Optimal CUR Matrix Decompositions

The CUR decomposition of an $m \times n$ matrix $A$ finds an $m \times c$ matrix $C$ with a subset of $c < n$ columns of $A,$ together with an $r \times n$ matrix $R$ with a subset of $r < m$ rows of $A,$ as well as a $c \times r$ low-rank matrix $U$ such that the matrix $C U R$ approximates the matrix $A,$ that is, $|| A - CUR ||_F^2 \le (1+\epsilon) || A - A_k||_F^2$, where $||.||_F$ denotes the Frobenius norm and $A_k$ is the best $m \times n$ matrix of rank $k$ constructed via the SVD. We present input-sparsity-time and deterministic algorithms for constructing such a CUR decomposition where $c=O(k/\epsilon)$ and $r=O(k/\epsilon)$ and rank$(U) = k$. Up to constant factors, our algorithms are simultaneously optimal in $c, r,$ and rank$(U)$.Comment: small revision in lemma 4.

Efficient Algorithms for CUR and Interpolative Matrix Decompositions

The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized library codes can be utilized for implementation. For certain applications, further acceleration can be attained by incorporating techniques based on randomized projections. Numerical experiments demonstrate advantageous performance compared to existing techniques for computing CUR factorizations

CUR Decompositions, Similarity Matrices, and Subspace Clustering

A general framework for solving the subspace clustering problem using the CUR decomposition is presented. The CUR decomposition provides a natural way to construct similarity matrices for data that come from a union of unknown subspaces $\mathscr{U}=\underset{i=1}{\overset{M}\bigcup}S_i$. The similarity matrices thus constructed give the exact clustering in the noise-free case. Additionally, this decomposition gives rise to many distinct similarity matrices from a given set of data, which allow enough flexibility to perform accurate clustering of noisy data. We also show that two known methods for subspace clustering can be derived from the CUR decomposition. An algorithm based on the theoretical construction of similarity matrices is presented, and experiments on synthetic and real data are presented to test the method. Additionally, an adaptation of our CUR based similarity matrices is utilized to provide a heuristic algorithm for subspace clustering; this algorithm yields the best overall performance to date for clustering the Hopkins155 motion segmentation dataset.Comment: Approximately 30 pages. Current version contains improved algorithm and numerical experiments from the previous versio

A DEIM Induced CUR Factorization

We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a low rank approximate decomposition of the form $A \approx C U R$, where $C$ and $R$ are subsets of the columns and rows of $A$, and $U$ is constructed to make $CUR$ a good approximation. Given a low-rank singular value decomposition $A \approx V S W^T$, the DEIM procedure uses $V$ and $W$ to select the columns and rows of $A$ that form $C$ and $R$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of $V$ and $W$. For large-scale problems, $V$ and $W$ can be approximated using an incremental QR algorithm that makes one pass through $A$. Numerical examples illustrate the favorable performance of the DEIM-CUR method, compared to CUR approximations based on leverage scores