Generalized Principal Component Analysis by René Vidal Yi Ma & S.S. Sastry

Author:René Vidal, Yi Ma & S.S. Sastry
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

7.5 Spectral Algebraic Subspace Clustering (SASC)

In this subsection, we present another global approach to building a subspace clustering affinity based on the ASC algorithm described in Chapter 5. Recall that ASC is based on fitting a set of polynomials to the data and computing the normals to each subspace from the gradients of these polynomials at n data points, one per subspace. As it turns out, we can use the normal vectors computed by ASC to define a subspace clustering affinity. The key idea is that instead of computing the normal vectors at n points only, we can compute them at each of the N data points. In this way, we assign to each data point a set of normal vectors. Then, we can define an affinity between two points using any affinity between the two subspaces spanned by the two sets of normal vectors.

Let us begin with the simple case of data lying in a union of hyperplanes. As discussed in Chapter 5, in this case there is a single polynomial that vanishes in the union of hyperplanes. Moreover, we can use to estimate the normal to the hyperplane passing through as (see Algorithm 5.3)

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