🔭 astrophysics

Comparison of symbolic regression algorithms in Star/galaxy/quasar separation

This study demonstrates that four state-of-the-art symbolic regression frameworks can effectively classify stars, galaxies, and quasars in SDSS DR17 data using compact, interpretable analytic functions that match traditional machine learning performance while providing transparent mathematical insights into astrophysical boundaries.

Original authors: Rachit Deshpande, Shantanu Desai

Published 2026-03-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rachit Deshpande, Shantanu Desai

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

Imagine you are a cosmic librarian trying to sort a massive, chaotic pile of books (astronomical objects) into three specific shelves: Stars (like our Sun), Galaxies (like our Milky Way), and Quasars (super-bright, distant monsters).

For a long time, librarians have used "Black Box" machines (like deep neural networks) to do this sorting. These machines are incredibly smart and accurate, but they are like a magic box: you put a book in, and it tells you which shelf it belongs to, but you have no idea how it made that decision. It's just a secret recipe of millions of numbers.

This paper asks a simple question: Can we find a simple, readable rule that does just as good a job as the magic box?

The authors tried using a technique called Symbolic Regression. Think of this not as a magic box, but as a detective trying to solve a puzzle by writing down simple math equations (like $y = 2x + 5$ ) instead of complex code.

Here is the story of their investigation, broken down into simple parts:

1. The Clue: Redshift (The "Cosmic Stretch")

The researchers decided to ignore all the complicated data (like the color of the light) and focus on just one clue: Redshift.

Analogy: Imagine a siren on a passing ambulance. As it moves away, the sound gets lower (stretches). In space, as objects move away from us, their light stretches.
The Pattern:
- Stars are our neighbors, so they barely move away (Redshift $\approx$ 0).
- Galaxies are further away, so they are moving at a medium pace.
- Quasars are the furthest away, zooming away very fast (High Redshift).
  The goal was to write a simple math formula that says: "If the redshift is this low, it's a star. If it's medium, it's a galaxy. If it's high, it's a quasar."

2. The Contestants: Four Different "Detectives"

The team set up a race between four different types of Symbolic Regression algorithms to see which one could find the best simple formula.

PySR (The Evolutionary Gardener):
- How it works: It acts like nature. It grows thousands of random math formulas, lets the "fittest" ones survive, mixes them up, and mutates them over generations.
- Result: It found a good formula, but it was a bit wobbly. It was like a gardener who found a great plant but kept changing the soil a little bit every time.
ESR (The Exhaustive Checker):
- How it works: It doesn't guess. It tries every single possible simple math combination (within a limit) and picks the one that fits the data best using a strict "Occam's Razor" rule (the simplest explanation is usually the right one).
- Result: Very reliable and stable, like a methodical accountant checking every receipt.
PhySO (The AI Reinforcement Learner):
- How it works: This is a robot that learns by trial and error, but with a safety net. It's trained like a video game character: it gets points for finding a formula that works and loses points if it creates a math disaster (like dividing by zero).
- Result: This was the most stable detective. Once it found a formula, it stuck to it perfectly every time. It was like a robot that never made a typo.
MvSR (The Specialist Team):
- How it works: This team realized that the universe isn't the same everywhere. So, they split the job into three teams: one for low redshift (stars), one for medium (galaxies), and one for high (quasars). They each found a piece of the puzzle and stitched them together.
- Result: The Winner! This approach found the most accurate formula. It was like having three experts, each handling the part of the job they knew best.

3. The Results: Simple vs. Complex

The researchers compared their simple math formulas against the "Black Box" machines (like Random Forests and Neural Networks).

The Shock: The simple math formulas (the "detectives") performed just as well as the complex, heavy-duty AI machines.
The Accuracy: The best simple formula got about 94% accuracy, which is almost the same as the best "Black Box" machine.
The Bonus: Because the formula is simple (e.g., $3.134 - \frac{2.320}{z + e^{-2.904z}}$ ), a human can read it, understand it, and trust it. We know why it works.

4. The "Wall" They Hit

The paper also discovered something fascinating: There is a limit to how well we can do.
Even the best AI and the best math formulas hit a wall around 90-94% accuracy.

Why? Because near a certain distance (Redshift $\approx$ 1.25), galaxies and quasars look exactly the same in this one-dimensional view. It's like trying to tell the difference between two identical twins wearing the same clothes in the dark. No amount of math can fix that; the data itself is ambiguous.

The Big Takeaway

This paper proves that we don't always need a "Black Box" to solve complex scientific problems. Sometimes, the universe follows simple, elegant rules that we can write down on a napkin.

Old Way: Use a giant, unexplainable AI to sort the stars.
New Way: Use Symbolic Regression to find a simple, human-readable math equation that does the same job, is faster to run, and tells us exactly how the universe is organized.

It's a victory for Explainable AI: getting the same results as the magic box, but with the lights turned on so we can see how the magic works.

1. Problem Statement

The classification of astronomical objects (stars, galaxies, and quasars) is fundamental to cosmology and galaxy evolution studies. While deep learning and gradient-boosted trees achieve high accuracy (>95%) on large surveys like the Sloan Digital Sky Survey (SDSS), they function as "black boxes," obscuring the underlying astrophysical relationships (e.g., redshift evolution and population density gradients).

This work addresses the need for interpretable machine learning by investigating Symbolic Regression (SR). The specific challenge is to derive compact, human-readable analytic functions that map a single physical parameter—spectroscopic redshift ( $z$ )—to continuous classification scores, thereby separating Galactic stars, galaxies, and extragalactic quasars without relying on high-dimensional color spaces.

2. Methodology

Dataset and Preprocessing

Source: SDSS Data Release 17 (DR17).
Sample: 100,000 spectroscopically confirmed objects (59.45% Galaxies, 21.59% Stars, 18.96% Quasars).
Input Feature: Spectroscopic redshift ( $z$ ) only.
Protocol:
- A representative 10% subset (10,000 objects) is used for the symbolic discovery phase to ensure computational efficiency.
- A 5-fold stratified cross-validation is performed on the full 100,000-object dataset for global validation.
- Decision Manifold: A continuous scoring function $s(z)$ $s (z)$ is generated, which is then discretized into three classes using two optimized thresholds ( $t_1, t_2$ $t_{1}, t_{2}$ ):
  - $s(z) < t_1 \rightarrow$ Galaxy
  - $t_1 \leq s(z) \leq t_2 \rightarrow$ Star
  - $s(z) > t_2 \rightarrow$ Quasar (QSO)

Symbolic Regression Frameworks Evaluated

The study compares four state-of-the-art SR algorithms, all constrained to a maximum expression complexity of 10 nodes:

PySR: An evolutionary island-model genetic programming approach. It uses parallel subpopulations with migration, tournament selection, and local BFGS optimization for numerical constants.
Exhaustive Symbolic Regression (ESR): A deterministic approach that enumerates all mathematically valid expressions within the complexity limit. It uses Minimum Description Length (MDL) (via BIC) for model selection to balance fit and parsimony.
PhySO (Physical Symbolic Optimization): A Deep Reinforcement Learning (DRL) method using an LSTM network. It treats expression generation as a sequential decision process, employing risk-seeking policy gradients and masking invalid mathematical structures (e.g., nested exponentials).
Multi-View Symbolic Regression (MvSR): A specialized approach that decomposes the redshift domain into three views (Low-z, Intermediate-z, High-z). It evolves a single global symbolic template but refines numerical parameters independently for each view to handle non-stationarity.

Baselines

To establish an Information-Theoretic Ceiling, three non-interpretable machine learning models were trained on the same 1D redshift feature space:

Random Forest (RF)
Support Vector Machine (SVM) with RBF kernel
Multi-Layer Perceptron (MLP)

Evaluation Metrics

Primary: Cohen's Kappa ( $\kappa$ ) to handle class imbalance.
Secondary: Balanced Accuracy, Macro-Averaged F1-Score, and parametric stability (standard deviation of thresholds $t_1, t_2$ across folds).
Physical Analysis: Error density profiles and Recall vs. Redshift curves to identify "Physical Transition Zones" where classes overlap.

3. Key Contributions

Systematic Benchmarking: The first comprehensive comparison of four distinct SR paradigms (Evolutionary, Exhaustive, DRL, and Multi-View) applied to the specific astrophysical problem of Star/Galaxy/Quasar separation.
Redshift-Centric Discovery: Demonstrating that compact analytic functions of a single variable ( $z$ ) can achieve near-optimal separation, isolating the intrinsic physical degeneracies of the dataset.
Transparency vs. Performance: Proving that interpretable symbolic models can match the performance of complex black-box baselines (RF, SVM, MLP) while providing explicit mathematical formulas.
Extension of Prior Work: Extending the analysis of Llorella and Cebrian (2025) by introducing more advanced SR frameworks and a rigorous cross-validation protocol.

4. Results

Performance Metrics

MvSR (Top Performer): Achieved the highest Cohen's $\kappa$ of 0.8948 and Balanced Accuracy of 92.51%. It showed superior robustness in classifying the underrepresented Quasar class ( $F1_{QSO} \approx 0.86$ ).
PhySO: Demonstrated exceptional parametric stability, with threshold standard deviations ( $\sigma < 0.002$ ) an order of magnitude lower than evolutionary methods. It achieved $\kappa \approx 0.875$ .
ESR: Achieved $\kappa \approx 0.8665$ , outperforming PySR in Quasar identification due to its deterministic MDL selection.
PySR: Achieved $\kappa \approx 0.8526$ .
Black-Box Baselines: The optimized RF, SVM, and MLP models converged at a performance ceiling of $\kappa \approx 0.895$ .

Key Findings

Performance Parity: The best symbolic model (MvSR) is statistically indistinguishable from the best black-box models (SVM/RF). This suggests that the "complexity deficit" of symbolic models is negligible for this specific 1D problem.
Physical Limits: The convergence of all models at $\kappa \approx 0.90$ indicates that the remaining error is due to intrinsic physical degeneracies in the SDSS data (specifically the spectral overlap between galaxies and quasars around $z \approx 1.25$ ), not model limitations.
Error Analysis: All models struggle most in the transition zone $z \in [0.8, 1.5]$ . MvSR successfully flattened the error spikes seen in single-view methods, shifting the peak failure rate to a lower magnitude ( $\approx 0.30$ ) and slightly different redshift ( $z \approx 0.9$ ).
Stability: PhySO and MvSR provided the most stable decision boundaries, making them ideal for automated pipelines where threshold consistency is critical.

5. Significance and Conclusion

This work establishes Symbolic Regression as a viable, transparent alternative to black-box machine learning in astroinformatics.

Scientific Impact: The derived analytic functions (e.g., $s(z) = 3.134 - \frac{2.320}{z + e^{-2.904z}}$ ) offer a mathematically concise characterization of astrophysical boundaries, allowing astronomers to directly inspect the logic of classification.
Operational Utility: Unlike neural networks, these compact expressions require no large-scale model persistence and can be easily integrated into automated spectroscopic workflows.
Future Directions: The study suggests that while redshift alone reaches a performance ceiling, future work should integrate photometric colors and multi-wavelength features into the symbolic discovery phase to resolve the high-redshift degeneracies identified in this analysis.

In summary, the paper demonstrates that for the specific task of separating stars, galaxies, and quasars using redshift, interpretability does not come at the cost of accuracy, and specialized SR frameworks like MvSR can outperform or match traditional ML baselines while providing physical insight.