Data Clustering with Quantum Mechanics

Scott, Tony; Therani, Madhusudan; Wang, Xing M.

doi:10.3390/math5010005

Cited by 14 publications

(15 citation statements)

References 44 publications

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“…As stated before, we limit ourselves to two dimensions. This method is more stable and precise than the standard K-means method [3].…”

Section: Statement Of the Problemmentioning

confidence: 95%

“…Consider a set of particles ( ) =1.. , the quantum clustering is a process that detects the clusters of the distributed set ( ) =1.. by finding the cluster centers. Those centers are the minima of the potential energy function defined by [2,3]:…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…(1) such that ∈ 2. This function results from injecting a Parzen window into the Schrödinger wave equation [2,3] and balancing the resulting energy. Other methods based on energy variation may also be instructive [19].…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…It has been shown that the quantum clustering method (QC) [2,3] can naturally cluster data originating from a number of sources whether they be: scientific (natural), engineering and even text. In particular, it is more stable and is often more accurate than the standard data clustering method known as K-means [3]. This method requires isolating the minima of a quantum potential and is equivalent to finding the roots of its gradients i.e.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Clustering Analysis: Minima of the Potential Energy Function

Maignan¹,

Scott²

2020

Computer Science &Amp; Information Technology (CS &Amp; IT)

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Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a 𝜎 value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size 𝜎, there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach “per block”. This technique decreases the number of particles (or samples) by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

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“…As stated before, we limit ourselves to two dimensions. This method is more stable and precise than the standard K-means method [3].…”

Section: Statement Of the Problemmentioning

confidence: 95%

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Statement Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum Clustering Analysis: Minima of the Potential Energy Function

Maignan¹,

Scott²

2020

Computer Science &Amp; Information Technology (CS &Amp; IT)

Self Cite

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“…Another way to accelerate the usually time-consuming training process in machine learning models is to make use of efficient quantum computing algorithms [9,12,33]. For supervised and unsupervised problems, several approaches are developed [3,46,49,50,64]. A recent overview for recently developed quantum-inspired approaches can be found in [26].…”

Section: Introductionmentioning

confidence: 99%

Quantum-inspired learning vector quantizers for prototype-based classification

Villmann

Engelsberger

Ravichandran

et al. 2020

Neural Comput & Applic

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Prototype-based models like the Generalized Learning Vector Quantization (GLVQ) belong to the class of interpretable classifiers. Moreover, quantum-inspired methods get more and more into focus in machine learning due to its potential efficient computing. Further, its interesting mathematical perspectives offer new ideas for alternative learning scenarios. This paper proposes a quantum computing-inspired variant of the prototype-based GLVQ for classification learning. We start considering kernelized GLVQ with real- and complex-valued kernels and their respective feature mapping. Thereafter, we explain how quantum space ideas could be integrated into a GLVQ using quantum bit vector space in the quantum state space $${\mathcal {H}}^{n}$$ H n and show the relations to kernelized GLVQ. In particular, we explain the related feature mapping of data into the quantum state space $${\mathcal {H}}^{n}$$ H n . A key feature for this approach is that $${\mathcal {H}}^{n}$$ H n is an Hilbert space with particular inner product properties, which finally restrict the prototype adaptations to be unitary transformations. The resulting approach is denoted as Qu-GLVQ. We provide the mathematical framework and give exemplary numerical results.

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