Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.29451

Linear stability analysis of the Lloyd algorithm on a circle

Silpi Saha·Sangita Jha·Mrinal Kanti Roychowdhury
View original source ↗Read source document ↗Export metadata ↓

Abstract

Lloyd algorithm is the standard iterative method for computing quantizers and codebooks in source coding and vector quantization. In this article, we study the dynamical and stability properties of the Lloyd map on the unit circle $\mathbb S^1$ using von Mises distributions. We construct the Lloyd iteration as a discrete dynamical system on the configuration space of ordered point sets modulo rotational symmetry. Also, we study the rotational equivarience of the Lloyd map. Further, we derive an explicit representation of the Jacobian matrix and prove that it possesses a circulant structure for the equally spaced configuration. Also, we study the bifurcation characteristics based on Lloyd map analysis. In the end, we provide the numerical algorithms for stability diagrams, Lyapunov spectrum estimation, and residue analysis, purely for empirical visualization. Our results provide a dynamical systems framework for Lloyd quantization on $\mathbb S^1$ for studying stability properties.

Explore related subjects

math.DSmath.OC

Keep this discovery

BibTeXRIS

Silpi Saha, Sangita Jha, Mrinal Kanti Roychowdhury. 2026-05-28. Linear stability analysis of the Lloyd algorithm on a circle. https://arxiv.org/abs/2605.29451

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗