Zero Convolutions in Deep Learning

• Zero Convolutions are computational techniques that skip operations on zero-valued elements in CNNs and FFT processes, thereby reducing unnecessary FLOPs and memory usage.
• They implement strategies like filter trimming, sparse-to-dense transformation, and implicit padding to optimize operations and enhance hardware performance.
• Empirical benchmarks on models like ResNet18 and VGG16 demonstrate speed-ups up to 1.81× while maintaining accuracy, showcasing the practical benefits in deep learning applications.

Zero convolutions are algorithmic and architectural approaches designed to eliminate computations involving zero-valued elements in convolutional operations. In the context of deep learning and signal processing, such methods target both spatial convolutions in neural networks and FFT-based spectral convolutions to reduce computational complexity, memory footprint, and hardware inefficiency. Key strategies include skipping zero multiplications, trimming filters to minimal support, partitioning sparse convolution patterns into multiple dense subproblems, and employing implicit padding to avoid explicit storage of zero blocks.

1. Foundational Problem: Zero Operations in Convolutional Workloads

Convolutional neural networks (CNNs) and spectral algorithms routinely introduce zeros into their computation—either as explicit padding for boundary conditions, stride/dilation expansion, or aliasing prevention in FFT-based convolutions. The direct implication is redundant arithmetic and excess memory usage. For instance, in standard 2D convolution, zero-padding of inputs results in multiply-accumulate operations where the kernel interacts with zero-valued regions near the boundaries, directly increasing FLOP count without contributing to the output. Similarly, in FFT-based pseudospatial convolutions, zero-padding for dealiasing requires unnecessary allocation and transformation of zero elements, artificially inflating memory bandwidth and computation time (Bowman et al., 2010).

2. Skipping Zero Computations: Algorithmic Techniques

A range of zero-convolution approaches have been developed for reducing or outright eliminating these redundant computations in both direct and FFT-based convolution:

• Filter Trimming (ConvV2, C-K-S): For forward convolutions, every output channel's filter is statically trimmed to its minimal nonzero support. The trimmed kernel $W'_{f,c}$ operates only over the necessary indices, and the convolution is formulated to maintain correct spatial offsets. This leads to a direct computational savings $\Delta$FLOPs proportional to the difference between total kernel elements $K_h K_w$ and the average support $|S_{\text{avg}}|$:

$$\Delta \text{FLOPs} = 2 N F H' W' C (K_h K_w - |S_{\text{avg}}|)$$

(Zhang et al., 2023).

• Sparse-to-Dense Transformation (KS-deconv, Sk-dilated): Deconvolution and dilated convolutions traditionally insert zeros in feature-maps or kernels, respectively. The KS-deconv method decomposes the kernel into $s_h s_w$ smaller, dense sub-kernels and computes a series of standard convolutions, followed by scattered composition into the gradient buffer. The Sk-dilated operation reformulates dilated convolutions by leveraging a "leaping" access pattern on the input feature-map, such that dense patches are convolved with nonzero kernel entries (Zhang et al., 2023).
• Implicit Padding in Spectral Convolution (FFT): Instead of explicit zero-padding, the implicit padding method decomposes the FFT computation so only physical, nonzero modes are ever touched. This is achieved through radix-$q$ partitioning and use of small scratch arrays, reducing memory and computation time, particularly for multidimensional and higher-order convolutions (Bowman et al., 2010).

3. Implementation: Practical Strategies and Hardware Considerations

Implementation of zero convolution methods is characterized by advanced tensor manipulation and hardware-specific optimizations:

• Memory Layout Transforms: Kernels and inputs are transposed to maximize contiguous access for vectorized multiply-accumulate instructions (SIMD), particularly critical on GPU architectures. In C-K-S, filter trimming and kernel-splitting are performed offline or at model load time, allowing main inference/execution kernels to operate only on dense, valid regions (Zhang et al., 2023).
• Micro-Kernel Design: Zero-memory-overhead convolution eliminates temporary buffers (e.g., $X^{col}$ in im2col+GEMM) by restructuring the inner loops to operate directly on register-blocked spatial tiles, maximizing cache reuse and minimizing bandwidth waste (Zhang et al., 2018).
• Shared Memory Buffering, Register Optimizations: For GPU execution, input patches are loaded into shared memory with double buffering, and threads are launched to process output tiles, further reducing global DRAM latency and maximizing throughput (Zhang et al., 2023).
• Branch Elimination and Index Precomputation: Algorithms avoid runtime dynamic branching by precomputing valid index ranges, storing small lookup tables for index mappings, and eliminating zero-padding checks in the innermost execution kernels.

4. Complexity Analysis and Theoretical Speed-ups

Zero convolution frameworks lead to substantive reductions in computational complexity:

• Forward Convolution: When filtering over $1 \times 1$ core in a $3 \times 3$ kernel, ConvV2 yields a speed-up of $9\times$ relative to standard convolution. General speed-ups scale as $\frac{K_h K_w}{|S_{\text{avg}}|}$ (Zhang et al., 2023).
• Deconvolution and Dilated Convolution: The KS-deconv and Sk-dilated approaches yield theoretical speed-ups proportional to stride or dilation values $(s_h s_w$ or $r_h r_w)$; for example, $2\times2$ deconvolution on $3\times3$ kernels exhibit up to $2.9\times$ empirical speed-up (Zhang et al., 2023).
• FFT-based Convolution: Implicit padding achieves a reduction in memory usage and runtime by a factor of up to $2$ in two dimensions and $4$ in three dimensions, asymptotically approaching optimal usage for large arrays (Bowman et al., 2010).

5. Experimental Results and Benchmarks

Empirical studies demonstrate that zero convolution strategies yield significant improvements in speed and efficiency without loss of accuracy:

Network	PyTorch Epoch (s)	Zero Conv Epoch (s)	Speedup
ResNet18 (CIFAR)	7.31	5.82	1.26×
ResNet34 (CIFAR)	14.15	11.53	1.23×
VGG16 (CIFAR)	11.04	9.71	1.14×
ResNet18 (ImageNet1000, RTX4090)	1202	665	1.81×
ResNet34 (ImageNet1000, RTX4090)	2219	1264	1.76×

Convergence curves between zero convolution implementations (C-K-S family) and standard PyTorch/cuDNN are essentially identical, with final top-1/top-5 accuracies matching to within $0.2\%$ (Zhang et al., 2023).

Zero-memory-overhead direct convolutions deliver throughput improvements of $10\%$–$400\%$ compared to im2col+SGEMM, with especially strong per-core scaling and reduced efficiency drop-off under multi-threaded execution (Zhang et al., 2018).

6. Limitations and Future Directions

Despite substantial efficiency gains, zero convolutions exhibit specific boundaries:

• Indexing Overhead: When kernel sparsity is low (i.e., nearly all entries are nonzero), the benefit of filter trimming is negated by the indexing overhead, potentially making standard convolution faster.
• Temporary Memory for Split Kernels: KS-deconv requires extra memory for split kernels, which can limit maximal batch size on devices with constrained memory.
• Assumptions of Uniform Stride/Dilation: Both KS-deconv and Sk-dilated methodologies currently presume uniform stride/dilation values; architectures with dynamic patterns require further generalization.
• Applicability to 3D/Volumetric Convolution: Extension to 3D is straightforward in principle, splitting kernels into subkernels for each spatial dimension, but optimal realization requires additional tuning.
• Hardware-Specific Optimizations: For CPU/FPGA platforms, sparse dot-product routines and reordering can provide similar zero-skipping benefits, but optimal implementation depends on architecture-specific features.

7. Summary and Significance

Zero convolution techniques systematically eliminate computational and memory inefficiencies arising from zero-valued regions in both direct CNN and FFT-based convolutional workloads. By employing filter-trimming, sparse-to-dense splitting, and implicit padding, these frameworks significantly reduce FLOPs, improve hardware utilization, and maintain accuracy. Their integration into deep learning frameworks and signal-processing toolchains offers empirically validated performance gains—even for large-scale settings typical of image classification and scientific computing, as evidenced on CIFAR and ImageNet datasets (Zhang et al., 2023, Zhang et al., 2018, Bowman et al., 2010). As model architectures and hardware platforms continue to evolve, further research in zero convolution paradigms will likely drive continued improvements in efficiency and scalability.