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 $CUR$ approximates the matrix $A,$ that is, $\|A - C U R\|_{F}^2 \leq (1+\varepsilon) {}\|A - A_k\|_{F}^2$, 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 $c=O(k/\varepsilon)$ and $r=O(k/\varepsilon)$ and rank$(U) = k$. Up to constant factors, our algorithms are simultaneously optimal in the values $c, r,$ and rank$(U)$.

Principal components analysis~(PCA) is the optimal linear encoder of data. Sparse linear encoders (e.g., sparse PCA) produce more interpretable features that can promote better generalization. (\rn{1}) Given a level of sparsity, what is the best approximation to PCA? (\rn{2}) Are there efficient algorithms which can achieve this optimal combinatorial tradeoff? We answer both questions by providing the first polynomial-time algorithms to construct \emph{optimal} sparse linear auto-encoders; additionally, we demonstrate the performance of our algorithms on real data.

This paper studies the Principal Component Analysis (PCA) problem in the distributed and streaming models of computation. Given a matrix A ∈ Rm×n, a rank parameter k<rank(A), and an accuracy parameter 0<ε<1, we want to output an m×k orthonormal matrix U for which ||A-UUTA||2F≤(1+ε)||A-Ak||2F where Ak∈Rm×n is the best rank-k approximation to A.

Spectral clustering is one of the most important algorithms in data mining and machine intelligence; however, its computational complexity limits its application to truly large scale data analysis. The computational bottleneck in spectral clustering is computing a few of the top eigenvectors of the (normalized) Laplacian matrix corresponding to the graph representing the data to be clustered. One way to speed up the computation of these eigenvectors is to use the “power method” from the numerical linear algebra literature. Although the power method has been empirically used to speed up spectral clustering, the theory behind this approach, to the best of our knowledge, remains unexplored. This paper provides the first such rigorous theoretical justification, arguing that a small number of power iterations suffices to obtain near-optimal partitionings using the approximate eigenvectors. Specifically, we prove that solving the k-means clustering problem on the approximate eigenvectors obtained via the power method gives an additive-error approximation to solving the k-means problem on the optimal eigenvectors.

We consider the online version of the well known Principal Component Analysis (PCA) problem. In standard PCA, the input to the problem is a set of d-dimensional vectors X = [x1, …, xn] and a target dimension k < d; the output is a set of k-dimensional vectors Y = [y1, …, yn] that minimize the reconstruction error: . Here, Φ ∊ ℝd × k is restricted to being isometric. The global minimum of this quantity, OPTk, is obtainable by offline PCA. In online PCA (OPCA) the setting is identical except for two differences: i) the vectors xt are presented to the algorithm one by one and for every presented xt the algorithm must output a vector yt before receiving xt+1; ii) the output vectors yt are ℓ dimensional with ℓ ≥ k to compensate for the handicap of operating online. To the best of our knowledge, this paper is the first to consider this setting of OPCA. Our algorithm produces yt ∊ ℝℓ with ℓ = O(k · poly(1/ε)) such that .

We study the topic of dimensionality reduction for k-means clustering. Dimensionality reduction encompasses the union of two approaches: 1) feature selection and 2) feature extraction. A feature selection-based algorithm for k-means clustering selects a small subset of the input features and then applies k-means clustering on the selected features. A feature extraction-based algorithm for k-means clustering constructs a small set of new artificial features and then applies k-means clustering on the constructed features. Despite the significance of k-means clustering as well as the wealth of heuristic methods addressing it, provably accurate feature selection methods for k-means clustering are not known. On the other hand, two provably accurate feature extraction methods for k-means clustering are known in the literature; one is based on random projections and the other is based on the singular value decomposition (SVD). This paper makes further progress toward a better understanding of dimensionality reduction for k-means clustering. Namely, we present the first provably accurate feature selection method for k-means clustering and, in addition, we present two feature extraction methods. The first feature extraction method is based on random projections and it improves upon the existing results in terms of time complexity and number of features needed to be extracted. The second feature extraction method is based on fast approximate SVD factorizations and it also improves upon the existing results in terms of time complexity. The proposed algorithms are randomized and provide constant-factor approximation guarantees with respect to the optimal k-means objective value.

We develop a fast algorithm for computing the “SVD-truncated” regularized solution to the least-squares problem: minx∥Ax - b∥2. Let Akof rank k be the best rank k matrix computed via the SVD of A. Then, the SVD-truncated regularized solution is: xk= Ak†b. If A is m × n, then, it takes O(mnmin{m, n}) time to compute xkusing the SVD of A. We give an approximation algorithm for xkwhich constructs a rank k approximation Ãkand computes x̃k= Ãk†in roughly O(nnz(A)k log n) time. Our algorithm uses a randomized variant of the subspace iteration method. We show that, with high probability: ∥Ax̃k- b∥2≈ ∥Axk- b∥2and ∥xk- x̃k∥2≈ 0.

We explain theoretically a curious empirical phenomenon: "Approximating a matrix by deterministically selecting a subset of its columns with the corresponding largest leverage scores results in a good low-rank matrix surrogate". In this work, we provide a novel theoretical analysis of deterministic leverage score sampling. We show that such sampling can be provably as accurate as its randomized counterparts, if the leverage scores follow a moderately steep power-law decay. We support this power-law assumption by providing empirical evidence that such decay laws are abundant in real-world data sets. We then demonstrate empirically the performance of deterministic leverage score sampling, which many times matches or outperforms the state-of-the-art techniques.

The CUR decomposition of an m×n matrix A finds an m×c matrix C with a small subset of c < n columns of A, together with an r×n matrix R with a small subset of r < m rows of A, as well as a c×r low rank matrix U such that the matrix CUR approximates the input matrix A, that is, ||A --- CUR||2F ≤ (1 + ε)||A --- Ak||2F, where ||.||F denotes the Frobenius norm, 0 < ε < 1 is an accuracy parameter, and Ak is the best m × n matrix of rank k constructed via the SVD of A. We present input-sparsity-time and deterministic algorithms for constructing such a CUR matrix decomposition of A where c = O(k/ε) and r = O(k/ε) and rank(U) = k. Up to constant factors, our construction is simultaneously optimal in c, r, and rank(U).

We consider low-rank reconstruction of a matrix using a subset of its columns and present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are (i) the use of fast approximate SVD-like decompositions for column-based matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [J. D. Batson, D. A. Spielman, and N. Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC), 2009, pp. 255--262].

Let X be a data matrix of rank ρ, whose rows represent n points in d -dimensional space. The linear support vector machine constructs a hyperplane separator that maximizes the 1-norm soft margin. We develop a new oblivious dimension reduction technique that is precomputed and can be applied to any input matrix X . We prove that, with high probability, the margin and minimum enclosing ball in the feature space are preserved to within ϵ-relative error, ensuring comparable generalization as in the original space in the case of classification. For regression, we show that the margin is preserved to ϵ-relative error with high probability. We present extensive experiments with real and synthetic data to support our theory.

Let X be a data matrix of rank ρ, representing n points in d-dimensional space. The linear support vector machine constructs a hyperplane separator that maximizes the 1-norm soft margin. We develop a new oblivious dimension reduction technique which is precomputed and can be applied to any input matrix X. We prove that, with high probability, the margin and minimum enclosing ball in the feature space are preserved to within ε-relative error, ensuring comparable generalization as in the original space. We present extensive experiments with real and synthetic data to support our theory.

We present a fast algorithm for approximate Canonical Correlation Analysis (CCA). Given a pair of tall-and-thin matrices, the proposed algorithm first employs a randomized dimensionality reduction transform to reduce the size of the input matrices, and then applies any standard CCA algorithm to the new pair of matrices. The algorithm computes an approximate CCA to the original pair of matrices with provable guarantees, while requiring asymptotically less operations than the state-of-the-art exact algorithms.

We study the (constrained) least-squares regression as well as multiple response least-squares regression and ask the question of whether a subset of the data, a coreset, suffices to compute a good approximate solution to the regression. We give deterministic, low-order polynomial-time algorithms to construct such coresets with approximation guarantees, together with lower bounds indicating that there is not much room for improvement upon our results.

We study feature selection for k-means clustering. Although the literature contains many methods with good empirical performance, algorithms with provable theoretical behavior have only recently been developed. Unfortunately, these algorithms are randomized and fail with, say, a constant probability. We present the first deterministic feature selection algorithm for k-means clustering with relative error guarantees. At the heart of our algorithm lies a deterministic method for decompositions of the identity and a structural result which quantifies some of the tradeoffs in dimensionality reduction.

We consider low-rank reconstruction of a matrix using a subset of its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our results are: (i) the use of fast approximate SVD-like decompositions for column-based matrix reconstruction, and (ii) two deterministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse representation theorem for decompositions of the identity that appeared in [1].

Principal Components Analysis~(PCA) is often used as a feature extraction procedure. Given a matrix $X \in \mathbb{R}^{n \times d}$, whose rows represent $n$ data points with respect to $d$ features, the top $k$ right singular vectors of $X$ (the so-called \textit{eigenfeatures}), are arbitrary linear combinations of all available features. The eigenfeatures are very useful in data analysis, including the regularization of linear regression. Enforcing sparsity on the eigenfeatures, i.e., forcing them to be linear combinations of only a \textit{small} number of actual features (as opposed to all available features), can promote better generalization error and improve the interpretability of the eigenfeatures. We present deterministic and randomized algorithms that construct such sparse eigenfeatures while \emph{provably} achieving in-sample performance comparable to regularized linear regression. Our algorithms are relatively simple and practically efficient, and we demonstrate their performance on several data sets.

This paper discusses the topic of dimensionality reduction for $k$-means clustering. We prove that any set of $n$ points in $d$ dimensions (rows in a matrix $A \in \RR^{n \times d}$) can be projected into $t = \Omega(k / \eps^2)$ dimensions, for any $\eps \in (0,1/3)$, in $O(n d \lceil \eps^{-2} k/ \log(d) \rceil )$ time, such that with constant probability the optimal $k$-partition of the point set is preserved within a factor of $2+\eps$. The projection is done by post-multiplying $A$ with a $d \times t$ random matrix $R$ having entries $+1/\sqrt{t}$ or $-1/\sqrt{t}$ with equal probability. A numerical implementation of our technique and experiments on a large face images dataset verify the speed and the accuracy of our theoretical results.

We present a novel feature selection algorithm for the $k$-means clustering problem. Our algorithm is randomized and, assuming an accuracy parameter $\epsilon \in (0,1)$, selects and appropriately rescales in an unsupervised manner $\Theta(k \log(k / \epsilon) / \epsilon^2)$ features from a dataset of arbitrary dimensions. We prove that, if we run any $\gamma$-approximate $k$-means algorithm ($\gamma \geq 1$) on the features selected using our method, we can find a $(1+(1+\epsilon)\gamma)$-approximate partition with high probability.

We consider the problem of selecting the “best” subset of exactly k columns from an m × n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn2, m2n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-chosen probability distribution that depends on information in the top-k right singular subspace of A. In the second stage (the deterministic stage), the algorithm applies a deterministic column-selection procedure to select and return exactly k columns from the set of columns selected in the first stage. Let C be the m × k matrix containing those k columns, let PC denote the projection matrix onto the span of those columns, and let Ak denote the “best” rank-k approximation to the matrix A as computed with the singular value decomposition. Then, we prove that with probability at least 0.7. This spectral norm bound improves upon the best previously-existing result (of Gu and Eisenstat [21]) for the spectral norm version of this Column ubset election Problem. We also prove that with the same probability. This Frobenius norm bound is only a factor of worse than the best previously existing existential result and is roughly better than the best previous algorithmic result (both of Deshpande et al. [11]) for the Frobenius norm version of this Column ubset election Problem.

Motivated by the enormous amounts of data collected in a large IT service provider organization, this paper presents a method for quickly and automatically summarizing and extracting meaningful insights from the data. Termed Clustered Subset Selection (CSS), our method enables program-guided data explorations of high-dimensional data matrices. CSS combines clustering and subset selection into a coherent and intuitive method for data analysis. In addition to a general framework, we introduce a family of CSS algorithms with different clustering components such as k-means and Close-to-Rank-One (CRO) clustering, and Subset Selection components such as best rank-one approximation and Rank-Revealing QR (RRQR) decomposition.

Principal Components Analysis (PCA) is the predominant linear dimensionality reduction technique, and has been widely applied on datasets in all scientific domains. We consider, both theoretically and empirically, the topic of unsupervised feature selection for PCA, by leveraging algorithms for the so-called Column Subset Selection Problem (CSSP). In words, the CSSP seeks the "best" subset of exactly k columns from an m x n data matrix A, and has been extensively studied in the Numerical Linear Algebra community. We present a novel two-stage algorithm for the CSSP. From a theoretical perspective, for small to moderate values of k, this algorithm significantly improves upon the best previously-existing results [24, 12] for the CSSP. From an empirical perspective, we evaluate this algorithm as an unsupervised feature selection strategy in three application domains of modern statistical data analysis: finance, document-term data, and genetics. We pay particular attention to how this algorithm may be used to select representative or landmark features from an object-feature matrix in an unsupervised manner. In all three application domains, we are able to identify k landmark features, i.e., columns of the data matrix, that capture nearly the same amount of information as does the subspace that is spanned by the top k "eigenfeatures."