Notes on Computational Hardness of Hypothesis Testing: Predictions Using the Low-Degree Likelihood Ratio

These notes survey and explore an emerging method, which we call the low-degree for understanding statistical-versus-computational tradeoffs in high-dimensional inference problems. In short, method posits that a certain quantity—the second moment of likelihood ratio—gives insight into how much computational time is required to solve given hypothesis testing problem, can turn be used predict hardness variety statistical tasks. While this originated study sum-of-squares (SoS) hierarchy convex programs, present self-contained introduction does not require knowledge SoS. addition showing carry out predictions using include discussion investigating both rigorous conjectural consequences these predictions. some new results, simplified proofs, refined conjectures. For instance, point formal connection between spectral methods ratio, give sharp lower bound against subexponential-time algorithms tensor PCA.