Quantum-inspired learning vector quantizers for prototype-based classification

Abstract

Abstract 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 $${\mathcal {H}}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> and show the relations to kernelized GLVQ. In particular, we explain the related feature mapping of data into the quantum state space $${\mathcal {H}}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> . A key feature for this approach is that $${\mathcal {H}}^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> 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.