Set Of Singular Values Research Articles

Learning discriminative singular value decomposition representation for face recognition

Face representation is a critical step in face recognition. Recently, singular value decomposition (SVD) based representation methods have attracted researchers׳ attentions for their power of alleviating the facial variations. The SVD representation reveals that the SVD basis set is important for the recognition purpose and the corresponding singular values (SVs) are regulated to form a more effective representation image. However, there exists a common problem in the existing SVD based representation methods: they all empirically make a rule to regulate the SVs, which is obviously not optimal in theory. To address this problem, in this paper, we propose a novel method named learning discriminative singular value decomposition representation (LDSVDR) for face recognition. We build an individual SVD basis set for each image and then learn a common set of SVs by taking account of the information in the basis sets according to a discriminant criterion across the training images. The proposed model is solved by sequential quadratic programming (SQP) method. Extensive experiments are conducted on three popular face databases and the results demonstrate the effectiveness of our method when dealing with variations of illumination, occlusion, disguise and face sketch recognition task.

A separation theorem for entire transcendental maps

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix $p\in\N$ and assume that all dynamic rays which are invariant under $f^p$ land. An interior $p$-periodic point is a fixed point of $f^p$ which is not the landing point of any periodic ray invariant under $f^p$. Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above we show that rays which are invariant under $f^p$, together with their landing points, separate the plane into finitely many regions, each containing exactly one interior $p-$periodic point or one parabolic immediate basin invariant under $f^p$. This result generalizes the Goldberg-Milnor Separation Theorem for polynomials, and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel discs; that "hidden components" of a bounded Siegel disc have to be either wandering domains or preperiodic to the Siegel disc itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.

A new characterisation of the Eremenko-Lyubich class

The Eremenko-Lyubich class of transcendental entire functions with a bounded set of singular values has been much studied. We give a new characterisation of this class of functions. We also give a new result regarding direct singularities which are not logarithmic.

On the exceptional set in a conditional theorem of Littlewood

In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, “most” preimages of almost all points are concentrated in a small subset of the plane. In 1988, Lewis and Wu proved Littlewood’s conjecture. Using techniques from complex dynamics, we construct entire functions of finite order with a bounded set of singular values for which the set of exceptional preimages is infinite, with logarithmically growing cardinality.

Signatura of magic and Latin integer squares: isentropic clans and indexing

The 2010 study of the Shannon entropy of order nine Sudoku and Latin square matrices by Newton and DeSalvo [Proc. Roy. Soc. A 2010] is extended to natural magic and Latin squares up to order nine. We demonstrate that decimal and integer measures of the Singular Value sets, here named SV clans, are a powerful way of comparing different integer squares. Several complete sets of magic and Latin squares are included, including the order eight Franklin subset which is of direct relevance to magic square line patterns on chess boards. While early examples suggested that lower rank specimens had lower entropy, sufficient data is presented to show that some full rank cases with low entropy possess a set of singular values separating into a dominant group with the remainder much weaker. An effective rank measure helps understand these issues. We also introduce a new measure for integer squares based on the sum of the fourth powers of the singular values which appears to give a useful method of indexing both Latin and magic squares. This can be used to begin cataloging a ”library” of magical squares. Based on a video presentation in celebration of George Styan’s 75th at LINSTAT2012 and IWMS-21 on 19 July, 2012 at B ↪ edlewo, Poland.

A landing theorem for dynamic rays of geometrically finite entire functions

A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective curve consisting of escaping points of f that connects z_0 to infinity. (This curve is called a dynamic ray.) In fact, the assumption of finite order can be weakened considerably; for example, it is sufficient to assume that f can be written as a finite composition of finite-order functions.

Imaging diffraction points using the local image matrices generated in prestack migration

Energy-angle distributions in the local image matrix, which is a function of local incident and receiving angles at a subsurface point, are different for a planar reflector and a diffraction point. The former exhibits a linear distribution along a certain dip direction, whereas the latter shows a scattered distribution in the entire matrix. Therefore, the singular values of the local image matrix in the local angle domain indicate the energy distribution along different dip directions. The difference between the auto- and crosscorrelation coefficients of the sets of singular values between adjacent image points can be used to distinguish these two types of targets and obtain prestack images which contain only diffraction points. Using three synthetic examples, we found that the method can effectively image diffraction points. Seismic images of diffraction points may provide important information about geological discontinuities and diffraction points can be taken as secondary sources for potential applications in imaging subsalt structures which are poorly illuminated by primary waves.

Hausdorff dimensions of escaping sets of transcendental entire functions

Let $f$ and $g$ be transcendental entire functions, each with a bounded set of singular values, and suppose that $f$ and $g$ are affinely equivalent (that is, $g \circ \phi= \psi\circ f$, where $\phi,\psi:\C\to\C$ are affine). We show that the escaping sets of $f$ and $g$ have the same Hausdorff dimension. Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one.

Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Boettcher's theorem for transcendental entire functions in the Eremenko-Lyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are *quasiconformally equivalent* in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that an Eremenko-Lyubich class function f has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f and g which belong to the same parameter space are conjugate on their sets of escaping points.

Balanced truncation model reduction of second-order systems

In this paper we consider structure-preserving model reduction of second-order systems using a balanced truncation approach. Several sets of singular values are introduced for such systems, which lead to different concepts of balancing and different second-order balanced truncation methods. A comparison of these methods with other second-order balanced truncation techniques is presented. We also show that, in general, none of the existing structure-preserving balanced truncation methods for second-order systems preserves stability in the reduced models. Numerical examples are given that demonstrate the properties of the new methods.

An entire function with a Baker domain and sparsely distributed singular values

AbstractWe present an example of an entire function with a Baker domain such that the complement of the set of singular values of the function contains annuli of arbitrarily large width. This is the first known example of a function with this property.

The spectra and singular values of multidimensional integral operators with bihomogeneous kernels

We consider the multidimensional integral operators with bihomogeneous kernel invariant under all rotations. For truncated operators of the type we describe the limit behavior of the set of singular values and in the case when these operators are selfadjoint we describe the limit behavior of their spectra.

Siegel disks and periodic rays of entire functions

Let f be an entire function whose set of singular values is bounded and suppose that f has a Siegel disk U such that f|∂U is a homeomorphism. We show that U is bounded. Using a result of Herman, we deduce that if additionally the rotation number of U is Diophantine, then ∂U contains a critical point of f.

On a question of Eremenko concerning escaping components of entire functions

Let f be an entire function with a bounded set of singular values, and suppose furthermore that the postsingular set of f is bounded. We show that every component of the escaping set I(f) is unbounded. This provides a partial answer to a question of Eremenko.

Determination of the regularization level of truncated singular‐value decomposition inversion: The case of 1D inversion of MT data

ABSTRACTThe total mean‐square error (MSE) of the estimated model, defined as the sum of the standard model variance and the bias variance, is used to define the truncation level of the singular‐value decomposition to give a reasonable balance between model resolution and model variance. This balance is determined largely by the data and no further assumptions are necessary except that the bias terms are estimated sufficiently well. This principle has been tested on the 1D magnetotelluric inverse problem with special emphasis on high‐frequency radio magnetotelluric (RMT) data. Simulations clearly demonstrate that the method provides a good balance between resolution and variance. Starting from a homogeneous half‐space, the best solution is sought for a fixed set of singular values. The model variance is estimated from the sum of the inverse eigenvalues squared, up to a certain threshold, and the bias variance is estimated from the model projections on the remaining eigenvectors. By varying the threshold, the minimum of the MSE is found for an increasing number of fixed singular values until the number of active singular values becomes greater than or equal to the estimated number. As a side‐effect, the depth of penetration of a given set of measurements can be estimated very efficiently by simply noting at which depth the final model deviates little from the starting homogeneous half‐space model. A suite of synthetic data is inverted and an example of inversion of one site is shown to illustrate how the truncation is carried out as the non‐linear inversion process proceeds. A field example with a profile across a plume of contaminated groundwater in the Netherlands shows good agreement with the electrical resistivity obtained in a nearby borehole.

Non-recurrent meromorphic functions

We consider a transcendental meromorphic function $f$ belonging to the class ${\mathcal B}$ (with bounded set of singular values). We show that if the Julia set $J(f)$ is the whole complex plane ${\mathbb C}$, and the closure of the postcritical set $P(f

Resolving seismic anisotropy: Sparse matrix methods for geophysical inverse problems

Two techniques for the singular value decomposition (SVD) of large sparse matrices are directly applicable to geophysical inverse problems: subspace iteration and the Lanczos method. Both methods require very little in‐core storage and efficiently compute a set of singular values and singular vectors. A comparison of the singular value and vector estimates of these iterative approaches with the results of a conventional in‐core SVD algorithm demonstrates their accuracy. Hence, it is possible to conduct an assessment of inversion results for larger sparse inverse problems such as those arising in seismic tomography. As an example, we examine the resolution matrix associated with a crosswell seismic inversion of first arrival times for lateral variations in anisotropy. The application to a set of first arrival times from a crosswell survey at the Grimsel Laboratory emphasizes the utility of including anisotropy in a traveltime inversion. The isotropic component of the estimated velocity structure appears to be well constrained even when anisotropy is included in the inversion. In the case of the Grimsel experiment, we are able to resolve a fracture zone, in agreement with borehole fracture intersections. Elements of the resolution matrix reveal moderate averaging among anisotropy coefficients as well as between anisotropy coefficients and source‐receiver static terms. The information on anisotropy, such as the directions of maximum velocity, appears sensitive to lateral variations in velocity and must be interpreted with some caution.

An inverse problem for singular values and the Jacobian of the elementary symmetric functions

Given a nonnegative diagonal matrix D, how do we find a column y such that the augmented matrix [ D∣ y] has a prescribed set of singular values? (Conditions for existence are known.) This question is shown to lead to a system of linear equations with an interesting matrix, some of whose properties are studied.

A note on invariance of the eigenvalues, singular values, and norms of matrix products involving generalized inverses

It is shown that the set of singular values of the product AB - C, as well as any norm of this product, is invariant with respect to the choice of a generalized inverse B - if and only if AB - C is invariant itself. This is in contrast with the corresponding invariance property of the set of eigenvalues of ⊙AB -⊙C, for which the invariance of AB - C is not necessary.

THEOREMS ON THE TOPOLOGICAL EQUISINGULARITY OF FAMILIES OF ALGEBRAIC VARIETIES AND FAMILIES OF POLYNOMIAL MAPPINGS

In this paper we consider families of complex or real algebraic varieties. We prove that for almost all values of the parameters both the topology of the variety and its position in space will be the same. The set of singular values of the parameters is calculated constructively. In this paper we also isolate a class of families of polynomial mappings. For such families we prove the topological equivalence of almost all the mappings included in them. These results are applied to a proof of Zariski's theorem on the fundamental group of the complement to an algebraic hypersurface.