Analysis of learned representations has a blind spot: it focuses on $similarity$, measuring how closely embeddings align with external references, but similarity reveals only what is represented, not whether that structure is robust. We introduce $geometric$ $stability$, a distinct dimension that quantifies how reliably representational geometry holds under perturbation, and present $Shesha$, a framework for measuring it. Across 2,463 configurations in seven domains, we show that stability and similarity are empirically uncorrelated ($ρ\approx 0.01$) and mechanistically distinct: similarity metrics collapse after removing the top principal components, while stability retains sensitivity to fine-grained manifold structure. This distinction yields actionable insights: for safety monitoring, stability acts as a functional geometric canary, detecting structural drift nearly 2$\times$ more sensitively than CKA while filtering out the non-functional noise that triggers false alarms in rigid distance metrics; for controllability, supervised stability predicts linear steerability ($ρ= 0.89$-$0.96$); for model selection, stability dissociates from transferability, revealing a geometric tax that transfer optimization incurs. Beyond machine learning, stability predicts CRISPR perturbation coherence and neural-behavioral coupling. By quantifying $how$ $reliably$ systems maintain structure, geometric stability provides a necessary complement to similarity for auditing representations across biological and computational systems.

Understanding the dynamic reorganization of brain networks is critical for predicting cognitive decline, neurological progression, and individual variability in clinical outcomes. This work proposes a multimodal graph neural network framework that integrates structural MRI, diffusion tensor imaging, and functional MRI to model spatiotemporal brain network reorganization. Brain regions are represented as nodes and structural and functional connectivity as edges, forming longitudinal brain graphs for each subject. Temporal evolution is captured via fractional stochastic differential operators embedded within graph-based recurrent networks, enabling the modeling of long-term dependencies and stochastic fluctuations in network dynamics. Attention mechanisms fuse multimodal information and generate interpretable biomarkers, including network energy entropy, graph curvature, fractional memory indices, and modality-specific attention scores. These biomarkers are combined into a composite prognostic index to quantify individual risk of network instability or cognitive decline. Experiments on longitudinal neuroimaging datasets demonstrate both predictive accuracy and interpretability. The results highlight the potential of mathematically rigorous, multimodal graph-based approaches for deriving clinically meaningful biomarkers from existing imaging data without requiring new data collection.

Phase transitions have been proposed as the origin of emergent abilities in large language models (LLMs), where new capabilities appear abruptly once models surpass critical thresholds of scale. Prior work, such as that of Wei et al., demonstrated these phenomena under model and data scaling, with transitions revealed after applying a log scale to training compute. In this work, we ask three complementary questions: (1) Are phase transitions unique to large models, or can they also be observed in small transformer-based language models? (2) Can such transitions be detected directly in linear training space, rather than only after log rescaling? and (3) Can these transitions emerge at early stages of training? To investigate, we train a small GPT-style transformer on a character-level corpus and analyze the evolution of vocabulary usage throughout training. We track the average word length, the number of correct versus incorrect words, and shifts in vocabulary diversity. Building on these measures, we apply Poisson and sub-Poisson statistics to quantify how words connect and reorganize. This combined analysis reveals a distinct transition point during training. Notably, these transitions are not apparent in standard loss or validation curves, but become visible through our vocabulary- and statistics-based probes. Our findings suggest that phase-transition reorganizations are a general feature of language model training, observable even in modest models, detectable directly in linear training space, and occurring surprisingly early as coherence emerges. This perspective provides new insight into the nonlinear dynamics of language model training and underscores the importance of tailored metrics for uncovering phase transition behaviors

There are thus no fair comparisons that can be made with such methods.

We list the statement, preprocessing time, number of iterations and cost per iteration for our algorithm and original Frank-Wolfe algorithm to converge.

We list the statement, preprocessing time, number of iterations and cost per iteration for our algorithm and original Frank-Wolfe algorithm to converge.

There are thus no fair comparisons that can be made with such methods.

How can we detect when global events fundamentally reshape public discourse? This study introduces a topological framework for identifying structural change in media narratives using persistent homology. Drawing on international news articles surrounding major events - including the Russian invasion of Ukraine (Feb 2022), the murder of George Floyd (May 2020), the U.S. Capitol insurrection (Jan 2021), and the Hamas-led invasion of Israel (Oct 2023) - we construct daily co-occurrence graphs of noun phrases to trace evolving discourse. Each graph is embedded and transformed into a persistence diagram via a Vietoris-Rips filtration. We then compute Wasserstein distances and persistence entropies across homological dimensions to capture semantic disruption and narrative volatility over time. Our results show that major geopolitical and social events align with sharp spikes in both H0 (connected components) and H1 (loops), indicating sudden reorganization in narrative structure and coherence. Cross-correlation analyses reveal a typical lag pattern in which changes to component-level structure (H0) precede higher-order motif shifts (H1), suggesting a bottom-up cascade of semantic change. An exception occurs during the Russian invasion of Ukraine, where H1 entropy leads H0, possibly reflecting top-down narrative framing before local discourse adjusts. Persistence entropy further distinguishes tightly focused from diffuse narrative regimes. These findings demonstrate that persistent homology offers a mathematically principled, unsupervised method for detecting inflection points and directional shifts in public attention - without requiring prior knowledge of specific events. This topological approach advances computational social science by enabling real-time detection of semantic restructuring during crises, protests, and information shocks.

OpenAI has filed a countersuit against Elon Musk, accusing him of staging press attacks and malicious campaigns on "the social media platform he controls," as well as of making "harassing legal claims" and a "sham bid for OpenAI's assets." In its filing, courtesy of TechCrunch, the ChatGPT-maker said Musk could not tolerate seeing such "success for an enterprise he had abandoned and declared doomed" and had made it his own project to take down the organization. It also said that Musk's efforts have ramped up in recent months after it announced its plans to restructure and become a for-profit entity with a non-profit division. Last year, Musk sued OpenAI, accusing it of ditching its nonprofit mission, becoming a "closed-source de facto subsidiary" Microsoft and of violating its foundational agreement to develop generative AI "for the benefit of humanity." But Musk, OpenAI said in its new lawsuit, is only pretending to represent the public and in truth is seeking to stop it from restructuring.

Recent advances in measuring hardness-wise properties of data guide language models in sample selection within low-resource scenarios. However, class-specific properties are overlooked for task setup and learning. How will these properties influence model learning and is it generalizable across datasets? To answer this question, this work formally initiates the concept of $\textit{class-wise hardness}$. Experiments across eight natural language understanding (NLU) datasets demonstrate a consistent hardness distribution across learning paradigms, models, and human judgment. Subsequent experiments unveil a notable challenge in measuring such class-wise hardness with instance-level metrics in previous works. To address this, we propose $\textit{GeoHard}$ for class-wise hardness measurement by modeling class geometry in the semantic embedding space. $\textit{GeoHard}$ surpasses instance-level metrics by over 59 percent on $\textit{Pearson}$'s correlation on measuring class-wise hardness. Our analysis theoretically and empirically underscores the generality of $\textit{GeoHard}$ as a fresh perspective on data diagnosis. Additionally, we showcase how understanding class-wise hardness can practically aid in improving task learning.