We introduce and analyze Target-Induced Loss Tilting (TILT) for unsupervised domain adaptation under covariate shift. It is based on a novel objective function that decomposes the source predictor as $f+b$, fits $f+b$ on labeled source data while simultaneously penalizing the auxiliary component $b$ on unlabeled target inputs. The resulting fit $f$ is deployed as the final target predictor. At the population level, we show that this target-side penalty implicitly induces relative importance weighting at the population level, but in terms of an estimand $b^*_f$ that is self-localized to the current error, and remains uniformly bounded for any source-target pair (even those with disjoint supports). We prove a general finite-sample oracle inequality on the excess risk, and use it to give an end-to-end guarantee for training with sparse ReLU networks. Experiments on controlled regression problems and shifted CIFAR-100 distillation show that TILT improves target-domain performance over source-only training, exact importance weighting, and relative density-ratio baselines, with a stable dependence on the regularization parameter.

This paper focuses on the problem of unbounded density ratio estimation -- an understudied yet critical challenge in statistical learning -- and its application to covariate shift adaptation. Much of the existing literature assumes that the density ratio is either uniformly bounded or unbounded but known exactly. These conditions are often violated in practice, creating a gap between theoretical guarantees and real-world applicability. In contrast, this work directly addresses unbounded density ratios and integrates them into importance weighting for effective covariate shift adaptation. We propose a three-step estimation method that leverages unlabeled data from both the source and target distributions: (1) estimating a relative density ratio; (2) applying a truncation operation to control its unboundedness; and (3) transforming the truncated estimate back into the standard density ratio. The estimated density ratio is then employed as importance weights for regression under covariate shift. We establish rigorous, non-asymptotic convergence guarantees for both the proposed density ratio estimator and the resulting regression function estimator, demonstrating optimal or near-optimal convergence rates. Our findings offer new theoretical insights into density ratio estimation and learning under covariate shift, extending classical learning theory to more practical and challenging scenarios.

The cortico-spinal neural pathway is fundamental for motor control and movement execution, and in humans it is typically studied using concurrent electroencephalography (EEG) and electromyography (EMG) recordings. However, current approaches for capturing high-level and contextual connectivity between these recordings have important limitations. Here, we present a novel application of statistical dependence estimators based on orthonormal decomposition of density ratios to model the relationship between cortical and muscle oscillations. Our method extends from traditional scalar-valued measures by learning eigenvalues, eigenfunctions, and projection spaces of density ratios from realizations of the signal, addressing the interpretability, scalability, and local temporal dependence of cortico-muscular connectivity. We experimentally demonstrate that eigenfunctions learned from cortico-muscular connectivity can accurately classify movements and subjects. Moreover, they reveal channel and temporal dependencies that confirm the activation of specific EEG channels during movement.

There are increasing cases where the class labels of test samples are unavailable, creating a significant need and challenge in measuring the discrepancy between training and test distributions. This distribution discrepancy complicates the assessment of whether the hypothesis selected by an algorithm on training samples remains applicable to test samples. We present a novel approach called Importance Divergence (I-Div) to address the challenge of test label unavailability, enabling distribution discrepancy evaluation using only training samples. I-Div transfers the sampling patterns from the test distribution to the training distribution by estimating density and likelihood ratios. Specifically, the density ratio, informed by the selected hypothesis, is obtained by minimizing the Kullback-Leibler divergence between the actual and estimated input distributions. Simultaneously, the likelihood ratio is adjusted according to the density ratio by reducing the generalization error of the distribution discrepancy as transformed through the two ratios. Experimentally, I-Div accurately quantifies the distribution discrepancy, as evidenced by a wide range of complex data scenarios and tasks.

The cortico-spinal neural pathway is fundamental for motor control and movement execution, and in humans it is typically studied using concurrent electroen-cephalography (EEG) and electromyography (EMG) recordings.

The assumption that data are independently and identically distributed (IID) is staple in statistical machine learning. It suggests that a hypothesis selected by an algorithm, after observing several training samples, should perform effectively on test samples from the same unknown distribution.

Efficient coding theory hasbeen used topredict howthe brain should allocate its limited resources in these scenarios by removing redundant information [1-4].