Learning vector quantization and relevances in complex coefficient space

Straat, Michiel; Kaden, Marika; Villmann, Thomas; Lampe, A.; Seiffert, Udo; Biehl, Michael; Melchert, Friedrich

doi:10.1007/s00521-019-04080-5

Cited by 6 publications

(9 citation statements)

References 16 publications

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“…For the complex-valued embedding (so far) a quite limited number of machine learning algorithm is available, like the complex-valued support vector machine (cSVM) [41,3], the complex-valued generalized learning vector quantization (cGMLVQ) [4,37], or a complex-valued neural network (cNN) [13,38,7]. Further, a nearest neighbor (NN) classifier by employing a standard norm operator can be used.…”

Section: Algorithm 1 Approximated Embedding Of Symmetric Proximitiesmentioning

confidence: 99%

Complex-Valued Embeddings of Generic Proximity Data

Münch

Straat

Biehl

et al. 2021

Lecture Notes in Computer Science

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Proximities are at the heart of almost all machine learning methods. If the input data are given as numerical vectors of equal lengths, euclidean distance, or a Hilbertian inner product is frequently used in modeling algorithms. In a more generic view, objects are compared by a (symmetric) similarity or dissimilarity measure, which may not obey particular mathematical properties. This renders many machine learning methods invalid, leading to convergence problems and the loss of guarantees, like generalization bounds. In many cases, the preferred dissimilarity measure is not metric, like the earth mover distance, or the similarity measure may not be a simple inner product in a Hilbert space but in its generalization a Krein space. If the input data are non-vectorial, like text sequences, proximity-based learning is used or ngram embedding techniques can be applied. Standard embeddings lead to the desired fixed-length vector encoding, but are costly and have substantial limitations in preserving the original data's full information. As an information preserving alternative, we propose a complex-valued vector embedding of proximity data. This allows suitable machine learning algorithms to use these fixed-length, complex-valued vectors for further processing. The complex-valued data can serve as an input to complex-valued machine learning algorithms. In particular, we address supervised learning and use extensions of prototype-based learning. The proposed approach is evaluated on a variety of standard benchmarks and shows strong performance compared to traditional techniques in processing non-metric or non-psd proximity data.

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Section: Algorithm 1 Approximated Embedding Of Symmetric Proximitiesmentioning

confidence: 99%

Complex-Valued Embeddings of Generic Proximity Data

Münch

Straat

Biehl

et al. 2021

Lecture Notes in Computer Science

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“…with Ω a linear projection matrix. This matrix can be learned as outlined in [9]. In our application, this matrix links the multiple perspectives of the time series to each other and permits an interpretable importance weighting of the single perspectives and the relevance of individual time series within a perspective 1 .…”

Section: Multi-perspective Embedding Of Non-psd Proximitiesmentioning

confidence: 99%

“…On own previous work on complex-valued embeddings in [8] and using ideas from [2], this can be done at low costs with moderate approximations. The recombined complex-valued data can be processed by an effective classifier model for complex-valued data suggested in [9].…”

Section: Introductionmentioning

confidence: 99%

Multi-perspective embedding for non-metric time series classification

Münch¹,

Heilig²,

Schleif³

2021

ESANN 2021 Proceedings

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The interest in time series analysis is rapidly increasing, providing new challenges for machine learning. Over many decades, Dynamic Time Warping (DTW) is referred to as the de facto standard distance measure for time series and the tool of choice when analyzing such data. Nevertheless, DTW has two major drawbacks: (a) it is non-metric and therefore hard to handle by standard machine learning techniques, and (b) it is not well suited for multi-dimensional time series. For this purpose, we propose a multi-perspective embedding of the time series into a complex-valued vector space and the evaluation by a model that is able to handle complex-valued data. The approach is evaluated on various multi-dimensional time series data and with different classifier techniques.

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“…Following [14,54], the derivatives according to (5) and (6) are obtained by the Wirtinger calculus (see ''Appendix 6'') applying the conjugate derivatives as…”

Section: Complex Variants Of Glvqmentioning

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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Learning vector quantization and relevances in complex coefficient space

Cited by 6 publications

References 16 publications

Complex-Valued Embeddings of Generic Proximity Data

Complex-Valued Embeddings of Generic Proximity Data

Multi-perspective embedding for non-metric time series classification

Quantum-inspired learning vector quantizers for prototype-based classification

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