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.

Deep neural networks trained on natural images are shown to produce outputs consistent with human observers for brightness illusions. While this phenomenon has been documented across architectures, all evidence, to date, is measured at the output level: restored pixels, decoded trajectories, or classification decisions. Whether these models actually represent illusions internally, and if so where and how, remains unknown. We show that denoising models develop illusion-sensitive representations at specific internal layers, across varied architectures. Specifically, we identify the layers and channels that discriminate illusory from physically matched control regions. We show that the denoising objective is a more important driver of the effect than the architecture. On domain-appropriate stimuli, these activations track a validated psychophysical model of human brightness perception (FLODOG; Spearman $ρ\geq 0.70$) and scale monotonically with parametric illusion strength. Leveraging these findings, we provide causal evidence via channel ablation showing that illusion-sensitive channels specifically and substantially affect the internal signal. Yet injecting these representations into the generation pipeline produces no measurable pixel shift across all tested architectures; we term such representations perceptual phantoms: active in internal processing yet invisible to any output-based evaluation. While related internal-output dissociations have been characterized in language models, this is the first such characterization for perceptual representations in denoising vision models.