Complex-valued embeddings of generic proximity data

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.Comment: Proximity learning, embedding, complex values, complex-valued embedding, learning vector quantizatio

Positive Definite Kernels in Machine Learning

This survey is an introduction to positive definite kernels and the set of methods they have inspired in the machine learning literature, namely kernel methods. We first discuss some properties of positive definite kernels as well as reproducing kernel Hibert spaces, the natural extension of the set of functions $\{k(x,\cdot),x\in\mathcal{X}\}$ associated with a kernel $k$ defined on a space $\mathcal{X}$. We discuss at length the construction of kernel functions that take advantage of well-known statistical models. We provide an overview of numerous data-analysis methods which take advantage of reproducing kernel Hilbert spaces and discuss the idea of combining several kernels to improve the performance on certain tasks. We also provide a short cookbook of different kernels which are particularly useful for certain data-types such as images, graphs or speech segments.Comment: draft. corrected a typo in figure

Adaptive spectrum transformation by topology preserving on indefinite proximity data

Similarity-based representation generates indefinite matrices, which are inconsistent with classical kernel-based learning frameworks. In this paper, we present an adaptive spectrum transformation method that provides a positive semidefinite ( psd ) kernel consistent with the intrinsic geometry of proximity data. In the proposed method, an indefinite similarity matrix is rectified by maximizing the Euclidian fac- tor ( EF ) criterion, which represents the similarity of the resulting feature space to Euclidean space. This maximization is achieved by modifying volume elements through applying a conformal transform over the similarity matrix. We performed several experiments to evaluate the performance of the proposed method in comparison with flip, clip, shift , and square spectrum transformation techniques on similarity matrices. Applying the resulting psd matrices as kernels in dimensionality reduction and clustering problems confirms the success of the proposed approach in adapting to data and preserving its topological information. Our experiments show that in classification applications, the superiority of the proposed method is considerable when the negative eigenfraction of the similarity matrix is significant