We present a simplified theory of generalized eigenfunction expansions for a commuting family of bounded operators and with finitely many unbounded operators. We also study the convergence of these expansions, giving an abstract type of uniform convergence result, and illustrate the theory by giving two examples: The Fourier transform on Hecke operators, and the Laplacian operators in hyperboli...

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis (J. Mech. Phys. Solids, 2004) concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in t...

In distributed learning, a central server trains model according to updates provided by nodes holding local data samples. the presence of one or more malicious servers sending incorrect information (a Byzantine adversary), standard algorithms for training such as stochastic gradient descent (SGD) fail converge. this paper, we present simplified convergence theory generic Resilient SGD method or...

FOCal Underdetermined System Solver (FOCUSS) is a powerful tool for sparse representation and underdetermined inverse problems, which is extremely easy to implement. In this paper, we provide a comprehensive convergence analysis on the FOCUSS algorithm towards establishing a systematic convergence theory for it. First, we give a rigorous derivation for this algorithm exploiting the auxiliary fu...