### Description

We are living in a time in which huge amounts of data are constantly generated. For instance around 300 hours of video material are uploaded to YouTube every minute.

This mass of data calls for intelligent methods to extract the relevant information from a data collection; this task is called data analysis.

Application areas include recommendation systems for users of

audio or video streaming services or the automatic detection of anomalies on surveillance cameras.

In the past 15 years, a novel and perhaps surprising

approach of data analysis has been developed:

the qualitative properties of a data set are studied with the mathematical theory of algebraic topology. The rough idea is that the data is transformed into a geometric shape and it is studied in what way this shape is connected.

As an example, a ball and a doughnut are connected differently because the latter has a hole in its center. Such a topological analysis often reveals valuable information about the data set, as demonstrated by a multitude of applications to realistic scenarios, for instance, in biology, physics, or computer graphics. However, the computation of topological properties

on a computer requires a lot of time for large data sets, restricting current uses of the technique to rather small data.

The goal of project is to change that: we want to develop computer programs that are able to process data sizes which are out-of-reach with current technology. In this way, we significantly expand the range of applications for which topological data analysis can provide new insights.

Such an advancement, however, requires novel ways to compute the relevant topological information from data. To prove the advantages of our novel methods, we will analyze them in the framework of theoretical computer science,

but also compare them with existing approaches on realistic data sets.

While the latter seems to be the natural measure of quality,

the former is equally important because it allows the comparison of computational methods over all possible future inputs,

while experiments can only measure the performance of currently available inputs. Designing solutions efficient in both way yields to long-lasting contributions to the research field.

This project has a remarkably interdisciplinary character,

bringing together algebraic topology, data analysis, geometry, theoretical computer science, and software engineering.

It establishes efficient computations as the bridge between abstract mathematics and real-world applications and paves the way of using topological methods as a standard tool in the context of data analytics.

This mass of data calls for intelligent methods to extract the relevant information from a data collection; this task is called data analysis.

Application areas include recommendation systems for users of

audio or video streaming services or the automatic detection of anomalies on surveillance cameras.

In the past 15 years, a novel and perhaps surprising

approach of data analysis has been developed:

the qualitative properties of a data set are studied with the mathematical theory of algebraic topology. The rough idea is that the data is transformed into a geometric shape and it is studied in what way this shape is connected.

As an example, a ball and a doughnut are connected differently because the latter has a hole in its center. Such a topological analysis often reveals valuable information about the data set, as demonstrated by a multitude of applications to realistic scenarios, for instance, in biology, physics, or computer graphics. However, the computation of topological properties

on a computer requires a lot of time for large data sets, restricting current uses of the technique to rather small data.

The goal of project is to change that: we want to develop computer programs that are able to process data sizes which are out-of-reach with current technology. In this way, we significantly expand the range of applications for which topological data analysis can provide new insights.

Such an advancement, however, requires novel ways to compute the relevant topological information from data. To prove the advantages of our novel methods, we will analyze them in the framework of theoretical computer science,

but also compare them with existing approaches on realistic data sets.

While the latter seems to be the natural measure of quality,

the former is equally important because it allows the comparison of computational methods over all possible future inputs,

while experiments can only measure the performance of currently available inputs. Designing solutions efficient in both way yields to long-lasting contributions to the research field.

This project has a remarkably interdisciplinary character,

bringing together algebraic topology, data analysis, geometry, theoretical computer science, and software engineering.

It establishes efficient computations as the bridge between abstract mathematics and real-world applications and paves the way of using topological methods as a standard tool in the context of data analytics.

Status | Active |
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Effective start/end date | 1/04/17 → 31/03/21 |