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Rewrite the Stars

Xu Ma, Xiyang Dai, Yue Bai, Yizhou Wang, Yun Fu

arXiv:2403.19967v1cs.CV

TL;DR

The paper addresses the limited foundational understanding of why element-wise multiplication performs well in network design. It analyzes the star operation as a kernel-like transformation into implicit high-dimensional nonlinear spaces and introduces StarNet as a compact implementation. StarNet reports strong efficiency and performance, including a 0.9% top-1 advantage over EdgeViT-XS while running faster on several platforms.

  • Problem

    The foundational rationale for the star operation’s strong performance and effective use in networks remains insufficiently examined despite intuitive explanations and growing adoption.

  • Method

    The paper rewrites and analyzes star operations as implicit high-dimensional nonlinear feature transformations, then validates the analysis with the compact proof-of-concept network StarNet.

  • Results

    StarNet-S4 surpasses EdgeViT-XS by 0.9% top-1 accuracy on ImageNet-1K validation while running 3× faster on iPhone13 and CPU and 2× faster on GPU.

  • Takeaways & Limitations

    Star operations provide a practical route to efficient network design by implicitly handling extremely high-dimensional features within compact feature spaces.

  • Takeaways & Limitations

    The analysis focuses on a one-output-channel transformation with a single-element input, though the authors state it extends to multiple channels and feature elements; StarNet also deliberately omits sophisticated design enhancements.

Abstract

from arXiv · show

Recent studies have drawn attention to the untapped potential of the "star operation" (element-wise multiplication) in network design. While intuitive explanations abound, the foundational rationale behind its application remains largely unexplored. Our study attempts to reveal the star operation's ability to map inputs into high-dimensional, non-linear feature spaces -- akin to kernel tricks -- without widening the network. We further introduce StarNet, a simple yet powerful prototype, demonstrating impressive performance and low latency under compact network structure and efficient budget. Like stars in the sky, the star operation appears unremarkable but holds a vast universe of potential. Our work encourages further exploration across tasks, with codes available at https://github.com/ma-xu/Rewrite-the-Stars.

1. Introduction

The paper investigates why element-wise multiplication can outperform summation in network blocks and argues that the star operation creates high-dimensional, nonlinear representations without widening the network. It uses this insight to motivate StarNet, a compact efficient network with strong reported accuracy and latency.

  • Motivation: Element-wise multiplication consistently outperforms summation in the paper’s DemoNet comparison, particularly at narrower widths.The comparison keeps other factors constant while varying the operation used to combine two branches.
  • Core analysis: The star operation maps inputs into an exceedingly high-dimensional, nonlinear feature space through feature multiplication across distinct channels.The authors compare this behavior to polynomial kernel functions rather than conventional width expansion.
  • Core analysis: Stacking star-operation layers produces exponential growth in implicit dimensional complexity while retaining a compact feature space.The paper presents this as the basis for achieving very large effective dimensions without increasing channel width.
  • StarNet: StarNet is a straightforward proof-of-concept efficient network that avoids sophisticated designs and extensive hyper-parameter tuning.Its design is intended to isolate and capitalize on the star operation’s contribution.
  • StarNet: StarNet-S4 exceeds EdgeViT-XS by 0.9% top-1 accuracy on ImageNet-1K validation while running 3× faster on iPhone13 and CPU and 2× faster on GPU.The authors present this comparison as empirical validation of the star operation’s practical value.
  • Future directions: The paper identifies further unexplored directions, including learning without activations and refining operations within implicit dimensions.The authors frame the analysis as a guide for more systematic star-operation network design.

2. Related Work

Related work studies element-wise multiplication, high-dimensional nonlinear representations, and efficient network design from separate perspectives. The paper connects these areas by treating the star operation as a compact route to implicit high-dimensional features.

  • Element-wise multiplication: Prior network designs use element-wise multiplication for feature aggregation and report advantages over summation.Examples include FocalNet, VAN, Conv2Former, and HorNet, with explanations involving modulation, high-order features, and attention.
  • High-dimensional nonlinear transformation: Deep networks typically obtain high-dimensional nonlinear features through linear projection followed by activation functions.This contrasts with kernel methods that combine dimensionality and nonlinearity through kernel constructions.
  • High-dimensional nonlinear transformation: Polynomial kernels can project inputs into high-dimensional nonlinear spaces, while Gaussian kernels can yield infinite-dimensional spaces through Taylor expansion.The paper uses these kernel behaviors as a conceptual reference for understanding star operations.
  • Paper’s connection: The paper demonstrates that star operations can obtain high-dimensional nonlinear feature spaces from low-dimensional inputs, akin to kernel tricks.A visualization experiment further illustrates the connection to polynomial kernel functions.
  • Efficient networks: Unlike prior efficiency techniques, star operations implicitly process extremely high-dimensional features while computing in a low-dimensional space.The authors argue that this property makes the operation suitable for efficient networks and allows straightforward designs to outperform heavily handcrafted models.

3. Rewrite the Stars

The star operation rewrites element-wise multiplication into many implicit nonlinear feature dimensions, whose complexity grows exponentially when layers are stacked. Experiments associate this high-dimensional behavior with improved performance over summation and motivate the compact StarNet design.

  • 3.1. Star Operation in One layer: A single star layer expands a d-dimensional computation into (d+2)(d+1)/2 distinct implicit items, many of which are nonlinear in the input.The operation increases implicit feature dimensions without additional single-layer computational overhead.
  • 3.3. Special Cases: Star operations resemble polynomial kernel functions because they pairwise multiply features across channels rather than widening the network.The paper distinguishes this mechanism from conventional width-based expansion and discusses special cases with fewer implicit dimensions.
  • 3.2. Generalized to multiple layers: Stacking l star layers increases implicit dimensionality exponentially, reaching approximately 90^1024 dimensions for a 10-layer, width-128 network.The paper describes this scale as reasonably approximating infinite dimensions.
  • 3.5. Open Discussions & Broader Impacts: The paper proposes StarNet as a concise, efficient network whose star blocks are integrated into a hierarchical architecture without sophisticated designs or carefully tuned hyper-parameters.Its architecture uses convolutional down-sampling and repeated star blocks for feature extraction.
  • 3.4.1 Empirical superiority of star operation: Across varying widths and depths, DemoNet with star operation consistently outperforms the corresponding summation models, although width-related gains diminish as width increases.The diminishing gain is not observed when varying depth, which the authors interpret as evidence of intrinsic dimensional expansion.
  • 3.4.2 Decision Boundary comparison: The star-based network produces a more precise decision boundary than summation and closely matches a polynomial-kernel SVM rather than a Gaussian-kernel SVM.Both compared network blocks include activation functions, so the reported boundary difference is attributed primarily to star operation’s high dimensionality.

4. Proof-of-Concept: StarNet

StarNet is a minimalist, efficient proof-of-concept network designed to test the utility of star operations. It achieves strong ImageNet accuracy and latency results, while ablations show that star operations and activation placement materially affect performance.

  • Architecture: StarNet uses a four-stage hierarchical architecture with convolutional down-sampling, modified demo blocks, batch normalization, and depth-wise convolutions.Its channel expansion factor is consistently set at 4.
  • Design philosophy: StarNet deliberately avoids sophisticated design elements and extensive hyper-parameter tuning to emphasize the role of star operations.The training recipe follows a standard DeiT-based setup for fair comparison.
  • ImageNet results: 73.5% top-1 accuracy is achieved in 0.7 seconds on iPhone 13, exceeding MobileOne-S0 by 2.1% at the same latency.The comparison is 73.5% versus 71.4%.
  • ImageNet results: At a 1G FLOPs budget, StarNet surpasses MobileOne-S2 by 1.0% and EdgeViT-XS by 0.9% while running three times faster than EdgeViT-XS.The reported EdgeViT-XS latency is 1.0 ms versus 3.5 ms.
  • Ablations: Replacing star operations with summation reduces accuracy, including a drop from 78.4% to 74.4% when only one branch is transformed.The ablation also underscores the importance of transforming both branches.
  • Ablations: Removing all star operations lowers accuracy by 3.1%, with the largest reported stage-wise gains occurring in the final two stages at 1.6% each.The text attributes limited early-stage impact to narrow width and zeroed features after ReLU6.
  • Latency: Star operations add no latency relative to summation on GPU and iPhone, while CPU summation is slightly faster.For StarNet-S4 on CPU, the reported latencies are 8.4 ms for summation and 9.4 ms for star operations.
  • Activation placement: Activating only one branch produces the highest accuracy at 78.4%, while removing activations except in the stem reduces accuracy by 2.8% to 75.6%.The activation-placement study compares four activation strategies.

5. Conclusion

The paper concludes that star operations derive their strong representational capacity from implicitly high-dimensional spaces and resemble polynomial kernel functions. This interpretation is supported through empirical, theoretical, and visual analyses.

  • Conclusion: Star operations provide strong representational capacity through implicitly high-dimensional spaces, in a way that mirrors polynomial kernel functions.The paper states that this account goes beyond earlier intuitive explanations.
  • Conclusion: The analysis is validated through empirical, theoretical, and visual methods.

Supplementary Material

The supplementary material provides implementation details, reproducibility code, training settings, and additional analyses for DemoNet, 2D-point experiments, and StarNet.

  • Supplementary scope: Supplementary sections document DemoNet implementation, the 2D-point decision-boundary network, StarNet architecture, and exploratory ultra-compact-model studies.
  • Implementation: The DemoNet code supports star or sum aggregation by splitting features into two branches and applying the selected operation.The 2D-point version uses a simplified linear architecture with two output classes.
  • Implementation: The image-based DemoNet applies depth-wise convolution, GELU, and either summation or element-wise multiplication inside repeated blocks.
  • StarNet reproduction: The StarNet supplementary material includes a separate reproduction file and variant-specific training settings.StarNet uses fewer augmentations and omits EMA and learnable layer-scale techniques.
  • Training: DemoNet variants use one consistent training recipe to enable fair comparisons rather than maximize each model's performance.The supplementary text notes that specialized recipes could improve performance for different model sizes.

A.3. Latency Benchmark Settings

Latency benchmarks evaluate models under batch-size-one conditions across CPU, GPU, and iPhone hardware, with repeated runs used to reduce measurement noise. The supplementary figures also include operation comparisons and decision-boundary analyses.

  • Benchmark protocol: CPU and GPU latency is measured after 50 warm-up iterations and averaged over 500 iterations at batch size one on the same device.
  • Benchmark protocol: iPhone latency follows MobileOne's CoreML benchmark settings, reporting the final of three runs to account for faster initial execution.Typical remaining latency variation is reported as less than 0.05 ms.
  • Additional analyses: Figure 5 reports four-run results comparing sum and star operations, while Figure 6 presents more comprehensive decision-boundary analyses.

B. Decision Boundary Visualization

The decision-boundary analyses show that star operations provide stronger and more robust representations than summation operations, with the largest benefits when used early in the network.

  • Star operations surpass summation operations in representational capability and exhibit greater robustness across runs.Additional runs without fixed seeds show minimal decision-boundary variance for star operations.
  • Using star operations in the network’s early blocks yields the most significant benefits in mixed-operation experiments.The study evaluates different combinations of summation and star operations across four blocks.
  • Hyper-parameter changes alter SVM visualizations, but the differences are not significant because polynomial and RBF kernels are intrinsically different.

C. Exploring Extremely Small Models

The study evaluates StarNet variants at approximately 0.5M, 1.0M, and 1.5M parameters by tuning architectural widths, depths, and MLP expansion ratios. The ultra-compact models remain promising, with StarNet-150 outperforming a larger, longer-trained MobileNetv2-050 on accuracy and mobile speed.

  • The extremely small StarNet variants target approximately 0.5M, 1.0M, and 1.5M parameters.
  • For ultra-compact variants, the study tunes block number, base embedding width, and MLP expansion ratio.
  • StarNet-150 outperforms MobileNetv2-050 in both top-1 accuracy and mobile-device speed despite using fewer parameters.StarNet-150 has 1.56M parameters versus MobileNetv2-050’s 1.97M, while MobileNetv2-050 is trained for 450 rather than 300 epochs.
  • The block-design analysis also controls parameter counts and FLOPs across variants through expansion-ratio adjustments for fair comparison.

D.1. Analysis on removing all activations

Removing all activation functions from star-operation networks usually causes a slight performance drop, but some depths maintain or improve performance, suggesting that star operations may supply substantial non-linearity. Among tested activations, ReLU6 performs best for StarNet-S4.

  • Removing all activations usually slightly reduces DemoNet performance, but depths 14, 16, 18, and 22 can achieve similar or better results.
  • The activation-removal analysis varies DemoNet width at fixed depth and depth at fixed width.
  • The results suggest that the star operation may inherently provide sufficient non-linearity to replace typical activation layers.The authors identify this as a direction for more thorough investigation.
  • StarNet adopts ReLU6 following MobileNetv2, and StarNet-S4 performs best with ReLU6 among the tested activation functions.

E. Exploring Block Designs

Five StarNet block implementations are compared under matched parameter and FLOP budgets. Standard and special-case-III designs perform strongly, while special-case-II designs perform worse, indicating that block formulation matters beyond computational complexity alone.

  • The five variants represent a standard star operation, two special-case-II instantiations, and two special-case-III implementations.
  • Block I, Block IV, and Block V show strong performance, whereas Block II and Block III perform worse.All variants are evaluated using the StarNet-S4 architecture.
  • All block variants are matched for parameter count and FLOPs by varying the expansion ratio, enabling fair comparison.

F. More Latency Analysis on StarNet

The section examines StarNet’s CPU-latency trade-off and tests latency stability across four iPhone devices. The supplementary results report average latency and variance to identify stable and unstable models.

  • Mobile device latency robustness: Latency stability is evaluated on iPhone12, iPhone12 Pro Max, iPhone13, and iPhone14.
  • CPU latency visualization: CPU latency is plotted against ImageNet accuracy for StarNet and other models, excluding low-accuracy or high-latency models for visualization.
  • Mobile device latency robustness: Table 19 reports average latency and variance, marking the most stable models in green and the most unstable models in red.
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