Results 1  10
of
19
Contractions in the 2Wasserstein Length Space and Thermalization of Granular Media
, 2004
"... An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical ..."
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Cited by 55 (20 self)
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An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinitedimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even nonconvexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.
On quantum statistical inference
 J. Roy. Statist. Soc. B
, 2001
"... [Read before The Royal Statistical Society at a meeting organized by the Research Section ..."
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Cited by 24 (5 self)
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[Read before The Royal Statistical Society at a meeting organized by the Research Section
Free transportation cost inequalities via random matrix approximation
 Probab. Theor. Rel. Fields
, 2004
"... Abstract. By means of random matrix approximation procedure, we reprove Biane and Voiculescu’s free analog of Talagrand’s transportation cost inequality for measures on R in a more general setup. Furthermore, we prove the free transportation cost inequality for measures on T as well by extending the ..."
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Cited by 14 (0 self)
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Abstract. By means of random matrix approximation procedure, we reprove Biane and Voiculescu’s free analog of Talagrand’s transportation cost inequality for measures on R in a more general setup. Furthermore, we prove the free transportation cost inequality for measures on T as well by extending the method to special unitary random matrices.
A free probability analogue of the Wasserstein metric on the tracestate space
 Geom. Funct. Anal
"... Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semicircle distribution is majorized by a modified free entropy quantity. 0 ..."
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Cited by 13 (0 self)
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Abstract. We define a free probability analogue of the Wasserstein metric, which extends the classical one. In dimension one, we prove that the square of the Wasserstein distance to the semicircle distribution is majorized by a modified free entropy quantity. 0
Free relative entropy for measures and a corresponding perturbation theory
 J. Math. Soc. Japan
, 2002
"... Voiculescu's single variable free entropy is generalized in two di erent ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their eq ..."
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Cited by 9 (0 self)
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Voiculescu's single variable free entropy is generalized in two di erent ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. Next, the perturbation theory for compactly supported probability measures via free relative entropy is developed on the analogy of the perturbation theory via relative entropy. When the perturbed measure via relative entropy is suitably arranged on the space of selfadjoint matrices and the matrix size goes to in nity, it is proven that the perturbation via relative entropy on the matrix space approaches asymptotically to that via free relative entropy. The whole theory can be adapted to probability measures on the unit circle.
Mismatched estimation and relative entropy
 IEEE Trans. Inf. Theory
, 2010
"... Abstract—A random variable with distribution is observed in Gaussian noise and is estimated by a mismatched minimum meansquare estimator that assumes that the distribution is, instead of. This paper shows that the integral over all signaltonoise ratios (SNRs) of the excess meansquare estimation e ..."
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Cited by 8 (2 self)
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Abstract—A random variable with distribution is observed in Gaussian noise and is estimated by a mismatched minimum meansquare estimator that assumes that the distribution is, instead of. This paper shows that the integral over all signaltonoise ratios (SNRs) of the excess meansquare estimation error incurred by the mismatched estimator is twice the relative entropy (in nats). This representation of relative entropy can be generalized to nonrealvalued random variables, and can be particularized to give new general representations of mutual information in terms of conditional means. Inspired by the new representation, we also propose a definition of free relative entropy which fills a gap in, and is consistent with, the literature on free probability. Index Terms—Divergence, free probability, minimum meansquare error (MMSE) estimation, mutual information, relative entropy, Shannon theory, statistics. I.
RANDOM REGULARIZATION OF BROWN SPECTRAL MEASURE
, 2001
"... Abstract. We generalize a recent result of Haagerup; namely we show that a convolution with a standard Gaussian random matrix regularizes the behavior of Fuglede–Kadison determinant and Brown spectral distribution measure. In this way it is possible to establish a connection between the limit eigenv ..."
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Cited by 6 (0 self)
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Abstract. We generalize a recent result of Haagerup; namely we show that a convolution with a standard Gaussian random matrix regularizes the behavior of Fuglede–Kadison determinant and Brown spectral distribution measure. In this way it is possible to establish a connection between the limit eigenvalues distributions of a wide class of random matrices and the Brown measure of the corresponding limits. 1.
D.: Free diffusions and matrix models with strictly convex interaction
 Geom. Funct. Anal
"... We study solutions to the free stochastic differential equation dXt = dSt − 1 2DV (Xt)dt, where V is a locally convex polynomial potential in m noncommuting variables. We show that for selfadjoint V, the law µV of a stationary solution is the limit law of a random matrix model, in which an mtuple ..."
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Cited by 4 (0 self)
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We study solutions to the free stochastic differential equation dXt = dSt − 1 2DV (Xt)dt, where V is a locally convex polynomial potential in m noncommuting variables. We show that for selfadjoint V, the law µV of a stationary solution is the limit law of a random matrix model, in which an mtuple of selfadjoint matrices are chosen according to the law exp(−NTr(V (A1,..., Am)))dA1 · · · dAm. We show that if V = Vβ depends on complex parameters β1,..., βk, then the law µV is analytic in β at least for those β for which Vβ is locally convex. In particular, this gives information on the region of convergence of the generating function for planar maps. We show that the solution dXt has nice convergence properties with respect to the operator norm. This allows us to derive several properties of C ∗ and W ∗ algebras generated by an mtuple with law µV. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II1 factor. We show that the microstates free entropy χ(τV) is finite. A corollary of these results is the fact that the support of the law of any selfadjoint polynomial in X1,...,Xn under the law µV is connected, vastly generalizing the case of a single random matrix. 1
A note on cyclic gradients
 Indiana Univ. Math. J
"... To the memory of GianCarlo Rota The cyclic derivative was introduced by G.C. Rota, B. Sagan and P. R. Stein in [3] as an extension of the derivative to noncommutative polynomials. Here we show that there are simple necessary and sufficient conditions for an ntuple of polynomials in n noncommuting ..."
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Cited by 4 (0 self)
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To the memory of GianCarlo Rota The cyclic derivative was introduced by G.C. Rota, B. Sagan and P. R. Stein in [3] as an extension of the derivative to noncommutative polynomials. Here we show that there are simple necessary and sufficient conditions for an ntuple of polynomials in n noncommuting indeterminates to be a cyclic gradient (see Theorem 1) and similarly for a polynomial to have vanishing cyclic gradient (see Theorem 2). Our interest in cyclic gradients stems from free probability theory and random matrices (see the Remark at the end) [1],[2],[4],[5],[6]. This note should also reduce the paucity of results on cyclic derivatives in several variables pointed out in [3, page 73]. Let K〈n 〉 = K〈X1,..., Xn 〉 be the ring of polynomials in noncommuting indeterminates X1,...,Xn with coefficients in the field K of characteristic zero. The partial generalized difference quotients are the derivations ∂j: K〈n 〉 → K〈n 〉 ⊗ K〈n〉 such that ∂jXk = 0 if j ̸ = k and ∂jXj = 1 ⊗ 1. The ⊗ here is over K and K〈n 〉 ⊗ K〈n 〉 is given the bimodule structure such that a(b ⊗ c) = ab ⊗ c, (b ⊗ c)d = b ⊗ cd. The partial cyclic derivatives are then δj = ˜µ ◦ ∂j: K〈n 〉 → K〈n〉 where ˜µ(a ⊗ b) = ba. We shall denote by N: K〈n 〉 → K〈n 〉 the “number operator”, i.e. the linear map so that N1 = 0, NXi1... Xik = kXi1...Xik. Also, CK〈n 〉 will denote the cyclic subspace, i.e. the vector subspace spanned by all cyclic symmetrizations of monomials