Mutal Subspace Method

Summary

The Mutual Subspace Method (MSM) is an extension of the Subspace Method (SM), where instead of having an input vector \(\mathbf{x}\), we use an input subspace \(\mathcal{P}\). MSM is commonly used for image set classification [SSF19].

../_images/MSM.png

Fig.1

Theory

The Subspace method assumes an input subspace and \(k\) class subspaces. Let us define the input subspace to be a \(d_p\)-dimensional subspace \(\mathcal{P}\) and the class subspaces to be \(d_q\)-dimensional subspaces \(\{\mathcal{Q}, \mathcal{R}, \mathcal{S}...\}\).

The similarity \(S\) between, for example, \(\mathcal{P}\) and \(\mathcal{Q}\) was originally defined as the minimum canonical angle \(\theta_1\). Canonical angles [Cha12] are uniquely defined as:

\[\begin{split}\cos^2 \theta_i = \max_{\substack{u_i \perp u_j(=1,...,i−1) \\ v_i \perp v_j(=1,...,i−1)}} \frac{|(u_1,v_i)|^2}{||u_i||^2||v_i||^2}\end{split}\]

Where \(\mathbf{u}_i \in \mathcal{P}\), \(\mathbf{v}_i \in \mathcal{Q}\), \(||\mathbf{u}_i|| \neq 0\), \(||\mathbf{v}_i|| \neq 0\).

We can also include the remaining canonical angles when calculating the similarity.

\[\tilde{S} = \frac{1}{t} \sum^{t}_{i=1}\cos{\theta_i}^2\]

This value \(\tilde{S}\) reflects the structural similarity between two subspaces. It is also defined on the \(t\) smallest canonical angles.

../_images/MSM_structural_sim.png

Fig.2 source

For practical applications, the canonical angles can be easily obtained by applying the SVD to the orthonormal basis vectors of the subspaces.

Learning Phase

  1. Generate \(k\) class subspaces from each class by using PCA.

Recognition Phase

  1. Calculate \(S\) (or \(\tilde{S}\)) between \(\mathcal{P}\) and each subspace \(\mathcal{Q}\).

  2. Classify \(\mathcal{P}\) into the class where \(S\) (or \(\tilde{S}\)) was calculated to be the highest.

Also, you may add a rejection thershold on \(S\) (or \(\tilde{S}\)) to reject classifications with low similarity.

Warning

How is the dimension of the input subspaces decided? Are they obtained using PCA?