A recent study on the interpretability of real-valued convolutional neural networks (CNNs) {Stankovic_Mandic_2023CNN} has revealed a direct and physically meaningful link with the task of finding features in data through matched filters. However, applying this paradigm to illuminate the interpretability of complex-valued CNNs meets a formidable obstacle: the extension of matched filtering to a general class of noncircular complex-valued data, referred to here as the widely linear matched filter (WLMF), has been only implicit in the literature. To this end, to establish the interpretability of the operation of complex-valued CNNs, we introduce a general WLMF paradigm, provide its solution and undertake analysis of its performance. For rigor, our WLMF solution is derived without imposing any assumption on the probability density of noise. The theoretical advantages of the WLMF over its standard strictly linear counterpart (SLMF) are provided in terms of their output signal-to-noise-ratios (SNRs), with WLMF consistently exhibiting enhanced SNR. Moreover, the lower bound on the SNR gain of WLMF is derived, together with condition to attain this bound. This serves to revisit the convolution-activation-pooling chain in complex-valued CNNs through the lens of matched filtering, which reveals the potential of WLMFs to provide physical interpretability and enhance explainability of general complex-valued CNNs. Simulations demonstrate the agreement between the theoretical and numerical results.

翻译：一项关于实值卷积神经网络（CNN）可解释性的近期研究{Stankovic_Mandic_2023CNN}揭示了其与通过匹配滤波器在数据中寻找特征这一任务之间直接且具有物理意义的联系。然而，将该范式应用于阐明复值CNN的可解释性时面临重大障碍：匹配滤波向一般非圆复值数据的扩展——本文称之为宽线性匹配滤波器（WLMF）——在文献中仅为隐含存在。为此，为确立复值CNN运算的可解释性，我们引入通用WLMF范式，给出其解并分析其性能。为保证严谨性，我们的WLMF解在未对噪声概率密度作任何假设的条件下推导得出。从输出信噪比（SNR）的角度阐明了WLMF相较于标准严格线性匹配滤波器（SLMF）的理论优势，WLMF始终展现出增强的SNR。此外，推导了WLMF的SNR增益下界及其达到该界的条件。这使我们得以通过匹配滤波的视角重新审视复值CNN中的卷积-激活-池化链，揭示了WLMF为通用复值CNN提供物理可解释性并增强可解释性的潜力。仿真结果验证了理论与数值结果的一致性。