No abstract

Digital histopathological images, high-resolution images of stained tissue samples, are a vital tool for clinicians to diagnose and stage cancers. The visual analysis of patient state based on these images are an important part of oncology workflow. Although pathology workflows have historically been conducted in laboratories under a microscope, the increasing digitization of histopathological images has led to their analysis on computers in the clinic. The last decade has seen the emergence of machine learning, and deep learning in particular, a powerful set of tools for the analysis of histopathological images. Machine learning models trained on large datasets of digitized histopathology slides have resulted in automated models for prediction and stratification of patient risk. In this review, we provide context for the rise of such models in computational histopathology, highlight the clinical tasks they have found success in automating, discuss the various machine learning techniques that have been applied to this domain, and underscore open problems and opportunities.

Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras [12]. In subsequent work with Kleinberg and Szegedy, they connected this to the search for combinatorial objects called strong uniquely solvable puzzles (strong USPs) [11]. We begin a systematic computer-aided search for these objects. We develop and implement constraint-based algorithms build on reductions to SAT and IP to verify that puzzles are strong USPs, and to search for large strong USPs. We produce tight bounds on the maximum size of a strong USP for width k ≤ 5, construct puzzles of small width that are larger than previous work, and improve the upper bounds on strong USP size for k ≤ 12. Although our work only deals with puzzles of small-constant width, the strong USPs we find imply matrix multiplication algorithms that run in O(n ω ) time with exponent ω ≤ 2.66. While our algorithms do not beat the fastest algorithms, our work provides evidence and, perhaps, a path to finding families of strong USPs that imply matrix multiplication algorithms that are more efficient than those currently known.