The Importance of Phase in Neural Representations: An Internal Oppenheim-Lim Test of Image Classifiers

Neural networks trained with gradient-based methods exhibit a strong simplicity bias: they learn simpler statistical features of their data before moving to more complex features. Previous analyses of this phenomenon have largely focused on settings with (quasi-)isotropic inputs. In this work, we study the simplicity bias from a Fourier perspective, which allows us to include two key features of natural images in the analysis: approximate translation-invariance and power-law spectra. We first show experimentally that simple neural networks trained on image classification tasks first rely on amplitude information -- related to pair-wise correlations between pixels -- before exploiting phase information, which encodes edges and higher-order correlations. In view of this, we introduce a synthetic data model for translation-invariant inputs that allows precise control over amplitudes and phases while remaining tractable. We rigorously establish that for isotropic and high-dimensional inputs, classification based on phase information alone is a genuinely hard task: online stochastic gradient descent (SGD) cannot distinguish the structured inputs from noise within $n \ll N^3$ steps, but needs at least $n \gg N^3 \log^2{N}$ steps. In contrast, we show both experimentally and theoretically that power-law spectra can dramatically accelerate the speed of learning phase information, even if the spectra do not help with classification. Simulations with two-layer networks trained on textures and with deep convolutional networks on ImageNet and CIFAR100 confirm this non-trivial interaction between amplitudes and phases, providing mechanistic insights into how deep neural networks can learn natural image distributions efficiently.

Few-shot fine-grained image classification (FSFGIC) aims to classify similar images with limited labeled examples. This work highlights the critical yet underutilized role of phase information in capturing structural relationships within an image. This study introduces a novel plug-and-play amplitude-phase integration (API) module that effectively combines local and global frequency amplitude and phase information for obtaining more comprehensive feature descriptors. Additionally, a dedicated network, named PSF-Net, is proposed that adaptively fuses phase-based spatial and frequency information for FSFGIS. The designed PSF-Net can be easily integrated into standard episodic training architectures for end-to-end training from scratch. Extensive experiments on five public datasets demonstrate that the method outperforms existing state-of-the-art benchmarks.

Vision models do not form a representation at once; each block revises it. We ask whether the resulting computation path contains evidence that the final representation discards, and whether that evidence improves OOD detection and image classification on clean and shifted data. Unlike approaches that treat intermediate layers as separate snapshots, we retain sample identity across depth and study the transformations connecting successive states. We separate class-coherent transport from input-specific innovation, and coordinate movement from relational reorganization. Across supervised, self-supervised, vision--language, hierarchical, and convolutional encoders, these paths show strong sample-specific continuity and architecture-specific depth profiles that recur across datasets. They are also practically useful. An ID-only transition-surprise score complements strong final-state detectors, reducing FPR95 in 131/152 non-saturated comparisons on a balanced OpenOOD grid; gains are largest for visually disruptive and semantically far shifts, and remain positive on near-OOD for most detectors. Frozen update probes improve 71/72 clean model--dataset cases, while shifted-data gains vary with architecture and corruption type. Computation paths therefore provide a broadly useful reliability signal whose value is determined jointly by model organization and the shift encountered.