Spatio-temporal models of the functional architecture of the visual cortex

In this work I tried to explore many aspects of cognitive visual science, each one based on different academic fields, proposing mathematical models capable to reproduce both neuro-physiological and phenomenological results that were described in the recent literature. The structure of my thesis is mainly composed of three chapters, corresponding to the three main areas of research on which I focused my work. The results of each work put the basis for the following, and their ensemble form an homogeneous and large-scale survey on the spatio-temporal properties of the architecture of the visual cortex of mammals.

Rigorous results in Spin Glasses and Monomer-Dimer systems

In this work I reported recent results in the field of Statistical Mechanics of Equilibrium, and in particular in Spin Glass models and Monomer Dimer models . We start giving the mathematical background and the general formalism for Spin (Disordered) Models with some of their applications to physical and mathematical problems. Next we move on general aspects of the theory of spin glasses, in particular to the Sherrington-Kirkpatrick model which is of fundamental interest for the work. In Chapter 3, we introduce the Multi-species Sherrington-Kirkpatrick model (MSK), we prove the existence of the thermodynamical limit and the Guerra's Bound for the quenched pressure together with a detailed analysis of the annealed and the replica symmetric regime. The result is a multidimensional generalization of the Parisi's theory. Finally we brie y illustrate the strategy of the Panchenko's proof of the lower bound. In Chapter 4 we discuss the Aizenmann-Contucci and the Ghirlanda-Guerra identities for a wide class of Spin Glass models. As an example of application, we discuss the role of these identities in the proof of the lower bound. In Chapter 5 we introduce the basic mathematical formalism of Monomer Dimer models. We introduce a Gaussian representation of the partition function that will be fundamental in the rest of the work. In Chapter 6, we introduce an interacting Monomer-Dimer model. Its exact solution is derived and a detailed study of its analytical properties and related physical quantities is performed. In Chapter 7, we introduce a quenched randomness in the Monomer Dimer model and show that, under suitable conditions the pressure is a self averaging quantity. The main result is that, if we consider randomness only in the monomer activity, the model is exactly solvable.

Soil enzyme activities and biochemical indexes to assess soil quality in agronomic and forested ecosystems

The soil carries out a wide range of functions and it is important study the effects of land use on soil quality in order to provide most sustainable practices. Three fields trial have been considered to assess soil quality and functionality after human alteration, and to determine the power of soil enzymatic activities, biochemical indexes and mathematical model in the evaluation of soil status. The first field was characterized by conventional and organic management in which were tested also tillage effects. The second was characterized by conventional, organic and agro-ecological management. Finally, the third was a beech forest where was tested the effects of N deposition on soil organic carbon sequestration. Results highlight that both enzyme activities and biochemical indexes could be valid parameters for soil quality evaluation. Conventional management and plowing negatively affected soil quality and functionality with intensive tillage that lead to the downturn of microbial biomass and activity. Both organic and agro-ecological management revealed to be good practices for the maintenance of soil functionality with better microbial activity and metabolic efficiency. This positively affected also soil organic carbon content. At the eutrophic forest, enzyme activities and biochemical indexes positively respond to the treatments but one year of experimentation resulted to be not enough to observe variation in soil organic carbon content. Mathematical models and biochemical indicators resulted to be valid tools for assess soil quality, nonetheless it would be better including the microbial component in the mathematical model and consider more than one index if the aim of the work is to evaluate the overall soil quality and functionality. Concluding, the forest site is the richest one in terms of organic carbon, microbial biomass and activity while, the organic and the agro-ecological management seem to be the more sustainable but without taking in consideration the yield.

Modelling, simulation and control of rotary-wing aircraft in multi-agent scenario

The topic of this thesis is the design and the implementation of mathematical models and control system algorithms for rotary-wing unmanned aerial vehicles to be used in cooperative scenarios. The use of rotorcrafts has many attractive advantages, since these vehicles have the capability to take-off and land vertically, to hover and to move backward and laterally. Rotary-wing aircraft missions require precise control characteristics due to their unstable and heavy coupling aspects. As a matter of fact, flight test is the most accurate way to evaluate flying qualities and to test control systems. However, it may be very expensive and/or not feasible in case of early stage design and prototyping. A good compromise is made by a preliminary assessment performed by means of simulations and a reduced flight testing campaign. Consequently, having an analytical framework represents an important stage for simulations and control algorithm design. In this work mathematical models for various helicopter configurations are implemented. Different flight control techniques for helicopters are presented with theoretical background and tested via simulations and experimental flight tests on a small-scale unmanned helicopter. The same platform is used also in a cooperative scenario with a rover. Control strategies, algorithms and their implementation to perform missions are presented for two main scenarios. One of the main contributions of this thesis is to propose a suitable control system made by a classical PID baseline controller augmented with L1 adaptive contribution. In addition a complete analytical framework and the study of the dynamics and the stability of a synch-rotor are provided. At last, the implementation of cooperative control strategies for two main scenarios that include a small-scale unmanned helicopter and a rover.

New perspectives in statistical mechanics and high-dimensional inference

The main purpose of this thesis is to go beyond two usual assumptions that accompany theoretical analysis in spin-glasses and inference: the i.i.d. (independently and identically distributed) hypothesis on the noise elements and the finite rank regime. The first one appears since the early birth of spin-glasses. The second one instead concerns the inference viewpoint. Disordered systems and Bayesian inference have a well-established relation, evidenced by their continuous cross-fertilization. The thesis makes use of techniques coming both from the rigorous mathematical machinery of spin-glasses, such as the interpolation scheme, and from Statistical Physics, such as the replica method. The first chapter contains an introduction to the Sherrington-Kirkpatrick and spiked Wigner models. The first is a mean field spin-glass where the couplings are i.i.d. Gaussian random variables. The second instead amounts to establish the information theoretical limits in the reconstruction of a fixed low rank matrix, the “spike”, blurred by additive Gaussian noise. In chapters 2 and 3 the i.i.d. hypothesis on the noise is broken by assuming a noise with inhomogeneous variance profile. In spin-glasses this leads to multi-species models. The inferential counterpart is called spatial coupling. All the previous models are usually studied in the Bayes-optimal setting, where everything is known about the generating process of the data. In chapter 4 instead we study the spiked Wigner model where the prior on the signal to reconstruct is ignored. In chapter 5 we analyze the statistical limits of a spiked Wigner model where the noise is no longer Gaussian, but drawn from a random matrix ensemble, which makes its elements dependent. The thesis ends with chapter 6, where the challenging problem of high-rank probabilistic matrix factorization is tackled. Here we introduce a new procedure called "decimation" and we show that it is theoretically to perform matrix factorization through it.

Meaningful insights: explainability techniques for black-box models on tabular data

Artificial Intelligence (AI) and Machine Learning (ML) are novel data analysis techniques providing very accurate prediction results. They are widely adopted in a variety of industries to improve efficiency and decision-making, but they are also being used to develop intelligent systems. Their success grounds upon complex mathematical models, whose decisions and rationale are usually difficult to comprehend for human users to the point of being dubbed as black-boxes. This is particularly relevant in sensitive and highly regulated domains. To mitigate and possibly solve this issue, the Explainable AI (XAI) field became prominent in recent years. XAI consists of models and techniques to enable understanding of the intricated patterns discovered by black-box models. In this thesis, we consider model-agnostic XAI techniques, which can be applied to Tabular data, with a particular focus on the Credit Scoring domain. Special attention is dedicated to the LIME framework, for which we propose several modifications to the vanilla algorithm, in particular: a pair of complementary Stability Indices that accurately measure LIME stability, and the OptiLIME policy which helps the practitioner finding the proper balance among explanations' stability and reliability. We subsequently put forward GLEAMS a model-agnostic surrogate interpretable model which requires to be trained only once, while providing both Local and Global explanations of the black-box model. GLEAMS produces feature attributions and what-if scenarios, from both dataset and model perspective. Eventually, we argue that synthetic data are an emerging trend in AI, being more and more used to train complex models instead of original data. To be able to explain the outcomes of such models, we must guarantee that synthetic data are reliable enough to be able to translate their explanations to real-world individuals. To this end we propose DAISYnt, a suite of tests to measure synthetic tabular data quality and privacy.