RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basis

# RecKAN: Kolmogorov–Arnold Networks with a Learnable Recursive Polynomial Basis
Source: [https://arxiv.org/html/2609.01729](https://arxiv.org/html/2609.01729)
Amirhosein AzarpourAffiliation:Department of Computer ScienceAffiliation:Shahid Beheshti UniversityAffiliation:Tehran, IranEmail:[am\.azarpour@mail\.sbu\.ac\.ir](mailto:)

###### Abstract

Kolmogorov–Arnold Networks \(KANs\) replace the fixed scalar weights of a standard network with learnable univariate functions on each edge, but existing variants still fix the*basis*that those functions are built from: B\-splines, Chebyshev polynomials, wavelets, or Jacobi polynomials, and learn only the combination weights over it\. We introduce RecKAN, which instead defines the basis itself by a second order polynomial recurrence,Rn\+1​\(x\)=\(a​x2\+b​x\+c\)​Rn​\(x\)\+\(d​x\+e\)​Rn−1​\(x\)R\_\{n\+1\}\(x\)=\(ax^\{2\}\+bx\+c\)R\_\{n\}\(x\)\+\(dx\+e\)R\_\{n\-1\}\(x\), whose five coefficients are learned jointly with the network\. We show this recurrence recovers several classical polynomial families including both kinds of Chebyshev polynomials, Fibonacci, Pell, and Jacobsthal polynomials as special cases, and prove that its degree grows linearly innnexactly on the sub\-family containing all of them, giving a concrete sense in which the learned basis can move beyond any fixed classical choice\. Across multiple benchmark datasets spanning image, text, biomedical time series classification, and time series forecasting, RecKAN outperforms three parameter\-matched KAN baselines \(Chebyshev, Jacobi, and spline based\) on all classification tasks and achieves the lowest MSE on the ETTh1 forecasting benchmark\. Additionally, when used as a classifier head with a convolutional backbone, RecKAN achieves higher accuracy than standard MLP heads on Fashion MNIST, CIFAR\-10, and SVHN\. On a synthetic function fitting benchmark it tracks a sharply oscillatory target that a parameter comparable MLP under fits\. We further show that the learned recurrence coefficients are interpretable: on the task requiring the most local structure, training moves the basis away from the linear degree growth regime that contains every classical family we identify, consistent with our theoretical analysis of what that structural shift enables\.

Keywords:Kolmogorov–Arnold Networks, Recursive Polynomial Basis, Learnable Basis Functions, Polynomial Recurrence, Function Approximation\.

## 1Introduction

Multilayer perceptrons are universal function approximators\[[1](https://arxiv.org/html/2609.01729#bib.bib1),[2](https://arxiv.org/html/2609.01729#bib.bib2)\], but they reach this expressiveness with a fixed architecture: every node applies a fixed nonlinearity, and all of the network’s adaptivity lives in the scalar weights on its edges\. Kolmogorov–Arnold Networks \(KANs\) invert this design\. Motivated by the Kolmogorov–Arnold representation theorem\[[3](https://arxiv.org/html/2609.01729#bib.bib3)\], KANs replace each scalar edge weight with a learnable univariate function, so that adaptivity lives on the edges rather than the nodes; the original KAN parametrizes these edge functions with B\-splines\. This single change is reported to bring two practical benefits alongside the architectural novelty: better parameter efficiency than comparably sized MLPs on a range of function fitting and scientific computing tasks, and edge functions that can be inspected and, in simple cases, read off directly as closed form expressions an interpretability property standard MLPs largely lack\[[5](https://arxiv.org/html/2609.01729#bib.bib5),[6](https://arxiv.org/html/2609.01729#bib.bib6)\]\.

This reframing has been followed by a rapid succession of variants that keep the edge function idea but swap out the spline basis for something else: Chebyshev polynomials\[[7](https://arxiv.org/html/2609.01729#bib.bib7)\], wavelets\[[8](https://arxiv.org/html/2609.01729#bib.bib8)\], fractional Jacobi polynomials\[[9](https://arxiv.org/html/2609.01729#bib.bib9)\], radial basis functions\[[12](https://arxiv.org/html/2609.01729#bib.bib12),[13](https://arxiv.org/html/2609.01729#bib.bib13)\], and piecewise\-linear ReLU\-based bases\[[14](https://arxiv.org/html/2609.01729#bib.bib14)\], among others, have each been proposed as a faster or more accurate substitute for the original spline; a recent survey counts dozens of such variants and application\-specific descendants published within roughly a year of the original architecture\[[17](https://arxiv.org/html/2609.01729#bib.bib17)\]\. The resulting architectures have been applied to image classification\[[16](https://arxiv.org/html/2609.01729#bib.bib16)\], time series analysis\[[15](https://arxiv.org/html/2609.01729#bib.bib15)\], and scientific computing more broadly\[[6](https://arxiv.org/html/2609.01729#bib.bib6)\]\. Despite this diversity, every one of these variants shares a structural assumption with the original KAN: the basis family the finite set of candidate univariate functions each edge function is built from is fixed by the choice of architecture before training begins, and only the linear combination weights over that fixed basis are learned\. Choosing a basis is therefore a modeling decision made once, up front, and left to the practitioner’s judgment or to trial and error across variants; the network itself has no mechanism to revise that choice in light of the data\.

We remove that assumption\. Rather than selecting a basis family in advance, we generate it with a second order polynomial recurrence whose five coefficients are themselves learned jointly with the rest of the network \(Section[3](https://arxiv.org/html/2609.01729#S3)\)\. As we show in Section[3\.3](https://arxiv.org/html/2609.01729#S3.SS3), this recurrence is expressive enough to contain several classical polynomial families as special cases including both kinds of Chebyshev polynomials, Fibonacci, Pell, and Jacobsthal polynomials precisely because those families share the property of having recurrence coefficients that do not depend on the polynomial degree, unlike families such as Legendre polynomials\. Because the coefficients are learned rather than fixed, training can recover one of these known, well\-understood bases when it fits the data well, or move away from all of them into a region of function space that, to our knowledge, no fixed basis KAN variant can reach and, as we show in Section[4\.3](https://arxiv.org/html/2609.01729#S4.SS3), which of these two things happens in practice depends measurably on the target task\.

Our contributions are fourfold\. First, we introduce a learnable recursive polynomial basis for KANs, together with a formal characterization of which classical polynomial families it contains as special cases and how its expressiveness \(concretely, the polynomial degree reachable at a given recursion depth\) depends on its parameters\. Second, we provide an empirical comparison against three representative fixed or alternatively parametrized KAN baselines\[[7](https://arxiv.org/html/2609.01729#bib.bib7),[9](https://arxiv.org/html/2609.01729#bib.bib9),[5](https://arxiv.org/html/2609.01729#bib.bib5)\], at matched parameter budgets, across multiple benchmark datasets spanning image, text, biomedical time series classification, and time series forecasting\. Third, we extend this evaluation by replacing standard MLP classifier heads with RecKAN on top of convolutional backbones, showing consistent improvements on three image classification datasets\. Fourth, we analyze the recurrence parameters the model actually learns in these experiments and connect them back to the theoretical characterization from our first contribution, giving a concrete, interpretable account of what the model changed about its own basis in order to fit each task, rather than treating the learned basis as a black box\.

## 2Related Work

#### Kolmogorov–Arnold Networks\.

KANs replace scalar edge weights with learnable univariate functions\[[5](https://arxiv.org/html/2609.01729#bib.bib5),[6](https://arxiv.org/html/2609.01729#bib.bib6)\], motivated by the Kolmogorov–Arnold representation theorem\[[3](https://arxiv.org/html/2609.01729#bib.bib3),[4](https://arxiv.org/html/2609.01729#bib.bib4)\]and its earlier critical reappraisal in the neural\-network literature\[[18](https://arxiv.org/html/2609.01729#bib.bib18)\]\. Subsequent work has studied their theoretical expressiveness\[[19](https://arxiv.org/html/2609.01729#bib.bib19)\], benchmarked them against MLPs and tree based models on tabular data\[[28](https://arxiv.org/html/2609.01729#bib.bib28)\], and surveyed the rapidly growing design space\[[17](https://arxiv.org/html/2609.01729#bib.bib17),[20](https://arxiv.org/html/2609.01729#bib.bib20),[29](https://arxiv.org/html/2609.01729#bib.bib29)\]\. Architectural extensions include convolutional variants\[[22](https://arxiv.org/html/2609.01729#bib.bib22)\], graph variants\[[27](https://arxiv.org/html/2609.01729#bib.bib27),[21](https://arxiv.org/html/2609.01729#bib.bib21)\], temporal variants\[[26](https://arxiv.org/html/2609.01729#bib.bib26),[15](https://arxiv.org/html/2609.01729#bib.bib15)\], segmentation backbones\[[25](https://arxiv.org/html/2609.01729#bib.bib25)\], activation function\-based reformulations\[[35](https://arxiv.org/html/2609.01729#bib.bib35)\], and parameter\-efficient designs\[[24](https://arxiv.org/html/2609.01729#bib.bib24),[23](https://arxiv.org/html/2609.01729#bib.bib23),[12](https://arxiv.org/html/2609.01729#bib.bib12)\]\. Applications include computer vision\[[16](https://arxiv.org/html/2609.01729#bib.bib16)\], sensor\-data processing\[[37](https://arxiv.org/html/2609.01729#bib.bib37)\], and permutation invariant function representation\[[34](https://arxiv.org/html/2609.01729#bib.bib34)\]\.

#### Physics\-informed and scientific KANs\.

A substantial line of work embeds KANs into physics informed learning pipelines\. KINN reformulates strong, energy, and inverse\-form PDE losses on top of KAN layers and reports consistent gains over MLP\-based PINNs across multiscale, singular, and heterogeneous problems\[[30](https://arxiv.org/html/2609.01729#bib.bib30)\]\. Related efforts include physics\-informed KANs for Navier Stokes flow\[[31](https://arxiv.org/html/2609.01729#bib.bib31)\], hybrid encoder\-decoder KAN GRU architectures with augmented\-Lagrangian losses for PDE solving\[[32](https://arxiv.org/html/2609.01729#bib.bib32)\], and physics\-informed KAN formulations for power system dynamics\[[33](https://arxiv.org/html/2609.01729#bib.bib33)\]\. These results are broadly consistent with the parameter efficiency and interpretability motivations underlying the original KAN proposal\[[5](https://arxiv.org/html/2609.01729#bib.bib5),[6](https://arxiv.org/html/2609.01729#bib.bib6)\]\.

#### Fixed and structured basis families\.

A dominant strategy replaces the original B\-spline edge functions with a different, still\-fixed basis\. Examples include Chebyshev polynomials\[[7](https://arxiv.org/html/2609.01729#bib.bib7)\], trainable fractional Jacobi bases\[[9](https://arxiv.org/html/2609.01729#bib.bib9)\], wavelets\[[8](https://arxiv.org/html/2609.01729#bib.bib8)\], radial\-basis formulations\[[13](https://arxiv.org/html/2609.01729#bib.bib13),[12](https://arxiv.org/html/2609.01729#bib.bib12),[36](https://arxiv.org/html/2609.01729#bib.bib36)\], ReLU\-based piecewise constructions\[[14](https://arxiv.org/html/2609.01729#bib.bib14)\], and Fourier representations\[[21](https://arxiv.org/html/2609.01729#bib.bib21)\]\. A broad comparative study surveys eighteen distinct polynomial families including orthogonal, hypergeometric,qq\-, Fibonacci\-related, combinatorial, and number theoretic polynomials as candidate KAN bases\[[29](https://arxiv.org/html/2609.01729#bib.bib29)\]\. Classical results on polynomial recurrences\[[10](https://arxiv.org/html/2609.01729#bib.bib10),[11](https://arxiv.org/html/2609.01729#bib.bib11)\]underlie several of these families, since Chebyshev, Fibonacci, Pell, and Jacobsthal polynomials all arise from constant coefficient second order recurrences\. Across this line of work, the generating family is fixed by architecture choice, and training adjusts only coefficients, scales, or within family parameters\[[7](https://arxiv.org/html/2609.01729#bib.bib7),[9](https://arxiv.org/html/2609.01729#bib.bib9),[8](https://arxiv.org/html/2609.01729#bib.bib8),[13](https://arxiv.org/html/2609.01729#bib.bib13),[14](https://arxiv.org/html/2609.01729#bib.bib14),[36](https://arxiv.org/html/2609.01729#bib.bib36)\]\.

#### Relation to RecKAN\.

RecKAN differs by learning the second order recurrence that generates the basis itself, rather than selecting one fixed family in advance\. This recurrence subsumes several constant coefficient polynomial families studied in the classical literature\[[10](https://arxiv.org/html/2609.01729#bib.bib10),[11](https://arxiv.org/html/2609.01729#bib.bib11)\]as special cases, while remaining compatible with the broader KAN design space explored in vision, graph, temporal, tabular, and physics informed settings\[[16](https://arxiv.org/html/2609.01729#bib.bib16),[27](https://arxiv.org/html/2609.01729#bib.bib27),[26](https://arxiv.org/html/2609.01729#bib.bib26),[28](https://arxiv.org/html/2609.01729#bib.bib28),[30](https://arxiv.org/html/2609.01729#bib.bib30),[32](https://arxiv.org/html/2609.01729#bib.bib32)\]\. Where prior KAN variants adapt parameters within a fixed basis family\[[7](https://arxiv.org/html/2609.01729#bib.bib7),[9](https://arxiv.org/html/2609.01729#bib.bib9),[8](https://arxiv.org/html/2609.01729#bib.bib8),[13](https://arxiv.org/html/2609.01729#bib.bib13),[14](https://arxiv.org/html/2609.01729#bib.bib14),[21](https://arxiv.org/html/2609.01729#bib.bib21),[36](https://arxiv.org/html/2609.01729#bib.bib36)\], RecKAN adapts the basis\-generating rule itself\.

## 3Method

### 3\.1Preliminaries: Kolmogorov–Arnold Networks

The Kolmogorov–Arnold representation theorem states that any continuous functionf:\[0,1\]n→ℝf:\[0,1\]^\{n\}\\to\\mathbb\{R\}can be written as a finite superposition of continuous univariate functions\[[3](https://arxiv.org/html/2609.01729#bib.bib3),[4](https://arxiv.org/html/2609.01729#bib.bib4)\]\. Kolmogorov–Arnold Networks \(KANs\) operationalize this result as a learning architecture in which every edge of the network, rather than every node, carries a learnable univariate function\[[5](https://arxiv.org/html/2609.01729#bib.bib5)\]\. Concretely, a KAN layer mappingℝdin→ℝdout\\mathbb\{R\}^\{d\_\{\\text\{in\}\}\}\\to\\mathbb\{R\}^\{d\_\{\\text\{out\}\}\}is defined edge\-wise as

yj=∑i=1dinϕi,j\(xi\),j=1,…,dout,y\_\{j\}=\\sum\_\{i=1\}^\{d\_\{\\text\{in\}\}\}\\phi\_\{i,j\}\(x\_\{i\}\),\\qquad j=1,\\dots,d\_\{\\text\{out\}\},\(1\)where eachϕi,j:ℝ→ℝ\\phi\_\{i,j\}:\\mathbb\{R\}\\to\\mathbb\{R\}is a learnable univariate function rather than the fixed scalar weight of a standard linear layer\. The original KAN formulation parameterizesϕi,j\\phi\_\{i,j\}with B\-splines\[[5](https://arxiv.org/html/2609.01729#bib.bib5)\]; subsequent work has replaced the spline basis with other fixed function families, including Chebyshev polynomials\[[7](https://arxiv.org/html/2609.01729#bib.bib7)\], wavelets\[[8](https://arxiv.org/html/2609.01729#bib.bib8)\], and fractional Jacobi polynomials\[[9](https://arxiv.org/html/2609.01729#bib.bib9)\]\. In these variants, the basis family is fixed before training, while only the linear combination coefficients over that basis are learned\. RecKAN removes this restriction by learning the recurrence that generates the basis itself\.

### 3\.2A Learnable Recursive Polynomial Basis

###### Definition 1\(Recursive polynomial basis\)\.

Leta,b,c,d,e∈ℝa,b,c,d,e\\in\\mathbb\{R\}\. The*recursive polynomial basis*of orderKKgenerated by\(a,b,c,d,e\)\(a,b,c,d,e\)is the sequence of functionsR0,R1,…,RK:ℝ→ℝR\_\{0\},R\_\{1\},\\dots,R\_\{K\}:\\mathbb\{R\}\\to\\mathbb\{R\}defined by

R0​\(x\)\\displaystyle R\_\{0\}\(x\)=0,\\displaystyle=0,\(2\)R1​\(x\)\\displaystyle R\_\{1\}\(x\)=1,\\displaystyle=1,\(3\)Rn\+1​\(x\)\\displaystyle R\_\{n\+1\}\(x\)=\(a​x2\+b​x\+c\)⏟α⁡\(x\)Rn\(x\)\+\(d​x\+e\)⏟β⁡\(x\)Rn−1\(x\),n=1,…,K−1\.\\displaystyle=\\underbrace\{\\left\(ax^\{2\}\+bx\+c\\right\)\}\_\{\\textstyle\\alpha\(x\)\}R\_\{n\}\(x\)\+\\underbrace\{\\left\(dx\+e\\right\)\}\_\{\\textstyle\\beta\(x\)\}R\_\{n\-1\}\(x\),\\qquad n=1,\\dots,K\-1\.\(4\)The five coefficients\(a,b,c,d,e\)\(a,b,c,d,e\)are shared across all recursion steps and are treated as learnable parameters rather than fixed constants\.

Equation \([4](https://arxiv.org/html/2609.01729#S3.E4)\) is a second order linear recurrence with input\-dependent but recursion index independent coefficientsα⁡\(x\)\\alpha\(x\)andβ⁡\(x\)\\beta\(x\); because they are shared across all basis indices, the basis has exactly five recurrence parameters regardless of its orderKK\. This distinguishes the construction from the general three term recurrences used by many classical orthogonal polynomial families, whose coefficients usually depend on the polynomial index\.

###### Proposition 1\(Degree growth\)\.

Fora≠0a\\neq 0,deg⁡Rn≤2​\(n−1\)\\deg R\_\{n\}\\leq 2\(n\-1\)forn≥1n\\geq 1\. Fora=0a=0,deg⁡Rn≤n−1\\deg R\_\{n\}\\leq n\-1\. Equality holds in either case whenever no leading term cancellation occurs\.

*Sketch\.*Withδn=deg⁡Rn\\delta\_\{n\}=\\deg R\_\{n\}, Eq\. \([4](https://arxiv.org/html/2609.01729#S3.E4)\) givesδn\+1≤max⁡\{deg⁡α\+δn,deg⁡β\+δn−1\}\\delta\_\{n\+1\}\\leq\\max\\\{\\deg\\alpha\+\\delta\_\{n\},\\,\\deg\\beta\+\\delta\_\{n\-1\}\\\}; sincedeg⁡α=2\\deg\\alpha=2whena≠0a\\neq 0\(else≤1\\leq 1\) anddeg⁡β≤1\\deg\\beta\\leq 1always, the two stated bounds follow by induction onδ1=0\\delta\_\{1\}=0\. Practically: the quadratic terma​x2ax^\{2\}raises the maximum degree gained per recursion step from one to two, so at a fixed orderKKthea≠0a\\neq 0regime can reach basis elements of up to twice the degree available on thea=0a=0slice a capacity statement only, since cancellation can keep the realized degree lower A complete proof is provided in Appendix[B](https://arxiv.org/html/2609.01729#A2)\.

### 3\.3Recovering Classical Polynomial Families

A typical three\-term recurrence for a classical orthogonal\-polynomial family has the form

pn\+1​\(x\)=\(An​x\+Bn\)​pn​\(x\)−Cn​pn−1​\(x\),p\_\{n\+1\}\(x\)=\(A\_\{n\}x\+B\_\{n\}\)p\_\{n\}\(x\)\-C\_\{n\}p\_\{n\-1\}\(x\),\(5\)whereAn,BnA\_\{n\},B\_\{n\}, andCnC\_\{n\}may depend onnn\. The recurrence in Definition[1](https://arxiv.org/html/2609.01729#Thmdefinition1)instead uses coefficients that are constant with respect tonn, so only classical families with degree\-independent recurrence coefficients are reachable as special cases among them both kinds of Chebyshev polynomials and several Fibonacci type polynomial sequences\[[10](https://arxiv.org/html/2609.01729#bib.bib10),[11](https://arxiv.org/html/2609.01729#bib.bib11)\]\.

###### Proposition 2\(Chebyshev polynomials as a special case\)\.

Setting\(a,b,c,d,e\)=\(0,2,0,0,−1\)\(a,b,c,d,e\)=\(0,2,0,0,\-1\)in Definition[1](https://arxiv.org/html/2609.01729#Thmdefinition1)yields the recurrence

Rn\+1​\(x\)=2​x​Rn​\(x\)−Rn−1​\(x\)\.R\_\{n\+1\}\(x\)=2xR\_\{n\}\(x\)\-R\_\{n\-1\}\(x\)\.\(6\)With the initial conditionsR0=0R\_\{0\}=0andR1=1R\_\{1\}=1, this recurrence producesRn​\(x\)=Un−1​\(x\)R\_\{n\}\(x\)=U\_\{n\-1\}\(x\)forn≥1n\\geq 1, whereUnU\_\{n\}denotes the Chebyshev polynomials of the second kind\. Replacing the initial conditions by\(R0,R1\)=\(1,x\)\(R\_\{0\},R\_\{1\}\)=\(1,x\)produces the Chebyshev polynomials of the first kind,Rn​\(x\)=Tn​\(x\)R\_\{n\}\(x\)=T\_\{n\}\(x\)\.

Because Eq\. \([4](https://arxiv.org/html/2609.01729#S3.E4)\) is linear and second order, its solutions for fixed\(a,b,c,d,e\)\(a,b,c,d,e\)form a two\-dimensional space, so the same coefficients can generate more than one named sequence depending on the initial condition ChebyshevTnT\_\{n\}andUnU\_\{n\}above being the standard example\. RecKAN fixes\(R0,R1\)=\(0,1\)\(R\_\{0\},R\_\{1\}\)=\(0,1\)throughout, which selects the Fibonacci\-type solution for whichever coefficients training settles on\.

Table 1:Named polynomial families recovered from the recursive basis of Definition[1](https://arxiv.org/html/2609.01729#Thmdefinition1)for selected settings of\(a,b,c,d,e\)\(a,b,c,d,e\)\. “Init\. match” indicates whether the standard initial condition agrees with\(R0,R1\)=\(0,1\)\(R\_\{0\},R\_\{1\}\)=\(0,1\)\.Every listed family hasa=0a=0, and hence belongs to the degree one per step sub\-family described by Proposition[1](https://arxiv.org/html/2609.01729#Thmproposition1)\. The learned recurrence can move beyond this slice by activating the quadratic term inα⁡\(x\)\\alpha\(x\), by introducing a linear term inβ⁡\(x\)\\beta\(x\), or by jointly using the remaining coefficients bases that need not coincide with any fixed classical family\. The Chebyshev setting of Proposition[2](https://arxiv.org/html/2609.01729#Thmproposition2)identifies the point at which the learnable recurrence coincides with the fixed basis used in ChebyKAN\[[7](https://arxiv.org/html/2609.01729#bib.bib7)\], giving a natural initialization from which optimization remains free to adapt the basis to the data\.

### 3\.4Edge Functions and the RecKAN Layer

Given the basisR0,…,RKR\_\{0\},\\dots,R\_\{K\}generated by Eq\. \([4](https://arxiv.org/html/2609.01729#S3.E4)\), the learnable univariate edge function in Eq\. \([1](https://arxiv.org/html/2609.01729#S3.E1)\) is parameterized as

ϕi,j​\(xi\)=∑n=0Kwi,j,n​Rn​\(xi\),\\phi\_\{i,j\}\(x\_\{i\}\)=\\sum\_\{n=0\}^\{K\}w\_\{i,j,n\}R\_\{n\}\(x\_\{i\}\),\(7\)wherewi,j,n∈ℝw\_\{i,j,n\}\\in\\mathbb\{R\}are learnable combination coefficients\. For an input𝐱∈ℝdin\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\_\{\\text\{in\}\}\}, define the basis tensor

𝐑⁡\(𝐱\)∈ℝdin×\(K\+1\),𝐑​\(𝐱\)i,n=Rn​\(xi\),\\mathbf\{R\}\(\\mathbf\{x\}\)\\in\\mathbb\{R\}^\{d\_\{\\text\{in\}\}\\times\(K\+1\)\},\\qquad\\mathbf\{R\}\(\\mathbf\{x\}\)\_\{i,n\}=R\_\{n\}\(x\_\{i\}\),\(8\)evaluated independently for each input coordinate\. With a weight tensor𝐖∈ℝdin×dout×\(K\+1\)\\mathbf\{W\}\\in\\mathbb\{R\}^\{d\_\{\\text\{in\}\}\\times d\_\{\\text\{out\}\}\\times\(K\+1\)\}, the layer output is

yj=∑i=1din∑n=0K𝐑​\(𝐱\)i,n​Wi,j,n,𝐲=𝐑⁡\(𝐱\)×1,3𝐖\.y\_\{j\}=\\sum\_\{i=1\}^\{d\_\{\\text\{in\}\}\}\\sum\_\{n=0\}^\{K\}\\mathbf\{R\}\(\\mathbf\{x\}\)\_\{i,n\}W\_\{i,j,n\},\\qquad\\mathbf\{y\}=\\mathbf\{R\}\(\\mathbf\{x\}\)\\mathbin\{\\\!\\times\_\{1,3\}\\\!\}\\mathbf\{W\}\.\(9\)A RecKAN layer withdind\_\{\\text\{in\}\}inputs,doutd\_\{\\text\{out\}\}outputs, and basis orderKKtherefore containsdin​dout​\(K\+1\)d\_\{\\text\{in\}\}d\_\{\\text\{out\}\}\(K\+1\)combination weights\.

### 3\.5Network Architecture and Parameter Sharing

A RecKAN network of depthLLis a composition of RecKAN layers,

F\(𝐱\)=\(Φ\(L\)∘Φ\(L−1\)∘⋯∘Φ\(1\)\)\(𝐱\),F\(\\mathbf\{x\}\)=\\left\(\\Phi^\{\(L\)\}\\circ\\Phi^\{\(L\-1\)\}\\circ\\cdots\\circ\\Phi^\{\(1\)\}\\right\)\(\\mathbf\{x\}\),\(10\)where eachΦ\(ℓ\)\\Phi^\{\(\\ell\)\}applies Eqs\. \([8](https://arxiv.org/html/2609.01729#S3.E8)\)–\([9](https://arxiv.org/html/2609.01729#S3.E9)\) with its own combination tensor𝐖\(ℓ\)\\mathbf\{W\}^\{\(\\ell\)\}\. A single set of recurrence coefficients\(a,b,c,d,e\)\(a,b,c,d,e\)is instantiated once and shared by every layer, so the learned basis contributes exactly five shared parameters to the network, independent of depth; only the combination weights are layer\-specific\. Pre\-activations entering the basis are mapped to a bounded range beforehand, elementwise, viah=tanh⁡\(z\)h=\\tanh\(z\), after whichxix\_\{i\}in Eq\. \([4](https://arxiv.org/html/2609.01729#S3.E4)\) is replaced byhih\_\{i\}\.

R0=0R\_\{0\}=0R1=1R\_\{1\}=1R2R\_\{2\}R3R\_\{3\}⋯\\cdotsRKR\_\{K\}β⁡\(x\)\\beta\(x\)β⁡\(x\)\\beta\(x\)β⁡\(x\)\\beta\(x\)β⁡\(x\)\\beta\(x\)α⁡\(x\)\\alpha\(x\)α⁡\(x\)\\alpha\(x\)α⁡\(x\)\\alpha\(x\)α⁡\(x\)\\alpha\(x\)\(a,b,c,d,e\)\(a,b,c,d,e\)shared across all stepsFigure 1:Computational graph of the recursive polynomial basis for a scalar inputxx\. Each basis function is computed from the two preceding functions through the shared coefficients\(a,b,c,d,e\)\(a,b,c,d,e\)\.𝐱∈ℝd0\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\_\{0\}\}LayerΦ\(1\)\\Phi^\{\(1\)\}𝐖\(1\)\\mathbf\{W\}^\{\(1\)\}LayerΦ\(2\)\\Phi^\{\(2\)\}𝐖\(2\)\\mathbf\{W\}^\{\(2\)\}⋯\\cdotsLayerΦ\(L\)\\Phi^\{\(L\)\}𝐖\(L\)\\mathbf\{W\}^\{\(L\)\}F⁡\(𝐱\)∈ℝdLF\(\\mathbf\{x\}\)\\in\\mathbb\{R\}^\{d\_\{L\}\}Shared recursive basis module\(a,b,c,d,e\)→R0,…,RK\(a,b,c,d,e\)\\to R\_\{0\},\\dots,R\_\{K\}Figure 2:RecKAN architecture\. All layers share one recursive basis generated by\(a,b,c,d,e\)\(a,b,c,d,e\), while each layer has separate combination weights\.
### 3\.6Numerical Stability and Training Scheme

The recurrence coefficients can in principle take arbitrary real values, and repeated multiplication through the recurrence can increase the magnitude of basis functions as the basis order grows\. We use four complementary controls\.

#### Bounded recurrence coefficients\.

Rather than optimizing\(a,b,c,d,e\)\(a,b,c,d,e\)directly, we optimize unconstrained parameters\(a~,b~,c~,d~,e~\)∈ℝ5\(\\tilde\{a\},\\tilde\{b\},\\tilde\{c\},\\tilde\{d\},\\tilde\{e\}\)\\in\\mathbb\{R\}^\{5\}and obtain the recurrence coefficients through

a\\displaystyle a=B​tanh⁡\(a~\),\\displaystyle=B\\tanh\(\\tilde\{a\}\),b\\displaystyle b=B​tanh⁡\(b~\),\\displaystyle=B\\tanh\(\\tilde\{b\}\),c\\displaystyle c=B​tanh⁡\(c~\),\\displaystyle=B\\tanh\(\\tilde\{c\}\),\(11\)d\\displaystyle d=B​tanh⁡\(d~\),\\displaystyle=B\\tanh\(\\tilde\{d\}\),e\\displaystyle e=B​tanh⁡\(e~\),\\displaystyle=B\\tanh\(\\tilde\{e\}\),for a fixedB\>0B\>0\. This ensures\(a,b,c,d,e\)∈\[−B,B\]5\(a,b,c,d,e\)\\in\[\-B,B\]^\{5\}while preserving end\-to\-end differentiability\. WhenB\>2B\>2, the exact Chebyshev point from Proposition[2](https://arxiv.org/html/2609.01729#Thmproposition2)lies in the interior of the feasible region\.

#### Magnitude normalization\.

Each newly generated basis value is rescaled by one scalar computed over the current batch before it is used in the following recurrence step:

R^n\+1​\(x\)=Rn\+1​\(x\)max⁡\(ε,maxx⁡\|Rn\+1​\(x\)\|\)\.\\widehat\{R\}\_\{n\+1\}\(x\)=\\frac\{R\_\{n\+1\}\(x\)\}\{\\max\\\!\\left\(\\varepsilon,\\max\_\{x\}\\left\|R\_\{n\+1\}\(x\)\\right\|\\right\)\}\.\(12\)Hereε\>0\\varepsilon\>0provides numerical safety, and the denominator is detached from the gradient computation, so normalization acts only as a forward pass stabilizer\. A single normalizing scalar is used per basis indexnn, not per feature, so relative scale between input coordinates is preserved\.

#### Input range control\.

Pre\-activations are passed throughtanh⁡\(⋅\)\\tanh\(\\cdot\)before evaluating the recurrence, keeping recursion arguments in a bounded interval and preventing uncontrolled growth from large inputs\.

#### Two\-timescale optimization and gradient clipping\.

The recurrence parameters affect every generated basis function, whereas the combination weights enter linearly at the layer output\. We therefore allow a separate learning rate for the recurrence parameters,

ηbasis=γ​ηweights,γ∈\(0,1\],\\eta\_\{\\text\{basis\}\}=\\gamma\\eta\_\{\\text\{weights\}\},\\qquad\\gamma\\in\(0,1\],\(13\)and optionally freeze the recurrence parameters during an initial warm\-up period, in addition to global gradient\-norm clipping at every optimization step\.

Algorithm 1Forward pass of a RecKAN layer1:input

𝐱∈ℝdin\\mathbf\{x\}\\in\\mathbb\{R\}^\{d\_\{\\text\{in\}\}\}; raw recurrence parameters

\(a~,b~,c~,d~,e~\)\(\\tilde\{a\},\\tilde\{b\},\\tilde\{c\},\\tilde\{d\},\\tilde\{e\}\); weight tensor

𝐖∈ℝdin×dout×\(K\+1\)\\mathbf\{W\}\\in\\mathbb\{R\}^\{d\_\{\\text\{in\}\}\\times d\_\{\\text\{out\}\}\\times\(K\+1\)\}; bound

BB; basis order

KK
2:

𝐡←tanh⁡\(𝐱\)\\mathbf\{h\}\\leftarrow\\tanh\(\\mathbf\{x\}\)⊳\\trianglerightinput range control

3:

\(a,b,c,d,e\)←B​tanh⁡\(a~\),B​tanh⁡\(b~\),B​tanh⁡\(c~\),B​tanh⁡\(d~\),B​tanh⁡\(e~\)\(a,b,c,d,e\)\\leftarrow B\\tanh\(\\tilde\{a\}\),B\\tanh\(\\tilde\{b\}\),B\\tanh\(\\tilde\{c\}\),B\\tanh\(\\tilde\{d\}\),B\\tanh\(\\tilde\{e\}\)
4:

R0←𝟎R\_\{0\}\\leftarrow\\mathbf\{0\},

R1←𝟏R\_\{1\}\\leftarrow\\mathbf\{1\}⊳\\trianglerightelementwise over𝐡\\mathbf\{h\}

5:for

n=1,…,K−1n=1,\\dots,K\-1do

6:

α←a​𝐡2\+b​𝐡\+c\\alpha\\leftarrow a\\,\\mathbf\{h\}^\{2\}\+b\\,\\mathbf\{h\}\+c;

β←d​𝐡\+e\\beta\\leftarrow d\\,\\mathbf\{h\}\+e
7:

Rn\+1←α⊙Rn\+β⊙Rn−1R\_\{n\+1\}\\leftarrow\\alpha\\odot R\_\{n\}\+\\beta\\odot R\_\{n\-1\}
8:

Rn\+1←Rn\+1/max⁡\(ε,max⁡\|Rn\+1\|\)R\_\{n\+1\}\\leftarrow R\_\{n\+1\}\\,/\\,\\max\(\\varepsilon,\\,\\max\|R\_\{n\+1\}\|\)⊳\\trianglerightEq\. \([12](https://arxiv.org/html/2609.01729#S3.E12)\), detached

9:endfor

10:

𝐑←stack⁡\(R0,…,RK\)\\mathbf\{R\}\\leftarrow\\operatorname\{stack\}\(R\_\{0\},\\dots,R\_\{K\}\)⊳\\trianglerightshapedin×\(K\+1\)d\_\{\\text\{in\}\}\\times\(K\+1\)

11:

𝐲←𝐑×1,3𝐖\\mathbf\{y\}\\leftarrow\\mathbf\{R\}\\times\_\{1,3\}\\mathbf\{W\}⊳\\trianglerightEq\. \([9](https://arxiv.org/html/2609.01729#S3.E9)\)

12:return

𝐲\\mathbf\{y\}

## 4Experiments and Results

### 4\.1Experimental Setup

We evaluate RecKAN against three fixed or alternatively parameterized KAN baselines: ChebyKAN\[[7](https://arxiv.org/html/2609.01729#bib.bib7)\], a Jacobi\-polynomial variant in the style of fKAN\[[9](https://arxiv.org/html/2609.01729#bib.bib9)\], and a B\-spline variant in the style of the original KAN\[[5](https://arxiv.org/html/2609.01729#bib.bib5)\]\. We also compare RecKAN against a standard multilayer perceptron \(MLP\) on the synthetic function\-approximation tasks\.

Every comparison uses a matched parameter protocol\. For a given task, all KAN variants use the same input hidden output shape\. The basis order for polynomial methods, or the spline grid size for the spline baseline, is chosen so that total parameter counts are closely matched across the compared KAN models\. The synthetic MLP comparisons use parameter comparable MLPs; the exact parameter counts are reported with the corresponding results\. All models use Adam optimization and gradient clipping\. Within each task, models are trained with the same epoch budget and no baseline is assigned a reduced optimization budget\.

### 4\.2Function Approximation

We first evaluate the learned basis on two synthetic regression targets with local, non stationary, and multi scale structure\. These tasks are designed to make differences in the representational behavior of a fixed basis and a learned recurrence directly visible\.

#### 1D piecewise oscillatory function\.

The first target is a noisy one dimensional function containing a high frequency oscillatory region together with a smooth, slowly varying region\. RecKAN contains 149 parameters and is compared against an MLP with 385 parameters\. Figure[3](https://arxiv.org/html/2609.01729#S4.F3)shows the fitted functions together with the noisy training observations\.

![Refer to caption](https://arxiv.org/html/2609.01729v1/piecewise_1d_comparison.png)Figure 3:Fitting a noisy piecewise 1D function\. RecKAN \(red\) tracks both the high frequency oscillatory region and the smooth region, whereas the parameter comparable MLP \(green\) under fits the oscillatory portion and approximates a local average there\.Table 2:Training loss \(MSE\) on the 1D piecewise function, recorded at epoch 33,000\.At the recorded epoch 33,000 checkpoint, RecKAN attains an MSE of 0\.0391, compared with 0\.1583 for the MLP, while using fewer parameters \(149 versus 385\)\. The visual fit in Figure[3](https://arxiv.org/html/2609.01729#S4.F3)is consistent with this difference: RecKAN follows the high frequency part of the target as well as the smooth portion of the domain, whereas the MLP does not recover the rapid local oscillations\.

#### 2D fractal like noisy function\.

The second target is a two dimensional multi\-scale surface constructed from sinusoids at several spatial frequencies with additive noise\. RecKAN has 1,445 parameters, using three layers of shape2→8→16→12\\to 8\\to 16\\to 1at basis order88plus five shared recurrence parameters\. It is compared against an MLP with 2,305 parameters and hidden\-layer widths 64 and 32\. Figure[4](https://arxiv.org/html/2609.01729#S4.F4)shows the predicted surfaces next to the noisy target\.

Figure 4:Fitting a noisy fractal like 2D function\. Left: MLP predictions\. Center: RecKAN predictions\. Right: noisy target surface\. Both models recover the coarse structure; RecKAN additionally recovers more fine scale, high frequency detail visible in the target\.Table 3:Training loss \(MSE\) on the 2D fractal like function, recorded at epoch 3,001\.At the recorded epoch\-3,001 checkpoint, RecKAN achieves a lower training MSE than the MLP, 0\.113 versus 0\.200, while using fewer parameters\. The predicted surface in Figure[4](https://arxiv.org/html/2609.01729#S4.F4)is consistent with the quantitative result: both models reproduce broad variations, while RecKAN recovers more of the target’s local high frequency structure\.

### 4\.3Analysis of the Learned Recurrence Parameters

Beyond predictive performance, the five learned recurrence parameters\(a,b,c,d,e\)\(a,b,c,d,e\)can be interpreted using the recurrence analysis in Section[3\.3](https://arxiv.org/html/2609.01729#S3.SS3)\. The two synthetic experiments were initialized at

\(a,b,c,d,e\)=\(0,2,0,−1,0\)\.\(a,b,c,d,e\)=\(0,2,0,\-1,0\)\.\(14\)This initialization is Chebyshev like because it hasα⁡\(x\)=2​x\\alpha\(x\)=2x, but it is not the exact Chebyshev setting from Proposition[2](https://arxiv.org/html/2609.01729#Thmproposition2): its far term coefficient isβ⁡\(x\)=−x\\beta\(x\)=\-x, rather than the constantβ⁡\(x\)=−1\\beta\(x\)=\-1required by the standard Chebyshev recurrence\.

Table 4:Learned recurrence parameters at the end of the synthetic runs\. Both runs use the initialization\(a,b,c,d,e\)=\(0,2,0,−1,0\)\(a,b,c,d,e\)=\(0,2,0,\-1,0\)\.#### 2D fractal\-like function\.

The 2D run remains close to its initialization\. In particular,a=0\.049a=0\.049remains near zero, while the largest change is a small constant offset inβ⁡\(x\)\\beta\(x\)throughe=−0\.101e=\-0\.101\. The learned basis therefore stays close to the degree one per step regime described in Proposition[1](https://arxiv.org/html/2609.01729#Thmproposition1)\. This is consistent with the visual result in Figure[4](https://arxiv.org/html/2609.01729#S4.F4), where RecKAN improves reconstruction of fine detail without requiring a large structural departure from its initial basis\.

#### 1D piecewise function\.

The 1D run moves more substantially in two directions\. First,β⁡\(x\)=d​x\+e\\beta\(x\)=dx\+echanges from an input dependent term dominated byd=−1d=\-1to a term dominated by the constant componente=−0\.864e=\-0\.864\. Second, the quadratic coefficient grows froma=0a=0toa=0\.335a=0\.335\. By Proposition[1](https://arxiv.org/html/2609.01729#Thmproposition1), this activates a recurrence whose basis functions can accumulate up to two degrees per recursion step, rather than one\. At basis orderK=8K=8, the degree bound consequently expands from seven on thea=0a=0slice to fourteen when the quadratic contribution is active\.

This provides a plausible interpretation of the fit in Figure[3](https://arxiv.org/html/2609.01729#S4.F3): the local high frequency portion of the target may benefit from the additional polynomial capacity provided by the learned quadratic term\. The interpretation is descriptive rather than causal; establishing causality would require controlled ablations, such as directly comparing the learned basis spectrum with a fixed degree\-8 Chebyshev basis on the same target\.

### 4\.4Benchmark Classification and Forecasting

We next compare RecKAN with the three KAN baselines on five real benchmark datasets: MNIST and CIFAR\-10 for image classification, AG News for text classification, ECG5000 for biomedical time series classification, and ETTh1 for time\-series forecasting\. The models are compared under the matched\-parameter protocol described in Section[4\.1](https://arxiv.org/html/2609.01729#S4.SS1)\. For each dataset, we report the mean best test accuracy \(for classification tasks\) or mean best test MSE \(for the forecasting task\) and standard deviation across multiple independent runs\.

Table 5:Mean best test performance and standard deviation across multiple runs on five benchmark datasets\. All models are closely parameter matched within each dataset; the largest parameter count difference is five\. The best mean performance in each dataset row is shown in bold\. For classification tasks, higher accuracy is better; for ETTh1 forecasting, lower MSE is better\.RecKAN achieves the highest mean test accuracy on all four classification benchmark datasets in Table[5](https://arxiv.org/html/2609.01729#S4.T5)\. On MNIST, RecKAN attains 97\.39% accuracy, outperforming the strongest baseline \(JacobiKAN\) by 0\.20 percentage points\. On CIFAR\-10, RecKAN reaches 54\.39%, surpassing the best baseline \(ChebyKAN\) by 0\.89 percentage points\. For AG News, RecKAN achieves 88\.40%, improving upon the best baseline \(ChebyKAN\) by 0\.13 percentage points\. On ECG5000, RecKAN obtains 93\.89%, exceeding the best baseline \(JacobiKAN\) by 0\.08 percentage points\.

For the ETTh1 forecasting task, RecKAN achieves the lowest mean test MSE of 0\.011172, outperforming SplineKAN \(0\.011879\), JacobiKAN \(0\.033786\), and ChebyKAN \(0\.036781\)\. The performance gap between RecKAN and SplineKAN is relatively small on this task, while both significantly outperform the other two baselines\. This suggests that RecKAN and SplineKAN are particularly well suited for time series forecasting tasks\.

Table 6:Improvement of RecKAN over baseline KAN variants across all datasets\. For classification tasks, values represent the relative gain in mean best test accuracy \(%\)\. For ETTh1, values represent the relative reduction in MSE\.The results demonstrate that RecKAN consistently achieves competitive or superior performance across diverse data modalities, including images, text, and time series signals\. The learned recursive basis proves to be an effective alternative to fixed polynomial bases, offering improved representational flexibility while maintaining a compact parameter footprint\. The standard deviations indicate that RecKAN exhibits stable performance across runs, with particularly low variance on AG News \(±\\pm0\.07\) and competitive stability on other datasets\. Notably, RecKAN shows the largest improvement on CIFAR\-10 \(\+3\.107% over SplineKAN and \+0\.894% over ChebyKAN\), a more challenging image classification task, while maintaining consistent gains across all benchmarks\. On average, RecKAN outperforms SplineKAN by 1\.11%, ChebyKAN by 0\.38%, and JacobiKAN by 0\.41% on the classification benchmarks\.

### 4\.5RecKAN Classifier Heads for Convolutional Networks

We further evaluate whether the learnable recursive basis can complement a convolutional feature extractor\. We compare a standard CNN with a CNN–RecKAN variant on Fashion\-MNIST, CIFAR\-10, and SVHN\. For each dataset, both models use the same convolutional backbone and differ only in their classifier head\. The standard CNN uses a two layer MLP head, whereas CNN–RecKAN replaces this head with a recursive polynomial classifier\.

The models are closely parameter matched\. For Fashion\-MNIST, the standard CNN contains 146,823 trainable parameters and CNN–RecKAN contains 146,863\. For CIFAR\-10 and SVHN, the corresponding counts are 295,111 and 295,151\. Thus, CNN–RecKAN uses only 40 additional trainable parameters in each comparison\. All paired models use the same input preprocessing, convolutional backbone, batch size, optimizer, learning rate schedule, weight decay, gradient clipping, and training epoch budget\.

Table 7:Mean best test accuracy \(%\) and standard deviation across multiple runs for standard CNN and CNN–RecKAN models with identical convolutional backbones and closely matched parameter counts\. The best mean accuracy in each dataset row is shown in bold\.Under the matched parameter protocol, CNN–RecKAN obtains higher mean best test accuracy on all three datasets: 93\.357% versus 92\.840% on Fashion\-MNIST, 88\.503% versus 88\.117% on CIFAR\-10, and 96\.150% versus 96\.070% on SVHN\. The corresponding absolute improvements are 0\.517, 0\.386, and 0\.080 percentage points, respectively\. These results demonstrate that replacing the MLP classifier head with a RecKAN classifier head consistently improves performance across different image classification tasks, achieving gains with only 40 additional trainable parameters\. The learned recursive basis effectively complements the convolutional feature extractor, providing a lightweight yet effective enhancement for CNN\-based architectures\.

Table 8:Learned recurrence coefficients at the best CNN–RecKAN checkpoint for each image classification dataset\.The learned recurrence parameters in Table[8](https://arxiv.org/html/2609.01729#S4.T8)are reported from one representative run; they differ across datasets\. In all three CNN experiments, the quadratic coefficientaabecomes nonzero, placing the learned basis outside the degree one per step sub family containing the classical special cases in Table[1](https://arxiv.org/html/2609.01729#S3.T1)\. At the same time, the learned values retain a positive coefficient near two for the linear termbband a negative constant feedback termee\. CIFAR\-10 learns the largest input\-dependent contribution inβ⁡\(x\)=d​x\+e\\beta\(x\)=dx\+e, withd=0\.6865d=0\.6865, whereas Fashion\-MNIST learns a nearly constantβ⁡\(x\)\\beta\(x\)withd=−0\.0990d=\-0\.0990\. These task\-dependent parameter settings show that the learned recurrence does not simply retain its initial Chebyshev like configuration\. They are descriptive observations rather than causal explanations of the accuracy differences; establishing causality would require controlled ablations\.

Across all three datasets, CNN–RecKAN consistently outperforms the standard CNN with MLP head, with average improvement of 0\.328 percentage points using only 40 additional parameters\. The most substantial gain is observed on Fashion\-MNIST \(\+0\.517%\), while the improvement on SVHN is marginal \(\+0\.080%\) and within the standard deviation\. The standard deviations indicate that CNN–RecKAN exhibits slightly higher variance on Fashion\-MNIST and SVHN compared to the standard CNN, but lower variance on CIFAR\-10\. Overall, these results suggest that replacing an MLP head with a RecKAN classifier head provides a lightweight and effective enhancement for CNN\-based architectures\.

## 5Conclusion

We introduced RecKAN, a Kolmogorov–Arnold Network in which the univariate basis is generated by a learnable second order polynomial recurrence\. Rather than selecting a spline, Chebyshev, Jacobi, wavelet, or other basis family before optimization, RecKAN learns the coefficients of the rule that produces the basis itself\. This construction recovers several familiar constant coefficient polynomial recurrences as special cases, including Chebyshev, Fibonacci, Pell, and Jacobsthal polynomial sequences, while also allowing the learned basis to move beyond these predefined families\.

The degree analysis shows that activating the quadratic recurrence coefficient changes the maximum degree growth from one to two degrees per recursion step\. This provides a direct and interpretable capacity mechanism: at a fixed basis order, RecKAN can construct polynomial basis elements of higher degree than the classicala=0a=0recurrence slice\. The learned coefficients further make this mechanism observable, allowing the resulting basis to be inspected rather than treated solely as an implicit representation\.

Across the five benchmark datasets \(four classification and one forecasting\), RecKAN achieves the highest reported test accuracy on all classification tasks and the lowest MSE on the ETTh1 forecasting benchmark among the compared Chebyshev, Jacobi, and spline\-based KAN variants under closely matched parameter budgets\. On the synthetic function approximation tasks, RecKAN obtains lower training error than the parameter comparable MLPs and better captures local high frequency structure\. The learned recurrence parameters vary substantially across these tasks, indicating that the model adapts its basis rather than retaining its Chebyshev like initialization\.

The CNN head experiments extend this observation beyond fully KAN based architectures\. With identical convolutional feature extractors and nearly identical parameter counts, CNN–RecKAN achieves higher reported test accuracy than matched MLP classifier heads on Fashion\-MNIST, CIFAR\-10, and SVHN\. Taken together, these results suggest that learning the basis generating recurrence is a useful alternative to selecting a fixed polynomial basis in advance\.

## 6Limitations and Future Work

RecKAN currently uses one set of five recurrence coefficients shared across all edges and layers of a network\. This global sharing makes the learned basis compact and easy to interpret, but it may limit expressiveness when different layers, channels, or regions of an input require substantially different univariate representations\. A natural extension is to study layer specific, group specific, or edge specific recurrences while retaining constraints that preserve numerical stability and interpretability\.

The present formulation uses a second order recurrence with a quadratic coefficient in the current term and a linear coefficient in the preceding term\. This choice provides a small, tractable parameterization and includes several well known polynomial sequences, but it does not cover all classical orthogonal polynomial families whose recurrence coefficients vary with the basis index\. Extending RecKAN to controlled index dependent recurrences, higher order recurrences, rational recurrences, or mixtures of recursive and non\-polynomial basis functions could enlarge the family of adaptive bases available to KANs\.

The stabilization scheme relies on bounded recurrence coefficients, bounded inputs, basis normalization, and separate optimization control for recurrence parameters\. These components make the current implementation practical, but their interactions with basis order, network depth, normalization design, and different data modalities remain open\. A more detailed analysis of the optimization geometry and numerical properties of learned recurrences may clarify when particular recurrence structures are preferable\.

Finally, the learned coefficients provide a compact description of the basis, but their task level interpretation remains descriptive\. Future work can connect coefficient trajectories to explicit spectral properties, study the effective polynomial degrees used by trained edge functions, and develop regularizers that encourage selected structural properties such as sparsity, proximity to known polynomial families, or controlled departures from a reference recurrence\. These directions may further strengthen RecKAN as both a flexible function approximator and an interpretable basis learning framework\.

## References

- \[1\]G\. Cybenko\.Approximation by superpositions of a sigmoidal function\.*Mathematics of Control, Signals and Systems*, 2\(4\):303–314, 1989\.doi:[https://doi\.org/10\.1007/BF02551274](https://doi.org/10.1007/BF02551274)\.
- \[2\]K\. Hornik, M\. Stinchcombe, and H\. White\.Multilayer feedforward networks are universal approximators\.*Neural Networks*, 2\(5\):359–366, 1989\.doi:[https://doi\.org/10\.1016/0893\-6080\(89\)90020\-8](https://doi.org/10.1016/0893-6080(89)90020-8)\.
- \[3\]A\. N\. Kolmogorov\.On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition\.*Doklady Akademii Nauk SSSR*, 114:953–956, 1957\.
- \[4\]V\. I\. Arnold\.On functions of three variables\.*Doklady Akademii Nauk SSSR*, 114:679–681, 1957\.
- \[5\]Z\. Liu, Y\. Wang, S\. Vaidya, F\. Ruehle, J\. Halverson, M\. Soljačić, T\. Y\. Hou, and M\. Tegmark\.KAN: Kolmogorov–Arnold networks\.arXiv preprint arXiv:2404\.19756, 2024\.
- \[6\]Z\. Liu, P\. Ma, Y\. Wang, W\. Matusik, and M\. Tegmark\.KAN 2\.0: Kolmogorov–Arnold networks meet science\.arXiv preprint arXiv:2408\.10205, 2024\.
- \[7\]S\. S\. Sidharth, R\. Gokul, K\. P\. Anas, and A\. R\. Keerthana\.Chebyshev polynomial\-based Kolmogorov–Arnold networks: An efficient architecture for nonlinear function approximation\.arXiv preprint arXiv:2405\.07200, 2024\.
- \[8\]Z\. Bozorgasl and H\. Chen\.Wav\-KAN: Wavelet Kolmogorov–Arnold networks\.arXiv preprint arXiv:2405\.12832, 2024\.
- \[9\]A\. A\. Aghaei\.fKAN: Fractional Kolmogorov–Arnold networks with trainable Jacobi basis functions\.arXiv preprint arXiv:2406\.07456, 2024\.
- \[10\]G\. Y\. Lee and M\. Asci\.Some properties of the\(p,q\)\(p,q\)\-Fibonacci and\(p,q\)\(p,q\)\-Lucas polynomials\.*Journal of Applied Mathematics*, 2012:264842, 2012\.doi:[https://doi\.org/10\.1155/2012/264842](https://doi.org/10.1155/2012/264842)\.
- \[11\]A\. Luzón and M\. A\. Morón\.Recurrence relations for polynomial sequences via Riordan matrices\.arXiv preprint arXiv:0904\.2672, 2009\.
- \[12\]Z\. Li\.FastKAN: Very fast implementation of Kolmogorov–Arnold networks\.GitHub repository, 2024\.[https://github\.com/ZiyaoLi/fast\-kan](https://github.com/ZiyaoLi/fast-kan)\.
- \[13\]Z\. Li\.Kolmogorov–Arnold networks are radial basis function networks\.arXiv preprint arXiv:2405\.06721, 2024\.
- \[14\]Q\. Qiu, T\. Zhu, H\. Gong, L\. Chen, and H\. Ning\.ReLU\-KAN: New Kolmogorov–Arnold networks that only need matrix addition, dot multiplication, and ReLU\.arXiv preprint arXiv:2406\.02075, 2024\.
- \[15\]C\. J\. Vaca\-Rubio, L\. Blanco, R\. Pereira, and M\. Caus\.Kolmogorov–Arnold networks \(KANs\) for time series analysis\.arXiv preprint arXiv:2405\.08790, 2024\.
- \[16\]B\. Azam and N\. Akhtar\.Suitability of KANs for computer vision: A preliminary investigation\.arXiv preprint arXiv:2406\.09087, 2024\.
- \[17\]S\. Somvanshi, S\. A\. Javed, M\. M\. Islam, D\. Pandit, and S\. Das\.A survey on Kolmogorov–Arnold network\.*ACM Computing Surveys*, 58\(2\):Article 55, 2025\.arXiv preprint arXiv:2411\.06078\.
- \[18\]F\. Girosi and T\. Poggio\.Representation properties of networks: Kolmogorov’s theorem is irrelevant\.*Neural Computation*, 1\(4\):465–469, 1989\.doi:[https://doi\.org/10\.1162/neco\.1989\.1\.4\.465](https://doi.org/10.1162/neco.1989.1.4.465)\.
- \[19\]Y\. Wang, J\. W\. Siegel, Z\. Liu, and T\. Y\. Hou\.On the expressiveness and spectral bias of KANs\.arXiv preprint arXiv:2410\.01803, 2024\.
- \[20\]T\. Ji, Y\. Hou, and D\. Zhang\.A comprehensive survey on Kolmogorov–Arnold networks \(KAN\)\.arXiv preprint arXiv:2407\.11075, 2024\.
- \[21\]J\. Xu, Z\. Chen, J\. Li, S\. Yang, W\. Wang, X\. Hu, and E\. C\.\-H\. Ngai\.FourierKAN\-GCF: Fourier Kolmogorov–Arnold network for graph collaborative filtering\.arXiv preprint arXiv:2406\.01034, 2024\.
- \[22\]A\. D\. Bodner, A\. S\. Tepsich, J\. N\. Spolski, and S\. Pourteau\.Convolutional Kolmogorov–Arnold networks\.arXiv preprint arXiv:2406\.13155, 2024\.
- \[23\]B\. C\. Koenig, S\. Kim, and S\. Deng\.LeanKAN: A parameter\-lean Kolmogorov–Arnold network layer with improved memory efficiency and convergence behavior\.arXiv preprint arXiv:2502\.17844, 2025\.
- \[24\]H\.\-T\. Ta, D\.\-Q\. Thai, A\. Tran, G\. Sidorov, and A\. Gelbukh\.PRKAN: Parameter\-reduced Kolmogorov–Arnold networks\.arXiv preprint arXiv:2501\.07032, 2025\.
- \[25\]C\. Li, X\. Liu, W\. Li, C\. Wang, H\. Liu, and Y\. Yuan\.U\-KAN makes strong backbone for medical image segmentation and generation\.arXiv preprint arXiv:2406\.02918, 2024\.
- \[26\]R\. Genet and H\. Inzirillo\.TKAN: Temporal Kolmogorov–Arnold networks\.arXiv preprint arXiv:2405\.07344, 2024\.
- \[27\]M\. Kiamari, M\. Kiamari, and B\. Krishnamachari\.GKAN: Graph Kolmogorov–Arnold networks\.arXiv preprint arXiv:2406\.06470, 2024\.
- \[28\]E\. Poeta, F\. Giobergia, E\. Pastor, T\. Cerquitelli, and E\. Baralis\.A benchmarking study of Kolmogorov–Arnold networks on tabular data\.arXiv preprint arXiv:2406\.14529, 2024\.
- \[29\]S\. T\. Seydi\.Exploring the potential of polynomial basis functions in Kolmogorov–Arnold networks: A comparative study of different groups of polynomials\.arXiv preprint arXiv:2406\.02583, 2024\.
- \[30\]Y\. Wang, J\. Sun, J\. Bai, C\. Anitescu, M\. S\. Eshaghi, X\. Zhuang, T\. Rabczuk, and Y\. Liu\.Kolmogorov–Arnold\-informed neural network: A physics\-informed deep learning framework for solving PDEs based on Kolmogorov–Arnold networks\.*Computer Methods in Applied Mechanics and Engineering*, 433:117518, 2025\.
- \[31\]K\. Shukla, J\. D\. Toscano, Z\. Wang, Z\. Zou, and G\. E\. Karniadakis\.Physics\-informed Kolmogorov–Arnold networks: Investigating architectures and hyperparameter impacts for solving Navier–Stokes equations\.*Physics of Fluids*, 37\(3\):037159, 2025\.
- \[32\]Z\. Zhang, Q\. Wang, Y\. Zhang, and others\.Physics\-informed neural networks with hybrid Kolmogorov–Arnold network and augmented Lagrangian function for solving partial differential equations\.*Scientific Reports*, 15:10523, 2025\.
- \[33\]H\. Shuai and others\.Physics\-informed Kolmogorov–Arnold networks for power system dynamics\.2025\.
- \[34\]H\. Shuai and others\.Permutation invariance and equivariance in Kolmogorov–Arnold network function representation\.arXiv preprint arXiv:2502\.16664, 2025\.
- \[35\]H\.\-T\. Ta and others\.AF\-KAN: Activation function\-based Kolmogorov–Arnold networks\.arXiv preprint arXiv:2503\.06112, 2025\.
- \[36\]A\. Dutta and others\.Kolmogorov–Arnold networks with adaptive radial basis functions\.arXiv preprint arXiv:2601\.07760, 2026\.
- \[37\]A\. M\. Martínez\-Heredia and others\.Kolmogorov–Arnold networks for sensor data processing\.*Sensors*, 26\(8\):2515, 2026\.

## Appendix AAdditional Visualizations of the Recursive Basis

In this appendix, we provide comprehensive visualizations of the recursive polynomial basis functionsRn​\(x\)R\_\{n\}\(x\)and the learned recurrence coefficients across all experiments discussed in Section[4\.3](https://arxiv.org/html/2609.01729#S4.SS3)\. These plots supplement the main text by offering a detailed view of how the model adapts its basis to different tasks\.

![Refer to caption](https://arxiv.org/html/2609.01729v1/coefficient_comparison.png)Figure 5:Comparison of learned recurrence coefficients across different experiments\. The bar chart illustrates the distinct values ofa,b,c,d,ea,b,c,d,elearned by the model, showing significant deviation from the initial Chebyshev like configuration, particularly in image\-based benchmarks\.![Refer to caption](https://arxiv.org/html/2609.01729v1/degree_analysis.png)Figure 6:Maximum polynomial degree of recursive basis functions as a function of the basis indexnn\. The plot confirms that the learned coefficients \(e\.g\., in the SVHN experiment\) allow the basis to reach higher polynomial degrees \(up to 14\) compared to the initialization \(up to 7\)\.![Refer to caption](https://arxiv.org/html/2609.01729v1/classical_vs_learned.png)Figure 7:Comparison of the recursive polynomial basis functions, including classical polynomial families \(Chebyshev 1st/2nd kind, Fibonacci, Pell, Jacobsthal\) and the learned bases for different experiments\. This plot highlights that RecKAN can recover known families or move beyond them depending on the learned coefficients\.
## Appendix BComplete Proof of Proposition 1 \(Degree Growth\)

Degree growth: Fora≠0a\\neq 0,deg⁡Rn≤2​\(n−1\)\\deg R\_\{n\}\\leq 2\(n\-1\)forn≥1n\\geq 1\. Fora=0a=0,deg⁡Rn≤n−1\\deg R\_\{n\}\\leq n\-1\. Equality holds in either case whenever no leading term cancellation occurs\.

###### Proof\.

Letδn=deg⁡Rn\\delta\_\{n\}=\\deg R\_\{n\}forn≥0n\\geq 0, with the convention thatdeg⁡0=−∞\\deg 0=\-\\infty\. From Definition[1](https://arxiv.org/html/2609.01729#Thmdefinition1), we have:

R0​\(x\)\\displaystyle R\_\{0\}\(x\)=0⇒δ0=−∞,\\displaystyle=0\\quad\\Rightarrow\\quad\\delta\_\{0\}=\-\\infty,\(15\)R1​\(x\)\\displaystyle R\_\{1\}\(x\)=1⇒δ1=0,\\displaystyle=1\\quad\\Rightarrow\\quad\\delta\_\{1\}=0,\(16\)Rn\+1​\(x\)\\displaystyle R\_\{n\+1\}\(x\)=α⁡\(x\)​Rn​\(x\)\+β⁡\(x\)​Rn−1​\(x\),n≥1,\\displaystyle=\\alpha\(x\)R\_\{n\}\(x\)\+\\beta\(x\)R\_\{n\-1\}\(x\),\\quad n\\geq 1,\(17\)whereα⁡\(x\)=a​x2\+b​x\+c\\alpha\(x\)=ax^\{2\}\+bx\+candβ⁡\(x\)=d​x\+e\\beta\(x\)=dx\+e\.

#### Case 1:a≠0a\\neq 0\.

In this case,deg⁡α=2\\deg\\alpha=2anddeg⁡β≤1\\deg\\beta\\leq 1\. Forn≥1n\\geq 1, we have:

deg⁡\(α​Rn\)=2\+δn,deg⁡\(β​Rn−1\)≤1\+δn−1\.\\deg\(\\alpha R\_\{n\}\)=2\+\\delta\_\{n\},\\quad\\deg\(\\beta R\_\{n\-1\}\)\\leq 1\+\\delta\_\{n\-1\}\.\(18\)Therefore, by the triangle inequality for polynomial degrees:

δn\+1≤max⁡\{2\+δn,1\+δn−1\}\.\\delta\_\{n\+1\}\\leq\\max\\\{2\+\\delta\_\{n\},\\;1\+\\delta\_\{n\-1\}\\\}\.\(19\)
We prove by induction thatδn≤2​\(n−1\)\\delta\_\{n\}\\leq 2\(n\-1\)for alln≥1n\\geq 1\.

*Base case:*Forn=1n=1,δ1=0≤2​\(1−1\)=0\\delta\_\{1\}=0\\leq 2\(1\-1\)=0, so the bound holds\.

*Inductive step:*Assumeδk≤2​\(k−1\)\\delta\_\{k\}\\leq 2\(k\-1\)for allk≤nk\\leq n\. Then:

δn\+1\\displaystyle\\delta\_\{n\+1\}≤max⁡\{2\+δn,1\+δn−1\}\\displaystyle\\leq\\max\\\{2\+\\delta\_\{n\},\\;1\+\\delta\_\{n\-1\}\\\}\(20\)≤max⁡\{2\+2​\(n−1\),1\+2​\(n−2\)\}\\displaystyle\\leq\\max\\\{2\+2\(n\-1\),\\;1\+2\(n\-2\)\\\}\(21\)=max⁡\{2​n,2​n−3\}\\displaystyle=\\max\\\{2n,\\;2n\-3\\\}\(22\)=2​n=2​\(\(n\+1\)−1\)\.\\displaystyle=2n=2\(\(n\+1\)\-1\)\.\(23\)Thus, by induction,δn≤2​\(n−1\)\\delta\_\{n\}\\leq 2\(n\-1\)for alln≥1n\\geq 1\.

#### Case 2:a=0a=0\.

In this case,deg⁡α≤1\\deg\\alpha\\leq 1\(sinceα⁡\(x\)=b​x\+c\\alpha\(x\)=bx\+c\) anddeg⁡β≤1\\deg\\beta\\leq 1\. Forn≥1n\\geq 1, we have:

deg⁡\(α​Rn\)≤1\+δn,deg⁡\(β​Rn−1\)≤1\+δn−1\.\\deg\(\\alpha R\_\{n\}\)\\leq 1\+\\delta\_\{n\},\\quad\\deg\(\\beta R\_\{n\-1\}\)\\leq 1\+\\delta\_\{n\-1\}\.\(24\)Therefore:

δn\+1≤1\+max⁡\{δn,δn−1\}\.\\delta\_\{n\+1\}\\leq 1\+\\max\\\{\\delta\_\{n\},\\;\\delta\_\{n\-1\}\\\}\.\(25\)
We prove by induction thatδn≤n−1\\delta\_\{n\}\\leq n\-1for alln≥1n\\geq 1\.

*Base case:*Forn=1n=1,δ1=0≤1−1=0\\delta\_\{1\}=0\\leq 1\-1=0, so the bound holds\.

*Inductive step:*Assumeδk≤k−1\\delta\_\{k\}\\leq k\-1for allk≤nk\\leq n\. Then:

δn\+1\\displaystyle\\delta\_\{n\+1\}≤1\+max⁡\{δn,δn−1\}\\displaystyle\\leq 1\+\\max\\\{\\delta\_\{n\},\\;\\delta\_\{n\-1\}\\\}\(26\)≤1\+max⁡\{n−1,n−2\}\\displaystyle\\leq 1\+\\max\\\{n\-1,\\;n\-2\\\}\(27\)=1\+\(n−1\)=n=\(n\+1\)−1\.\\displaystyle=1\+\(n\-1\)=n=\(n\+1\)\-1\.\(28\)Thus, by induction,δn≤n−1\\delta\_\{n\}\\leq n\-1for alln≥1n\\geq 1\.

#### Equality conditions\.

Equality holds when no cancellation occurs between the dominant terms ofα⁡\(x\)​Rn​\(x\)\\alpha\(x\)R\_\{n\}\(x\)andβ⁡\(x\)​Rn−1​\(x\)\\beta\(x\)R\_\{n\-1\}\(x\)\. In thea≠0a\\neq 0case, this requires:

a⋅lc​\(Rn\)\+d⋅lc​\(Rn−1\)≠0a\\cdot\\text\{lc\}\(R\_\{n\}\)\+d\\cdot\\text\{lc\}\(R\_\{n\-1\}\)\\neq 0\(29\)for eachnn, wherelc​\(Rn\)\\text\{lc\}\(R\_\{n\}\)denotes the leading coefficient ofRnR\_\{n\}\. In thea=0a=0case, equality requires:

b⋅lc​\(Rn\)\+d⋅lc​\(Rn−1\)≠0\.b\\cdot\\text\{lc\}\(R\_\{n\}\)\+d\\cdot\\text\{lc\}\(R\_\{n\-1\}\)\\neq 0\.\(30\)

#### Interpretation\.

The boundδn≤2​\(n−1\)\\delta\_\{n\}\\leq 2\(n\-1\)fora≠0a\\neq 0means that the basis functions can grow by up to two degrees per recursion step, while fora=0a=0, they grow by at most one degree per step\. This is because the quadratic terma​x2ax^\{2\}inα⁡\(x\)\\alpha\(x\)doubles the degree contribution at each step\. For example, at basis orderK=8K=8:

- •Ifa=0a=0, the maximum achievable degree is77\.
- •Ifa≠0a\\neq 0, the maximum achievable degree is1414\.

This demonstrates that by activating the quadratic term, we can construct basis functions with higher polynomial degrees without increasing the basis orderKK\. ∎