742 research outputs found
On solutions of matrix equation AXB + CYD = F
AbstractIn this paper, the matrix equation with two unknown matrices X, Y of form AXB + CYD = F is discussed. By applying the canonical correlation decomposition (CCD) of matrix pairs, we obtain expressions of the least-squares solutions of the matrix equation, and sufficient and necessary conditions for the existence and uniqueness of the solutions. We also derive a general form of the solutions. We also study the least-squares Hermitian (skew-Hermitian) solutions of equation AXAH + CYCH = F
Dibaryons from Exceptional Collections
We discuss aspects of the dictionary between brane configurations in del
Pezzo geometries and dibaryons in the dual superconformal quiver gauge
theories. The basis of fractional branes defining the quiver theory at the
singularity has a K-theoretic dual exceptional collection of bundles which can
be used to read off the spectrum of dibaryons in the weakly curved dual
geometry. Our prescription identifies the R-charge R and all baryonic U(1)
charges Q_I with divisors in the del Pezzo surface without any Weyl group
ambiguity. As one application of the correspondence, we identify the cubic
anomaly tr R Q_I Q_J as an intersection product for dibaryon charges in large-N
superconformal gauge theories. Examples can be given for all del Pezzo surfaces
using three- and four-block exceptional collections. Markov-type equations
enforce consistency among anomaly equations for three-block collections.Comment: 47 pages, 11 figures, corrected ref
Comparison theorems for conjugate points in sub-Riemannian geometry
We prove sectional and Ricci-type comparison theorems for the existence of
conjugate points along sub-Riemannian geodesics. In order to do that, we regard
sub-Riemannian structures as a special kind of variational problems. In this
setting, we identify a class of models, namely linear quadratic optimal control
systems, that play the role of the constant curvature spaces. As an
application, we prove a version of sub-Riemannian Bonnet-Myers theorem and we
obtain some new results on conjugate points for three dimensional
left-invariant sub-Riemannian structures.Comment: 33 pages, 5 figures, v2: minor revision, v3: minor revision, v4:
minor revisions after publicatio
Geometric methods on low-rank matrix and tensor manifolds
In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors
Integrate the GM(1,1) and Verhulst models to predict software stage effort
This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ 2009 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.Software effort prediction clearly plays a crucial role in software project management. In keeping with more dynamic approaches to software development, it is not sufficient to only predict the whole-project effort at an early stage. Rather, the project manager must also dynamically predict the effort of different stages or activities during the software development process. This can assist the project manager to reestimate effort and adjust the project plan, thus avoiding effort or schedule overruns. This paper presents a method for software physical time stage-effort prediction based on grey models GM(1,1) and Verhulst. This method establishes models dynamically according to particular types of stage-effort sequences, and can adapt to particular development methodologies automatically by using a novel grey feedback mechanism. We evaluate the proposed method with a large-scale real-world software engineering dataset, and compare it with the linear regression method and the Kalman filter method, revealing that accuracy has been improved by at least 28% and 50%, respectively. The results indicate that the method can be effective and has considerable potential. We believe that stage predictions could be a useful complement to whole-project effort prediction methods.National Natural Science Foundation of
China and the Hi-Tech Research
and Development Program of Chin
Optimal approximate matrix product in terms of stable rank
We prove, using the subspace embedding guarantee in a black box way, that one
can achieve the spectral norm guarantee for approximate matrix multiplication
with a dimensionality-reducing map having
rows. Here is the maximum stable rank, i.e. squared ratio of
Frobenius and operator norms, of the two matrices being multiplied. This is a
quantitative improvement over previous work of [MZ11, KVZ14], and is also
optimal for any oblivious dimensionality-reducing map. Furthermore, due to the
black box reliance on the subspace embedding property in our proofs, our
theorem can be applied to a much more general class of sketching matrices than
what was known before, in addition to achieving better bounds. For example, one
can apply our theorem to efficient subspace embeddings such as the Subsampled
Randomized Hadamard Transform or sparse subspace embeddings, or even with
subspace embedding constructions that may be developed in the future.
Our main theorem, via connections with spectral error matrix multiplication
shown in prior work, implies quantitative improvements for approximate least
squares regression and low rank approximation. Our main result has also already
been applied to improve dimensionality reduction guarantees for -means
clustering [CEMMP14], and implies new results for nonparametric regression
[YPW15].
We also separately point out that the proof of the "BSS" deterministic
row-sampling result of [BSS12] can be modified to show that for any matrices
of stable rank at most , one can achieve the spectral norm
guarantee for approximate matrix multiplication of by deterministically
sampling rows that can be found in polynomial
time. The original result of [BSS12] was for rank instead of stable rank. Our
observation leads to a stronger version of a main theorem of [KMST10].Comment: v3: minor edits; v2: fixed one step in proof of Theorem 9 which was
wrong by a constant factor (see the new Lemma 5 and its use; final theorem
unaffected
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