1. Copy of your passport

2. Copy of your personal identification

3. A proof of your bank account

4. The bill of the plane tickets

5. A proof that you paid the plane tickets

6. Taxes' certification

Topological Data Analysis and Neural Networks in Computer Vision

https://www.mdpi.com/journal/mathematics/special_issues/Computational_algebraic_topology_neural_networks_computer_vision

Guest Editors: Rocio Gonzalez Diaz and Matthias Zeppelzauer

Deadline: June 30, 2020

MATTHIAS ZEPPELZAUER, matthias.zeppelzauer@fhstp.ac.at St. Pölten University of Applied Sciences, Austria

Representations of Persistence Homology for Machine Learning in Computer Vision and Beyond (part i).

In my talk, I will present two contributions from my previous research. The first part will address the automatic classification and segmentation of 3D surfaces with topological data analysis. The second part will focus on a novel vectorial representation for persistence diagrams and its application to different types of data and classification problems. Common to both parts are the central question of how to best represent the information obtained by persistent homology in a suitable way for machine learning. A further question investigated is to which degree information extracted by persistent homology is complementary to information obtained by traditional feature extraction techniques. In our experiments we could show that topological data analysis reveals complementary information, which can be beneficial e.g. for computer vision tasks. Furthermore, our proposed representation yields state-of-the-art performance and beyond in one to two orders of magnitude less time than related approaches.

MATTEO RUCCO, matteo.rucco@utrc.utc.com Staff Research Scientist United Technologies Research Center. Trento (TN), Italy.

Characterization of trained Artificial Neural Network via Topological Data Analysis

When a trained neural network is applied to a new unseen sample, it returns a prediction (e.g. 0/1 label) and it might return a probability for the sample to belong to one or more label. However, the network does not have the capability to inform the user if the prediction is a False Positive or a False Negative. False Positive or a False Negative information is obtained during the supervised testing of the network but not during its deployment on new data. Adding a monitoring system that informs the user if the prediction is reliable or not can help for 1) increase the trust on the network; 2) boost the certification of ANN component in safety critical domains (e.g. medical, industrial, etc...). In this talk, I will introduce a new approach based on Topological Data Analysis for the characterization of a Multi-Layer Perceptron trained as classifier. The results of the talk are twofold: the former are numerical evidences that the topological characterization of a trained MLP via persistent entropy is suitable for detecting when the network fails, the latter is a set of open questions: how to position persistent entropy w.r.t. Boltzmann and Shannon entropies? Is it on the edge between them? Does it capture emerging behaviors? Can we use persistent entropy for detecting whenever a system is in a deterministic regime or it is in a stochastic regime? The answers to these questions might contribute to decode the behaviors of complex systems, e.g. the human brain [1, 2, and 3].

[1] Liang, X. "Entropy evolution and uncertainty estimation with dynamical systems." Entropy 16.7 (2014): 3605-3634.

[2] Collell, Guillem, and Jordi Fauquet. "Brain activity and cognition: a connection from thermodynamics and information theory." Frontiers in psychology 6 (2015): 818.

[3] Merelli, Emanuela, et al. "Topological characterization of complex systems: Using persistent entropy." Entropy 17.10 (2015): 6872-6892.

MATTEO CAORSI, m.caorsi@l2f.ch PhD Research Scientist at L2F EPFL Innovation Park Lausanne, Suiza IMUS

Topological data analysis for time series

The talk will start with a review of the standard approaches to extract point clouds from a (multivariate) time series: e.g. takes embedding and ordinal partition graphs. Afterwards we will discuss topological feature extraction procedures and finally how to integrate such features in predictive pipelines (via classical ML techniques and neural networks). I will present the results of these pipelines on both real and simulated datasets.

MARÍA ALBERICH, maria.alberich@upc.edu UPC, Barcelona, España

Configuration spaces and cloth simulation tailored to robotic manipulation

In this talk I will explain the framework and recent work of my collaboration to the CLOTHILDE project (Cloth manipulation learning from demonstrations, https://clothilde.iri.upc.edu). I am working within

a team which aims to establish a theory of versatile cloth manipulation

by robots. First, I will explain previous contributions on topological

stratifications of the configuration space of parallel robots. Next, I

will present some recent developments on the dynamics of cloth simulation.

MATTIA BERGOMI, mattia.bergomi@neuro.fchampalimaud.org Champalimaud Research, Champalimaud Center for the Unknown - Lisbon, Portugal

Deep learning and Group equivariant (non-expansive) operators

AUTHORS: Mattia Bergomi (speaker), Pietro Vertechi, Patrizio Frosini

Artificial intelligence and deep learning are among the most successful strategies to tackle scientific questions and develop technical applications. However, if deep networks outclass humans in finding optimal features to solve a huge variety of tasks, their architectures are growing more and more complex and oftentimes as task-specific as hand-crafted features used to be. Furthermore, the representation of the data learnt by these models is increasingly complex, making their internal functionalities unintelligible to human eyes. One strategy to better control and understand artificial neural networks is to constrain them. Generally, this is done by forcing the solutions adopted by the machine to respect symmetries. We will show how the theory developed in [1] can be used to inject knowledge in state-of-the-art deep learning models, and give a first example of a deep networks, whose features are constrained to different subgroups of the group of isometries.

ANIBAL MEDINA anibal.medinamardones@epfl.ch PhD Research Scientist at L2F EPFL Innovation Park Lausanne, Suiza

New computable signals in persistence homology

After the groundbreaking work of Zomorodian, Carlsson and others, the use of topological techniques in data analysis has exploded in recent years. The main tool in this growing field is called persistence homology. In this talk, I will revisit the theoretical foundations of this tool and propose an enhancement based on secondary signals coming from higher order self-intersections. Unlike most other signals coming from theoretical topology, this added information is efficiently computable. I will describe algorithms with this purpose and examples illustrating their use.

GUILLAUME TAUZIN g.tauzin@l2f.ch PhD Research Scientist at L2F EPFL Innovation Park Lausanne, Suiza

Topological data analysis for images

We present a way to use our topological data analysis open source library Giotto, focussing on the image-analysis submodule. We apply persistent homology to generate a wide range of topological features using a point cloud obtained from an image, the filtration obtained from the grayscale and different filtrations defined on the binarized image. We show that this topological machine learning pipeline can be used as a highly relevant dimensionality reduction by applying it to the MNIST digits dataset. We conduct feature selection and study their correlations while providing an intuitive interpretation of their importance, which is relevant in both machine learning and TDA. Finally, we show that we can classify digit images while reducing the size of the feature set by an order of magnitude compared to the grayscale pixel value features and maintain similar accuracy.

PIETRO VERTECHI, pietro.vertechi@neuro.fchampalimaud.org Champalimaud Research, Champalimaud Center for the Unknown - Lisbon, Portugal

Small machines: an agile framework for artificial intelligence

AUTHORS: Pietro Vertechi (speaker), Mattia G. Bergomi, Patrizio Frosini

Using tools from category theory, we provide a framework where artificial neural networks (ANNs), and their architectures, can be formally described. We first define a category of reservoirs, whose endomorphisms (small machines) correspond to dynamics of neural networks. This category has several useful properties: completeness, cocompleteness, and a closed symmetric monoidal structure. We establish the key ingredients of neural networks---pointwise nonlinearity, partial symmetry, and architecture---in this general categorical setting. In the specific case of reservoirs of Banach spaces, we recover deep neural networks and deep kernel networks as finite products of elementary reservoirs. Applying the formalism of partial symmetries, we show how to design deep networks of group equivariant operators and validate their performance in image classification.

MATEUSZ JUDA mateusz.juda@uj.edu.pl Uniwersytet Łódzki , Krakow, Poland

Unsupervised Features Learning for Sampled Vector Fields

In this talk we introduce a new approach to computing hidden features of sampled vector fields. The basic idea is to convert the vector field data to a graph structure and use tools designed for automatic, unsupervised analysis of graphs. Using synthetic data sets we show that the collected features of the vector fields are correlated with the dynamics known for analytic models which generates the data. In particular the method may be useful in analysis of data sets where the analytic model is poorly understood or not known.

ROCIO GONZÁLEZ rogodi@us.es Computational image analysis research group, Department of Applied Math, University of Seville, Seville, Spain

Topological Data Analysis and Neural Networks in Computer Vision

Algebraic topology uses tools from abstract algebra to study topological spaces with the aim of defining algebraic invariants that classify topological spaces up to homeomorphism. Computational algebraic topology (CAT) provides methods to compute these invariants. The main topics in (Computational) Algebraic topology are simplicial and CW complexes, chain complexes, (co)homology and exact sequences. The recent field of Topological data analysis (TDA) is an approach to the analysis of datasets using techniques mainly from computational algebraic topology, being its leading tool persistent homology. In recent years. The development of methods based on neural networks is ubiquitous in computer vision. Nevertheless, there is a lack of mathematical proofs of robustness of such methods that could be compensated by the used of tools from CAT and TDA since they are robust by nature. We will explore future research lines with the aim to develop novel topology-based approaches for computer vision and/or to apply topology-based approaches to improve the current state-of-the-art in computer vision. Since as said before neural-network methods in computer vision are more and more used. This uses also pretend to cope the problems that computer-vision neural-network-based methods need to face with topology as main tool.

EDUARDO PALUZO epaluzo@us.es Computational image analysis research group, Department of Applied Math, University of Seville, Seville, Spain

Universal approximation theorem: a computational topology approach.

There is a theorem of existence that guarantees that any continuous function can be approximated by a one-hidden-layer feedforward network, but it is not constructive. Our aim here is to provide an algorithm to construct such network based on the simplicial approximation theorem.

MANUEL SORIANO TRIGUEROS. msoriano4@us.es Computational image analysis research group, Department of Applied Math, University of Seville, Seville, Spain

Matching of persistence modules.

The idea behind this point is to study how morphisms between persistence modules may induce a partial matching between the corresponding persistence barcodes. There exist some results in this direction, but no algorithmic approach to solve this problem has been given so far. The problem arises when filtrations depend not only on one parameter (as the ones introduced before) but also on two (or even more) parameters. If this is the case, the persistence modules will not be defined in the real line but in the real plane. In fact, persistence modules may be defined in any poset [1], but no general decomposition theorem is known [2]. Answering partial questions in this direction may help to understand the problem.

[1] F Chazal, V de Silva, M Glisse, S Oudot. The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics (2016)

[2] S Oudot. Persistence Theory: From Quiver Representations to Data Analysis. Mathematical Surveys and Monographs, American Mathematical Society (2015)