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Cross-encoder models, which jointly encode and score a query-item pair, areprohibitively expensive for direct k-nearest neighbor (k-NN) search.Consequently, k-NN search typically employs a fast approximate retrieval (e.g.using BM25 or dual-encoder vectors), followed by reranking with across-encoder; however, the retrieval approximation often has detrimentalrecall regret. This problem is tackled by ANNCUR (Yadav et al., 2022), a recentwork that employs a cross-encoder only, making search efficient using arelatively small number of anchor items, and a CUR matrix factorization. WhileANNCUR's one-time selection of anchors tends to approximate the cross-encoderdistances on average, doing so forfeits the capacity to accurately estimatedistances to items near the query, leading to regret in the crucial end-task:recall of top-k items. In this paper, we propose ADACUR, a method thatadaptively, iteratively, and efficiently minimizes the approximation error forthe practically important top-k neighbors. It does so by iteratively performingk-NN search using the anchors available so far, then adding these retrievednearest neighbors to the anchor set for the next round. Empirically, onmultiple datasets, in comparison to previous traditional and state-of-the-artmethods such as ANNCUR and dual-encoder-based retrieve-and-rerank, our proposedapproach ADACUR consistently reduces recall error-by up to 70% on the importantk = 1 setting-while using no more compute than its competitors.

Efficient k-NN Search with Cross-Encoders using Adaptive Multi-Round CUR Decomposition