SparseKAN: Compressing Kolmogorov--Arnold Networks Across Basis Functions, Neurons, and Bits

# SparseKAN: Compressing Kolmogorov–Arnold Networks Across Basis Functions, Neurons, and Bits
Source: [https://arxiv.org/html/2608.00859](https://arxiv.org/html/2608.00859)
###### Abstract

Kolmogorov–Arnold Networks \(KANs\) replace scalar edge weights with learnable univariate functions parameterized by multiple basis coefficients\. This introduces a source of redundancy that conventional neural\-network compression does not directly expose\. We presentSparseKAN, a unified approach that compresses KANs along three complementary axes:*basis functions*,*neurons/channels*, and*numerical precision*\. SparseKAN equips the base branch, nonlinear basis branch, and individual basis terms with hierarchical learnable gates trained under a differentiable active\-cost objective\. The learned importance structure is subsequently hardened under explicit basis and width budgets, recovered in full or low precision, and physically compacted into smaller dense tensors rather than retained as sparse masks\. Experiments on MNIST, CIFAR\-10, and CIFAR\-100 across spline, polynomial, RBF, wavelet, and convolutional KAN variants show that the structural axes compose predictably in cost\. We also find strong basis\-dependent differences in term importance: coefficient\-based selection outperforms matched low\-order truncation by up to 15\.25 accuracy points in the evaluated Gram\-polynomial settings\. Eight\-bit quantization is broadly robust, whereas 4\-bit convolutional KANs require quantization\-aware adaptation\. Physical compaction removes up to 73\.0% of parameters without accuracy loss on MNIST and reduces large\-batch CUDA latency to as little as0\.51×0\.51\\timesdense execution\. On a ZCU104 FPGA, the resulting sparse low\-bit models achieve up to23\.63×23\.63\\timeslower inference latency, demonstrating that SparseKAN converts functional redundancy into measurable software and hardware efficiency\. The SparseKAN implementation is available athttps://github\.com/OSU\-STARLAB/SparseKAN\.

## 1Introduction

Kolmogorov–Arnold Networks \(KANs\) replace the scalar weight on an MLP connection with a learnable univariate function\(Liuet al\.[2025](https://arxiv.org/html/2608.00859#bib.bib1)\)\. In spline\-based KANs, each edge combines a residual base activation with a basis expansion, so one connection carries multiple coefficients rather than a scalar\. This added functional capacity is also a source of cost\. For input widthdind\_\{\\mathrm\{in\}\}, output widthdoutd\_\{\\mathrm\{out\}\}, andKKbasis functions per edge, the dominant coefficient tensor scales asdin​dout​Kd\_\{\\mathrm\{in\}\}d\_\{\\mathrm\{out\}\}K\. KAN redundancy can therefore occur not only*across*edges and neurons, but also*inside*each learned function\.

Standard pruning granularities miss this\. A retained edge may need only a subset of its basis functions, while deleting the whole edge can remove useful computation together with redundant terms\. The surviving structure may also tolerate lower precision\. KAN compression thus exposes three complementary axes:*basis functions*,*neurons/channels*, and*precision/bits*\. Existing work addresses important parts of this space through efficient basis parameterizations\(Li[2024](https://arxiv.org/html/2608.00859#bib.bib3); SSet al\.[2024](https://arxiv.org/html/2608.00859#bib.bib26); Bozorgasl and Chen[2024](https://arxiv.org/html/2608.00859#bib.bib24); Merin\-Martinezet al\.[2026](https://arxiv.org/html/2608.00859#bib.bib10); Pooleet al\.[2025](https://arxiv.org/html/2608.00859#bib.bib5)\), coefficient sharing or vector quantization\(Raffelet al\.[2026](https://arxiv.org/html/2608.00859#bib.bib29)\), low\-precision inference\(Fuad and Chen[2026](https://arxiv.org/html/2608.00859#bib.bib11); Errabiiet al\.[2026b](https://arxiv.org/html/2608.00859#bib.bib17)\), learned architectural sparsity\(Bagrow and Bongard[2025](https://arxiv.org/html/2608.00859#bib.bib8)\), and specialized hardware\(Hoanget al\.[2026](https://arxiv.org/html/2608.00859#bib.bib16); Errabiiet al\.[2026a](https://arxiv.org/html/2608.00859#bib.bib12); Ouet al\.[2025](https://arxiv.org/html/2608.00859#bib.bib13),[2026](https://arxiv.org/html/2608.00859#bib.bib14); Sudarshanet al\.[2026](https://arxiv.org/html/2608.00859#bib.bib32)\)\. However, what remains less explored is a general mechanism that exposes the internal basis dimension, coordinates it with width and precision, and converts the resulting structure into physically smaller executable tensors\.

We introduceSparseKAN, a basis\-aware compression framework for this purpose\. SparseKAN attaches hierarchical gates to the base branch, nonlinear basis branch, and individual basis terms of each KAN edge, and optimizes them with a normalized active\-cost objective\. The soft\-gating stage is used mainly for*structure discovery and adaptation*; explicit compression is imposed during hardening through a basis and a neuron/channel budget\. The surviving model is then recovered in full precision or with low\-bit quantization\-aware training \(QAT\)\. When supports are structurally compatible, SparseKAN gathers retained basis terms and slices dead hidden dimensions, turning learned sparsity into smaller dense tensors rather than zeros in the original model\.

The experiments show why these axes should remain distinct\. Basis and width reductions compose predictably: on MNIST EfficientKAN, factors0\.5560\.556and0\.46510\.4651predict active cost0\.25850\.2585, versus0\.25840\.2584measured\. Basis identity is also family\-dependent\. In the evaluated Gram\-polynomial settings, coefficient\-based selection exceeds matched low\-order truncation by up to15\.2515\.25accuracy points, whereas the tested B\-spline models are nearly insensitive to support identity\. Precision introduces a different boundary: 8\-bit QAT is broadly benign, while naive 4\-bit PTQ falls to50\.19±11\.78%50\.19\\pm 11\.78\\%on CIFAR\-10 and9\.04±2\.43%9\.04\\pm 2\.43\\%on CIFAR\-100; QAT restores useful low\-bit operating points\. Our five\-seed audit further shows joint gate/QAT training is a robust single\-pipeline option, not a universal accuracy improvement over sufficiently recovered staged compression\.

Crucially, SparseKAN evaluates whether sparsity survives contact with the system\. Across 54 compacted checkpoints, top\-1 agreement with the corresponding masked model is at least0\.99970\.9997\. Physical compaction removes up to73\.0%73\.0\\%of parameters without degradation on MNIST, and compact convolutional models reduce large\-batch CUDA latency to as little as0\.51×0\.51\\timesdense, whereas masked models remain near dense latency\. On a ZCU104 HLS implementation, sparse 4\-bit EfficientKAN and Gram\-polynomial MLPs reach23\.63×23\.63\\timesand18\.21×18\.21\\timeslower inference latency, respectively; sparse 4\-bit KAGN\-conv reduces DSP occupancy from96\.2%96\.2\\%to29\.1%29\.1\\%for0\.090\.09accuracy points\. Our contributions are as follows\.

- •We formulate KAN compression along three native axes: basis functions, neurons/channels, and precision\. We provide a common cost\-gated interface for exposing their importance across multiple KAN parameterizations\.
- •We introduce per\-edge and shared basis hardening and combine it with width reduction so compatible supports can be physically compacted fromdℓ−1​dℓ​Kℓd\_\{\\ell\-1\}d\_\{\\ell\}K\_\{\\ell\}todℓ−1′​dℓ′​Kℓ′d^\{\\prime\}\_\{\\ell\-1\}d^\{\\prime\}\_\{\\ell\}K^\{\\prime\}\_\{\\ell\}, rather than merely masked\.
- •We validate the resulting operating points across MNIST, CIFAR\-10, and CIFAR\-100 using matched pruning baselines, five\-seed comparative audits, physical checkpoint size, CUDA latency, and an 18\-design FPGA/HLS study\. The result is a basis\-aware path from learned redundancy to compact, low\-bit, executable KANs\.

## 2Related Work

#### KAN architectures and efficient parameterizations\.

KANs originally parameterize edge functions with B\-splines\(Liuet al\.[2025](https://arxiv.org/html/2608.00859#bib.bib1)\)\. Later variants replace or restructure this basis: FastKAN uses Gaussian radial basis functions\(Li[2024](https://arxiv.org/html/2608.00859#bib.bib3)\), ChebyKAN uses Chebyshev polynomials\(SSet al\.[2024](https://arxiv.org/html/2608.00859#bib.bib26)\), Wav\-KAN uses wavelets\(Bozorgasl and Chen[2024](https://arxiv.org/html/2608.00859#bib.bib24)\), and KAN convolutions extend such representations to vision models\(Drokin[2024](https://arxiv.org/html/2608.00859#bib.bib27)\)\. LSS\-SKAN, LTBs\-KAN, and P\-KAN instead reduce cost through simpler or lower\-dimensional functional parameterizations\(Chenet al\.[2026](https://arxiv.org/html/2608.00859#bib.bib28); Merin\-Martinezet al\.[2026](https://arxiv.org/html/2608.00859#bib.bib10); Pooleet al\.[2025](https://arxiv.org/html/2608.00859#bib.bib5)\), while recent differentiable architecture optimization learns compact KAN structures directly\(Bagrow and Bongard[2025](https://arxiv.org/html/2608.00859#bib.bib8)\)\. SparseKAN is complementary: given a KAN family, it asks how much of its available basis, width, and precision is actually required\.

#### KAN compression\.

The original KAN already uses regularization and pruning for model simplification\(Liuet al\.[2025](https://arxiv.org/html/2608.00859#bib.bib1)\)\. MetaCluster clusters coefficient vectors\(Raffelet al\.[2026](https://arxiv.org/html/2608.00859#bib.bib29)\), SHARe\-KAN applies post\-training vector quantization\(Smith[2026](https://arxiv.org/html/2608.00859#bib.bib30)\), and QuantKAN and KANtize study low\-precision KAN inference\(Fuad and Chen[2026](https://arxiv.org/html/2608.00859#bib.bib11); Errabiiet al\.[2026b](https://arxiv.org/html/2608.00859#bib.bib17)\)\. SparseKAN differs in structural granularity: it can remove individual basis terms, regularize them into shared supports, and combine that basis dimension with neuron/channel pruning and QAT\. Our matched\-cost edge\-L0L\_\{0\}results are therefore intentionally a scope condition: basis sparsity is not universally more accurate than edge sparsity; its value is finer functional control and compactable structure\.

#### Joint neural compression\.

Pruning and quantization are well established for conventional networks\. Deep Compression combines pruning and quantization\(Hanet al\.[2016](https://arxiv.org/html/2608.00859#bib.bib35)\); DJPQ jointly optimizes structured pruning and mixed precision\(Wanget al\.[2021](https://arxiv.org/html/2608.00859#bib.bib21)\); and Bayesian Bits unifies pruning with mixed\-precision selection\(van Baalenet al\.[2020](https://arxiv.org/html/2608.00859#bib.bib22)\)\. SparseKAN also draws on differentiableL0L\_\{0\}gating and structured channel pruning\(Louizoset al\.[2018](https://arxiv.org/html/2608.00859#bib.bib23); Liuet al\.[2017](https://arxiv.org/html/2608.00859#bib.bib36)\), but applies these ideas to a dimension absent from an ordinary MLP: the basis terms inside a functional edge\.

#### KAN hardware and systems\.

KAN accelerators include FPGA LUT mapping with pruning and quantization\(Hoanget al\.[2026](https://arxiv.org/html/2608.00859#bib.bib16)\), systolic arrays\(Errabiiet al\.[2026a](https://arxiv.org/html/2608.00859#bib.bib12)\), reconfigurable sparse pipelines\(Ouet al\.[2025](https://arxiv.org/html/2608.00859#bib.bib13),[2026](https://arxiv.org/html/2608.00859#bib.bib14)\), and compute\-in\-memory designs\(Sudarshanet al\.[2026](https://arxiv.org/html/2608.00859#bib.bib32)\)\. FlashKAT further shows that memory traffic and implementation details can dominate nominal KAN arithmetic cost\(Raffel and Chen[2026](https://arxiv.org/html/2608.00859#bib.bib34)\), and surveys of sparse transformer accelerators reach a similar conclusion: unstructured zeros rarely translate into hardware gains without explicit architectural support\(Fuad and Chen[2023](https://arxiv.org/html/2608.00859#bib.bib31)\)\. These results motivate our separation of active cost, physical model size, and measured execution cost: SparseKAN compacts compatible supports so hardware receives a smaller dense problem rather than a dense problem containing zeros\.

## 3SparseKAN

![Refer to caption](https://arxiv.org/html/2608.00859v1/fig_method.png)Figure 1:SparseKAN compresses KANs along three complementary axes\.\(a\) Each edge contains a base branch and a basis branch, controlled by base, branch, and term\-level gates\. \(b\) Cost\-gated training first exposes redundancy; hardening then reduces basis functions and hidden units/channels\. The surviving model is adapted in full precision or with low\-bit QAT, and structured supports are compacted into smaller tensors\. The result uses fewer*basis functions*, fewer*neurons*, and fewer*bits*\.### 3\.1From Functional Redundancy to Structured Compression

Unlike an MLP connection, which carries a scalar weight, a KAN edge represents a learnable univariate function using multiple basis coefficients\. This expressiveness creates redundancy within basis expansions, across hidden units/channels, and in the precision of the surviving parameters\. SparseKAN targets these three sources jointly, as illustrated in Figure[1](https://arxiv.org/html/2608.00859#S3.F1)\.

The central idea is to separate*structure discovery*from*structure realization*\. We first train differentiable gates that expose the relative importance of the computation inside each KAN edge\. The learned soft structure is then converted into explicit compression along two structural dimensions \(basis functions and neurons/channels\), and the remaining parameters are quantized\. Structured masks can finally be physically compacted so that the selected sparsity changes the model dimensions rather than merely introducing zeros\.

### 3\.2Cost\-Gated KANs

Consider edge\(i→o\)\(i\\\!\\rightarrow\\\!o\)of a KAN layer withKKbasis functions\. We write its gated form as:

ϕ~o,i​\(xi\)=go,ib​wo,ib​a​\(xi\)\+go,ir​∑k=1Kgo,i,kt​co,i,k​ψk​\(xi\),\\widetilde\{\\phi\}\_\{o,i\}\(x\_\{i\}\)=g^\{\\mathrm\{b\}\}\_\{o,i\}w^\{\\mathrm\{b\}\}\_\{o,i\}a\(x\_\{i\}\)\+g^\{\\mathrm\{r\}\}\_\{o,i\}\\sum\_\{k=1\}^\{K\}g^\{\\mathrm\{t\}\}\_\{o,i,k\}c\_\{o,i,k\}\\psi\_\{k\}\(x\_\{i\}\),\(1\)wherea​\(⋅\)a\(\\cdot\)is the base activation,ψk​\(⋅\)\\psi\_\{k\}\(\\cdot\)is thekk\-th basis function, andco,i,kc\_\{o,i,k\}is its coefficient\. The gatesgbg^\{\\mathrm\{b\}\},grg^\{\\mathrm\{r\}\}, andgtg^\{\\mathrm\{t\}\}control the base branch, nonlinear basis branch, and individual basis terms, respectively\. This formulation is basis\-agnostic and applies to the spline, polynomial, RBF, wavelet, and convolutional KAN families considered in our experiments\. We use sigmoid\-relaxed learnable gates by default; alternative relaxations are described in Appendix[A\.1](https://arxiv.org/html/2608.00859#A1.SS1)\.

To bias the model toward cheaper structure, we optimize a normalized active\-cost penalty,

ℛcost=λb​Cb​∑gb\+λr​Cr​∑gr\+λt​Ct​∑gr​gtCb​Nb\+Cr​Nr\+Ct​Nt,\\mathcal\{R\}\_\{\\mathrm\{cost\}\}=\\frac\{\\lambda\_\{\\mathrm\{b\}\}C\_\{\\mathrm\{b\}\}\\sum g^\{\\mathrm\{b\}\}\+\\lambda\_\{\\mathrm\{r\}\}C\_\{\\mathrm\{r\}\}\\sum g^\{\\mathrm\{r\}\}\+\\lambda\_\{\\mathrm\{t\}\}C\_\{\\mathrm\{t\}\}\\sum g^\{\\mathrm\{r\}\}g^\{\\mathrm\{t\}\}\}\{C\_\{\\mathrm\{b\}\}N\_\{\\mathrm\{b\}\}\+C\_\{\\mathrm\{r\}\}N\_\{\\mathrm\{r\}\}\+C\_\{\\mathrm\{t\}\}N\_\{\\mathrm\{t\}\}\},\(2\)whereC∙C\_\{\\bullet\}andN∙N\_\{\\bullet\}denote the relative cost and dense count of each component type\. The productgr​gtg^\{\\mathrm\{r\}\}g^\{\\mathrm\{t\}\}reflects the hierarchy of Eq\. \([1](https://arxiv.org/html/2608.00859#S3.E1)\): a term contributes only when its parent basis branch is active\. Training minimizes

ℒ=ℒtask\+ℛcost\+λH​ℛH,\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{task\}\}\+\\mathcal\{R\}\_\{\\mathrm\{cost\}\}\+\\lambda\_\{\\mathrm\{H\}\}\\mathcal\{R\}\_\{\\mathrm\{H\}\},\(3\)with an optional entropy penaltyℛH\\mathcal\{R\}\_\{\\mathrm\{H\}\}to encourage decisive gates\. Sparsity pressure is introduced after a warm\-up and increased gradually\. Under our default setting this stage primarily learns and adapts the model to a useful importance structure; the desired compression level is imposed explicitly during hardening\. The complete schedule and a direct cost\-budget formulation are given in Appendix[A\.2](https://arxiv.org/html/2608.00859#A1.SS2)\.

### 3\.3Two Structural Axes: Basis Functions and Neurons

#### Basis\-function sparsification\.

SparseKAN first reduces redundancy*within*functional edges\. After cost\-gated training, our default term score is

so,i,k=\|co,i,k\|​go,i,kt\.s\_\{o,i,k\}=\|c\_\{o,i,k\}\|\\,g^\{\\mathrm\{t\}\}\_\{o,i,k\}\.\(4\)A flexible per\-edge top\-kkrule retains thekkhighest\-scoring terms of each active edge\. Different edges may therefore use different basis subsets\. Because this creates a ragged support, we additionally introduce*shared\-kk*structure: all edges originating from inputiishare the samekkterms, selected by aggregatingso,i,ks\_\{o,i,k\}over output units\. A layer\-shared variant uses one support for the entire layer\. Per\-edge top\-kktherefore prioritizes functional flexibility, whereas shared\-kkexposes the regular term dimension required for compact execution\. Alternative scoring rules are evaluated in Appendix[A\.3](https://arxiv.org/html/2608.00859#A1.SS3)\.

#### Neuron and channel sparsification\.

Basis pruning reduces the complexity of individual edges but leaves network width unchanged\. SparseKAN therefore removes complete hidden units, or channels in convolutional layers\. Removing outputooof layerℓ\\ellalso removes the corresponding input dimension of layerℓ\+1\\ell\+1, so dead features do not persist as unused columns in subsequent layers\. The output layer is kept intact\. Details of unit scoring and the additional block\-structured variant appear in Appendix[A\.4](https://arxiv.org/html/2608.00859#A1.SS4)\.

These two structural axes act on different dimensions of the dominant coefficient tensor\. After compression, its first\-order cost changes from

dℓ−1​dℓ​Kℓ⟶dℓ−1′​dℓ′​Kℓ′,d\_\{\\ell\-1\}d\_\{\\ell\}K\_\{\\ell\}\\quad\\longrightarrow\\quad d^\{\\prime\}\_\{\\ell\-1\}d^\{\\prime\}\_\{\\ell\}K^\{\\prime\}\_\{\\ell\},\(5\)so basis and width reductions compound rather than compete\.

### 3\.4Physical Compaction and Low\-Bit Representation

A mask alone does not make a model physically smaller\. For structurally compatible supports, SparseKAN therefore slices dead input/output dimensions and gathers the retained shared terms into a coefficient tensor of shapedℓ′×dℓ−1′×Kℓ′d^\{\\prime\}\_\{\\ell\}\\times d^\{\\prime\}\_\{\\ell\-1\}\\times K^\{\\prime\}\_\{\\ell\}\. A smallkept\_term\_idxtable records which original basis functions correspond to the compact term dimension\. Thus the resulting model contains smaller tensors rather than full\-size tensors populated with zeros\. Detailed indexing and the compact forward expression are given in Appendix[A\.5](https://arxiv.org/html/2608.00859#A1.SS5)\.

The third compression axis reduces the precision of the surviving parameters\. SparseKAN applies weight\-only, symmetric per\-output\-channel QAT to both base weights and basis coefficients, using88\- and44\-bit operating points\. The structural and precision reductions are summarized by the normalized joint bit\-cost

ρbit=ρstruct​b32,\\rho\_\{\\mathrm\{bit\}\}=\\rho\_\{\\mathrm\{struct\}\}\\frac\{b\}\{32\},\(6\)whereρstruct\\rho\_\{\\mathrm\{struct\}\}is the final active structural cost andbbis the weight precision\. Thus basis and neuron sparsification determine*how much*computation survives, while quantization determines how many bits represent each survivor\.

### 3\.5Training and Compression Pipeline

SparseKAN proceeds in four conceptual steps\.\(1\) Cost\-gated training:model and gate parameters are optimized jointly using Eq\. \([3](https://arxiv.org/html/2608.00859#S3.E3)\)\.\(2\) Structural hardening:continuous gates are binarized and the requested per\-edge/shared\-kkand neuron/channel constraints are imposed\.\(3\) Recovery:the discrete support is fixed and the surviving parameters are fine\-tuned; low\-bit models enable QAT during this stage\.\(4\) Physical realization:compatible structured supports are compacted by slicing dead dimensions and gathering retained terms\. The resulting operating point is controlled independently by the basis budgetkk, neuron keep ratiornr\_\{\\mathrm\{n\}\}, and precisionbb\. Implementation variants and the full algorithm are provided in Appendix[A](https://arxiv.org/html/2608.00859#A1)\.

## 4Experiments and Analysis

We evaluate SparseKAN around four questions:\(i\)where does compressible redundancy occur inside a KAN,\(ii\)do basis and neuron compression provide complementary operating points,\(iii\)how robust is the surviving model to low\-bit representation, and\(iv\)do these reductions translate into physically smaller and faster implementations? Throughout, we distinguish*active cost*,*physical model size*, and*measured execution cost*\. A sparse mask may reduce the first without changing the latter two\.

### 4\.1Setup and Statistical Protocol

Our primary benchmarks are MNIST, CIFAR\-10, and CIFAR\-100\. The main MLP families are EfficientKAN with a B\-spline basis and KAGN with a Gram\-polynomial basis; convolutional experiments use spatial\-kernel KAGN convolutions\. Additional KAN families and tabular benchmarks are reported in Appendix[B\.4](https://arxiv.org/html/2608.00859#A2.SS4)\.

Two complementary campaigns support the results\. A three\-seed frontier study contains 380 completed runs over seeds\{42,43,44\}\\\{42,43,44\\\}and maps the structural and precision operating space\. A separate five\-seed paired ablation campaign over seeds4242–4646tests comparative claims involving selection, sparsity granularity, gates, and low\-bit training\. For these comparisons we emphasize paired effect sizes, seed consistency, and confidence intervals\. With only five non\-zero pairs, the minimum attainable two\-sided exact Wilcoxon signed\-rankpp\-value is0\.06250\.0625; we therefore do not interpret a small nominal difference as significant when its confidence interval contains zero\.

Efficiency is reported using normalized active costρstruct\\rho\_\{\\mathrm\{struct\}\}, where11denotes dense computation, physical parameter counts measured directly from compact checkpoints, and joint bit\-costρbit=ρstruct​\(b/32\)\\rho\_\{\\mathrm\{bit\}\}=\\rho\_\{\\mathrm\{struct\}\}\(b/32\)\. Hardware experiments target a ZCU104 using Vitis HLS 2025\.2\.1 and a 200 MHz clock target\. Hardware methodology is detailed in Sec\.[4\.8](https://arxiv.org/html/2608.00859#S4.SS8)and Appendix[C](https://arxiv.org/html/2608.00859#A3)\.

### 4\.2Where Does the Redundancy Lie?

The structure\-discovery stage itself incurs essentially no accuracy penalty: MNIST EfficientKAN changes from97\.05±0\.28%97\.05\\pm 0\.28\\%to97\.07±0\.24%97\.07\\pm 0\.24\\%, CIFAR\-10 KAGN\-conv from81\.73±0\.62%81\.73\\pm 0\.62\\%to81\.41±0\.58%81\.41\\pm 0\.58\\%, and wide CIFAR\-100 KAGN\-conv from52\.72±0\.45%52\.72\\pm 0\.45\\%to53\.24±0\.52%53\.24\\pm 0\.52\\%\. The near\-dense cost under the default schedule is deliberate: a direct regularization sweep moves cost from approximately0\.9960\.996to0\.520\.52while MNIST accuracy changes by less than0\.30\.3points\. Stage 1 is therefore used as a common adapted checkpoint, with the desired structure imposed explicitly during hardening\.

Table[1](https://arxiv.org/html/2608.00859#S4.T1)makes the two structural axes explicit\. Basis reduction is often the gentler first step\. MNIST EfficientKAN retains97\.14±0\.11%97\.14\\pm 0\.11\\%with input\-sharedk=4k=4at cost0\.5530\.553, while KAGN retains only two of four Gram terms at cost0\.6000\.600and reaches98\.20±0\.21%98\.20\\pm 0\.21\\%\. The same axis remains useful on harder tasks: shared\-k=3k=3gives80\.67±0\.42%80\.67\\pm 0\.42\\%at cost0\.7990\.799on CIFAR\-10 and53\.39±0\.91%53\.39\\pm 0\.91\\%at cost0\.7960\.796on CIFAR\-100\.

Table 1:Representative structural frontier\(accuracy in %, three seeds\)\. Basis and neuron budgets expose distinct cost–accuracy trade\-offs\. The CIFAR\-10 Gram model shows the term\-axis cliff: shared\-k=2k=2is dominated by neuron keep\-0\.750\.75at similar cost\.ModelOperating pointCostAccuracyMNIST EffKANdense1\.00097\.05±0\.2897\.05\\pm 0\.28shared\-k=4k=40\.55397\.14±0\.1197\.14\\pm 0\.11neuron keep 0\.500\.46595\.69±0\.2095\.69\\pm 0\.20MNIST KAGNdense1\.00098\.07±0\.1198\.07\\pm 0\.11shared\-k=2k=20\.60098\.20±0\.2198\.20\\pm 0\.21neuron keep 0\.500\.46597\.70±0\.1597\.70\\pm 0\.15CIFAR\-10 convdense1\.00081\.73±0\.6281\.73\\pm 0\.62shared\-k=3k=30\.79980\.67±0\.4280\.67\\pm 0\.42shared\-k=2k=20\.60075\.32±4\.4775\.32\\pm 4\.47neuron keep 0\.750\.58378\.65±0\.5978\.65\\pm 0\.59CIFAR\-100 convdense1\.00052\.72±0\.4552\.72\\pm 0\.45shared\-k=3k=30\.79653\.39±0\.9153\.39\\pm 0\.91shared\-k=2k=20\.59752\.04±0\.7552\.04\\pm 0\.75

Width reduction is more architecture\-dependent because deleting a channel removes every function carried by that feature, rather than simplifying one function\. The CIFAR\-10 comparison is especially informative: shared\-k=2k=2falls to75\.32±4\.47%75\.32\\pm 4\.47\\%at cost0\.6000\.600, whereas neuron keep\-0\.750\.75reaches a slightly*lower*cost \(0\.5830\.583\) with78\.65±0\.59%78\.65\\pm 0\.59\\%\. Once the already\-small four\-term Gram basis is compressed too far, another removed term eliminates useful function classes rather than redundancy\. This is why SparseKAN keeps basis and width as separate controls instead of assuming one universal sparsity ratio\.

### 4\.3Which Basis Functions Survive Matters—Sometimes

Reducing the basis raises a second question: is the number of surviving functions enough, or does their identity matter? We compare coefficient\-based selection with random selection and low\-order truncation at identicalkkand recovery budgets\.

Table 2:Basis\-selection controls\.Coefficient\-based selection minus matched low\-order truncation and random selection in accuracy points \(five\-seed campaign; representative rows\)\.For several Gram\-polynomial models, learned selection is critical: low\-order truncation trails by15\.2515\.25points on MNIST KAGN,8\.478\.47on CIFAR\-10 KAGN\-conv, and2\.422\.42on CIFAR\-100 KAGN\-conv\. Against random selection, the corresponding MNIST and CIFAR\-10 gains are0\.590\.59and3\.573\.57points\. The effect is not universal: ChebyKAN is essentially tied and one MNIST convolutional setting reverses\. The supported conclusion is therefore*basis\-dependent selection sensitivity*, not a universal polynomial rule\. In the tested B\-spline models, all selectors stay within0\.330\.33points\.

Raw gate magnitude is also a poor post\-training selector\. At identical cost, MNIST KAGN obtains97\.67%97\.67\\%with coefficient scoring but only82\.75%82\.75\\%with gate scoring; CIFAR\-10 KAGN\-conv gives75\.22%75\.22\\%versus63\.92%63\.92\\%\. Gate saturation therefore makes the learned gate value itself unsuitable as the final importance score\.

### 4\.4Comparison with Conventional Pruning and Matched MLPs

Table[3](https://arxiv.org/html/2608.00859#S4.T3)summarizes the external controls\. At matched cost, basis\-level sparsity reaches essentially the same frontier as edge\-levelL0L\_\{0\}sparsity: the gap changes sign across datasets and remains within seed variation\. We therefore claim*feasibility rather than accuracy dominance*: moving the decision inside the functional edge preserves the conventional frontier while exposing shared basis supports and basis\-specific selection\.

A tuned KANL1L\_\{1\}\+entropy node\-pruning baseline is also competitive\. SparseKAN’s shared\-kkor neuron operating points match or modestly improve representative cost–accuracy points on CIFAR\-10 and CIFAR\-100 while using the same framework that later supports basis compaction and quantization\. The matched\-parameter MLP control provides an important scope condition: KAN wins decisively on CIFAR\-10 at the same parameter count, but the MLP wins on MNIST\. SparseKAN is therefore useful where the KAN representation first earns its additional functional complexity; it is not an argument for universal KAN superiority\.

Table 3:Baseline comparisons\.Edge\-L0L\_\{0\}: five paired seeds at matched achieved cost\. Node pruning: representative frontier points\. Matched MLP: identical dense parameter counts\. Accuracy in %\.
### 4\.5The Structural Axes Compose and Physically Compact

The two structural controls compound predictably\. For MNIST EfficientKAN, neuron pruning contributes a cost factor0\.46510\.4651and top\-kkbasis hardening a factor0\.5560\.556, predicting0\.4651×0\.556=0\.25850\.4651\\times 0\.556=0\.2585; the measured composed cost is0\.25840\.2584\. For CIFAR\-10 KAGN\-conv,0\.2783×0\.602=0\.16750\.2783\\times 0\.602=0\.1675predicts0\.16760\.1676\. The axes therefore modify distinct dimensions of the dominant coefficient tensor, consistent with Eq\. \([5](https://arxiv.org/html/2608.00859#S3.E5)\)\. More importantly, the resulting support can be realized as a genuinely smaller dense model\.

Table 4:Physical compaction\.Operating points are written skkk\+nrr, denoting shared\-kkbasis hardening combined with neuron keep ratiorr\. Reduction is measured from the compact checkpoint rather than inferred from a mask\.Δ\\Deltais relative to the dense anchor of the same architecture\.MNIST EfficientKAN removes73\.0%73\.0\\%of its parameters with no degradation, and KAGN removes70\.3%70\.3\\%for only0\.100\.10points\. Wide CIFAR\-100 removes58\.4%58\.4\\%for a1\.271\.27\-point difference\. CIFAR\-10 is the less redundant case:59\.8%59\.8\\%reduction costs2\.922\.92points, and more aggressive channel removal degrades rapidly\. Extending recovery from 20 to 40 epochs raises one79\.2%79\.2\\%\-reduction point from67\.48%67\.48\\%to71\.13%71\.13\\%, indicating that part of the severe\-cut loss is adaptation difficulty\.

Physical realization also changes the runtime conclusion\. Masked CUDA models remain at0\.950\.95–1\.03×1\.03\\timesdense latency because their tensor shapes are unchanged\. In contrast, physically compacted convolutional models become faster once the batch is large enough to amortize launch and basis\-evaluation overhead \(Table[5](https://arxiv.org/html/2608.00859#S4.T5)\)\.

Table 5:Physical sparsity translates to CUDA speedup\.Compact/dense latency ratio; values below one are faster\. “Kept” is the fraction of dense parameters remaining\. Masked models stay near1×1\\timesand are omitted\.The result also exposes the remaining systems bottleneck\. Parameter count falls faster than latency: the CIFAR\-10 sk3\+n0\.5 model keeps only20\.8%20\.8\\%of parameters yet still requires0\.54×0\.54\\timesdense latency at batch 1024\. Basis construction precedes the compressed contraction and therefore remains a target for fused kept\-basis kernels\. The MLP row shows the complementary limitation: even after73\.0%73\.0\\%parameter removal, small dense linear\-algebra kernels remain comparatively memory/launch bound\.

### 4\.6Low Precision: Eight Bits Are Robust, Four Bits Need Adaptation

Across the QAT grid, 8\-bit weights are effectively free: the largest observed decrease is0\.510\.51accuracy points, and several sparse cells slightly improve\. Four bits expose a harder regime\. MNIST loses only0\.120\.12–0\.430\.43points relative to corresponding FP32 sparse configurations, whereas the evaluated CIFAR\-10 settings lose1\.391\.39–3\.703\.70points and CIFAR\-100 loses approximately2\.42\.4–2\.92\.9points\.

Table 6:The 4\-bit regime\(accuracy in %, five seeds\)\. Dense PTQ uses the unpruned model; two\-stage and joint configurations use aggressive sparse operating points\. Joint\-versus\-two\-stage paired confidence intervals include zero\.Dense 4\-bit PTQ remains usable on MNIST but collapses on convolutional KANs:50\.19±11\.78%50\.19\\pm 11\.78\\%on CIFAR\-10 and9\.04±2\.43%9\.04\\pm 2\.43\\%on CIFAR\-100\. Quantization\-aware adaptation restores useful low\-bit operating points\. Joint gate/QAT training is notably stable on CIFAR\-10 \(79\.72±0\.30%79\.72\\pm 0\.30\\%versus75\.26±4\.19%75\.26\\pm 4\.19\\%for the matched two\-stage schedule\), but its paired confidence interval includes zero and extending the two\-stage recovery budget closes most of the mean gap\. We therefore claim a robust single\-pipeline option, not intrinsic mean\-accuracy superiority\.

### 4\.7Robustness Checks: What Survives Stronger Controls?

The expanded ablation campaign tests the main findings under five\-seed paired controls, separating the effects that survive from those that were artifacts of the smaller exploratory study\. Table[7](https://arxiv.org/html/2608.00859#S4.T7)summarizes the checks most likely to change the interpretation of the main results\.

Table 7:Robustness audit\.Five\-seed paired controls are used for comparative accuracy claims; cost\-model and compaction checks use the completed checkpoint set\.Two points are particularly important\. First, the earlier three\-seed\+5\.69\+5\.69\-point joint\-QAS difference and\+4\.77\+4\.77\-point training\-time\-hardening difference are not retained as accuracy claims after five\-seed validation\. Second, the conclusions are not an artifact of the nominal cost accountant: reweighting 683 completed runs using empirically fitted component costs preserves operating\-point order with Spearman correlation0\.9360\.936–1\.0001\.000across the non\-degenerate families\. Physical compaction is similarly faithful: across 54 compacted checkpoints, top\-1 agreement with the corresponding masked model is at least0\.99970\.9997\(at most three flips per 10,000 examples in the worst case\)\. These checks narrow the paper’s claims, but make the remaining ones substantially stronger\.

### 4\.8FPGA/HLS Realization

We finally instantiate the compact structures in hardware rather than leaving them as software masks\. All designs target a ZCU104 \(xczu7ev\-ffvc1156\-2\-e\) using Vitis HLS 2025\.2\.1 with a 5 ns target\. Within each family, FPGA part, clock target, AXI interface, weight\-storage policy, and HLS pragma policy are fixed across six designs: dense FP32, compact FP32, sparse\+int8, sparse\+int4, and dense int8/int4 controls\.

Hard gates are folded into the exported tensors, so the accelerator contains no gate evaluation or dynamic zero\-skipping\. Neuron pruning shortens compile\-time layer dimensions\. Shared\-kkcompression reduces the basis\-loop bound, with the retained identities stored in a smallkept\_term\_idxtable\. The accelerator therefore executes a smaller dense problem rather than a dense problem containing zeros\. EfficientKAN and GRAM exceed on\-chip memory and stream weights from DDR; KAGN\-conv is small enough to store its weights in on\-chip ROM\. All designs meet timing, with achieved clocks between approximately4\.404\.40and4\.714\.71ns\.

Table 8:FPGA/HLS inference ladder\.MLP latency is from RTL co\-simulation; KAGN\-conv†uses the HLS synthesis worst\-case cycle estimate\. Accuracy is evaluated on the exported MNIST hardware checkpoints\.FamilyDesignAcc\.LatencySpeedupEffKANFP3297\.482\.872 ms1\.00×1\.00\\timesSparse98\.000\.872 ms3\.29×3\.29\\timesSparse\+int897\.960\.195 ms14\.69×14\.69\\timesSparse\+int497\.920\.122 ms23\.63×23\.63\\timesDense int496\.830\.366 ms7\.85×7\.85\\timesGRAMFP3298\.141\.898 ms1\.00×1\.00\\timesSparse98\.220\.659 ms2\.88×2\.88\\timesSparse\+int898\.110\.151 ms12\.55×12\.55\\timesSparse\+int498\.220\.104 ms18\.21×18\.21\\timesDense int498\.060\.262 ms7\.25×7\.25\\timesKAGN\-convFP3298\.4586\.677 ms1\.00×1\.00\\timesSparse\+int498\.3657\.219 ms1\.51׆1\.51\\times^\{\\dagger\}

#### MLPs: compression becomes latency and bandwidth reduction\.

EfficientKAN improves from2\.8722\.872ms in FP32 to0\.1220\.122ms under sparse int4, a23\.63×23\.63\\timesreduction; GRAM reaches18\.21×18\.21\\times\. Dense int4 controls reach only7\.85×7\.85\\timesand7\.25×7\.25\\times, respectively, isolating the contribution of structural compaction\. At int8, the measured stagewise factors compose to rounding \(3\.29×4\.46≈14\.693\.29\\times 4\.46\\approx 14\.69for EfficientKAN and2\.88×4\.36≈12\.552\.88\\times 4\.36\\approx 12\.55for GRAM\)\. Because these MLPs stream weights from DDR, the same transformation reduces per\-image weight traffic from approximately8\.458\.45to0\.270\.27MB for EfficientKAN and4\.704\.70to0\.160\.16MB for GRAM\. The low\-bit MLP designs sometimes consume*more*LUT/DSP resources than FP32 because the HLS implementation trades parallel arithmetic for latency; their hardware result is therefore latency/bandwidth reduction, not area reduction\.

#### KAGN\-conv: compression becomes resource headroom\.

The convolutional accelerator exposes a different bottleneck\. Its one\-MAC\-per\-cycle engine remains dominated by spatial basis construction and convolution, so sparse int4 improves serial latency by only1\.51×1\.51\\times\. Device occupancy, however, falls sharply\.

Table 9:KAGN\-conv post\-synthesis utilization\.Sparse\+int4 reduces DSP occupancy from96\.2%96\.2\\%to29\.1%29\.1\\%with only−0\.09\-0\.09points of accuracy change\.Sparse\+int4 reduces DSP count from 1,663 to 502, LUT utilization from71\.0%71\.0\\%to34\.8%34\.8\\%, and BRAM from75\.8%75\.8\\%to43\.9%43\.9\\%, while accuracy changes from98\.45%98\.45\\%to98\.36%98\.36\\%\. Dense int4 reaches similar area but requires16\.516\.5M rather than12\.712\.7M cycles per image, showing that quantization supplies most of the arithmetic\-area reduction while structural sparsity additionally shortens the executed contraction\. The result is primarily*capacity headroom*: the compressed engine leaves room for parallel replication or larger models, although such replication is not implemented here\.

#### Verification\.

All 18 designs are checked against NumPy references generated from the exported artifacts\. MLP designs pass RTL co\-simulation; integer implementations using the same arithmetic are bit\-exact where applicable, while FP32 differences are limited to small floating\-point reassociation errors\. Full\-image RTL co\-simulation is impractical for the 12–19M\-cycle KAGN\-conv models, so their latency is synthesis\-estimated and each design is additionally checked on 64 C\-simulation images\. All six convolutional designs obtain 64/64 argmax agreement\.

### 4\.9Discussion and Limitations

Three conclusions emerge from the study\. First, KAN redundancy is*multi\-axis and basis\-dependent*: basis functions, neurons/channels, and precision expose distinct operating points, and the importance of individual terms depends on the basis family and task\. Second, sparsity improves deployment only when it is physically realized; masks leave dense tensor shapes unchanged, whereas structured hardening and compaction reduce storage, CUDA latency, and hardware loop bounds\. Third, precision interacts with task difficulty: 8\-bit QAT is broadly benign, while 4\-bit convolutional KANs require quantization\-aware recovery\.

The robustness audit also limits the claims appropriately\. Basis\-level sparsity does not consistently outperform matched\-cost edge\-L0L\_\{0\}pruning, joint QAS does not establish a mean\-accuracy advantage over a sufficiently recovered two\-stage pipeline, and gate training is not itself an accuracy contribution\. SparseKAN’s value is instead the common interface it provides for discovering, controlling, and physically realizing functional, structural, and precision compression across KAN families\.

## 5Conclusion

SparseKAN addresses a compression dimension specific to KANs: redundancy can occur not only across edges and neurons, but also within the basis expansion of each functional edge\. The framework exposes basis functions, neurons/channels, and precision as separately controllable axes, then converts the selected structure into compact tensors through hardening, recovery, and physical compaction\. Experiments across multiple KAN families show that basis and width reductions compose predictably, while basis selection can be critical in Gram\-polynomial models\. Compaction removes up to73\.0%73\.0\\%of parameters without loss on MNIST and reduces large\-batch CUDA latency to0\.51×0\.51\\timesdense execution\. On a ZCU104, sparse low\-bit MLPs achieve up to23\.63×23\.63\\timeslower latency, while KAGN\-conv reduces DSP occupancy from96\.2%96\.2\\%to29\.1%29\.1\\%with negligible accuracy change\. These results establish SparseKAN as a basis\-aware route from learned KAN redundancy to compact, low\-bit software and hardware implementations\.

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## Appendix AAdditional SparseKAN Method Details

This appendix provides the training, hardening, compaction, and quantization details omitted from Sec\.[3](https://arxiv.org/html/2608.00859#S3)for space\.

### A\.1Gate Relaxations and Hardening

The default SparseKAN gate is a sigmoid\-relaxed learnable scalar

g=σ​\(α\),g∈\(0,1\),g=\\sigma\(\\alpha\),\\qquad g\\in\(0,1\),\(7\)whereα\\alphais the gate logit\. The implementation additionally supports hard\-concrete gates\(Louizoset al\.[2018](https://arxiv.org/html/2608.00859#bib.bib23)\)and a binary Gumbel straight\-through relaxation\. These choices modify the stochastic training\-time surrogate but not the downstream compression interface\. At hardening, each soft gate is mapped to a deterministic binary decision using thresholdτ\\tau; the corresponding logits are then saturated to large positive or negative values so that the structure remains fixed during recovery\.

For convolutional KANs, the same base/branch/term hierarchy is used over spatial coefficient kernels\. A term decision applies to the complete spatial kernel associated with that basis function rather than independently to each kernel position\.

### A\.2Cost Scheduling and Direct Budget Control

SparseKAN begins with a warm\-up interval in which no sparsity pressure is applied\. Each cost coefficient is then increased linearly toward its target:

λq​\(e\)=λqmax​\{0,e≤Ew,min⁡\(1,e−EwEr\),e\>Ew,q∈\{b,r,t\},\\lambda\_\{q\}\(e\)=\\lambda\_\{q\}^\{\\max\}\\begin\{cases\}0,&e\\leq E\_\{\\mathrm\{w\}\},\\\\\[2\.84526pt\] \\min\\\!\\left\(1,\\frac\{e\-E\_\{\\mathrm\{w\}\}\}\{E\_\{\\mathrm\{r\}\}\}\\right\),&e\>E\_\{\\mathrm\{w\}\},\\end\{cases\}\\qquad q\\in\\\{\\mathrm\{b\},\\mathrm\{r\},\\mathrm\{t\}\\\},\(8\)whereEwE\_\{\\mathrm\{w\}\}andErE\_\{\\mathrm\{r\}\}are the warm\-up and ramp durations\.

The framework also supports direct cost\-budget control\. Letρsoft\\rho\_\{\\mathrm\{soft\}\}denote the normalized soft active cost evaluated without the fixedλq\\lambda\_\{q\}multipliers\. Given targetθ\\theta, we optimize

ℛbudget=μ​\[ρsoft−θ\]\+,\\mathcal\{R\}\_\{\\mathrm\{budget\}\}=\\mu\[\\rho\_\{\\mathrm\{soft\}\}\-\\theta\]\_\{\+\},\(9\)where\[z\]\+=max⁡\(0,z\)\[z\]\_\{\+\}=\\max\(0,z\)and the dual variable is updated by projected ascent,

μ←max⁡\(0,μ\+ημ​\(ρsoft−θ\)\)\.\\mu\\leftarrow\\max\\\!\\left\(0,\\,\\mu\+\\eta\_\{\\mu\}\(\\rho\_\{\\mathrm\{soft\}\}\-\\theta\)\\right\)\.\(10\)This mode is used when a desired computational budget should be specified directly rather than through a manually chosen regularization coefficient\.

Under the default schedule, Stage 1 is intentionally conservative and often finishes close to the dense soft\-cost point\. It should therefore be understood as a*structure\-discovery and adaptation stage*, not as the final compressed operating point\. Explicit compression is imposed by the hardening operations below\.

### A\.3Basis Selection Rules

#### Per\-edge top\-kk\.

Given the default score from Eq\. \([4](https://arxiv.org/html/2608.00859#S3.E4)\),

𝒮o,i\(k\)=TopKk⁡\{so,i,1,…,so,i,K\},\\mathcal\{S\}^\{\(k\)\}\_\{o,i\}=\\operatorname\{TopK\}\_\{k\}\\\{s\_\{o,i,1\},\\ldots,s\_\{o,i,K\}\\\},\(11\)and

g^o,i,kt=𝕀​\[k∈𝒮o,i\(k\)\]​𝕀​\[go,ir\>τ\]\.\\hat\{g\}^\{\\mathrm\{t\}\}\_\{o,i,k\}=\\mathbb\{I\}\[k\\in\\mathcal\{S\}^\{\(k\)\}\_\{o,i\}\]\\,\\mathbb\{I\}\[g^\{\\mathrm\{r\}\}\_\{o,i\}\>\\tau\]\.\(12\)

#### Input\-shared top\-kk\.

All outgoing edges associated with inputiishare one support:

s¯i,k=∑oso,i,k,𝒮i\(k\)=TopKk⁡\{s¯i,1,…,s¯i,K\}\.\\bar\{s\}\_\{i,k\}=\\sum\_\{o\}s\_\{o,i,k\},\\qquad\\mathcal\{S\}^\{\(k\)\}\_\{i\}=\\operatorname\{TopK\}\_\{k\}\\\{\\bar\{s\}\_\{i,1\},\\ldots,\\bar\{s\}\_\{i,K\}\\\}\.\(13\)

#### Layer\-shared top\-kk\.

The most regular variant uses

s¯k=∑i∑oso,i,k,𝒮\(k\)=TopKk⁡\{s¯1,…,s¯K\}\.\\bar\{s\}\_\{k\}=\\sum\_\{i\}\\sum\_\{o\}s\_\{o,i,k\},\\qquad\\mathcal\{S\}^\{\(k\)\}=\\operatorname\{TopK\}\_\{k\}\\\{\\bar\{s\}\_\{1\},\\ldots,\\bar\{s\}\_\{K\}\\\}\.\(14\)

#### Alternative scores\.

For controlled ablations the implementation also provides:

so,i,kgate\\displaystyle s^\{\\mathrm\{gate\}\}\_\{o,i,k\}=go,i,kt,\\displaystyle=g^\{\\mathrm\{t\}\}\_\{o,i,k\},\(15\)so,i,kocc\\displaystyle s^\{\\mathrm\{occ\}\}\_\{o,i,k\}=ρi,k​\|co,i,k\|​go,i,kt,\\displaystyle=\\rho\_\{i,k\}\|c\_\{o,i,k\}\|g^\{\\mathrm\{t\}\}\_\{o,i,k\},\(16\)so,i,kquant\\displaystyle s^\{\\mathrm\{quant\}\}\_\{o,i,k\}=ρi,k​\|Qb​\(co,i,k\)\|​go,i,kt,\\displaystyle=\\rho\_\{i,k\}\|Q\_\{b\}\(c\_\{o,i,k\}\)\|g^\{\\mathrm\{t\}\}\_\{o,i,k\},\(17\)whereρi,k\\rho\_\{i,k\}denotes empirical basis occupancy andQbQ\_\{b\}is thebb\-bit fake\-quantization operator\. These scores are experimental alternatives rather than prerequisites of the core framework\.

### A\.4Structured Neuron, Channel, and Block Pruning

SparseKAN supports neuron/channel sparsification in both the in\-training structured ablation and the final compaction path\. These use related but slightly different rankings\.

During generic structured hardening, an output unit is scored from the row\-wise activity of its base, branch, and term gates\. After shared\-kkhardening, however, many gate values can become saturated and nearly indistinguishable across units\. For final physical compaction we therefore rank outputoousing the magnitude of its surviving computation:

uo=∑igo,ib​\|wo,ib\|\+∑i,kgo,ir​go,i,kt​\|co,i,k\|\.u\_\{o\}=\\sum\_\{i\}g^\{\\mathrm\{b\}\}\_\{o,i\}\|w^\{\\mathrm\{b\}\}\_\{o,i\}\|\+\\sum\_\{i,k\}g^\{\\mathrm\{r\}\}\_\{o,i\}g^\{\\mathrm\{t\}\}\_\{o,i,k\}\|c\_\{o,i,k\}\|\.\(18\)For convolutional KANs, the coefficient magnitude is additionally summed over the spatial kernel dimensions\.

Given keep ratiornr\_\{\\mathrm\{n\}\},

dℓ′=max⁡\(1,round⁡\(rn​dℓ\)\)d^\{\\prime\}\_\{\\ell\}=\\max\\\!\\left\(1,\\operatorname\{round\}\(r\_\{\\mathrm\{n\}\}d\_\{\\ell\}\)\\right\)\(19\)highest\-scoring hidden units are retained\. Removing output unitoofrom layerℓ\\ellsimultaneously removes its corresponding input dimension from layerℓ\+1\\ell\+1\. The final network output is protected\.

The implementation also supports block\-structured pruning, in which groups of neighboring coefficient connections are hardened together\. Block sparsity is used as a regularity ablation and is not required by the main SparseKAN pipeline\.

### A\.5Physical Compaction

Letℐℓ\\mathcal\{I\}\_\{\\ell\}and𝒪ℓ\\mathcal\{O\}\_\{\\ell\}denote the surviving input and output indices of layerℓ\\ell\. The compact base matrix is

𝐖~b,ℓ=𝐖b,ℓ​\[𝒪ℓ,ℐℓ\]\.\\widetilde\{\\mathbf\{W\}\}^\{\\mathrm\{b\},\\ell\}=\\mathbf\{W\}^\{\\mathrm\{b\},\\ell\}\[\\mathcal\{O\}\_\{\\ell\},\\mathcal\{I\}\_\{\\ell\}\]\.\(20\)The same index sets slice the corresponding coefficient, gate, and normalization tensors\.

For input\-shared basis sparsity, let𝒦ℓ,i\(k\)\\mathcal\{K\}\_\{\\ell,i\}^\{\(k\)\}denote the retained basis indices associated with inputii\. The compact coefficient tensor is formed as

𝐂^:,i,:ℓ=𝐂𝒪ℓ,i,𝒦ℓ,i\(k\)ℓ,\\widehat\{\\mathbf\{C\}\}^\{\\ell\}\_\{:,i,:\}=\\mathbf\{C\}^\{\\ell\}\_\{\\mathcal\{O\}\_\{\\ell\},\\,i,\\,\\mathcal\{K\}\_\{\\ell,i\}^\{\(k\)\}\},\(21\)giving shapedℓ′×dℓ−1′×kd^\{\\prime\}\_\{\\ell\}\\times d^\{\\prime\}\_\{\\ell\-1\}\\times k\. The original basis identities are stored in

𝐑ℓ∈\{1,…,Kℓ\}dℓ−1′×k,\\mathbf\{R\}^\{\\ell\}\\in\\\{1,\\ldots,K\_\{\\ell\}\\\}^\{d^\{\\prime\}\_\{\\ell\-1\}\\times k\},\(22\)implemented bykept\_term\_idx\. Layer\-shared sparsity is the special case in which every row of𝐑ℓ\\mathbf\{R\}^\{\\ell\}is identical\.

The compact forward computation is

y~o=∑i=1dℓ−1′w~o,ib​a​\(xi\)\+∑i=1dℓ−1′∑j=1kc^o,i,j​ψRi,jℓ​\(xi\)\.\\widetilde\{y\}\_\{o\}=\\sum\_\{i=1\}^\{d^\{\\prime\}\_\{\\ell\-1\}\}\\widetilde\{w\}^\{\\mathrm\{b\}\}\_\{o,i\}a\(x\_\{i\}\)\+\\sum\_\{i=1\}^\{d^\{\\prime\}\_\{\\ell\-1\}\}\\sum\_\{j=1\}^\{k\}\\widehat\{c\}\_\{o,i,j\}\\psi\_\{R^\{\\ell\}\_\{i,j\}\}\(x\_\{i\}\)\.\(23\)Consequently, the dominant coefficient storage changes fromdℓ−1​dℓ​Kℓd\_\{\\ell\-1\}d\_\{\\ell\}K\_\{\\ell\}to approximatelydℓ−1′​dℓ′​kd^\{\\prime\}\_\{\\ell\-1\}d^\{\\prime\}\_\{\\ell\}k, plus the much smaller retained\-index table\. Arbitrary per\-edge top\-kkdoes not share a common basis dimension and is therefore not physically gathered by this regular compact representation\.

After aggressive structural cuts, a recovery or “healing” stage may be used so that the surviving parameters adapt to the compact architecture\.

### A\.6Quantization\-Aware Training

SparseKAN uses weight\-only symmetric fake quantization of the base weights and basis coefficients\. For target precisionb<32b<32,

qmax=2b−1−1\.q\_\{\\max\}=2^\{b\-1\}\-1\.\(24\)For output channeloo, the per\-channel scale is

so=max𝐰∈𝐖o⁡\|𝐰\|qmax,s\_\{o\}=\\frac\{\\max\_\{\\mathbf\{w\}\\in\\mathbf\{W\}\_\{o\}\}\|\\mathbf\{w\}\|\}\{q\_\{\\max\}\},\(25\)and

Qb​\(𝐖o\)=so​clip⁡\(round⁡\(𝐖oso\),−qmax,qmax\)\.Q\_\{b\}\(\\mathbf\{W\}\_\{o\}\)=s\_\{o\}\\,\\operatorname\{clip\}\\left\(\\operatorname\{round\}\\left\(\\frac\{\\mathbf\{W\}\_\{o\}\}\{s\_\{o\}\}\\right\),\-q\_\{\\max\},q\_\{\\max\}\\right\)\.\(26\)The underlying FP32 parameter is retained during optimization using the straight\-through construction

𝐖~=𝐖\+sg⁡\(Qb​\(𝐖\)−𝐖\),\\widetilde\{\\mathbf\{W\}\}=\\mathbf\{W\}\+\\operatorname\{sg\}\\left\(Q\_\{b\}\(\\mathbf\{W\}\)\-\\mathbf\{W\}\\right\),\(27\)so the forward pass observes fake\-quantized weights while gradients propagate through the full\-precision parameter\. Activations remain unquantized in the software SparseKAN experiments\.

### A\.7Quantization\-Aware Sparsification Extensions

The default pipeline first establishes structural sparsity and then enables QAT during recovery\. We additionally implement two tighter couplings between structure and precision\.

#### Joint gate learning and QAT\.

Fake quantization may be activated during Stage 1 so that gates are optimized under quantized forward semantics rather than being learned entirely in FP32\. We evaluate this as an alternative training schedule rather than a required part of SparseKAN\.

#### Learned layer\-wise precision\.

Each sparse layer can instead carry a continuous bit variableβℓ∈\[bmin,bmax\]\\beta\_\{\\ell\}\\in\[b\_\{\\min\},b\_\{\\max\}\], optimized jointly with the network using

ℛbits=∑ℓCℓsoft​βℓCdense​32\.\\mathcal\{R\}\_\{\\mathrm\{bits\}\}=\\frac\{\\sum\_\{\\ell\}C\_\{\\ell\}^\{\\mathrm\{soft\}\}\\beta\_\{\\ell\}\}\{C\_\{\\mathrm\{dense\}\}\\,32\}\.\(28\)After the structural training stage,βℓ\\beta\_\{\\ell\}is rounded to an integer and frozen\. This provides an optional mixed\-precision search mechanism\. The main\-paper results, however, use fixed 8\- and 4\-bit settings unless stated otherwise\.

### A\.8Complete Training Procedure

Algorithm 1SparseKAN Training and Compression1:Dataset

𝒟\\mathcal\{D\}, KAN

fθf\_\{\\theta\}, term budget

kk, neuron keep ratio

rnr\_\{\\mathrm\{n\}\}, bit\-width

bb, hardening threshold

τ\\tau
2:Attach hierarchical gates

\{gb,gr,gt\}\\\{g^\{\\mathrm\{b\}\},g^\{\\mathrm\{r\}\},g^\{\\mathrm\{t\}\}\\\}to each sparse KAN layer

3:for

e=1,…,Ee=1,\\ldots,Edo

4:Update cost coefficients using Eq\. \([8](https://arxiv.org/html/2608.00859#A1.E8)\)

5:Optimize model and gates using Eq\. \([3](https://arxiv.org/html/2608.00859#S3.E3)\)

6:endfor

7:Harden continuous gates at threshold

τ\\tau
8:ifbasis sparsification is enabledthen

9:Apply per\-edge top\-

kk, input\-shared

kk, or layer\-shared

kk
10:endif

11:ifneuron/channel compression is enabledthen

12:Retain the selected hidden dimensions and propagate removals across adjacent layers

13:endif

14:if

b<32b<32then

15:Enable per\-output\-channel

bb\-bit fake quantization

16:endif

17:Fine\-tune the surviving model with structural support fixed

18:ifthe selected support is compactablethen

19:Slice dead dimensions and gather shared retained terms

20:endif

21:returncompressed SparseKAN

## Appendix BAdditional Experimental Results

This section provides the complete evidence supporting Sec\.[4](https://arxiv.org/html/2608.00859#S4)\. We separate the three\-seed frontier study, which maps available operating points, from the five\-seed paired ablation campaign used to evaluate comparative claims\. We further distinguish analytical active cost, compact parameter count, measured CUDA latency, and FPGA/HLS results\.

### B\.1Experimental Protocol

The primary frontier consists of 380 completed runs using seeds\{42,43,44\}\\\{42,43,44\\\}\. Unless stated otherwise, frontier results are mean±\\pmsample standard deviation over these seeds\.

The follow\-up ablation campaign uses five paired seeds\{42,43,44,45,46\}\\\{42,43,44,45,46\\\}for comparisons involving selection controls, edge\-level baselines, gate necessity, PTQ, and joint versus staged low\-bit training\. Paired seeds are shared between arms wherever possible\. Atn=5n=5, an exact two\-sided Wilcoxon signed\-rank test cannot attain app\-value below0\.06250\.0625for five non\-zero paired differences\. We therefore report effect size, sign consistency, confidence intervals, and seed spread rather than usingp<0\.05p<0\.05as the sole criterion\.

The normalized active\-cost ratio is denotedρstruct\\rho\_\{\\mathrm\{struct\}\}\. For bit\-widthbb, the joint bit\-cost isρbit=ρstruct​\(b/32\)\\rho\_\{\\mathrm\{bit\}\}=\\rho\_\{\\mathrm\{struct\}\}\(b/32\)\. Physical reductions are computed directly from compact checkpoints rather than inferred from binary masks\.

### B\.2Five\-Seed Claim Audit

Because several exploratory three\-seed differences were large, we explicitly re\-tested the corresponding comparative claims with five paired seeds\. Table[10](https://arxiv.org/html/2608.00859#A2.T10)records the resulting evidence hierarchy\. This table is intended to make clear which conclusions are used as positive claims in the paper and which are retained only as parity, stability, or scope observations\.

Table 10:Five\-seed audit of the main comparative claims\.Δ\\Deltais the paired SparseKAN/joint\-minus\-control difference in accuracy points unless stated otherwise\. “Parity” means the observed difference is within seed variation or its confidence interval includes zero\.The audit changes the interpretation of two exploratory observations\. First, the original\+5\.69\+5\.69\-point joint\-QAS headline is not retained: a matched five\-seed comparison is not separable, and extending the recovery budget of the staged baseline closes most of the mean gap\. Second, the original\+4\.77\+4\.77\-point training\-time\-versus\-post\-hoc hardening observation also falls within seed variation\. Conversely, the basis\-selection result survives the expanded controls and becomes stronger mechanistically because the same family dependence appears against both low\-order truncation and random selection\. The engineering conclusions are also checked independently: reweighting the cost model preserves the frontier ordering, and physical compaction retains at least0\.99970\.9997top\-1 agreement with the masked source model across all 54 checked checkpoints\.

### B\.3Dense Anchors

Table 11:Dense and Stage\-1 cost\-gated anchors\.Accuracy in %, three seeds\. Stage 1 uses the conservative default schedule; explicit compression is imposed during subsequent hardening\.DatasetVariantParamsDense acc\.Sparse\-trained acc\.CostMNISTEfficientKAN4,460,28897\.05±0\.2897\.05\\pm 0\.2897\.07±0\.2497\.07\\pm 0\.240\.9959MNISTKAGN2,583,07298\.07±0\.1198\.07\\pm 0\.1198\.18±0\.1098\.18\\pm 0\.101\.0000MNISTKAGN\-conv33,12097\.59±0\.2597\.59\\pm 0\.2597\.75±0\.1597\.75\\pm 0\.150\.9969MNISTChebyKAN2,582,27297\.59±0\.1797\.59\\pm 0\.1797\.41±0\.1097\.41\\pm 0\.100\.9534MNISTFastKAN4,463,01897\.49±0\.2197\.49\\pm 0\.2197\.51±0\.1197\.51\\pm 0\.110\.9964MNISTPyKAN4,695,04098\.20±0\.2698\.20\\pm 0\.2698\.28±0\.1498\.28\\pm 0\.140\.9997MNISTReLUKAN4,460,28896\.71±0\.2196\.71\\pm 0\.2196\.89±0\.1796\.89\\pm 0\.170\.9961MNISTWavKAN4,462,64896\.78±0\.3596\.78\\pm 0\.3596\.83±0\.4196\.83\\pm 0\.410\.9982CIFAR\-10EfficientKAN16,251,58489\.96±0\.3889\.96\\pm 0\.3890\.30±0\.3590\.30\\pm 0\.350\.9995CIFAR\-10KAGN\-conv592,93281\.73±0\.6281\.73\\pm 0\.6281\.41±0\.5881\.41\\pm 0\.580\.9960CIFAR\-100KAGN\-conv \(narrow\)719,83249\.04±0\.7149\.04\\pm 0\.7149\.24±0\.6949\.24\\pm 0\.690\.9751CIFAR\-100KAGN\-conv \(wide\)2,586,32852\.72±0\.4552\.72\\pm 0\.4553\.24±0\.5253\.24\\pm 0\.520\.9683

### B\.4Variant Transfer

Table 12:Transfer of post\-hoc term sparsification across MNIST KAN families\(accuracy in %, three seeds\)\. The result demonstrates substantial family dependence rather than a universal polynomial\-versus\-spline rule\.
### B\.5Selection Controls

Table 13:Coefficient selection versus low\-order truncation and random selection\(five\-seed campaign\)\. Large truncation failures occur in several Gram\-polynomial models, while the tested B\-spline models are largely insensitive to support identity\.For MNIST KAGN, coefficient selection exceeds random selection on all five paired seeds; the reported paired interval is\[\+0\.41,\+0\.77\]\[\+0\.41,\+0\.77\]points\. For CIFAR\-100 KAGN\-conv, the coefficient\-versus\-random effect is effectively zero\. Thus the data supports basis\-dependent selection sensitivity, not a universal selector advantage\.

The post\-hoc scoring sweep further shows that raw gate value is a poor criterion after structured gate saturation\. MNIST KAGN falls from97\.67%97\.67\\%under coefficient scoring to82\.75%82\.75\\%under gate scoring at the same cost, and CIFAR\-10 KAGN\-conv falls from75\.22%75\.22\\%to63\.92%63\.92\\%\. A quantization\-aware coefficient score is also worse than ordinary coefficient magnitude in seven of nine evaluated cells and is not used as the default\.

### B\.6Matched\-Cost Edge\-Level Baseline

The original edge\-level control did not produce non\-trivial sparsity and was therefore re\-specified using the direct Lagrangian budget mode\. The replacement baseline gates edges across all evaluated runs, with budget points chosen to bracket the SparseKAN operating point\. The edge frontier is interpolated to the exact SparseKAN cost\.

Table 14:Basis\-function sparsity versus matched\-cost edge\-L0L\_\{0\}\(accuracy in %, five paired seeds\)\. The gaps change sign by dataset and remain within the observed seed spread\.
### B\.7KAN Node\-Pruning Baseline

The KANL1L\_\{1\}\+entropy baseline is trained with three penalty strengths and evaluated over eight node\-retention levels, followed by healing\. Representative CIFAR\-10 points are80\.82%80\.82\\%at cost0\.9240\.924,80\.01%80\.01\\%at0\.8540\.854,76\.23%76\.23\\%at0\.7080\.708, and64\.79%64\.79\\%at0\.4270\.427\. SparseKAN reaches80\.67±0\.42%80\.67\\pm 0\.42\\%at0\.7990\.799using shared\-k=3k=3and78\.65±0\.59%78\.65\\pm 0\.59\\%at0\.5830\.583using neuron keep\-0\.750\.75\.

For CIFAR\-100, representative baseline points are53\.57%53\.57\\%at0\.9310\.931,52\.36%52\.36\\%at0\.8640\.864,48\.06%48\.06\\%at0\.7310\.731, and34\.12%34\.12\\%at0\.4780\.478\. SparseKAN shared\-k=3k=3gives53\.39±0\.91%53\.39\\pm 0\.91\\%at0\.7960\.796\. The baseline frontiers are similar across penalty strengths, so these results are used to establish competitiveness rather than a large accuracy advantage\.

### B\.8Gate\-Training Ablation

Gate\-trained and gate\-free versions use the same downstream hardening procedure\. Five of six evaluated arms are not separable at five seeds: representative gated\-minus\-ungated effects are\+0\.13±0\.16\+0\.13\\pm 0\.16points on MNIST FP32,\+0\.03±0\.12\+0\.03\\pm 0\.12on MNIST 4\-bit, and\+0\.16±0\.30\+0\.16\\pm 0\.30on CIFAR\-10 FP32\.

The largest mean effect appears on the CIFAR\-100 4\-bit arm \(approximately\+1\.15\+1\.15points; reported interval\[\+0\.03,\+2\.27\]\[\+0\.03,\+2\.27\]\), although the sign reverses for one seed\. We therefore do not claim that differentiable gate training is necessary for ordinary coefficient hardening\. Its value is the common optimization interface it provides for structure discovery, cost budgets, and optional joint structural/precision training\.

### B\.9Matched\-Parameter MLP Control

On CIFAR\-10, the parameter\-matched MLP and KAN each begin with 16,251,584 parameters\. The MLP reaches40\.51±1\.22%40\.51\\pm 1\.22\\%when dense and44\.39±0\.58%44\.39\\pm 0\.58\\%in its best evaluated compressed configuration, whereas the corresponding KAN pipeline reaches90\.73±0\.13%90\.73\\pm 0\.13\\%\. The 4\-bit MLP arm reaches23\.75±3\.56%23\.75\\pm 3\.56\\%\.

The result reverses on MNIST: at the matched 4,460,288\-parameter scale, the MLP reaches98\.22±0\.06%98\.22\\pm 0\.06\\%versus97\.44±0\.15%97\.44\\pm 0\.15\\%for the KAN\. This experiment therefore scopes the deployment motivation: KAN compression is useful where the KAN representation first provides a task\-level benefit\.

### B\.10Full Post\-Hoc Selection Frontier

Table 15:Global keep\-ratio and per\-edge top\-kkat approximately matched cost\(accuracy in %, three seeds\)\. Global importance does not consistently dominate per\-edge top\-kk; the more reproducible observation is the failure of raw gate scoring after hardening\.
### B\.11Structured Frontiers

Table 16:Representative in\-training structured operating points\(accuracy in %, three seeds\)\.The three\-seed training\-time\-versus\-post\-hoc comparison initially suggested advantages of\+0\.42\+0\.42,\+1\.94\+1\.94, and\+4\.77\+4\.77points on MNIST KAGN, CIFAR\-10 KAGN\-conv, and wide CIFAR\-100 KAGN\-conv\. The subsequent five\-seed study places these timing differences within the observed seed variation\. Accordingly, training\-time hardening is treated as an adaptation option rather than an accuracy contribution\.

### B\.12Structural Composition

Table 17:Composition of the two structural cost factors\.The corresponding composed accuracies are96\.18±0\.43%96\.18\\pm 0\.43\\%for MNIST and67\.96±1\.19%67\.96\\pm 1\.19\\%for CIFAR\-10\. The key observation is the cost factorization; accuracy itself is not assumed to factorize\.

### B\.13Complete Physical\-Compaction Results

Table 18:Physical compact checkpoints\.Sizes assume FP32 storage\. All parameter reductions are measured from the actual compact checkpoint\.For CIFAR\-10, extending recovery to 40 epochs raises sk3\+n0\.5 from67\.48%67\.48\\%to71\.13%71\.13\\%and sk2\+n0\.5 from64\.45%64\.45\\%to69\.73%69\.73\\%without changing model size\.

Across 54 compacted checkpoints, prediction agreement with their masked source models is at least0\.99970\.9997\. Small residual logit differences remain because weakly active structure is physically removed, so compaction is described as prediction\-preserving rather than bit\-exact\.

### B\.14Fixed\-Bit QAT Frontier

Table 19:Fixed\-bit QAT grid\(accuracy in %, three seeds\)\. The final column is joint bit\-cost\.
### B\.15Joint QAS versus Two\-Stage Training

Table 20:Joint QAS versus budget\-matched two\-stage QAT at 4 bits\(five paired seeds, accuracy in %\)\. The paired confidence intervals include zero, so no mean\-accuracy superiority is claimed\.On CIFAR\-10, joint training has standard deviation0\.300\.30compared with4\.194\.19for the matched two\-stage schedule\. However, extending the recovery budget of the two\-stage model reduces the joint\-minus\-two\-stage differences to approximately−0\.12\-0\.12,\+2\.66\+2\.66, and−0\.27\-0\.27points on MNIST, CIFAR\-10, and CIFAR\-100\. Thus the earlier three\-seed\+5\.69\+5\.69\-point observation is not treated as a headline contribution\.

### B\.16Dense PTQ Reference

Table 21:Dense post\-training quantization control\(accuracy in %, five seeds\)\.
### B\.17Learned Layer\-Wise Precision

Table 22:Learned layer\-wise precisionon MNIST EfficientKAN, single seed\.The experiment demonstrates that the formulation can learn heterogeneous precision, but the single\-seed setting is insufficient for a primary claim\.

### B\.18Cost\-Model Validation

The analytical active\-cost model is intended to rank structures rather than predict exact cycle counts\. Fitting effective component weights from measured latency produces base\-to\-term relative costs spanning approximately0\.860\.86–2\.162\.16across families\. Re\-costing 683 completed operating points with these measured weights preserves their ranking: Spearman correlation lies between0\.9360\.936and1\.0001\.000for every non\-degenerate family with more than three points\. CIFAR\-100 KAGN\-conv gives approximately0\.999970\.99997across 177 runs, while CIFAR\-10 KAGN\-conv gives approximately0\.99980\.9998across 181 runs\.

The regularization sweep also confirms that the near\-dense Stage 1 point is a schedule choice\. Across the tested grid, cost falls from approximately0\.9960\.996to0\.5050\.505while accuracy stays within roughly0\.30\.3points\.

### B\.19CUDA Latency

Masked models retain the dense tensor shape and show no meaningful speedup: their measured ratios remain approximately0\.950\.95–1\.031\.03across batch sizes\. Table[23](https://arxiv.org/html/2608.00859#A2.T23)gives the complete batch\-size sweep underlying the representative CUDA results reported in Table[5](https://arxiv.org/html/2608.00859#S4.T5)of the main paper\.

Table 23:Compact/dense CUDA latency ratio\.Values below one indicate an execution speedup\.The difference between parameter reduction and latency reduction exposes the basis\-construction residual\. For example, CIFAR\-10 sk3\+n0\.5 retains only approximately20\.8%20\.8\\%of parameters but still requires approximately54%54\\%of dense latency at batch 1024\.

### B\.20Supporting Tabular Results

Table 24:Supporting tabular/time\-series experiments\.Traffic uses a standardized target and is not directly comparable to published numbers on a differently scaled target\. Wine has a 36\-example test split and is not used for a primary claim\.
### B\.21Additional Supporting Ablations

#### Spatial kernels\.

On MNIST KAGN\-conv, replacing the full spatial basis kernel with box mode reduces accuracy from97\.59±0\.25%97\.59\\pm 0\.25\\%to55\.40%55\.40\\%while reducing the parameter count from 33,120 to 9,456\. EfficientKAN\-conv box mode reaches65\.16%65\.16\\%\. These single\-seed controls demonstrate that convolutional KAN compression must preserve genuine spatial structure\.

#### Scale\.

A 38\.37M\-parameter Eight\-KAGN\-v2 model trains and sparsifies successfully on CIFAR\-100, reaching50\.83±0\.46%50\.83\\pm 0\.46\\%\. It underperforms the 2\.59M\-parameter wide KAGN\-conv anchor, so the experiment is used only as evidence that the training pipeline executes at the larger scale\.

#### Symbolic regression\.

On Feynman I\.6\.2, spline PyKAN RMSE changes from approximately0\.00010\.0001dense to0\.0609±0\.01650\.0609\\pm 0\.0165after sparsification and0\.0885±0\.00970\.0885\\pm 0\.0097at 4 bits\. The corresponding KAGN experiment is non\-informative because its polynomial branch is removed entirely and the base branch already matches the dense solution\.

## Appendix CComplete FPGA/HLS Evaluation

### C\.1Platform and Implementation

All hardware designs target the Xilinx ZCU104 \(xczu7ev\-ffvc1156\-2\-e\) with Vitis HLS 2025\.2\.1\. The clock target is 5\.00 ns with 12% clock uncertainty\. The available resources are 230,400 LUTs, 460,800 FFs, 1,728 DSPs, and 624 BRAM\-18K blocks\.

Within each KAN family, part, target clock, interface, storage strategy, and HLS pragma policy are fixed across all six designs\. Only model dimensions and arithmetic precision change\.

After structural hardening, gate values are folded into the exported tensors\. The accelerator therefore contains no gate evaluation, mask multiplication, or runtime zero\-skipping logic\. Neuron/channel pruning changes compile\-time input/output dimensions, while shared\-kkcompression changes the basis\-loop bound and stores the original basis identities inkept\_term\_idx\.

The EfficientKAN and GRAM MLPs are too large to store their dense weights in BRAM and therefore stream the packed weight tensors through AXI\-MM from DDR\. All rungs within each MLP family use the same storage architecture\. KAGN\-conv has a substantially smaller weight footprint and stores weights in on\-chip ROM\.

For the quantized MLPs, surviving low\-bit weights are combined with fixed\-point activation arithmetic, while the basis and normalization operations retain the precision needed by their respective implementations\. The convolutional Gram basis contains transcendental operations and is not bit\-identical to a purely integer golden model; this distinction affects verification tolerance but not the reported argmax accuracy\.

### C\.2Complete Hardware Ladder

Table 25:Complete 18\-design FPGA/HLS ladder\.Accuracy is in % on the full MNIST test set using the exported hardware checkpoint\. MLP latency is RTL co\-simulation; KAGN\-conv latency is the post\-synthesis worst\-case estimate\. Resources are post\-synthesis estimates\.
### C\.3Hardware Gain Decomposition

For EfficientKAN, structured compaction alone gives a3\.29×3\.29\\timeslatency reduction\. Sparse\+int8 reaches14\.69×14\.69\\times, and sparse\+int4 reaches23\.63×23\.63\\times\. Dense int4 reaches only7\.85×7\.85\\times\. At int8, the stagewise factors satisfy3\.29×4\.46≈14\.693\.29\\times 4\.46\\approx 14\.69\. The weight stream correspondingly decreases from approximately 8\.45 MB/image for dense FP32 to 0\.27 MB/image for sparse int4\.

For GRAM, the analogous reductions are2\.88×2\.88\\timesfor sparse FP32,12\.55×12\.55\\timesfor sparse int8, and18\.21×18\.21\\timesfor sparse int4\. Dense int4 gives7\.25×7\.25\\times, while weight traffic falls from approximately 4\.70 to 0\.16 MB/image\.

The integer MLP implementations use more arithmetic resources than the FP32 baseline in some configurations because the HLS design spends additional parallel resources to minimize latency\. Hence resource reduction is not the appropriate interpretation for these MLPs; their principal benefits are latency and memory traffic\.

For KAGN\-conv, sparse\+int4 reaches only1\.51×1\.51\\timesserial latency reduction but reduces DSP count by3\.31×3\.31\\times, FF count by approximately3\.05×3\.05\\times, LUT count by approximately2\.04×2\.04\\times, and BRAM use by1\.73×1\.73\\times\. It is therefore primarily a capacity result\.

### C\.4Device Utilization

Table 26:Post\-synthesis resource utilizationas a percentage of the ZU7EV device budget\.
### C\.5Functional Verification

Every exported hardware design is checked against a NumPy golden reference\. All twelve MLP configurations pass RTL co\-simulation\. Their checked predictions agree with the golden reference by argmax\. Integer EfficientKAN implementations using the identical integer arithmetic are bit\-exact on the 1000\-image verification set\. Floating\-point variants exhibit only small logit differences from reassociation, with observed maximum errors on the order of10−510^\{\-5\}\.

Full\-image RTL co\-simulation is infeasible for KAGN\-conv because one inference requires approximately 12–19 million cycles\. Each of the six convolutional designs is therefore checked using a 64\-image C\-simulation spot test, and all six obtain 64/64 argmax agreement\.

The quantized KAGN\-conv kernels evaluate the Gram basis using floating\-point transcendental operations, so their logits are not expected to be bit\-exact to a purely integer golden reference\. One testbench therefore emits a strict bit\-level failure despite perfect argmax agreement; this is a tolerance artifact rather than a classification failure\.

### C\.6Scope of the Hardware Evidence

The FPGA study intentionally separates two deployment regimes\. For the DDR\-streaming MLPs, structural compaction and low precision both shrink the dominant weight\-transfer workload and therefore translate into large batch\-1 latency reductions\.

For KAGN\-conv, the one\-MAC\-per\-cycle implementation remains dominated by spatial convolution and basis evaluation\. Compression instead frees device resources\. The resulting headroom could be used to replicate multiple compressed engines or to deploy a larger model, but such replication is not implemented or measured here\.

Accordingly, we describe the results as FPGA/HLS realization evidence\. They are not board\-level measurements of wall\-clock latency, energy, or power\. MLP latency comes from RTL co\-simulation and convolutional latency from HLS synthesis estimates, as labeled throughout\.