Semi-Supervised Learning for Lensed Quasar Detection

This paper addresses the scarcity of confirmed lensed quasars by developing two semi-supervised machine learning models—a variational autoencoder combined with a dense neural network and a convolutional neural network using virtual adversarial training—that effectively leverage unlabelled data to identify high-quality candidates, exemplified by the discovery of GRALJ140833.73+042229.98.

The Gist

Imagine the universe as a giant, cosmic library filled with billions of books (stars and galaxies). Among these, there are a few very special, rare books called lensed quasars. These aren't just ordinary books; they are like "magic mirrors" in space.

A quasar is a super-bright lighthouse powered by a black hole at the center of a distant galaxy. Sometimes, a massive galaxy sits directly between us and that lighthouse. Its gravity acts like a giant magnifying glass, bending the light and creating multiple copies of the same lighthouse. You might see two, or even four, identical blue-white dots arranged in a perfect pattern around a red galaxy.

The Problem: Finding a Needle in a Haystack
Astronomers want to find these "magic mirrors" because they help us measure the size of the universe and understand how galaxies form. But finding them is incredibly hard.

• They are rare: For every 1,000 to 10,000 quasars, only one is lensed.
• The data is messy: The photos from telescopes are often grainy, noisy, or distorted, like trying to read a book through a dirty window.
• The "Training" is small: We only have a few hundred confirmed examples to show a computer what to look for. It's like trying to teach a child to recognize a specific type of dog when you only have three photos of it, but you have to search through a million photos of other animals.

The Solution: The "Semi-Supervised" Detective
The authors of this paper built a computer program (a machine learning model) to act as a detective. They used a clever trick called Semi-Supervised Learning.

Think of it like this:

1. The Teacher (Labelled Data): You have a small stack of flashcards with pictures of the "magic mirrors" (lensed quasars) and a small stack of "not magic mirrors" (regular stars).
2. The Library (Unlabelled Data): You have a massive warehouse full of millions of photos, but you don't know what's in them.
3. The Trick: Instead of just teaching the computer with the few flashcards, you let it study the massive warehouse too. You ask the computer: "Look at all these millions of photos. Even though you don't know exactly what they are, can you learn the general 'vibe' of what a galaxy or a star looks like?"

By studying the millions of unknown photos, the computer gets much smarter about the background noise and the general shapes of space objects. This helps it understand the few flashcards much better.

Two Different Detective Styles
The paper tested two different ways to build this detective:

1. The "Compression" Detective (Autoencoder):
   Imagine you have a messy room (a noisy image). You try to describe the room using only a few words (compressing the data). If you can describe a "magic mirror" room with very few words, but a "regular star" room needs a huge, complicated description, the computer learns to spot the difference.

   • How it worked: They trained a computer to shrink millions of images down to their "essence" and then asked a second computer to guess if the essence was a lensed quasar. This method was very good at spotting the patterns in clean data.

2. The "Stress-Test" Detective (Virtual Adversarial Training):
   Imagine you are teaching a student to spot a fake painting. You show them a real one, then you slightly smudge the paint or change the lighting (a tiny "adversarial" change) and ask, "Is this still real?" If the student says "No!" too easily, you teach them to be more robust.

   • How it worked: This model was trained to look at the millions of unknown photos and make sure that even if the image was slightly noisy or changed, it wouldn't get confused. This helped it handle the messy, real-world data better.

The Result: A New Discovery!
The team used these computer detectives to scan millions of images. They picked the top candidates and sent them to a giant telescope (the Keck Observatory) for a real-life check-up.

• The Success: They confirmed one brand new lensed quasar, which they nicknamed "The Snowman" because the two images of the quasar and the galaxy in the middle looked like a snowman.
• The Reality Check: They also found that the computers sometimes got tricked by "asterisms" (random stars that happen to line up looking like a lens) or crowded star fields. But, the success rate was competitive with human experts, and the computers could do it millions of times faster.

Why This Matters
This paper shows that by letting computers "read" the millions of unknown books in the cosmic library, we can find the rare, special ones much faster. As new telescopes like the LSST start taking photos of the entire sky every night (generating terabytes of data), we can't rely on humans to look at every picture. We need these smart, semi-supervised detectives to help us find the universe's hidden treasures.

Technical Summary

1. Problem Statement

The detection of gravitationally lensed quasars is critical for cosmology, galaxy structure studies, and measuring the Hubble constant. However, identifying them is extremely challenging due to:

• Extreme Class Imbalance: Lensed quasars are rare (approx. 1 in 1,000 to 1 in 10,000 quasars).
• Scarcity of Labeled Data: Only ~250 confirmed lensed quasars exist, with perhaps 400 more identified but unpublished. This is insufficient for training deep learning models that typically require millions of labeled examples.
• Data Quality and Heterogeneity: Data comes from different surveys (Pan-STARRS for the northern sky, DESI for the southern sky) with varying noise levels, artifacts, and band coverage.
• Distribution Shift: Undiscovered lensed quasars likely possess different characteristics (e.g., smaller separations, reddened images) than the known training set, violating the standard machine learning assumption that training and test data are independent and identically distributed (i.i.d.).
• Ambiguity: Visual identification is difficult; even expert astronomers achieve only 5–30% success rates in follow-up observations due to confusion with stars or interacting galaxies.

2. Methodology

The authors propose two semi-supervised learning approaches to leverage the small set of labeled lensed quasars alongside a massive pool of unlabeled quasar candidates (from the Milliquas catalogue).

Data Preparation

• Labeled Data: A curated set of ~650 lensed quasars (confirmed and high-confidence candidates) and ~1,000 hand-labeled non-lensed quasars.
• Unlabeled Data: Millions of quasar images from Pan-STARRS and DESI surveys.
• Preprocessing: Images were converted to 64x64 pixel 3-channel (g, r, i) JPEGs. For DESI data, missing bands were zero-filled with a flag indicating the null value.

Approach A: Autoencoder-Classifier Pipeline

This is a two-stage model designed to reduce dimensionality and extract features before classification.

1. $\beta$-Variational Autoencoder ($\beta$-VAE):
   • Trained on millions of unlabeled quasars to learn a compressed latent representation of quasar images.
   • Uses a bottleneck layer (4–64 neurons) to force the network to learn a meaningful latent space.
   • Loss Function: Combines Mean Squared Error (MSE) for reconstruction and Kullback-Leibler (KL) divergence to regularize the latent distribution. A hyperparameter $\beta$ weights the KL term.
   • Noise Metric: A Fourier transform-based metric was added to the input to help the classifier distinguish between genuine information loss (complex lensing) and image noise.
2. Traditional Classifier:
   • The latent space vectors (and reconstruction errors/noise metrics) from the VAE are fed into traditional classifiers (Random Forest, Gradient Boosting, Neural Networks, SVMs).
   • Best Configuration: A $\beta$-VAE with $\beta=0.0001$ and a bottleneck of 32, combined with a densely connected Artificial Neural Network (ANN).

Approach B: Virtual Adversarial Training (VAT)

This is an end-to-end Convolutional Neural Network (CNN) approach.

• Mechanism: VAT regularizes the model by penalizing it if its predictions change when the input image is subjected to a small "adversarial perturbation."
• Semi-Supervised Aspect: The model is trained on both labeled data (using cross-entropy loss) and unlabeled data. For unlabeled data, the model is penalized if the perturbed image changes the predicted class, encouraging the decision boundary to lie in low-density regions of the data space.
• Architecture: A CNN with four convolutional layers (8, 16, 32, 64 channels) followed by fully connected layers. It uses Batch Normalization and Leaky ReLU activations.

3. Key Contributions

• Novel Application of Semi-Supervised Learning: Successfully applied semi-supervised techniques to the specific domain of lensed quasar detection, addressing the "small data" bottleneck.
• Two Distinct Architectures: Developed and compared a feature-extraction pipeline ($\beta$-VAE + Classifier) and a robust end-to-end CNN (VAT).
• Noise Handling: Introduced a Fourier-based noise metric to help classifiers calibrate reconstruction errors in noisy astronomical images.
• Real-World Discovery: The models were used to select candidates for telescope time, leading to the confirmation of a new lensed quasar.

4. Results

• Performance on Clean Data:
  • The Autoencoder-Classifier model achieved an F1 score of 0.897 on the test set.
  • The VAT model achieved an F1 score of 0.58 on the same clean test set.
  • Interpretation: On relatively clean, labeled data, the traditional classifier trained on VAE features outperformed the end-to-end CNN.
• Performance on Unlabeled Candidates (Discovery Phase):
  • When ranking millions of unlabeled images for follow-up, both models performed similarly in terms of candidate quality.
  • Robustness: The VAT model proved superior in handling "crowded stellar fields," correctly identifying them as poor candidates, whereas the Autoencoder model falsely ranked them highly due to coincidental asterisms.
• Observational Confirmation:
  • Five candidates selected by the models were observed using the W.M. Keck Observatory.
  • Success: One new lensed quasar, GRALJ140833.73+042229.98 (internally named "the Snowman"), was confirmed. It features a $z=2.998$ quasar lensed by a $z=0.542$ early-type galaxy.
  • Failures: Three candidates were identified as interlopers (quasar + star), and one remained unresolved. This success rate is competitive with state-of-the-art methods.

5. Significance

• Scalability: The approach demonstrates that machine learning can effectively parse massive astronomical surveys (like Pan-STARRS and DESI) to find rare objects where human inspection is impossible.
• Complementarity: The image-based classifiers are independent of existing methods (e.g., quantum annealing or photometric/astrometric approaches). Combining these distinct data modalities is expected to significantly boost overall discovery rates.
• Future Potential: The paper outlines a path for future improvements, such as incorporating simulated data, adding more spectral bands (z-band), and using cross-survey regularization (leveraging overlapping Pan-STARRS/DESI regions).
• Broader Impact: This work serves as a blueprint for using semi-supervised learning to extract rare signals from colossal datasets in other fields of astronomy and science.