Consider a high-dimensional data set, such that for every data-point there is incomplete information. Each object in the data set represents a real entity, which models as a point in high-dimensional space. We assume that all real entities are embedded in the same space, which means they have the same dimension. We model the lack of information for a given object as  affine subspace in R^d with dimension k . Our goal in this paper is to find clusters of objects . The main problem is to cope with the partial information . We studied a simple algorithm we callData clustering using flats minimum distances , using the following assumptions: 1) There are m clusters. 2) Each cluster is modeled as a  ball in \mathbb{R}^d  . 3) Each cluster contains a \frac{n{m}, k-dimensional affine subspaces . 4) All k-dimensional affine subspaces, which belong in the same cluster, are intersected with the ball of the cluster. 5) Each k-dimensional affine subspace, that belong to a cluster, is selected uniformly among all k-dimensional affine subspaces that intersect the ball's cluster. A data set that satisfy these assumptions will be called }\textit{separable data . Our suggested algorithm  calculates pair-wise projection of the data  . We use probabilistic considerations to prove the algorithm  correctness. These probabilistic results are of independent interest, as can serve to better understand the geometry of high dimensional objects.
Consider a high-dimensional data set, in which for every data-point there is incomplete information. Each object in the data set represents a real entity, which is described by a point in high-dimensional space. We model the lack of information for a given object as an affine subspace in R^d whose dimension k is the number of missing features . Our goal in this study is to find clusters of objects where the main problem is to cope with partial information and high dimension. Assuming the data set is separable, namely, its emergence from clusters that can be modeled as a set of disjoint ball in \mathbb{R}^d , we suggest a simple data clustering algorithm . {m}, k-dimensional affine subspaces . 4) All k-dimensional affine subspaces, which belong in the same cluster, are intersected with the ball of the cluster. 5) Each k-dimensional affine subspace, that belong to a cluster, is selected uniformly among all k-dimensional affine subspaces that intersect the ball's cluster. A data set that satisfy these assumptions will be called }\textit{ Our suggested algorithm use the affine subspaces minimum distance and calculates pair-wise projection of the data achieving poly-logarithmic time complexity . We use probabilistic considerations to prove the algorithm 's correctness. These probabilistic results are of independent interest, and can serve to better understand the geometry of high dimensional objects.
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