Regularizing neural networks is important for anticipating model behavior in regions of the data space that are not well represented. In this work, we propose a regularization technique for enforcing a level of smoothness in the mapping between the data input space and the loss value. We specify the level of regularity by requiring that the loss of the network satisfies an elliptic operator over the data domain. To do this, we modify the usual empirical risk minimization objective such that we instead minimize a new objective that satisfies an elliptic operator over points within the domain. This allows us to use existing theory on elliptic operators to anticipate the behavior of the error for points outside the training set. We propose a tractable computational method that approximates the behavior of the elliptic operator while being computationally efficient. Finally, we analyze the properties of the proposed regularization to understand the performance on common problems of distribution shift and group imbalance. Numerical experiments confirm the utility of the proposed regularization technique.

McKean-Vlasov stochastic differential equations (MV-SDEs) provide a mathematical description of the behavior of an infinite number of interacting particles by imposing a dependence on the particle density. As such, we study the influence of explicitly including distributional information in the parameterization of the SDE. We propose a series of semi-parametric methods for representing MV-SDEs, and corresponding estimators for inferring parameters from data based on the properties of the MV-SDE. We analyze the characteristics of the different architectures and estimators, and consider their applicability in relevant machine learning problems. We empirically compare the performance of the different architectures and estimators on real and synthetic datasets for time series and probabilistic modeling. The results suggest that explicitly including distributional dependence in the parameterization of the SDE is effective in modeling temporal data with interaction under an exchangeability assumption while maintaining strong performance for standard It\^o-SDEs due to the richer class of probability flows associated with MV-SDEs.

Transformers incorporating copula structures have demonstrated remarkable performance in time series prediction. However, their heavy reliance on self-attention mechanisms demands substantial computational resources, thus limiting their practical utility across a wide range of tasks. In this work, we present a model that combines the perceiver architecture with a copula structure to enhance time-series forecasting. By leveraging the perceiver as the encoder, we efficiently transform complex, high-dimensional, multimodal data into a compact latent space, thereby significantly reducing computational demands. To further reduce complexity, we introduce midpoint inference and local attention mechanisms, enabling the model to capture dependencies within imputed samples effectively. Subsequently, we deploy the copula-based attention and output variance testing mechanism to capture the joint distribution of missing data, while simultaneously mitigating error propagation during prediction. Our experimental results on the unimodal and multimodal benchmarks showcase a consistent 20\% improvement over the state-of-the-art methods, while utilizing less than half of available memory resources.

Understanding individual treatment effects in extreme regimes is important for characterizing risks associated with different interventions. This is hindered by the fact that extreme regime data may be hard to collect, as it is scarcely observed in practice. In addressing this issue, we propose a new framework for estimating the individual treatment effect in extreme regimes (ITE$_2$). Specifically, we quantify this effect by the changes in the tail decay rates of potential outcomes in the presence or absence of the treatment. Subsequently, we establish conditions under which ITE$_2$ may be calculated and develop algorithms for its computation. We demonstrate the efficacy of our proposed method on various synthetic and semi-synthetic datasets.

Point processes often have a natural interpretation with respect to a continuous process. We propose a point process construction that describes arrival time observations in terms of the state of a latent diffusion process. In this framework, we relate the return times of a diffusion in a continuous path space to new arrivals of the point process. This leads to a continuous sample path that is used to describe the underlying mechanism generating the arrival distribution. These models arise in many disciplines, such as financial settings where actions in a market are determined by a hidden continuous price or in neuroscience where a latent stimulus generates spike trains. Based on the developments in It\^o's excursion theory, we propose methods for inferring and sampling from the point process derived from the latent diffusion process. We illustrate the approach with numerical examples using both simulated and real data. The proposed methods and framework provide a basis for interpreting point processes through the lens of diffusions.

We propose a new generative modeling technique for learning multidimensional cumulative distribution functions (CDFs) in the form of copulas. Specifically, we consider certain classes of copulas known as Archimedean and hierarchical Archimedean copulas, popular for their parsimonious representation and ability to model different tail dependencies. We consider their representation as mixture models with Laplace transforms of latent random variables from generative neural networks. This alternative representation allows for computational efficiencies and easy sampling, especially in high dimensions. We describe multiple methods for optimizing the network parameters. Finally, we present empirical results that demonstrate the efficacy of our proposed method in learning multidimensional CDFs and its computational efficiency compared to existing methods.