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This paper introduces fast R updating algorithms designed for statistical applications, including regression, filtering, and model selection, where data structures change frequently. Although traditional QR decomposition is essential for matrix operations, it becomes computationally intensive when dynamically updating the design matrix in statistical models. The proposed algorithms efficiently update the R matrix without recalculating Q, significantly reducing computational costs. These algorithms provide a scalable solution for high-dimensional regression models, enhancing the feasibility of large-scale statistical analyses and model selection in data-intensive fields. Comprehensive simulation studies and real-world data applications reveal that the methods significantly reduce computational time while preserving accuracy. An extensive discussion highlights the versatility of fast R updating algorithms, illustrating their benefits across a wide range of models and applications in statistics and machine learning.

Accelerated failure time (AFT) models are frequently used to model survival data, providing a direct quantification of the relationship between event times and covariates. These models allow for the acceleration or deceleration of failure times through a multiplicative factor that accounts for the effect of covariates. While existing literature provides numerous methods for fitting AFT models with time-fixed covariates, adapting these approaches to scenarios involving both time-varying covariates and partly interval-censored data remains challenging. Motivated by a randomised clinical trial dataset on advanced melanoma patients, we propose a maximum penalised likelihood approach for fitting a semiparametric AFT model to survival data with partly interval-censored failure times. This method also accommodates both time-fixed and time-varying covariates. We utilise Gaussian basis functions to construct a smooth approximation of the non-parametric baseline hazard and fit the model using a constrained optimisation approach. The effectiveness of our method is demonstrated through extensive simulations. Finally, we illustrate the relevance of our approach by applying it to a dataset from a randomised clinical trial involving patients with advanced melanoma.

Background. The rapid growth of Language Models (LMs), particularly in code generation, requires substantial computational resources, raising concerns about energy consumption and environmental impact. Optimizing LMs inference for energy efficiency is crucial, and Small Language Models (SLMs) offer a promising solution to reduce resource demands. Aim. Our goal is to analyze the impact of deep learning runtime engines and execution providers on energy consumption, execution time, and computing-resource utilization from the point of view of software engineers conducting inference in the context of code SLMs. Method. We conducted a technology-oriented, multi-stage experimental pipeline using twelve code generation SLMs to investigate energy consumption, execution time, and computing-resource utilization across the configurations. Results. Significant differences emerged across configurations. CUDA execution provider configurations outperformed CPU execution provider configurations in both energy consumption and execution time. Among the configurations, TORCH paired with CUDA demonstrated the greatest energy efficiency, achieving energy savings from 37.99% up to 89.16% compared to other serving configurations. Similarly, optimized runtime engines like ONNX with the CPU execution provider achieved from 8.98% up to 72.04% energy savings within CPU-based configurations. Also, TORCH paired with CUDA exhibited efficient computing-resource utilization. Conclusions. Serving configuration choice significantly impacts energy efficiency. While further research is needed, we recommend the above configurations best suited to software engineers' requirements for enhancing serving efficiency in energy and performance.

Large language models (LLMs) demonstrate an impressive ability to utilise information within the context of their input sequences to appropriately respond to data unseen by the LLM during its training procedure. This ability is known as in-context learning (ICL). Humans and non-human animals demonstrate similar abilities, however their neural architectures differ substantially from LLMs. Despite this, a critical component within LLMs, the attention mechanism, resembles modern associative memory models, widely used in and influenced by the computational neuroscience community to model biological memory systems. Using this connection, we introduce an associative memory model capable of performing ICL. We use this as inspiration for a novel residual stream architecture which allows information to directly flow between attention heads. We test this architecture during training within a two-layer Transformer and show its ICL abilities manifest more quickly than without this modification. We then apply our architecture in small language models with 8 million parameters, focusing on attention head values, with results also indicating improved ICL performance at this larger and more naturalistic scale.

This paper presents an innovative method for predicting shape errors in 5-axis machining using graph neural networks. The graph structure is defined with nodes representing workpiece surface points and edges denoting the neighboring relationships. The dataset encompasses data from a material removal simulation, process data, and post-machining quality information. Experimental results show that the presented approach can generalize the shape error prediction for the investigated workpiece geometry. Moreover, by modelling spatial and temporal connections within the workpiece, the approach handles a low number of labels compared to non-graphical methods such as Support Vector Machines.

We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the "vanilla" PG method, achieving the best-known iteration complexity for finding an approximate stationary point of the problem. We then develop an "auto-conditioned" projected gradient (AC-PG) variant that achieves the same iteration complexity without requiring the input of the Lipschitz constant of the gradient or any line search procedure. The key idea is to estimate the Lipschitz constant using first-order information gathered from the previous iterations, and to show that the error caused by underestimating the Lipschitz constant can be properly controlled. We then generalize the PG methods to the stochastic setting, by proposing a stochastic projected gradient (SPG) method and a variance-reduced stochastic gradient (VR-SPG) method, achieving new complexity bounds in different oracle settings. We also present auto-conditioned stepsize policies for both stochastic PG methods and establish comparable convergence guarantees.

One of the questions in Rigidity Theory is whether a realization of the vertices of a graph in the plane is flexible, namely, if it allows a continuous deformation preserving the edge lengths. A flexible realization of a connected graph in the plane exists if and only if the graph has a so called NAC-coloring, which is surjective edge coloring by two colors such that for each cycle either all the edges have the same color or there are at least two edges of each color. The question whether a graph has a NAC-coloring, and hence also the existence of a flexible realization, has been proven to be NP-complete. We show that this question is also NP-complete on graphs with maximum degree five and on graphs with the average degree at most $4+\varepsilon$ for every fixed $\varepsilon >0$. The existence of a NAC-coloring is fixed parameter tractable when parametrized by treewidth. Since the only existing implementation of checking the existence of a NAC-coloring is rather naive, we propose new algorithms along with their implementation, which is significantly faster. We also focus on searching all NAC-colorings of a graph, since they provide useful information about its possible flexible realizations.

We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer solutions, and in particular such sets are generally not computably enumerable. And so this gives the first extension of the second incompleteness theorem to non classically computable formal systems. Let's motivate this with a somewhat physical application. Let $\mathcal{H} $ be the suitable infinite time limit (stabilization in the sense of the paper) of the mathematical output of humanity, specializing to first order sentences in the language of arithmetic (for simplicity), and understood as a formal system. Suppose that all the relevant physical processes in the formation of $\mathcal{H} $ are Turing computable. Then as defined $\mathcal{H} $ may \emph{not} be computably enumerable, but it is stably computably enumerable. Thus, the classical G\"odel disjunction applied to $\mathcal{H} $ is meaningless, but applying our incompleteness theorems to $\mathcal{H} $ we then get a sharper version of G\"odel's disjunction: assume $\mathcal{H} \vdash PA$ then either $\mathcal{H} $ is not stably computably enumerable or $\mathcal{H} $ is not 1-consistent (in particular is not sound) or $\mathcal{H} $ cannot prove a certain true statement of arithmetic (and cannot disprove it if in addition $\mathcal{H} $ is 2-consistent).

The stability number of a graph, defined as the cardinality of the largest set of pairwise non-adjacent vertices, is NP-hard to compute. The exact subgraph hierarchy (ESH) provides a sequence of increasingly tighter upper bounds on the stability number, starting with the Lov\'asz theta function at the first level and including all exact subgraph constraints of subgraphs of order $k$ into the semidefinite program to compute the Lov\'asz theta function at level $k$. In this paper, we investigate the ESH for Paley graphs, a class of strongly regular, vertex-transitive graphs. We show that for Paley graphs, the bounds obtained from the ESH remain the Lov\'asz theta function up to a certain threshold level, i.e., the bounds of the ESH do not improve up to a certain level. To overcome this limitation, we introduce the local ESH for the stable set problem for vertex-transitive graphs such as Paley graphs. We prove that this new hierarchy provides upper bounds on the stability number of vertex-transitive graphs that are at least as tight as those obtained from the ESH. Additionally, our computational experiments reveal that the local ESH produces superior bounds compared to the ESH for Paley graphs.