864 research outputs found
Optimal vector quantization in terms of Wasserstein distance
The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the R\'enyi--entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained and memory-size constrained quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by Gy\"{o}rgy and Linder
Constructive quantization: approximation by empirical measures
In this article, we study the approximation of a probability measure on
by its empirical measure interpreted as a
random quantization. As error criterion we consider an averaged -th moment
Wasserstein metric. In the case where , we establish refined upper and
lower bounds for the error, a high-resolution formula. Moreover, we provide a
universal estimate based on moments, a so-called Pierce type estimate. In
particular, we show that quantization by empirical measures is of optimal order
under weak assumptions.Comment: 22 page
Learning Probability Measures with respect to Optimal Transport Metrics
We study the problem of estimating, in the sense of optimal transport
metrics, a measure which is assumed supported on a manifold embedded in a
Hilbert space. By establishing a precise connection between optimal transport
metrics, optimal quantization, and learning theory, we derive new probabilistic
bounds for the performance of a classic algorithm in unsupervised learning
(k-means), when used to produce a probability measure derived from the data. In
the course of the analysis, we arrive at new lower bounds, as well as
probabilistic upper bounds on the convergence rate of the empirical law of
large numbers, which, unlike existing bounds, are applicable to a wide class of
measures.Comment: 13 pages, 2 figures. Advances in Neural Information Processing
Systems, NIPS 201
Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
We introduce a numerical method for extracting minimal geodesics along the
group of volume preserving maps, equipped with the L2 metric, which as observed
by Arnold solve Euler's equations of inviscid incompressible fluids. The method
relies on the generalized polar decomposition of Brenier, numerically
implemented through semi-discrete optimal transport. It is robust enough to
extract non-classical, multi-valued solutions of Euler's equations, for which
the flow dimension is higher than the domain dimension, a striking and
unavoidable consequence of this model. Our convergence results encompass this
generalized model, and our numerical experiments illustrate it for the first
time in two space dimensions.Comment: 21 pages, 9 figure
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