In this exposition paper we present the optimal transport problem of Monge-Ampère-Kantorovitch(MAK in short)and its approximative entropical regularization.Contrary to the MAK optimal transport problem,the soluti...In this exposition paper we present the optimal transport problem of Monge-Ampère-Kantorovitch(MAK in short)and its approximative entropical regularization.Contrary to the MAK optimal transport problem,the solution of the entropical optimal transport problem is always unique,and is characterized by the Schrödinger system.The relationship between the Schrödinger system,the associated Bernstein process and the optimal transport was developed by Léonard[32,33](and by Mikami[39]earlier via an h-process).We present Sinkhorn’s algorithm for solving the Schrödinger system and the recent results on its convergence rate.We study the gradient descent algorithm based on the dual optimal question and prove its exponential convergence,whose rate might be independent of the regularization constant.This exposition is motivated by recent applications of optimal transport to different domains such as machine learning,image processing,econometrics,astrophysics etc..展开更多
The objective of this paper concerns at first the motivation and the method of Shor’s algorithm including remarks on quantum computing introducing an algorithmic description of the method.The corner stone of the Shor...The objective of this paper concerns at first the motivation and the method of Shor’s algorithm including remarks on quantum computing introducing an algorithmic description of the method.The corner stone of the Shor’s algorithm is the modular exponentiation that is themost computational component(in time and space).A linear depth unit based on phase estimation is introduced and a description of a generic version of a modular multiplier based on phases is introduced to build block of a gates to efficient modular exponentiation circuit.Our proposal includes numerical experiments achieved on both the IBM simulator using the Qiskit library and on quantum physical optimizers provided by IBM.The shor’s algorithm based on phase estimation succeeds in factoring integer numbers with more than 35 digits using circuits with about 100 qubits.展开更多
Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic Sobolev inequality in the radial case.We try to fill this gap here using different methods:Bobkov's ar...Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic Sobolev inequality in the radial case.We try to fill this gap here using different methods:Bobkov's argument and super-Poincaréinequalities,direct approach via L_(1)-logarithmic Sobolev inequalities.We also give various examples where the obtained bounds are quite sharp.Recent bounds obtained by Lee–Vempala in the log-concave bounded case are refined for radial measures.展开更多
文摘In this exposition paper we present the optimal transport problem of Monge-Ampère-Kantorovitch(MAK in short)and its approximative entropical regularization.Contrary to the MAK optimal transport problem,the solution of the entropical optimal transport problem is always unique,and is characterized by the Schrödinger system.The relationship between the Schrödinger system,the associated Bernstein process and the optimal transport was developed by Léonard[32,33](and by Mikami[39]earlier via an h-process).We present Sinkhorn’s algorithm for solving the Schrödinger system and the recent results on its convergence rate.We study the gradient descent algorithm based on the dual optimal question and prove its exponential convergence,whose rate might be independent of the regularization constant.This exposition is motivated by recent applications of optimal transport to different domains such as machine learning,image processing,econometrics,astrophysics etc..
文摘The objective of this paper concerns at first the motivation and the method of Shor’s algorithm including remarks on quantum computing introducing an algorithmic description of the method.The corner stone of the Shor’s algorithm is the modular exponentiation that is themost computational component(in time and space).A linear depth unit based on phase estimation is introduced and a description of a generic version of a modular multiplier based on phases is introduced to build block of a gates to efficient modular exponentiation circuit.Our proposal includes numerical experiments achieved on both the IBM simulator using the Qiskit library and on quantum physical optimizers provided by IBM.The shor’s algorithm based on phase estimation succeeds in factoring integer numbers with more than 35 digits using circuits with about 100 qubits.
基金Supported by ANR(Grant No.EFI ANR-17-CE40-0030)。
文摘Poincaréinequality has been studied by Bobkov for radial measures,but few are known about the logarithmic Sobolev inequality in the radial case.We try to fill this gap here using different methods:Bobkov's argument and super-Poincaréinequalities,direct approach via L_(1)-logarithmic Sobolev inequalities.We also give various examples where the obtained bounds are quite sharp.Recent bounds obtained by Lee–Vempala in the log-concave bounded case are refined for radial measures.
