This paper establishes the theoretical limits of graph clustering under the Popularity-Adjusted Block Model (PABM), addressing limitations of existing models. In contrast to the Stochastic Block Model (SBM), which assumes uniform vertex degrees, and to the Degree-Corrected Block Model (DCBM), which applies uniform degree corrections across clusters, PABM introduces separate popularity parameters for intra- and inter-cluster connections. Our main contribution is the characterization of the optimal error rate for clustering under PABM, which provides novel insights on clustering hardness: we demonstrate that unlike SBM and DCBM, cluster recovery remains possible in PABM even when traditional edge-density signals vanish, provided intra- and inter-cluster popularity coefficients differ. This highlights a dimension of degree heterogeneity captured by PABM but overlooked by DCBM: local differences in connectivity patterns can enhance cluster separability independently of global edge densities. Finally, because PABM exhibits a richer structure, its expected adjacency matrix has rank between $k$ and $k^2$, where $k$ is the number of clusters. As a result, spectral embeddings based on the top $k$ eigenvectors may fail to capture important structural information. Our numerical experiments on both synthetic and real datasets confirm that spectral clustering algorithms incorporating $k^2$ eigenvectors outperform traditional spectral approaches.

Machine Learning Yearning is about structuring the development of machine learning projects. The book contains practical insights that are difficult to find somewhere else, in a format that is easy to share with teammates and collaborators. Most technical AI courses will explain to you how the different ML algorithms work under the hood, but this book teaches you how to actually use them. If you aspire to be a technical leader in AI, this book will help you on your way. Historically, the only way to learn how to make strategic decisions about AI projects was to participate in a graduate program or to gain experience working at a company.

Andrew NG is a computer scientist, executive, investor, entrepreneur, and one of the leading experts in Artificial Intelligence. He is the former Vice President and Chief Scientist of Baidu, an adjunct professor at Stanford University, the creator of one of the most popular online courses for machine learning, the co-founder of Coursera.com At Baidu, he was significantly involved in expanding their AI team into several thousand people. The book starts with a little story. Imagine, you want to build the leading cat detector system as a company.

Machine learning methods have shown promise in predicting molecular properties, and given sufficient training data ML may surpass density functional theory in computational speed and chemical accuracy. However, the most accurate machine learning methods require optimized 3D molecular geometries, limiting their applicability for high-throughput screening. We show that near-optimal results for large polymeric molecules can be obtained without optimized 3D geometry, and that trained model weights can be used to improve performance on related tasks.