This is the in-depth technical version of Phase, light, and the geometry of data: three signals about where representation actually lives.
What part of an image actually tells a classifier what it is looking at? Not which pixels — that is the easy answer — but which component of the signal carries identity once you decompose it. This is a precise question, and a 1981 result by Oppenheim and Lim gave a precise answer for natural images: reconstruct an image from its Fourier phase alone, discard the magnitude, and the image stays recognizable; do the reverse and you get noise. Phase carries identity. Magnitude is mostly energy distribution.
The interesting move in The Importance of Phase in Neural Representations is to stop treating this as a fact about images and start treating it as a causal hypothesis about networks. The author runs an internal Oppenheim-Lim test: take two images, compute hidden-layer representations, transplant the phase of one onto the magnitude of the other at a chosen layer, push it through the rest of the network, and record which image the prediction follows. That is a clean intervention — not a correlational probe, but a transplant. The prediction is the readout.
what the transplant tells you in practice
Across PRISM2D, GFNet, and ViT-B/16 the prediction follows the phase (or sign) donor, and deleting all image-specific magnitude barely moves accuracy (Yıldırım, 2026). In practice this means something concrete for anyone shipping a vision model: the channel-wise magnitude statistics you so carefully normalize, augment, and worry about are, for identity, largely dispensable to the readout. The information that decides the class rides on phase structure.
ResNet-50 looks like a counterexample, and the way it fails to be one is the most useful part of the paper. Transplant the sign after its ReLUs and nothing happens — which is exactly what you'd expect, because a ReLU has already thrown away the sign bit on the negative side. Intervene before the ReLU and a strong latent sign code shows up in the late blocks. The phase/sign identity code is there; rectification just hides it in a different basis. A DC-only control then shows the readout is consuming a channel-wise spatial average. The architectures share the same identity code but expose it in different bases, set by rectification and readout geometry (Yıldırım, 2026).
That single sentence is the mechanistic account I'd been missing for the texture–shape gap between CNNs and attention models. We have known empirically that CNNs lean on texture and ViTs lean more on shape. Here the cause is not mystical: it is where in the algebra the identity code lives, and what the nonlinearity does to it before the readout reads it. For a practitioner doing adversarial hardening or domain-shift work, the takeaway is direct — if you want to perturb or protect identity, you operate on phase/sign, and you must intervene at the right side of the rectifier, or your intervention is a no-op that you'll misread as robustness.
if the operator is linear, does it need to be digital?
Now flip from vision to time series, and a substrate-level question appears. Simple linear and frequency-domain models stay stubbornly competitive in long-horizon forecasting, and mechanistic evidence suggests the standard benchmarks may not require the dense superposed representations that make transformers powerful elsewhere (HAMON, 2026). If the core forecasting operator is often low-complexity and approximately linear, why implement it as learned digital temporal mixing at all?
HAMON answers by building the mixer out of light. Historical values are encoded onto an optical aperture, future positions are left dark, and cascaded trainable phase masks with free-space diffraction shape the forecast directly in the output field. At inference there is no trainable digital sequence-mixing layer — prediction is a single passive optical propagation pass (HAMON, 2026). Read that against the phase paper and the rhyme is hard to miss: in both, phase is the trainable, identity-bearing degree of freedom. One paper finds phase already doing the work inside digital networks; the other builds a forecaster whose only learned parameters are phase masks.
Does it work? On the numbers reported, HAMON outperforms the strongest digital baselines on ETTm2 at all horizons and on ETTh2 at all but the longest horizon, improving MSE by up to 14% — and it does so consistently across horizons rather than at isolated points. It is competitive on Weather and trails on the remaining ETT settings and on the high-channel-count Traffic and Electricity datasets (HAMON, 2026). That trade-off is the honest part. A passive Fourier-optics core is naturally low-channel; the moment you need to mix hundreds of correlated channels, free-space diffraction is not where the work wants to live. But for the regime where the operator is genuinely near-linear, the energy story is the point — propagation is passive, the multiply happens in physics, and a phase-scrambling ablation plus a TorchOptics cross-simulator check confirm the forecast comes from the data-bearing field, not a hidden digital head.
The applied reading: HAMON is not a product, it is a target. It defines a concrete spec for optical hardware and passive physical sequence mixing. If your forecasting workload is the low-complexity kind — and a lot of industrial telemetry is — the question stops being "which transformer" and becomes "does this even need a digital tensor core."
the wider frame: computation as a landscape, not a feedforward graph
Step back and both papers sit inside a larger shift that the energy-based dynamical models tutorial maps out well. The argument there is that dynamical systems compute — continuous-time Hopfield networks, Boltzmann machines, dense associative memories, oscillator networks for optimization, proximal-descent dynamics — by encoding information in gradient flows and energy landscapes rather than in feedforward backprop (Montanari et al., 2026). An oscillator network stores its answer in phase relationships. A diffractive optical stack stores its computation in a phase profile. A trained classifier, it turns out, stores identity in a phase/sign code. The unifying object is phase, and the unifying claim is that the substrate — light, oscillators, gradient flow — can be the computer, not just the simulator of one.
This is also where the geometry comes in. The shape space review makes the case that real geometric data lives on nonlinear manifolds, and that flat machine-learning methods are frequently ill-equipped to respect that structure (Choi et al., 2026). Phase is, literally, a coordinate on a circle — a curved space, not a vector space. When the Oppenheim-Lim paper says ResNet exposes the same identity code "in a different basis," it is making a geometric statement: the same manifold, read through a different chart set by the rectifier. The forecasting and vision results are not separate curiosities; they are two readings of the same lesson, that identity and computation live in the geometry of the representation, and that geometry is often phase.
What I'd push back on, as someone who has to ship: none of this is free. The phase result is an analysis tool, not yet a robustness recipe — knowing identity rides on phase tells you where to intervene, not how to do it without breaking calibration. HAMON's wins are real but bounded to the near-linear, low-channel regime, and "competitive on Weather, trails on Traffic" is the sentence a deployment review actually reads. The honest version of the excitement is narrower and more useful: phase is the load-bearing coordinate in more systems than we treated it as, and the substrate that manipulates phase does not have to be a GPU. That is a design constraint worth taking seriously before it is a revolution worth announcing.
Sources
- The Importance of Phase in Neural Representations: An Internal Oppenheim-Lim Test of Image Classifiers — arXiv · cs.AI · 2026-06-15
- HAMON: Passive Optical Sequence Mixing for Long-Horizon Forecasting — arXiv · cs.AI · 2026-06-15
- Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis — arXiv · cs.LG · 2026-06-15
- [2604.05042] Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization — Manual / ad-hoc · 2026-06-17