⚛️ phenomenology

A method to challenge symmetries in data with self-supervised learning

This paper presents a self-supervised learning method that tests suspected symmetries in data by training models to distinguish real data from symmetrically transformed alternatives, effectively handling filtered data to determine if such symmetries are truly realized in nature.

Original authors: Rupert Tombs, Christopher G. Lester

Published 2026-10-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rupert Tombs, Christopher G. Lester

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of the physical world, scientists often build models based on the idea that nature follows certain rules of balance. These rules, known as symmetries, suggest that if you change a system in a specific way—like flipping it like a mirror or spinning it around an axis—the underlying laws governing it should remain exactly the same. For decades, physicists have relied on these symmetries to design experiments and interpret the chaotic data streaming in from particle detectors and telescopes. However, a fundamental question remains: does nature actually obey these rules, or are they just convenient assumptions we have made? If a symmetry is broken, it means the universe is more complex than our current theories allow, potentially pointing toward new physics. The challenge has always been distinguishing a genuine break in nature's laws from the messy, imperfect reality of how we collect data. Sensors fail, equipment has blind spots, and researchers often have to discard large chunks of information to make sense of it all. These imperfections can create the illusion of asymmetry where none exists, or hide a real break in the pattern, making it incredibly difficult to trust what the data is saying.

Two researchers at the University of Cambridge have developed a new, practical way to test these suspected symmetries without needing to clean up the data or rely on outside labels to tell them what is real. They call their approach "which is real?" and it uses a form of artificial intelligence known as self-supervised learning. Instead of teaching a computer model with a list of correct answers, they let the model learn by playing a game of its own making. The researchers take a piece of real data and create a "fake" version of it by applying the symmetry rule they want to test. For example, if they are testing rotational symmetry, they might take a particle collision event and rotate it to a new angle. They then pair the original real event with this rotated fake event and ask the computer to guess which one is the original. If the computer can consistently tell the difference between the real data and the rotated copy, it proves that the symmetry is broken. If the symmetry were perfect, the real and fake versions would be indistinguishable, and the computer would be forced to guess randomly.

The brilliance of this method lies in how it handles the inevitable messiness of real-world experiments. In many scientific fields, data is filtered; some events are missed because of sensor gaps, and others are discarded because they don't meet strict quality criteria. Usually, this filtering creates a bias that looks like a break in symmetry. Traditional methods try to mathematically correct for these missing pieces, often by assigning heavy weights to the few data points that survive, which can lead to unstable and unreliable results. The "which is real?" method avoids this trap entirely. Instead of trying to undo the filtering, the researchers apply the same filtering rules to the fake data they generate. They take a real event, create a fake version, and then run that fake version through the same imperfect sensor simulation that the real data went through. Because both the real and the fake data suffer from the exact same losses and gaps, the computer model learns to ignore the filtering errors and focus only on the genuine differences in the underlying physics. It is a bit like trying to spot a difference between two identical twins, but both are wearing the same pair of sunglasses that obscure half their faces; the observer learns to look past the glasses to see the true face.

To demonstrate that this approach works, the team tested it on two different scenarios using computer simulations. The first example involved a cylindrical particle detector, similar to those used in high-energy physics, which had large holes in it and some sections that were much less sensitive than others. They created a dataset where the particles were distributed in a way that looked symmetric but had a subtle, hidden shift that broke the rotational symmetry. When they ran their "which is real?" model, it successfully identified the asymmetry, pinpointing exactly where the data was shifted and how the detector's gaps were influencing the results. The model produced a score that strongly indicated the symmetry was broken, even though the data was heavily filtered. In a second test, they applied the method to a topographical map, checking for a different kind of symmetry where the landscape should look the same if shifted horizontally within certain flat areas. Again, the model detected when the data violated this rule, learning the specific patterns of the violation while ignoring the artificial borders where data was removed.

The researchers found that when the data truly obeyed a symmetry, the model failed to distinguish between real and fake, resulting in a score that indicated no evidence of a break. Conversely, when the symmetry was broken, the model learned to predict the correct label with high accuracy, providing a clear, quantitative measure of the violation. Crucially, the method worked even when the data was sparse or when the filtering was severe, proving that it does not require perfect, clean data to function. The team showed that by simply asking the computer to choose between a real event and a transformed copy, they could uncover hidden asymmetries that might otherwise be mistaken for experimental errors. This approach offers a robust tool for scientists to test their theories against the raw, unpolished data of the universe, ensuring that when they claim a symmetry is broken, they are seeing a feature of nature itself, not an artifact of their instruments. The code and data used for these tests are publicly available, allowing other researchers to apply this method to their own datasets and challenge the symmetries that underpin our understanding of the physical world.

Technical Summary: A Method to Challenge Symmetries in Data with Self-Supervised Learning

Problem Statement
Physical models and experimental designs often rely on specific symmetries, yet it is not always certain whether nature adheres to these symmetries. Refuting a symmetry invalidates all models implying it. However, testing suspected symmetries is complicated by "filtering"—data loss or selection effects arising from hardware inefficiencies, gaps, or deliberate analysis cuts. Traditional approaches that attempt to correct for filtering via data weighting often suffer from statistical instability, particularly when weights become unbounded due to tight filtering. Furthermore, standard supervised learning requires external labels, which are unavailable when testing fundamental symmetries where the ground truth is the question itself.

Methodology: "Which is Real?"
The authors introduce a self-supervised learning method named "which is real?" designed to test candidate symmetries without external labels and robustly handle filtered data.

Core Concept: The method constructs pairs of data points $(x, x')$ , where $x$ is a real observation and $x'$ is a "fake" copy generated by applying a candidate symmetry transformation $\tau_S$ to $x$ . The learning task is to predict which entry in the pair is the real datum and which is the transformed fake.
Self-Supervision: Unlike supervised learning, no external labels are provided. The labels are intrinsic to the construction: for every pair, one ordering is "real–fake" and the other is "fake–real." The model learns to distinguish the original distribution $p(x)$ from the transformed distribution $p_S(x)$ .
Handling Filtering: A critical innovation is the treatment of data filtering (represented by a filter function $L(x)$ $L (x)$ ).
- In standard classification, one might try to re-weight data to undo filtering, leading to unstable, unbounded weights.
- In the "which is real?" method, the filter is applied twice: once to the real data (as it naturally occurs) and once to the fake data after the symmetry transformation.
- Mathematically, if the fake is sampled from $L(x')\tau_S(x'|x)$ , the filter terms $L(x)$ and $L(x')$ appear in both the numerator and denominator of the log-odds ratio. They cancel out, allowing the model to learn the underlying asymmetry without needing to know or invert the filter function explicitly, provided the filter is known and applied consistently to both real and fake generation.
Orbital Symmetries and Function Learning: For "orbital symmetries" (which include group symmetries common in physics), the problem simplifies. Instead of learning a function on the joint space of pairs $(x, x')$ , the model learns a scalar function $\zeta(x)$ on the original data space. The log-odds for the pair is derived from the difference $\zeta(x) - \zeta(x')$ .
Training and Evaluation:
- The model is trained using a binary cross-entropy loss function (Equation 15) on the log-odds difference.
- Performance is quantified by a "quality" function $Q$ (Equation 8), defined as the mean log-likelihood ratio between the learned asymmetric model and the symmetric hypothesis (which assigns probability 0.5 to all labels).
- A positive $Q$ indicates the model successfully distinguishes real from fake, implying the symmetry is broken. A negative or near-zero $Q$ suggests the data is consistent with the symmetry (or that the model failed to find a signal).

Key Contributions

General Symmetry Testing: A framework applicable to any computerized data and any candidate symmetry, requiring only the data, the filter function, and the transformation rule.
Robustness to Filtering: The method avoids the statistical pitfalls of data weighting by applying the filter twice, ensuring that filtering effects do not create artificial asymmetries or dominate the training signal.
Self-Supervised Formulation: It eliminates the need for external labels, deriving the learning objective entirely from the data structure and the proposed symmetry transformation.
Interpretability: The learned function $\zeta(x)$ provides a map of where asymmetries exist in the data space, allowing researchers to visualize the nature of the symmetry breaking.

Results and Demonstrations
The paper validates the method using two examples:

Cylindrical Particle Detector: A simulated detector with gaps and varying efficiencies (filtering). The method successfully detected a misalignment (rotational symmetry breaking) in the data. When the filter model was corrected to include a known hardware gap, the detected asymmetry reduced. When the detector was perfectly aligned in a subsequent simulation, the method correctly returned a negative $Q$ , indicating no asymmetry.
Height Map (Topographical Symmetry): A test of translational symmetry within contours of constant brightness.
- In a symmetric case (uniform density within orbits), the method yielded a negative $Q$ , correctly failing to find asymmetry.
- In an asymmetric case (density modulated by sinusoidal waves), the method yielded a large positive $Q$ , and the learned $\zeta(x)$ function successfully visualized the wave pattern breaking the symmetry.

Significance and Claims
The authors claim this method provides a practical tool for "challenging" symmetries in data. It does not claim to discover symmetries from first principles but rather to test suspected symmetries proposed by theoretical models or experimental designs.

Refutation vs. Confirmation: A successful prediction (positive $Q$ ) refutes the symmetry. A failure to predict (negative or low $Q$ ) does not strictly prove the symmetry holds (as the model might lack capacity), but indicates the symmetry is not refuted by the current data and algorithm.
Diagnostic Value: If asymmetry is detected, it signals either that nature is asymmetric or that the experimental setup/filtering needs recalibration. Both outcomes are scientifically valuable.
Readiness: The authors state that the method is ready to challenge theoretically or experimentally proposed symmetries in real applications, having demonstrated success on non-trivial examples with minimal tuning.

The paper concludes that by leveraging self-supervision and a specific handling of filtering, this approach offers a rigorous, label-free way to test the fundamental assumptions of physical models against experimental data.

Technical Summary: A Method to Challenge Symmetries in Data with Self-Supervised Learning

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