Abstract
A lattice tiling decomposition induces dual operations: quantisation and wrapping, which map the Euclidean space to the lattice and to one of its fundamental domains, respectively. Applying such decomposition to random variables over the Euclidean space produces quantised and wrapped random variables. In studying the characteristic function of those, we show a ‘frequency domain’ characterisation for deterministic quantisation, which is dual to the known ‘frequency domain’ characterisation of uniform wrapping. In a second part, we apply the tiling decomposition to describe dithered quantisation, which consists in adding noise during quantisation to improve its perceived quality. We propose a non-collaborative type of dithering and show that, in this case, a wrapped dither minimises the Kullback-Leibler divergence to the original distribution. Numerical experiments illustrate this result.
Supported by the Brazilian National Council for Scientific and Technological Development (CNPq) grants 141407/2020-4 and 32441/2021-2.
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Notes
- 1.
Wrapping can be thought of as the canonical projection \(\mathbb {R}^n \rightarrow \mathbb {R}^n/\varLambda \) to the quotient space (isomorphic to an n-torus), composed with a choice of representatives.
- 2.
The classical definition of characteristic function in statistics books does not include the factor \(-2\pi \) in the exponent, e.g. [13, § 3.3]. Here, we follow [18, § 4.2] and define the characteristic function as in (1) to simplify relations. There are slightly different definitions for the Fourier transform too: the form we adopt (as in [18]) has the advantage of simplifying the expression for the inverse Fourier transform [10, § 6.2].
- 3.
A uniform ‘dither’ N of this form has been used to find discrete entropy upper bounds in Massey-type inequalities [15].
- 4.
We note that the Gaussian dither is not confined to the fundamental region. Its variance was optimised by grid search.
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The authors thank the anonymous reviewers for their careful reading and valuable comments, which have improved the original manuscript.
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Meneghetti, F.C.C., Miyamoto, H.K., Costa, S.I.R., Costa, M.H.M. (2023). Revisiting Lattice Tiling Decomposition and Dithered Quantisation. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14071. Springer, Cham. https://doi.org/10.1007/978-3-031-38271-0_31
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