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Register a new userFast Statistical Leverage Score Approximation in Kernel Ridge Regression
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Nystrom approximation is a fast randomized method that rapidly solves kernel ridge regression (KRR) problems through sub-sampling the n-by-n empirical kernel matrix appearing in the objective function. However, the performance of such a sub-sampling method heavily relies on correctly estimating the statistical leverage scores for forming the sampling distribution, which can be as costly as solving the original KRR. In this work, we propose a linear time (modulo poly-log terms) algorithm to accurately approximate the statistical leverage scores in the stationary-kernel-based KRR with theoretical guarantees. Particularly, by analyzing the first-order condition of the KRR objective, we derive an analytic formula, which depends on both the input distribution and the spectral density of stationary kernels, for capturing the non-uniformity of the statistical leverage scores. Numerical experiments demonstrate that with the same prediction accuracy our method is orders of magnitude more efficient than existing methods in selecting the representative sub-samples in the Nystrom approximation.
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Building a sketch of an n-by-n empirical kernel matrix is a common approach to accelerate the computation of many kernel methods. In this paper, we propose a unified framework of constructing sketching methods in kernel ridge regression (KRR), which views the sketching matrix S as an accumulation of m rescaled sub-sampling matrices with independent columns. Our framework incorporates two commonly used sketching methods, sub-sampling sketches (known as the Nystrom method) and sub-Gaussian sketches, as special cases with m=1 and m=infinity respectively. Under the new framework, we provide a unified error analysis of sketching approximation and show that our accumulation scheme improves the low accuracy of sub-sampling sketches when certain incoherence characteristic is high, and accelerates the more accurate but computationally heavier sub-Gaussian sketches. By optimally choosing the number m of accumulations, we show that a best trade-off between computational efficiency and statistical accuracy can be achieved. In practice, the sketching method can be as efficiently implemented as the sub-sampling sketches, as only minor extra matrix additions are needed. Our empirical evaluations also demonstrate that the proposed method may attain the accuracy close to sub-Gaussian sketches, while is as efficient as sub-sampling-based sketches.
We propose a penalized likelihood method to jointly estimate multiple precision matrices for use in quadratic discriminant analysis and model based clustering. A ridge penalty and a ridge fusion penalty are used to introduce shrinkage and promote similarity between precision matrix estimates. Block-wise coordinate descent is used for optimization, and validation likelihood is used for tuning parameter selection. Our method is applied in quadratic discriminant analysis and semi-supervised model based clustering.
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.
The divide-and-conquer method has been widely used for estimating large-scale kernel ridge regression estimates. Unfortunately, when the response variable is highly skewed, the divide-and-conquer kernel ridge regression (dacKRR) may overlook the underrepresented region and result in unacceptable results. We develop a novel response-adaptive partition strategy to overcome the limitation. In particular, we propose to allocate the replicates of some carefully identified informative observations to multiple nodes (local processors). The idea is analogous to the popular oversampling technique. Although such a technique has been widely used for addressing discrete label skewness, extending it to the dacKRR setting is nontrivial. We provide both theoretical and practical guidance on how to effectively over-sample the observations under the dacKRR setting. Furthermore, we show the proposed estimate has a smaller asymptotic mean squared error (AMSE) than that of the classical dacKRR estimate under mild conditions. Our theoretical findings are supported by both simulated and real-data analyses.
This paper carries out a large dimensional analysis of a variation of kernel ridge regression that we call emph{centered kernel ridge regression} (CKRR), also known in the literature as kernel ridge regression with offset. This modified technique is obtained by accounting for the bias in the regression problem resulting in the old kernel ridge regression but with emph{centered} kernels. The analysis is carried out under the assumption that the data is drawn from a Gaussian distribution and heavily relies on tools from random matrix theory (RMT). Under the regime in which the data dimension and the training size grow infinitely large with fixed ratio and under some mild assumptions controlling the data statistics, we show that both the empirical and the prediction risks converge to a deterministic quantities that describe in closed form fashion the performance of CKRR in terms of the data statistics and dimensions. Inspired by this theoretical result, we subsequently build a consistent estimator of the prediction risk based on the training data which allows to optimally tune the design parameters. A key insight of the proposed analysis is the fact that asymptotically a large class of kernels achieve the same minimum prediction risk. This insight is validated with both synthetic and real data.
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