Automatic program repair seeks to generate correct code from buggy programs, with most approaches searching the correct program in a discrete, symbolic space of source code tokens. This symbolic search is fundamentally limited by its inability to directly reason about program behavior. We introduce Gradient-Based Program Repair (GBPR), a new paradigm that reframes program repair as continuous optimization in a differentiable numerical program space. Our core insight is to compile symbolic programs into differentiable numerical representations, enabling search in the numerical program space directly guided by program behavior. To evaluate GBPR, we present RaspBugs, a new benchmark of 1,466 buggy symbolic RASP programs and their respective numerical representations. Our experiments demonstrate that GBPR can effectively repair buggy symbolic programs by gradient-based optimization in the numerical program space, with convincing repair trajectories. To our knowledge, we are the first to state program repair as continuous optimization in a numerical program space. Our work establishes a new direction for program repair research, bridging two rich worlds: continuous optimization and program behavior.

Recent research has extensively studied how large language models manipulate integers in specific arithmetic tasks, and on a more fundamental level, how they represent numeric values. These previous works have found that language model embeddings can be used to reconstruct the original values, however, they do not evaluate whether language models actually model continuous values as continuous. Using expected properties of the embedding space, including linear reconstruction and principal component analysis, we show that language models not only represent numeric spaces as non-continuous but also introduce significant noise. Using models from three major providers (OpenAI, Google Gemini and Voyage AI), we show that while reconstruction is possible with high fidelity ($R^2 \geq 0.95$), principal components only explain a minor share of variation within the embedding space. This indicates that many components within the embedding space are orthogonal to the simple numeric input space. Further, both linear reconstruction and explained variance suffer with increasing decimal precision, despite the ordinal nature of the input space being fundamentally unchanged. The findings of this work therefore have implications for the many areas where embedding models are used, in-particular where high numerical precision, large magnitudes or mixed-sign values are common.

This paper leverages microscaling floating-point formats, a novel technique designed to address these challenges by reducing the storage and computational overhead associated with numerical representations in LLMs. Unlike traditional floating-point representations that allocate a dedicated scale for each value, microscaling employs a shared scale across a block of values, enabling compact one-byte floating-point representations while maintaining an extended dynamic range. We explore the application of microscaling in the context of 8-bit floating-point formats to significantly reduce memory footprint and computational costs. We tested several configurations of microscaling floats within the GPT -2 LLM architecture, demonstrating that microscaling data formats can achieve competitive accuracy during training and inference, proving its efficacy as a resource-efficient alternative for deploying LLMs at scale. The source code is publicly available at: https://github.com/

With the rapid proliferation of streaming services, network load exhibits highly time-varying and bursty behavior, posing serious challenges for maintaining Quality of Service (QoS) in Crowdsourced Cloud-Edge Platforms (CCPs). While CCPs leverage Predict-then-Schedule architecture to improve QoS and profitability, accurate load forecasting remains challenging under traffic surges. Existing methods either minimize mean absolute error, resulting in underprovisioning and potential Service Level Agreement (SLA) violations during peak periods, or adopt conservative overprovision-ing strategies, which mitigate SLA risks at the expense of increased resource expenditure. To address this dilemma, we propose HRS, a H ybrid R epresentation framework with S cheduling awareness that integrates numerical and image-based representations to better capture extreme load dynamics. We further introduce a Scheduling-A ware Loss (SAL) that captures the asymmetric impact of prediction errors, guiding predictions that better support scheduling decisions. Extensive experiments on four real-world datasets demonstrate that HRS consistently outperforms ten baselines and achieves state-of-the-art performance, reducing SLA violation rates by 63.1% and total profit loss by 32.3%. Our code is available at [28].

This study presents a fascinating linguistic property related to the number of letters in words and their corresponding numerical values. By selecting any arbitrary word, counting its constituent letters, and subsequently spelling out the resulting count and tallying the letters anew, an unanticipated pattern is observed. Remarkably, this iterative sequence, conducted on a dataset of 100,000 random words, invariably converges to the numeral four (4), termed the "Linguistic Loop (LL) constant". Examining 73 languages utilizing the Latin alphabet, this research reveals distinctive patterns. Among them, 28 languages exhibit LL-positive behavior adhering to the established property, while 31 languages deviate as LL-negative. Additionally, 13 languages display nuanced tendencies: eight feature two LL constants (bi-positivity), and five feature three constants (tri-positivity). This discovery highlights a linguistic quirk within Latin alphabet-based language number-word representations, uncovering an intriguing facet across diverse alphabetic systems. It also raises questions about the underlying linguistic and cognitive mechanisms responsible for this phenomenon.

Humans are believed to perceive numbers on a logarithmic mental number line, where smaller values are represented with greater resolution than larger ones. This cognitive bias, supported by neuroscience and behavioral studies, suggests that numerical magnitudes are processed in a sublinear fashion rather than on a uniform linear scale. Inspired by this hypothesis, we investigate whether large language models (LLMs) exhibit a similar logarithmic-like structure in their internal numerical representations. By analyzing how numerical values are encoded across different layers of LLMs, we apply dimensionality reduction techniques such as PCA and PLS followed by geometric regression to uncover latent structures in the learned embeddings. Our findings reveal that the model's numerical representations exhibit sublinear spacing, with distances between values aligning with a logarithmic scale. This suggests that LLMs, much like humans, may encode numbers in a compressed, non-uniform manner.

With the advancement of large language models (LLMs), an increasing number of student models have leveraged LLMs to analyze textual artifacts generated by students to understand and evaluate their learning. These student models typically employ pre-trained LLMs to vectorize text inputs into embeddings and then use the embeddings to train models to detect the presence or absence of a construct of interest. However, how reliable and robust are these models at processing language with different levels of complexity? In the context of learning where students may have different language backgrounds with various levels of writing skills, it is critical to examine the robustness of such models to ensure that these models work equally well for text with varying levels of language complexity. Coincidentally, a few (but limited) research studies show that the use of language can indeed impact the performance of LLMs. As such, in the current study, we examined the robustness of several LLM-based student models that detect student self-regulated learning (SRL) in math problem-solving. Specifically, we compared how the performance of these models vary using texts with high and low lexical, syntactic, and semantic complexity measured by three linguistic measures.

In this paper, we introduce the proper latent decomposition (PLD) as a generalization of the proper orthogonal decomposition (POD) on manifolds. PLD is a nonlinear reduced-order modeling technique for compressing high-dimensional data into nonlinear coordinates. First, we compute a reduced set of intrinsic coordinates (latent space) to accurately describe a flow with fewer degrees of freedom than the numerical discretization. The latent space, which is geometrically a manifold, is inferred by an autoencoder. Second, we leverage tools from differential geometry to develop numerical methods for operating directly on the latent space; namely, a metric-constrained Eikonal solver for distance computations. With this proposed numerical framework, we propose an algorithm to perform PLD on the manifold. Third, we demonstrate results for a laminar flow case and the turbulent Kolmogorov flow. For the laminar flow case, we are able to identify a semi-analytical expression for the solution of Navier-Stokes; in the Kolmogorov flow case, we are able to identify a dominant mode that exhibits physical structures, which are compared with POD. This work opens opportunities for analyzing autoencoders and latent spaces, nonlinear reduced-order modeling and scientific insights into the structure of high-dimensional data.

In recent years, there has been considerable interest in developing machine learning models on graphs in order to account for topological inductive biases. In particular, recent attention was given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types.