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Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical…

Statistical learning relies upon data sampled from a distribution, and we usually do not care what actually generated it in the first place. From the point of view of causal modeling, the structure of each distribution is induced by…

We investigate the emergence of non-linear diffusivity in kinetically constrained, one-dimensional symmetric exclusion processes satisfying the gradient condition. Recent developments introduced new gradient dynamics based on the Bernstein…

The exponential random graph model (ERGM) is a central object in the study of clustering properties in social networks as well as canonical ensembles in statistical physics. Despite some breakthrough works in the mathematical understanding…

In order to describe the firing activity of a homogenous assembly of neurons, we consider time elapsed models, which give mathematical descriptions of the probability density of neurons structured by the distribution of times elapsed since…

The stability-plasticity dilemma is a major challenge in continual learning, as it involves balancing the conflicting objectives of maintaining performance on previous tasks while learning new tasks. In this paper, we propose the…

We study stochastic optimization in the context of performative shifts, where the data distribution changes in response to the deployed model. We demonstrate that naive retraining can be provably suboptimal even for simple distribution…

Prior interpretability research studying narrow distributions has preliminarily identified self-repair, a phenomena where if components in large language models are ablated, later components will change their behavior to compensate. Our…

We study self-programming in recurrent neural networks where both neurons (the `processors') and synaptic interactions (`the programme') evolve in time simultaneously, according to specific coupled stochastic equations. The interactions are…

Generative compressed sensing uses the range of a pretrained generator as a nonlinear model for recovering structured signals from limited measurements. We study a conditional version of this problem for image recovery from subsampled…

The use of synthetically generated data for training models is becoming a common practice. While generated data can augment the training data, repeated training on synthetic data raises concerns about distribution drift and degradation of…

Diffusion models have revolutionized various application domains, including computer vision and audio generation. Despite the state-of-the-art performance, diffusion models are known for their slow sample generation due to the extensive…

While generative models have become increasingly prevalent across various domains, fundamental concerns regarding their reliability persist. A crucial yet understudied aspect of these models is the uncertainty quantification surrounding…

This paper studies multiparty learning, aiming to learn a model using the private data of different participants. Model reuse is a promising solution for multiparty learning, assuming that a local model has been trained for each party.…

Diffusion models, which leverage stochastic processes to capture complex data distributions effectively, have shown their performance as generative models, achieving notable success in image-related tasks through iterative denoising…

In this paper, a single population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method…

Fine-tuning large language models (LLMs) typically relies on producing large sets of input-output pairs. Yet for a given question, there can be many valid outputs. In practice, these outputs are often derived by distilling knowledge from…

The remarkable generalization performance of large-scale models has been challenging the conventional wisdom of the statistical learning theory. Although recent theoretical studies have shed light on this behavior in linear models and…

We consider the problem of learning from training data obtained in different contexts, where the underlying context distribution is unknown and is estimated empirically. We develop a robust method that takes into account the uncertainty of…

When studying the dynamics of trait distribution of populations in a heterogeneous environment, classical models from quantitative genetics choose to look at its system of moments, specifically the first two ones. Additionally, in order to…