We study the problem of computing a rank-\(k\) approximation of a matrix using randomized block Krylov iteration. Prior work has shown that, for block size \(b = 1\) or \(b = k\), a \((1+\varepsilon)\)-factor approximation to the best rank-\(k\) approximation can be obtained after \(\tilde{O}(k/\sqrt{\varepsilon})\) matrix-vector products with the target matrix. On the other hand, when \(b\) is between \(1\) and \(k\), the best known bound on the number of matrix-vector products scales with \(b(k-b)\), which could be as large as \(O(k^2)\). Nevertheless, in practice, the performance of block Krylov methods is often optimized by choosing a block size \(1 \ll b \ll k\). We address this theory-practice gap by proving that randomized block Krylov iteration produces a \((1+\varepsilon)\)-factor approximate rank-\(k\) approximation using \(\tilde{O}(k/\sqrt{\varepsilon})\) matrix-vector products for any block size \(1 \le b \le k\). Our analysis relies on new bounds for the minimum singular value of a random block Krylov matrix, which may be of independent interest. Similar bounds are central to recent breakthroughs on faster algorithms for sparse linear systems [Peng & Vempala 2021; SODA 2021; Nie, STOC 2022].

Object detection in biomedical settings is fundamentally constrained by the scarcity of labeled data and the frequent emergence of novel or rare categories. We present FSP-DETR, a unified detection framework that enables robust few-shot detection, open-set recognition, and generalization to unseen biomedical tasks within a single model. Built upon a class-agnostic DETR backbone, our approach constructs class prototypes from original support images and learns an embedding space using augmented views and a lightweight transformer decoder. Training jointly optimizes a prototype matching loss, an alignment-based separation loss, and a KL divergence regularization to improve discriminative feature learning and calibration under scarce supervision. Unlike prior work that tackles these tasks in isolation, FSP-DETR enables inference-time flexibility to support unseen class recognition, background rejection, and cross-task adaptation without retraining. We also introduce a new ova species detection benchmark with 20 parasite classes and establish standardized evaluation protocols. Extensive experiments across ova, blood cell, and malaria detection tasks demonstrate that FSP-DETR significantly outperforms prior few-shot and prototype-based detectors, especially in low-shot and open-set scenarios.