InKAN: B-Spline KANs via Truncated Power Form
cs.LG · math.NA
Quick verdict
Exact B-spline KAN basis via classical truncated-power form plus clamp: 2.8–3.5× faster forwards; math sound, novelty modest, compile-fairness unstated.
Summary
The paper replaces the Cox-de Boor recursion used to evaluate B-spline basis functions in Kolmogorov-Arnold Networks (KANs) with the classical truncated power form, which expresses each uniform cubic B-spline as five max(0,·)^3 terms. The motivation is efficiency: basis computation accounts for 91% of forward time in the profiled configuration (Table 1). The contributions are (1) an elementwise expression structured for torch.compile fusion, (2) a bounded-coordinate stabilization that clamps the normalized coordinate to [0, k+1], bounding cancellation error that otherwise grows as O(ε·u³) and produces partition-of-unity errors up to 0.88 at grid size 100 without the clamp (Table 2), and (3) an open-source pip package. On H100, forward latency is 2.8–3.5× lower than Efficient-KAN, though FastKAN remains 3–11% faster (Table 3). Ten-epoch classification on MNIST/FashionMNIST/CIFAR-10 and 3,000-epoch synthetic regression (Tables 4–5) show task performance comparable to other B-spline KANs; InKAN has the lowest mean accuracy on MNIST and FashionMNIST among all tested methods, and FastKAN baselines are missing for five of seven regression functions. The evaluation is restricted to cubic, uniform grids, and reports forward-pass latency only; eager-versus-compiled fairness is deferred to future work. The mathematics is classical and correct; the novelty is modest, positioning the work as an implementation and stabilization contribution.
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Assessment at a glance
5
risk flags identified
● high 0
● medium 3
● low 2
5 strengths · 5 weaknesses · 5 author questions
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Key claims under review
- The truncated power form computes exactly the same uniform cubic B-spline basis values as the Cox-de Boor recursion, but as five elementwise max(0,·)³ terms without sequential passes (Eq. 6–7).
- Clamping the normalized coordinate to [0, k+1] eliminates growing cancellation error: partition-of-unity error stays below 1e-5 for grid sizes up to 200, versus 0.88 unclamped at grid size 100 (Table 2).
- The compiler-friendly implementation achieves 2.8–3.5× lower H100 forward-pass latency than Efficient-KAN, while remaining 3–11% slower than FastKAN (Table 3).
- Under the tested configurations, task performance (10-epoch classification; 3,000-epoch synthetic regression) is comparable to other B-spline KAN implementations (Tables 4–5).
- Basis evaluation dominates KAN layer forward time, accounting for 91% in the profiled configuration (Table 1).
Strengths
Numerically sound stabilization with quantified evidence
Clamping u to [0, k+1] (Eq. 8) is exactly zero-preserving outside the support in exact arithmetic; Table 2 shows out-of-support residual drops to 0 and PU error stays ≤8.6e-6 up to G=200, versus 0.88 PU error unclamped at G=100, with growth consistent with the stated O(ε·u³) model (3.8e-5 → 4.2e-2 residual from G=5 to 64).
Honest comparative reporting
Table 3 explicitly shows FastKAN is still 3–11% faster than InKAN; Table 4 shows InKAN's mean accuracy is lowest on MNIST (96.05%) and FashionMNIST (86.92%); §6 candidly lists cubic-only, uniform-only, and forward-only scope.
Exactness preserved where prior work traded it away
Strong reproducibility provisions
Correct, pedagogical derivation
Weaknesses
Compile-status ambiguity undercuts the headline speedup
Incomplete regression baseline
Unexplained accuracy gap despite claimed mathematical equivalence
Forward-pass only; profile claim from one configuration
Small unexplained gaps in the numerical study
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Scope & Claims
InKAN (arXiv 2609.01956v3, single author) targets the dominant cost in spline-based Kolmogorov-Arnold Networks (KANs): evaluating B-spline basis functions. The paper claims that (1) the classical truncated power form (Eq. 6–7) evaluates the exact same uniform cubic B-spline basis as the Cox-de Boor recursion, but as five elementwise max(0,·)³ terms with no sequential passes, structured for torch.compile fusion; (2) clamping the normalized coordinate to [0, k+1] (Eq. 8) eliminates cancellation-error growth outside the support, keeping partition-of-unity (PU) error below 1e-5 up to grid size 200 versus 0.88 unclamped at G=100 (Table 2); (3) forward latency is 2.8–3.5× lower than Efficient-KAN on H100 (Table 3). The positioning is honest: this is an implementation and stabilization contribution on top of a classical identity already derived by Southworth et al. (2026) and approached from other angles by TruKAN, PowerMLP, and MatrixKAN.
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- 5 risk flags with severity, section & reviewer questions
- Methods & statistical evidence assessment
- 5 questions a peer reviewer would ask
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Frequently asked questions
What is the truncated power form of a B-spline?
A classical representation (Curry & Schoenberg, 1940s) that writes each B-spline as a finite alternating sum of max(0, x−t)^k terms — for a uniform cubic B-spline, five such terms with coefficients {1,−4,6,−4,1}/6. It evaluates the same function as the Cox–de Boor recursion but without sequential passes; its historical drawback is floating-point cancellation error far from the basis support.
How much faster is InKAN than Efficient-KAN?
In the paper's H100 benchmarks (batch 256, four input widths 784–3072), forward-pass latency is 2.8–3.5× lower than Efficient-KAN's Cox–de Boor implementation (e.g., 0.253 vs 0.722 ms at dim=784). FastKAN, which replaces splines with Gaussian RBFs, is still 3–11% faster than InKAN. Only forward-pass latency is reported; backward timing and memory are not.
Why does InKAN clamp the input coordinate?
Far outside a basis function's support the true value is exactly zero, but the truncated-power alternating sum of cubed terms accumulates cancellation error growing like ε·u³ in float32. Clamping the normalized coordinate to [0, k+1] bounds every intermediate term without changing the correct output (the spline is zero there anyway), keeping partition-of-unity error below 1e-5 up to grid size 200, versus 0.88 unclamped at grid size 100.
Does InKAN improve accuracy over other KANs?
No — the claim is mathematical exactness at the basis level, not accuracy gains. In the reported 10-epoch experiments, InKAN's mean accuracy (96.05% MNIST, 86.92% FashionMNIST, 45.20% CIFAR-10) is slightly below Efficient-KAN on the first two datasets, and regression MSE is comparable to Efficient-KAN, with roughly 100× lower MSE than MLP on the Gaussian function (1.9e-5 vs 2.0e-3).
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This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
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