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Large language models (LLMs) have shown significant progress in reasoning tasks. However, recent studies show that transformers and LLMs fail catastrophically once reasoning problems exceed modest complexity. We revisit these findings…
Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term…
Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…
This paper discusses basic results and recent developments on variational regularization methods, as developed for inverse problems. In a typical setup we review basic properties needed to obtain a convergent regularization scheme and…
The Retrieval-Augmented Language Model (RALM) has shown remarkable performance on knowledge-intensive tasks by incorporating external knowledge during inference, which mitigates the factual hallucinations inherited in large language models…
Despite the recent advancement in Retrieval-Augmented Generation (RAG) systems, most retrieval methodologies are often developed for factual retrieval, which assumes query and positive documents are semantically similar. In this paper, we…
Tiny Recursive Models (TRMs) have recently demonstrated remarkable performance on ARC-AGI, showing that very small models can compete against large foundation models through a two-step refinement mechanism that updates an internal reasoning…
We develop a behavioural theory of reflective parallel algorithms (RAs), i.e. synchronous parallel algorithms that can modify their own behaviour. The theory comprises a set of postulates defining the class of RAs, an abstract machine…
Human recursive numeral systems (i.e., counting systems such as English base-10 numerals), like many other grammatical systems, are highly regular. Following prior work that relates cross-linguistic tendencies to biases in learning, we ask…
Large reasoning models (LRMs) often consume excessive tokens, inflating computational cost and latency. More broadly, in goal reaching sequential decision problems we often want to reach the goal quickly, and LRM reasoning can be viewed…
Improving the multi-step reasoning ability of Large Language Models (LLMs) is a critical yet challenging task. The dominant paradigm, outcome-supervised reinforcement learning (RLVR), rewards only correct final answers, often propagating…
Although large language models (LLMs) have been touted for their ability to generate natural-sounding text, there are growing concerns around possible negative effects of LLMs such as data memorization, bias, and inappropriate language.…
Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of inter-dependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier…
The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is…
We consider a Reinforcement Learning setup where an agent interacts with an environment in observation-reward-action cycles without any (esp.\ MDP) assumptions on the environment. State aggregation and more generally feature reinforcement…
We provide new theoretical results in the field of inverse regression methods for dimension reduction. Our approach is based on the study of some empirical processes that lie close to a certain dimension reduction subspace, called the…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
Reduced Rank Extrapolation (RRE) is a polynomial type method used to accelerate the convergence of sequences of vectors $\{\boldsymbol{x}_m\}$. It is applied successfully in different disciplines of science and engineering in the solution…
Do large datasets provide value to psychologists? Without a systematic methodology for working with such datasets, there is a valid concern that analyses will produce noise artifacts rather than true effects. In this paper, we offer a way…
Large reasoning models (e.g., R1, o3) have demonstrated remarkable mathematical problem-solving abilities. However, the high reported accuracy of these advanced models on popular datasets, reliance on purely numerical evaluation and…