Data assimilation (DA) integrates observational information with model predictions to improve state estimation in complex systems. While filtering provides the basis for online forecasts by using only past and present observations, it can exhibit delays and biases when the underlying dynamics evolve rapidly or undergo regime transitions. Smoothing, which additionally incorporates future observations, provides a natural pipeline for hindcasting and reanalysis that yields an uncertainty reduction beyond the filter. This paper introduces an ensemble Kalman-Bucy smoother (EnKBS) for continuous-time DA of nonlinear dynamical systems, where the smoother's conditional distributions are reconstructed using ensemble moments. The result is a derivative-free framework that does not require explicit computation of tangent-linear or adjoint models, which converges to the exact smoother solution at the infinite-ensemble limit for a wide class of complex systems. Incorporating standard regularization techniques for high-dimensional systems, such as covariance localization and inflation, the skill of the EnKBS is demonstrated in various important scientific problems. By integrating future observations, which reveal the underlying causal mechanisms for retrospective state updates, the EnKBS is used for Bayesian-based inference of causal relationships and their temporal influence range in a dyadic trigger-feedback model and the development of a causality-driven iterative learning algorithm that identifies the structure and recovers the hidden parameters of a nonlinear reduced-order model mimicking midlatitude atmospheric circulation. Notably, both tasks remain effective with an ensemble size of $O(10)$ under partial observations, suggesting that EnKBS can support the instantaneous discovery of high-dimensional complex systems over time.

We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only $O(n^d)$ and respectively $\tilde O(1/\varepsilon^d)$ upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and $k$-Sum problems. Our reductions yield matching lower bounds of $\tildeΩ(n^d)$ and respectively $\tildeΩ(1/\varepsilon^d)$ based on Affine Degeneracy testing, and $\tildeΩ(n^{d/2})$ and respectively $\tildeΩ(1/\varepsilon^{d/2})$ conditioned on $k$-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Byconstruction0 < h h < 1 implies q h(x) q h(x). Wealsoacknowledgesupportofthe European Research Council (grant SEQUOIA 724063).

We consider the online setting, where the input arrives over time, and irrevocable decisions must be made without knowledge of the future. For all these problems, any online algorithm must incur a cost that is approximately log|I| times the optimal cost in the worst-case, where |I| is the length of theinput.

This thesis develops methods for causal inference and causal representation learning (CRL) in high-dimensional, time-varying data. The first contribution introduces the Causal Dynamic Variational Autoencoder (CDVAE), a model for estimating Individual Treatment Effects (ITEs) by capturing unobserved heterogeneity in treatment response driven by latent risk factors that affect only outcomes. CDVAE comes with theoretical guarantees on valid latent adjustment and generalization bounds for ITE error. Experiments on synthetic and real datasets show that CDVAE outperforms baselines, and that state-of-the-art models greatly improve when augmented with its latent substitutes, approaching oracle performance without access to true adjustment variables. The second contribution proposes an efficient framework for long-term counterfactual regression based on RNNs enhanced with Contrastive Predictive Coding (CPC) and InfoMax. It captures long-range dependencies under time-varying confounding while avoiding the computational cost of transformers, achieving state-of-the-art results and introducing CPC into causal inference. The third contribution advances CRL by addressing how latent causes manifest in observed variables. We introduce a model-agnostic interpretability layer based on the geometry of the decoder Jacobian. A sparse self-expression prior induces modular, possibly overlapping groups of observed features aligned with shared latent influences. We provide recovery guarantees in both disjoint and overlapping settings and show that meaningful latent-to-observed structure can be recovered without anchor features or single-parent assumptions. Scalable Jacobian-based regularization techniques are also developed.