# One-dimensional subspace clustering

Consider the inner product space $$(\mathbb{R}^n, \langle\cdot,\cdot\rangle)$$ and suppose that there are one-dimensional orthogonal subspaces $$\{V_i\}_{i=1}^n$$ such that $$\mathbb{R^n} = \oplus_{i=1}^n V_i$$.

Now suppose that you observe data $$X \subseteq \mathbb{R}^n$$, and each data point belongs to one of the $$n$$ subspaces. The aim is to group the data into $$n$$ clusters, one cluster for each subspace. Are there any efficient algorithms to determine such clustering?

The figure below shows an example of a clustering when $$n=2$$:

• With your picture, replacing the two dimensions with two principal components does the job of differentiating between the two clouds. In effect, each PC designates its "own" cloud. Jan 31 at 14:28
• In fact even a Gaussian mixture with flexible covariance matrices should do this properly. (Although you may have a more general problem in mind where point clouds look less Gaussian.) Jun 23 at 10:17