The interface between stochastic analysis and machine learning is a rapidly evolving field, with path signatures - iterated integrals that provide faithful, hierarchical representations of paths - offering a principled and universal feature map for sequential and structured data. Rooted in rough path theory, path signatures are invariant to reparameterization and well-suited for modelling evolving dynamics, long-range dependencies, and irregular sampling - common challenges in real-world time series and graph data. This thesis investigates how to harness the expressive power of path signatures within scalable machine learning pipelines. It introduces a suite of models that combine theoretical robustness with computational efficiency, bridging rough path theory with probabilistic modelling, deep learning, and kernel methods. Key contributions include: Gaussian processes with signature kernel-based covariance functions for uncertainty-aware time series modelling; the Seq2Tens framework, which employs low-rank tensor structure in the weight space for scalable deep modelling of long-range dependencies; and graph-based models where expected signatures over graphs induce hypo-elliptic diffusion processes, offering expressive yet tractable alternatives to standard graph neural networks. Further developments include Random Fourier Signature Features, a scalable kernel approximation with theoretical guarantees, and Recurrent Sparse Spectrum Signature Gaussian Processes, which combine Gaussian processes, signature kernels, and random features with a principled forgetting mechanism for multi-horizon time series forecasting with adaptive context length. We hope this thesis serves as both a methodological toolkit and a conceptual bridge, and provides a useful reference for the current state of the art in scalable, signature-based learning for sequential and structured data.

We develop a principled framework for discovering causal structure in partial differential equations (PDEs) using physics-informed neural networks and counterfactual perturbations. Unlike classical residual minimization or sparse regression methods, our approach quantifies operator-level necessity through functional interventions on the governing dynamics. We introduce causal sensitivity indices and structural deviation metrics to assess the influence of candidate differential operators within neural surrogates. Theoretically, we prove exact recovery of the causal operator support under restricted isometry or mutual coherence conditions, with residual bounds guaranteeing identifiability. Empirically, we validate the framework on both synthetic and real-world datasets across climate dynamics, tumor diffusion, and ocean flows. Our method consistently recovers governing operators even under noise, redundancy, and data scarcity, outperforming standard PINNs and DeepONets in structural fidelity. This work positions causal PDE discovery as a tractable and interpretable inference task grounded in structural causal models and variational residual analysis.

Meta's chief technology officer has called it "the great honor of my life" to be enlisted in a new US army corps that defence chiefs set up to better integrate military and tech industry expertise, including senior figures from top tech firms that also include Palantir and OpenAI. Andrew Bosworth, a long-term lieutenant to Mark Zuckerberg known widely as "Boz", is one of several senior Silicon Valley executives commissioned to the rank of lieutenant colonel in the corps, called Detachment 201, which the US army says will "fuse cutting-edge tech expertise with military innovation". Bosworth, who joined Facebook in 2006, was sworn into the army reserves earlier this month alongside Shyam Sankar, the chief technology officer of Palantir, a technology firm with extensive defence contracts, Kevin Weil, chief product officer of OpenAI, and Bob McGrew, an adviser at Thinking Machines Lab, a 10bn AI company. They wore military fatigues at the swearing-in ceremony but will not be full-time soldiers. The recruitment is a sign of the increasing importance of technology in modern warfare and growing commercial and research links between some of the largest tech firms and the military.

A fragile ceasefire remains in place between Israel and Iran, one day after US President Donald Trump announced a truce, ending 12 days of fighting that erupted following Israeli strikes on Tehran's nuclear and military sites. An analysis of data from the Armed Conflict Location and Event Data Project (ACLED) shows that between October 7, 2023, and just before Israel attacked Iran on June 13, 2025, Israel carried out nearly 35,000 recorded attacks across five countries: the occupied Palestinian territory, Lebanon, Syria, Yemen, and Iran. These attacks include air and drone strikes, shelling and missile attacks, remote explosives, and property destruction. The majority of attacks have been on Palestinian territory with at least 18,235 recorded incidents, followed by Lebanon (15,520), Syria (616), Iran (58) and Yemen (39). While the bulk of Israel's attacks have concentrated on nearby Gaza, the occupied West Bank, and Lebanon, its military operations have also reached far beyond its immediate borders.

Automatic Speech Recognition (ASR) has reached impressive accuracy for high-resource languages, yet its utility in linguistic fieldwork remains limited. Recordings collected in fieldwork contexts present unique challenges, including spontaneous speech, environmental noise, and severely constrained datasets from under-documented languages. In this paper, we benchmark the performance of two fine-tuned multilingual ASR models, MMS and XLS-R, on five typologically diverse low-resource languages with control of training data duration. Our findings show that MMS is best suited when extremely small amounts of training data are available, whereas XLS-R shows parity performance once training data exceed one hour. We provide linguistically grounded analysis for further provide insights towards practical guidelines for field linguists, highlighting reproducible ASR adaptation approaches to mitigate the transcription bottleneck in language documentation.

Cloud cover in multispectral imagery (MSI) poses significant challenges for early season crop mapping, as it leads to missing or corrupted spectral information. Synthetic aperture radar (SAR) data, which is not affected by cloud interference, offers a complementary solution, but lack sufficient spectral detail for precise crop mapping. To address this, we propose a novel framework, Time-series MSI Image Reconstruction using Vision Transformer (ViT), to reconstruct MSI data in cloud-covered regions by leveraging the temporal coherence of MSI and the complementary information from SAR from the attention mechanism. Comprehensive experiments, using rigorous reconstruction evaluation metrics, demonstrate that Time-series ViT framework significantly outperforms baselines that use non-time-series MSI and SAR or time-series MSI without SAR, effectively enhancing MSI image reconstruction in cloud-covered regions.

In [104] a powerful fully lifted (fl) probabilistic blirp interpolating mechanism was introduced. It arrived as a strong upgrade on partially lifted concepts from [100, 101] and the basic ones from [49, 84] (see also, e.g., [31, 32, 60, 106] for early considerations as well as [5, 64, 67, 101, 107] for a brief history, relevance, and development overview). While the range of applicability in a variety of scientific fields is rather wide, applications in random optimizations are of our prevalent interest. They became particularly fruitful over the last two decades (some of the most prominent examples include, compressed sensing, machine learning, and neural network statistical studies; see, e.g., [50, 72-75, 86-91, 108]). Characterizing typical behavior of their various features ranging from standard optimization metrics (objective values, optimal solutions, relations between optimizing variables) to associated algorithmic ones (accuracy, speed, convergence) became possible in large part due to a strong progress made in understanding and developing powerful comparison mechanisms. For example, many of the above performance metrics often exhibit the so-calledphase-transition (PT) phenomenon where they undergo an abrupt change as one moves from one region of system parameters to another.

We consider \emph{bilinearly indexed random processes} (blirp) and study their interpolating comparative mechanisms. Generic introduction of the \emph{fully lifted} (fl) blirp interpolation in [105] was followed by a corresponding stationarization counterpart in [103]. A \emph{large deviation} upgrade of [105] introduced in companion paper [106] is complemented here with the corresponding one of [103]. Similarly to [106], the mechanism that we introduce extends the range of [103]'s applicability so that it encompasses random structures \emph{atypical} features. Among others these include the \emph{local entropies} (LE) which explain atypical solutions clusterings in hard random optimization problems believed to be directly responsible for the presumable existence of the so-called \emph{computational gaps}. Moreover (and similar to [105]), despite on occasion somewhat involved technical considerations, the final forms of the uncovered fundamental interpolating parameters relations are rather elegant and as such provide a valuable tool readily available for further use.

We study classical asymmetric binary perceptron (ABP) and associated \emph{local entropy} (LE) as potential source of its algorithmic hardness. Isolation of \emph{typical} ABP solutions in SAT phase seemingly suggests a universal algorithmic hardness. Paradoxically, efficient algorithms do exist even for constraint densities $α$ fairly close but at a finite distance (\emph{computational gap}) from the capacity. In recent years, existence of rare large dense clusters and magical ability of fast algorithms to find them have been posited as the conceptual resolution of this paradox. Monotonicity or breakdown of the LEs associated with such \emph{atypical} clusters are predicated to play a key role in their thinning-out or even complete defragmentation. Invention of fully lifted random duality theory (fl RDT) [90,93,94] allows studying random structures \emph{typical} features. A large deviation upgrade, sfl LD RDT [96,97], moves things further and enables \emph{atypical} features characterizations as well. Utilizing the machinery of [96,97] we here develop a generic framework to study LE as an ABP's atypical feature. Already on the second level of lifting we discover that the LE results are closely matching those obtained through replica methods. For classical zero threshold ABP, we obtain that LE breaks down for $α$ in $(0.77,0.78)$ interval which basically matches $α\sim 0.75-0.77$ range that currently best ABP solvers can handle and effectively indicates that LE's behavior might indeed be among key reflections of the ABP's computational gaps presumable existence.

We present an application of a foundation model for small- to medium-sized tabular data (TabPFN), to sub-national yield forecasting task in South Africa. TabPFN has recently demonstrated superior performance compared to traditional machine learning (ML) models in various regression and classification tasks. We used the dekadal (10-days) time series of Earth Observation (EO; FAPAR and soil moisture) and gridded weather data (air temperature, precipitation and radiation) to forecast the yield of summer crops at the sub-national level. The crop yield data was available for 23 years and for up to 8 provinces. Covariate variables for TabPFN (i.e., EO and weather) were extracted by region and aggregated at a monthly scale. We benchmarked the results of the TabPFN against six ML models and three baseline models. Leave-one-year-out cross-validation experiment setting was used in order to ensure the assessment of the models capacity to forecast an unseen year. Results showed that TabPFN and ML models exhibit comparable accuracy, outperforming the baselines. Nonetheless, TabPFN demonstrated superior practical utility due to its significantly faster tuning time and reduced requirement for feature engineering. This renders TabPFN a more viable option for real-world operation yield forecasting applications, where efficiency and ease of implementation are paramount.