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Abstract A spectral minimal partition of a manifold is its decomposition into disjoint open sets that minimizes a spectral energy functional. It is known that bipartite spectral minimal partitions coincide with nodal partitions of Courant‐sharp Laplacian eigenfunctions. However, almost all minimal partitions are non‐bipartite. To study those, we define a modified Laplacian operator and prove that the nodal partitions of its Courant‐sharp eigenfunctions are minimal within a certain topological class of partitions. This yields new results in the non‐bipartite case and recovers the above known result in the bipartite case. Our approach is based on tools from algebraic topology, which we illustrate by a number of examples where the topological types of partitions are characterized by relative homology. 

Proceedings of The 28th International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research #258 

Automated region of interest detection in histopathological image analysis is a challenging and important topic with tremendous potential impact on clinical practice. The deep learning methods used in computational pathology may help us to reduce costs and increase the speed and accuracy of cancer diagnosis. We started with the UNC Melanocytic Tumor Dataset cohort which contains 160 hematoxylin and eosin whole slide images of primary melanoma (86) and nevi (74). We randomly assigned 80% (134) as a training set and built an in-house deep learning method to allow for classification, at the slide level, of nevi and melanoma. The proposed method performed well on the other 20% (26) test dataset; the accuracy of the slide classification task was 92.3% and our model also performed well in terms of predicting the region of interest annotated by the pathologists, showing excellent performance of our model on melanocytic skin tumors. Even though we tested the experiments on a skin tumor dataset, our work could also be extended to other medical image detection problems to benefit the clinical evaluation and diagnosis of different tumors. 

High intratumoral heterogeneity is thought to be a poor prognostic indicator. However, the source of heterogeneity may also be important, as genomic heterogeneity is not always reflected in histologic or ‘visual’ heterogeneity. We aimed to develop a predictor of histologic heterogeneity and evaluate its association with outcomes and molecular heterogeneity. We used VGG16 to train an image classifier to identify unique, patient-specific visual features in 1655 breast tumors (5907 core images) from the Carolina Breast Cancer Study (CBCS). Extracted features for images, as well as the epithelial and stromal image components, were hierarchically clustered, and visual heterogeneity was defined as a greater distance between images from the same patient. We assessed the association between visual heterogeneity, clinical features, and DNA-based molecular heterogeneity using generalized linear models, and we used Cox models to estimate the association between visual heterogeneity and tumor recurrence. Basal-like and ER-negative tumors were more likely to have low visual heterogeneity, as were the tumors from younger and Black women. Less heterogeneous tumors had a higher risk of recurrence (hazard ratio = 1.62, 95% confidence interval = 1.22–2.16), and were more likely to come from patients whose tumors were comprised of only one subclone or had a TP53 mutation. Associations were similar regardless of whether the image was based on stroma, epithelium, or both. Histologic heterogeneity adds complementary information to commonly used molecular indicators, with low heterogeneity predicting worse outcomes. Future work integrating multiple sources of heterogeneity may provide a more comprehensive understanding of tumor progression.