Towards Stratification Learning through Homology Inference

A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions; some geometric intuition for these topological conditions is also provided. Our correctness result is then given a probabilistic flavor: we give bounds on the minimum number of sample points required to infer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering, prove its correctness, and apply it to some simulated data.Comment: 48 page

Homology and Robustness of Level and Interlevel Sets

Given a function f: \Xspace \to \Rspace on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of $f$. In addition, we quantify the robustness of the homology classes under perturbations of $f$ using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case \Xspace = \Rspace^3 has ramifications in the fields of medical imaging and scientific visualization

Imaging-Based Prevalence of Superior Labral Anterior-Posterior Tears Significantly Increases in the Aging Shoulder.

BackgroundSuperior labral anterior-posterior (SLAP) tears can be associated with pain and shoulder dysfunction. Relatively little is known about the age-related prevalence of SLAP tears.PurposeTo investigate the age-related prevalence of imaging-diagnosed SLAP tears in a heterogeneous grouping of shoulder conditions in a large cohort at a single institution with multiple blinded reviewers.Study designCross-sectional study; Level of evidence, 3.MethodsA total of 281 shoulder magnetic resonance imaging (MRI) scans obtained over 8 months were reviewed by a musculoskeletal radiologist and an orthopaedic surgeon. The mean ± SD age of the group was 49.6 ± 15.5 years, and 107 of the patients were female (38.1%). Patients were divided into 4 age groups: 35 years or younger, 36 to 50 years, 51 to 65 years, and older than 65 years. Statistical analyses were completed by use of the Fisher exact test to compare proportions of SLAP tears between age groups, odds ratios to determine the likelihood of having a SLAP tear in each age group, and a logistic regression to control for associated abnormalities.ResultsThere was a significant diffference in the proportion of SLAP tears found on the MRIs for each age group (P < .001). Patients were significantly more likely to have SLAP tears if aged 51 to 65 years (66.7%; odds ratio [OR], 2.00; 95% CI, 1.27-3.15) and if older than 65 years (81.2%; OR, 4.31; 95% CI, 2.36-7.88). No increased prevalence was observed in patients aged 35 years or younger (47.5%; OR, 0.91; 95% CI, 0.55-1.50) or 36 to 50 years (51.8%; OR, 1.08; 95% CI, 0.70-1.67). Logistic regression demonstrated that age was the only significant predictor for having a SLAP tear (P < .001). Kappa values were 0.46 to 0.65 between reviewers, indicating moderate to substantial agreement.ConclusionAn increasing prevalence of MRI-based SLAP tears was observed with increasing patient age. Patients older than 50 years were significantly more likely to have superior labral abnormalities regardless of other shoulder injury or disease

Mitochondrial DNA, chloroplast DNA and the origins of development in eukaryotic organisms

<p>Abstract</p> <p>Background</p> <p>Several proposals have been made to explain the rise of multicellular life forms. An internal environment can be created and controlled, germ cells can be protected in novel structures, and increased organismal size allows a "division of labor" among cell types. These proposals describe advantages of multicellular versus unicellular organisms at levels of organization at or above the individual cell. I focus on a subsequent phase of evolution, when multicellular organisms initiated the process of development that later became the more complex embryonic development found in animals and plants. The advantage here is realized at the level of the mitochondrion and chloroplast.</p> <p>Hypothesis</p> <p>The extreme instability of DNA in mitochondria and chloroplasts has not been widely appreciated even though it was first reported four decades ago. Here, I show that the evolutionary success of multicellular animals and plants can be traced to the protection of organellar DNA. Three stages are envisioned. <it>Sequestration </it>allowed mitochondria and chloroplasts to be placed in "quiet" germ line cells so that their DNA is not exposed to the oxidative stress produced by these organelles in "active" somatic cells. This advantage then provided <it>Opportunity</it>, a period of time during which novel processes arose for signaling within and between cells and (in animals) for cell-cell recognition molecules to evolve. <it>Development </it>then led to the enormous diversity of animals and plants.</p> <p>Implications</p> <p>The potency of a somatic stem cell is its potential to generate cell types other than itself, and this is a systems property. One of the biochemical properties required for stemness to emerge from a population of cells might be the metabolic quiescence that protects organellar DNA from oxidative stress.</p> <p>Reviewers</p> <p>This article was reviewed by John Logsdon, Arcady Mushegian, and Patrick Forterre.</p