[论文解读] A theoretical argument for complex-valued convolutional networks

Joan Bruna, Soumith Chintala+4 more|arXiv (Cornell University)|Mar 11, 2015

Image and Signal Denoising Methods参考文献 18被引用 3

一句话总结

本文提出复值卷积神经网络（复值卷积网络）作为计算多尺度窗函数功率谱、绝对谱及非线性多小波包的端到端数据驱动框架。通过应用复值卷积、绝对值运算和局部平均，该模型生成层次化、自适应的表征，其结果与时间序列和图像数据中随机过程的类小波变换完全对应。

ABSTRACT

A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with several complex-valued vectors followed by (2) taking the absolute value of every entry of the resulting vectors followed by (3) local averaging. For processing real-valued random vectors, complex-valued convnets can be viewed as “data-driven multiscale windowed power spectra,” “data-driven multiscale windowed absolute spectra,” “data-driven multiwavelet absolute values,” or (in their most general configuration) “data-driven nonlinear multiwavelet packets.” Indeed, complex-valued convnets can calculate multiscale windowed spectra when the convnet filters are windowed complex-valued exponentials. Standard real-valued convnets, using rectified linear units (ReLUs), sigmoidal (for example, logistic or tanh) nonlinearities, max. pooling, etc., do not obviously exhibit the same exact correspondence with data-driven wavelets (whereas for complex-valued convnets, the correspondence is much more than just a vague analogy). This note develops “data-driven multiscale windowed spectra” for certain stochastic processes that are common in the modeling of time series (such as audio) and natural images (including patterns and textures). We motivate the construction of such multiscale spectra in the form of “local averages of multiwavelet absolute values” or (in the most general configuration) “nonlinear multiwavelet packets” and connect these to certain “complex-valued convolutional networks.” A textbook treatment of all concepts and terms used above and below is given by [12]. Further information is available in the original work of [7], [15], [5], [4], [19], [16], [9], [20], and [18], for example. The work of [8], [13], [17], [2], and [3] also develops complex-valued convolutional networks (convnets). Renormalization group theory and its connection to convnets is discussed by [14]; this connection is incredibly insightful, though we leave further discussion to the cited work. Our exposition relies on nothing but the basic signal processing treated by [12]. For simplicity, we first limit consideration to the special case of a doubly infinite sequence of nonnegative random variables Xk, where k ranges over the integers. This input data will be the result of convolving an unmeasured independent and identically distributed (i.i.d.) sequence Zk, where k ranges over the integers, with an unknown sequence of real numbers fk, where k ranges over the integers (this latter sequence is known as a “filter,” whereas the i.i.d. sequence is known as “white noise”):

研究动机与目标

• 建立复值卷积网络作为数据驱动多尺度窗函数谱表征的理论基础。
• 证明复值卷积网络自然地源于将信号处理原理应用于白噪声与未知滤波器卷积的随机过程。
• 阐明复值卷积网络与基于小波的变换之间的对应关系，特别是多小波绝对值与非线性多小波包。
• 将复值卷积网络与标准实值卷积网络（如ReLU、最大池化）进行对比，展示其在理论上与小波理论的更优契合度。
• 仅使用基础信号处理知识，提供一个自包含的阐述，无需超出标准教科书的高级先决知识。

提出的方法

• 该模型处理由独立同分布的白噪声与未知实值滤波器卷积生成的双无限非负实值随机变量序列。
• 其应用由三个操作组成的复合结构：(1) 与复值滤波器进行卷积，(2) 元素级绝对值运算，(3) 局部平均。
• 复值滤波器被解释为加窗复指数函数，从而实现多尺度谱分量的提取。
• 所得架构被证明可计算数据驱动的多尺度窗函数功率谱或绝对谱，具体取决于滤波器配置。
• 当滤波器不限于简单窗函数复指数时，该方法可推广至非线性多小波包。
• 该框架建立在信号处理理论基础上，虽暗示与重整化群理论的联系，但未作深入展开。

实验结果

研究问题

• RQ1 复值卷积结合绝对值与池化操作，如何生成类似于多尺度窗函数谱的表征？
• RQ2 复值卷积网络与小波变换之间存在何种理论关系，特别是与多小波绝对值的关系？
• RQ3 为何复值卷积网络相较于使用ReLU或最大池化的标准实值卷积网络，与小波理论具有更精确的对应关系？
• RQ4 复值卷积网络在何种意义上自然地成为过滤白噪声等随机过程的表征？
• RQ5 卷积、绝对值与局部平均的组合如何生成层次化、数据驱动的谱特征？

主要发现

• 当滤波器为加窗复指数时，复值卷积网络可生成数据驱动的多尺度窗函数功率谱。
• 该架构计算多小波绝对值的局部平均，提供一种非线性、自适应的谱表征。
• 在最一般配置下，该框架可推广至非线性多小波包，支持更丰富的特征层次。
• 复值卷积网络与类小波变换之间的对应关系是精确且理论严谨的，而非仅类比关系。
• 使用ReLU或最大池化的标准实值卷积网络缺乏与小波理论之间这种直接且精确的联系。
• 该模型仅基于基础信号处理原理推导得出，无需超出教科书水平的高级理论工具。