned models for specific use, but does not adapt well to new contexts. By contrast, in meta-learning, the data is associated with numerous tasks, and we seek a model that may perform well on all tasks simultaneously, in pursuit of greater generalization. One challenge in meta-learning is how to exploit relationships between tasks and classes, which is overlooked by commonly used random or cyclic passes through data. In this work, we propose actively selecting samples on which to train by discerning covariates inside and between meta-training sets. Specifically, we cast the problem of selecting a sample from a number of meta-training sets as either a multi-armed bandit or a Markov Decision Process (MDP), depending on how one encapsulates correlation across tasks. We develop scheduling schemes based on Upper Confidence Bound (UCB), Gittins Index and tabular Markov Decision Problems (MDPs) solved with linear programming, where the reward is the scaled statistical accuracy to ensure it is a time-invariant function of state and action. Across a variety of experimental contexts, we observe significant reductions in sample complexity of active selection scheme relative to cyclic or i.i.d. sampling, demonstrating the merit of exploiting covariates in practice.

In many applications, we are given access to noisy modulo samples of a smooth function with the goal being to robustly unwrap the samples, i.e., to estimate the original samples of the function. In a recent work, Cucuringu and Tyagi proposed denoising the modulo samples by first representing them on the unit complex circle and then solving a smoothness regularized least squares problem -- the smoothness measured w.r.t the Laplacian of a suitable proximity graph $G$ -- on the product manifold of unit circles. This problem is a quadratically constrained quadratic program (QCQP) which is nonconvex, hence they proposed solving its sphere-relaxation leading to a trust region subproblem (TRS). In terms of theoretical guarantees, $\ell_2$ error bounds were derived for (TRS). These bounds are however weak in general and do not really demonstrate the denoising performed by (TRS). In this work, we analyse the (TRS) as well as an unconstrained relaxation of (QCQP). For both these estimators we provide a refined analysis in the setting of Gaussian noise and derive noise regimes where they provably denoise the modulo observations w.r.t the $\ell_2$ norm. The analysis is performed in a general setting where $G$ is any connected graph.

Many modern applications involve the acquisition of noisy modulo samples of a function $f$, with the goal being to recover estimates of the original samples of $f$. For a Lipschitz function $f:[0,1]^d \to \mathbb{R}$, suppose we are given the samples $y_i = (f(x_i) + \eta_i)\bmod 1; \quad i=1,\dots,n$ where $\eta_i$ denotes noise. Assuming $\eta_i$ are zero-mean i.i.d Gaussian's, and $x_i$'s form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples $f(x_i)$ with a uniform error rate $O((\frac{\log n}{n})^{\frac{1}{d+2}})$ holding with high probability. The first stage involves embedding the points on the unit complex circle, and obtaining denoised estimates of $f(x_i)\bmod 1$ via a $k$NN (nearest neighbor) estimator. The second stage involves a sequential unwrapping procedure which unwraps the denoised mod $1$ estimates from the first stage. Recently, Cucuringu and Tyagi proposed an alternative way of denoising modulo $1$ data which works with their representation on the unit complex circle. They formulated a smoothness regularized least squares problem on the product manifold of unit circles, where the smoothness is measured with respect to the Laplacian of a proximity graph $G$ involving the $x_i$'s. This is a nonconvex quadratically constrained quadratic program (QCQP) hence they proposed solving its semidefinite program (SDP) based relaxation. We derive sufficient conditions under which the SDP is a tight relaxation of the QCQP. Hence under these conditions, the global solution of QCQP can be obtained in polynomial time.

When modelling censored observations, a typical approach in current regression methods is to use a censored-Gaussian (i.e. Tobit) model to describe the conditional output distribution. In this paper, as in the case of missing data, we argue that exploiting correlations between multiple outputs can enable models to better address the bias introduced by censored data. To do so, we introduce a heteroscedastic multi-output Gaussian process model which combines the non-parametric flexibility of GPs with the ability to leverage information from correlated outputs under input-dependent noise conditions. To address the resulting inference intractability, we further devise a variational bound to the marginal log-likelihood suitable for stochastic optimization. We empirically evaluate our model against other generative models for censored data on both synthetic and real world tasks and further show how it can be generalized to deal with arbitrary likelihood functions. Results show how the added flexibility allows our model to better estimate the underlying non-censored (i.e. true) process under potentially complex censoring dynamics.

Sparse polynomial chaos expansions (PCEs) are an efficient and widely used surrogate modeling method in uncertainty quantification. Among the many contributions aiming at computing an accurate sparse PCE while using as few model evaluations as possible, basis adaptivity is a particularly interesting approach. It consists in starting from an expansion with a small number of polynomial terms, and then parsimoniously adding and removing basis functions in an iterative fashion. We describe several state-of-the-art approaches from the recent literature and extensively benchmark them on a large set of computational models representative of a wide range of engineering problems. We investigate the synergies between sparse regression solvers and basis adaptivity schemes, and provide recommendations on which of them are most promising for specific classes of problems. Furthermore, we explore the performance of a novel cross-validation-based solver and basis adaptivity selection scheme, which consistently provides close-to-optimal results.

Under the framework of reproducing kernel Hilbert space (RKHS), we consider the penalized least-squares of the partially functional linear models (PFLM), whose predictor contains both functional and traditional multivariate part, and the multivariate part allows a divergent number of parameters. From the non-asymptotic point of view, we focus on the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form under a more general assumption known as the effective dimension to the model, by which we also show the prediction consistency when the number of multivariate covariates $p$ slightly increases with the sample size $n$. Our new finding implies a trade-off between the number of non-functional predictors and the effective dimension of the kernel principal components to ensure the prediction consistency in the increasing-dimensional setting. The analysis in our proof hinges on the spectral condition of the sandwich operator of the covariance operator and the reproducing kernel, and on the concentration inequalities for the random elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower bound under the regularity assumption of Kullback-Leibler divergence of the models.

The Classification on high-dimension low-sample-size data (HDLSS) is a challenging problem and it is common to have class-imbalanced data in most application fields. We term this as Imbalanced HDLSS (IHDLSS). Recent theoretical results reveal that the classification criterion and tolerance similarity are crucial to HDLSS, which emphasizes the maximization of within-class variance on the premise of class separability. Based on this idea, a novel linear binary classifier, termed Population Structure-learned Classifier (PSC), is proposed. The proposed PSC can obtain better generalization performance on IHDLSS by maximizing the sum of inter-class scatter matrix and intra-class scatter matrix on the premise of class separability and assigning different intercept values to majority and minority classes. The salient features of the proposed approach are: (1) It works well on IHDLSS; (2) The inverse of high dimensional matrix can be solved in low dimensional space; (3) It is self-adaptive in determining the intercept term for each class; (4) It has the same computational complexity as the SVM. A series of evaluations are conducted on one simulated data set and eight real-world benchmark data sets on IHDLSS on gene analysis. Experimental results demonstrate that the PSC is superior to the state-of-art methods in IHDLSS.

Adversarial training, especially projected gradient descent (PGD), has been the most successful approach for improving robustness against adversarial attacks. After adversarial training, gradients of models with respect to their inputs are meaningful and interpretable by humans. However, the concept of interpretability is not mathematically well established, making it difficult to evaluate it quantitatively. We define interpretability as the alignment of the model gradient with the vector pointing toward the closest point of the support of the other class. We propose a method for measuring this alignment for binary classification problems, using generative adversarial model training to produce the smallest residual needed to change the class present in the image. We show that PGD-trained models are more interpretable than the baseline according to our definition, and our metric presents higher alignment values than a competing metric formulation. We also show that enforcing this alignment increases the robustness of models without adversarial training.