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Matrix sketching is a powerful tool for reducing the size of large data matrices. Yet there are fundamental limitations to this size reduction when we want to recover an accurate estimator for a task such as least square regression. We show that these limitations can be circumvented in the distributed setting by designing sketching methods that minimize the bias of the estimator, rather than its error. In particular, we give a sparse sketching method running in optimal space and current matrix multiplication time, which recovers a nearly-unbiased least squares estimator using two passes over the data. This leads to new communication-efficient distributed averaging algorithms for least squares and related tasks, which directly improve on several prior approaches. Our key novelty is a new bias analysis for sketched least squares, giving a sharp characterization of its dependence on the sketch sparsity. The techniques include new higher moment restricted Bai-Silverstein inequalities, which are of independent interest to the non-asymptotic analysis of deterministic equivalents for random matrices that arise from sketching. 

ABSTRACT Many microscopic images and simulations of cells give results in different kinds of formats, making it difficult for people lacking computational skills to visualize and interact with them. Minecraft—known for its three-dimensional, open-world, voxel-based environment—offers a unique solution by allowing the direct insertion of voxel-based cellular structures from light microscopy and simulations into its worlds without modification. This integration enables Minecraft players to explore the ultrastructure of cells in a highly immersive and interactive environment. Here, we demonstrate several workflows that can convert images and simulation results into Minecraft worlds. Using the workflows, students can easily import and interact with a variety of cellular content, including bacteria, yeast, and cancer cells. This approach not only opens new avenues for science education but also demonstrates the potential of combining scientific visualization with interactive gaming platforms for facilitating research and improving appreciation of cellular structure for a broad audience.