arXiv subjects

Dimitrios Bachtis

Publications and source records attributed to Dimitrios Bachtis.

17 recordsLinked to original sources

arXiv2025

Modeling financial time series with $\phi^{4}$ quantum field theory

We use a $\phi^{4}$ quantum field theory with inhomogeneous couplings and explicit symmetry-breaking to model an ensemble of financial time series from the S$\&$P 500 index. The continuum nature of the $\phi^4$ theory avoids the inaccuracies that occur in Ising-based models which require a discretization of the time series. We demonstrate this using the example of the 2008 global financial crisis. The $\phi^{4}$ quantum field theory is expressive enough to reproduce the higher-order statistics such as the market kurtosis, which can serve as an indicator of possible market shocks. Accurate reproduction of high kurtosis is absent in binarized models. Therefore Ising models, despite being widely employed in econophysics, are incapable of fully representing empirical financial data, a limitation not present in the generalization of the $\phi^{4}$ scalar field theory. We then investigate the scaling properties of the $\phi^{4}$ machine learning algorithm and extract exponents which govern the behavior of the learned couplings (or weights and biases in ML language) in relation to the number of stocks in the model. Finally, we use our model to forecast the price changes of the AAPL, MSFT, and NVDA stocks. We conclude by discussing how the $\phi^{4}$ scalar field theory could be used to build investment strategies and the possible intuitions that the QFT operations of dimensional compactification and renormalization can provide for financial modelling.

q-fin.ST↗

arXiv2025

Cascade of phase transitions in the training of Energy-based models

In this paper, we investigate the feature encoding process in a prototypical energy-based generative model, the Restricted Boltzmann Machine (RBM). We start with an analytical investigation using simplified architectures and data structures, and end with numerical analysis of real trainings on real datasets. Our study tracks the evolution of the model's weight matrix through its singular value decomposition, revealing a series of phase transitions associated to a progressive learning of the principal modes of the empirical probability distribution. The model first learns the center of mass of the modes and then progressively resolve all modes through a cascade of phase transitions. We first describe this process analytically in a controlled setup that allows us to study analytically the training dynamics. We then validate our theoretical results by training the Bernoulli-Bernoulli RBM on real data sets. By using data sets of increasing dimension, we show that learning indeed leads to sharp phase transitions in the high-dimensional limit. Moreover, we propose and test a mean-field finite-size scaling hypothesis. This shows that the first phase transition is in the same universality class of the one we studied analytically, and which is reminiscent of the mean-field paramagnetic-to-ferromagnetic phase transition.

cs.LG↗

arXiv2024

Lattice $ϕ^{4}$ field theory as a multi-agent system of financial markets

We introduce a $ϕ^{4}$ lattice field theory with frustrated dynamics as a multi-agent system to reproduce stylized facts of financial markets such as fat-tailed distributions of returns and clustered volatility. Each lattice site, represented by a continuous degree of freedom, corresponds to an agent experiencing a set of competing interactions which influence its decision to buy or sell a given stock. These interactions comprise a cooperative term, which signifies that the agent should imitate the behavior of its neighbors, and a fictitious field, which compels the agent instead to conform with the opinion of the majority or the minority. To introduce the competing dynamics we exploit the Markov field structure to pursue a constructive decomposition of the $ϕ^{4}$ probability distribution which we recompose with a Ferrenberg-Swendsen acceptance or rejection sampling step. We then verify numerically that the multi-agent $ϕ^{4}$ field theory produces behavior observed on empirical data from the FTSE 100 London Stock Exchange index. We conclude by discussing how the presence of continuous degrees of freedom within the $ϕ^{4}$ lattice field theory enables a representational capacity beyond that possible with multi-agent systems derived from Ising models.

cond-mat.dis-nn↗

arXiv2024

Inverse renormalization group of spin glasses

We propose inverse renormalization group transformations to construct approximate configurations for lattice volumes that have not yet been accessed by supercomputers or large-scale simulations in the study of spin glasses. Specifically, starting from lattices of volume $V=8^{3}$ in the case of the three-dimensional Edwards-Anderson model we employ machine learning algorithms to construct rescaled lattices up to $V'=128^{3}$, which we utilize to extract two critical exponents. We conclude by discussing how to incorporate numerical exactness within inverse renormalization group methods of disordered systems, thus opening up the opportunity to explore a sustainable and energy-efficient generation of exact configurations for increasing lattice volumes without the use of dedicated supercomputers.

cond-mat.stat-mech↗

arXiv2024

Disordered Lattice Glass $ϕ^{4}$ Quantum Field Theory

We study numerically the three-dimensional $ϕ^{4}$ spin glass, a prototypical disordered and discretized Euclidean field theory that manifests inhomogeneities in space and time but considers a homogeneous squared mass and lambda terms. The $ϕ^{4}$ lattice glass field theory is a conceptual generalization of spin glasses to continuous degrees of freedom and we discuss the existence of a limit under which it formally reduces to the Edwards-Anderson model. By defining four variants of an order parameter which are suitable for continuous spin glasses, we verify numerically the emergence of an overlap in absence of the magnetization thus confirming the presence of a spin glass phase transition for a value of a critical squared mass. We conclude by discussing how the $ϕ^{4}$ spin glass can be utilized to address assumptions of complete homogeneity in space or time and how, in parallel to statistical physics, provides a suitable framework to investigate the nonperturbative dynamics of machine learning algorithms from the perspective of disordered lattice and constructive field theory.

hep-lat↗

arXiv2024

Generating configurations of increasing lattice size with machine learning and the inverse renormalization group

We review recent developments of machine learning algorithms pertinent to the inverse renormalization group, which was originally established as a generative numerical method by Ron-Swendsen-Brandt via the implementation of compatible Monte Carlo simulations. Inverse renormalization group methods enable the iterative generation of configurations for increasing lattice size without the critical slowing down effect. We discuss the construction of inverse renormalization group transformations with the use of convolutional neural networks and present applications in models of statistical mechanics, lattice field theory, and disordered systems. We highlight the case of the three-dimensional Edwards-Anderson spin glass, where the inverse renormalization group can be employed to construct configurations for lattice volumes that have not yet been accessed by dedicated supercomputers.

hep-lat↗

arXiv2024

Overlap renormalization group transformations for disordered systems

We establish a renormalization group approach which is implemented on the degrees of freedom defined by the overlap of two replicas to determine the critical fixed point and to extract four critical exponents for the phase transition of the three-dimensional Edwards-Anderson model. In addition, we couple the overlap order parameter to a fictitious field and introduce it within the two-replica Hamiltonian of the system to study its explicit symmetry-breaking with the renormalization group. Overlap transformations do not require a renormalization of the random couplings of a system to extract the critical exponents associated with the relevant variables of the renormalization group. We conclude by discussing the applicability of such transformations in the study of any phase transition which is fully characterized by an overlap order parameter.

cond-mat.stat-mech↗

arXiv2024

Reducing finite-size effects with reweighted renormalization group transformations

We combine histogram reweighting techniques with the two-lattice matching Monte Carlo renormalization group method to conduct computationally efficient calculations of critical exponents on systems with moderately small lattice sizes. The approach, which relies on the construction of renormalization group mappings between two systems of identical lattice size to partially eliminate finite-size effects, and the use of histogram reweighting to obtain computationally efficient results in extended regions of parameter space, is utilized to explicitly determine the renormalized coupling parameters of the two-dimensional $ϕ^{4}$ scalar field theory and to extract multiple critical exponents. We conclude by quantifying the computational benefits of the approach and discuss how reweighting opens up the opportunity to extend Monte Carlo renormalization group methods to systems with complex-valued actions.

hep-lat↗

arXiv2023

Phase Transitions in Particle Physics -- Results and Perspectives from Lattice Quantum Chromo-Dynamics

Phase transitions in a non-perturbative regime can be studied by ab initio Lattice Field Theory methods. The status and future research directions for LFT investigations of Quantum Chromo-Dynamics under extreme conditions are reviewed, including properties of hadrons and of the hypothesized QCD axion as inferred from QCD topology in different phases. We discuss phase transitions in strong interactions in an extended parameter space, and the possibility of model building for Dark Matter and Electro-Weak Symmetry Breaking. Methodological challenges are addressed as well, including new developments in Artificial Intelligence geared towards the identification of different phases and transitions.

hep-lat↗

arXiv2022

Quantum field theories, Markov random fields and machine learning

The transition to Euclidean space and the discretization of quantum field theories on spatial or space-time lattices opens up the opportunity to investigate probabilistic machine learning within quantum field theory. Here, we will discuss how discretized Euclidean field theories, such as the $ϕ^{4}$ lattice field theory on a square lattice, are mathematically equivalent to Markov fields, a notable class of probabilistic graphical models with applications in a variety of research areas, including machine learning. The results are established based on the Hammersley-Clifford theorem. We will then derive neural networks from quantum field theories and discuss applications pertinent to the minimization of the Kullback-Leibler divergence for the probability distribution of the $ϕ^{4}$ machine learning algorithms and other probability distributions.

cs.LG↗

arXiv2022

Inverse Renormalization Group in Quantum Field Theory

We propose inverse renormalization group transformations within the context of quantum field theory that produce the appropriate critical fixed point structure, give rise to inverse flows in parameter space, and evade the critical slowing down effect in calculations pertinent to criticality. Given configurations of the two-dimensional $ϕ^{4}$ scalar field theory on sizes as small as $V=8^{2}$, we apply the inverse transformations to produce rescaled systems of size up to $V'=512^{2}$ which we utilize to extract two critical exponents. We conclude by discussing how the approach is generally applicable to any method that successfully produces configurations from a statistical ensemble and how it can give novel insights into the structure of the renormalization group.

hep-lat↗

arXiv2021

Interpreting machine learning functions as physical observables

We propose to interpret machine learning functions as physical observables, opening up the possibility to apply "standard" statistical-mechanical methods to outputs from neural networks. This includes histogram reweighting and finite-size scaling, to analyse phase transitions quantitatively. In addition we incorporate predictive functions as conjugate variables coupled to an external field within the Hamiltonian of a system, allowing to induce order-disorder phase transitions in a novel manner. A noteworthy feature of this approach is that no knowledge of the symmetries in the Hamiltonian is required.

hep-lat↗

arXiv2021

Machine learning with quantum field theories

The precise equivalence between discretized Euclidean field theories and a certain class of probabilistic graphical models, namely the mathematical framework of Markov random fields, opens up the opportunity to investigate machine learning from the perspective of quantum field theory. In this contribution we will demonstrate, through the Hammersley-Clifford theorem, that the $ϕ^{4}$ scalar field theory on a square lattice satisfies the local Markov property and can therefore be recast as a Markov random field. We will then derive from the $ϕ^{4}$ theory machine learning algorithms and neural networks which can be viewed as generalizations of conventional neural network architectures. Finally, we will conclude by presenting applications based on the minimization of an asymmetric distance between the probability distribution of the $ϕ^{4}$ machine learning algorithms and target probability distributions.

cs.LG↗

arXiv2021

Quantum field-theoretic machine learning

We derive machine learning algorithms from discretized Euclidean field theories, making inference and learning possible within dynamics described by quantum field theory. Specifically, we demonstrate that the $ϕ^{4}$ scalar field theory satisfies the Hammersley-Clifford theorem, therefore recasting it as a machine learning algorithm within the mathematically rigorous framework of Markov random fields. We illustrate the concepts by minimizing an asymmetric distance between the probability distribution of the $ϕ^{4}$ theory and that of target distributions, by quantifying the overlap of statistical ensembles between probability distributions and through reweighting to complex-valued actions with longer-range interactions. Neural network architectures are additionally derived from the $ϕ^{4}$ theory which can be viewed as generalizations of conventional neural networks and applications are presented. We conclude by discussing how the proposal opens up a new research avenue, that of developing a mathematical and computational framework of machine learning within quantum field theory.

hep-lat↗

arXiv2021

Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration

We present a physical interpretation of machine learning functions, opening up the possibility to control properties of statistical systems via the inclusion of these functions in Hamiltonians. In particular, we include the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field within the Hamiltonian of a system. Results in the two-dimensional Ising model evidence that the field can induce an order-disorder phase transition by breaking or restoring the symmetry, in contrast with the field of the conventional order parameter which causes explicit symmetry breaking. The critical behavior is then studied by proposing a Hamiltonian-agnostic reweighting approach and forming a renormalization group mapping on quantities derived from the neural network. Accurate estimates of the critical point and of the critical exponents related to the operators that govern the divergence of the correlation length are provided. We conclude by discussing how the method provides an essential step toward bridging machine learning and physics.

hep-lat↗

arXiv2020

Extending machine learning classification capabilities with histogram reweighting

We propose the use of Monte Carlo histogram reweighting to extrapolate predictions of machine learning methods. In our approach, we treat the output from a convolutional neural network as an observable in a statistical system, enabling its extrapolation over continuous ranges in parameter space. We demonstrate our proposal using the phase transition in the two-dimensional Ising model. By interpreting the output of the neural network as an order parameter, we explore connections with known observables in the system and investigate its scaling behaviour. A finite size scaling analysis is conducted based on quantities derived from the neural network that yields accurate estimates for the critical exponents and the critical temperature. The method improves the prospects of acquiring precision measurements from machine learning in physical systems without an order parameter and those where direct sampling in regions of parameter space might not be possible.

cond-mat.stat-mech↗

arXiv2020

Mapping distinct phase transitions to a neural network

We demonstrate, by means of a convolutional neural network, that the features learned in the two-dimensional Ising model are sufficiently universal to predict the structure of symmetry-breaking phase transitions in considered systems irrespective of the universality class, order, and the presence of discrete or continuous degrees of freedom. No prior knowledge about the existence of a phase transition is required in the target system and its entire parameter space can be scanned with multiple histogram reweighting to discover one. We establish our approach in q-state Potts models and perform a calculation for the critical coupling and the critical exponents of the $ϕ^{4}$ scalar field theory using quantities derived from the neural network implementation. We view the machine learning algorithm as a mapping that associates each configuration across different systems to its corresponding phase and elaborate on implications for the discovery of unknown phase transitions.

cond-mat.stat-mech↗