Disentangling Language Roles in Multilingual LLM Task Execution

Multilingual LLMs are increasingly used when instruction, source content, and required response languages do not coincide. Existing benchmarks have expanded multilingual instruction-following evaluation, but they rarely isolate these three roles within a fully crossed design. We introduce MTM-Bench, a controlled benchmark for language-conditioned task execution in which each instance is defined by a triplet \((L_{\text{instr}}, L_{\text{content}}, L_{\text{resp}})\). Across English, Spanish, and Chinese, MTM-Bench enumerates all 27 triplets and contains 2{,}430 instances per model across semantic reversal, final-state extraction, and language purity with update realization. We evaluate 20 frontier and open-weight LLMs using decomposed metrics for semantic correctness, target-language adherence, constraint satisfaction, contamination ratio, and joint success, with scoring validated by a targeted human audit. The fully crossed design reveals that degradation is organized by the role a language occupies in the task structure, not merely by mismatch count. The response-language role is the dominant axis of variation, and a single response-slot mismatch accounts for most degradation. The response-only and full-mismatch comparison suggests that mismatch count is not a monotonic predictor of difficulty, with model-level ordering varying across systems. Task families fail through distinct channels, showing that semantic correctness alone does not capture reliable multilingual task execution.

A Unified Theoretic and Algorithmic Framework for Solving Multivariate Linear Model with $\ell^1$-norm Optimization

It is a challenging problem that solving the \textit{multivariate linear model} (MLM) $\mathbf{A}\mathbf{x}=\mathbf{b}$ with the $\ell_1 $-norm approximation method such that $||\mathbf{A}\mathbf{x}-\mathbf{b}||_1$, the $\ell_1$-norm of the \textit{residual error vector} (REV), is minimized. In this work, our contributions lie in two aspects: firstly, the equivalence theorem for the structure of the $\ell_1$-norm optimal solution to the MLM is proposed and proved; secondly, a unified algorithmic framework for solving the MLM with $\ell_1$-norm optimization is proposed and six novel algorithms (L1-GPRS, L1-TNIPM, L1-HP, L1-IST, L1-ADM, L1-POB) are designed. There are three significant characteristics in the algorithms discussed: they are implemented with simple matrix operations which do not depend on specific optimization solvers; they are described with algorithmic pseudo-codes and implemented with Python and Octave/MATLAB which means easy usage; and the high accuracy and efficiency of our six new algorithms can be achieved successfully in the scenarios with different levels of data redundancy. We hope that the unified theoretic and algorithmic framework with source code released on GitHub could motivate the applications of the $\ell_1$-norm optimization for parameter estimation of MLM arising in science, technology, engineering, mathematics, economics, and so on.