Simpler proof for non-basic sets in 2D.
problem Proving non-basic sets in 2D.
method Defining Sternfeld arrays and proving non-basic sets.
result Simpler proof of non-basic sets in 2D.
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48 results for “Kolmogorov Superposition Theorem”
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.
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.
Optimized neural network approximates high-dimensional functions with minimal parameters.
problem Achieving optimal approximation of high-dimensional continuous functions with minimal parameters.
method Developed a neural network with a specific activation function and architecture to achieve super approximation property.
result A composed network with at most 10889d + 10887 nonzero parameters achieves super approximation property, suggesting optimality in parameter growth.
New method for explaining neural network activation functions.
problem Transparency in black-box deep learning algorithms.
method Symbolic explanation of activation functions using adaptive Gaussian Processes.
result Achieved partially explainable learning model with scalable topology.
It is shown that superpositions of path integrals with arbitrary Hamiltonians and different scaling parameters v ("variances") obey the Chapman-Kolmogorov relation for Markovian processes if and only if the corresponding smearing distributions for v have a specific functional form. Ensuing "smearing" distributions subs…
Revisits the connection between neural networks and the Kolmogorov-Arnold theorem.
problem Explains the limitations of using the Kolmogorov-Arnold theorem to explain neural networks with multiple hidden layers.
method Derives modifications of the Kolmogorov-Arnold representation that transfer smoothness properties to the outer function and can be well approximated by ReLU networks.
result Shows that a deep neural network with most layers approximating the interior function is a more natural interpretation of the Kolmogorov-Arnold representation.
Kolmogorov-Arnold Networks promise scalable performance in high dimensions.
problem Curse of dimensionality in multilayer perceptrons.
method Kolmogorov-Arnold representation theorem and interpolation methods.
result Kolmogorov-Arnold Networks achieve true freedom from the curse of dimensionality.
Mixed superposition rules, i.e., functions describing the general solution of a system of first-order differential equations in terms of a generic family of particular solutions of first-order systems and some constants, are studied. The main achievement is a generalization of the celebrated Lie-Scheffers Theorem, char…
P1-KAN: an effective Kolmogorov-Arnold network with application to hydraulic valley optimizationcs.LG
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.
We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, general…
This note is purely expository. In the course of the Kolmogorov-Arnold solution of Hilbert's 13th problem on superpositions there appeared the notion of basic embedding. A subset K of R^2 is basic if for each continuous function f:K->R there exist continuous functions g,h:R->R such that f(x,y)=g(x)+h(y) for each point …
This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. Wh…
No free lunch theorems suggest inductive biases are needed, but we show neural networks prefer low-complexity data.
problem The need for inductive biases in machine learning.
method Analysis of Kolmogorov complexity and neural network behavior on various datasets.
result Neural networks prefer low-complexity data, suggesting inductive biases are not always necessary.
Superposition accelerates training to a universal power-law exponent.
problem Training dynamics in neural networks.
method Teacher-student framework and analytic theory.
result Superposition leads to a universal power-law exponent of ~1, independent of data and channel statistics.
Presented are two neural network architectures for convex functions, demonstrating competitive performance.
problem Approximating convex functions efficiently and accurately.
method Developed two neural network architectures: one based on linear-by-part representation and the other on cubic splines.
result Cubic ICKAN networks produce results similar to classical ICNNs in solving convex approximation problems.
Proposes a new neural network architecture combining MLP and basis functions.
problem Function approximation and operator learning in scientific machine learning.
method Combines robust MLP inner functions with flexible basis functions outer functions.
result KKAN outperforms MLPs and KANs in function approximation and operator learning tasks.
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.
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.
Proves a generalized isoperimetric inequality for spheres in dimensions 4 and above.
problem Proving a generalized isoperimetric inequality for spheres in dimensions 4 and above.
method Reduced to a theorem about thick embeddings of graphs, proved using Kolmogorov-Barzdin theorem and max-flow min-cut theorem. Counterexample in dimension 3 uses coarea inequality and winding number computation.
result A generalized isoperimetric inequality for spheres in dimensions 4 and above.
A new model explains asset returns with a single factor, improving cross-sectional performance.
problem Understanding the cross-section of asset returns with complex models.
method Proposes a non-linear single-factor asset pricing model with a nonparametric link function estimated jointly with sieve-based estimators.
result The model delivers superior cross-sectional performance with a low-dimensional approximation of the link function.
Study harmonic surfaces in 3D space, proving superposition principle.
problem Understanding harmonic surfaces in .
method Using harmonic Enneper immersions and superposition principle.
result Minimal and maximal surfaces can be decomposed into harmonic components.
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.
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations …
Kolmogorov neural networks can represent various types of functions.
problem Representing different types of functions with neural networks.
method Continuous, discontinuous bounded or unbounded activation functions in a two hidden layer model.
result Kolmogorov neural networks can represent continuous, discontinuous bounded and all unbounded multivariate functions.
Develops methods to answer counterfactual questions in temporal point processes.
problem Lack of counterfactual analysis in temporal point process models.
method Causal model of thinning based on Gumbel-Max structural causal model, superposition theorem, and sampling algorithm.
result Simulation of counterfactual realizations provides valuable insights for targeted interventions.
In many learning tasks, structural models usually lead to better interpretability and higher generalization performance. In recent years, however, the simple structural models such as lasso are frequently proved to be insufficient. Accordingly, there has been a lot of work on "superposition-structured" models where mul…
Researchers use estimated Kolmogorov complexity for better link prediction in graphs.
problem Improving link prediction accuracy in complex networks.
method Regularization based on an approximation of Kolmogorov complexity, which is differentiable and compatible with recent link prediction algorithms.
result The regularization method shows good performance on diverse real-world networks, but the success is likely due to an aggregation method rather than actual estimation of Kolmogorov complexity.
The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.
problem Finding general solutions for Lie systems.
method Develops mixed superposition rules for Lie systems with imprimitive Lie algebras and semidirect sums.
result Extends coalgebra method to Lie systems of partial differential equations.
Quantum machine learning uses superposition to create a large ensemble of classifiers.
problem Improving machine learning efficiency on quantum computers.
method Using superposition to create an exponentially large ensemble of classifiers, trained with an optimization-free learning algorithm.
result Adding an optimization step improves the performance of quantum ensembles of classifiers.
We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
problem Finding solutions for specific types of equations.
method Providing superposition formulae for Bäcklund transformations.
result Algebraically obtain infinitely many solutions after first integration.
New algorithm extracts features from superpositions in machine learning models.
problem Challenges in extracting interpretable features from complex models in superposition.
method An efficient query algorithm that identifies non-degenerate feature directions and reconstructs the function.
result Identifies all feature directions whose responses are non-degenerate and reconstructs the function \( f \) in a general superposition setting.
Observing a linear superposition principle, a family of new minimal hypersurfaces in Euclidean space is found, as well as that linear combinations of generalized helicoids induce new algebraic minimal cones of arbitrarily high degree.
High dimensional superposition models characterize observations using parameters which can be written as a sum of multiple component parameters, each with its own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated sparse vectors, etc. In this paper, we consider general superposition models …
Superposition in autoencoders leads to loss in simple models.
problem Mechanistic interpretability of neural networks
method Analyzing mathematical basis for superposition and providing bounds for reconstruction loss
result Upper and lower bounds for L2 reconstruction loss in the sparse regime
Paper introduces Manifold Probe for discovering representation manifolds in superposition.
problem Discovering representation manifolds in complex superposition representations.
method Generalizes linear regression probes to learn feature spaces and directions in superposition representations.
result Demonstrates Manifold Probe on Llama 2-7b representations, finding causally involved manifolds in model behaviour.
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of …
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.
Efficient algorithm approximates discrete random variables with minimal Kolmogorov distance.
problem Estimating the probability of missing deadlines in series-parallel schedules.
method An efficient algorithm that computes a random variable with minimal Kolmogorov distance to a given discrete random variable.
result The algorithm efficiently approximates the probability of missing deadlines with minimal Kolmogorov distance.
Fokker-Planck and Chapman-Kolmogorov equations for Ito processes with finite memorycond-mat.stat-mech
The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed in Kolmogorov's two pdes, K1 (the backward time pde) and K2 (the Fokker…
We study the solution to Kolmogorov-Feller equation and by using it provide pricing formulas of well known some options under jump-diffusion model.
New methods combine model predictions to avoid linear mixtures' limitations.
problem Combining predictions from different models to avoid linear mixtures' limitations.
method Log-linear pooling (locking) and quantum superposition (quacking) to optimise model weights.
result Demonstrated locking method with illustrative example and practical application.
Study finds conjugate points in geodesics of Kolmogorov flows on torus.
problem Characterizing pairs of integers (m,n) for which geodesics have conjugate points.
method Analysis of geodesics in the group of volume-preserving diffeomorphisms of a torus using stream functions.
result Existence of conjugate points for all pairs of strictly positive integers (m,n).
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
We consider the learning of multi-agent Hawkes processes, a model containing multiple Hawkes processes with shared endogenous impact functions and different exogenous intensities. In the framework of stochastic maximum likelihood estimation, we explore the associated risk bound. Further, we consider the superposition o…
The study examines Euclid's Book I, focusing on area applications and construction methods.
problem Exploring Euclid's geometric constructions and proofs, particularly those involving area calculations.
method Summarizing medieval editions and ancient commentaries, comparing constructions and proofs.
result Medieval editions often avoid Euclid's use of superposition in area proofs, offering alternative constructions.
Paper proposes an algorithm to reconstruct optimal model structure from graph adjacency matrix.
problem Optimal model structure reconstruction from weighted colored graph adjacency matrix.
method Uses prize-collecting Steiner tree algorithm to reconstruct minimum spanning tree.
result Demonstrates the effectiveness of the prize-collecting Steiner tree algorithm for model structure reconstruction.