Linear regression is a classical paradigm in statistics. A new look at it is provided via the lens of universal learning. In applying universal learning to linear regression the hypotheses class represents the label @math as a linear combination of the feature vector @math where @math , within a Gaussian error. The Predictive Normalized Maximum Likelihood (pNML) solution for universal learning of individual data can be expressed analytically in this case, as well as its associated learnability measure. Interestingly, the situation where the number of parameters @math may even be larger than the number of training samples @math can be examined. As expected, in this case learnability cannot be attained in every situation; nevertheless, if the test vector resides mostly in a subspace spanned by the eigenvectors associated with the large eigenvalues of the empirical correlation matrix of the training data, linear regression can generalize despite the fact that it uses an over-parametrized'' model. We demonstrate the results with a simulation of fitting a polynomial to data with a possibly large polynomial degree.
In order to evaluate a pointwise confidence measure for linear regression, several methods were proposed. Leverage values are employed to identify outliers with respect to their feature values @cite_3 . A leverage value is a measure of the distance between an observation and the center of the data Whenever a matrix is inverted it is assumed that the matrix is invertible. If needed, @math with small @math is added to assure invertibility where @math is the (unnormalized) correlation matrix of the training set and @math is the feature value which is examined. If the leverage value @math of observation is large, the observation is considered as an outlier. Using the pNML, in section we get a confidence measure for the prediction of the next label which is similar to the leverage measure.
@cite_3
2129566246 2616345629 2615055317 2601789736
The statistical leverage scores of a data matrix are the squared row-norms of any matrix whose columns are obtained by orthogonalizing the columns of the data matrix; and, the coherence is the largest leverage score. These quantities play an important role in several machine learning algorithms because they capture the key structural nonuniformity of the data matrix that must be dealt with in developing efficient randomized algorithms. Our main result is a randomized algorithm that takes as input an arbitrary n × d matrix A, with n ≫ d, and returns, as output, relative-error approximations to all n of the statistical leverage scores. The proposed algorithm runs in O(nd log n) time, as opposed to the O(nd2) time required by the naive algorithm that involves computing an orthogonal basis for the range of A. This resolves an open question from (, 2006) and (Mohri & Talwalkar, 2011); and our result leads to immediate improvements in coreset-based l2-regression, the estimation of the coherence of a matrix, and several related low-rank matrix problems. Interestingly, to achieve our result we judiciously apply random projections on both sides of A. The statistical leverage scores of a matrix A are the squared row-norms of the matrix containing its (top) left singular vectors and the coherence is the largest leverage score. These quantities are of interest in recently-popular problems such as matrix completion and Nystrom-based low-rank matrix approximation as well as in large-scale statistical data analysis applications more generally; moreover, they are of interest since they define the key structural nonuniformity that must be dealt with in developing fast randomized matrix algorithms. Our main result is a randomized algorithm that takes as input an arbitrary n × d matrix A, with n ≫ d, and that returns as output relative-error approximations to all n of the statistical leverage scores. The proposed algorithm runs (under assumptions on the precise values of n and d) in O(nd logn) time, as opposed to the O(nd2) time required by the naive algorithm that involves computing an orthogonal basis for the range of A. Our analysis may be viewed in terms of computing a relative-error approximation to an underconstrained least-squares approximation problem, or, relatedly, it may be viewed as an application of Johnson-Lindenstrauss type ideas. Several practically-important extensions of our basic result are also described, including the approximation of so-called cross-leverage scores, the extension of these ideas to matrices with n ≈ d, and the extension to streaming environments. One popular method for dealing with large-scale data sets is sampling. For example, by using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows columns of data matrices to reduce the data size before performing computations on the subproblem. This method has been successful in improving computational efficiency of algorithms for matrix problems such as least-squares approximation, least absolute deviations approximation, and low-rank matrix approximation. Existing work has focused on algorithmic issues such as worst-case running times and numerical issues associated with providing high-quality implementations, but none of it addresses statistical aspects of this method. In this paper, we provide a simple yet effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model with a fixed number of predictors. In particular, for several versions of leverage-based sampling, we derive results for the bias and variance, both conditional and unconditional on the observed data. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms: one constructs a smaller least-squares problem with "shrinkage" leverage scores (SLEV), and the other solves a smaller and unweighted (or biased) least-squares problem (LEVUNW). A detailed empirical evaluation of existing leverage-based methods as well as these two new methods is carried out on both synthetic and real data sets. The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance. For example, with the same computation reduction as in the original algorithmic leveraging approach, our proposed SLEV typically leads to improved biases and variances both unconditionally and conditionally (on the observed data), and our proposed LEVUNW typically yields improved unconditional biases and variances. By stacking layers of convolution and nonlinearity, convolutional networks (ConvNets) effectively learn from lowlevel to high-level features and discriminative representations. Since the end goal of large-scale recognition is to delineate complex boundaries of thousands of classes, adequate exploration of feature distributions is important for realizing full potentials of ConvNets. However, state-of-theart works concentrate only on deeper or wider architecture design, while rarely exploring feature statistics higher than first-order. We take a step towards addressing this problem. Our method consists in covariance pooling, instead of the most commonly used first-order pooling, of highlevel convolutional features. The main challenges involved are robust covariance estimation given a small sample of large-dimensional features and usage of the manifold structure of covariance matrices. To address these challenges, we present a Matrix Power Normalized Covariance (MPNCOV) method. We develop forward and backward propagation formulas regarding the nonlinear matrix functions such that MPN-COV can be trained end-to-end. In addition, we analyze both qualitatively and quantitatively its advantage over the well-known Log-Euclidean metric. On the ImageNet 2012 validation set, by combining MPN-COV we achieve over 4 , 3 and 2.5 gains for AlexNet, VGG-M and VGG-16, respectively; integration of MPN-COV into 50-layer ResNet outperforms ResNet-101 and is comparable to ResNet-152. The source code will be available on the project page: http: www.peihuali.org MPN-COV.