Deep learning models predict option prices from 3D tensor data.
problem Predicting option prices for risk management and trading.
method 3D tensor representation of financial data, deep learning models (2D tensors in 3 channels).
result Proposed models outperform traditional methods like B-S model and vector-based LSTM.
Researchers found all homogeneous structure tensors on two specific 3D manifolds.
problem Classifying homogeneous structure tensors on specific 3D manifolds.
method Determined all homogeneous structure tensors on S2imesR and H2imesR. result Complete classification of homogeneous structure tensors on three-dimensional homogeneous Riemannian manifolds.
DeepTensor uses deep networks to efficiently decompose tensors with improved performance and robustness.
problem Efficiently decomposing tensors with deep learning to capture nonlinear structures.
method Low-rank tensor decomposition using deep generative networks trained to minimize approximation error.
result DeepTensor outperforms classical methods like SVD and PCA in various applications, including image denoising and 3D MRI.
New method uses scalar-based models to approximate spherical tensors efficiently.
problem Efficiently approximating spherical tensors with equivariant functions.
method Expressing equivariant functions as the product of a scalar function and a small tensor basis.
result Approximations are fast, simple to implement, and accurate in practical settings.
Tensor neural network improves human pose classification from 3D skeleton data.
problem Efficiently processing spatiotemporal data for human pose classification.
method Proposes a tensor-based neural network with three components: spatiotemporal feature construction, tensor fusion, and tensor-based neural network processing.
result Achieves state-of-the-art performance in human pose classification.
Study classifies 3D Einstein manifolds with cyclic Ricci tensor.
problem Classifying Einstein manifolds with specific tensor properties.
method Derived integral formula involving tensor D for classification.
result Obtained rigidity results for 3D manifolds.
Study on 3D Lie groups finds all generalized Einstein metrics.
problem Classifying generalized Einstein metrics on 3D Lie groups.
method Developed theory of left-invariant generalized pseudo-Riemannian metrics, computed Ricci tensor, determined all metrics.
result Determined all generalized Einstein metrics on three-dimensional Lie groups.
Study on 3D manifolds with circulant structures and their properties.
problem Characterizing 3D almost Einstein manifolds with circulant structures.
method Analyzing the curvature tensor and properties of the Levi-Civita connection.
result Determined geometric characteristics and examples of such manifolds.
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N=2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,Z) Weil representations, quantum mo…
Study of 3D vacuum static spaces with specific curvature properties.
problem Classifying 3D vacuum static spaces with certain curvature conditions.
method Used generalized maximum principle to classify 3D spaces.
result Gave a complete classification of 3D complete vacuum static spaces.
Advanced 3D metrology technologies such as Coordinate Measuring Machine (CMM) and laser 3D scanners have facilitated the collection of massive point cloud data, beneficial for process monitoring, control and optimization. However, due to their high dimensionality and structure complexity, modeling and analysis of point…
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
Equivariant networks improve geometric prediction without scalar approximations.
problem Efficiently predicting geometric tensors in real-world scenarios.
method Equivariant networks for geometric prediction.
result Equivariant networks can generalize to unseen systems for geometric prediction.
We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of t…
A modular functor is constructed from non-semisimple 3d TFTs.
problem Constructing modular functors from non-semisimple 3d topological field theories.
method Using a 3d TFT defined in [arXiv:1912.02063], a symmetric monoidal 2-functor is constructed from a 2-category of bordisms to a 2-category of finite linear categories.
result A modular functor is explicitly described as a symmetric monoidal 2-functor.
We present a convolutional network that is equivariant to rigid body motions. The model uses scalar-, vector-, and tensor fields over 3D Euclidean space to represent data, and equivariant convolutions to map between such representations. These SE(3)-equivariant convolutions utilize kernels which are parameterized as a …
Defines a new 3D TQFT from non-semisimple categories.
problem Developing a TQFT from non-semisimple categories.
method Generators and relations framework, decomposition of 3-manifolds.
result TQFT values on 3-manifolds match known invariants.
The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvatur…
3D topological order linked to Seifert manifolds and gauge groups.
problem Classifying 3D topological orders using Seifert manifolds and gauge groups.
method Correspondence between topological order, Seifert manifolds, and ADE gauge groups.
result Construction of modular fusion categories from Seifert manifolds and gauge groups.
Compact 3D Cotton-parallel manifolds are always conformally flat.
problem Understanding the properties of compact 3D Cotton-parallel manifolds.
method Analyzing the Cotton tensor and its parallelism condition.
result Compact 3D Cotton-parallel manifolds are conformally flat.
New quantum algebra connects 3D gravity to complex plane.
problem Quantize 3D gravity with positive cosmological constant.
method Introduced quantum pseudo-Kähler plane and studied its representations.
result Found new operators for 3D gravity quantization.
The study characterizes and proves properties of 3D Poisson quasi-Nijenhuis manifolds.
problem Characterizing and understanding 3D Poisson quasi-Nijenhuis manifolds.
method Characterization through deformation and application of Haantjes structures.
result Every 3D Poisson quasi-Nijenhuis manifold is a Haantjes manifold.
3D ConvNets improved with Project & Excite for medical imaging segmentation.
problem Improving segmentation performance in 3D medical imaging.
method Proposed Project & Excite (PE) modules for 3D F-CNNs, extending 2D recalibration methods.
result Project & Excite modules boost segmentation performance up to 0.3 in Dice Score.
The study characterizes constant curvature manifolds using ruled surfaces.
problem Characterizing manifolds of constant curvature using ruled surfaces.
method Investigating ruled surfaces in 3d Riemannian manifolds, finding stiction curve, distribution parameter, and fundamental forms.
result Identifies necessary and sufficient conditions for extrinsically flat surfaces to be ruled and proves manifold properties.
We define a symmetric monoidal (4,3)-category with duals whose objects are certain enriched multi-fusion categories. For every modular tensor category C, there is a self enriched multi-fusion category C giving rise to an object of this symmetric monoidal (4,3)-category. We conjecture that the e…
Improves tensor networks for classifying medical images.
problem Classifying 2D and 3D medical images efficiently.
method Develops LoTeNet, a tensor network that treats small image regions as orderless and aggregates local representations hierarchically.
result LoTeNet achieves comparable or superior performance to other methods with less computational resources.
Study odd generalized Einstein metrics on 3D Lie groups.
problem Classify odd generalized Einstein metrics on 3D Lie groups.
method Left-invariant generalized connections, divergence operators, and Ricci tensors.
result Describe all odd generalized Einstein metrics on all 3D Lie groups.
Proposes polynomial neural networks for improved function approximation in various tasks.
problem Improving function approximation in various tasks like image generation, face verification, and 3D mesh representation learning.
method Introduces polynomial neural networks (Π-Nets) and three tensor decompositions to reduce parameter count and enhance expressiveness. result Demonstrates that Π-Nets can produce state-of-the-art results in challenging tasks without non-linear activation functions. 3D CNNs interpret brain MRI differences between men and women.
problem Interpreting 3D CNNs for voxel-wise brain MRI analysis.
method Three interpretation methods: Meaningful Perturbations, Grad CAM, and Guided Backpropagation.
result Voxel-wise 3D CNN interpretation of brain MRI data.
New signs and gradings enable detailed comparison in Heegaard Floer theory.
problem Comparing decategorified Heegaard Floer theory with modern TQFTs.
method Added signs and gradings to interval gluing theorem over Z.
result Detailed comparison possible with modern TQFTs.
Mathematical study supports connection between 3D manifolds and modular tensor categories.
problem Connecting geometric topology and quantum topology using Chern-Simons invariants and Reidemeister torsions.
method Developed an algorithm to generate modular T-matrices and quantum dimensions from Seifert fibered spaces and torus bundles over the circle. result Mathematically constructed premodular categories from Seifert fibered spaces and torus bundles over the circle, conjecturing their modularity under specific conditions.
We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhof…
Principal component analysis (PCA) is an unsupervised method for learning low-dimensional features with orthogonal projections. Multilinear PCA methods extend PCA to deal with multidimensional data (tensors) directly via tensor-to-tensor projection or tensor-to-vector projection (TVP). However, under the TVP setting, i…
Study on 3D trans-Sasakian manifolds with η-Einstein solitons.
problem Characterizing 3D trans-Sasakian manifolds with η-Einstein solitons.
method Analyzing properties of Codazzi type and cyclic parallel Ricci tensors on 3D trans-Sasakian manifolds.
result Examples and properties of 3D trans-Sasakian manifolds with η-Einstein solitons.
Improved tensor GLM estimation for complex data.
problem Complex tensor data in GLMs leads to high-dimensional, ill-posed estimation.
method Proposed LSRTR-M algorithm using Muon updates for faster convergence and lower errors.
result LSRTR-M converges faster and achieves lower errors than LSRTR.
New insights into 3D PDEs via Einstein-Weyl geometry.
problem Understanding second-order PDEs in 3D with Einstein-Weyl conformal structure.
method Analyzing solutions of second-order dispersionless integrable PDEs in 3D, relating them to Einstein-Weyl geometry.
result The covector w can be expressed in terms of the equation for generic second-order PDEs, providing a dispersionless integrability test.
We develop a new approach on the (1+3) threading of spacetime (M,g) with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial tensor fields and on the Riemannian spatial connection ∇⋆, which behave as 3D geometric objects. We obtain new formul…
The paper generalizes knot invariants and their connections to quivers and ideals.
problem Understanding knot complements and their invariants.
method Generalizing FK invariants, knots-quivers correspondence, and A-polynomials; associating FK to branch of A-polynomial; quiver generating series; R-matrices; quantum a-deformed A-polynomial; 3d-5d theory. result Explicit expressions for FK invariants and their quiver representations for several simple knots. We devise an algorithm which allows one to count the number of Killing vectors for a Lorentzian manifold of dimension 3. Our algorithm relies on the principal traces of powers of the Ricci tensor and branches intricately according to the values of differential invariants arising from the compatibility conditions of the…
Paper proves conditions for 3D submanifolds to embed in 4D space.
problem Conditions for 3D Riemannian submanifolds to embed in R4. method Used symbolic method from classical invariant theory.
result Two known intrinsic conditions are sufficient for embedding.
Paper proves rigidity of metrics near hyperbolic ones in 3D.
problem Proving rigidity of metrics near hyperbolic ones in 3D.
method Introducing marked Poincaré determinant and proving local rigidity.
result Lichnerowicz Laplacian is injective in negative curvature.
Training deep neural networks with spatio-temporal (i.e., 3D) or multidimensional convolutions of higher-order is computationally challenging due to millions of unknown parameters across dozens of layers. To alleviate this, one approach is to apply low-rank tensor decompositions to convolution kernels in order to compr…
Ancient Ricci flows with bounded girth found in 3D and higher.
problem Finding ancient Ricci flows with bounded girth in dimensions 3 and higher.
method Invariant conditions on curvature and its derivatives under O(2)imesO(n−1) symmetry, proving Ricci flow invariance. result Construction of new ancient Ricci flows with positive curvature operator and bounded girth.
We study knots in 3d Chern-Simons theory with complex gauge group SL(N,C), in the context of its relation with 3d N=2 theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d (2,0) theory, which is compactified on a 3-manifold M^. …
The study finds Ricci solitons on a specific type of 3D Lorentzian Walker manifold.
problem Existence of non-trivial Ricci solitons on specific manifolds.
method Defined Ricci solitons on pseudo-Riemannian manifolds and applied to a family of 3D Lorentzian Walker manifolds.
result Existence of non-trivial Ricci solitons on a family of 3D Lorentzian Walker manifolds.
Study of 3d-3d correspondence involving q-Weyl algebra and 3d-index.
problem Understanding the action of a q-Weyl algebra on the 3d-index of knots. method Investigation of the q-Weyl algebra's module action on the 3d-index, conjecturing structural properties. result Bilinear factorization, pair of linear q-difference equations, and rational function matrix for the 3d-index determination. The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a definite amount of time under three-dimensional Ricci flow, and leads to local C/t deca…
3D dual field theories for Virasoro minimal models constructed using Seifert fiber spaces.
problem Constructing 3D dual field theories for Virasoro minimal models.
method 3D-3D correspondence and Seifert fiber spaces.
result 3D dual field theories constructed for Virasoro minimal models.