Multi-view diffusion maps

Abstract

In this paper, a reduced dimensionality representation is learned from multiple views of the processed data. These multiple views can be obtained, for example, when the same underlying process is observed using several different modalities, or measured with different instrumentation. The goal is to effectively utilize the availability of such multiple views for various purposes such as non-linear embedding, manifold learning, spectral clustering, anomaly detection and non-linear system identification. The proposed method, which is called multi-view, exploits the intrinsic relation within each view as well as the mutual relations between views. This is achieved by defining a cross-view model in which an implied random walk process is restrained to hop between objects in the different views. This multi-view method is robust to scaling and it is insensitive to small structural changes in the data. Within this framework, new diffusion distances are defined to analyze the spectra of the implied kernels. The applicability of the multi-view approach is demonstrated for clustering, classification and manifold learning using both artificial and real data. 2 The problem of learning from two views has been studied in the field of spectral clustering. Most of these studies have been focused on classification and clustering that are based on spectral characteristics of the data while using two or more sampled views. Some approaches, which address this problem, are Bilinear Model [9], Partial Least Squares [10] and Canonical Correlation Analysis [11]. These methods are powerful for learning the relation between different views but do not provide separate insights or combined into the low dimensional geometry or structure of each view. Recently, a few kernel based methods (e.g [12]) propose a model of co-regularizing kernels in both views in a way that resembles joint diagonalization. It is done by searching for an orthogonal transformation that maximizes the diagonal terms of the kernel matrices obtained from all views. A penalty term, which incorporates the disagreement between clusters from the views, was added. Their algorithm is based on alternating maximization procedure. A mixture of Markov chains is proposed in [13] to model multiple views in order to apply spectral clustering. It deals with two cases in graph theory: directed and undirected graph where the second case is related to our work. This approach converges the undirected graph problem to a Markov chains averaging where each is constructed separately within the views. A way to incorporate a given multiple metrics for the same data using a cross diffusion process is described in [14]. They define a new diffusion distance which is useful for classification, clustering or retrieval tasks. However, the proposed process is not symmetrical thus does not allow to compute an embedding. An iterative algorithm for spectral clustering is proposed in [15]. The idea is to iteratively modify each view using the representation of the other view. The problem of two manifolds, ...

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