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• W2525451309 abstract "We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let $$mathcal {V}$$ be a finite set system defined over an n-point set X; we view $$mathcal {V}$$ as a set of indicator vectors over the n-dimensional unit cube. A $$delta $$ -separated set of $$mathcal {V}$$ is a subcollection $$mathcal {W}$$ , s.t. the Hamming distance between each pair $$mathbf{u}, mathbf{v}in mathcal {W}$$ is greater than $$delta $$ , where $$delta > 0$$ is an integer parameter. The $$delta $$ -packing number is then defined as the cardinality of a largest $$delta $$ -separated subcollection of $$mathcal {V}$$ . Haussler showed an asymptotically tight bound of $$Theta ((n/delta )^d)$$ on the $$delta $$ -packing number if $$mathcal {V}$$ has VC-dimension (or primal shatter dimension) d. We refine this bound for the scenario where, for any subset, $$X' subseteq X$$ of size $$m le n$$ and for any parameter $$1 le k le m$$ , the number of vectors of length at most k in the restriction of $$mathcal {V}$$ to $$X'$$ is only $$O(m^{d_1} k^{d-d_1})$$ , for a fixed integer $$d > 0$$ and a real parameter $$1 le d_1 le d$$ (this generalizes the standard notion of bounded primal shatter dimension when $$d_1 = d$$ ). In this case when $$mathcal {V}$$ is “k-shallow” (all vector lengths are at most k), we show that its $$delta $$ -packing number is $$O(n^{d_1} k^{d-d_1}/delta ^d)$$ , matching Haussler’s bound for the special cases where $$d_1=d$$ or $$k=n$$ . We present two proofs, the first is an extension of Haussler’s approach, and the second extends the proof of Chazelle, originally presented as a simplification for Haussler’s proof." @default.
• W2525451309 created "2016-10-07" @default.
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• W2525451309 date "2016-09-27" @default.
• W2525451309 modified "2023-09-25" @default.
• W2525451309 title "Two Proofs for Shallow Packings" @default.
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• W2525451309 doi "https://doi.org/10.1007/s00454-016-9824-0" @default.
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