## António Leitão

### NOVA University

I’m a mathematician currently doing my Master’s thesis at ISI Foundation under the supervision of Giovanni Petri. My work focuses mainly on the Topology of spaces in learning systems.

I like to condense interesting parts of my work into over-illustrated, easily-digestible chunks.

### Interests

• Topology
• Neural Networks
• Machine Learning

### Education

• MSc in Data Science, 2020

NOVA University, Lisbon

NOVA University, Lisbon

• BSc in Mathematics, 2017

NOVA University, Lisbon

# Recent Posts

The MNIST dataset is a collection of images of handwritten digits. Before the birth of Convolutional Neural Networks most machine learning approaches arranged each 28x28 pixel image into a 758-dimensional vector. Does this make sense? Does natural data have these coordinates? What if I told you this is seen as standard procedure in “machine learning”. Are we cramming vectors, coordinates and metrics where they have no business in?

### Approaching the Edge of Decision

Complex data requires complex models, right? But does it really? Classifying two concentric circles is challenging not because they are circles but because they are concentric. The complexity of the decision boundary measures how the classes are entangled and is the closest approximation to the intrinsic difficulty of a classification problem. But it all starts by sampling the decision boundary. (Or is it a decision boundary?)

### Architecture's Expressiveness

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Adipiscing at in tellus integer. Ac feugiat sed lectus vestibulum mattis ullamcorper velit sed ullamcorper. Risus viverra adipiscing at in tellus integer feugiat. Vel facilisis volutpat est velit egestas dui id ornare. Volutpat ac tincidunt vitae semper quis lectus nulla. Cras adipiscing enim eu turpis. Lorem ipsum dolor sit amet consectetur.

### Redrawing Persistent Homology

Imagine that you wanted to compare the persistent homology of two different metric spaces. The filtration parameter of standard Vietoris-Rips filtration is the metric, but each space has its own metric. So, how do we compare their persistent homology?

Coming Soon

# Publications

### Topological Expressive Power of Neural Networks

We propose a topological description of neural network expressive power. We adopt the topology of the space of decision boundaries …

### Topological Complexity of Decision Boundaries

Complex data requires complex models, or so the saying goes. However, the reason why classifying two concentric circles is challenging …