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This paper argues that systolic arrays can efficiently accelerate Kolmogorov-Arnold Network inference by replacing recursive B-spline evaluation with a tabulated cardinal B-spline unit and exploiting the guaranteed N:M sparsity of B-splines

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2026-08-03 21:10 UTC pith:BMNMSOVD

load-bearing objection First systolic-array KAN accelerator with a sound core idea and credible synthesis results; the accuracy validation is thin but the flaws are addressable, not fatal. the 4 major comments →

arxiv 2512.00055 v1 pith:BMNMSOVD submitted 2025-11-20 cs.AR cs.AI

KAN-SAs: Efficient Acceleration of Kolmogorov-Arnold Networks on Systolic Arrays

Sohaib Errabii (TARAN) , Olivier Sentieys (TARAN) , Marcello Traiola (TARAN) This is my paper
classification cs.AR cs.AI
keywords Kolmogorov-Arnold Networkssystolic arraysB-splineshardware accelerationN:M sparsityinference acceleratortabular basis functionprocessing element
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper targets a bottleneck in Kolmogorov-Arnold Networks: the recursive, data-dependent computation of B-spline basis functions, which makes KAN inference poorly suited to systolic arrays. It shows that once the B-spline values are computed, a KAN layer is just a matrix multiply, so the real problem is generating the activation matrix cheaply. The authors' solution is a look-up-table unit that stores half of a cardinal B-spline, exploits translation and scaling invariance to align arbitrary inputs, and uses the local-support property to output only the P+1 nonzero basis values. A vectorized N:M processing element then performs multiplies only for those nonzero values. Synthesized on a 28nm FD-SOI process, the resulting KAN-SAs design reaches up to 100% processing-element utilization and cuts clock cycles by up to 50% versus a conventional systolic array of comparable area, with less than 1% accuracy loss from 8-bit quantization.

Core claim

The central claim is that the obstacle to running KANs on systolic arrays is not the GEMM but the B-spline activation stage, and that this stage can be made systolic-array-friendly. For a uniform grid, a B-spline is a translated and scaled copy of a single cardinal B-spline, so only half of one basis function needs to be tabulated; from that table, the unit reconstructs all P+1 nonzero B-spline values in a single cycle using an aligned, quantized address and a bitwise complement. Because every input x in the interval [t_k, t_{k+1}) activates exactly the P+1 basis functions indexed by k-P..k, the activation matrix has a dynamic but guaranteed N:M structured sparsity. The paper's N:M processin

What carries the argument

A basis function unit that tabulates half a cardinal B-spline in a 256-entry ROM and derives all P+1 non-zero values by affine alignment (Eq. 4) plus address inversion (Eq. 5), paired with an N:M vector processing element. The unit streams only the non-zero activations and the index k into each row of the weight-stationary systolic array; a M-to-N multiplexer in the PE routes the correct coefficients to the adders. This combination converts the recursive Cox-de Boor evaluation into a single-cycle lookup and turns the KAN's inherent local-support sparsity from a utilization killer into a speed advantage.

Load-bearing premise

The load-bearing premise is that a uniform-grid, 8-bit tabulated cardinal B-spline recovers all nonzero B-spline values faithfully enough that KAN accuracy drops less than 1%, and that any non-uniform KAN can be converted to a fine uniform grid without retraining.

What would settle it

Run a KAN model with a non-uniform knot sequence, convert it to a uniform grid per the paper's least-squares procedure, quantize the B-spline table to 8 bits, and measure accuracy on a standard benchmark: if the drop exceeds 1%, the tabulation premise fails. Also, feed inputs concentrated in the grid-extension regions (k<P or k>G+P-1) where fewer than P+1 B-splines are nonzero; if utilization falls well below the reported near-100% for such workloads, the N:M assumption does not cover boundary inputs.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If this design is right, KAN inference can be accelerated on the same spatial-array hardware that already accelerates conventional DNNs, preserving generality.
  • The 72x B-spline throughput improvement suggests tabulation-based inference makes KANs practical on edge-class arrays, not just GPUs.
  • The N:M sparsity is guaranteed by the math, not by pruning, so utilization gains apply to any KAN with a uniform grid, regardless of training specifics.
  • Because the PE still handles scalar workloads with 1:1, the accelerator remains useful for standard MLP/convolution layers, making it a drop-in enhancement rather than a KAN-only device.
  • The <1% accuracy drop on the tested models implies 8-bit integer inference is viable for KANs, matching the quantization path used for conventional DNNs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: applying the same tabulated cardinal B-spline + N:M PE idea to KAN training, where gradients pass through B-spline values, would require the table to be differentiable or the LUT to be updated; the paper's inference-only assumption leaves this open.
  • The worst-case utilization on small workloads is still limited by imperfect tiling; a natural extension is a dataflow or output-stationary variant that tiles the B matrix more flexibly.
  • The accuracy claim rests on a small set of models; running the same 8-bit tabulation on larger KANs (e.g., deep vision or language models) would test whether <1% drop holds beyond the reported benchmarks.
  • The comparison to prior recursive implementations uses an estimated area; a cycle-accurate RTL co-simulation with real workloads would sharpen the 72x figure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies systolic-array acceleration of Kolmogorov-Arnold Network inference. It observes that a KAN layer is a GEMM once B-splines are evaluated, and that the local support of B-splines induces an N:M structured sparsity in the intermediate activation matrix B. The proposed KAN-SAs adds a tabulated, non-recursive B-spline unit and a vectorized N:M processing element that skips zero B-spline contributions. The authors report post-synthesis results on 28nm FD-SOI showing up to 100% PE utilization, about 39.9% average utilization improvement and 50% average clock-cycle reduction versus a conventional systolic array of equivalent area, plus a 72x B-spline-evaluation speedup over the prior ArKANe work. The evaluation spans multiple KAN workloads with different depths, grid sizes, and degrees.

Significance. The architectural insight is timely and well motivated: using affine invariance, translation invariance, and local support to reduce B-spline evaluation to small LUT lookups and to turn KAN sparsity from a liability into a structured N:M pattern is a genuine contribution. The paper includes a post-synthesis delay/power table, a multi-workload design-space exploration, and area-versus-utilization Pareto plots, which are concrete strengths. The central cycle-count and utilization derivations follow from the B-spline properties and are convincing. The main weakness is that the accuracy of the 8-bit uniform-grid LUT is validated on only one model, and the generality of the uniform-grid assumption is asserted rather than demonstrated. If these points are resolved, the work would be a solid contribution to KAN hardware acceleration.

major comments (4)
  1. [Section V, accuracy validation, and Eq. (5)] The claim 'less than 1% in accuracy drops for all the models' is supported only by a single example (MNIST-KAN, 96.58% to 96.0%). The B-spline unit uses an 8-bit LUT without interpolation, and Eq. (5) yields only about 255/(G+2P) distinct xa levels per grid interval; for G=10, P=3 this is about 16 levels. This coarse quantization of the B-spline argument could produce nontrivial approximation errors that differ across the applications in Table II (e.g., U-KAN, Prefetcher, 5G-STARDUST). Since the entire efficiency argument rests on the LUT being both small and accurate, please report per-model accuracy for every benchmark and include a sensitivity analysis over LUT depth / input bit-width.
  2. [Section II-B and III-B (uniform-grid assumption)] The statement that non-uniform KAN grids can be approximated by finer uniform grids without changing the model is not established. Least-squares grid refinement as in [1] is exact only when the original spline space is nested in the refined space, which is not generally true for arbitrary non-uniform knots. If KAN-SAs assumes uniform grids, this should be stated as a limitation, and the approximation error for non-uniform models should be measured or bounded. Without this, the 'generality' of the B-spline unit is overstated.
  3. [Section V-B (72x speedup vs. ArKANe)] The 72x speedup is an area-normalized estimate, not a measured comparison. It combines an ideal cycle model of ArKANe with the area of an FPMax FMA unit, and ignores AIE tile overheads, control logic, memory, and the fact that ArKANe uses floating-point arithmetic for training. The same-area argument also counts only FMA area. Please reframe this as an upper-bound estimate and, if possible, include a cycle-accurate or measured comparison with ArKANe on a common workload.
  4. [Table I (normalized energy)] The normalized energy in Table I assumes that a 1:1 scalar PE takes G+P times more cycles than an N:M PE for a KAN workload. However, an N:M PE has larger area and power than a scalar PE, so a per-PE energy comparison can overstate the benefit for a full array at equivalent area. Please clarify the comparison basis: is the energy per useful MAC, per output, or per PE? Including area-normalized energy would make the table more directly comparable with the equal-area claims in Figures 7 and 8.
minor comments (4)
  1. [Eq. (5) and Section III-B2] The quantization notation is unclear: please define xq, tq0, and the affine quantization scale explicitly, and show the algebraic derivation from Eq. (4) to Eq. (5). The text says 'using affine integer quantization scheme' but does not state the scale factor.
  2. [Section IV-A] The case where the input lies in the grid extension (k < P or k > G+P-1) is mentioned as having fewer than P+1 non-zero B-splines, but the proposed PE always handles N=P+1 values. Please specify whether zero-padding or masking is used for these boundary cases.
  3. [Figure 7] The labels for the square SA configurations overlap in places, making it difficult to map data points to array sizes. Consider using distinct markers or a legend.
  4. [Section V-B] The sentence 'in the same estimated area for ArKANe, i.e., 4 × 0.0081mm2' should clarify that this is only the FMA-area estimate and does not include ArKANe's control or interconnect overhead.

Circularity Check

0 steps flagged

No circular dependency found; the hardware derivation is self-contained and validated against external results.

full rationale

The paper's central design uses standard B-spline properties (translation/scaling invariance, local support, symmetry), citing external spline-theory references. The Eq. (4)-(5) address generation follows from the affine normalization identity; the N:M sparsity exploitation follows from local support and is not fitted to the measured outcomes. Utilization and cycle improvements (Figs. 7-8, Table I) come from post-synthesis hardware measurements and workload tiling arithmetic, not from a parameter fitted to the claimed result. The only accuracy evidence is a single MNIST-KAN number while the text claims '<1% in accuracy drops for all the models' (Sec. V); this is an unsupported generalization and a correctness risk, but it is not circular because the claim is empirical, not derived from its own conclusions. The non-uniform-to-uniform grid justification cites the original KAN paper [1], not the present authors' prior work, and is therefore external support rather than a self-citation chain. No load-bearing step reduces to its own input by definition, and no fitted value is renamed as a prediction. Score 0.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted to data; the only hand-chosen number is the LUT depth. The core axioms are standard spline mathematics plus the uniform-grid and quantization assumptions stated in the paper.

free parameters (1)
  • B-spline LUT depth (256 entries per half-support) = 256
    Chosen by hand as the int8 quantization of B0,P; the central efficiency and accuracy claims depend on this table resolution, calibrated to keep the accuracy drop below 1%.
axioms (5)
  • standard math Cox-de Boor recursion and B-spline local support (Eq. 2-3)
    Used to derive that only P+1 B-splines are non-zero per input, which is the basis of the N:M sparsity and the vector PE.
  • standard math B-spline affine invariance and symmetry (Sec. III-B1, [19])
    Allows tabulating only half of B0,P on a uniform grid and recovering all P+1 non-zero values by shifting and inverting the address.
  • domain assumption Uniform-grid assumption
    The tabulation and Eq. 4 require a uniform knot sequence; the paper assumes non-uniform grids can be approximated by finer uniform grids via least-squares coefficient recomputation [1].
  • domain assumption Integer-only quantization preserves KAN accuracy
    The architecture uses 8-bit inputs and 32-bit accumulation following [18]; the reported <1% accuracy drops are claimed in Sec. V without full per-model detail.
  • domain assumption Weight-stationary GEMM formulation of a KAN layer
    The KAN layer is reformulated as multiplication of the B-spline activation matrix by the coefficient matrix (Sec. II-A), requiring coefficients to be preloaded as stationary weights.

pith-pipeline@v1.3.0-alltime-deepseek · 11820 in / 11710 out tokens · 98473 ms · 2026-08-03T21:10:23.957590+00:00 · methodology

0 comments
read the original abstract

Kolmogorov-Arnold Networks (KANs) have garnered significant attention for their promise of improved parameter efficiency and explainability compared to traditional Deep Neural Networks (DNNs). KANs' key innovation lies in the use of learnable non-linear activation functions, which are parametrized as splines. Splines are expressed as a linear combination of basis functions (B-splines). B-splines prove particularly challenging to accelerate due to their recursive definition. Systolic Array (SA)based architectures have shown great promise as DNN accelerators thanks to their energy efficiency and low latency. However, their suitability and efficiency in accelerating KANs have never been assessed. Thus, in this work, we explore the use of SA architecture to accelerate the KAN inference. We show that, while SAs can be used to accelerate part of the KAN inference, their utilization can be reduced to 30%. Hence, we propose KAN-SAs, a novel SA-based accelerator that leverages intrinsic properties of B-splines to enable efficient KAN inference. By including a nonrecursive B-spline implementation and leveraging the intrinsic KAN sparsity, KAN-SAs enhances conventional SAs, enabling efficient KAN inference, in addition to conventional DNNs. KAN-SAs achieves up to 100% SA utilization and up to 50% clock cycles reduction compared to conventional SAs of equivalent area, as shown by hardware synthesis results on a 28nm FD-SOI technology. We also evaluate different configurations of the accelerator on various KAN applications, confirming the improved efficiency of KAN inference provided by KAN-SAs.

Figures

Figures reproduced from arXiv: 2512.00055 by Marcello Traiola (TARAN), Olivier Sentieys (TARAN), Sohaib Errabii (TARAN).

Figure 1
Figure 1. Figure 1: A single layer of dimensions [2, 1] for MLP (left) and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: B-spline basis functions of degree P = 3 on a uniform grid of size G = 3 for the input domain [t3, t6] that is extended on both ends to account for the contributions of B-spline functions that are non-zero within this input domain. to AMD-Xilinx FPGA boards, as it relies on Xilinx’s recent spatial computing architecture (the Adaptive Intelligent Engine, AIE), for deployment on AMD-Xilinx Versal ASoCs. Fina… view at source ↗
Figure 3
Figure 3. Figure 3: GEMM weight stationary (WS) systolic array with [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Simplified version of the B-Spline unit. The address is [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Proposed KAN-SAs PE architecture showing how the [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: PE Utilization(%) for KAN workloads. early stages, most applications have small dimensions. This leads to a noticeable imperfect tiling effect in these applications. Also, the values of G impact the utilization. When G has high values, two effects are present: (i) the overall number of parameters also increases, and (ii) the sparsity grows, as shown in Sec. IV. The first effect intrinsically reduces the im… view at source ↗
Figure 7
Figure 7. Figure 7: Average PE utilization (a) and runtime in clock cycles [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

discussion (0)

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