Adjoint Inversion Reveals Holographic Superposition and Destructive Interference in CNN Classifiers

Interpreting a neural network requires understanding what its internal features extract from a particular input. Feature inversion seeks to express a selected feature in the input domain, but canonical iterative methods search for an input whose re-encoded representation matches the target. Because many inputs can satisfy this constraint, target matching alone does not specify the inverse associated with the sample that generated the feature. We formulate source-grounded feature inversion by conditioning the inverse on the source-local network geometry at the target-generating input. At each boundary of the computational DAG, backpropagation provides the correct reverse dependencies but transports an adjoint signal rather than an upstream-state estimate. We locally repair this signal with a closed-form matrix Wiener map from a mean-seed VJP to the upstream state, followed by a second Wiener map for the JVP forward-consistency residual, and compose the repaired states through the same DAG in one finite reverse pass. One calibrated zero-intercept map family supports new inputs, depths, channels, and channel groups across diverse CNN and Transformer architectures, tensor components, and visual distributions without query-specific optimisation. Matched target and source controls verify that each inverse depends on the selected feature and the local operators of the sample being explained, rather than a target-independent image template. Prediction-conditioned feature atlases align these visualisations with independent interventions on the corresponding internal features. Together, source-grounded feature inversion opens the model's hidden feature hierarchy to inspection at the level of individual layers and channels, linking what the network extracts from an input to the internal evidence that shapes its decision.

Modern image classifiers widely adopt global average pooling (GAP) followed by a linear classification head. This linearity ensures that the image-level logits equal the average of logits obtained by applying the classification head pointwise to the feature grid prior to GAP. Consequently, standard classifiers may inherently retain spatial class evidence that remains recoverable even when the image-level prediction is incorrect. This structure naturally suggests a multiple-instance learning (MIL) interpretation, where an image is viewed as a bag of spatial instances. Within this formulation, we demonstrate that standard classifiers trained with a single label per image can still learn the intended classification task in multi-object scenes. We further exploit this property to decompose image-level logits into a prediction grid, providing a post-hoc diagnostic to extract spatial class evidence that GAP otherwise obscures. Our systematic evaluation reveals that off-the-shelf models consistently recover the ground-truth class within foreground regions. The MIL interpretation further suggests that common classifier failures reflect known limitations of mean aggregation.

In this paper, we attempt to enhance the theoretical understanding of convolutional neural networks (CNNs) as feature extractors in classification tasks by analyzing them through the lens of Cover's function-counting theory. Specifically, our focus lies on the notion of separation capacity, a combinatorial quantity derived from counting the number of realizable dichotomies (i.e., binary label assignments). Our contributions are threefold. First, we extend Cover's framework by establishing a conceptually insightful and practically useful formulation for the separation capacity. Second, leveraging this formulation, we identify the factors governing the separation capacity of feature extractors that employ a specific CNN architecture, so-called scattering networks, in terms of their network building blocks. Third, we provide practical insights for scattering network design.