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This work introduces a generalized kernel perspective for spectral dimensionality reduction approaches. Firstly, an elegant matrix view of kernel principal component analysis (peA) is described. We show the relationship between kernel peA, and conventional peA using a parametric distance. Secondly, we introduce a weighted kernel peA framework followed from least squares support vector machines (LS-SVM). This approach starts with a latent variable that allows to write a relaxed LS-SVM problem. Such a problem is addressed by a primal-dual formulation. As a result, we provide kernel alternatives to spectral methods for dimensionality reduction such as multidimensional scaling, locally linear embedding, and laplacian eigenmaps; as well as a versatile framework to explain weighted peA approaches. Experimentally, we prove that the incorporation of a SVM model improves the performance of kernel peA.

Dimensionality reduction aims at providing faithful low-dimensional representations of high-dimensional data. Its general principle is to attempt to reproduce in a low-dimensional space the salient characteristics of data, such as proximities. A large variety of methods exist in the literature, ranging from principal component analysis to deep neural networks with a bottleneck layer. In this cornucopia, it is rather difficult to find out why a few methods clearly outperform others. This paper identifies two important properties that enable some recent methods like stochastic neighborhood embedding and its variants to produce improved visualizations of high-dimensional data. The first property is a low sensitivity to the phenomenon of distance concentration. The second one is plasticity, that is, the capability to forget about some data characteristics to better reproduce the other ones. In a manifold learning perspective, breaking some proximities typically allow for a better unfolding of data. Theoretical developments as well as experiments support our claim that both properties have a strong impact. In particular, we show that equipping classical methods with the missing properties significantly improves their results.

Dimensionality reduction techniques aim at representing high-dimensional data in low-dimensional spaces. To be faithful and reliable, the representation is usually required to preserve proximity relationships. In practice, methods like multidimensional scaling try to fulfill this requirement by preserving pairwise distances in the low-dimensional representation. However, such a simplification does not easily allow for local scalings in the representation. It also makes these methods suboptimal with respect to recent quality criteria that are based on distance rankings. This paper addresses this issue by introducing a dimensionality reduction method that works with ranks. Appropriate hypotheses enable the minimization of a rank-based cost function. In particular, the scale indeterminacy that is inherent to ranks is circumvented by representing data on a space with a spherical topology., Anglais

Unsupervised dimensionality reduction aims at representing high-dimensional data in lower-dimensional spaces in a faithful way. Dimensionality reduction can be used for compression or denoising purposes, but data visualization remains one its most prominent applications. This paper attempts to give a broad overview of the domain. Past developments are briefly introduced and pinned up on the time line of the last eleven decades. Next, the principles and techniques involved in the major methods are described. A taxonomy of the methods is suggested, taking into account various properties. Finally, the issue of quality assessment is briefly dealt with.

Denoising is a key step in the processing of medical images. It aims at improving both the interpretability and visual aspect of the images. Yet, designing a robust and efficient denoising tool remains an unsolved challenge and a specific issue concerns the noise model. Many filters typically assume that noise is additive and Gaussian, with uniform variance. In contrast, noise in medical images often has more complex properties. This paper considers images with Poissonian noise and the patch-based bilateral filters, that is, filters that involve a tonal kernel and pair wise comparisons between shifted blocks of the images. The main aim is then to integrate two variance stabilizing transformations that allow the filters to work with Gaussianized noise. The performances of these filters are compared to those of the classical bilateral filter with the same transformations. The experiments include an artificial benchmark as well as a positron emission tomography image., Anglais

Blind source separation (BSS) consists in recovering unobserved signals from observed mixtures of them. In most cases the whole set of mixtures is used for the separation, possibly after a dimension reduction by PCA. This paper aims to show that in many applications the quality of the separation can be improved by first selecting a subset of some mixtures among the available ones, possibly by an information content criterion, and performing PCA and BSS afterwards. The benefit of this procedure is shown on simulated electrocardiographic data by extracting the fetal electrocardiogram signal from mixtures recorded on the abdomen of a pregnant woman., Anglais