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Register a new userEstimating leverage scores via rank revealing methods and randomization
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Added by
Aleksandros Sobczyk
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We study algorithms for estimating the statistical leverage scores of rectangular dense or sparse matrices of arbitrary rank. Our approach is based on combining rank revealing methods with compositions of dense and sparse randomized dimensionality reduction transforms. We first develop a set of fast novel algorithms for rank estimation, column subset selection and least squares preconditioning. We then describe the design and implementation of leverage score estimators based on these primitives. These estimators are also effective for rank deficient input, which is frequently the case in data analytics applications. We provide detailed complexity analyses for all algorithms as well as meaningful approximation bounds and comparisons with the state-of-the-art. We conduct extensive numerical experiments to evaluate our algorithms and to illustrate their properties and performance using synthetic and real world data sets.
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We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. In particular, we study $s$-sparse functions of the form $f(x) = sum_{j=1}^s a_j e^{i lambda_j x}$ for coefficients $a_j in mathbb{C}$ and frequencies $lambda_j in mathbb{R}$. Bounding Fourier sparse leverage scores under various measures is of pure mathematical interest in approximation theory, and our work extends existing results for the uniform measure [Erd17,CP19a]. Practically, our bounds are motivated by two important applications in machine learning: 1. Kernel Approximation. They yield a new random Fourier features algorithm for approximating Gaussian and Cauchy (rational quadratic) kernel matrices. For low-dimensional data, our method uses a near optimal number of features, and its runtime is polynomial in the $statistical dimension$ of the approximated kernel matrix. It is the first oblivious sketching method with this property for any kernel besides the polynomial kernel, resolving an open question of [AKM+17,AKK+20b]. 2. Active Learning. They can be used as non-uniform sampling distributions for robust active learning when data follows a Gaussian or Laplace distribution. Using the framework of [AKM+19], we provide essentially optimal results for bandlimited and multiband interpolation, and Gaussian process regression. These results generalize existing work that only applies to uniformly distributed data.
Low-rank tensor decomposition generalizes low-rank matrix approximation and is a powerful technique for discovering low-dimensional structure in high-dimensional data. In this paper, we study Tucker decompositions and use tools from randomized numerical linear algebra called ridge leverage scores to accelerate the core tensor update step in the widely-used alternating least squares (ALS) algorithm. Updating the core tensor, a severe bottleneck in ALS, is a highly-structured ridge regression problem where the design matrix is a Kronecker product of the factor matrices. We show how to use approximate ridge leverage scores to construct a sketched instance for any ridge regression problem such that the solution vector for the sketched problem is a $(1+varepsilon)$-approximation to the original instance. Moreover, we show that classical leverage scores suffice as an approximation, which then allows us to exploit the Kronecker structure and update the core tensor in time that depends predominantly on the rank and the sketching parameters (i.e., sublinear in the size of the input tensor). We also give upper bounds for ridge leverage scores as rows are removed from the design matrix (e.g., if the tensor has missing entries), and we demonstrate the effectiveness of our approximate ridge regressioni algorithm for large, low-rank Tucker decompositions on both synthetic and real-world data.
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. To obtain provable guarantees, previous work requires randomized sampling of the columns with probabilities proportional to their leverage scores. In this work, we provide a novel theoretical analysis of deterministic leverage score sampling. We show that such deterministic 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.
Since being analyzed by Rokhlin, Szlam, and Tygert and popularized by Halko, Martinsson, and Tropp, randomized Simultaneous Power Iteration has become the method of choice for approximate singular value decomposition. It is more accurate than simpler sketching algorithms, yet still converges quickly for any matrix, independently of singular value gaps. After $tilde{O}(1/epsilon)$ iterations, it gives a low-rank approximation within $(1+epsilon)$ of optimal for spectral norm error. We give the first provable runtime improvement on Simultaneous Iteration: a simple randomized block Krylov method, closely related to the classic Block Lanczos algorithm, gives the same guarantees in just $tilde{O}(1/sqrt{epsilon})$ iterations and performs substantially better experimentally. Despite their long history, our analysis is the first of a Krylov subspace method that does not depend on singular value gaps, which are unreliable in practice. Furthermore, while it is a simple accuracy benchmark, even $(1+epsilon)$ error for spectral norm low-rank approximation does not imply that an algorithm returns high quality principal components, a major issue for data applications. We address this problem for the first time by showing that both Block Krylov Iteration and a minor modification of Simultaneous Iteration give nearly optimal PCA for any matrix. This result further justifies their strength over non-iterative sketching methods. Finally, we give insight beyond the worst case, justifying why both algorithms can run much faster in practice than predicted. We clarify how simple techniques can take advantage of common matrix properties to significantly improve runtime.
In this paper, we develop a novel procedure for low-rank tensor regression, namely emph{underline{I}mportance underline{S}ketching underline{L}ow-rank underline{E}stimation for underline{T}ensors} (ISLET). The central idea behind ISLET is emph{importance sketching}, i.e., carefully designed sketches based on both the responses and low-dimensional structure of the parameter of interest. We show that the proposed method is sharply minimax optimal in terms of the mean-squared error under low-rank Tucker assumptions and under randomized Gaussian ensemble design. In addition, if a tensor is low-rank with group sparsity, our procedure also achieves minimax optimality. Further, we show through numerical study that ISLET achieves comparable or better mean-squared error performance to existing state-of-the-art methods while having substantial storage and run-time advantages including capabilities for parallel and distributed computing. In particular, our procedure performs reliable estimation with tensors of dimension $p = O(10^8)$ and is $1$ or $2$ orders of magnitude faster than baseline methods.
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