Photometric redshift estimation via deep learning

• Photometric redshifts support processing large source catalogs when direct spectroscopic measurements are costly, despite lower accuracy than spectroscopy.Precise redshifts matter for cosmology and missions such as Euclid.Source: §1 · p. 1
• The proposed deep convolutional network–mixture density network predicts redshift PDFs directly from images, avoiding pre-extracted features.The convolutional network extracts image information and the mixture density network produces the distributions.Source: §1 · p. 2
• The study evaluates the image-based model against a modified random forest and a plain mixture density network using SDSS data.PDF quality is assessed with CRPS and PIT alongside traditional scores.Source: §1 · p. 2

2.1. Random forest

• The random forest reference fits a Gaussian mixture to its tree predictions to produce PDFs from 15-dimensional magnitude-and-color feature vectors.The experiments use a 256-tree model with five ugriz magnitudes and pairwise color combinations.Source: §2.1 · p. 2

2.2. Mixture density network

• The mixture density network interprets network outputs as Gaussian-mixture means, standard deviations, and weights, and uses CRPS as its loss.Unlike the commonly used log-likelihood loss, CRPS emphasizes distribution location linearly with distance from the reference value.Source: §2.2 · p. 3

2.3. Deep convolutional mixture density network

• The DCN used five SDSS ugriz image bands and pixel-wise differences; convolutional layers extracted feature maps from these inputs, which were flattened and supplied to the fully connected part.Selected convolution filters had dimensions 3 × 3 and 2 × 2.Source: §2.3 · pp. 3–4
• The DCMDN replaces the fully connected output with an MDN to predict PDFs directly from images, without preprocessing or pre-classification.Its CRPS training loss and automated feature extraction and selection are intended to use image information beyond condensed, general-purpose features.Source: §2.3 · pp. 4–5, Figure 2 · p. 4
• Dropout, image augmentation, and early stopping are described as safeguards against overfitting in the high-parameter network.Network choices were explored extensively, but the selected configuration was obtained through manual engineering; GPU training was used because CPU training was too time-consuming.Source: §2.3 · pp. 4–5

3.1. Data

• The data comprise SDSS quasar catalogs from DR7 and DR9, plus DR9 samples of 200,000 galaxies and 200,000 stars.Combining the quasar catalogs improves redshift-space coverage; the stars are assigned redshift zero and included to test performance on objects that may be contaminated with stellar sources.Source: §3.1 · p. 5

3. Experiments

• The models are compared across three experiments: galaxies, quasars, and a mixed catalog, with the DCMDN trained on images and reference models on photometric features.The mixed experiment is intended to represent a more realistic use case with uncertain source classification.Source: §3.2 · pp. 5–7

3.2. Setup and general configuration

• The reference models use 15 magnitudes and pairwise colors; the RF has 256 trees, while the MDN uses five Gaussian components, selected based on BIC experiments.The DCMDN architecture was chosen through manual and empirical testing; it uses early stopping and 60% dropout to limit overfitting.Source: §3.2 · pp. 5–6
• Evaluation reports RMSE, MAD, bias, and σ2 alongside CRPS and PIT; only a simple holdout strategy is used because DCMDN training is computationally demanding.CRPS and PIT capture PDF location and shape, whereas the common scores use PDF means or the first mode; absolute results may vary with the training and test sets.Source: §3.2 · p. 6
• The galaxy-only experiment uses 100,000 training and 100,000 test objects, retaining 28 × 28 pixel images because galaxies are extended.Its redshift range is mostly z ∈ [0..1], and its models depend on correct pre-classification.Source: §3.2 · p. 6
• The quasar experiment uses 100,000 training and 85,000 test objects with 16 × 16 pixel cutouts, chosen to capture compact sources while saving computation.The sample spans z ∈ [0..6], and its models depend on correct pre-classification.Source: §3.2 · p. 7
• The mixed-catalog experiment combines quasars, galaxies, and stars to test predictions without pre-classifying source type, assigning stars redshift z = 0.It uses 300,000 training and 285,000 test objects, with 16 × 16 pixel inputs.Source: §3.2 · p. 7

4.1. Experiment 1 - Galaxies

• For galaxies, RF performs slightly better on PIT, while MDN and DCMDN show more uniform PITs and cleaner alignment with the ideal diagonal.Source: §4.1 · p. 7, Figure 5 · p. 9

4.2. Experiment 2 - Quasars

• For quasars, DCMDN performs best across all three point-estimate representations and has superior catastrophic-outlier and probabilistic CRPS performance.Multimodal behavior is especially visible around z ≈ 0.5–0.9 and z ≈ 1.5–2.5, where the first mode may not be the best estimate.Source: §4.2 · pp. 7, 9, Figure 7 · p. 11

4.3. Experiment 3 - Mixed

• In the mixed-catalog test, DCMDN significantly outperforms RF and MDN using mean or first-mode estimates, approaches ideal performance on unambiguous PDFs, and achieves the best CRPS.The same experiment identifies overdispersed PDFs and high-redshift outliers as shortcomings of DCMDN.Source: §4.3 · pp. 9, 11
• In the mixed-catalog experiment, DCMDN PDFs are overdispersed, and high-redshift outliers are associated with underrepresentation of those objects in training.Source: §4.3 · p. 11

5. Conclusions

• The method uses CRPS and PIT alongside traditional scores to assess probabilistic predictions, whose multimodal and asymmetric shapes traditional scores may not capture.The experiments compare the image-based model against a modified random forest and a mixture density network.Source: §5 · p. 11
• The DCMDN automatically extracts and selects image features and estimates redshift without requiring objects to be pre-classified.Its PDFs represent uncertainty in both source classification and redshift estimation, reducing the effect of selection biases associated with hard classification.Source: §5 · p. 13
• The DCMDN outperforms both feature-based reference models, and performs best when using PDFs to identify and exclude ambiguous objects.The comparison summarizes all experiments; the mixed experiment also tests performance with stellar contamination.Source: Table 6 · p. 13, §5 · p. 13
• An SDSS-trained model could be applied to SDSS sources regardless of source type, but it would retain selection biases from spectroscopic target selection.The authors relate the DCMDN's performance boost to better use of information and automatic selection of useful features.Source: §5 · p. 13

Appendix A: CRPS and PIT

• CRPS is the integral over x of the squared difference between the predicted and observation CDFs.For a Gaussian mixture model, the score can be calculated in closed form.Source: Appendix A · p. 15
• The CRPS closed-form expression uses the standard-normal PDF and CDF evaluated at the normalized prediction error.The cited equation introduces A(μ, σ²) for the Gaussian-mixture calculation.Source: Appendix A · p. 15
• PIT histograms use the predicted CDF evaluated at each observed value to diagnose PDF calibration.A uniform histogram indicates widths consistent with deviations from the true measurements; a central peak indicates overdispersion, while a U-shape indicates underdispersion.Source: Appendix A · p. 15, Figure A.1 · p. 15

Appendix B: CRPS vs. log-likelihood as loss-function

• CRPS-trained MDNs produce a well-calibrated PIT after 500 epochs, while the log-likelihood-trained model's PIT begins degrading again at 10,000 epochs.The experiment used independent quasar-data subsets of 100,000 training objects and 85,000 test objects.Source: Appendix B · p. 15
• Log-likelihood training improves log-likelihood scores relative to the same MDN trained with CRPS.Conversely, CRPS training improves CRPS scores relative to log-likelihood training.Source: Appendix B · p. 15
• The authors favor CRPS training because it accounts for calibration and sharpness of the predicted PDFs.Source: Appendix B · p. 15

Appendix C: Number of Gaussian components

• The component-count experiment uses the random forest on quasar data and averages BIC scores across redshift bins.Scores are calculated for different numbers of Gaussian components.Source: Appendix C · p. 15
• n = 3 and n = 5 perform comparably well, while n = 1 and n = 7 perform poorly in multiple redshift regions.The figure shows that different regions of the redshift range demand different numbers of components.Source: Figure C.1 · p. 16, Appendix C · p. 16
• The authors choose n = 5 as reasonable because BIC is based on log-likelihood, whereas their model is trained using CRPS.Source: Appendix C · p. 16

Appendix E: Data

• The supplementary identifiers list the SDSS IDs of the randomly extracted stars and galaxies, with quasar IDs supplied as a plain ASCII file.Source: Appendix E · p. 16
• DCMDN output files include test-object spectroscopic redshifts and 15 values—five means, five sigmas, and five weights—used to calculate GMM parameters and reproduce model performance.Source: Appendix E · p. 16
• Separate files provide the identifiers for galaxies, quasars, and stars, and DCMDN predictions for the galaxy, quasar, and mixed experiments.Source: Appendix E · p. 16

Appendix F: SQL-queries

• The galaxy query randomly selects up to 200,000 objects with spectroscopic redshifts from 0 to 6 and class GALAXY.Source: Appendix F · p. 16
• Both queries retrieve coordinates, five-band photometry, PSF and model magnitudes, spectroscopic object IDs, classes, and redshifts.Source: Appendix F · p. 16
• The stellar-source query randomly selects up to 200,000 objects classified as STAR, without specifying a redshift range.Source: Appendix F · p. 16