Yongshun Liang | Searcharxiv

arXiv2026

The Assouad Dimension of Graphs: An Affirmative Answer to Fraser's Question

This article primarily investigates the Assouad dimension of graphs of continuous functions. Initially, the Assouad spectrum of the graph of a Hölder and count-Hölder continuous function has been derived. It has been shown that the Assouad dimension of the graph of a continuous function $f:[a,b]\to \mathbb{R}$ with the count-Hölder condition is at least $ 1 + \sup \left\{ \overline{\dim}_{\mathrm{B}}L_f(y) : y \in \mathscr{R}_f[a,b]\right\} $ where $\overline{\dim}_{\mathrm{B}}$ denotes the upper box dimension, $L_f(y)$ is the level set of $f$ at level $y$, and $\mathscr{R}_f[a,b]$ is the range of $f$ on $[a,b]$. As a consequence, non-trivial lower bounds are obtained for the Assouad dimension of the graphs of classical functions such as the Weierstrass and Takagi functions. We have also employed a unified projection solely in the direction of the $y$-axis to establish lower and upper bounds for the Assouad dimension. On this basis, we have shown that the lower bound is sharp in terms of level sets, namely the upper box dimension cannot be replaced by the Assouad dimension. This provides a partial positive answer to Question 17.11.1 posed by Fraser in~\cite{Fraser2021}, and to Question 2.2 posed by Yu in~\cite{Yu2020}. Building on this result, the analysis has been further extended to a broader class of Weierstrass-type functions and some self-affine functions like fractal interpolation functions. Moreover, we have discussed certain circumstances under which the Assouad dimension can reach 2 with several examples constructed. A supplementary explanation of why Brownian motion has the full Assouad dimension has also been provided based on the preceding work of Howroyd and Yu \cite{Howroyd2019}.

math.DS↗

arXiv2026

MiMo-V2.6: Scaling Reinforcement Learning Towards Self-Improvement

Reinforcement learning (RL) is the central training paradigm for advancing large foundation models towards self-improvement. This report introduces the MiMo-V2.6 series, an omni-modal family that pushes the frontier of model intelligence by scaling RL compute. Prior to RL, we conduct mid-training on a broad multimodal corpus to provide ample exploration space, and build a solid infrastructure on the pretrained hybrid-SWA architecture to support subsequent scale-up. We scale RL compute along three dimensions: (1) larger batches and higher throughput, with an asynchronous training that consumes 1,568 samples and 2.7-3.7B tokens per step at context lengths of up to 1M; (2) more diverse and complex environments, spanning code, general, visual, and cyber domains under a mixture of agent harnesses; and (3) more grader compute, via groupwise agentic grading that yields more accurate reward signals for long-horizon tasks and steers the model towards shorter, more token-efficient solutions. To keep training stable at scale, we freeze the MoE router and establish a multi-layer defense against reward hacking. We further build infrastructure for mixed-task agentic RL, including a unified trajectory representation, high-concurrency multi-framework rollout, decoupled control and data planes, and training-inference consistency. We open-source the training dynamics, RL environments, and RL framework to facilitate reproduction and further research on scaled RL and model self-improvement.

cs.CL↗