Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces

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 present \textbf{SparseKAN}, 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 as $0.51\times$ dense execution. On a ZCU104 FPGA, the resulting sparse low-bit models achieve up to $23.63\times$ lower inference latency, demonstrating that SparseKAN converts functional redundancy into measurable software and hardware efficiency. The SparseKAN implementation is available at https://github.com/OSU-STARLAB/SparseKAN.

Kolmogorov--Arnold Networks (KANs) replace fixed node activations with learned one-dimensional edge functions, offering an explicit interface for interpretation and a possible alternative to transformer feed-forward networks. We test these claims separately. In a six-layer, 10M-parameter B-spline KAN, we reconstruct all 884,736 feed-forward edges: 87.8% exceed (NLS>0.1) and 0.4% are inactive. Pruning the lowest-activity 20--25% causes negligible loss increase, although structured MLP neuron pruning tolerates comparable sparsity. The audit replicates on BabyLM, but grid-size sweeps show that near-total fPCA compression and high closed-form-fit coverage are properties of the low-capacity grid-2 basis, not universal KAN behavior. For replacement, we evaluate MLP, SwiGLU, grouped Chebyshev, and rational GR-KAN networks on BabyLM. The KAN-family and gated variants improve validation loss over the GELU MLP, but this ordering does not transfer to standardized benchmarks: across ten seeds and 59,875 BLiMP pairs, accuracies span 62.4--63.1%, EWoK remains at chance, and a (+0.7)-point GR-KAN effect on BLiMP reverses on the supplement. Larger tests are also cautionary: parameter-matched MLPEdge underperforms the MLP on Wikitext-103, and 286M-parameter GR-KAN remains below a SwiGLU ClimbMix baseline after stabilization. Thus, small-basis KANs provide a practical, corpus-transferable interface for auditing learned scalar transformations, but the tested replacements show no consistent benchmark, quality, or latency advantage over strong MLP baselines.

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 \emph{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, $R_{n+1}(x) = (ax^2+bx+c)R_n(x) + (dx+e)R_{n-1}(x)$, 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 in $n$ exactly 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.