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Giuseppe Genovese

Publications and source records attributed to Giuseppe Genovese.

At least 19 recordsLinked to original sources
arXiv2026

Derivation of the AMP equations from belief propagation for the $\ell_2$ minimisation problem

We consider the $\ell_p$-minimisation, which consists of finding the vector $x\in\mathbb{R}^N$ which minimises $\|x\|_p$ subject to the linear constraint $y=Ax$, where $y\in\mathbb{R}^m$ is given and $A$ is a $m\times N$ random matrix with i.i.d. sub-Gaussian centred entries ($m<N$). This can be viewed as the zero temperature version of a statistical mechanics problem, in which one introduces a suitable Gibbs measure on $\mathbb{R}^N$. To such a Gibbs measure there are associated belief propagation equations. We prove in the easiest case $p=2$ that the means of the distributions obtained by the belief propagation iteration satisfy asymptotically the approximate message passing equations.

math.PR↗
arXiv2025

Variational formula for the logarithmic potential of free additive convolutions

We establish a general variational formula for the logarithmic potential of the free additive convolution of two compactly supported probability measure on $\R$. The formula is given in terms of the $R$-transform of the first measure, and the logarithmic potential of second measure. The result applies in particular to the additive convolution with the semicircle or Marchenko-Pastur laws, for which the formula simplifies. The logarithmic potential of additive convolutions appears for instance in estimates of the determinant of sums of independent random matrices.

math.PR↗
arXiv2024

On the determinant in Bray-Moore's TAP complexity formula

In the computation of the TAP complexity, originally carried out by Bray and Moore, a fundamental step is to calculate the determinant of a random Hessian. As the replica method does not give a clear prescription, physicists debated how to perform this computation and its consequences on the TAP complexity for a long time. In this paper we prove the original Bray and Moore formula for the behaviour of the determinant at exponential scale to be correct, and compute an important prefactor coming from a small outlier in the spectrum.

math.PR↗
arXiv2023

Pattern reconstruction with restricted Boltzmann machines

Restricted Boltzmann machines are energy models made of a visible and a hidden layer. We identify an effective energy function describing the zero-temperature landscape on the visible units and depending only on the tail behaviour of the hidden layer prior distribution. Studying the location of the local minima of such an energy function, we show that the ability of a restricted Boltzmann machine to reconstruct a random pattern depends indeed only on the tail of the hidden prior distribution. We find that hidden priors with strictly super-Gaussian tails give only a logarithmic loss in pattern retrieval, while an efficient retrieval is much harder with hidden units with strictly sub-Gaussian tails; if the hidden prior has Gaussian tails, the retrieval capability is determined by the number of hidden units (as in the Hopfield model).

math.PR↗
arXiv2023

Universal approximation of credit portfolio losses using Restricted Boltzmann Machines

We introduce a new portfolio credit risk model based on Restricted Boltzmann Machines (RBMs), which are stochastic neural networks capable of universal approximation of loss distributions. We test the model on an empirical dataset of default probabilities of 1'012 US companies and we show that it outperforms commonly used parametric factor copula models -- such as the Gaussian or the t factor copula models -- across several credit risk management tasks. In particular, the model leads to better fits for the empirical loss distribution and more accurate risk measure estimations. We introduce an importance sampling procedure which allows risk measures to be estimated at high confidence levels in a computationally efficient way and which is a substantial improvement over the Monte Carlo techniques currently available for copula models. Furthermore, the statistical factors extracted by the model admit an interpretation in terms of the underlying portfolio sector structure and provide practitioners with quantitative tools for the management of concentration risk. Finally, we show how to use the model for stress testing by estimating stressed risk measures (e.g. stressed VaR) under various macroeconomic stress test scenarios, such as those specified by the FRB's Dodd-Frank Act stress test.

q-fin.CP↗
arXiv2022

Minimax formula for the replica symmetric free energy of deep restricted Boltzmann machines

We study the free energy of a most used deep architecture for restricted Boltzmann machines, where the layers are disposed in series. Assuming independent Gaussian distributed random weights, we show that the error term in the so-called replica symmetric sum rule can be optimised as a saddle point. This leads us to conjecture that in the replica symmetric approximation the free energy is given by a min max formula, which parallels the one achieved for two-layer case.

cond-mat.dis-nn↗
arXiv2022

Transport of Gaussian measures with exponential cut-off for Hamiltonian PDEs

We show that introducing an exponential cut-off on a suitable Sobolev norm facilitates the proof of quasi-invariance of Gaussian measures with respect to Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for the density. We exploit this idea in two different contexts, namely the periodic fractional Benjamin-Bona-Mahony (BBM) equation with dispersion~$γ>1$ and the periodic one dimensional quintic defocussing nonlinear Schrödinger equation (NLS). For the BBM equation we study the transport of the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS equation we study the measures based on the modified energies introduced by Planchon-Visciglia and the third author. Moreover for the BBM equation we also show almost sure global well-posedness for data in~$C^\a(\T)$ for arbitrarily small~$\a>0$ and invariance of the Gaussian measure associated with the $H^{γ/2}(\T)$ norm.

math.AP↗
arXiv2022

A remark on the spherical bipartite spin glass

Auffinger and Chen proved a variational formula for the free energy of the spherical bipartite spin glass in terms of a global minimum over the overlaps. We show that a different optimisation procedure leads to a saddle point, similar to the one achieved for models on the vertices of the hypercube.

cond-mat.dis-nn↗
arXiv2022

Quasi-invariance of Gaussian measures for the periodic Benjamin-Ono-BBM equation

The BBM equation is a Hamiltonian PDE which revealed to be a very interesting test-model to study the transformation property of Gaussian measures along the flow. In this paper we study the BBM equation with critical dispersion (which is a Benjamin-Ono type model). We prove that the image of the Gaussian measures supported on fractional Sobolev spaces of increasing regularity are absolutely continuous, but we cannot identify the density, for which new ideas are needed.

math.PR↗
arXiv2020

Long time behaviour of a local perturbation in the isotropic XY chain under periodic forcing

We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic problem can be mapped into a highly resonant Floquet-Schrödinger equation, for which, under a diophantine-like assumption on the frequency, we show the existence of a periodic solution. The proof is based on a KAM-type renormalisation. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing also at low frequencies.

math-ph↗
arXiv2020

Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS

The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on $L^2(\T)$ with covariance $[1+(-\D)^{s}]^{-1}$ under these transformations for any $s>\frac12$. This extends previous achievements by Nahmod, Ray-Bellet, Sheffield and Staffilani (2011) and Genovese, Lucà and Valeri (2018), who proved the result for integer values of the regularity parameter $s$.

math.PR↗
arXiv2020

Legendre Equivalences of Spherical Boltzmann Machines

We study either fully visible and restricted Boltzmann machines with sub-Gaussian random weights and spherical or Gaussian priors. We prove the free energies of the spherical and Gaussian models are related by a Legendre transformation. Incidentally our analysis brings also a new purely variational derivation of the free energy of the spherical models.

cond-mat.dis-nn↗
arXiv2019

Local Central Limit Theorem for a Random Walk Perturbed in One Point

We consider a symmetric random walk on the $ν$-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction to diffusive behaviour appears in any dimension along with a long-range correction in the one-dimensional case.

math.PR↗
arXiv2018

Invariant measures for the periodic derivative nonlinear Schrödinger equation

We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in $L^2$ and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of Gaussian measures on $L^2$.

math-ph↗
arXiv2017

Overlap synchronisation in multipartite random energy models

In a multipartite random energy model, made of a number of coupled GREMs, we determine the joint law of the overlaps in terms of the ones of the single GREMs. This provides the simplest example of the so-called overlap synchronisation.

cond-mat.dis-nn↗
arXiv2017

Phase transitions in Restricted Boltzmann Machines with generic priors

We study Generalised Restricted Boltzmann Machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as Generalised Hopfield models. We underline the role of the retrieval phase for both inference and learning processes and we show that retrieval is robust for a large class of weight and unit priors, beyond the standard Hopfield scenario. Furthermore we show how the paramagnetic phase boundary is directly related to the optimal size of the training set necessary for good generalisation in a teacher-student scenario of unsupervised learning.

cond-mat.dis-nn↗
arXiv2017

Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors

Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn corresponds to the one of a generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of retrieval capabilities, both at low and high load. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e. weight) distribution and spin (i.e. unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region is larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.

cond-mat.dis-nn↗