arXiv subjects

Alessandro Laio

Publications and source records attributed to Alessandro Laio.

At least 55 records · Page 3Linked to original sources

arXiv2021

Distributional Results for Model-Based Intrinsic Dimension Estimators

Modern datasets are characterized by a large number of features that may conceal complex dependency structures. To deal with this type of data, dimensionality reduction techniques are essential. Numerous dimensionality reduction methods rely on the concept of intrinsic dimension, a measure of the complexity of the dataset. In this article, we first review the TWO-NN model, a likelihood-based intrinsic dimension estimator recently introduced in the literature. The TWO-NN estimator is based on the statistical properties of the ratio of the distances between a point and its first two nearest neighbors, assuming that the points are a realization from an homogeneous Poisson point process. We extend the TWO-NN theoretical framework by providing novel distributional results of consecutive and generic ratios of distances. These distributional results are then employed to derive intrinsic dimension estimators, called Cride and Gride. These novel estimators are more robust to noisy measurements than the TWO-NN and allow the study of the evolution of the intrinsic dimension as a function of the scale used to analyze the dataset. We discuss the properties of the different estimators with the help of simulation scenarios.

stat.ME↗

arXiv2021

Statistically Unbiased Free Energy Estimates from Biased Simulations

Estimating the free energy in molecular simulation requires, implicitly or explicitly, counting how many times the system is observed in a finite region. If the simulation is biased by an external potential, the weight of the configurations within the region can vary significantly, and this can make the estimate numerically unstable. We introduce an approach to estimate the free energy as a simultaneous function of several collective variables starting from data generated in a statically-biased simulation. The approach exploits the property of a free energy estimator recently introduced by us which provides by construction the estimate in a region of infinitely small size. We show that this property allows removing the effect of the external bias in a simple and rigorous manner. The approach is validated on model systems for which the free energy is known analytically and on a small peptide for which the ground truth free energy is estimated in an independent unbiased run. In both cases the free energy obtained with our approach is an unbiased estimator of the ground-truth free energy, with an error whose magnitude is also predicted by the model.

physics.chem-ph↗

arXiv2021

Automatic topography of high-dimensional data sets by non-parametric Density Peak clustering

Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach providing this description in the form of a topography of the data, namely a human-readable chart of the probability density from which the data are harvested. The approach is based on an unsupervised extension of Density Peak clustering and a non-parametric density estimator that measures the probability density in the manifold containing the data. This allows finding automatically the number and the height of the peaks of the probability density, and the depth of the "valleys" separating them. Importantly, the density estimator provides a measure of the error, which allows distinguishing genuine density peaks from density fluctuations due to finite sampling. The approach thus provides robust and visual information about the density peaks' height, their statistical reliability, and their hierarchical organization, offering a conceptually powerful extension of the standard clustering partitions. We show that this framework is particularly useful in the analysis of complex data sets.

stat.ML↗

arXiv2021

Dynamical Landscape and Multistability of a Climate Model

We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model and analyze their interplay. First, drawing from the theory of quasipotentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states, and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, specifically manifold learning, we characterize the data landscape of the simulation output to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two climate models we consider. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth's climate.

physics.ao-ph↗

arXiv2020

Proton strings and rings in atypical nucleation of ferroelectricity in ice

Ordinary ice has a proton-disordered phase which is kinetically metastable, unable to reach, spontaneously, the ferroelectric (FE) ground state at low temperature where a residual Pauling entropy persists. Upon light doping with KOH at low temperature, the transition to FE ice takes place, but its microscopic mechanism still needs clarification. We introduce a lattice model based on dipolar interactions plus a competing, frustrating term that enforces the ice rule (IR). In the absence of IR-breaking defects, standard Monte Carlo (MC) simulation leaves this ice model stuck in a state of disordered proton ring configurations with the correct Pauling entropy. A replica exchange accelerated MC sampling strategy succeeds, without open path moves, interfaces, or off-lattice configurations, in equilibrating this defect-free ice, reaching its low-temperature FE order through a well-defined first-order phase transition. When proton vacancies mimicking the KOH impurities are planted into the IR-conserving lattice, they enable standard MC simulation to work, revealing the kinetics of evolution of ice from proton disorder to partial FE order below the transition temperature. Replacing ordinary nucleation, each impurity opens up a proton ring generating a linear string, an actual FE hydrogen bond wire that expands with time. Reminiscent of those described for spin ice, these impurity-induced strings are proposed to exist in doped water ice too, where IRs are even stronger. The emerging mechanism yields a dependence of the long-time FE order fraction upon dopant concentration, and upon quenching temperature, that compares favorably with that known in real-life KOH doped ice.

cond-mat.mtrl-sci↗

arXiv2020

Data segmentation based on the local intrinsic dimension

One of the founding paradigms of machine learning is that a small number of variables is often sufficient to describe high-dimensional data. The minimum number of variables required is called the intrinsic dimension (ID) of the data. Contrary to common intuition, there are cases where the ID varies within the same data set. This fact has been highlighted in technical discussions, but seldom exploited to analyze large data sets and obtain insight into their structure. Here we develop a robust approach to discriminate regions with different local IDs and segment the points accordingly. Our approach is computationally efficient and can be proficiently used even on large data sets. We find that many real-world data sets contain regions with widely heterogeneous dimensions. These regions host points differing in core properties: folded vs unfolded configurations in a protein molecular dynamics trajectory, active vs non-active regions in brain imaging data, and firms with different financial risk in company balance sheets. A simple topological feature, the local ID, is thus sufficient to achieve an unsupervised segmentation of high-dimensional data, complementary to the one given by clustering algorithms.

stat.ML↗

arXiv2020

Hierarchical nucleation in deep neural networks

Deep convolutional networks (DCNs) learn meaningful representations where data that share the same abstract characteristics are positioned closer and closer. Understanding these representations and how they are generated is of unquestioned practical and theoretical interest. In this work we study the evolution of the probability density of the ImageNet dataset across the hidden layers in some state-of-the-art DCNs. We find that the initial layers generate a unimodal probability density getting rid of any structure irrelevant for classification. In subsequent layers density peaks arise in a hierarchical fashion that mirrors the semantic hierarchy of the concepts. Density peaks corresponding to single categories appear only close to the output and via a very sharp transition which resembles the nucleation process of a heterogeneous liquid. This process leaves a footprint in the probability density of the output layer where the topography of the peaks allows reconstructing the semantic relationships of the categories.

cs.CV↗

arXiv2020

PARCE: Protocol for Amino acid Refinement through Computational Evolution

The in silico design of peptides and proteins as binders is useful for diagnosis and therapeutics due to their low adverse effects and major specificity. To select the most promising candidates, a key matter is to understand their interactions with protein targets. In this work, we present PARCE, an open source Protocol for Amino acid Refinement through Computational Evolution that implements an advanced and promising method for the design of peptides and proteins. The protocol performs a random mutation in the binder sequence, then samples the bound conformations using molecular dynamics simulations, and evaluates the protein-protein interactions from multiple scoring. Finally, it accepts or rejects the mutation by applying a consensus criterion based on binding scores. The procedure is iterated with the aim to explore efficiently novel sequences with potential better affinities toward their targets. We also provide a tutorial for running and reproducing the methodology.

q-bio.BM↗

arXiv2019

Intrinsic dimension of data representations in deep neural networks

Deep neural networks progressively transform their inputs across multiple processing layers. What are the geometrical properties of the representations learned by these networks? Here we study the intrinsic dimensionality (ID) of data-representations, i.e. the minimal number of parameters needed to describe a representation. We find that, in a trained network, the ID is orders of magnitude smaller than the number of units in each layer. Across layers, the ID first increases and then progressively decreases in the final layers. Remarkably, the ID of the last hidden layer predicts classification accuracy on the test set. These results can neither be found by linear dimensionality estimates (e.g., with principal component analysis), nor in representations that had been artificially linearized. They are neither found in untrained networks, nor in networks that are trained on randomized labels. This suggests that neural networks that can generalize are those that transform the data into low-dimensional, but not necessarily flat manifolds.

cs.LG↗

arXiv2018

Estimating the intrinsic dimension of datasets by a minimal neighborhood information

Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number of coordinates. Such manifold is generally twisted and curved, in addition points on it will be non-uniformly distributed: two factors that make the identification of the ID and its exploitation really hard. Here we propose a new ID estimator using only the distance of the first and the second nearest neighbor of each point in the sample. This extreme minimality enables us to reduce the effects of curvature, of density variation, and the resulting computational cost. The ID estimator is theoretically exact in uniformly distributed datasets, and provides consistent measures in general. When used in combination with block analysis, it allows discriminating the relevant dimensions as a function of the block size. This allows estimating the ID even when the data lie on a manifold perturbed by a high-dimensional noise, a situation often encountered in real world data sets. We demonstrate the usefulness of the approach on molecular simulations and image analysis.

stat.ML↗

arXiv2017

METAGUI 3: a graphical user interface for choosing the collective variables in molecular dynamics simulations

Molecular dynamics (MD) simulations allow the exploration of the phase space of biopolymers through the integration of equations of motion of their constituent atoms. The analysis of MD trajectories often relies on the choice of collective variables (CVs) along which the dynamics of the system is projected. We developed a graphical user interface (GUI) for facilitating the interactive choice of the appropriate CVs. The GUI allows: defining interactively new CVs; partitioning the configurations into microstates characterized by similar values of the CVs; calculating the free energies of the microstates for both unbiased and biased (metadynamics) simulations; clustering the microstates in kinetic basins; visualizing the free energy landscape as a function of a subset of the CVs used for the analysis. A simple mouse click allows one to quickly inspect structures corresponding to specific points in the landscape.

physics.comp-ph↗

arXiv2017

Sampling molecular conformers in solution with quantum mechanical accuracy at a nearly molecular mechanics cost

We introduce a method to evaluate the relative populations of different conformers of molecular species in solution, aiming at quantum mechanical accuracy, while keeping the computational cost at a nearly molecular-mechanics level. This goal is achieved by combining long classical molecular-dynamics simulations to sample the free-energy landscape of the system, advanced clustering techniques to identify the most relevant conformers, and thermodynamic perturbation theory to correct the resulting populations, using quantum-mechanical energies from density-functional theory. A quantitative criterion for assessing the accuracy thus achieved is proposed. The resulting methodology is demonstrated in the specific case of cyanin (cyanidin-3-glucoside) in water solution.

physics.comp-ph↗

arXiv2016

Metadynamics Surfing on Topology Barriers: the $CP^{N-1}$ Case

As one approaches the continuum limit, $QCD$ systems, investigated via numerical simulations, remain trapped in sectors of field space with fixed topological charge. As a consequence the numerical studies of physical quantities may give biased results. The same is true in the case of two dimensional $CP^{N-1}$ models. In this paper we show that metadynamics, when used to simulate $CP^{N-1}$, allows to address efficiently this problem. By studying $CP^{20}$ we show that we are able to reconstruct the free energy of the topological charge $F(Q)$ and compute the topological susceptibility as a function of the coupling and of the volume. This is a very important physical quantity in studies of the dynamics of the $θ$ vacuum and of the axion. This method can in principle be extended to $QCD$ applications.

hep-lat↗

arXiv2016

Modelling the impact of financialization on agricultural commodity markets

We propose a stylized model of production and exchange in which long-term investors set their production decision over a horizon τ , the "time to produce", and are liquidity constrained, while financial investors trade over a much shorter horizon δ (<< τ ) and are therefore more duly informed on the exogenous shocks affecting the production output. The equilibrium solution proves that: (i) long-term producers modify their production decisions to anticipate the impact of short-term investors allocations on prices; (ii) short-term investments return a positive expected profit commensurate to the informational advantage. While the presence of financial investors improves the efficiency of risk allocation in the short-term and reduces price volatility, the model shows that the aggregate effect of commodity market financialization results in rising the volatility of both farms' default risk and production output.

q-fin.MF↗

arXiv2014

Shape and area fluctuation effects on nucleation theory

In standard nucleation theory, the nucleation process is characterized by computing $ΔΩ(V)$, the reversible work required to form a cluster of volume $V$ of the stable phase inside the metastable mother phase. However, other quantities besides the volume could play a role in the free energy of cluster formation, and this will in turn affect the nucleation barrier and the shape of the nucleus. Here we exploit our recently introduced mesoscopic theory of nucleation to compute the free energy cost of a nearly-spherical cluster of volume $V$ and a fluctuating surface area $A$, whereby the maximum of $ΔΩ(V)$ is replaced by a saddle point in $ΔΩ(V,A)$. Compared to the simpler theory based on volume only, the barrier height of $ΔΩ(V,A)$ at the transition state is systematically larger by a few $k_BT$. More importantly, we show that, depending on the physical situation, the most probable shape of the nucleus may be highly non spherical, even when the surface tension and stiffness of the model are isotropic. Interestingly, these shape fluctuations do not influence or modify the standard Classical Nucleation Theory manner of extracting the interface tension from the logarithm of the nucleation rate near coexistence.

cond-mat.stat-mech↗

arXiv2013

A fingerprint of surface-tension anisotropy in the free-energy cost of nucleation

We focus on the Gibbs free energy $ΔG$ for nucleating a droplet of the stable phase (e.g. solid) inside the metastable parent phase (e.g. liquid), close to the first-order transition temperature. This quantity is central to the theory of homogeneous nucleation, since it superintends the nucleation rate. We recently introduced a field theory describing the dependence of $ΔG$ on the droplet volume $V$, taking into account besides the microscopic fuzziness of the droplet-parent interface, also small fluctuations around the spherical shape whose effect, assuming isotropy, was found to be a characteristic logarithmic term. Here we extend this theory, introducing the effect of anisotropy in the surface tension, and show that in the limit of strong anisotropy $ΔG(V)$ once more develops a term logarithmic on $V$, now with a prefactor of opposite sign with respect to the isotropic case. Based on this result, we argue that the geometrical shape that large solid nuclei mostly prefer could be inferred from the prefactor of the logarithmic term in the droplet free energy, as determined from the optimization of its near-coexistence profile.

cond-mat.stat-mech↗

arXiv2012

Systematic Improvement of Classical Nucleation Theory

We reconsider the applicability of classical nucleation theory (CNT) to the calculation of the free energy of solid cluster formation in a liquid and its use to the evaluation of interface free energies from nucleation barriers. Using two different freezing transitions (hard spheres and NaCl) as test cases, we first observe that the interface-free-energy estimates based on CNT are generally in error. As successive refinements of nucleation-barrier theory, we consider corrections due to a non-sharp solid-liquid interface and to a non-spherical cluster shape. Extensive calculations for the Ising model show that corrections due to a non-sharp and thermally fluctuating interface account for the barrier shape with excellent accuracy. The experimental solid nucleation rates that are measured in colloids are better accounted for by these non-CNT terms, whose effect appears to be crucial in the interpretation of data and in the extraction of the interface tension from them.

cond-mat.stat-mech↗

arXiv2011

Finite temperature properties of clusters by replica exchange metadynamics: the water nonamer

We introduce an approach for the accurate calculation of thermal properties of classical nanoclusters. Based on a recently developed enhanced sampling technique, replica exchange metadynamics, the method yields the true free energy of each relevant cluster structure, directly sampling its basin and measuring its occupancy in full equilibrium. All entropy sources, whether vibrational, rotational anharmonic and especially configurational -- the latter often forgotten in many cluster studies -- are automatically included. For the present demonstration we choose the water nonamer (H2O)9, an extremely simple cluster which nonetheless displays a sufficient complexity and interesting physics in its relevant structure spectrum. Within a standard TIP4P potential description of water, we find that the nonamer second relevant structure possesses a higher configurational entropy than the first, so that the two free energies surprisingly cross for increasing temperature.

cond-mat.stat-mech↗