arXiv subjects

Gabriel Turinici

Publications and source records attributed to Gabriel Turinici.

At least 37 records · Page 2Linked to original sources

arXiv2024

Transformer for Times Series: an Application to the S&P500

The transformer models have been extensively used with good results in a wide area of machine learning applications including Large Language Models and image generation. Here, we inquire on the applicability of this approach to financial time series. We first describe the dataset construction for two prototypical situations: a mean reverting synthetic Ornstein-Uhlenbeck process on one hand and real S&P500 data on the other hand. Then, we present in detail the proposed Transformer architecture and finally we discuss some encouraging results. For the synthetic data we predict rather accurately the next move, and for the S&P500 we get some interesting results related to quadratic variation and volatility prediction.

cs.AI↗

arXiv2023

Diversity in deep generative models and generative AI

The decoder-based machine learning generative algorithms such as Generative Adversarial Networks (GAN), Variational Auto-Encoders (VAE), Transformers show impressive results when constructing objects similar to those in a training ensemble. However, the generation of new objects builds mainly on the understanding of the hidden structure of the training dataset followed by a sampling from a multi-dimensional normal variable. In particular each sample is independent from the others and can repeatedly propose same type of objects. To cure this drawback we introduce a kernel-based measure quantization method that can produce new objects from a given target measure by approximating it as a whole and even staying away from elements already drawn from that distribution. This ensures a better diversity of the produced objects. The method is tested on classic machine learning benchmarks.

cs.CV↗

arXiv2023

Deep Conditional Measure Quantization

Quantization of a probability measure means representing it with a finite set of Dirac masses that approximates the input distribution well enough (in some metric space of probability measures). Various methods exists to do so, but the situation of quantizing a conditional law has been less explored. We propose a method, called DCMQ, involving a Huber-energy kernel-based approach coupled with a deep neural network architecture. The method is tested on several examples and obtains promising results.

stat.ML↗

arXiv2023

A cell-based population control of Monte Carlo particles for the global variance reduction for transport equations

We present a population control method with sampling and regulation steps for Monte Carlo particles involved in the numerical simulation of a transport equation. We recall in the first section the difficulties related to the variance reduction methods in the general framework of transport equations; we continue with a brief presentation of the mathematical tools invoked when solving the radiative transport equations and we focus on the importance of the emission and control of existing Monte Carlo particles. The next part discusses several novel methods based on the cell-based population control method proposed earlier. To this end, we analyze theoretically two types of splitting: one is conservative in energy (at the particle level) and the other is not. Thanks to these results, a new algorithm is introduced that uses the cell-based population control method and a spatial distribution. A numerical comparison of the different types of splitting is proposed in a simplified framework, then the various algorithms presented are compared against two benchmarks : the propagation of a Marshak wave and the propagation of two waves having different intensity and speed scales. To carry out these last tests, we use the multi-physics code FCI2.

math.NA↗

arXiv2022

Algorithms that get old : the case of generative deep neural networks

Generative deep neural networks used in machine learning, like the Variational Auto-Encoders (VAE), and Generative Adversarial Networks (GANs) produce new objects each time when asked to do so with the constraint that the new objects remain similar to some list of examples given as input. However, this behavior is unlike that of human artists that change their style as time goes by and seldom return to the style of the initial creations. We investigate a situation where VAEs are used to sample from a probability measure described by some empirical dataset. Based on recent works on Radon-Sobolev statistical distances, we propose a numerical paradigm, to be used in conjunction with a generative algorithm, that satisfies the two following requirements: the objects created do not repeat and evolve to fill the entire target probability distribution.

stat.ML↗

arXiv2022

COVID-19 adaptive humoral immunity models: weakly neutralizing versus antibody-disease enhancement scenarios

The interplay between the virus, infected cells and the immune responses to SARS-CoV-2 is still under debate. Extending the basic model of viral dynamics we propose here a formal approach to describe the neutralizing versus weakly (or non-)neutralizing scenarios and compare with the possible effects of antibody-dependent enhancement (ADE). The theoretical model is consistent with data available from the literature; we show that weakly neutralizing antibodies or ADE can both give rise to either final virus clearance or disease progression, but the immuno-dynamic is different in each case. Given that a significant part of the world population is already naturally immunized or vaccinated, we also discuss the implications on secondary infections infections following vaccination or in presence of immune system dysfunctions.

math.DS↗

arXiv2021

Radon Sobolev Variational Auto-Encoders

The quality of generative models (such as Generative adversarial networks and Variational Auto-Encoders) depends heavily on the choice of a good probability distance. However some popular metrics like the Wasserstein or the Sliced Wasserstein distances, the Jensen-Shannon divergence, the Kullback-Leibler divergence, lack convenient properties such as (geodesic) convexity, fast evaluation and so on. To address these shortcomings, we introduce a class of distances that have built-in convexity. We investigate the relationship with some known paradigms (sliced distances - a synonym for Radon distances -, reproducing kernel Hilbert spaces, energy distances). The distances are shown to possess fast implementations and are included in an adapted Variational Auto-Encoder termed Radon Sobolev Variational Auto-Encoder (RS-VAE) which produces high quality results on standard generative datasets. Keywords: Variational Auto-Encoder; Generative model; Sobolev spaces; Radon Sobolev Variational Auto-Encoder;

cs.LG↗

arXiv2021

Convergence dynamics of Generative Adversarial Networks: the dual metric flows

Fitting neural networks often resorts to stochastic (or similar) gradient descent which is a noise-tolerant (and efficient) resolution of a gradient descent dynamics. It outputs a sequence of networks parameters, which sequence evolves during the training steps. The gradient descent is the limit, when the learning rate is small and the batch size is infinite, of this set of increasingly optimal network parameters obtained during training. In this contribution, we investigate instead the convergence in the Generative Adversarial Networks used in machine learning. We study the limit of small learning rate, and show that, similar to single network training, the GAN learning dynamics tend, for vanishing learning rate to some limit dynamics. This leads us to consider evolution equations in metric spaces (which is the natural framework for evolving probability laws) that we call dual flows. We give formal definitions of solutions and prove the convergence. The theory is then applied to specific instances of GANs and we discuss how this insight helps understand and mitigate the mode collapse. Keywords: GAN; metric flow; generative network

stat.ML↗

arXiv2020

Architectures of epidemic models: accommodating constraints from empirical and clinical data

Deterministic compartmental models have been used extensively in modeling epidemic propagation. These models are required to fit available data and numerical procedures are often implemented to this end. But not every model architecture is able to fit the data because the structure of the model imposes hard constraints on the solutions. We investigate in this work two such situations: first the distribution of transition times from a compartment to another may impose a variable number of intermediary states; secondly, a non-linear relationship between time-dependent measures of compartments sizes may indicate the need for structurations (i.e., considering several groups of individuals of heterogeneous characteristics).

q-bio.PE↗

arXiv2020

Heterogeneous social interactions and the COVID-19 lockdown outcome in a multi-group SEIR model

We study variants of the SEIR model for interpreting some qualitative features of the statistics of the Covid-19 epidemic in France. Standard SEIR models distinguish essentially two regimes: either the disease is controlled and the number of infected people rapidly decreases, or the disease spreads and contaminates a significant fraction of the population until herd immunity is achieved. After lockdown, at first sight it seems that social distancing is not enough to control the outbreak. We discuss here a possible explanation, namely that the lockdown is creating social heterogeneity: even if a large majority of the population complies with the lockdown rules, a small fraction of the population still has to maintain a normal or high level of social interactions, such as health workers, providers of essential services, etc. This results in an apparent high level of epidemic propagation as measured through re-estimations of the basic reproduction ratio. However, these measures are limited to averages, while variance inside the population plays an essential role on the peak and the size of the epidemic outbreak and tends to lower these two indicators. We provide theoretical and numerical results to sustain such a view.

q-bio.PE↗

arXiv2020

Contact rate epidemic control of COVID-19: an equilibrium view

We consider the control of the COVID-19 pandemic through a standard SIR compartmental model. This control is induced by the aggregation of individuals' decisions to limit their social interactions: when the epidemic is ongoing, an individual can diminish his/her contact rate in order to avoid getting infected, but this effort comes at a social cost. If each individual lowers his/her contact rate, the epidemic vanishes faster, but the effort cost may be high. A Mean Field Nash equilibrium at the population level is formed, resulting in a lower effective transmission rate of the virus. We prove theoretically that equilibrium exists and compute it numerically. However, this equilibrium selects a sub-optimal solution in comparison to the societal optimum (a centralized decision respected fully by all individuals), meaning that the cost of anarchy is strictly positive. We provide numerical examples and a sensitivity analysis, as well as an extension to a SEIR compartmental model to account for the relatively long latent phase of the COVID-19 disease. In all the scenarii considered, the divergence between the individual and societal strategies happens both before the peak of the epidemic, due to individuals' fears, and after, when a significant propagation is still underway.

physics.soc-ph↗

arXiv2020

Stochastic Runge-Kutta methods and adaptive SGD-G2 stochastic gradient descent

The minimization of the loss function is of paramount importance in deep neural networks. On the other hand, many popular optimization algorithms have been shown to correspond to some evolution equation of gradient flow type. Inspired by the numerical schemes used for general evolution equations we introduce a second order stochastic Runge Kutta method and show that it yields a consistent procedure for the minimization of the loss function. In addition it can be coupled, in an adaptive framework, with a Stochastic Gradient Descent (SGD) to adjust automatically the learning rate of the SGD, without the need of any additional information on the Hessian of the loss functional. The adaptive SGD, called SGD-G2, is successfully tested on standard datasets.

cs.LG↗

arXiv2019

Stochastic learning control of inhomogeneous quantum ensembles

In quantum control, the robustness with respect to uncertainties in the system's parameters or driving field characteristics is of paramount importance and has been studied theoretically, numerically and experimentally. We test in this paper stochastic search procedures (Stochastic gradient descent and the Adam algorithm) that sample, at each iteration, from the distribution of the parameter uncertainty, as opposed to previous approaches that use a fixed grid. We show that both algorithms behave well with respect to benchmarks and discuss their relative merits. In addition the methodology allows to address high dimensional parameter uncertainty; we implement numerically, with good results, a 3D and a 6D case.

math.NA↗

arXiv2012

Control through operators for quantum chemistry

We consider the problem of operator identification in quantum control. The free Hamiltonian and the dipole moment are searched such that a given target state is reached at a given time. A local existence result is obtained. As a by-product, our works reveals necessary conditions on the laser field to make the identification feasible. In the last part of this work, some algorithms are proposed to compute effectively these operators.

math.OC↗

arXiv2012

Critical points of the optimal quantum control landscape: a propagator approach

Numerical and experimental realizations of quantum control are closely connected to the properties of the mapping from the control to the unitary propagator. For bilinear quantum control problems, no general results are available to fully determine when this mapping is singular or not. In this paper we give suffcient conditions, in terms of elements of the evolution semigroup, for a trajectory to be non-singular. We identify two lists of "way-points" that, when reached, ensure the non-singularity of the control trajectory. It is found that under appropriate hypotheses one of those lists does not depend on the values of the coupling operator matrix.

math.NA↗

arXiv2011

Hamiltonian identification through enhanced observability utilizing quantum control

This paper considers Hamiltonian identification for a controllable quantum system with non-degenerate transitions and a known initial state. We assume to have at our disposal a single scalar control input and the population measure of only one state at an (arbitrarily large) final time T. We prove that the quantum dipole moment matrix is locally observable in the following sense: for any two close but distinct dipole moment matrices, we construct discriminating controls giving two different measurements. Such discriminating controls are constructed to have three well defined temporal components, as inspired by Ramsey interferometry. This result suggests that what may appear at first to be very restrictive measurements are actually rich for identification, when combined with well designed discriminating controls, to uniquely identify the complete dipole moment of such systems. The assessment supports the employment of quantum control as a promising means to achieve high quality identification of a Hamiltonian.

quant-ph↗

arXiv2010

A monotonic method for solving nonlinear optimal control problems

Initially introduced in the framework of quantum control, the so-called "monotonic algorithms" have demonstrated excellent numerical performance when dealing with bilinear optimal control problems. This paper presents a unified formulation that can be applied to more nonlinear settings compatible with the hypothesis detailed below. In this framework, we show that the well-posedness of the general algorithm is related to a nonlinear evolution equation. We prove the existence of the solution to this equation and give important properties of the optimal control functional. Finally we show how the algorithm works for selected models from the literature and compare it with the gradient algorithm.

math.OC↗

arXiv2010

Analysis of the Toolkit method for the time-dependant Schrödinger equation

The goal of this paper is to provide an analysis of the "toolkit" method used in the numerical approximation of the time-dependent Schrödinger equation. The "toolkit" method is based on precomputation of elementary propagators and was seen to be very efficient in the optimal control framework. Our analysis shows that this method provides better results than the second order Strang operator splitting. In addition, we present two improvements of the method in the limit of low and large intensity control fields.

math.OC↗