Nonasymptotic bounds for vector quantization in Hilbert spaces
doi:10.1214/14-AOS1293arXiv:1405.6672
Margin conditions in quantization
- Use
conceptual template for the DS14 slab-margin assumption (M4) and for a future fast-rate refinement of the bridge. Levrard's (mass) and (separation) are the published codification of what (M2)/(M5) buy; their positivity at population optima is inherited from Graf–Luschgy, never assumed of empirical optima.
- Does not transfer
fast-rate mechanics rely on a Pythagorean identity for squared Euclidean distortion with no known log-det/Schur analogue; the margin condition is a hypothesis on the law, never proven automatic.
- Sources
Levrard (2015, Ann. Statist. 43:592–619, k-means margin condition bounding mass near optimal cell boundaries, fast rates; survey 2018 J. SFdS 159:1–48); Mammen & Tsybakov (1999, Ann. Statist. 27:1808–1829); Antos, Györfi & György (2005, individual rates).
AOS 43(2):592-619. Definition 2.1 margin condition: p(t) <= (B p_min / 128 M^2) t on the mass of t-neighborhoods of optimal-cell boundaries - a hypothesis on the LAW at population optima, never proven at empirical optima; buys finite optimal-codebook sets, quadratic growth, and 1/n rates; Gaussian mixtures satisfy it. (arXiv 1405.6672; the earlier arXiv 1310.7138 "Margin conditions for vector quantization" is the 2013 precursor.)