Latent Smooth Skeleton Embedding

Existing methods mostly rely on distances (or similarities) In many fields of science and engineering, one is often to model the intrinsic structure of data. They either provide confronted with the problem of dimensionality reduction a similarity matrix as a prior (Belkin and Niyogi 2001; (Burges 2009; Van der Maaten, Postma, and van den Herik Schölkopf, Smola, and Muller 1999), or learn a similarity 2009). The problem aims to extract low-dimensional structures measurement based on a subset of distances in a local from high-dimensional datasets, which are generally region (Elhamifar and Vidal 2011; Saul and Roweis characterized by much fewer degrees of freedom than actual 2003), or directly learn a kernel matrix from data (Weinberger, number of features. Packer, and Saul 2005; Xiao, Sun, and Boyd 2006; In this paper, we are particularly interested in unveiling a Mao and Tsang 2010). These distances become unreliable if smooth skeleton structure in a latent space from data with the data is noisy. Moreover, they lack the ability to model a noise. Figure 1 illustrates an intuitive example in which synthetic smooth skeleton from noisy data. As shown in Figure 1, the data points are drawn from a smooth circle with noises strict distance preservation in maximum variance unfolding in two-dimensional space. It is challenging to recover the (MVU) (Weinberger, Sha, and Saul 2004) fails to capture the circle (Figures 1(c) and 1(d)) from the noisy data without smooth circle from the data (see Figure 1 (b)).