  While recently many designs have been proposed to improve the model
efficiency of convolutional neural networks (CNNs) on a fixed resource budget,
theoretical understanding of these designs is still conspicuously lacking. This
paper aims to provide a new framework for answering the question: Is there
still any remaining model redundancy in a compressed CNN? We begin by
developing a general statistical formulation of CNNs and compressed CNNs via
the tensor decomposition, such that the weights across layers can be summarized
into a single tensor. Then, through a rigorous sample complexity analysis, we
reveal an important discrepancy between the derived sample complexity and the
naive parameter counting, which serves as a direct indicator of the model
redundancy. Motivated by this finding, we introduce a new model redundancy
measure for compressed CNNs, called the $K/R$ ratio, which further allows for
nonlinear activations. The usefulness of this new measure is supported by
ablation studies on popular block designs and datasets.
