Spaces Of Rank Research Articles

HALOC: Hardware-Aware Automatic Low-Rank Compression for Compact Neural Networks

Low-rank compression is an important model compression strategy for obtaining compact neural network models. In general, because the rank values directly determine the model complexity and model accuracy, proper selection of layer-wise rank is very critical and desired. To date, though many low-rank compression approaches, either selecting the ranks in a manual or automatic way, have been proposed, they suffer from costly manual trials or unsatisfied compression performance. In addition, all of the existing works are not designed in a hardware-aware way, limiting the practical performance of the compressed models on real-world hardware platforms. To address these challenges, in this paper we propose HALOC, a hardware-aware automatic low-rank compression framework. By interpreting automatic rank selection from an architecture search perspective, we develop an end-to-end solution to determine the suitable layer-wise ranks in a differentiable and hardware-aware way. We further propose design principles and mitigation strategy to efficiently explore the rank space and reduce the potential interference problem. Experimental results on different datasets and hardware platforms demonstrate the effectiveness of our proposed approach. On CIFAR-10 dataset, HALOC enables 0.07% and 0.38% accuracy increase over the uncompressed ResNet-20 and VGG-16 models with 72.20% and 86.44% fewer FLOPs, respectively. On ImageNet dataset, HALOC achieves 0.9% higher top-1 accuracy than the original ResNet-18 model with 66.16% fewer FLOPs. HALOC also shows 0.66% higher top-1 accuracy increase than the state-of-the-art automatic low-rank compression solution with fewer computational and memory costs. In addition, HALOC demonstrates the practical speedups on different hardware platforms, verified by the measurement results on desktop GPU, embedded GPU and ASIC accelerator.

Cohomogeneity one actions on symmetric spaces of noncompact type and higher rank

We develop a new structural result for cohomogeneity one actions on (not necessarily irreducible) symmetric spaces of noncompact type and arbitrary rank. We apply this result to classify cohomogeneity one actions on the spaces SLn(R)/SOn, n≥2, up to orbit equivalence. We also reduce the classification problem on a reducible space to the classification on each one of its irreducible factors, which in particular allows to classify cohomogeneity one actions on any finite product of hyperbolic spaces.

Classification of homogeneous hypersurfaces in some noncompact symmetric spaces of rank two

We classify, up to isometric congruence, the homogeneous hypersurfaces in the Riemannian symmetric spaces textrm{SL}(3,mathbb {H})/textrm{Sp}(3), textrm{SO}(5,mathbb {C})/textrm{SO}(5), and textrm{Gr}^*(2,mathbb {C}^{n+4}) = textrm{SU}(n+2,2)/textrm{S}(textrm{U}(n+2)textrm{U}(2)), , n geqslant 1.

The Climate Toolbox—A Microsoft® Excel® Based Tool for Assessing and Comparing the Effects of Internal Climates on Museum Artefacts

This paper describes the Climate Toolbox—a set of utilities for assessing and comparing real internal climates in terms of hazards they pose to museum collections. The Toolbox is a Microsoft® Excel® workbook with complex VBA codes optimised to process large datasets efficiently and quickly. The Toolbox takes user-provided climate data (up to 8000 sets of temperature, T, and relative humidity, rh, data) and characterises the climates in terms of critical parameters for stresses and mechanical damage, risk and magnitude of mould, relative permanence compared to (selectable) reference specifications, the proportion of data lying within and without selectable specification ranges, and the proportion of rh data swings exceeding a given magnitude. The interface is easily customisable, allowing users to input desired specification ranges, insert opening and closing times (to allow for different temperature specifications for periods when a space is open and occupied to when it is closed and empty), selectively change material critical strain parameters, and adjust the cycle periods for stress analysis. Results are summarised in a range of different graphical and tabular outputs and can be processed further to compare and rank spaces for their suitability to house different collections.

Packings and Steiner systems in polar spaces

A finite classical polar space of rank \(n\) consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that \(n\) is the maximal dimension of such a subspace. A \(t\)-Steiner system in a finite classical polar space of rank \(n\) is a collection \(Y\) of totally isotropic \(n\)-spaces such that each totally isotropic \(t\)-space is contained in exactly one member of \(Y\). Nontrivial examples are known only for \(t=1\) and \(t=n-1\). We give an almost complete classification of such \(t\)-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces.Mathematics Subject Classifications: 51E23, 05E30, 33C80Keywords: Association schemes, codes, polar spaces, Steiner systems

An explicit lifting construction of CAP forms on O(1,5)

In this paper, we explicitly construct nontempered cusp forms on the orthogonal group O(1,5) of signature [Formula: see text]. Given a definite quaternion algebra [Formula: see text] over [Formula: see text], the orthogonal group is attached to the indefinite quadratic space of rank 6 with the anisotropic part defined by the reduced norm of [Formula: see text]. Our construction can be viewed as a generalization of the previous work by the first two authors joint with Masanori Muto to the case of any definite quaternion algebras, for which we note that the work just mentioned takes up the case where the discriminant of [Formula: see text] is two. Unlike the previous work the method of the construction is to consider the theta lifting from Maass cusp forms to O(1,5), following the formulation by Borcherds. The cuspidal representations generated by our cusp forms are studied in detail. We determine all local components of the cuspidal representations and show that our cusp forms are CAP forms.

Ternary algebras associated with irreducible tensor representations of SO(3) and the quark model

We show that each irreducible tensor representation of weight 2 of the rotation group in the space of rank 3 tensors over three-dimensional space gives rise to an associative algebra with unity. We find the algebraic conditions to be satisfied by the generators of these algebras. Part of these relations is of binary, and another part is of ternary type. The structure of the latter one is based on the use of the cyclic group [Formula: see text] generated by the primitive cubic root of unity given by [Formula: see text]. The subspace of each algebra spanned by triple products of generators is five-dimensional (5D) and is identical with the space of irreducible tensor representation of weight 2 of the rotation group [Formula: see text]. We define a Hermitian scalar product in this 5D subspace and construct its orthonormal basis in terms of triple products of generators. Then we find an explicit formula for the Lie algebra homomorphism [Formula: see text]. We suggest that the algebras constructed in this way, with binary and ternary constitutive relations, may find applications in the quark model and the Grand Unified Theories.

Decompositions of moduli spaces of vector bundles and graph potentials

Abstract We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau–Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.

The cohomology of free loop spaces of rank $2$ flag manifolds

The cohomology of free loop spaces of rank $2$ flag manifolds

Monads and moduli components for stable rank 2 bundles with odd determinant on the projective space

We propose a three-step program for the classification of stable rank 2 bundles on the projective space $\mathbb{P}^3$ inspired by an article by Hartshorne and Rao. While this classification program has been successfully completed for stable rank 2 bundles with even determinant and $c_2\le5$, much less is known for bundles with odd determinant. After revising the known facts about these objects, we list all possible spectra and minimal monads for stable rank 2 bundles with odd determinant and $c_2\le8$. We provide a full classification of all bundles with positive minimal monads, provide a negative answer to a question raised by Hartshorne and Rao, and describe new irreducible components of the moduli spaces of stable rank 2 bundles with odd determinant and $c_2=6,8$.

Hitchin WKB-problem andP = Wconjecture in lowest degree for rank 2 over the 5-punctured sphere

AbstractWe use abelianization of Higgs bundles away from the ramification divisor and fiducial solutions to analyze the large-scale behavior of Fenchel–Nielsen co-ordinates on the moduli space of rank 2 Higgs bundles on the Riemann sphere with five punctures. We solve the related Hitchin WKB problem and prove the lowest degree weighted pieces of the P = W conjecture in this case.

Foliations by curves on threefolds

AbstractWe study the conormal sheaves and singular schemes of one‐dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ‐stable whenever the tangent bundle is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on and on a smooth quadric hypersurface . Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q3.

Moduli space of irregular rank two parabolic bundles over the Riemann sphere and its compactification

In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the moduli spaces of refined parabolic bundles when the dimensions of the moduli spaces are two. These are related to geometry of some weak del Pezzo surfaces.

Foliated Hopf hypersurfaces in complex hyperbolic quadrics

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kähler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kähler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kähler manifolds.

The moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points

We explicitly describe the moduli space M s ( X , 3 ) M^s(X,3) of stable rank 2 parabolic bundles over an elliptic curve X X with trivial determinant bundle and 3 marked points. Specifically, we exhibit M s ( X , 3 ) M^s(X,3) as a blow-up of an embedded elliptic curve in ( C P 1 ) 3 (\mathbb {CP}^1)^3 . The moduli space M s ( X , 3 ) M^s(X,3) can also be interpreted as the S U ( 2 ) SU(2) character variety of the 3-punctured torus. Our description of M s ( X , 3 ) M^s(X,3) reproduces the known Poincaré polynomial for this space.

Geometry of some moduli of bundles over a very general sextic surface for small second Chern classes and Mestrano-Simpson conjecture

Let S⊂P3 be a very general sextic surface over complex numbers. Let M(H,c2) be the moduli space of rank 2 stable bundles on S with fixed first Chern class H and second Chern class c2. In this article we study the configuration of certain locally complete intersection zero-dimensional subschemes on S satisfying Cayley-Bacharach property, which leads to the existence of non-trivial sections of a general member of each component of the moduli space for small c2. Using this study we will make an attempt to prove Mestrano-Simpson conjecture on the number of irreducible components of M(H,11) and prove the conjecture partially. We will also show that M(H,c2) is irreducible for c2≤10.

Laplace Eigenfunctions on Riemannian Symmetric Spaces and Borel–Weil Theorem

AbstractWe identify a geometric relation between the Laplace–Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel–Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating with each compact Riemannian symmetric space, via Marsden–Weinstein reduction, a generalized flag manifold that covers the space parametrizing all of its maximal totally geodesic tori. In the process, we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in detail the examples of the classical simply connected spaces of rank one and the space $SU(3)/SO(3)$. In the 2nd part of the paper, with the aid of harmonic polynomials, we induce Laplace–Beltrami eigenfunctions on the symmetric space from holomorphic sections of the associated line bundle on the generalized flag manifold. In the examples we consider, we show that our construction provides all of the eigenfunctions.

Tate’s thesis in the de Rham setting

We calculate the category of D D -modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3 d 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures.

A non-parametric cause-effect testing for environmental variables - method and application.

Traditional similarity or resemblance indexes are insufficient to directly reveal the cause-effect relations between environmental variables. Even the typical regression methods are not persuasive enough, since they rely on the assumptions about the data distribution and thus they are not really suitable for small amount of data. In this research, we devise a method to measure the strength of cause and effect (SCE), which is then turned into a non-parametric statistic. By analysing the empirical environmental data from the European Union, we calculate the SCE of these related variables. In addition, by constructing the ranking space and calculating the statistic distribution, we further specify the critical levels and values to conduct the cause-effect testing of these variables. The results show some sectoral activities do, to some degree, directly affect the quality of water and air. Moreover, there is a very clear-cut cause-effect relation between water quality and biodiversity. These results shall provide the policy makers with some ideas regarding the relations between environmental variables.Supplementary InformationThe online version contains supplementary material available at 10.1007/s11356-022-21423-3.

Hodge conjecture for the moduli space of semi-stable sheaves over a nodal curve

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We also compute the algebraic Poincaré polynomial of the associated cohomology ring.