Fast Convolution Structures: Algorithms & Architectures

• Fast convolution structures are advanced algorithmic techniques that reduce multiplication counts through algebraic transformations and hardware-specific optimizations.
• They encompass methods such as Winograd, Cook–Toom, FFT-based, and structured kernel schemes to achieve efficiency in performing discrete convolutions.
• These techniques enable practical improvements in digital signal processing, deep learning model compression, and large-scale scientific computing applications.

Fast convolution structures comprise a diverse class of algorithmic and architectural techniques designed to accelerate the computation of discrete convolution, particularly in applications spanning digital signal processing, deep learning, and scientific computing. These structures achieve constant-factor or asymptotic improvements by systematically reducing multiplication counts, exploiting algebraic structure (e.g., polynomial transforms, symmetry), or adapting data representation for hardware efficiency. The modern landscape includes Winograd and Cook–Toom algorithms, FFT-based and symbolic Fourier convolutions, structured kernel schemes, cascaded FIR factorizations, hardware-adaptive tiling, scalable multidimensional mappings, and domain-specific structured sparsity. This article provides a comprehensive technical account of these fast convolution architectures, detailing their formalism, complexity, stability, hardware integration, and application domains.

1. Algebraic Foundations and Bilinear Formalism

Fast convolution algorithms are best understood through the formalism of bilinear maps and tensor factorizations. A discrete convolution of input $x$ and kernel $h$ over a fixed domain can be recast as a bilinear transformation: $y = \mathcal{F}(x, h)$ By algebraic decomposition, classical approaches structure the computation around three linear operators, producing so-called bilinear algorithms: $y = A^\mathsf{T} \left[ (G h) \odot (B x) \right]$ where $A$, $B$, $G$ are explicit matrices encoding interpolation, input transform, and kernel transform, and $\odot$ denotes Hadamard (elementwise) product. In the Winograd minimal-filter setting, these matrices are constructed so that the number of required multiplications matches the minimal bilinear complexity for the given kernel and tile size. The Cook–Toom algorithm generalizes this by leveraging evaluation–interpolation at carefully chosen points to minimize the multiplication count for polynomial multiplication, yielding subquadratic complexity when iterated (Parhi, 1 Dec 2025).

For multidimensional convolution, tensor product embeddings are used, and the complexity bounds generalize as $O(\prod_{j=1}^d (m_j + r_j -1))$ multiplications for producing an $m_1 \times \cdots \times m_d$ output from an $r_1 \times \cdots \times r_d$ kernel.

2. Principal Fast Convolution Algorithms

2.1 Winograd and Cook–Toom Minimal Filtering

Winograd's algorithms transform the convolution problem via polynomial evaluation at $t=m+r-1$ points, elementwise multiplication, and interpolation. For 1D convolutions: $$d' = B d, \quad g' = G g, \quad Y' = g' \odot d', \quad y = A^\mathsf{T} Y'$$ Here $B$, $G$, $A$ are explicit matrices, often Vandermonde or Toeplitz, constructed for each $(m, r)$ pair. The multiplication count reduces from $m r$ (naïve) to $m + r - 1$ (Winograd), e.g., for $F(2,3)$, only 4 multiplications are needed versus 6 naively (Tong et al., 2021).

Cook–Toom designs achieve similar reduction for long convolutions by block recursive application and evaluation-interpolation at $2r-1$ distinct points, enabling fast modular multiplication, cyclic convolution, and parallel FIR filtering (Parhi, 1 Dec 2025).

2.2 FFT-Based and Symbolic Convolution

FFT convolution exploits the convolution theorem, mapping spatial convolution into frequency domain pointwise products. For sufficiently large kernels, FFT-based methods achieve $O(n\log n)$ scaling; however, for small kernel sizes, the overhead is prohibitive compared to Winograd. Recent work introduces Symbolic Fourier Convolution (SFC), which extends the DFT with symbolic computation at special transform points, rendering all transforms and inverse transforms as pure additions (no irrational multiplies) (He et al., 3 Jul 2024). SFC further improves multiplication reduction and quantization compatibility, yielding a $3.68\times$ reduction in multiplies for $3\times 3$ convolution and lower quantization-induced error than Winograd.

2.3 Fast FIR Factorization for State Space Models

Beylkin's FIR-cascade approach for linear time-invariant systems factors the resolvent as an infinite product: $$(I - z^{-1}A)^{-1} = \prod_{n=0}^{\infty}[I + (z^{-1}A)^{2^n}]$$ Truncating after $N + 1$ stages yields a matrix polynomial of degree $2^{N + 1} - 1$, which allows time-domain implementation via a cascade of $N + 1$ shift-and-multiply stages, independent of output length $L$. The algorithm guarantees unconditional numerical stability and permits use of structured matrix approximations (PLR, wavelet, Toeplitz/FFT) for $O(m)$ or $O(m \log m)$ per-step cost (Beylkin, 22 Nov 2024).

3. Structured, Sparse, and Circulant Convolutions

3.1 Structured Convolution and Composite Kernels

Structured Convolutions impose composite or block structure on weights, enabling decomposition into sum-pooling followed by small convolutions. For kernel $W\in\mathbb{R}^{C\times N\times N}$, structured decomposition writes $W = \sum_{m=1}^M \alpha_m \beta_m$ with binary tensors $\beta_m$ encoding cuboid/shifts. Convolution becomes: $X*W = \sum_m \alpha_m (X * \beta_m)$ This reduces parameters and multiplications by the compression factor $(c n^2)/(C_{\rm in} N^2)$, with practical $2\times$–$8\times$ reductions and negligible loss in accuracy (Bhalgat et al., 2020).

3.2 Circulant Convolutional Structures

CircConv enforces block-circulant constraints along the input/output channel axes of each convolutional weight tensor. Each $N\times N$ block is specified by a single generator vector, and the multiplication is performed via batched 1D FFTs: $$\mathcal{Y}(w_2, h_2, c_2) = \sum_{w_1, h_1, j} \text{FFT}(\mathcal{X}(...)) * \text{FFT}(\mathcal{W}') \to \text{iFFT}(\cdot)$$ Parameter and operation counts are reduced by $O(N)$, with up to $8\times$ net compression in ResNet and Wide ResNet models. Backpropagation operates directly on generator vectors (Liao et al., 2019).

3.3 Sparse Fast Convolution

Fine-grained sparse convolution operators, as in FSCNN, utilize custom node-based sparse data structures that extend LIBLINEAR formats to multi-dimensional tensors. This approach skips zero weights, reducing arithmetic but incurring pointer and memory-access overhead. Speedups up to $6\times$ are observed only at ultra-high sparsity ($<5\%$ density); otherwise, structured pruning (coarse-grained) is preferred for hardware compatibility (Ji et al., 2022).

4. Hardware-Adaptive, Scalable, and Irregular-Domain Structures

4.1 FPGA, SOC, and DPRT-Based Architectures

Scalable architectures map 2D convolution to collections of 1D convolutions using the Discrete Periodic Radon Transform (DPRT). For $N\times N$ blocks, forward DPRT is: $$F(m,d) = \sum_{i=0}^{N-1} f(i, (d + mi) \!\!\!\!\mod N)$$ All $N+1$ DPRT directions are computed in parallel, followed by collections of $N+1$ 1D circular convolutions, and reconstruction via inverse DPRT. Hardware design uses $H$ parallel rows and $J$ 1D convolution engines, allowing latency to scale from $O(P)$ to $O(P^2)$. For low-rank kernels, SVD-LU decompositions permit separation into $r$ 1D row and column convolutions, further reducing resources (Carranza et al., 2021).

4.2 Fast Mesh and Irregular-Domain Convolutions

SpiralNet++ defines fast mesh convolution on triangle meshes by precomputing a spiral ordering of each vertex, gathering and concatenating neighbor features, and fusing via a single dense MLP. The computational complexity is $O(N\,l\,F_{\mathrm{in}}\,F_{\mathrm{out}})$ per layer ($l$ is spiral length), outperforming ChebyNet and MoNet in both speed and accuracy for dense correspondence and 3D facial expression tasks (Gong et al., 2019).

5. Complexity, Stability, and Numerical Analysis

Algorithmic complexity reductions are central to fast convolution structures. Winograd minimal filtering achieves $O((m + r - 1)^2)$ multiplications per tile versus $O(m^2 r^2)$ naively (Tong et al., 2021). Symbolic Fourier Convolution outperforms Winograd under low-precision constraints, maintaining lower condition numbers (e.g., $3.3$ vs. $20$) and minimal accuracy degradation under quantization (He et al., 3 Jul 2024). FIR-cascade methodologies maintain unconditional stability even for spectral radii $\geq 1$ and permit arbitrary-structured $A$ matrices as long as the degree $N$ meets a chosen accuracy target (Beylkin, 22 Nov 2024).

Quantization, pruning, and resource constraints further impact the practicality of these methods. Symbolic and structured approaches maintain accuracy at int8-int4 quantization with negligible loss, outperforming traditional FFT and Winograd techniques in low-precision regimes.

6. Hardware Optimization: Tiling, Caching, and Parallelization

Advanced implementations on CPUs exploit shared caches (L3 Fusion) and tile-level fusion to maximize arithmetic intensity and minimize memory bandwidth bottlenecks. For tiled Winograd convolution, operation counts are: $\text{FLOPs per task} = 2 R C C' T^2$ Arithmetic intensity at L3 is $R/2$, and fusion parameters are tuned to saturate compute throughput. Empirical performance on modern CPUs confirms $2\times$–$4\times$ acceleration over vendor-optimized 3-stage pipelines, especially for layers with moderate channel dimensions and large shared caches (Gelashvili et al., 2019).

FPGAs and hardware accelerators exploit scalable FastConv/DPRT methods, with resource–throughput trade-off tables guiding instantiation based on DSP, BRAM, and LUT availability (Carranza et al., 2021). SFC achieves higher DSP efficiency and lower LUT consumption versus Winograd and NTT designs in FPGA synthesis trials (He et al., 3 Jul 2024).

7. Domain-Specific Convolution: Filtering, Fractional Derivatives, and Scientific Computing

Fast convolution quadrature for fractional derivatives employs sum-of-exponential approximations, reducing the complexity of Riemann–Liouville derivative discretization from $O(N^2)$ to $O(N N_p)$ ($N_p$ quadrature points), maintaining first- or second-order time-accuracy without additional regularity assumptions and achieving up to $10\times$ speedups in numerical simulations (Sun et al., 2019).

In scientific computing, far-field smooth approximations split singular convolution kernels into regular and singular integrals, resolving both via FFT and trapezoidal quadrature, and attaining spectral accuracy with $O(N^d \log N)$ complexity and minimum memory (Liu et al., 28 Apr 2025). The Fast Free Memory method leverages descent-only, kernel-independent low-rank compression and on-the-fly NUFFT for generic and oscillatory kernels, scaling linearly in storage and quasi-linearly (or log-squared for oscillatory cases) in compute for billion-scale unknowns (Aussal et al., 2019).

Summary Table: Representative Fast Convolution Structures

Structure/Algorithm	Complexity Reduction	Key Hardware Feature	Stability/Precision	Typical Application Domains
Winograd Minimal Filtering	$O((m+r-1)^2)$ mults/tile	SIMD, tile-based, cache fusion	Sensitive to tile size	CNNs, vision, FIR filtering
Cook–Toom/Toom–Cook	Subquadratic for long convolutions	Modular recursion, polynomial evaluation	Stable (careful interpolation)	Signal processing, cryptography
FFT/Symbolic Fourier Convolution	$O(n \log n)$ for large $n$	Batched FFTs, symbolic transforms	SFC robust to quantization	Deep learning, quantized inference
Structured/Composite Convolution	$2\times$–$8\times$ compression	Sum-pooling + small convolution	Minimal accuracy loss	Model compression in DNNs
CircConv (Circulant)	$O(N \log N)$ per block	1D FFTs, generator parameterization	Stable, flexible initialization	Low-complexity DNNs, mobile inference
FSCNN (Sparse)	$3\times$–$6\times$ if ultra-sparse	Node format, custom inner-products	Pointer overhead limits benefit	Pruned/sparse networks (CPU)
DPRT/SVD-LU (FastConv)	$O(P)$–$O(P^2)$ clock cycles	Parallel 1D engines, SRAM blocks	High-throughput, scalable	FPGA/SOC acceleration, 2D/3D signal, image
FIR-Cascade (Beylkin)	$O(m L + f(m) \log(1/\epsilon))$	Structured $A$, precomputed powers	Unconditionally stable	Long-range dependency, SSMs, HiPPO
FFM/FFT (Scientific Kernels)	$O(N \log N)$–$O(N \log^2 N)$	Octree, ACA, NUFFT	Kernel-independent	Massive integral equations, BIEs, field solve

Fast convolution structures are central to contemporary computational practice in both algorithmic and hardware contexts. Ongoing research advances the interplay among algebraic minimality, hardware efficiency, precision/stability constraints, and domain-specific requirements, yielding a rich toolkit for efficient implementation in wide-ranging signal, learning, and scientific applications.