Paris Dauphine University

About: Paris Dauphine University is a education organization based out in Paris, France. It is known for research contribution in the topics: Population & Approximation algorithm. The organization has 1766 authors who have published 6909 publications receiving 162747 citations. The organization is also known as: Paris Dauphine & Dauphine.

Model Selection with Low Complexity Priors

Abstract: Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied regularization models with various types of low-dimensional structures. In such settings, the general approach is to solve a regularized optimization problem, which combines a data fidelity term and some regularization penalty that promotes the assumed low-dimensional/simple structure. This paper provides a general framework to capture this low-dimensional structure through what we coin partly smooth functions relative to a linear manifold. These are convex, non-negative, closed and finite-valued functions that will promote objects living on low-dimensional subspaces. This class of regularizers encompasses many popular examples such as the L1 norm, L1-L2 norm (group sparsity), as well as several others including the Linfty norm. We also show that the set of partly smooth functions relative to a linear manifold is closed under addition and pre-composition by a linear operator, which allows to cover mixed regularization, and the so-called analysis-type priors (e.g. total variation, fused Lasso, finite-valued polyhedral gauges). Our main result presents a unified sharp analysis of exact and robust recovery of the low-dimensional subspace model associated to the object to recover from partial measurements. This analysis is illustrated on a number of special and previously studied cases, and on an analysis of the performance of Linfty regularization in a compressed sensing scenario.

Weighted Node Coloring: When Stable Sets Are Expensive

Abstract: A version of weighted coloring of a graph is introduced: each node v of a graph G = (V, E) is provided with a positive integer weight w(v) and the weight of a stable set S of G is w(S) = max{w(v) : v ? V ? S} A k-coloring S = (S1, , Sk) of G is a partition of V into k stable sets S1, , Sk and the weight of S is w(S1) + + w(Sk) The objective then is to find a coloring S = (S1, , Sk) of G such that w(S1) + + w(Sk) is minimized Weighted node coloring is NP-hard for general graphs (as generalization of the node coloring problem) We prove here that the associated decision problems are NP-complete for bipartite graphs, for line-graphs of bipartite graphs and for split graphs We present approximation results for general graphs For the other families of graphs dealt, properties of optimal solutions are discussed and complexity and approximability results are presented

Near fairness in matroids

Abstract: This article deals with the fair allocation of indivisible goods and its generalization to matroids. The notions of fairness under consideration are equitability, proportionality and envy-freeness. It is long known that some instances fail to admit a fair allocation. However, an almost fair solution may exist if an appropriate relaxation of the fairness condition is adopted. This article deals with a matroid problem which comprises the allocation of indivisible goods as a special case. It is to find a base of a matroid and to allocate it to a pool of agents. We first adapt the aforementioned fairness concepts to matroids. Next we propose a relaxed notion of fairness said to be near to fairness. Near fairness respects the fairness up to one element. We show that a nearly fair solution always exists and it can be constructed in polynomial time in the general context of matroids.

Progressive enlargement of ltrations and Backward SDEs with jumps

Abstract: This work deals with backward stochastic dierential equation (BSDE) with random marked jumps, and their applications to default risk. We show that these BSDEs are linked with Brownian BSDEs through the decomposition of processes with respect to the progressive enlargement of ltrations. We prove that the equations have solutions if the associated Brownian BSDEs have solutions. We also provide a uniqueness theorem for BSDEs with jumps by giving a comparison theorem based on the comparison for Brownian BSDEs. We give in particular some results for quadratic BSDEs. As applications, we study the pricing and the hedging of a European option in a market with a single jump, and the utility maximization problem in an incomplete market with a nite number of jumps.

Mean Field Game of Controls and An Application To Trade Crowding

Abstract: In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of "extended MFG", we hence provide generic results to address these "MFG of controls", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of "heterogenous preferences" (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can "learn" it day after day, observing others' behaviors.