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Article Dans Une Revue Journal of Machine Learning Research Année : 2020

Convergence Rate of Optimal Quantization and Application to the Clustering Performance of the Empirical Measure

Résumé

We study the convergence rate of the optimal quantization for a probability measure sequence (µn) n∈N* on R^d converging in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal quantizer x (n) ∈ (R d) K of µn at level K; the other one is the convergence rate of the distortion function valued at x^(n), called the "performance" of x^(n). Moreover, we also study the mean performance of the optimal quantization for the empirical measure of a distribution µ with finite second moment but possibly unbounded support. As an application, we show that the mean performance for the empirical measure of the multidimensional normal distribution N (m, Σ) and of distributions with hyper-exponential tails behave like O(log n √ n). This extends the results from [BDL08] obtained for compactly supported distribution. We also derive an upper bound which is sharper in the quantization level K but suboptimal in n by applying results in [FG15].
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Dates et versions

hal-02484426 , version 1 (19-02-2020)

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  • HAL Id : hal-02484426 , version 1

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Yating Liu, Gilles Pagès. Convergence Rate of Optimal Quantization and Application to the Clustering Performance of the Empirical Measure. Journal of Machine Learning Research, 2020, 21, pp.1-36. ⟨hal-02484426⟩
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