Novel multigraph network improves chemical classification tasks.
problem Learning from variable graphs with multiple relationships.
method Proposed a multigraph network using Chebyshev GCNs to handle variable graphs and learned edges.
result Achieved competitive results on chemical classification benchmarks.
For node level graph encoding, a recent important state-of-art method is the graph convolutional networks (GCN), which nicely integrate local vertex features and graph topology in the spectral domain. However, current studies suffer from several drawbacks: (1) graph CNNs relies on Chebyshev polynomial approximation whi…
A Chebyshev knot is a knot which admits a parametrization of the form x(t)=Ta(t); y(t)=Tb(t); z(t)=Tc(t+φ), where a,b,c are pairwise coprime, Tn(t) is the Chebyshev polynomial of degree n, and $φ\in \RR .$ Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that the…
Chebyshev steps improve convergence in deep-unfolded gradient descent.
problem Improving convergence speed in iterative algorithms.
method Introducing Chebyshev steps to bound convergence rate of gradient descent.
result Chebyshev steps lead to asymptotically optimal convergence rate.
The paper connects Chebyshev polynomials and Gram determinants on Möbius bands.
problem Exploring the relationship between Chebyshev polynomials and Gram determinants on Möbius bands.
method Analyzing Mersenne numbers and Chebyshev polynomials, proving conjectures, and developing algorithms.
result A factor of the Gram determinant supports a conjecture about its closed formula involving Chebyshev polynomials.
A Chebyshev knot C(a,b,c,φ) is a knot which has a parametrization of the form x(t)=Ta(t);y(t)=Tb(t);z(t)=Tc(t+φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ∈R. We show that any two-bridge knot is a Chebyshev knot with a=3 and also with a=4. For e…
We report on the works of Euler and Chebyshev on the drawing of geographical maps. We point out relations with questions about the fitting of garments that were studied by Chebyshev.
This paper studies the Riley polynomial of 2-bridge knots using Chebyshev polynomials.
problem Understanding the Riley polynomial of 2-bridge knots and its splitting property.
method Introducing ε-Chebyshev polynomials to express and split the Riley polynomial.
result Explicit formula for the splitting polynomial as ε-Chebyshev polynomials.
Efficiently calibrates volatility models using Chebyshev Tensors.
problem Calibrating pricing models efficiently.
method Used Chebyshev Tensors to speed up calibration of the rough Bergomi volatility model.
result Chebyshev Tensors can calibrate the rough Bergomi volatility model 40,000 times more efficiently than brute-force methods.
Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
problem Exploring the Chebyshev-Frobenius homomorphism in 3-manifold skein modules.
method Generalization of splitting homomorphism for stated skein modules of 3-manifolds.
result Existence and properties of Chebyshev-Frobenius homomorphism for 3-manifold skein modules.
New technique reduces FRTB IMA capital calculation burden by over 90%.
problem Reduction of computational burden in FRTB IMA capital calculation.
method Orthogonal Chebyshev Sliding Technique based on high-dimensional Chebyshev Tensors.
result Reduction of computational burden by more than 90%.
The study bounds positive bases of skein algebras using Chebyshev polynomials.
problem Finding positive bases in skein algebras of surfaces.
method Using Chebyshev polynomials to establish bounds.
result Normalized Chebyshev polynomials of type one give the only positive basis for the closed torus.
This paper categorifies Chebyshev polynomials using diagrammatic algebra.
problem Categorifying two-variable Chebyshev polynomials of the second kind.
method Using A2 spider and Karoubi envelope of A2 spider, the recursive formula is shown. result A q-deformation of the two-variable Chebyshev polynomials is defined. New CFNN architecture approximates functions with machine accuracy.
problem Function approximation with high precision.
method Chebyshev Feature Neural Network (CFNN) with learnable frequencies.
result Achieves machine accuracy in function approximation.
We solve for functions from their truncated Hilbert transforms using Chebyshev series.
problem Finding functions from their truncated Hilbert transforms.
method Express functions in Chebyshev series and numerically estimate coefficients.
result Numerical methods work well for extrapolating functions from truncated Hilbert transforms.
Paper uses Chebyshev Tensors for accurate dynamic sensitivities and ISDA SIMM computation.
problem Computing dynamic sensitivities and initial margin for financial instruments.
method Uses Chebyshev Tensors in Monte Carlo simulations to compute dynamic sensitivities and ISDA SIMM.
result High accuracy and computational gains for FX swaps and Spread Options.
Some results on existence of global Chebyshev coordinates on a Riemannian manifold or, more generally, on Aleksandrov surface are proved. For instance, if the positive and the negative parts of integral curvature of a Riemannian manifold M are less than 2πeach, then there exist global Chebyshev coordinates on M. These …
Using Chebyshev polynomials, C. Frohman and R. Gelca introduce a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones-Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which em…
Efficiently trains GCNs with reduced time and memory usage.
problem Hard training of GCNs over large graph datasets.
method Layer-wise and learned efficient training framework (L2-GCN). result Significantly reduces training time and memory usage.
New view: Deep GCNs learn to anti-oversmooth during training.
problem Performance drop in deep GCNs due to oversmoothing.
method Interpreted GCN as MLP + graph regularization, analyzed training process.
result Deep GCNs learn to anti-oversmooth during training, not over-smooth.
We introduce a new activation function using Chebyshev-Lagrange polynomials for improved neural network performance.
problem Improving data efficiency and accuracy of neural networks.
method Parameterized piece-wise polynomial activation functions based on Chebyshev nodes and Lagrangian interpolation.
result Significant improvements in model capacity and accuracy, especially in linear extrapolation.
Cluster-GCN efficiently trains deep GCNs on large graphs without memory or computational constraints.
problem Training large-scale GCNs is computationally and memory-intensive.
method Cluster-GCN exploits graph clustering to restrict neighborhood search to dense subgraphs, reducing memory and computational requirements.
result Cluster-GCN achieves comparable test accuracy to previous algorithms while being faster and using less memory.
GCNs favor high-degree nodes, leading to biased performance; a new method mitigates this.
problem Degree-related biases in GCNs, especially for low-degree nodes.
method Developed a novel SL-DSGC that reduces model and data biases.
result SL-DSGC improves GCN accuracy significantly for low-degree nodes.
A Chebyshev curve C(a,b,c,φ) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ\in \RR. When C(a,b,c,φ) has no double points, it defines a polynomial knot. We determine all possible knots when a, b and c are given.
GCNs struggle with learning graph moments, but modular designs improve their performance.
problem GCNs' limitations in learning graph moments.
method Investigated through graph moments, analyzed expressiveness, designed modular GCNs.
result Modular GCNs using different propagation rules can distinguish graphs from various models.
The paper computes groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
problem Computing groups and modules for wheel graphs.
method Utilized Fibonacci and Chebyshev polynomials to compute the Reduced Fox Coloring Group and Alexander-Burau-Fox Module.
result Computed groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.
We show that every two-bridge knot K of crossing number N admits a polynomial parametrization x=T3(t),y=Tb(t),z=C(t) where Tk(t) are the Chebyshev polynomials and b+°C=3N. If C(t)=Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots …
New bound improves on weighted majority vote risk estimation.
problem Improving risk estimation for weighted majority vote.
method Novel Chebyshev-Cantelli inequality and PAC-Bayes-Bennett inequality.
result New bounds improve on existing methods.
We show that every rational knot K of crossing number N admits a polynomial parametrization x=Ta(t),y=Tb(t),z=C(t) where Tk(t) are the Chebyshev polynomials, a=3 and b+°C=3N. We show that every rational knot also admits a polynomial parametrization with a=4. If C(t)=Tc(t) is a Chebyshev p…
The study estimates the expressiveness of GCNs with bounds on the number of linear regions.
problem Characterizing the expressiveness of graph convolutional networks (GCNs).
method Estimates the number of linear regions for one-layer and multi-layer GCNs.
result GCNs with multiple layers have exponentially more expressivity per parameter than one-layer GCNs.
RR-GCN uses random transformations instead of learned weights for node embeddings.
problem Learning node embeddings in KGs.
method Random Relational Graph Convolutional Network (RR-GCN) with untrained parameters.
result RR-GCN can compete with fully trained R-GCNs in node classification and link prediction.
Paper tackles over-smoothing in deep GCNs, proposing DropEdge to improve performance.
problem Over-smoothing reduces expressivity in deep GCNs, especially affecting node classification.
method Theoretical analysis of GCN behavior with depth, proposing DropEdge to alleviate over-smoothing.
result DropEdge improves performance on various GCNs, shallow and deep.
Proposes ML-GCN for multi-label graph node classification using GCN and relaxed skip-gram model.
problem Loss of label correlations in multi-label graph node classification.
method Uses a GCN to embed node features and graph topology, generates random label vectors, and detects correlations using a skip-gram model.
result Significantly outperforms state-of-the-art methods on graph classification datasets.
SStaGCN improves GCN by stacking and aggregation for better node feature extraction.
problem Mitigating over-smoothing in GCN for heterogeneous graph data.
method SStaGCN combines stacking and aggregation to improve GCN performance.
result SStaGCN effectively mitigates over-smoothing and enhances node feature extraction.
Adding node feature kernels improves GCN robustness to graph perturbations.
problem GCNs' robustness to graph perturbations is a concern.
method Introduced random GCN and added node feature kernels to message passing.
result Perturbations of the graph structure can significantly degrade GCN performance.
DeeperGCN tackles deep GCNs by overcoming vanishing gradient and over-smoothing issues.
problem Vanishing gradient, over-smoothing, and over-fitting issues in deep GCNs.
method DeeperGCN uses differentiable generalized aggregation functions and a novel normalization layer (MsgNorm) to train deep GCNs.
result DeeperGCN significantly boosts performance on large-scale graph learning tasks.
STAR-GCN improves recommender systems by learning node representations.
problem Cold start problem in recommender systems.
method Stacked and reconstructed Graph Convolutional Networks (GCN) with intermediate supervision and node embedding reconstruction.
result Significant improvements in predicting ratings, especially in the cold start scenario.
GCNs help in diagnosing label scarcity and feature quality on graphs.
problem Understanding when GCNs improve node classification.
method Simulated label scarcity, feature ablation, and per-class analysis.
result GCNs provide largest gains under extreme label scarcity, matching original performance with noisy features, but hurt when homophily is low and features are strong.
Graph Convolutional Networks (GCNs) have shown significant improvements in semi-supervised learning on graph-structured data. Concurrently, unsupervised learning of graph embeddings has benefited from the information contained in random walks. In this paper, we propose a model: Network of GCNs (N-GCN), which marries th…
Enhances GCNs using VAT for better node classification.
problem Limited use of unlabeled data in GCNs.
method Virtual Adversarial Training (VAT) on GCN supervised loss.
result Improves GCN generalization performance.
I-GCN improves GCNs' robustness against adversarial attacks.
problem Adversarial attacks degrade GCNs' performance in security-critical applications.
method Influence mechanism divides node effects into introverted and extroverted influences.
result I-GCN achieves higher accuracy rates than state-of-the-art methods in defending against adversarial attacks.
Advances deep network embedding through multi-filtering GCN.
problem Existing attribute embedding methods fail to capture different aspects of node features.
method Multi-filtering Graph Convolution Neural Network (GCN) framework.
result Significant improvement in link prediction and node classification tasks with limited training data.
Many interesting problems in machine learning are being revisited with new deep learning tools. For graph-based semisupervised learning, a recent important development is graph convolutional networks (GCNs), which nicely integrate local vertex features and graph topology in the convolutional layers. Although the GCN mo…
Paper proposes efficient GCN learning method for limited data.
problem Learning GCNs from data with extremely limited annotations.
method Adaptive sampling strategy and model compression.
result Cut down annotation requirement by 90% and compress parameters 6x.
Chebyshev polynomials analyze Czech enterprises' stock dynamics.
problem Analyzing stock dynamics of enterprises not following normal distribution.
method Chebyshev polynomial decomposition of stock time series.
result Allows effective analysis of stock dynamics without variance and correlation.
In this paper we study the skein algebras of marked surfaces and the skein modules of marked 3-manifolds. Muller showed that skein algebras of totally marked surfaces may be embedded in easy to study algebras known as quantum tori. We first extend Muller's result to permit marked surfaces with unmarked boundary compone…
T-GCN predicts traffic using neural networks for spatial and temporal data.
problem Accurate real-time traffic forecasting in urban networks.
method Combines GCN for spatial and GRU for temporal data analysis.
result T-GCN outperforms state-of-the-art baselines on real-world traffic datasets.
This paper explains GCNs using NTKs and improves their performance.
problem GCNs' performance degrades with depth, and skip connections marginally improve it.
method Derive NTKs for GCNs, validate with simulations, propose NTK as a surrogate model.
result Suitable normalisation can prevent drastic performance drop with depth.