UltraLIF：通过超离散化和最大-加法代数实现完全可微的脉冲神经网络
UltraLIF: Fully Differentiable Spiking Neural Networks via Ultradiscretization and Max-Plus Algebra
AI总结：
UltraLIF通过超离散化和最大-加法代数实现完全可微的脉冲神经网络，提升神经形态和时间数据集的性能。
摘要：
脉冲神经网络（SNNs）提供了高效且生物合理的计算，但受到非可微的脉冲生成的限制，需要依赖启发式替代梯度。本文介绍了UltraLIF，一个原理性的框架，用超离散化替代替代梯度，超离散化是来自热带几何的数学形式化，提供离散动态的连续松弛。核心见解是超离散化背后的max-plus半环自然地建模神经元阈值动态：log-sum-exp函数作为可微的软最大值，随着可学习的温度参数ε→0，收敛到硬阈值化。两种神经元模型来自不同的动力学系统：UltraLIF来自LIF常微分方程（时间动态）和UltraDLIF来自扩散方程，建模神经元群体之间的间隙连接（空间动态）。两者均产生可完全微分的SNNs，可通过标准反向传播训练，无前向-后向不匹配。理论分析证明在点wise收敛到经典LIF动态，具有定量误差界和有界非消失梯度。在六个基准测试上进行实验，涵盖静态图像、神经形态视觉和音频，证明优于替代梯度基线，收益在单步（T=1）设置上最显著，在神经形态和时间数据集上。可选的稀疏性惩罚可实现显著的能耗减少，同时保持竞争性的准确性。
英文摘要：
Spiking Neural Networks (SNNs) offer energy-efficient, biologically plausible computation but suffer from non-differentiable spike generation, necessitating reliance on heuristic surrogate gradients. This paper introduces UltraLIF, a principled framework that replaces surrogate gradients with ultradiscretization, a mathematical formalism from tropical geometry providing continuous relaxations of discrete dynamics. The central insight is that the max-plus semiring underlying ultradiscretization naturally models neural threshold dynamics: the log-sum-exp function serves as a differentiable soft-maximum that converges to hard thresholding as a learnable temperature parameter $\eps \to 0$. Two neuron models are derived from distinct dynamical systems: UltraLIF from the LIF ordinary differential equation (temporal dynamics) and UltraDLIF from the diffusion equation modeling gap junction coupling across neuronal populations (spatial dynamics). Both yield fully differentiable SNNs trainable via standard backpropagation with no forward-backward mismatch. Theoretical analysis establishes pointwise convergence to classical LIF dynamics with quantitative error bounds and bounded non-vanishing gradients. Experiments on six benchmarks spanning static images, neuromorphic vision, and audio demonstrate improvements over surrogate gradient baselines, with gains most pronounced in single-timestep ($T{=}1$) settings on neuromorphic and temporal datasets. An optional sparsity penalty enables significant energy reduction while maintaining competitive accuracy.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
