Introduces the random Stinespring superchannel to convert channel queries into isometry queries, yielding a channel analogue of Uhlmann's theorem and proving optimal channel learning query complexity of Θ(d_A d_B r).
Metric Entropy of Homogeneous Spaces
3 Pith papers cite this work. Polarity classification is still indexing.
3
Pith papers citing it
abstract
For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em metric entropy}, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous spaces of unitary (or orthogonal) groups with respect to some natural metrics, most notably the one induced by the operator norm. This generalizes earlier author's results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. In the process we give a characterization of geodesics in $U(n)$ (or $SO(m)$) for a class of non-Riemannian metric structures.
verdicts
UNVERDICTED 3representative citing papers
Proposes pointwise Riemannian Dimension from feature eigenvalues to derive tighter, representation-aware generalization bounds for deep networks in the nonlinear regime.
Tripartite Haar-random states with balanced subsystems exhibit no distillable bipartite EPR entanglement, with doubly-exponential probability suppression, and imply no non-trivial logical operators in the associated quantum error-correcting code.
citing papers explorer
-
Random Stinespring superchannel: converting channel queries into dilation isometry queries
Introduces the random Stinespring superchannel to convert channel queries into isometry queries, yielding a channel analogue of Uhlmann's theorem and proving optimal channel learning query complexity of Θ(d_A d_B r).
-
Pointwise Generalization in Deep Neural Networks
Proposes pointwise Riemannian Dimension from feature eigenvalues to derive tighter, representation-aware generalization bounds for deep networks in the nonlinear regime.
-
Tripartite Haar random state has no bipartite entanglement
Tripartite Haar-random states with balanced subsystems exhibit no distillable bipartite EPR entanglement, with doubly-exponential probability suppression, and imply no non-trivial logical operators in the associated quantum error-correcting code.