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Convex Analysisの二,三の進展について

2005년 대한 생화학·분자생물학회 하계학술대회를 다녀와서

Associate DIETRICH BELITZ, University of Oregon Editors: Condensed Matter Physics (Theoretical) J. IGNACIO CIRAC, Max-Planck-Institut für Quantenoptik Quantum Information RAYMOND E. GOLDSTEIN, University of Cambridge Biological Physics ARTHUR F. HEBARD, University of Florida Condensed Matter Physics (Experimental) RANDALL D. KAMIEN, University of Pennsylvania Soft Condensed Matter DANIEL KLEPPNER, Massachusetts Institute of Technology Atomic, Molecular, and Optical Physics (Experimental) PAUL G. LANGACKER, Institute for Advanced Study, Princeton University Particle Physics (Theoretical) VERA LÜTH, Stanford University Particle Physics (Experimental) DAVID D. MEYERHOFER, University of Rochester Physics of Plasmas and Matter at High-Energy Density WITOLD NAZAREWICZ, University of Tennessee, Oak Ridge National Laboratory Nuclear Physics JOHN H. SCHWARZ, California Institute of Technology Mathematical Physics FRIEDRICH-KARL THIELEMANN, Universität Basel Astrophysics Senior Assistant Editor: DEBBIE BRODBAR, APS Editorial Office American Physical Society

Reviews of Modern Physics

Optimal Transport: A Semi-Discrete Approach

This work is devoted to the study of rates of convergence of the empirical measures μn = 1 n ∑n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) or [ E(W p p (μn, μ)) ]1/p in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one 1 √ n to slower rates, and both lower and upperbounds on E(Wp(μn, μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, logconcavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances

We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending the results first obtained by A. Figalli in [19]. Our main results are: a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions; and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed jointly with L. Ambrosio in [6], to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations for commutators between smoothing operators and diffusion generators.

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Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients

We establish, in a rather general setting, an analogue of DiPerna–Lions theory on well-posedness of flows of ODEs associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into ℝ∞. When specialized to the setting of Euclidean or infinite-dimensional (e.g., Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of RCD(K,∞) metric measure spaces, introduced by Ambrosio, Gigli and Savare [Duke Math. J. 163:7 (2014) 1405–1490] and the object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context.

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Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by S.\ Caracciolo et al.\ (Phys. Rev. E, {\bf 90} 012118, 2014) that "linearise" the Monge-Amp\`ere equation.

A PDE approach to a 2-dimensional matching problem

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by Caracciolo et al. (Phys Rev E 90:012118, 2014) that linearizes the Monge–Ampere equation.

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of $n$ qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton’s transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.

Dario Trevisan

Papers

Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

A PDE approach to a 2-dimensional matching problem

A PDE approach to a 2-dimensional matching problem

The Quantum Wasserstein Distance of Order 1