Topic: sketching

Alex Townsend examines what greedy pivoting can guarantee in rank-revealing factorizations, which remain important alongside randomized sampling and sketching. A local maximum-volume viewpoint gives sharp criteria for reliable rank revelation by pivoted Gaussian elimination and QR. The comparison with pivoted Cholesky on smooth-kernel matrices shows that greedy pivoting there cannot exhibit Kahan-like behavior. These results clarify the theoretical strengths and limitations of deterministic steps in matrix approximation.

José Ramón Pareja Monturiol presents a method that converts black-box functions into tensor-train representations using sketching, cross interpolation and a small sample of the relevant domain. For neural networks, this enables privacy-oriented obfuscation and interpretable estimates of topological phases. The method also supplies an efficient initialisation for tensor-train optimisation and improves the memory–computation trade-off in model compression.

Aaron Sidford introduces recent advances towards nearly linear-time algorithms for fundamental computational problems. Examples include matching, maximum flow, minimum cuts, linear systems and linear programming. The lecture explains how continuous optimisation, dynamic data structures and sketching interact, often transferring techniques between apparently different settings.