Lipschitz regularized gradient flows and latent generative particles

Lipschitz regularized f-divergences are constructed by imposing a bound on the Lipschitz constant of the discriminator in the variational representation. They interpolate between the Wasserstein metric and f-divergences and provide a flexible family of loss functions for non-absolutely continuous (e.g. empirical) distributions, possibly with heavy tails. We construct Lipschitz regularized gradient flows on the space of probability measures based on these divergences. Examples of such gradient flows are Lipschitz regularized Fokker-Planck and porous medium partial differential equations (PDEs) for the Kullback-Leibler and alpha-divergences, respectively. The regularization corresponds to imposing a Courant-Friedrichs-Lewy numerical stability condition on the PDEs. For empirical measures, the Lipschitz regularization on gradient flows induces a numerically stable transporter/discriminator particle algorithm, where the generative particles are transported along the gradient of the discriminator. The gradient structure leads to a regularized Fisher information (particle kinetic energy) used to track the convergence of the algorithm. The Lipschitz regularized discriminator can be implemented via neural network spectral normalization and the particle algorithm generates approximate samples from possibly high-dimensional distributions known only from data. Notably, our particle algorithm can generate synthetic data even in small sample size regimes. A new data processing inequality for the regularized divergence allows us to combine our particle algorithm with representation learning, e.g. autoencoder architectures. The resulting algorithm yields markedly improved generative properties in terms of efficiency and quality of the synthetic samples. From a statistical mechanics perspective the encoding can be interpreted dynamically as learning a better mobility for the generative particles

Artificial Intelligence, Mathematical Modeling and Magnetic Resonance Imaging for Precision Medicine in Neurology and Neuroradiology

La tesi affronta la possibilità di utilizzare metodi matematici, tecniche di simulazione, teorie fisiche riadattate e algoritmi di intelligenza artificiale per soddisfare le esigenze cliniche in neuroradiologia e neurologia al fine di descrivere e prevedere i patterns e l’evoluzione temporale di una malattia, nonché di supportare il processo decisionale clinico. La tesi è suddivisa in tre parti. La prima parte riguarda lo sviluppo di un workflow radiomico combinato con algoritmi di Machine Learning al fine di prevedere parametri che favoriscono la descrizione quantitativa dei cambiamenti anatomici e del coinvolgimento muscolare nei disordini neuromuscolari, con particolare attenzione alla distrofia facioscapolo-omerale. Il workflow proposto si basa su sequenze di risonanza magnetica convenzionali disponibili nella maggior parte dei centri neuromuscolari e, dunque, può essere utilizzato come strumento non invasivo per monitorare anche i più piccoli cambiamenti nei disturbi neuromuscolari oltre che per la valutazione della progressione della malattia nel tempo. La seconda parte riguarda l’utilizzo di un modello cinetico per descrivere la crescita tumorale basato sugli strumenti della meccanica statistica per sistemi multi-agente e che tiene in considerazione gli effetti delle incertezze cliniche legate alla variabilità della progressione tumorale nei diversi pazienti. L'azione dei protocolli terapeutici è modellata come controllo che agisce a livello microscopico modificando la natura della distribuzione risultante. Viene mostrato come lo scenario controllato permetta di smorzare le incertezze associate alla variabilità della dinamica tumorale. Inoltre, sono stati introdotti metodi di simulazione numerica basati sulla formulazione stochastic Galerkin del modello cinetico sviluppato. La terza parte si riferisce ad un progetto ancora in corso che tenta di descrivere una porzione di cervello attraverso la teoria quantistica dei campi e di simularne il comportamento attraverso l'implementazione di una rete neurale con una funzione di attivazione costruita ad hoc e che simula la funzione di risposta del modello biologico neuronale. E’ stato ottenuto che, nelle condizioni studiate, l'attività della porzione di cervello può essere descritta fino a O(6), i.e, considerando l’interazione fino a sei campi, come un processo gaussiano. Il framework quantistico definito può essere esteso anche al caso di un processo non gaussiano, ovvero al caso di una teoria di campo quantistico interagente utilizzando l’approccio della teoria wilsoniana di campo efficace.The thesis addresses the possibility of using mathematical methods, simulation techniques, repurposed physical theories and artificial intelligence algorithms to fulfill clinical needs in neuroradiology and neurology. The aim is to describe and to predict disease patterns and its evolution over time as well as to support clinical decision-making processes. The thesis is divided into three parts. Part 1 is related to the development of a Radiomic workflow combined with Machine Learning algorithms in order to predict parameters that quantify muscular anatomical involvement in neuromuscular diseases, with special focus on Facioscapulohumeral dystrophy. The proposed workflow relies on conventional Magnetic Resonance Imaging sequences available in most neuromuscular centers and it can be used as a non-invasive tool to monitor even fine change in neuromuscular disorders and to evaluate longitudinal diseases’ progression over time. Part 2 is about the description of a kinetic model for tumor growth by means of classical tools of statistical mechanics for many-agent systems also taking into account the effects of clinical uncertainties related to patients’ variability in tumor progression. The action of therapeutic protocols is modeled as feedback control at the microscopic level. The controlled scenario allows the dumping of uncertainties associated with the variability in tumors’ dynamics. Suitable numerical methods, based on Stochastic Galerkin formulation of the derived kinetic model, are introduced. Part 3 refers to a still-on going project that attempts to describe a brain portion through a quantum field theory and to simulate its behavior through the implementation of a neural network with an ad-hoc activation function mimicking the biological neuron model response function. Under considered conditions, the brain portion activity can be expressed up to O(6), i.e., up to six fields interaction, as a Gaussian Process. The defined quantum field framework may also be extended to the case of a Non-Gaussian Process behavior, or rather to an interacting quantum field theory in a Wilsonian Effective Field theory approach

On quantitative hypocoercivity estimates based on Harris-type theorems

This review concerns recent results on the quantitative study of convergence towards the stationary state for spatially inhomogeneous kinetic equations. We focus on analytical results obtained by means of certain probabilistic techniques from the ergodic theory of Markov processes. These techniques are sometimes referred to as Harris-type theorems. They provide constructive proofs for convergence results in the $L^1$ (or total variation) setting for a large class of initial data. The convergence rates can be made explicit (both for geometric and sub-geometric rates) by tracking the constants appearing in the hypotheses. Harris-type theorems are particularly well-adapted for equations exhibiting non-explicit and non-equilibrium steady states since they do not require prior information on the existence of stationary states. This allows for significant improvements of some already-existing results by relaxing assumptions and providing explicit convergence rates. We aim to present Harris-type theorems, providing a guideline on how to apply these techniques to the kinetic equations at hand. We discuss recent quantitative results obtained for kinetic equations in gas theory and mathematical biology, giving some perspectives on potential extensions to nonlinear equations.Comment: 40 pages, typos are corrected, new references are added and structure of the paper has change

Hypocoercivity of linear kinetic equations via Harris's Theorem

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space $(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem

Exact Time-Dependent Solutions and Information Geometry of a Rocking Ratchet

The noise-induced transport due to spatial symmetry-breaking is a key mechanism for the generation of a uni-directional motion by a Brownian motor. By utilising an asymmetric sawtooth periodic potential and three different types of periodic forcing G(t) (sinusoidal, square and sawtooth waves) with period T and amplitude A, we investigate the performance (energetics, mean current, Stokes efficiency) of a rocking ratchet in light of thermodynamic quantities (entropy production) and the path-dependent information geometric measures. For each G(t), we calculate exact time-dependent probability density functions under different conditions by varying T, A and the strength of the stochastic noise D in an unprecedentedly wide range. Overall similar behaviours are found for different cases of G(t). In particular, in all cases, the current, Stokes efficiency and the information rate normalised by A and D exhibit one or multiple local maxima and minima as A increases. However, the dependence of the current and Stokes efficiency on A can be quite different, while the behaviour of the information rate normalised by A and D tends to resemble that of the Stokes efficiency. In comparison, the irreversibility measured by a normalised entropy production is independent of A. The results indicate the utility of the information geometry as a proxy of a motor efficiency