We classify six-dimensional Lie groups which admit a left-invariant half-flat SU(3)-structure and which split in a direct product of three-dimensional factors. Moreover, a complete list of those direct products is obtained which admit a left-invariant half-flat SU(3)-structure such that the three-dimensional factors ar…
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48 results for “orthogonal direct product”
Equivalent tests for SGD batch size selection found.
problem Finding equivalent tests for adaptive batch size selection in SGD.
method Norm and inner product/orthogonality tests equivalence demonstration.
result Norm and inner product/orthogonality tests are equivalent under specific conditions.
Characterizes non-degenerate cyclic metric Lie algebras.
problem Understanding the structure of non-solvable cyclic metric Lie algebras.
method Using sufficient conditions, cyclic quadruples, and double extension method.
result Complete characterization of non-degenerate cyclic metric Lie algebras.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
New differential geometry perspective on orthogonal RNNs.
problem Mitigating exploding and vanishing gradients in RNNs.
method Using tools from differential geometry, parameterizing vector fields via directional derivatives of scalar functions.
result Our approach achieves comparable or better results on benchmark tasks.
Based on the orthogonal Labastida-Mari{ñ}o-Ooguri-Vafa conjecture made by L. Chen & Q. Chen [5], we derive an infinite product formula for Chern-Simons partition functions, which generalizes the Liu-Peng's [19] recent results to the orthogonal case. Symmetry property of this new infinite product structure is also discu…
The paper proves rigidity for submanifolds in warped product manifolds.
problem Rigidity of submanifolds in warped product manifolds.
method Dihedral extremality and rigidity theorem for submanifolds with polyhedral boundary.
result Dihedral rigidity results for hyperbolic polyhedra in flat warped product spaces.
In every dimension we introduce a class of orthogonal graph-manifolds and prove that the fundamental group of any orthogonal graph-manifold quasi-isometrically embeds into a product of trees. As a consequence, we obtain that asymptotic and linearly-controlled asymptotic dimensions of such group are equal t…
Orthogonal almost complex structures on the Riemannian products of even-dimensional round spheresmath.DG
We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.
Geodesics in symmetrized bidisc have intrinsic orthogonality and distinguished directions.
problem Understanding orthogonality and distinguished geodesics in the symmetrized bidisc under the Carathéodory metric.
method Analyzing the complex tangent bundle of the symmetrized bidisc, splitting it into sharp and flat bundles, and defining orthogonality and geodesics.
result Geodesics in the symmetrized bidisc are orthogonal to flat geodesics, and the space is foliated by geodesics orthogonal to a fixed flat geodesic.
The twistor space of the sphere S^{2n} is an isotropic Grassmannian that fibers over S^{2n}. An orthogonal complex structure on a subdomain of S^{2n} (a complex structure compatible with the round metric) determines a section of this fibration with holomorphic image. In this paper, we use this correspondence to prove t…
We develop variation formulas for the quantities of extrinsic geometry for adapted variations of metrics on almost-product (e.g. foliated) Riemannian manifolds, and apply them to study the total mixed scalar curvature of a distribution -- analogue of the classical Einstein-Hilbert action. The mixed scalar curvature ${\…
Explores tensor products in hyperdimensional computing.
problem Understanding tensor products in hyperdimensional computing.
method Generalized results from graph embeddings to vector symbolic architectures and hyperdimensional computing.
result Tensor product is the most general and expressive representation with errorless unbinding and detection.
We present an algorithm for the decomposition of periodic financial return data into orthogonal factors of expected return and "systemic", "productive", and "nonproductive" risk. Generally, when the number of funds does not exceed the number of periods, the expected return of a portfolio is an affine function of its pr…
A new algorithm POGO optimizes thousands of orthogonal matrices efficiently.
problem Optimizing thousands of orthogonal constraints at scale is computationally expensive.
method Revisits Landing algorithm, uses modern adaptive optimizers, reduces hyperparameters.
result POGO optimizes thousands of orthogonal matrices in minutes, outperforming alternatives.
In optimization, the negative gradient of a function denotes the direction of steepest descent. Furthermore, traveling in any direction orthogonal to the gradient maintains the value of the function. In this work, we show that these orthogonal directions that are ignored by gradient descent can be critical in equilibri…
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
Calibration in 16D disproves Federer's product question.
problem Proving Federer's product question in 16D.
method Adapted Cayley calibration in R^16, constructing a specific calibration Φ.
result Product of two orthogonally supported calibrations is not always a calibration.
Inverted file and asymmetric distance computation (IVFADC) have been successfully applied to approximate nearest neighbor search and subsequently maximum inner product search. In such a framework, vector quantization is used for coarse partitioning while product quantization is used for quantizing residuals. In the ori…
Let be a Riemannian homogeneous space. For an orthogonal representation of on the Euclidean space , there corresponds the vector bundle with fiberwise inner product. Provided that is the direct sum of at most two representations which are either …
The paper generalizes the number of complex structures on metric Lie algebras.
problem How many orthogonal bi-invariant complex structures exist on metric Lie algebras?
method Developed a unique orthogonal decomposition into irreducible factors for metric Lie algebras.
result There are either 0 or 2^k such complex structures, with k the number of irreducible factors.
In higher dimensions, Schottky spaces have unique topological properties.
problem Characterize the topology of Schottky spaces in higher dimensions.
method Analyzing the fundamental group and homotopy properties of Schottky spaces in the borderline dimension.
result In the borderline dimension, the space is simply connected but has a dense open part with fundamental group a product of cyclic groups of order two.
Integral formulas for a Riemannian manifold with several orthogonal complementary distributionsmath.DG
The paper proves integral formulas for manifolds with multiple orthogonal distributions.
problem Understanding geometric properties of manifolds with multiple orthogonal distributions.
method Develops integral formulas for Riemannian manifolds with orthogonal complementary distributions.
result Generalizes known formulas for and applies to manifold splitting and immersions.
A new knot invariant measures crossings in three orthogonal directions.
problem Defining a new knot invariant for certain knot diagrams.
method Defining the simultaneous crossing number for knots with doubly transvergent diagrams.
result The limit of the ratio of the new invariant to the usual crossing number is at most 8.
Study Finsler metrics with vanishing Landsberg curvature.
problem Characterize Finsler metrics with specific curvature properties.
method Derive expressions for curvatures, solve PDEs, construct examples.
result Construct non-regular Landsberg metrics not of Berwald type.
An algorithm for efficient computation of equivariant neural network layers.
problem Efficiently computing with Brauer's group equivariant neural network layers.
method Category theoretic constructions and Kronecker product matrices.
result Significant reduction in computational cost compared to naive implementation.
Let and be Riemannian symmetric spaces and be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces with for such that . As a starting point, we describe how…
The paper studies geometric properties of Grassman manifolds within Euclidean spaces.
problem Understanding the geometric structure of Grassman manifolds.
method Analyzing Grassman manifold as a subset of Euclidean space and orthogonal projections.
result Explicit formulas for differential geometry of as a submanifold.
The paper introduces polarizations in symplectic and orthogonal settings.
problem Understanding polarizations in symplectic and orthogonal contexts.
method Exposition of symplectic and orthogonal polarizations, emphasizing symmetry.
result Grassmannians of polarizations in symplectic and orthogonal settings.
A new pruning method reduces neural network computation without retraining.
problem Efficiently reduce neural network computation while maintaining accuracy.
method Structured directional pruning via perturbation orthogonal projection.
result Achieves state-of-the-art pruned accuracy without retraining.
This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. S…
Deep networks without non-linearities are equivalent to shallow ones.
problem Training deep orthogonal linear networks with no non-linearity.
method Riemannian gradient descent and gradient descent on factorization.
result Training deep overparametrized networks is equivalent to shallow ones.
Efficient Orthogonal Tensor Decomposition, with an Application to Latent Variable Model Learningstat.ML
Decomposing tensors into orthogonal factors is a well-known task in statistics, machine learning, and signal processing. We study orthogonal outer product decompositions where the factors in the summands in the decomposition are required to be orthogonal across summands, by relating this orthogonal decomposition to the…
Holomorphic torsion invariant for log-Enriques surfaces derived from Borcherds products.
problem Holomorphic torsion invariant for log-Enriques surfaces.
method Introduced a holomorphic torsion invariant using Borcherds products.
result The invariant is given by the Petersson norm of an explicit Borcherds product.
Introduces new curvature concept for Kähler manifolds.
problem Optimizing curvature constraints for projective Kähler manifolds.
method Introduces weighted orthogonal Ricci curvature and proves vanishing theorems.
result Proves optimal curvature constraints for projective Kähler manifolds.
We study the structure group of a canonical algebraic curvature tensor built from a symmetric bilinear form, and show that in most cases it coincides with the isometry group of the symmetric form from which it is built. Our main result is that the structure group of the direct sum of such canonical algebraic curvature …
MOBO-OSD optimizes multi-objective functions using orthogonal search directions.
problem Challenging multi-objective optimization problem.
method Solves multiple constrained optimization problems along orthogonal search directions.
result Consistently outperforms state-of-the-art algorithms.
A new algorithm avoids retractions to optimize orthogonal matrices efficiently.
problem Optimizing functions over the manifold of orthogonal matrices efficiently.
method Landing algorithm that avoids retractions using potential energy.
result The landing algorithm is faster and less prone to numerical errors than retraction-based methods.
By this short preface we show the main idea and we will bring some definitions and concepts in each section.
We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham ope…
New conditions for circular orderability of direct products, linking to left-orderability of groups.
problem Conditions for circular orderability of direct products .
method Cohomological conditions and characterizations for left-orderability.
result New characterization for left-orderability of fundamental groups of rational homology 3-spheres.
Introduces MSW distances to improve SW metrics.
problem Redundant projections in SW distance.
method Imposes Markov structure on projecting directions.
result MSW distances improve SW metrics.
If one could assume that local coordinates in a Riemannian manifold were orthogonal, then local expressions for differential operators, and curvature computations, would be simplified. It is always possible on 2-manifolds, using geometric normal coordinates or isothermal coordinates. In 1984, Dennis DeTurck and Dean Ya…
The study explores globally defined eigenfamilies on closed manifolds, providing existence and orthogonality results.
problem Existence and properties of globally defined eigenfamilies on closed Riemannian manifolds.
method Analyzes topological properties, provides non-/existence results, and uses combinatorial identities.
result Highly rigid orthogonality relations for eigenfunction powers in .
We study smooth complex hypersurfaces in direct products of closed hyperbolic Riemann surfaces and give a classification in terms of their fundamental groups. This answers a question of Delzant and Gromov on subvarieties of products of Riemann surfaces in the smooth codimension one case. We also answer Delzant and Grom…
New framework detects directional influence in multivariate time series.
problem Detecting directional influence in multivariate time series.
method Order-constrained spectral non-invariance.
result Unique diagnostic functional for directional influence.
We show that coarse property C is preserved by finite coarse direct products. We also show that the coarse analog of Dydak's countable asymptotic dimension is equivalent to the coarse version of straight finite decomposition complexity and is therefore preserved by direct products.