Zador Theorem for optimal quantization with respect to Bregman divergences
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914441634775040 |
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| author | Boutoille, Guillaume Pagès, Gilles |
| author_facet | Boutoille, Guillaume Pagès, Gilles |
| contents | We establish a Zador like theorem for $L^r$-optimal vector quantization when the similarity measure is a twice differentiable Bregman divergence of a strictly convex function. On our way we also prove a similar result when the Bregman divergence is replaced by a continuous matrix-valued vector field having values in the set of positive definite matrices. We adopt the strategy of the first fully rigorous proof of the original Zador' theorem (when the similarity measure is the power of a norm). We have to overcome several difficulties which are specific to this framework especially concerning the so-called firewall lemma. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02354 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Zador Theorem for optimal quantization with respect to Bregman divergences Boutoille, Guillaume Pagès, Gilles Functional Analysis Probability 65C20, 65N50, 90C39, 93E35, 91B28 We establish a Zador like theorem for $L^r$-optimal vector quantization when the similarity measure is a twice differentiable Bregman divergence of a strictly convex function. On our way we also prove a similar result when the Bregman divergence is replaced by a continuous matrix-valued vector field having values in the set of positive definite matrices. We adopt the strategy of the first fully rigorous proof of the original Zador' theorem (when the similarity measure is the power of a norm). We have to overcome several difficulties which are specific to this framework especially concerning the so-called firewall lemma. |
| title | Zador Theorem for optimal quantization with respect to Bregman divergences |
| topic | Functional Analysis Probability 65C20, 65N50, 90C39, 93E35, 91B28 |
| url | https://arxiv.org/abs/2604.02354 |