A major effort in modern high-dimensional statistics has been devoted to the analysis of linear predictors trained on nonlinear feature embeddings via empirical risk minimization (ERM). Gaussian equivalence theory (GET) has emerged as a powerful universality principle in this context: it states that the behavior of high-dimensional, complex features can be captured by Gaussian surrogates, which are more amenable to analysis. Despite its remarkable successes, numerical experiments show that this equivalence can fail even for simple embeddings -- such as polynomial maps -- under general scaling regimes. We investigate this breakdown in the setting of random feature (RF) models in the quadratic scaling regime, where both the number of features and the sample size grow quadratically with the data dimension. We show that when the target function depends on a low-dimensional projection of the data, such as generalized linear models, GET yields incorrect predictions. To capture the correct asymptotics, we introduce a Conditional Gaussian Equivalent (CGE) model, which can be viewed as appending a low-dimensional non-Gaussian component to an otherwise high-dimensional Gaussian model. This hybrid model retains the tractability of the Gaussian framework and accurately describes RF models in the quadratic scaling regime. We derive sharp asymptotics for the training and test errors in this setting, which continue to agree with numerical simulations even when GET fails. Our analysis combines general results on CLT for Wiener chaos expansions and a careful two-phase Lindeberg swapping argument. Beyond RF models and quadratic scaling, our work hints at a rich landscape of universality phenomena in high-dimensional ERM.

Electromagnetic Navigation Systems (eMNS) can be used to control a variety of multiscale devices within the human body for remote surgery. Accurate modeling of the magnetic fields generated by the electromagnets of an eMNS is crucial for the precise control of these devices. Existing methods assume a linear behavior of these systems, leading to significant modeling errors within nonlinear regions exhibited at higher magnetic fields. In this paper, we use a random forest (RF) and an artificial neural network (ANN) to model the nonlinear behavior of the magnetic fields generated by an eMNS. Both machine learning methods outperformed the state-of-the-art linear multipole electromagnet method (LMEM). The RF and the ANN model reduced the root mean squared error of the LMEM when predicting the field magnitude by around 40% and 80%, respectively, over the entire current range of the eMNS. At high current regions, especially between 30 and 35 A, the field-magnitude RMSE improvement of the ANN model over the LMEM was over 35 mT. This study demonstrates the feasibility of using machine learning methods to model an eMNS for medical applications, and its ability to account for complex nonlinear behavior at high currents. The use of machine learning thus shows promise for improving surgical procedures that use magnetic navigation.

Abstract--Recent work has demonstrated the utility of Random Forest (RF) proximities for various supervised machine learning tasks, including outlier detection, missing data imputation, and visualization. However, the utility of the RF proximities depends upon the success of the RF model, which itself is not the ideal model in all contexts. RF proximities have recently been extended to time series by means of the distance-based Proximity Forest (PF) model, among others, affording time series analysis with the benefits of RF proximities. In this work, we introduce the generalized PF model, thereby extending RF proximities to all contexts in which supervised distance-based machine learning can occur . Additionally, we introduce a variant of the PF model for regression tasks. We also introduce the notion of using the generalized PF model as a meta-learning framework, extending supervised imputation capability to any pre-trained classifier . We experimentally demonstrate the unique advantages of the generalized PF model compared with both the RF model and the k-nearest neighbors model.

Efficiently and meaningfully estimating prediction uncertainty is important for exploration in active learning campaigns in materials discovery, where samples with high uncertainty are interpreted as containing information missing from the model. In this work, the effect of different uncertainty estimation and calibration methods are evaluated for active learning when using ensembles of ALIGNN, eXtreme Gradient Boost, Random Forest, and Neural Network model architectures. We compare uncertainty estimates from ALIGNN deep ensembles to loss landscape uncertainty estimates obtained for solubility, bandgap, and formation energy prediction tasks. We then evaluate how the quality of the uncertainty estimate impacts an active learning campaign that seeks model generalization to out-of-distribution data. Uncertainty calibration methods were found to variably generalize from in-domain data to out-of-domain data. Furthermore, calibrated uncertainties were generally unsuccessful in reducing the amount of data required by a model to improve during an active learning campaign on out-of-distribution data when compared to random sampling and uncalibrated uncertainties. The impact of poor-quality uncertainty persists for random forest and eXtreme Gradient Boosting models trained on the same data for the same tasks, indicating that this is at least partially intrinsic to the data and not due to model capacity alone. Analysis of the target, in-distribution uncertainty, out-of-distribution uncertainty, and training residual distributions suggest that future work focus on understanding empirical uncertainties in the feature input space for cases where ensemble prediction variances do not accurately capture the missing information required for the model to generalize.

Rectified Flow (RF) has been widely used as an effective generative model. Although RF is primarily based on probability flow Ordinary Differential Equations (ODE), recent studies have shown that injecting noise through reverse-time Stochastic Differential Equations (SDE) for sampling can achieve superior generative performance. Inspired by Positive-incentive Noise (pi-noise), we propose an innovative generative algorithm to train pi-noise generators, namely Rectified Noise (RN), which improves the generative performance by injecting pi-noise into the velocity field of pre-trained RF models. After introducing the Rectified Noise pipeline, pre-trained RF models can be efficiently transformed into pi-noise generators. We validate Rectified Noise by conducting extensive experiments across various model architectures on different datasets. Notably, we find that: (1) RF models using Rectified Noise reduce FID from 10.16 to 9.05 on ImageNet-1k. (2) The models of pi-noise generators achieve improved performance with only 0.39% additional training parameters.

Gradient information is widely useful and available in applications, and is therefore natural to include in the training of neural networks. Yet little is known theoretically about the impact of Sobolev training -- regression with both function and gradient data -- on the generalization error of highly overparameterized predictive models in high dimensions. In this paper, we obtain a precise characterization of this training modality for random feature (RF) models in the limit where the number of trainable parameters, input dimensions, and training data tend proportionally to infinity. Our model for Sobolev training reflects practical implementations by sketching gradient data onto finite dimensional subspaces. By combining the replica method from statistical physics with linearizations in operator-valued free probability theory, we derive a closed-form description for the generalization errors of the trained RF models. For target functions described by single-index models, we demonstrate that supplementing function data with additional gradient data does not universally improve predictive performance. Rather, the degree of overparameterization should inform the choice of training method. More broadly, our results identify settings where models perform optimally by interpolating noisy function and gradient data.

Rectified Flow offers a simple and effective approach to high-quality generative modeling by learning a velocity field. However, we identify a limitation in directly modeling the velocity with an unconstrained neural network: the learned velocity often fails to satisfy certain boundary conditions, leading to inaccurate velocity field estimations that deviate from the desired ODE. This issue is particularly critical during stochastic sampling at inference, as the score function's errors are amplified near the boundary. T o mitigate this, we propose a Boundary-enforced Rectified Flow Model (Boundary RF Model), in which we enforce boundary conditions with a minimal code modification. Boundary RF Model improves performance over vanilla RF model, demonstrating 8.01% improvement in FID score on ImageNet using ODE sampling and 8.98% improvement using SDE sampling.