  In this work we use the persistent homology method, a technique in
topological data analysis (TDA), to extract essential topological features from
the data space and combine them with deep learning features for classification
tasks. In TDA, the concepts of complexes and filtration are building blocks.
Firstly, a filtration is constructed from some complex. Then, persistent
homology classes are computed, and their evolution along the filtration is
visualized through the persistence diagram. Additionally, we applied
vectorization techniques to the persistence diagram to make this topological
information compatible with machine learning algorithms. This was carried out
with the aim of classifying images from multiple classes in the MNIST dataset.
Our approach inserts topological features into deep learning approaches
composed by single and two-streams neural networks architectures based on a
multi-layer perceptron (MLP) and a convolutional neral network (CNN) taylored
for multi-class classification in the MNIST dataset. In our analysis, we
evaluated the obtained results and compared them with the outcomes achieved
through the baselines that are available in the TensorFlow library. The main
conclusion is that topological information may increase neural network accuracy
in multi-class classification tasks with the price of computational complexity
of persistent homology calculation. Up to the best of our knowledge, it is the
first work that combines deep learning features and the combination of
topological features for multi-class classification tasks.
