The use of mini-batches of data in training artificial neural networks is nowadays very
common. Despite its broad usage, theories explaining quantitatively how large or small the …

We consider the unlabeled linear regression reading as $\mathbf {Y}=\mathbf
{\Pi}^{*}\mathbf {X}\mathbf {B}^*+\mathbf {W} $, where $\mathbf {\Pi}^{*},\mathbf {B}^* $ and …

This chapter delves into the realm of computational complexity, exploring the world of
challenging combinatorial problems and their ties with statistical physics. Our exploration …

We derive, within the replica formalism, a generalisation of the Crisanti–Sommers formula to
describe the large deviation function (LDF) L (e) for the speed-N atypical fluctuations of the …

For random combinatorial optimization problems, there has been much progress in
establishing laws of large numbers and computing limiting constants for the optimal values …

We investigate the behavior of the rare fluctuations of the free energy in the p-spin spherical
model, evaluating the corresponding rate function via the Gärtner-Ellis theorem. This …

We study the phase transition phenomenon inherent in the shuffled (permuted) regression
problem, which has found numerous applications in databases, privacy, data analysis, etc …

We study the random-link matching problem on random regular graphs, along with two
relaxed versions of the problem, namely the fractional matching and the so …

In this thesis, we apply methods from replica theory to deal with two contemporary problems
in the study of complex systems. In Part I, we discuss the behavior of the rare fluctuations of …

This paper considers the permuted linear regression, ie, ${\mathbf {Y}}={\mathbf
{\Pi}}^{\natural}{\mathbf {X}}{\mathbf {B}}^{\natural}+{\mathbf {W}} $, where ${\mathbf …