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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 $n=2$:

enter image description here

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    $\begingroup$ 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. $\endgroup$
    – ttnphns
    Jan 31, 2022 at 14:28
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    $\begingroup$ 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.) $\endgroup$
    – Christian Hennig
    Jun 23, 2022 at 10:17

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