Missing Data and Matrix Completion

Luogo: Polo Scientifico e Tecnologico "Fabio Ferrari" Povo1, via Sommarive, 9 - Povo (TN) - Aula A207 
ore 11.00

Relatore:

• Gilbert Strang Professore di Matematica al Massachussets Institute of Technology dove insegna Introduction to Linear Algebra e Computational Science and Engineering. E' membro della National Academy of Sciences (USA) e nella sua carriera ha ricevuto numerosissimi premi.
  La sua ricerca riguarda soprattutto problemi di algebra lineare numerica, approssimazione di equazioni a derivate parziali, trasformate wavelet. Il Prof. Strang e' noto anche per i suoi contributi all'educazione matematica: ha infatti pubblicato numerosi libri di testo e le sue lezioni si possono vedere liberamente sull'MIT OpenCourseWare.

Abstract:
Start with a banded matrix B. The entries outside that band are missing. Possibly B is the start of a covariance matrix C, but covariances outside the band are unknown or too expensive to compute. It is a poor idea to assume those numbers are zero. Much better to complete B to C by a rule of maximum entropy: for Gaussians this rule meansmaximum determinant.
Statisticians discovered that the optimally completed matrix is the inverse of a banded matrix. It has a special form with low rank submatrices except near the main diagonal.
Best of all, the matrix actually needed in computations is that banded C^-1 (which is not B!). C^-1 comes quickly from a "local inverse formula."
A very special set of banded matrices contains those whose inverses are also banded (unusual!). These arise in wavelets: a transform and its inverse are both FIR. We describe a way to construct them.
We will begin with some general thoughts about teaching linear algebra.

Il seminario presuppone solo conoscenze di base di algebra lineare, ed e' quindi consigliato per tutti gli studenti dal secondo anno in poi.

Referente: Andrea Pugliese