Generalized Singular Value Decompositions for Tensors and Their Applications

In this paper, we consider the generalized singular value decompositions for two tensors via the T-product. We investigate and discuss in detail the structures of the quotient singular value decomposition (T-QSVD) and product singular value decomposition (T-PSVD) for two tensors. The algorithms are presented with numerical examples illustrating the results. For applications, we consider color image watermarking processing with T-QSVD and T-PSVD. There are two advantages to T-QSVD and T-PSVD approaches on color watermark processing: two color watermarks can be processed simultaneously and only one key needs to be saved.     

Text Indexing and Dictionary Matching with One Error

The indexing problem is where a text is preprocessed and subsequent queries of the form “Find all occurrences of pattern P in the text” are answered in time proportional to the length of the query and the number of occurrences. In the dictionary matching problem a set of patterns is preprocessed and subsequent queries of the form “Find all occurrences of dictionary patterns in text T” are answered in time proportional to the length of the text and the number of occurrences.There exist efficient worst-case solutions for the indexing problem and the dictionary matching problem, but none that find approximate occurrences of the patterns, i.e., where the pattern is within a bound edit (or Hamming) distance from the appropriate text location.In this paper we present a uniform deterministic solution to both the indexing and the general dictionary matching problem with one error. We preprocess the data in time O(n log² n), where n is the text size in the indexing problem and the dictionary size in the dictionary matching problem. Our query time for the indexing problem is O(m log n log log n + tocc), where m is the query string size and tocc is the number of occurrences. Our query time for the dictionary matching problem is O(n log³ d log log d + tocc), where n is the text size and d the dictionary size. The time bounds above apply to both bounded and unbounded alphabets.     

Regression Modeling and File Matching Using Possibly Erroneous Matching Variables

Many analyses require linking records from two databases comprising overlapping sets of individuals. In the absence of unique identifiers, the linkage procedure often involves matching on a set of categorical variables, such as demographics, common to both files. Typically, however, the resulting matches are inexact: some cross-classifications of the matching variables do not generate unique links across files. Further, the variables used for matching can be subject to reporting errors, which introduce additional uncertainty in analyses. We present a Bayesian file matching methodology designed to estimate regression models and match records simultaneously when categorical variables used for matching are subject to errors. The method relies on a hierarchical model that includes (1) the regression of interest involving variables from the two files given a vector indicating the links, (2) a model for the linking vector given the true values of the variables used for matching, (3) a model for reported values of the variables used for matching given their true values, and (4) a model for the true values of the variables used for matching. We describe algorithms for sampling from the posterior distribution of the model. We illustrate the methodology using artificial data and data from education records in the state of North Carolina.     

Approximation of High-Dimensional Rank One Tensors

Many real world problems are high-dimensional in that their solution is a function which depends on many variables or parameters. This presents a computational challenge since traditional numerical techniques are built on model classes for functions based solely on smoothness. It is known that the approximation of smoothness classes of functions suffers from the so-called ‘curse of dimensionality’. Avoiding this curse requires new model classes for real world functions that match applications. This has led to the introduction of notions such as sparsity, variable reduction, and reduced modeling. One theme that is particularly common is to assume a tensor structure for the target function. This paper investigates how well a rank one function f(x ₁,…,x _d )=f ₁(x ₁)?f _d (x _d ), defined on Ω=[0,1] ^d can be captured through point queries. It is shown that such a rank one function with component functions f _j in $W^{r}_{\infty}([0,1])$ can be captured (in L _∞) to accuracy O(C(d,r)N ^?r ) from N well-chosen point evaluations. The constant C(d,r) scales like d ^dr . The queries in our algorithms have two ingredients, a set of points built on the results from discrepancy theory and a second adaptive set of queries dependent on the information drawn from the first set. Under the assumption that a point z∈Ω with nonvanishing f(z) is known, the accuracy improves to O(dN ^?r ).     

Classification Using Generalized Partial Least Squares

Advances in computational biology have made simultaneous monitoring of thousands of features possible. The high throughput technologies not only bring about a much richer information context in which to study various aspects of gene function, but they also present the challenge of analyzing data with a large number of covariates and few samples. As an integral part of machine learning, classification of samples into two or more categories is almost always of interest to scientists. We address the question of classification in this setting by extending partial least squares (PLS), a popular dimension reduction tool in chemometrics, in the context of generalized linear regression, based on a previous approach, iteratively reweighted partial least squares, that is, IRWPLS. We compare our results with two-stage PLS and with other classifiers. We show that by phrasing the problem in a generalized linear model setting and by applying Firth's procedure to avoid (quasi)separation, we often get lower classification error rates.     

Using Signature Hashing for Approximate String Matching

The objective is to demonstrate the expediency of using compressed inverted files with a signature-hashing dictionary for approximate string matching. A comparison of different types of dictionaries is performed and a method based on keyword signature hashing is described. In conclusion, we report the comparative characteristics (search speed, index size, indexing speed) for our search system using compressed inverted files with keyword signature hashing and the Glimpse search freeware.     

Linear preservers of orthogonal decomposable Tensors

Let G be a subgroup of the symmetric group S_m and V be an n-dimensional unitary space where nm. Let V(G) be the symmetry class of tensors over V associated with G and the identity character. Let D(G) be the set of all decomposable elements of V(G) and O(G) be its subset consisting of all nonzero decomposable tensors x ₁ ^?…^? x_m such that {x ₁,…,x_m } is an orthogonal set. In this paper we study the structure of linear mappings on V(G) that preserve one of the following subsets: (i)O(G), (ii) D(G)\(O(G)?{0}).     

Smoothing Inversion of Fourier Series Using Generalized Cross-Validation

Let f be a 1-periodic, smooth function and $\hat f_k(k\in {\rm Z})$ its Fourier coefficients. The reconstruction of f from a given sequence of perturbed values of $\hat f_k$ is an ill-posed problem. Applying Tihonov’s regularization method, this problem can be solved in a stable manner. The efficiency of this method is closely related to the optimal choice of the smoothing parameter. In this paper we compute an optimal smoothing parameter by the method of generalized cross-validation. The related algorithm is tested in some numerical examples. Finally the results are extended to the bivariate case.     

Local Stereology of Tensors of Convex Bodies

In this paper, we present local stereological estimators of Minkowski tensors defined on convex bodies in ? ^d . Special cases cover a number of well-known local stereological estimators of volume and surface area in ?³, but the general set-up also provides new local stereological estimators of various types of centres of gravity and tensors of rank two. Rank two tensors can be represented as ellipsoids and contain information about shape and orientation. The performance of some of the estimators of centres of gravity and volume tensors of rank two is investigated by simulation.     

Codazzi Tensors and the Topology of Surfaces

We introduce the notion of (0,m)-Codazzi tensors relative to an affine connection which extends the well known concept for m = 2. On compact Riemannian surfaces of genus we determine the dimension of the R-vector space of traceless Codazzi tensors, which depends on m and only, additionally we extend these result for genus zero. We give some exemplary applications to submanifolds in affine and in Riemannian geometry and to Weyl geometries.     

Structural Properties of Tensors and Complementarity Problems

In this paper, one of our main purposes is to prove the boundedness of the solution set of tensor complementarity problems such that the specific bounds depend only on the structural properties of such a tensor. To achieve this purpose, firstly, we prove that this class of structured tensors is strictly semi-positive. Subsequently, the strictly lower and upper bounds of operator norms are given for two positively homogeneous operators. Finally, with the help of the above upper bounds, we show that the solution set of tensor complementarity problems has the strictly lower bound. Furthermore, the upper bounds of spectral radius are obtained, which depends only on the principal diagonal entries of tensors.     

可合对称张量相等的条件

本文讨论了由一般对称化算子诱导的两非零可合对称张量相等的必要充分条件与可合对称张量为零的必要充分条件。     

A Unified Calculus Using the Generalized Bernoulli Polynomials

We introduce an α-calculus with the help of the generalized Bernoulli polynomials. The parameter α is the order of a Bessel function of the first kind. The differential α-calculus can be put in a general context where the concept of supporting function is an important tool for practical purposes. Our somewhat more restrictive point of view has the advantage of permitting a consistent definition of an α-integral with several interesting properties. It results in the possibility of expressing a remainder, in the aforementioned context, in a completely new form in our case.     

Composite Nonsmooth Optimization Using Approximate Generalized Gradient Vectors

A new approximation method is presented for directly minimizing a composite nonsmooth function that is locally Lipschitzian. This method approximates only the generalized gradient vector, enabling us to use directly well-developed smooth optimization algorithms for solving composite nonsmooth optimization problems. This generalized gradient vector is approximated on each design variable coordinate by using only the active components of the subgradient vectors; then, its usability is validated numerically by the Pareto optimum concept. In order to show the performance of the proposed method, we solve four academic composite nonsmooth optimization problems and two dynamic response optimization problems with multicriteria. Specifically, the optimization results of the two dynamic response optimization problems are compared with those obtained by three typical multicriteria optimization strategies such as the weighting method, distance method, and min–max method, which introduces an artificial design variable in order to replace the max-value cost function with additional inequality constraints. The comparisons show that the proposed approximation method gives more accurate and efficient results than the other methods.     

The Dimensions of Certain Symmetry Classes of Tensors

Commutative Rings, by Irving Kaplansky. Revised edition. The University of Chicago Press, Chicago and London, 1974, ix+182pp.($9.75)

Symmetry Groups and their Applications, by Willard Miller, Jr. Academic Press.