[Paper Review] Finding two-dimensional peaks

З. К. Силагадзе|ArXiv.org|Feb 17, 2004

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a two-dimensional extension of a quantum-inspired peak-finding algorithm based on Markov chains with quantum-like tunneling properties. By simulating a random walk on a 2D histogram using transition probabilities derived from local intensity differences, the method amplifies global peaks in noisy or overlapping data, achieving stable invariant distributions that highlight true maxima after sufficient iterations.

ABSTRACT

Two-dimensional generalization of the original peak finding algorithm suggested earlier is given. The ideology of the algorithm emerged from the well known quantum mechanical tunneling property which enables small bodies to penetrate through narrow potential barriers. We further merge this ``quantum'' ideology with the philosophy of Particle Swarm Optimization to get the global optimization algorithm which can be called Quantum Swarm Optimization. The functionality of the newborn algorithm is tested on some benchmark optimization problems.

Motivation & Objective

• To extend a one-dimensional quantum-inspired peak-finding algorithm to two dimensions for improved global optimization in multidimensional data.
• To address the challenge of locating global maxima in overlapping or irregularly shaped peaks common in experimental data such as gamma-ray spectra.
• To develop a robust, non-derivative optimization method that does not require gradient information or prior knowledge of the function's shape.
• To test the algorithm's performance on benchmark functions with known global minima, validating its convergence and accuracy.

Proposed method

• Adapts a one-dimensional Markov chain model to two dimensions, where states represent grid points (i,j) in a histogram.
• Defines transition probabilities between neighboring cells (up, down, left, right) based on a quantum-like tunneling mechanism using intensity differences.
• Uses a generalized Q-matrix formulation with exponential weighting based on the ratio of neighboring bin counts to determine transition likelihoods.
• Employs a stopping criterion based on relative change in probability distributions across iterations to detect convergence.
• Simulates the Markov process iteratively starting from a uniform initial distribution until the invariant distribution stabilizes.
• Leverages the invariant distribution to identify sharp peaks corresponding to global maxima in the original data.

Experimental results

Research questions

• RQ1 Can a quantum-inspired Markov process effectively locate global maxima in two-dimensional, noisy, or overlapping data distributions?
• RQ2 How does the penetrating ability parameter (m) affect convergence speed and peak resolution in the invariant distribution?
• RQ3 To what extent does the algorithm outperform traditional optimization methods in multimodal benchmark functions with complex landscapes?
• RQ4 Can the algorithm reliably detect global optima without requiring gradient information or prior knowledge of the function's structure?
• RQ5 How does the algorithm perform on functions with narrow, deep global minima surrounded by broad local minima?

Key findings

• The algorithm successfully identifies global maxima in 2D histograms with overlapping Gaussians, producing sharp peaks in the invariant distribution even when the original data has broad, indistinct maxima.
• For a 100×100 histogram with three overlapping Gaussians, convergence to within ε = 10⁻³ was achieved in 113 iterations with m = 30, compared to 258 iterations with m = 3.
• The invariant distribution develops pronounced peaks at locations of global maxima, confirming the method's ability to amplify true peaks while suppressing side lobes.
• The algorithm converges slowly for low penetrating ability (m = 3), but eventually suppresses weaker side peaks in favor of the dominant global maximum.
• The method was tested on 18 standard benchmark functions including Rastrigin, Ackley, Griewank, and Goldstein-Price, all of which had their known global optima correctly located.
• The algorithm's performance is robust across diverse function types, including those with multiple local minima, narrow valleys, and high dimensionality (in the 2D case).