SVD-RAG: Efficient Tree-Organized Retrieval-Augmented Generation via Singular Value Decomposition

Retrieval-Augmented Generation (RAG) systems enhance large language models by retrieving relevant documents from external knowledge bases. Recent work by Sarthi et al. (2024) introduced RAPTOR, which organizes documents into hierarchical tree structures for efficient retrieval, but requires expensive LLM-based abstractive summarization at each internal node -- making large-scale deployment prohibitively costly. We present SVD-RAG, the first method to apply Singular Value Decomposition (SVD) on dense sentence embedding matrices for extractive summarization in hierarchical RAG. Unlike classical LSA which operates on sparse TF-IDF matrices, SVD-RAG exploits the rich semantic representations of modern embedding models, identifying the most informative sentences through their energy contribution in the principal components. Our approach is (1) deterministic -- unlike LLM-based summarization, SVD produces identical results for the same input; (2) cost-efficient -- tree construction requires no additional API calls beyond the initial embedding, reducing token consumption by ~85%; and (3) content-adaptive -- the energy-ratio threshold tau automatically adjusts compression based on content complexity. In a controlled head-to-head comparison using identical corpora, clustering, and beam search, SVD-RAG achieves retrieval quality within 1-5% of RAPTOR with LLM summarization (MRR 0.867 vs. 0.875, Recall@1 0.483 vs. 0.458) while building the tree 317x faster (0.1s vs. 31.7s). On a scaled multi-topic benchmark with 205 chunks and 100 queries across 20 topic variations, SVD-RAG achieves a 4.2x improvement in Recall@1 and 3.1x improvement in MRR over flat embedding retrieval. We provide a detailed cost analysis and parameter sensitivity study. Our implementation is released as an open-source Python package.