[Paper Review] Modelling multimodal photometric redshift regression with noisy observations

S. D. Kügler, Nikolaos Gianniotis|arXiv (Cornell University)|Jul 20, 2016

Remote Sensing in Agriculture1 references3 citations

TL;DR

This paper proposes a Bayesian photometric redshift regression model that treats spectra as the primary input, using a constrained PCA to model uncertainty and missing data, enabling multimodal redshift predictions with posterior distributions. The method outperforms standard models by naturally handling input uncertainty and predicting outside training redshifts, crucial for detecting high-redshift quasars.

ABSTRACT

In this work, we are trying to extent the existing photometric redshift regression models from modeling pure photometric data back to the spectra themselves. To that end, we developed a PCA that is capable of describing the input uncertainty (including missing values) in a dimensionality reduction framework. With this "spectrum generator" at hand, we are capable of treating the redshift regression problem in a fully Bayesian framework, returning a posterior distribution over the redshift. This approach allows therefore to approach the multimodal regression problem in an adequate fashion. In addition, input uncertainty on the magnitudes can be included quite naturally and lastly, the proposed algorithm allows in principle to make predictions outside the training values which makes it a fascinating opportunity for the detection of high-redshifted quasars.

Motivation & Objective

• Address the multimodal nature of photometric redshift regression, where a single color can correspond to multiple redshifts.
• Incorporate input uncertainty and missing values in photometric data directly into the regression framework.
• Develop a method that can predict redshifts beyond the training range, supporting high-redshift quasar discovery.
• Replace inadequate metrics like RMS with posterior-based measures that reflect multimodal uncertainty.
• Provide a fully Bayesian framework that returns a full posterior distribution over redshifts, rather than point estimates.

Proposed method

• Use synthetic photometry derived directly from quasar spectra to avoid calibration errors and ensure data consistency.
• Apply a dimensionality reduction via PCA on rest-frame spectra, constrained to avoid overfitting by using reconstructed spectra as discrete prototypes.
• Model input uncertainty (including missing values) by incorporating flux errors into the PCA reconstruction process.
• Formulate the redshift regression as a Bayesian inference problem, computing the posterior distribution over redshifts given the PCA coefficients.
• Use a discrete prototype-based representation of the PCA space to stabilize learning and prevent unphysical spectra from high-dimensional flexibility.
• Introduce likelihood-based evaluation metrics that assess both the accuracy of capturing the true redshift and the model's ability to represent multimodal solutions.

Experimental results

Research questions

• RQ1 Can a PCA-based spectrum generator effectively model photometric data uncertainty and missing values in a Bayesian regression framework?
• RQ2 How does the proposed method handle the multimodal nature of photometric redshifts, where one color corresponds to multiple redshifts?
• RQ3 Can the model make reliable predictions for redshifts outside the training range, enabling high-redshift quasar detection?
• RQ4 Are traditional metrics like RMS and MAD inadequate for evaluating multimodal redshift regression, and what better alternatives exist?
• RQ5 Can a discrete prototype-based PCA model avoid overfitting compared to a continuous PCA while preserving predictive performance?

Key findings

• The proposed model successfully captures the multimodal nature of photometric redshifts by returning a full posterior distribution over redshifts, unlike standard models that assume one-to-one mappings.
• The model naturally incorporates input uncertainty and missing values through error-weighted PCA reconstruction, which is not feasible in most data-driven approaches.
• The method enables out-of-sample predictions beyond the training redshift range, offering a promising path for detecting high-redshift quasars.
• Traditional metrics like RMS and MAD are shown to be inadequate for multimodal problems, and the authors introduce new likelihood-based metrics that better reflect model performance.
• A 10-dimensional continuous PCA led to overfitting and unphysical spectra, but using reconstructed spectra as discrete prototypes stabilized the model and improved generalization.
• The model’s performance is robust even when photometry is derived from noiseless, calibrated spectra, suggesting strong potential for real-world application with proper calibration.