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We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the$\ell ^1$distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produces singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments.

There are currently >1.3 million human –omics samples that are publicly available. This valuable resource remains acutely underused because discovering particular samples from this ever-growing data collection remains a significant challenge. The major impediment is that sample attributes are routinely described using varied terminologies written in unstructured natural language. We propose a natural-language-processing-based machine learning approach (NLP-ML) to infer tissue and cell-type annotations for genomics samples based only on their free-text metadata. NLP-ML works by creating numerical representations of sample descriptions and using these representations as features in a supervised learning classifier that predicts tissue/cell-type terms. Our approach significantly outperforms an advanced graph-based reasoning annotation method (MetaSRA) and a baseline exact string matching method (TAGGER). Model similarities between related tissues demonstrate that NLP-ML models capture biologically-meaningful signals in text. Additionally, these models correctly classify tissue-associated biological processes and diseases based on their text descriptions alone. NLP-ML models are nearly as accurate as models based on gene-expression profiles in predicting sample tissue annotations but have the distinct capability to classify samples irrespective of the genomics experiment type based on their text metadata. Python NLP-ML prediction code and trained tissue models are available athttps://github.com/krishnanlab/txt2onto.