Topological methods for dimensionality reduction with application to simulation and privacy

PhD program in Pure and Applied Mathematics (Interateneo UNITO – POLITO)

Supervisors

Francesco Vaccarino – francesco.vaccarino@polito.it
Giovanni Petri – giovanni.petri@isi.it

Context of the research activity

One of the characteristic of the so called Big Data is the occurrence of high dimensional data. The analysis of high dimensional data is affected by the so-called curse of dimensionality. As an example, one of the aspects of this phenomenon is the impossibility of sampling efficiently point from neighborhood of a data set.
There are several methods used to overcome this difficulty, many of them based on projecting data onto smaller dimensional subspaces. Besides deterministic methods, as e.g. PCA, there techniques based on random matrices and the Johnson-Lindenstrauss Theorem which have become quite popular both for their computational efficiency and also because it has been recently shown that they can be efficiently used in differential privacy.
Furthermore, very recently it has been started the study of the use of topological data analysis to understand the behavior of data under random projections from a topological point of view.

Objectives

To study the behavior of data, real or generated in silico, under random projections by means of techniques of topological data analysis. To apply the mathematical findings, also by developing ad-hoc algorithms and codes, to high dimensional data and structures arisng in simulations and privacy issues, in particular those one coming from Progetto DISMA, Dipartimento di eccellenza 2018-2022.

Skills and competencies for the development of the activity

The candidate is required to have very good competences in basic machine learning, topology/geometry, experience in algorithm design/analysis and good programming skills

 

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