We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems.

The theory of Koopman operators allows to deploy non-parametric machine learning algorithms to predict and analyze complex dynamical systems.

Learning the Koopman operator and its spectral decomposition from data is enabled by a number of algorithms.

Conditional independence (CI) is central to causal inference, feature selection, and graphical modeling, yet it is untestable in many settings without additional assumptions. Existing CI tests often rely on restrictive structural conditions, limiting their validity on real-world data. Kernel methods using the partial covariance operator offer a more principled approach but suffer from limited adaptivity, slow convergence, and poor scalability. In this work, we explore whether representation learning can help address these limitations. Specifically, we focus on representations derived from the singular value decomposition of the partial covariance operator and use them to construct a simple test statistic, reminiscent of the Hilbert-Schmidt Independence Criterion (HSIC). We also introduce a practical bi-level contrastive algorithm to learn these representations. Our theory links representation learning error to test performance and establishes asymptotic validity and power guarantees. Preliminary experiments suggest that this approach offers a practical and statistically grounded path toward scalable CI testing, bridging kernel-based theory with modern representation learning.

We address the problem of causal effect estimation in the presence of hidden confounders using nonparametric instrumental variable (IV) regression. An established approach is to use estimators based on learned spectral features, that is, features spanning the top singular subspaces of the operator linking treatments to instruments. While powerful, such features are agnostic to the outcome variable. Consequently, the method can fail when the true causal function is poorly represented by these dominant singular functions. To mitigate, we introduce Augmented Spectral Feature Learning, a framework that makes the feature learning process outcome-aware. Our method learns features by minimizing a novel contrastive loss derived from an augmented operator that incorporates information from the outcome. By learning these task-specific features, our approach remains effective even under spectral misalignment. We provide a theoretical analysis of this framework and validate our approach on challenging benchmarks.

Sparse GNN aggregations (CSR SpMM/SDDMM) vary widely in performance with degree skew, feature width, and GPU micro-architecture. We present AutoSAGE, an input-aware CUDA scheduler that chooses tiling and mapping per input using a lightweight estimate refined by on-device micro-probes, with a guardrail that safely falls back to vendor kernels and a persistent cache for deterministic replay. AutoSAGE covers SpMM and SDDMM and composes into a CSR attention pipeline (SDDMM -> row-softmax -> SpMM). On Reddit and OGBN-Products, it matches vendor baselines at bandwidth-bound feature widths and finds gains at small widths; on synthetic sparsity and skew stress tests it achieves up to 4.7x kernel-level speedups. We release CUDA sources, Python bindings, a reproducible harness, and replayable cache logs.