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We study a high-dimensional regression setting under the assumption of known covariate distribution. We aim at estimating the amount of explained variation in the response by the best linear function of the covariates (the signal level). In our setting, neither sparsity of the coefficient vector, nor normality of the covariates or linearity of the conditional expectation are assumed. We present an unbiased and consistent estimator and then improve it by using a zero-estimator approach, where a zero-estimator is a statistic whose expected value is zero. More generally, we present an algorithm based on the zero estimator approach that in principle can improve any given estimator. We study some asymptotic properties of the proposed estimators and demonstrate their finite sample performance in a simulation study. • High-dimensional regression setting under the assumption of known covariate distribution. • No assumptions regarding the sparsity of the coefficient vector, normality of the covariates, or linearity of the conditional expectation. • Estimating the amount of explained variation in the response (the signal level) by the best linear function of the covariates. • Zero-estimators are used to improve initial estimators of the signal and noise levels. [ABSTRACT FROM AUTHOR]

We work over an algebraically closed field of characteristic zero. For a nondegenerate projective variety X ⊆ P N , the locus of points from which X is projected nonbirationally onto its image, is called the Segre locus of X. The purpose here is to give an upper bound of the number of the irreducible components of the Segre locus of a projective variety in terms of its invariants. [ABSTRACT FROM AUTHOR]

In recent years, there have been several L1-norm-based linear projection methods and max–min-based dimensionality reduction methods, which show robustness to outliers and noises and show large margin for discrimination. In this paper, we propose L1-norm-based max–min large margin (MLM-L1) for linear projection-based dimensionality reduction. It makes use of the robustness of L1-norm to outliers and noises and the max–min idea for large margin. A non-greedy iterative algorithm (NMLM-L1) is proposed to solve the optimization problem of the proposed MLM-L1. Experiments on several face image databases show that the proposed method has better classification performance than its closely related methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a linear supervised dimension reduction method for classification settings: stochastic discriminant analysis (SDA). This method matches similarities between points in the projection space with those in a response space. The similarities are represented by transforming distances between points to joint probabilities using a transformation which resembles Student’s t-distribution. The matching is done by minimizing the Kullback–Leibler divergence between the two probability distributions. We compare the performance of our SDA method against several state-of-the-art methods for supervised linear dimension reduction. In our experiments, we found that the performance of the SDA method is often better and typically at least equal to the compared methods. We have made experiments with various types of data sets having low, medium, or high dimensions and quite different numbers of samples, and with both sparse and dense data sets. If there are several classes in the studied data set, the low-dimensional projections computed using our SDA method provide often higher classification accuracies than the compared methods. [ABSTRACT FROM AUTHOR]

In this paper, a new linear dimensionality reduction method named Two-Dimensional Discriminant Locality Preserving Projection Based on ℓ 1 -norm Maximization (2DDLPP-L1) is proposed for preprocessing of image data. 2DDLPP-L1 makes full use of the robustness of ℓ 1 -norm to noises and outliers. Furthermore, 2DDLPP-L1 is a 2D-based method which extracts image features directly from image matrices, avoiding instability and high complexity of matrix computation. Two graphs, separation graph and cohesiveness graph, are constructed with feature vectors as vertices to represent the inter-class separation and intra-class cohesiveness. An iterative algorithm with proof of convergence is proposed to solve the optimal projection matrix. Experiments on several face image databases demonstrate that the performance and robustness of 2DDLPP-L1 are better than its related methods. [ABSTRACT FROM AUTHOR]

Abstract: We associate with a k-tuple of hermitian N × N matrices a probability measure on supported on their joint numerical range: The joint numerical shadow of these matrices. When k =2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices. We apply this material to the theory of quantum information. Thus, we show that quantum maps on the set of quantum states defined by Kraus operators satisfying the identity resolution assumption shrink joint numerical ranges. [Copyright &y& Elsevier]

Abstract: Manifold learning have attracted considerable attention over the last decade. The most frequently used functional is the l 2-norm of the gradient of the function. In this paper, we consider the linear manifold learning problem by minimizing this functional with appropriate constraint. We provide theoretical analysis on both the functional and the constraint, which shows the affine hulls of the manifold and the connected components are essential to linear manifold learning problem. Based on the theoretical analysis, we introduce a novel linear manifold learning algorithm called approximately harmonic projection (AHP). Unlike canonical linear methods such as principal component analysis, our method is sensitive to the connected components. This makes our method especially applicable to data clustering. We conduct several experimental results on three real data sets to demonstrate the effectiveness of our proposed method. [Copyright &y& Elsevier]

Abstract: We consider the problem of locality preserving projections (LPP) in two-dimensional sense. Recently, LPP was proposed for dimensionality reduction, which can detect the intrinsic manifold structure of data and preserve the local information. As far as matrix data, such as images, are concerned, they are often vectorized for LPP algorithm to find the intrinsic manifold structure. While the dimension of matrix data is usually very high, LPP cannot be implemented because of the singularity of matrix. In this paper, we propose a method called two-dimensional locality preserving projections (2D-LPP) for image recognition, which is based directly on 2D image matrices rather than 1D vectors as conventional LPP does. From an algebraic procedure, we induce that 2D-LPP is related to two other linear projection methods, which are based directly on image matrix: 2D-PCA and 2D-LDA. 2D-PCA and 2D-LDA preserve the Euclidean structure of image space, while 2D-LPP finds an embedding that preserves local information and detects the intrinsic image manifold structure. To evaluate the performance of 2D-LPP, several experiments are conducted on the ORL face database, the Yale face database and a digit dataset. The high recognition rates and speed show that 2D-LPP achieves better performance than 2D-PCA and 2D-LDA. Experiments even show that conducting PCA after 2D-LPP achieves higher recognition than LPP at the same dimension of feature spaces. [Copyright &y& Elsevier]

In this paper, a novel method for palmprint recognition, called Fisherpalms, is proposed. In this method, each pixel of a palmprint image is considered as a coordinate in a high-dimensional image space. A linear projection based on Fisher’s linear discriminant is used to project palmprints from this high-dimensional original palmprint space to a significantly lower dimensional feature space (Fisherpalm space), in which the palmprints from the different palms can be discriminated much more efficiently. The relationship between the recognition accuracy and the resolution of the palmprint image is also investigated. The experimental results show that, in the proposed method, the palmprint images with resolution 32 × 32 are optimal for medium security biometric systems while those with resolution 64 × 64 are optimal for high security biometric systems. High accuracies (>99%) have been obtained by the proposed method and the speed of this method (responding time⩽0.4 s) is rapid enough for real-time palmprint recognition. [Copyright &y& Elsevier]

The miRNA plays a key role in the biological process and it has close relationship with disease. The wet experiment to test the association between miRNA and disease is time-consuming and costly, so semi-supervised learning based computational methods have been proposed to predict potential miRNAs associated with disease. However, these methods cannot fully utilize the local structure and global structure information of miRNA-disease association data, and the prediction performance can be improved further. In this work, we propose a novel approach for miRNA-disease association prediction by using multi-layer linear projection (MLPMDA). Firstly, we use the top n neighbors of miRNA and disease to update miRNA-disease association matrix, respectively, to employ the local structure and reduce the sparsity effect. Secondly, we define a computing layer to which a heterogeneous matrix composing of the updated association matrix and integrated miRNA similarity and integrated disease similarity is fed, and output predicted scores for miRNA-disease associations by linear projection method. Thirdly, we design multiple computing layers where the heterogeneous matrix input for the current layer is constructed based on the predicted miRNA-disease association scores from the last layer. Finally, we capture possible missing miRNA-disease associations by integrating prediction scores from each single view. We obtain AUC values 0.9847, 0.9883, 0.9899 for one dataset under 2-fold, 5-fold and 10-fold cross-validations, respectively, and their corresponding AUPR values are 0.7777, 0.7806 and 0.7518, which outperforms seven state-of-the-art methods. At last, we predict the potential miRNAs associated with three diseases, most of which are verified with some evidence. [ABSTRACT FROM AUTHOR]