Research talks
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Malago, Luigi - A review of different geometries for the training of neural networks
http://library.cirm-math.fr/Record.htm?record=19283006124910012889
Neural networks consist of a variegate class of computational models, used in machine learning for both supervised and unsupervised learning. Several topologies of networks have been proposed in the literature, since the preliminary work from the late 50s, including models based on undirected probabilistic graphical models, such as (Restricted) Boltzmann Machines, and on multi-layer feed-forward computational graphs. The training of a neural network is usually performed by the minimization of a cost function, such as the negative log-likelihood. During the talk we will review alternative geometries used to describe the space of the functions encoded by a neural network, parametrized by its connection weights, and the implications on the optimization of the cost function during training, from the perspective of Riemannian optimization. In the first part of the presentation, we will introduce a probabilistic interpretation for neural networks, which goes back to the work of Amari and coauthors from the 90s, and which is based on the use of the Fisher-Rao metric studied in Information Geometry. In this framework, the weights of a Boltzmann Machine, and similarly for feed-forward neural networks, are interpreted as the parameters of a (joint) statistical model for the observed, and possibly latent, variables. In the second part of the talk, we will review other approaches, motivated by invariant principles in neural networks and not explicitly based on probabilistic models, to the definition of alternative geometries for the space of the parameters of a neural network. The use of alternative non-Euclidean geometries has direct impact on the training algorithms, indeed by modeling the space of the functions associated to a neural network as a Riemannian manifold determines a dependence of the gradient on the choice of metric tensor. We conclude the presentation by reviewing some recently proposed training algorithm for neural networks, based on Riemannian optimization algorithms.Malago, LuigiMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283006124910012889Pennec, Xavier Trouvé, Alain - Minicourse shape spaces and geometric statistics
http://library.cirm-math.fr/Record.htm?record=19283002124910012849
Pennec, Xavier ; Trouvé, AlainMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283002124910012849Zyczkowski, Karol - Geometry of quantum entanglement
http://library.cirm-math.fr/Record.htm?record=19283000124910012829
Zyczkowski, KarolMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283000124910012829Bensoam, Joël - Multisympletic geometry and covariant formalism for mechanical systems with a Lie group as configuration space
http://library.cirm-math.fr/Record.htm?record=19283097124910012799
Bensoam, JoëlMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283097124910012799Madiman, Mokshay - Minicourse on information-theoretic geometry of metric measure
http://library.cirm-math.fr/Record.htm?record=19283091124910012739
Madiman, MokshayMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283091124910012739Gobet, Emmanuel - Forward and backward simulation of Euler scheme
http://library.cirm-math.fr/Record.htm?record=19283099124910012719
We analyse how reverting Random Number Generator can be efficiently used to save memory in solving dynamic programming equation. For SDEs, it takes the form of forward and backward Euler scheme. Surprisingly the error induced by time reversion is of order 1.Gobet, EmmanuelMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283099124910012719Mertz, Laurent - Stochastic variational inequalities for random mechanics
http://library.cirm-math.fr/Record.htm?record=19283086124910012689
The mathematical framework of variational inequalities is a powerful tool to model problems arising in mechanics such as elasto-plasticity where the physical laws change when some state variables reach a certain threshold [1]. Somehow, it is not surprising that the models used in the literature for the hysteresis effect of non-linear elasto-plastic oscillators submitted to random vibrations [2] are equivalent to (finite dimensional) stochastic variational inequalities (SVIs) [3]. This presentation concerns (a) cycle properties of a SVI modeling an elasto-perfectly-plastic oscillator excited by a white noise together with an application to the risk of failure [4,5]. (b) a set of Backward Kolmogorov equations for computing means, moments and correlation [6]. (c) free boundary value problems and HJB equations for the control of SVIs. For engineering applications, it is related to the problem of critical excitation [7]. This point concerns what we are doing during the CEMRACS research project. (d) (if time permits) on-going research on the modeling of a moving plate on turbulent convection [8]. This is a mixture of joint works and / or discussions with, amongst others, A. Bensoussan, L. Borsoi, C. Feau, M. Huang, M. Laurière, G. Stadler, J. Wylie, J. Zhang and J.Q. Zhong.Mertz, LaurentMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283086124910012689Laurière, Mathieu - Mean field type control with congestion
http://library.cirm-math.fr/Record.htm?record=19283963124910011459
The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. They describe respectively the evolution of the distribution of the agents' states and the evolution of the value function. Since it comes from a control problem, this PDE system differs in general from the one arising in mean field games.
Recently, this kind of model has been applied to crowd dynamics. More precisely, in this talk we will be interested in modeling congestion effects: the agents move but try to avoid very crowded regions. One way to take into account such effects is to let the cost of displacement increase in the regions where the density of agents is large. The cost may depend on the density in a non-local or in a local way. We will present one class of models for each case and study the associated PDE systems. The first one has classical solutions whereas the second one has weak solutions. Numerical results based on the Newton algorithm and the Augmented Lagrangian method will be presented.
This is joint work with Yves Achdou.Laurière, MathieuMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283963124910011459Silva Álvarez, Francisco José - On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications
http://library.cirm-math.fr/Record.htm?record=19283960124910011429
In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the non-negativity of the solution, conserves the mass and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. This convergence result is proved by assuming only that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, under these assumptions, we obtain a new proof of existence of solutions for such equations.
We apply our results to several examples, including Mean Field Games systems and variations of the Hughes model for pedestrian dynamics.Silva Álvarez, Francisco JoséMulti anglehttp://library.cirm-math.fr/Record.htm?record=19283960124910011429Zubelli, Jorge P. - Project evaluation under uncertainty
http://library.cirm-math.fr/Record.htm?record=19283957124910011399
Industrial strategic decisions have evolved tremendously in the last decades towards a higher degree of quantitative analysis. Such decisions require taking into account a large number of uncertain variables and volatile scenarios, much like financial market investments. Furthermore, they can be evaluated by comparing to portfolios of investments in financial assets such as in stocks, derivatives and commodity futures. This revolution led to the development of a new field of managerial science known as Real Options.
The use of Real Option techniques incorporates also the value of flexibility and gives a broader view of many business decisions that brings in techniques from quantitative finance and risk management. Such techniques are now part of the decision making process of many corporations and require a substantial amount of mathematical background. Yet, there has been substantial debate concerning the use of risk neutral pricing and hedging arguments to the context of project evaluation. We discuss some alternatives to risk neutral pricing that could be suitable to evaluation of projects in a realistic context with special attention to projects dependent on commodities and non-hedgeable uncertainties. More precisely, we make use of a variant of the hedged Monte-Carlo method of Potters, Bouchaud and Sestovic to tackle strategic decisions. Furthermore, we extend this to different investor risk profiles. This is joint work with Edgardo Brigatti, Felipe Macias, and Max O. de Souza.
If time allows we shall also discuss the situation when the historical data for the project evaluation is very limited and we can make use of certain symmetries of the problem to perform (with good estimates) a nonintrusive stratified resampling of the data. This is joint work with E. Gobet and G. Liu.Zubelli, Jorge P.Multi anglehttp://library.cirm-math.fr/Record.htm?record=19283957124910011399