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Abstract We consider two notions of functions of bounded variation in complete metric measure spaces,one due to Martio and the other due to Miranda Jr. We show that these two notionscoincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.
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From Poincaré inequalities to nonlinear matrix concentration
This paper deduces exponential matrix concentration from a Poincaré inequality via a short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of a uniformly log-concave random vector. The proof relies on the subadditivity of Poincaré inequalities and a chain rule inequality for the trace of the matrix Dirichlet form. It also uses a symmetrization technique to avoid difficulties associated with a direct extension of the classic scalar argument.
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- PAR ID:
- 10286833
- Date Published:
- Journal Name:
- Bernoulli
- Volume:
- 23
- Issue:
- 3
- ISSN:
- 1350-7265
- Page Range / eLocation ID:
- 1724–1744
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/https://doi.org/10.3150/20-BEJ1289
