HYDRA: Hyperbolic Dynamic Representation Architecture for Kolmogorov-Arnold Networks

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) have recently emerged as a promising alternative to traditional multilayer perceptrons by replacing linear weights with learnable univariate functions. Despite their theoretical advantages in interpretability and expressiveness, practical research of KANs remains difficult due to high computational costs and inconsistent feature support across existing frameworks. This paper introduces KANLib, a modular, extensible, and computationally efficient framework for developing and evaluating KAN architectures. KANLib unifies core concepts from existing implementations, including PyKAN, EfficientKAN, and FastKAN, within a consistent software architecture that emphasizes flexibility, feature parity, and high performance. The framework supports two basis function types, adaptive grid rescaling, grid extension, and fine-grained architectural customization while maintaining compatibility with standard PyTorch workflows. Experimental evaluation on the California Housing benchmark demonstrates that KANLib reproduces the predictive behavior of established reference KAN implementations while achieving competitive computational efficiency. Furthermore, the framework enables the exploration of architectural variations beyond standard KAN formulations with only minor impacts on predictive performance. Overall, KANLib provides a robust foundation for future research on scalable and extensible KAN architectures.

We prove that any continuous function f from [0,1]^n to R representable by a finite computation tree with N internal nodes and compositional sparsity s = O(1) admits a deep Kolmogorov-Arnold Network (KAN) representation. Each internal node is realised by a primitive KAN block with controlled block depth and Lipschitz product. The layer-wise Lipschitz product satisfies the primary domain-sensitive bound independent of the input dimension n. It simplifies to P(KAN_f) <= max(C*,1)^L_f with L_f <= c_max * N. For the standard operations {+,-,x,sin,cos} with x nodes on [0,1]-bounded inputs we obtain P(KAN) <= 1. Layer widths satisfy n_l <= n + 2 w_max * N. The uniform approximation error is bounded by N * max(C*,1)^d(f) * epsilon_Op (simplifies when C* <=1). For f in C^m we obtain optimal B-spline rates. Range bounds are also derived (B_f <= N+1 for additive trees). This addresses the gap on Lipschitz control in deep KAN stacks noted by Liu et al. (2024). Experiments confirm P(KAN)=1.0 for several compositionally structured functions.