arXiv:2603.19465v2 Announce Type: replace-cross
Abstract: We analyze a fixed-point iteration $v leftarrow phi(v)$ arising in the optimization of a regularized nuclear norm objective involving the Hadamard product structure, posed in DMR+22 in the context of an optimization problem over the space of algorithms in private machine learning. We prove that the iteration $v^(k+1) = textdiag((D_v^(k)^1/2 M D_v^(k)^1/2)^1/2)$ converges monotonically to the unique global optimizer of the potential function $J(v) = 2 textTr((D_v^1/2 M D_v^1/2)^1/2) – sum v_i$, closing a problem left open there.
The bulk of this proof was provided by Gemini 3, subject to some corrections and interventions. Gemini 3 also sketched the initial version of this note. Thus, it represents as much a commentary on the practical use of AI in mathematics as it represents the closure of a small gap in the literature. As such, we include a small narrative description of the prompting process, and some resulting principles for working with AI to prove mathematics.
Depression subtype classification from social media posts: few-shot prompting vs. fine-tuning of large language models
BackgroundSocial media provides timely proxy signals of mental health, but reliable tweet-level classification of depression subtypes remains challenging due to short, noisy text, overlapping symptomatology,