Estimating Kalton's constant: from twisted sums of Banach spaces to fine-tuning algorithms
Tomasz Kania (Institute of Mathematics, Czech Academy of Sciences, Czech Republic)
Abstract: We shall try to draw a quite unexpected connection between Kalton and Roberts' work in the theory of twisted sums of Banach spaces (and quasi-linear maps) that originated with the construction of twisted Hilbert spaces and fine-tuning certain optimisation algorithms. Kalton and Roberts [Trans. Amer. Math. Soc. 1983] proved a stability result for 1-additive maps asserting that there exists a universal constant K not smaller than 44.5 such that for any set algebra F, for every scalar-valued 1-additive map f defined thereon, there is a 0-additive map (a finitely additive signed measure) whose distance to f is at most K. Pawlik [Colloq. Math. 1987] noticed that in general K cannot be smaller than 1.5. We shall present a class of positive 1-additive maps, which witnesses that K cannot be smaller than 3. If time permits, we shall mention certain results due to Feige, Feldman, and Talgam-Cohen [SIAM J. Comput. 2020] illustrating the sensitivity of certain machine-learning algorithms to estimates for K.
This is joint work with M. Gnacik (UoP) and M. Guzik (UBS) [Proc. Amer. Math. Soc. 2020+].
functional analysisgeneral mathematics
Audience: researchers in the topic
( paper )
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