**Project leader**: Professor Alex Dunlap

**Project manager**: Haotian Gu

**Team members**: Ben Greene, Kaden McLaughlin, Shuhuai Yu

Clustering, the task of dividing a set of points into different groups, has broad applications in machine learning, pattern recognition, and statistics. A well-known approach, k-means clustering, involves clustering points into k clusters and is a non-convex optimization problem. Since k-means clustering is a non-convex problem, finding the global minimum solution can be challenging. In this project, we investigated an alternative clustering scheme known as Sum-of-Norms (SON), which formulates the problem as a convex optimization problem. One key advantage of SON is that it does not require specifying the number of clusters in advance. We have obtained several interesting results regarding the application of SON in both discrete and continuous configurations.

More specifically, we are interested in critical values of lambda, a tuning parameter in SON that determines the clustering behaviours. We proposed a condition to determine whether three arbitrary points would cluster together. Additionally, we studied SON under continuous configurations where points are sampled from different measures. We obtained results for clustering behaviours in (1) 1-d continuous measures, (2) concentric circles, and some partial results for (3) a unit disk. Although in some cases the precise critical lambda values cannot be solved exactly, some bounds are proposed to better understand SON behaviours in these configurations.