One-dimensional disordered Ising models by replica and cavity methods

Using a formalism based on the spectral decomposition of the replicated transfer matrix for disordered Ising models, we obtain several results that apply both to isolated one-dimensional systems and to locally tree-like graph and factor graph (p-spin) ensembles. We present exact analytical expressions, which can be efficiently approximated numerically, for many types of correlation functions and for the average free energies of open and closed finite chains. All the results achieved, with the exception of those involving closed chains, are then rigorously derived without replicas, using a probabilistic approach with the same flavour of cavity method

One-loop diagrams in the Random Euclidean Matching Problem

The matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community. Here we analyze the Euclidean version of the problem, i.e. the optimal matching problem between points randomly distributed on a $d$-dimensional Euclidean space, where the cost to minimize depends on the points' pairwise distances. Using Mayer's cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation. We give the diagrammatic rules for each term of the expansion, and we analyze in detail the one-loop diagrams. A characteristic feature of the theory, when diagrams are perturbatively computed around the mean field part of the action, is the vanishing of the mass at zero momentum. In the non-Euclidean case of uncorrelated costs instead, we predict and numerically verify an anomalous scaling for the sub-sub-leading correction to the asymptotic average cost.Comment: 17 pages, 7 figure

Generalized Approximate Survey Propagation for High-Dimensional Estimation

In Generalized Linear Estimation (GLE) problems, we seek to estimate a signal that is observed through a linear transform followed by a component-wise, possibly nonlinear and noisy, channel. In the Bayesian optimal setting, Generalized Approximate Message Passing (GAMP) is known to achieve optimal performance for GLE. However, its performance can significantly degrade whenever there is a mismatch between the assumed and the true generative model, a situation frequently encountered in practice. In this paper, we propose a new algorithm, named Generalized Approximate Survey Propagation (GASP), for solving GLE in the presence of prior or model mis-specifications. As a prototypical example, we consider the phase retrieval problem, where we show that GASP outperforms the corresponding GAMP, reducing the reconstruction threshold and, for certain choices of its parameters, approaching Bayesian optimal performance. Furthermore, we present a set of State Evolution equations that exactly characterize the dynamics of GASP in the high-dimensional limit

Scaling hypothesis for the Euclidean bipartite matching problem

We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large $N$ limit of the average cost in dimension $d=1,2$ and of the subleading correction in higher dimension. A non-trivial scaling exponent, $\gamma_d=\frac{d-2}{d}$, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension $d>2$.Comment: 11 page

Experience of Material Tinkering from Waste in the Year 3-Year 5 Primary School Age Range as an Introduction to Design and Sustainability

A possibility increasingly experimented in the field of design to improve end-of-life sustainability is integration of agro-waste into materials. Design can “upcycle” waste, offering perceptive and aesthetical acceptance to objects produced from it. On-field experimentations can foster awareness over the possibility to modify objects lifecycle, avoiding a “use-and-throw” perspective, and exploring the identity of materials in their prospected field of application. This will be aimed at creating a bond between user and objects, based both on function and on affection, also considering the modifications the material can undergo over time, making it adapted to different uses. This practice can have a significant educational value: in particular, this study moves from an experimentation carried out at July and September 2017 by design and materials engineering researchers, with several groups of 15 children, age 8-11, at Museo dei Bambini-Explora in Rome, investigating their “experience” and “creativity” on the topic. Conclusions are also drawn on possible modifications of the step-like procedure to introduce students of all ages to the knowledge of experimental method through the production of DIY bioplastics from waste. This “trial and error” procedure allows reflecting from a play-like point of view on aspects essential for the success of this operation, such as mode of coloration, effect of texture, possibility of obtaining curved or complex shapes, mechanical workability, optimal cooking and aromatization. The consequence of the exercise is customizing materials obtained from waste, with the idea of teaching how an expressive and functional success of the objects produced can make these “resilient” over time, therefore inherently sustainable. This would lead, through a combined and simple teaching of elements of experimental procedure and sustainability, to an awareness of their respective importance in design. The tool for this result is the development of DIY materials. The significance of this experience, which could be applied also in other age ranges, appeared to be to communicate the role of design to improve the perception of materials by transforming them into valuable objects. This proved to be better explained in the challenging case of waste, hence a substance bearing no longer any emotionalor functional relation with us, with which this “bond” needs to be rebuilt through an experiential path

Experience of Material Tinkering from Waste in the Year 3-Year 5 Primary School Age Range as an Introduction to Design and Sustainability

A possibility increasingly experimented in the field of design to improve end-of-life sustainability is integration of agro-waste into materials. Design can “upcycle” waste, offering perceptive and aesthetical acceptance to objects produced from it. On-field experimentations can foster awareness over the possibility to modify objects lifecycle, avoiding a “use-and-throw” perspective, and exploring the identity of materials in their prospected field of application. This will be aimed at creating a bond between user and objects, based both on function and on affection, also considering the modifications the material can undergo over time, making it adapted to different uses. This practice can have a significant educational value: in particular, this study moves from an experimentation carried out at July and September 2017 by design and materials engineering researchers, with several groups of 15 children, age 8-11, at Museo dei Bambini-Explora in Rome, investigating their “experience” and “creativity” on the topic. Conclusions are also drawn on possible modifications of the step-like procedure to introduce students of all ages to the knowledge of experimental method through the production of DIY bioplastics from waste. This “trial and error” procedure allows reflecting from a play-like point of view on aspects essential for the success of this operation, such as mode of coloration, effect of texture, possibility of obtaining curved or complex shapes, mechanical workability, optimal cooking and aromatization. The consequence of the exercise is customizing materials obtained from waste, with the idea of teaching how an expressive and functional success of the objects produced can make these “resilient” over time, therefore inherently sustainable. This would lead, through a combined and simple teaching of elements of experimental procedure and sustainability, to an awareness of their respective importance in design. The tool for this result is the development of DIY materials. The significance of this experience, which could be applied also in other age ranges, appeared to be to communicate the role of design to improve the perception of materials by transforming them into valuable objects. This proved to be better explained in the challenging case of waste, hence a substance bearing no longer any emotional or functional relation with us, with which this “bond” needs to be rebuilt through an experiential path.  Keywords: design education; material tinkering; sustainability for childre

The Exponential Capacity of Dense Associative Memories

Recent generalizations of the Hopfield model of associative memories are able to store a number $P$ of random patterns that grows exponentially with the number $N$ of neurons, $P=\exp(\alpha N)$. Besides the huge storage capacity, another interesting feature of these networks is their connection to the attention mechanism which is part of the Transformer architectures widely applied in deep learning. In this work, we consider a generic family of pattern ensembles, and thanks to the statistical mechanics analysis of an auxiliary Random Energy Model, we are able to provide exact asymptotic thresholds for the retrieval of a typical pattern, $\alpha_1$, and lower bounds for the maximum of the load $\alpha$ for which all patterns can be retrieved, $\alpha_c$. Additionally, we characterize the size of the basins of attractions. We discuss in detail the cases of Gaussian and spherical patterns, and show that they display rich and qualitatively different phase diagrams.Comment: preliminary version with no Appendi

Finite size corrections to disordered Ising models on Random Regular Graphs

We derive the analytical expression for the first finite size correction to the average free energy of disordered Ising models on random regular graphs. The formula can be physically interpreted as a weighted sum over all non self-intersecting loops in the graph, the weight being the free-energy shift due to the addition of the loop to an infinite tree