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SHORT GEODESICS OF UNITARIES IN THE L2 METRIC∗
"... Let M be a type II1 von Neumann algebra, τ a trace in M, and L2(M, τ) the GNS Hilbert space of τ. We regard the unitary group UM as a subset of L2(M, τ), and characterize the shortest smooth curves joining two fixed unitaries, in the L2 metric. As a consequence of this we obtain that UM, though a co ..."
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Cited by 1 (1 self)
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Let M be a type II1 von Neumann algebra, τ a trace in M, and L2(M, τ) the GNS Hilbert space of τ. We regard the unitary group UM as a subset of L2(M, τ), and characterize the shortest smooth curves joining two fixed unitaries, in the L2 metric. As a consequence of this we obtain that UM, though a
CURVATURE OF THE L2METRIC ON THE DIRECT IMAGE OF A FAMILY OF HERMITIANEINSTEIN VECTOR BUNDLES
"... of the L2metric on the direct image of a family of HermitianEinstein vector bundles ..."
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of the L2metric on the direct image of a family of HermitianEinstein vector bundles
NAKANO POSITIVITY AND THE L 2METRIC ON THE DIRECT IMAGE OF AN ADJOINT POSITIVE LINE BUNDLE
, 1999
"... Abstract. We prove that the L 2 metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers o ..."
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Abstract. We prove that the L 2 metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers
Policy gradient methods for reinforcement learning with function approximation.
 In NIPS,
, 1999
"... Abstract Function approximation is essential to reinforcement learning, but the standard approach of approximating a value function and determining a policy from it has so far proven theoretically intractable. In this paper we explore an alternative approach in which the policy is explicitly repres ..."
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Cited by 439 (20 self)
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;actorcritic" or policyiteration architectures (e.g., Policy Gradient Theorem We consider the standard reinforcement learning framework (see, e.g., Sutton and Barto, 1998), in which a learning agent interacts with a Markov decision process (MDP). The state, action, and reward at each time t ∈ {0, 1, 2
On the geometry of metric measure spaces
 II, ACTA MATH
, 2004
"... We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Amo ..."
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Cited by 247 (9 self)
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We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d
Some properties of Noether charge and a proposal for dynamical black hole entropy
 Phys. Rev
, 1994
"... We consider a general, classical theory of gravity with arbitrary matter fields in n dimensions, arising from a diffeomorphism invariant Lagrangian, L. We first show that L always can be written in a “manifestly covariant ” form. We then show that the symplectic potential current (n − 1)form, Θ, an ..."
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Cited by 326 (4 self)
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We consider a general, classical theory of gravity with arbitrary matter fields in n dimensions, arising from a diffeomorphism invariant Lagrangian, L. We first show that L always can be written in a “manifestly covariant ” form. We then show that the symplectic potential current (n − 1)form, Θ
On the Surprising Behavior of Distance Metrics in High Dimensional Space
 Lecture Notes in Computer Science
, 2001
"... In recent years, the effect of the curse of high dimensionality has been studied in great detail on several problems such as clustering, nearest neighbor search, and indexing. In high dimensional space the data becomes sparse, and traditional indexing and algorithmic techniques fail from a efficienc ..."
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Cited by 200 (2 self)
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preferable than the Euclidean distance metric (L2 norm) for high dimensional data mining applications. Using the intuition derived from our analysis, we introduce and examine a natural extension of the Lk norm to fractional distance metrics. We show that the fractional distance metric provides more
Submersions and equivariant Quillen metrics
 Ann. Inst. Fourier (Grenoble
, 2000
"... Let ξ be a Hermitian vector bundle on a compact Hermitian complex manifold X. Let λ(ξ) be the inverse of the determinant of the cohomology of ξ. Quillen defined first a metric on λ(ξ) in the case that X is a Riemann surface. Quillen metric is the product of the L2 metric on λ(ξ) by the ..."
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Cited by 11 (0 self)
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Let ξ be a Hermitian vector bundle on a compact Hermitian complex manifold X. Let λ(ξ) be the inverse of the determinant of the cohomology of ξ. Quillen defined first a metric on λ(ξ) in the case that X is a Riemann surface. Quillen metric is the product of the L2 metric on λ(ξ) by the
Results 1  10
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