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This thesis is a corpus-based, quantitative, and typological analysis of the functions of Early Slavic participle constructions and their finite competitors ($jegda$-'when'-clauses). The first part leverages detailed linguistic annotation on Early Slavic corpora at the morphosyntactic, dependency, information-structural, and lexical levels to obtain indirect evidence for different potential functions of participle clauses and their main finite competitor and understand the roles of compositionality and default discourse reasoning as explanations for the distribution of participle constructions and $jegda$-clauses in the corpus. The second part uses massively parallel data to analyze typological variation in how languages express the semantic space of English $when$, whose scope encompasses that of Early Slavic participle constructions and $jegda$-clauses. Probabilistic semantic maps are generated and statistical methods (including Kriging, Gaussian Mixture Modelling, precision and recall analysis) are used to induce cross-linguistically salient dimensions from the parallel corpus and to study conceptual variation within the semantic space of the hypothetical concept WHEN.

This paper achieves noteworthy progress in the realm of abstract reasoning, particularly in addressing Raven's Progressive Matrices (RPM) and Bongard-Logo challenges. Initially, we introduce Lico-Net, a novel baseline model that resolves RPM problems with remarkable accuracy. Leveraging this foundation, we advance with the D3C approach, which advocates representing the underlying concepts in abstract reasoning problems through distributions. This perspective enhances the performance of both Lico-Net and a baseline model excelling in Bongard-Logo tasks. To bolster the computational efficiency of D3C, we present the D3C-cos variant, offering a streamlined yet precise solution. Furthermore, we propose the D2C method, redefining conceptual boundaries within these domains and bridging the divide between high-level abstractions and their lower-dimensional counterparts. Finally, we extend our methodology to D4C, employing adversarial techniques to refine conceptual boundaries further and demonstrate substantial improvements in both RPM and Bongard-Logo challenges. Overall, our contributions present a fresh outlook and practical advancements in the field of abstract reasoning.

We introduce a two-player game played on undirected graphs called Trail Trap, which is a variant of a game known as Partizan Edge Geography. One player starts by choosing any edge and moving a token from one endpoint to the other; the other player then chooses a different edge and does the same. Alternating turns, each player moves their token along an unused edge from its current vertex to an adjacent vertex, until one player cannot move and loses. We present an algorithm to determine which player has a winning strategy when the graph is a tree, and partially characterize the trees on which a given player wins. Additionally, we show that Trail Trap is NP-hard, even for connected bipartite planar graphs with maximum degree $4$ as well as for disconnected graphs. We determine which player has a winning strategy for certain subclasses of complete bipartite graphs and grid graphs, and we propose several open problems for further study.

In this work, we analyze the convergence rate of randomized quasi-Monte Carlo (RQMC) methods under Owen's boundary growth condition [Owen, 2006] via spectral analysis. Specifically, we examine the RQMC estimator variance for the two commonly studied sequences: the lattice rule and the Sobol' sequence, applying the Fourier transform and Walsh--Fourier transform, respectively, for this analysis. Assuming certain regularity conditions, our findings reveal that the asymptotic convergence rate of the RQMC estimator's variance closely aligns with the exponent specified in Owen's boundary growth condition for both sequence types. We also provide guidance on choosing the importance sampling density to minimize RQMC estimator variance.

We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's mean square error in a closed form. This is in contrast to conventional MC and MLMC covariance estimators, which are based on biased mean square errors defined solely by upper bounds, particularly within the MLMC. The numerical results of the algorithms are demonstrated by estimating the covariance of the stochastic response of a simple 1D stochastic elliptic PDE such as Poisson's model.

The Gromov--Hausdorff distance measures the difference in shape between metric spaces and poses a notoriously difficult problem in combinatorial optimization. We introduce its quadratic relaxation over a convex polytope whose solutions provably deliver the Gromov--Hausdorff distance. The optimality guarantee is enabled by the fact that the search space of our approach is not constrained to a generalization of bijections, unlike in other relaxations such as the Gromov--Wasserstein distance. We suggest conditional gradient descent for solving the relaxation in cubic time per iteration, and demonstrate its performance on metric spaces of hundreds of points. In particular, we use it to obtain a new bound of the Gromov--Hausdorff distance between the unit circle and the unit hemisphere equipped with Euclidean metric. Our approach is implemented as a Python package dGH.

The Robbins estimator is the most iconic and widely used procedure in the empirical Bayes literature for the Poisson model. On one hand, this method has been recently shown to be minimax optimal in terms of the regret (excess risk over the Bayesian oracle that knows the true prior) for various nonparametric classes of priors. On the other hand, it has been long recognized in practice that Robbins estimator lacks the desired smoothness and monotonicity of Bayes estimators and can be easily derailed by those data points that were rarely observed before. Based on the minimum-distance distance method, we propose a suite of empirical Bayes estimators, including the classical nonparametric maximum likelihood, that outperform the Robbins method in a variety of synthetic and real data sets and retain its optimality in terms of minimax regret.

Despite notable advancements in automatic speech recognition (ASR), performance tends to degrade when faced with adverse conditions. Generative error correction (GER) leverages the exceptional text comprehension capabilities of large language models (LLM), delivering impressive performance in ASR error correction, where N-best hypotheses provide valuable information for transcription prediction. However, GER encounters challenges such as fixed N-best hypotheses, insufficient utilization of acoustic information, and limited specificity to multi-accent scenarios. In this paper, we explore the application of GER in multi-accent scenarios. Accents represent deviations from standard pronunciation norms, and the multi-task learning framework for simultaneous ASR and accent recognition (AR) has effectively addressed the multi-accent scenarios, making it a prominent solution. In this work, we propose a unified ASR-AR GER model, named MMGER, leveraging multi-modal correction, and multi-granularity correction. Multi-task ASR-AR learning is employed to provide dynamic 1-best hypotheses and accent embeddings. Multi-modal correction accomplishes fine-grained frame-level correction by force-aligning the acoustic features of speech with the corresponding character-level 1-best hypothesis sequence. Multi-granularity correction supplements the global linguistic information by incorporating regular 1-best hypotheses atop fine-grained multi-modal correction to achieve coarse-grained utterance-level correction. MMGER effectively mitigates the limitations of GER and tailors LLM-based ASR error correction for the multi-accent scenarios. Experiments conducted on the multi-accent Mandarin KeSpeech dataset demonstrate the efficacy of MMGER, achieving a 26.72% relative improvement in AR accuracy and a 27.55% relative reduction in ASR character error rate, compared to a well-established standard baseline.

Formalizing creativity-related concepts has been a long-term goal of Computational Creativity. To the same end, we explore Formal Learning Theory in the context of creativity. We provide an introduction to the main concepts of this framework and a re-interpretation of terms commonly found in creativity discussions, proposing formal definitions for novelty and transformational creativity. This formalisation marks the beginning of a research branch we call Formal Creativity Theory, exploring how learning can be included as preparation for exploratory behaviour and how learning is a key part of transformational creative behaviour. By employing these definitions, we argue that, while novelty is neither necessary nor sufficient for transformational creativity in general, when using an inspiring set, rather than a sequence of experiences, an agent actually requires novelty for transformational creativity to occur.

We consider modal logic extended with the well-known temporal operator `eventually' and provide a cut-elimination procedure for a cyclic sequent calculus that captures this fragment. The work showcases an adaptation of the reductive cut-elimination method to cyclic calculi. Notably, the proposed algorithm applies to a cyclic proof and directly outputs a cyclic cut-free proof without appealing to intermediate machinery for regularising the end proof.