Operator learning is an emerging area of machine learning which aims to learn mappings between infinite dimensional function spaces. Here we uncover a connection between operator learning architectures and conditioned neural fields from computer vision, providing a unified perspective for examining differences between popular operator learning models. We find that many commonly used operator learning models can be viewed as neural fields with conditioning mechanisms restricted to point-wise and/or global information. Motivated by this, we propose the Continuous Vision Transformer (CViT), a novel neural operator architecture that employs a vision transformer encoder and uses cross-attention to modulate a base field constructed with a trainable grid-based positional encoding of query coordinates. Despite its simplicity, CViT achieves state-of-the-art results across challenging benchmarks in climate modeling and fluid dynamics. Our contributions can be viewed as a first step towards adapting advanced computer vision architectures for building more flexible and accurate machine learning models in physical sciences.

High-dimensional linear bandits with low-dimensional structure have received considerable attention in recent studies due to their practical significance. The most common structure in the literature is sparsity. However, it may not be available in practice. Symmetry, where the reward is invariant under certain groups of transformations on the set of arms, is another important inductive bias in the high-dimensional case that covers many standard structures, including sparsity. In this work, we study high-dimensional symmetric linear bandits where the symmetry is hidden from the learner, and the correct symmetry needs to be learned in an online setting. We examine the structure of a collection of hidden symmetry and provide a method based on model selection within the collection of low-dimensional subspaces. Our algorithm achieves a regret bound of $ O(d_0^{1/3} T^{2/3} \log(d))$, where $d$ is the ambient dimension which is potentially very large, and $d_0$ is the dimension of the true low-dimensional subspace such that $d_0 \ll d$. With an extra assumption on well-separated models, we can further improve the regret to $ O(d_0\sqrt{T\log(d)} )$.

We introduce LatentTimePFN (LaT-PFN), a foundational Time Series model with a strong embedding space that enables zero-shot forecasting. To achieve this, we perform in-context learning in latent space utilizing a novel integration of the Prior-data Fitted Networks (PFN) and Joint Embedding Predictive Architecture (JEPA) frameworks. We leverage the JEPA framework to create a prediction-optimized latent representation of the underlying stochastic process that generates time series and combines it with contextual learning, using a PFN. Furthermore, we improve on preceding works by utilizing related time series as a context and introducing a normalized abstract time axis. This reduces training time and increases the versatility of the model by allowing any time granularity and forecast horizon. We show that this results in superior zero-shot predictions compared to established baselines. We also demonstrate our latent space produces informative embeddings of both individual time steps and fixed-length summaries of entire series. Finally, we observe the emergence of multi-step patch embeddings without explicit training, suggesting the model actively learns discrete tokens that encode local structures in the data, analogous to vision transformers.

Traditionally, the task of sampling from un-normalized densities is largely delegated to MCMC methods. However, modern machine learning developments in optimal transport and generative modeling have greatly expanded the toolbox we have available for performing such tasks, which can further benefit from powerful advancements in deep learning. By reducing the problem to performing empirical risk minimization with neural networks, they hold promise especially for sampling from high-dimensional and multimodal distributions compared to MCMC-based alternatives. In this work, we propose novel training objectives that are amenable to tractable approximations for sampling (without access to data from the target, as typically considered in the MCMC setup), and demonstrate their numerical advantages compared to several related methods that are also built around forward-backward SDEs through time-reversals. Before elaborating on these connections and synthesis in Section 2, we briefly summarize our contributions below. We present a transport/control based sampler using Schrödinger bridge (SB) idea (generalizable to extended state space and nonlinear prior), which compared to MCMC-based approaches has two benefits: (1) alleviate metastability without being confined to make local moves in the state space; (2) based on interpolating path, it comes with the ability to provide low-variance, unbiased normalizing constant estimates using the trajectory information (c.f.

We study min-max algorithms to solve zero-sum differentiable games on Riemannian manifold. The notions of differentiable Stackelberg equilibrium and differentiable Nash equilibrium in Euclidean space are generalized to Riemannian manifold, through an intrinsic definition which does not depend on the choice of local coordinate chart of manifold. We then provide sufficient conditions for the local convergence of the deterministic simultaneous algorithms $\tau$-GDA and $\tau$-SGA near such equilibrium, using a general methodology based on spectral analysis. These algorithms are extended with stochastic gradients and applied to the training of Wasserstein GAN. The discriminator of GAN is constructed from Lipschitz-continuous functions based on Stiefel manifold. We show numerically how the insights obtained from the local convergence analysis may lead to an improvement of GAN models.

The recently introduced TabPFN pretrains an In-Context Learning (ICL) transformer on synthetic data to perform tabular data classification. As synthetic data does not share features or labels with real-world data, the underlying mechanism that contributes to the success of this method remains unclear. This study provides an explanation by demonstrating that ICL-transformers acquire the ability to create complex decision boundaries during pretraining. To validate our claim, we develop a novel forest dataset generator which creates datasets that are unrealistic, but have complex decision boundaries. Our experiments confirm the effectiveness of ICL-transformers pretrained on this data. Furthermore, we create TabForestPFN, the ICL-transformer pretrained on both the original TabPFN synthetic dataset generator and our forest dataset generator. By fine-tuning this model, we reach the current state-of-the-art on tabular data classification.

In multivariate spline regression, the number and locations of knots influence the performance and interpretability significantly. However, due to non-differentiability and varying dimensions, there is no desirable frequentist method to make inference on knots. In this article, we propose a fully Bayesian approach for knot inference in multivariate spline regression. The existing Bayesian method often uses BIC to calculate the posterior, but BIC is too liberal and it will heavily overestimate the knot number when the candidate model space is large. We specify a new prior on the knot number to take into account the complexity of the model space and derive an analytic formula in the normal model. In the non-normal cases, we utilize the extended Bayesian information criterion to approximate the posterior density. The samples are simulated in the space with differing dimensions via reversible jump Markov chain Monte Carlo. We apply the proposed method in knot inference and manifold denoising. Experiments demonstrate the splendid capability of the algorithm, especially in function fitting with jumping discontinuity.

The cosine router in sparse Mixture of Experts (MoE) has recently emerged as an attractive alternative to the conventional linear router. Indeed, the cosine router demonstrates favorable performance in image and language tasks and exhibits better ability to mitigate the representation collapse issue, which often leads to parameter redundancy and limited representation potentials. Despite its empirical success, a comprehensive analysis of the cosine router in sparse MoE has been lacking. Considering the least square estimation of the cosine routing sparse MoE, we demonstrate that due to the intrinsic interaction of the model parameters in the cosine router via some partial differential equations, regardless of the structures of the experts, the estimation rates of experts and model parameters can be as slow as $\mathcal{O}(1/\log^{\tau}(n))$ where $\tau > 0$ is some constant and $n$ is the sample size. Surprisingly, these pessimistic non-polynomial convergence rates can be circumvented by the widely used technique in practice to stabilize the cosine router -- simply adding noises to the $\mathbb{L}_{2}$ norms in the cosine router, which we refer to as \textit{perturbed cosine router}. Under the strongly identifiable settings of the expert functions, we prove that the estimation rates for both the experts and model parameters under the perturbed cosine routing sparse MoE are significantly improved to polynomial rates. Finally, we conduct extensive simulation studies in both synthetic and real data settings to empirically validate our theoretical results.

We initiate the study of locally differentially private (LDP) learning with public features. We define semi-feature LDP, where some features are publicly available while the remaining ones, along with the label, require protection under local differential privacy. Under semi-feature LDP, we demonstrate that the mini-max convergence rate for non-parametric regression is significantly reduced compared to that of classical LDP. Then we propose HistOfTree, an estimator that fully leverages the information contained in both public and private features. Theoretically, HistOfTree reaches the mini-max optimal convergence rate. Empirically, HistOfTree achieves superior performance on both synthetic and real data. We also explore scenarios where users have the flexibility to select features for protection manually. In such cases, we propose an estimator and a data-driven parameter tuning strategy, leading to analogous theoretical and empirical results.

Given an unconditional diffusion model $\pi(x, y)$, using it to perform conditional simulation $\pi(x \mid y)$ is still largely an open question and is typically achieved by learning conditional drifts to the denoising SDE after the fact. In this work, we express conditional simulation as an inference problem on an augmented space corresponding to a partial SDE bridge. This perspective allows us to implement efficient and principled particle Gibbs and pseudo-marginal samplers marginally targeting the conditional distribution $\pi(x \mid y)$. Contrary to existing methodology, our methods do not introduce any additional approximation to the unconditional diffusion model aside from the Monte Carlo error. We showcase the benefits and drawbacks of our approach on a series of synthetic and real data examples.