User interaction data (e.g., click data) has proven to be a powerful signal for learning-to-rank models in web search. However, such models require observing multiple interactions across many users for the same query-document pair to achieve statistically meaningful gains. Therefore, utilizing user interaction data for improving search over personal, rather than public, content is a challenging problem. First, the documents (e.g., emails or private files) are not shared across users. Second, user search queries are of personal nature (e.g., [alice's address]) and may not generalize well across users. In this paper, we propose a solution to these challenges, by projecting user queries and documents into a multi-dimensional space of fine-grained and semantically coherent attributes. We then introduce a novel parameterization technique to overcome sparsity in the multi-dimensional attribute space. Attribute parameterization enables effective usage of cross-user interactions for improving personal search quality -- which is a first such published result, to the best of our knowledge. Experiments with a dataset derived from interactions of users of one of the worlds' largest personal search engines demonstrate the effectiveness of the proposed attribute parameterization technique.