We consider the problem of recovering a common latent source with independent components from multiple views. This applies to settings in which a variable is measured with multiple experimental modalities, and where the goal is to synthesize the disparate measurements into a single unified representation. We consider the case that the observed views are a nonlinear mixing of component-wise corruptions of the sources. When the views are considered separately, this reduces to nonlinear Independent Component Analysis (ICA) for which it is provably impossible to undo the mixing. We present novel identifiability proofs that this is possible when the multiple views are considered jointly, showing that the mixing can theoretically be undone using function approximators such as deep neural networks. In contrast to known identifiability results for nonlinear ICA, we prove that independent latent sources with arbitrary mixing can be recovered as long as multiple, sufficiently different noisy views are available.
CCA can be interpretated probabilistically @cite_33 and is equivalent to maximum likelihood estimation in a graphical model which is a special case of that depicted in Figure . The differences compared to our setting are (i) the latent components retrieved in CCA are forced to be uncorrelated, whereas our method is retrieves independent components; (ii) in CCA, mappings between the sources @math and @math are linear, whereas our method allows for nonlinear mappings.
@cite_33
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We give a probabilistic interpretation of canonical correlation (CCA) analysis as a latent variable model for two Gaussian random vectors. Our interpretation is similar to the probabilistic interpretation of principal component analysis (Tipping and Bishop, 1999, Roweis, 1998). In addition, we cast Fisher linear discriminant analysis (LDA) within the CCA framework. Building a common representation for several related data sets is an important problem in multi-view learning. CCA and its extensions have shown that they are effective in finding the shared variation among all data sets. However, these models generally fail to exploit the common structure of the data when the views are with private information. Recently, methods explicitly modeling the information into shared part and private parts have been proposed, but they presume to know the prior knowledge about the latent space, which is usually impossible to obtain. In this paper, we propose a probabilistic model, which could simultaneously learn the structure of the latent space whilst factorize the information correctly, therefore the prior knowledge of the latent space is unnecessary. Furthermore, as a probabilistic model, our method is able to deal with missing data problem in a natural way. We show that our approach attains the performance of state-of-art methods on the task of human pose estimation when the motion capture view is completely missing, and significantly improves the inference accuracy with only a few observed data. In this paper we introduce a new underlying probabilistic model for principal component analysis (PCA). Our formulation interprets PCA as a particular Gaussian process prior on a mapping from a latent space to the observed data-space. We show that if the prior's covariance function constrains the mappings to be linear the model is equivalent to PCA, we then extend the model by considering less restrictive covariance functions which allow non-linear mappings. This more general Gaussian process latent variable model (GPLVM) is then evaluated as an approach to the visualisation of high dimensional data for three different data-sets. Additionally our non-linear algorithm can be further kernelised leading to 'twin kernel PCA' in which a mapping between feature spaces occurs. This paper investigates the problem of modeling Internet images and associated text or tags for tasks such as image-to-image search, tag-to-image search, and image-to-tag search (image annotation). We start with canonical correlation analysis (CCA), a popular and successful approach for mapping visual and textual features to the same latent space, and incorporate a third view capturing high-level image semantics, represented either by a single category or multiple non-mutually-exclusive concepts. We present two ways to train the three-view embedding: supervised, with the third view coming from ground-truth labels or search keywords; and unsupervised, with semantic themes automatically obtained by clustering the tags. To ensure high accuracy for retrieval tasks while keeping the learning process scalable, we combine multiple strong visual features and use explicit nonlinear kernel mappings to efficiently approximate kernel CCA. To perform retrieval, we use a specially designed similarity function in the embedded space, which substantially outperforms the Euclidean distance. The resulting system produces compelling qualitative results and outperforms a number of two-view baselines on retrieval tasks on three large-scale Internet image datasets.