Singular Vector Spaces Research Articles

Fast computation of error bounds for all eigenpairs of a Hermitian and all singular pairs of a rectangular matrix with emphasis on eigen- and singular value clusters

We present verification methods to compute error bounds for all eigenvectors of a Hermitian matrix as well as for all singular vectors of a rectangular real or complex matrix. In case of clusters these are bounds for an orthonormal basis of the invariant subspace or singular vector space, respectively. Individual error bounds for all eigenvalues and singular values including clustered and/or multiple ones are computed as well. The computed bounds do contain the true result with mathematical certainty, and the algorithms apply to interval data as well. In that case the computed bounds are true for every real/complex matrix within the tolerances. The computational complexity to compute inclusions of all eigen/singular pairs of an n×n matrix or m×n matrix is O(n3) or O(mn2) operations, respectively.

Verified inclusions for a nearest matrix of specified rank deficiency via a generalization of Wedin’s $$\sin (\theta )$$ theorem

For an $$m \times n$$ matrix A, the mathematical property that the rank of A is equal to r for $$0< r < \min (m,n)$$ is an ill-posed problem. In this note we show that, regardless of this circumstance, it is possible to solve the strongly related problem of computing a nearby matrix with at least rank deficiency k in a mathematically rigorous way and using only floating-point arithmetic. Given an integer k and a real or complex matrix A, square or rectangular, we first present a verification algorithm to compute a narrow interval matrix $$\varDelta $$ with the property that there exists a matrix within $$A-\varDelta $$ with at least rank deficiency k. Subsequently, we extend this algorithm for computing an inclusion for a specific perturbation E with that property but also a minimal distance with respect to any unitarily invariant norm. For this purpose, we generalize Wedin’s $$\sin (\theta )$$ theorem by removing its orthogonality assumption. The corresponding result is the singular vector space counterpart to Davis and Kahan’s generalized $$\sin (\theta )$$ theorem for eigenspaces. The presented verification methods use only standard floating-point operations and are completely rigorous including all possible rounding errors and/or data dependencies.

A method for selecting the relevant dimensions for high-dimensional classification in singular vector spaces

In this paper, we give a new feature selection algorithm for the binary class classification problem in sparse high-dimensional spaces. Singular value decomposition (SVD) is a popular dimension reduction method in higher-dimensional classification. The traditional SVD method begins by ranking the Singular Dimensions (SDs) from largest singular value to the smallest. However, when the number of signals is fewer than the number of noise, the first few ranked SDs are not necessarily the best for classification. We demonstrate, theoretically and empirically, that our method efficiently selects the SDs most appropriate for classification and significantly reduces the misclassification error. We also apply our method to real data text mining applications.

Structural Convergence Results for Approximation of Dominant Subspaces from Block Krylov Spaces

This paper is concerned with approximating the dominant left singular vector space of a real matrix $A$ of arbitrary dimension, from block Krylov spaces generated by the matrix ${A}{A}^T$ and the b...

Identifying Multi-Dimensional Co-Clusters in Tensors Based on Hyperplane Detection in Singular Vector Spaces.

Co-clustering, often called biclustering for two-dimensional data, has found many applications, such as gene expression data analysis and text mining. Nowadays, a variety of multi-dimensional arrays (tensors) frequently occur in data analysis tasks, and co-clustering techniques play a key role in dealing with such datasets. Co-clusters represent coherent patterns and exhibit important properties along all the modes. Development of robust co-clustering techniques is important for the detection and analysis of these patterns. In this paper, a co-clustering method based on hyperplane detection in singular vector spaces (HDSVS) is proposed. Specifically in this method, higher-order singular value decomposition (HOSVD) transforms a tensor into a core part and a singular vector matrix along each mode, whose row vectors can be clustered by a linear grouping algorithm (LGA). Meanwhile, hyperplanar patterns are extracted and successfully supported the identification of multi-dimensional co-clusters. To validate HDSVS, a number of synthetic and biological tensors were adopted. The synthetic tensors attested a favorable performance of this algorithm on noisy or overlapped data. Experiments with gene expression data and lineage data of embryonic cells further verified the reliability of HDSVS to practical problems. Moreover, the detected co-clusters are well consistent with important genetic pathways and gene ontology annotations. Finally, a series of comparisons between HDSVS and state-of-the-art methods on synthetic tensors and a yeast gene expression tensor were implemented, verifying the robust and stable performance of our method.

Feigin–Frenkel center in types B, C and D

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).

Classification of Bipartite and Tripartite Qutrit Entanglement Under SLOCC

We classify biqutrit and triqutrit pure states under stochastic local operations and classical communication. By investigating the right singular vector spaces of the coefficient matrices of the states, we obtain explicitly two equivalent classes of biqutrit states and twelve equivalent classes of triqutrit states respectively.

Video summarization and retrieval using singular value decomposition

In this paper, we propose novel video summarization and retrieval systems based on unique properties from singular value decomposition (SVD). Through mathematical analysis, we derive the SVD properties that capture both the temporal and spatial characteristics of the input video in the singular vector space. Using these SVD properties, we are able to summarize a video by outputting a motion video summary with the user-specified length. The motion video summary aims to eliminate visual redundancies while assigning equal show time to equal amounts of visual content for the original video program. On the other hand, the same SVD properties can also be used to categorize and retrieve video shots based on their temporal and spatial characteristics. As an extended application of the derived SVD properties, we propose a system that is able to retrieve video shots according to their degrees of visual changes, color distribution uniformities, and visual similarities.

Singular Dimensions of the¶N= 2 Superconformal Algebras. I

Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N= 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of singular vector. As a consequence we prove the conjecture of Gato-Rivera and Rosado on the possible existing types of topological singular vectors (4 in chiral Verma modules and 29 in complete Verma modules). Interestingly, we have found two-dimensional spaces of singular vectors at level 1. Finally, by using the topological twists and the spectral flows, we also obtain the maximal dimensions of the singular vector spaces for the Neveu–Schwarz N= 2 algebra (0, 1 or 2) and for the Ramond N= 2 algebra (0, 1, 2 or 3).

Singular Dimensions¶of the N= 2 Superconformal Algebras II:¶The Twisted N= 2 Algebra

We introduce a suitable adapted ordering for the twisted N=2 superconformal algebra (i.e. with mixed boundary conditions for the fermionic fields). We show that the ordering kernels for complete Verma modules have two elements and the ordering kernels for G-closed Verma modules just one. Therefore, spaces of singular vectors may be two-dimensional for complete Verma modules whilst for G-closed Verma modules they can only be one-dimensional. We give all singular vectors for the levels 1/2, 1, and 3/2 for both complete Verma modules and G-closed Verma modules. We also give explicit examples of degenerate cases with two-dimensional singular vector spaces in complete Verma modules. General expressions are conjectured for the relevant terms of all (primitive) singular vectors, i.e. for the coefficients with respect to the ordering kernel. These expressions allow to identify all degenerate cases as well as all G-closed singular vectors. They also lead to the discovery of subsingular vectors for the twisted N=2 superconformal algebra. Explicit examples of these subsingular vectors are given for the levels 1/2, 1, and 3/2. Finally, the multiplication rules for singular vector operators are derived using the ordering kernel coefficients. This sets the basis for the analysis of the twisted N=2 embedding diagrams.

Singular Dimensions of the¶

Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N=2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of singular vector. As a consequence we prove the conjecture of Gato-Rivera and Rosado on the possible existing types of topological singular vectors (4 in chiral Verma modules and 29 in complete Verma modules). Interestingly, we have found two-dimensional spaces of singular vectors at level 1. Finally, by using the topological twists and the spectral flows, we also obtain the maximal dimensions of the singular vector spaces for the Neveu-Schwarz N=2 algebra (0, 1 or 2) and for the Ramond N=2 algebra (0, 1, 2 or 3).

Image restoration in singular vector space by the method of convex projections

In singular vector space produced by singular value decomposition, the effect of noise on the expansion coefficients of an object is clear. A new image restoration method estimates the expansion coefficients corrupted by noise from the expansion coefficients corresponding to large singular values by incorporating a priori information about the noise and the object. This information can be described as closed convex sets. Iterative projections onto the convex sets are performed to obtain a restored image. The effectiveness of the method in singular vector space is made clear through computer simulations for 2-D bandlimited degraded images.

Image restoration by iterative estimation of the expansion coefficients of an object in singular vector space

The singular-value-decomposition (SVD) method has been utilized in an algebraic image-reconstruction problem. The effect of noise on a reconstructed image can be understood from the expansion coefficients of an object in singular vector space calculated by the SVD algorithm. We propose a method in which the coefficients corresponding to small singular values are estimated by using a priori information about the object. Results of computer simulations for iterative restoration of an image subjected to low-pass frequency filtering show the utility of the method.