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.

We make progress towards characterizing the algebraic matroid of the determinantal variety defined by the minors of fixed size of a matrix of variables. Our main result is a novel family of base sets of the matroid, which characterizes the matroid in special cases. Our approach relies on the combinatorial notion of relaxed supports of linkage matching fields that we introduce, our interpretation of the problem of completing a matrix of bounded rank from a subset of its entries as a linear section problem on the Grassmannian, and a connection that we draw with a class of local coordinates on the Grassmannian described by Sturmfels and Zelevinsky in 1993.