This paper proposes a computationally efficient mechanism for multi-dimensional matching markets where agents report preferences over object features rather than complete utility assessments. We use Singular Value Decomposition (SVD) to identify the principal direction of variation in feature space and match agents to objects along this dimension, reducing a complex multi-dimensional problem to an effectively one-dimensional problem solvable in $O(N \log N)$ time. We show that when data exhibit low effective dimensionality, our mechanism approximately maximizes Nash Social Welfare, satisfies distributional truthfulness, and achieves symmetry. We establish a novel connection between Nash Social Welfare and Geometric Distributionally Robust Optimization, providing robustness guaranties. Numerical experiments demonstrate that our approach achieves 99\% optimal welfare while running three orders of magnitude faster than direct optimization. The framework applies naturally to school choice, labor markets, and course allocation, where feature-based elicitation reduces the cognitive burden on agents.

翻译：本文提出了一种计算高效的多维匹配市场机制，其中主体报告对物品特征的偏好，而非完整的效用评估。我们利用奇异值分解(SVD)识别特征空间中的主变化方向，并沿该维度将主体与物品进行匹配，从而将复杂的多维问题降维为可在$O(N \log N)$时间内求解的有效一维问题。研究表明，当数据呈现低有效维度时，我们的机制能近似最大化纳什社会福利，满足分布真实性，并实现对称性。我们建立了纳什社会福利与几何分布鲁棒优化之间的新颖联系，提供了鲁棒性保证。数值实验表明，该方法在实现99%最优福利的同时，运行速度比直接优化快三个数量级。该框架可自然应用于学校选择、劳动力市场和课程分配等领域，其中基于特征的偏好提取能降低主体的认知负担。