What spatial frequency information do humans and neural networks use to recognize objects? In neuroscience, critical band masking is an established tool that can reveal the frequency-selective filters used for object recognition. Critical band masking measures the sensitivity of recognition performance to noise added at each spatial frequency. Existing critical band masking studies show that humans recognize periodic patterns (gratings) and letters by means of a spatial-frequency filter (or "channel") that has a frequency bandwidth of one octave (doubling of frequency). Here, we introduce critical band masking as a task for network-human comparison and test 14 humans and 76 neural networks on 16-way ImageNet categorization in the presence of narrowband noise.

In this appendix, as a complementary to Theorems 1-2, we provide additional theorems, Theorems 3-4, which further illustrate the two advantages of the discretization process by considering an abstract model with the discretization bottleneck. For the advantage on the sensitivity, the error due to potential noise and perturbation without discretization -- the third term ξ(w,r0,M0,d) >0 in Theorem 4 -- is shown to be minimized to zero with discretization in Theorems 3. See Appendix C.1 for a simple comparison between the bound of Theorem 3 and that of Theorem 4 when the metric spaces (M,d) and (M0,d0) are chosen to be Euclidean spaces. We now introduce the notation used in Theorems 3-4. Here, ϕw represents a deep neural network with weight parameters w W RD, qe is the discretization process with the codebook e E RL m, and hθ represents a deep neural network with parameters θ Θ Rζ. Thus, the tuple of all learnable parameters are (w,e,θ).

Deep learning has advanced from fully connected architectures to structured models organized into components, e.g., the transformer composed of positional elements, modular architectures divided into slots, and graph neural nets made up of nodes. The nature of structured models is that communication among the components has a bottleneck, typically achieved by restricted connectivity and attention. In this work, we further tighten the bottleneck via discreteness of the representations transmitted between components. We hypothesize that this constraint serves as a useful form of inductive bias. Our hypothesis is motivated by past empirical work showing the benefits of discretization in non-structured architectures as well as our own theoretical results showing that discretization increases noise robustness and reduces the underlying dimensionality of the model.

Representations learned by pre-training a neural network on a large dataset are increasingly used successfully to perform a variety of downstream tasks. In this work, we take a closer look at how features are encoded in such pre-trained representations. We find that learned representations in a given layer exhibit a degree of diffuse redundancy, i.e., any randomly chosen subset of neurons in the layer that is larger than a threshold size shares a large degree of similarity with the full layer and is able to perform similarly as the whole layer on a variety of downstream tasks. For example, a linear probe trained on 20% of randomly picked neurons from the penultimate layer of a ResNet50 pre-trained on ImageNet1k achieves an accuracy within 5% of a linear probe trained on the full layer of neurons for downstream CIFAR10 classification. We conduct experiments on different neural architectures (including CNNs and Transformers) pretrained on both ImageNet1k and ImageNet21k and evaluate a variety of downstream tasks taken from the VTAB benchmark. We find that the loss & dataset used during pre-training largely govern the degree of diffuse redundancy and the "critical mass" of neurons needed often depends on the downstream task, suggesting that there is a task-inherent redundancy-performance Pareto frontier. Our findings shed light on the nature of representations learned by pre-trained deep neural networks and suggest that entire layers might not be necessary to perform many downstream tasks. We investigate the potential for exploiting this redundancy to achieve efficient generalization for downstream tasks and also draw caution to certain possible unintended consequences.

Post-hoc gradient-based interpretability methods [1, 2] that provide instancespecific explanations of model predictions are often based on assumption (A): magnitude of input gradients--gradients of logits with respect to input--noisily highlight discriminative task-relevant features. In this work, we test the validity of assumption (A) using a three-pronged approach: 1. We develop an evaluation framework, DiffROAR, to test assumption (A) on four image classification benchmarks. Our results suggest that (i) input gradients of standard models (i.e., trained on original data) may grossly violate (A), whereas (ii) input gradients of adversarially robust models satisfy (A) reasonably well.