Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have Q-factorial terminal singularities. Q-Fano varieties are of fundamental importance in geometry as they are'atomic pieces' of more complex shapes - the process of breaking a shape into simpler pieces in this sense is called the Minimal Model Programme. Despite their importance, the classification of Q-Fano varieties remains unknown. In this paper we demonstrate that machine learning can be used to understand this classification.

Hyponatremia: Predict whether a hyponatremia lab comes back as normal (>=135 mmol/L), mild (>=130 and <135 mmol/L), moderate (>=125 and <130 mmol/L), or severe (<125 mmol/L). We consider all lab results coded as LOINC/LG11363-5, LOINC/2951-2, or LOINC/2947-0. Anemia: Predict whether an anemia lab comes back as normal (>=120 g/L), mild (>=110 and <120 g/L), moderate (>=70 and <110 g/L), or severe (<70 g/L). We consider all lab results coded as LOINC/LP392452-1. Please note that for the results of our baseline experiments in Section 5, we reframe these lab value tasks as binary classification tasks, where a label is "negative" if the result is normal and "positive" otherwise.

As the scale of machine learning models increases, trends such as scaling laws anticipate consistent downstream improvements in predictive accuracy. However, these trends take the perspective of a single model-provider in isolation, while in reality providers often compete with each other for users. In this work, we demonstrate that competition can fundamentally alter the behavior of these scaling trends, even causing overall predictive accuracy across users to be non-monotonic or decreasing with scale. We define a model of competition for classification tasks, and use data representations as a lens for studying the impact of increases in scale. We find many settings where improving data representation quality (as measured by Bayes risk) decreases the overall predictive accuracy across users (i.e., social welfare) for a marketplace of competing model-providers. Our examples range from closed-form formulas in simple settings to simulations with pretrained representations on CIFAR-10. At a conceptual level, our work suggests that favorable scaling trends for individual model-providers need not translate to downstream improvements in social welfare in marketplaces with multiple model providers.

We show that the RANDOMCOORDINATECUT algorithm gives the optimal competitive ratio for explainable k-medians in ℓ1. The problem of explainable k-medians was introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian in 2020. Several groups of authors independently proposed a simple polynomial-time randomized algorithm for the problem and showed that this algorithm is O(logkloglogk) competitive. We provide a tight analysis of the algorithm and prove that its competitive ratio is upper bounded by 2lnk +2. This bound matches the Ω(logk)lower bound by Dasgupta et al (2020).