In this section, we compare the symmetric solutions found in erf [2] and ReLU networks [5] to our one-neuron solution (n =1). The main difference is that both earlier studies constrain the search space to the symmetric subspace whereas we first prove that the non-trivial critical points are contained in this subspace in Theorem 5.1 for a broad class of activation functions, including erf and ReLU. Solving the low-dimensional loss, we recover the same solution for ReLU and erf as in [2, 5] for unit-orthonormal teachers.

Any continuous function f can be approximated arbitrarily well by a neural network with sufficiently many neurons k. We consider the case when f itself is a neural network with one hidden layer and k neurons. Approximating f with a neural network with n < k neurons can thus be seen as fitting an under-parameterized "student" network with nneurons to a "teacher" network with k neurons. As the student has fewer neurons than the teacher, it is unclear, whether each of the n student neurons should copy one of the teacher neurons or rather average a group of teacher neurons. For shallow neural networks with erf activation function and for the standard Gaussian input distribution, we prove that "copy-average" configurations are critical points if the teacher's incoming vectors are orthonormal and its outgoing weights are unitary. Moreover, the optimum among such configurations is reached when n 1student neurons each copy one teacher neuron and the n-th student neuron averages the remaining k n+1 teacher neurons. For the student network with n = 1 neuron, we provide additionally a closed-form solution of the non-trivial critical point(s) for commonly used activation functions through solving an equivalent constrained optimization problem. Empirically, we find for the erf activation function that gradient flow converges either to the optimal copy-average critical point or to another point where each student neuron approximately copies a different teacher neuron. Finally, we find similar results for the ReLU activation function, suggesting that the optimal solution of underparameterized networks has a universal structure.

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Global information is essential for dense prediction problems, whose goal is to compute adiscrete or continuous label for each pixel in the images. Traditional convolutional layers in neural networks, initially designed for image classification, are restrictive in these problems since the filter size limits their receptive fields. In this work, we propose to replace any traditional convolutional layer with an autoregressivemoving-average (ARMA) layer,anovelmodule with an adjustable receptive field controlled by the learnable autoregressive coefficients.

Understanding what sparse auto-encoder (SAE) features in vision transformers truly represent is usually done by inspecting the patches where a feature's activation is highest. However, self-attention mixes information across the entire image, so an activated patch often co-occurs with--but does not cause--the feature's firing. W e propose Causal F eature Explanation (CaFE), which levarages Effective Receptive Field (ERF). W e consider each activation of an SAE feature to be a target and apply input-attribution methods to identify the image patches that causally drive that activation. Across CLIP-ViT features, ERF maps frequently diverge from naive activation maps, revealing hidden context dependencies (e.g., a "roaring face" feature that requires the co-occurrence of eyes and nose, rather than merely an open mounth.). Patch insertion tests confirm that our CaFE more effectively recovers or suppresses feature activations than activation-ranked patches. Our results show that CaFE yields more faithful and semantically precise explanations of vision-SAE features, highlighting the risk of misinterpretation when relying solely on activation location.