Kan extensions help in data science extrapolation and learning.
problem Generalizing functions over larger sets in data science.
method Kan extensions in category theory applied to data science problems.
result Kan extensions can be used to derive classification and clustering algorithms.
SVGP KAN integrates sparse variational GP with KANs for scalable probabilistic inference.
problem Lack of probabilistic outputs in standard KANs and cubic scaling of Gaussian Process methods.
method Sparse Variational GP-KAN combines KAN topology with sparse variational inference and permutation-based importance analysis.
result Enables probabilistic KANs to handle larger datasets with linear computational complexity.
HaKAN uses Hahn-KAN blocks to forecast multivariate time series.
problem Long-term time series forecasting challenges with high complexity and spectral bias.
method HaKAN integrates channel independence, patching, and a stack of Hahn-KAN blocks with residual connections. It uses Hahn polynomial-based learnable activation functions.
result HaKAN consistently outperforms state-of-the-art methods on various forecasting benchmarks.
Wavelet Kolmogorov-Arnold Networks improve federated learning performance.
problem Improving performance in federated learning with heterogeneous data.
method Implemented Wav-KAN with CWT and DWT for multiresolution capability, integrating wavelet-based activation functions.
result Significant improvements in computational efficiency, robustness, and accuracy in federated learning.
GP-KAN uses Gaussian Processes in KANs for robust, parameter-efficient non-linear modeling.
problem Non-linear modeling with limited parameters and uncertainty estimates.
method Integrates Gaussian Processes into Kolmogorov Arnold Networks (KANs) for robust non-linear modeling.
result GP-KAN achieves 98.5% accuracy on MNIST with 80k parameters compared to 1.5M for state-of-the-art models.
We establish a functor Kan from local Kan simplicial manifolds to weak Kan simplicial manifolds. It gives a solution to the problem of extending local Lie groupoids to Lie 2-groupoids.
KANs replace fixed MLP weights with learnable edge functions, improving accuracy and interpretability.
problem Lack of interpretability and scalability in MLPs.
method KANs use learnable activation functions on edges instead of fixed weights, replacing weights with spline functions.
result KANs outperform MLPs in accuracy and interpretability with smaller models.
Wav-KAN improves neural network interpretability and performance.
problem Challenges in interpretability, training speed, robustness, and performance of traditional neural networks.
method Integrates wavelet functions into the Kolmogorov-Arnold network structure for efficient data representation.
result Enhanced accuracy, faster training speeds, and increased robustness compared to existing methods.
Smooth KANs improve model reliability in computational biomedicine.
problem Limited convergence of KANs in representing generic smooth functions.
method Introducing smooth, structurally informed KANs that can approximate MLPs in specific function classes.
result Smooth KANs can achieve equivalence to MLPs in specific function classes, enhancing model reliability and performance.
Adaptive RBF-KAN improves KANs by dynamically adjusting kernel parameters.
problem Efficiently approximating multivariate functions using learnable univariate edge functions.
method Integrates LOOCV-based kernel scale estimation with adaptive kernel learning.
result Adaptive RBF-KAN outperforms fixed kernel KANs on various benchmark functions.
Local Kan conditions enable differentiation of simplicial manifolds.
problem Differentiating simplicial manifolds into Lie algebroids.
method Expanding a technique for higher Lie groupoids to simplicial manifolds.
result Derivation of a method to differentiate simplicial manifolds into higher Lie algebroids.
T-KAN improves HFT LOB forecasting with learnable splines.
problem Alpha decay in HFT LOB forecasting models.
method T-KAN uses learnable B-spline activation functions to model market signals.
result 19.1% relative improvement in F1-score at k = 100 horizon.
Temporal Functional Circuits explain KAN forecasts with interpretable edge functions.
problem Lack of mechanistic explanations in KAN forecasting.
method Transform KAN edge functions into faithful, temporally grounded explanations using a gated residual KAN.
result Gated KAN achieves lower MSE than linear-only models on regime-switching signals.
A new Kolmogorov-Arnold network improves function approximation and optimization.
problem Approximating potentially irregular functions in high dimensions.
method Proposes a new Kolmogorov-Arnold network (KAN) and provides error bounds and universal approximation theorems.
result Outperforms multilayer perceptrons in accuracy and convergence speed for irregular functions.
GC-KAN uses KANs to detect Granger causality in time series data.
problem Detecting causal relationships in nonlinear time series data.
method Developed GC-KAN framework using Kolmogorov-Arnold networks for Granger causality detection.
result KANs outperform MLPs in identifying sparse Granger causal relationships.
DKL-KAN combines deep learning and kernel methods for scalable, expressive models.
problem Combining deep learning's depth with kernel methods' flexibility for scalable models.
method DKL-KAN uses Kolmogorov-Arnold Networks (KAN) to optimize kernel attributes within a Gaussian process framework.
result DKL-KAN outperforms DKL-MLP on datasets with a low number of observations and DKL-MLP on large datasets.
Curvature penalties improve interpretability of KANs without sacrificing accuracy.
problem Pathologically high-curvature oscillations in KANs activations make them hard to interpret.
method Derived a curvature penalty and proved an upper bound on model curvature.
result KANs with curvature penalties achieve substantially smoother activations while maintaining accuracy.
Bayesian KANs achieve near-minimax posterior contraction rates in anisotropic Besov spaces.
problem Statistical foundation for Bayesian Kolmogorov-Arnold networks in anisotropic Besov spaces.
method Sparse Bayesian KANs with spike-and-slab priors, hyperprior on model size, and approximation complexity bounds.
result Posterior contraction rates depend on intrinsic anisotropic smoothness and effective dimension of the compositional structure.
KAN-PCA improves asset return analysis by capturing more variance than classical PCA during market crises.
problem Inefficient classical PCA during market crises when correlations between assets change dramatically.
method KAN-PCA uses KAN (Kolmogorov-Arnold Networks) with B-spline functions to learn nonlinear projections.
result KAN-PCA achieves a higher reconstruction R^2 (66.57%) compared to classical PCA (62.99%) on 20 S&P 500 stocks.
This study compares MLPs and KANs in low-data regimes, finding MLPs with personalized activation functions outperform KANs.
problem Comparing MLPs and KANs in low-data regimes.
method Introduced an effective technique for designing MLPs with unique, parameterized activation functions for each neuron.
result MLPs with personalized activation functions achieve significantly higher predictive accuracy with only a modest increase in parameters, especially in low-data regimes.
Proposes a new neural network architecture inspired by biology to improve learning and information flow.
problem Improving artificial neural networks to match biological neuron properties like multidirectional propagation and probabilistic modeling.
method Extends KAN approach with joint distribution neurons that can propagate values and distributions, including variance and higher-order moments.
result Proposed architecture can predict and propagate distributions, including expected values and variances.
Kolmogorov-Arnold Networks offer interpretable models for energy applications.
problem Lack of interpretability in modern machine learning methods for sensitive industries.
method Symbolic regression with Kolmogorov-Arnold Networks compared to traditional feedforward neural networks.
result Kolmogorov-Arnold Networks yield perfectly interpretable models and learn real, physical relations.
Novel KAN-based autoencoder improves asset pricing models' accuracy and interpretability.
problem Improving asset pricing models' accuracy and interpretability.
method Kolmogorov-Arnold Networks (KANs) inspired autoencoder for latent factor exposures.
result Outperforms Multilayer Perceptrons in both accuracy and interpretability.
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
Kolmogorov-Arnold Networks achieve optimal convergence rates in nonparametric regression.
problem Nonparametric function approximation in multivariate settings.
method Structured additive and multiplicative KANs using B-splines.
result Achieve minimax-optimal convergence rate O(n−2r/(2r+1)) for Sobolev space functions. Enhances KANs for accuracy and interpretability with multi-exit architecture.
problem Unclear optimal depth for KANs and difficulty in optimization and interpretation.
method Introduces multi-exit KANs with each layer having its own prediction branch.
result Multi-exit KANs outperform single-exit versions on various datasets.
KANOP uses KANs to efficiently price American options.
problem Efficiently pricing American options with limited data.
method Combines KANs with LSMC to estimate continuation value.
result KANOP provides more accurate option value estimates.
DecompKAN improves time series forecasting accuracy and transparency.
problem Accurate and transparent time series forecasting in scientific domains.
method Combines decomposition, patching, normalization, and B-spline KAN edge functions.
result Achieves best or tied-best MSE on 20 of 36 comparisons across 9 datasets.
Kolmogorov-Arnold Networks improve deep learning adaptivity and can approximate Besov functions optimally.
problem Improving deep learning adaptivity and understanding approximation rates.
method Analyzing Besov norms and using Res-KANs for approximation.
result KANs can optimally approximate Besov functions at the optimal rate.
The paper constructs automorphisms of Lie groupoids and applies them to symplectic reductions on orbifolds.
problem Formulating Hamiltonian actions of Lie 2-groups on orbifolds.
method Constructing automorphisms of Lie groupoids, using 2-group actions and Kan fibrations.
result Symplectic reductions of Lie 2-group actions on orbifolds are Lie 2-groupoids under certain conditions.
New architectures improve KANs, making them more interpretable and accurate.
problem Improving Kolmogorov-Arnold networks while maintaining interpretability.
method Overprovisioned architectures combined with sparsification, deep supervision, and depth selection, optimized with a minimum description length objective.
result Combining sparsification with depth selection achieves competitive or superior accuracy while discovering smaller models.
A semisimplicial set has face maps but not degeneracies. A basic fact, due to Rourke and Sanderson, is that a semisimplicial set satisfying the Kan condition can be given a simplicial structure. The present paper gives a combinatorial proof of this fact and a generalization to multisemisimplicial sets.
Geometric models for representations up to homotopy using simplicial vector bundles.
problem Geometric models for representations up to homotopy of Lie groupoids.
method Application of higher analogs of cleavages in simplicial fibrations to geometric models.
result An equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage.
Lie's third theorem proven for Lie ∞-algebras.
problem Integrating finite-type Lie ∞-algebras to Lie ∞-groups.
method Local minimal models for Kan simplicial manifolds.
result Every finite-type Lie ∞-algebra integrates to a finite-dimensional Lie ∞-group.
BEKAN uses RBFs and evolutionary methods to solve PDEs with boundary conditions.
problem Enforcing boundary conditions in neural networks for PDE solutions.
method Boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (BEKAN) with radial basis functions (RBFs). Incorporates Dirichlet, periodic, and Neumann conditions.
result BEKAN outperforms MLP and B-splines KAN in solving PDEs with boundary conditions.
KAPLAN-HR models survival data without manual interactions, outperforming existing methods.
problem Survival analysis challenges with complex covariates and time-varying effects.
method Kolmogorov-Arnold Networks (KAN) for nonparametric hazard estimation.
result KAPLAN-HR matches or exceeds existing methods in clinical survival data.
In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids, we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids, we use inner Kan fibra…
Kolmogorov-Arnold Networks enable ultrafast online learning with fixed-point quantization.
problem Efficient online learning for high-frequency systems with strict memory constraints.
method Fixed-point online training on FPGAs exploiting B-spline locality in KANs.
result Kolmogorov-Arnold Networks are more efficient and expressive than MLPs for low-latency tasks.
Kolmogorov-Arnold Networks offer improved interpretability and parsimony in science tasks.
problem Improving interpretability and parsimony in science-oriented tasks.
method Theoretical analysis of Kolmogorov-Arnold Networks (KAN) with generalization bounds and model complexity.
result Generalization bounds for KAN with various activation functions, scaling with the l1 norm of coefficient matrices and Lipschitz constants. Lie ∞-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie ∞-groupoids called `Lie ∞-groups' by integrating finite type Lie n-algebras. In order to study the compatibility between this…
Paper develops a neural model to assess cascading extreme events.
problem Risk assessment of domino effects like earthquakes and tsunamis.
method Develops a Kolmogorov-Arnold neural network (KANE) framework.
result Estimates the probability of one extreme event triggering another.
Improved KAN model explains brain dynamics through edge learning and synaptic strength.
problem Explaining brain dynamics and frequencies in different brain regions.
method ELKAN (Edge Learning KNN) model with edge learning and trimming, inspired by brain science.
result ELKAN model outperforms KAN in explaining brain frequencies and dynamics.
Study compares various ANN models for SPX and NDX options pricing.
problem Approximating complex multivariate functions for accurate option pricing.
method Hybrid RNN models (LSTM-GRU, TDNN, MLP, KAN) with attention mechanisms.
result LSTM-GRU hybrid RNN with attention outperforms other models.
SVGP KAN integrates uncertainty quantification into Kolmogorov-Arnold networks.
problem Uncertainty quantification in scientific machine learning models.
method Sparse variational Gaussian process inference with Kolmogorov-Arnold topology.
result Demonstrated ability to distinguish aleatoric and epistemic uncertainty in various scientific applications.
A new KAN variant uses sinusoidal activations to approximate functions.
problem Approximating multivariable functions using neural networks.
method Replacing inner and outer functions in Kolmogorov-Arnold representation with weighted sinusoidal functions.
result The new KAN variant outperforms fixed-frequency Fourier transform and achieves comparable performance to MLPs.
New bounds for KANs trained with DP-SGD, addressing correlated noise.
problem Risk bounds for Kolmogorov-Arnold Networks trained by DP-SGD with correlated noise.
method Established new optimization and population risk analysis for KANs trained with DP-SGD, addressing correlated noise.
result First optimization and population risk analysis of correlated-noise mechanisms for DP training in non-convex settings, including neural networks.
Let Y be a CW-complex with a single 0-cell, let K be its Kan group, a free simplicial group whose realization is a model for the space ΩY of based loops on Y, and let G be a Lie group, not necessarily connected. By means of simplicial techniques involving fundamental results of {\smc Kan's} and the standard $…
GKAN extends KAN to graphs, improving graph-based learning.
problem Graph-based learning tasks, especially semi-supervised.
method Learnable spline-based functions applied to graph data.
result GKAN achieves higher accuracy in graph semi-supervised learning.