Intrinsic Dimension Estimation for Radio Galaxy Zoo using Diffusion Models

Intrinsic Dimension Estimation for Radio Galaxy Zoo using Diffusion Models Joan Font-Quer Roset Department of Computer Science The University of Manchester Manchester, United Kingdom [email protected] Devina Mohan Department of Physics and Astronomy The University of Manchester Manchester, United Kingdom [email protected] Anna Scaife∗ Department of Physics and Astronomy The University of Manchester Manchester, United Kingdom [email protected] Abstract In this work, we estimate the intrinsic dimension (iD) of the Radio Galaxy Zoo (RGZ) dataset using a score-based diffusion model. We examine how the iD estimates vary as a function of Bayesian neural network (BNN) energy scores, which measure how similar the radio sources are to the MiraBest subset of the RGZ dataset. We find that out-of-distribution sources exhibit higher iD values, and that the overall iD for RGZ exceeds those typically reported for natural image datasets. Furthermore, we analyse how iD varies across Fanaroff-Riley (FR) morphological classes and as a function of the signal-to-noise ratio (SNR). While no relationship is found between FR I and FR II classes, a weak trend toward higher SNR at lower iD. Future work using the RGZ dataset could make use of the relationship between iD and energy scores to quantitatively study and improve the representations learned by various self-supervised learning algorithms. 1 Introduction The field of astronomy has experienced a deluge of data from large-scale surveys from new observational facilities such as Euclid [ 1 ] and the James Webb Space Telescope [JWST; 2 ]. This has led to the increased adoption of deep learning models to classify or analyse data [ 3 ]. However, these models are constrained by the scarcity of labelled data. This has motivated the development of techniques that can analyse large unlabelled datasets, such as the Radio Galaxy Zoo [RGZ; 4]. According to the manifold hypothesis, high-dimensional data often lies in low dimensional submanifolds [ 5 ], whose intrinsic dimensions (iD) reflects the effective number of degrees of freedom in a data distribution. Estimating iD then provides a useful tool to not only analyse data, but also to analyse models trained on the data. Recent work has shown that diffusion models encode the iD of data manifolds [ 6 ] through their score functions. We use this method to estimate the intrinsic dimension of the RGZ dataset using a score-based diffusion model, and compare the resulting estimates against traditional statistical methods (LPCA [7][8], PPCA [9], and MLE [10]). ∗The Alan Turing Institute, 96 Euston Rd, London, UK [email protected] Machine Learning and the Physical Sciences Workshop, NeurIPS 2025. We relate the iD estimates to Bayesian neural network (BNN) energy scores obtained from a Hamiltonian Monte Carlo BNN trained on the MiraBest dataset, which is a labelled subset of the RGZ data. These energy scores provide a measure of how in or out of distribution a radio source is relative to the labelled benchmark [ 11 ]. Furthermore, we examine how the estimated iD differs across Fanaroff-Riley (FR) morphological classes, using existing labels from recent work on foundation model based radio galaxy classification [ 12 ]. The FR classification divides radio galaxies into two morphological types: FR I sources show peak brightness near the core and fade with distance, FR II sources exhibit bright outer lobes separated by a fainter core [13]. 2 Estimating intrinsic dimension of data with diffusion models In a diffusion model, the data distribution p0 is perturbed using a series of noise-corrupted distributions pt using a stochastic differential equation [ 14 ]. Diffusion models are trained to approximate the score vector ∇xln(ptx) with a neural network sθ(xt, t) . Once the score function is approximated, it can be used to generate new samples from the data distribution. The recently proposed method provides a theoretical foundation, showing that the approximate local dimensionality of the data manifold can be recovered from the score vector [ 6 ]. In experiments with the MNIST dataset, it is found that digits with different geometric complexity have different iDs. For example the digit 1 has iD = 66 and the digit 9 has iD = 152 . Previous work by [ 15 ] also estimated the iD of several image datasets using a maximum likelihood estimation (MLE) based method, but later work showed that this severely underestimates iD. 2.1 Theoretical Foundation Algorithm 1 Estimate the Intrinsic Dimension at x0[6] Require: sθ– trained diffusion model (score), t0– sampling time, K– number of score vectors. Sample x0∼p0(x)from the dataset d←dim(x0) S←empty matrix for i= 1, . . . , K do Sample x(i) t0∼ N(x0, σ2 t0I) Append sθ(x(i) t0, t0)as a new column to S end for (si)d i=1,(vi)d i=1,(wi)d i=1 ←SVD(S) ˆ k(x0)←d−arg maxi=1,...,d−1(si−si+1) return ˆ k(x0) (si)d i=1,(vi)d i=1,(wi)d i=1 denote singular values, left and right singular vectors respectively. Theorem 2.1 For any point x∈Rd sufficiently close to a compact embedded manifold M , and a sufficiently small diffusion time t , the score vector ∇xln(pt(x)) points directly at the projection of x on the manifold. Corollary 2.1.1 The ratio of the projection of the score vector ∇xln(ptx) on the tangent space of the manifold Tπ(x)Mto the normal space of the manifold Nπ(x)Mapproaches 0 as tapproaches 0. Theorem 2.1 only works under certain assumptions: (i) The data lies on a compact, smooth, embedded manifold; (ii) p0 has a smooth (differentiable) density function defined over the manifold; and (iii) the density function is strictly positive on the manifold (i.e. there are no zero-probability regions). Theorem 2.1 and Corollary 2.1.1 underpin the method’s central idea: the score vectors of perturbed points near the manifold will concentrate on the normal space to the manifold at x0 as t→0 . Therefore, its projection to tangent space vanishes as t→0 , and so the number of vanishing singular values corresponds to the dimension of the tangent space. The iD is then the ambient dimension minus the rank of the normal space. Algorithm 1 describes the method proposed in the paper for estimating the iD at a point x0 using a trained diffusion model. For each i∈ {1,...K} , a sample 2 Figure 1: Example images from the Radio Galaxy Zoo dataset taken from different intervals of the mean energy distribution. x(i) t0 is drawn from the forward process transition kernel p0(x(i) t0|x0) = N(x0, σ2 t0I) , and its score vector sθ(x(i) t0) is computed. Each of these score vectors forms a column of a d×K matrix S , where d is the ambient dimension of x0 . Singular value decomposition (SVD) is applied to the matrix S , and the iD is estimated as d−i∗ , where i∗ corresponds to the index of the largest gap between singular values arg maxi=1,...,d−1(si−si+1). 3 Experimental Setup Data We use the Radio Galaxy Zoo Data Release 1 [ 4 ] as our unlabelled dataset of radio sources to train the diffusion model. Details of the MiraBest dataset [ 16 ] which is used to train the BNN is provided in Appendix A. Radio Galaxy Zoo Data Release 1 The Radio Galaxy Zoo (RGZ) is a citizen science project which contains extended radio sources from the FIRST and ATLAS radio surveys cross-matched with their host galaxies in the infrared from the WISE and SpitzerSpace Telescope, respectively [ 4 ]. Instead of classifying morphologies based on the FR scheme, RGZ classifies sources into the number of components, which are discrete patches of emission enclosed by a contour of constant brightness, and the number of peaks of maximum brightness within a component. The catalogue provides 100,185 classifications along with a weighted user consensus level. The user consensus level is calibrated by comparing each individual users’ classifications to that of an expert for a smaller sample of 20 sources. We use RGZ Data Release 1 in this work which contains classifications with a user consensus level greater than or equal to 0.65 . FIRST-based catalogues make up 99.4% of DR1. We select radio sources with angular size >20 arcsecond. The images are of size 72x72. Bayesian neural network To impose some structure in our analysis of the iD estimates of RGZ, we create subsets of the dataset using a BNN. We use the BNN based on Hamiltonian Monte Carlo (HMC) from the benchmark provided for radio galaxy classification in [ 11 ]. This BNN is trained using the MiraBest dataset to classify radio galaxies into FRI and FRII and performs the best in terms of being able to clearly detect distribution shifted radio galaxies based on energy scores [ 17 ]. We pass the RGZ dataset through the trained BNN and obtain a distribution of the scalar energy scores, E(x;f) := −T. log PK iefi(x)/T , for all the sources in the dataset, x , using the logit values, fi(x) , for each class, i . The temperature term, T , is set to 1 . We use N= 200 posterior samples from HMC to construct the energy score distributions and calculate the mean and standard deviation of the distribution for each RGZ source. We then fit a log-normal distribution to the distribution of the mean and standard deviation estimates of the entire RGZ dataset and calculate an interval label for each source based on how many standard deviations away a source is from the mean of the mean 3 (a) (b) Figure 2: Intrinsic dimension estimates for the RGZ dataset as a function of the (a) mean interval and (b) standard deviation interval of the energy distribution from the Hamiltonian Monte Carlo based Bayesian neural network trained on the MiraBest dataset. Table 1: Standard deviation interval estimates across benchmarks. “N/A” indicates the method was not applicable due to insufficient samples. Method 1 2 3 4 5 6 MLE (m= 5) 9.318 9.172 17.546 19.157 18.394 20.681 MLE (m= 20) 8.182 7.954 15.667 17.207 18.277 20.353 Local PCA 10.571 11.892 16.907 19.144 20.586 21.556 PPCA 4977 3777 N/A N/A N/A N/A energy values and the mean of the standard deviation values of the energy distributions. We find that the RGZ sources fall into 8 mean intervals and 6 standard deviation intervals. Examples of galaxies from each mean and standard deviation interval are shown in Figure 1 and Figure 4, respectively. Diffusion Intrinsic Dimension Estimation We train a score-based diffusion model on the RGZ data using the weighted denoising score matching objective [ 14 ] and a Denoising diffusion probabilistic model (DDPM) architecture [ 18 ]. The diffusion model is trained for 400 epochs, selecting the model with the lowest validation loss. We use the method described in Section 2 to estimate the intrinsic dimension of data from different intervals of the mean and standard deviation of energy values. We perturb all the samples k= 328 times to calculate the score vector for each noisy point using the diffusion model. The score vectors are collected into a matrix and SVD is performed to estimate iD. The score spectrum plots are shown in Figure 5. Baseline Intrinsic Dimension Estimation We estimate the iD for each sample in the RGZ dataset using the MLE method with two neighbouring settings: m= 5 and m= 20 . In addition, we compute the per-sample estimates using local PCA via the Fukunaga-Olsen method [ 19 ], setting alpha to 0.05 . Probabilistic PCA produces only global estimates, so iD values were instead computed per label. 4 Results & Conclusion Figure 2 and Tables 2-1 show that the estimated iD increase with interval number for both the mean and standard deviations. This suggests that the object identified with the BNN as out-of-distribution have a higher iD than the objects that are in distribution. Comparisons with the classical estimators (MLE, LPCA, and PPCA) show that, while they all broadly follow the same trend, the diffusion estimates are much larger in magnitude. This could be due to the 4 Figure 3: Signal to noise ratio versus the intrinsic dimension for a subset of galaxies labelled by Fanaroff-Riley class. Both axes are shown in logarithmic scale. Table 2: Mean interval estimates across benchmarks. “N/A” indicates the method was not applicable due to insufficient samples. Method 1 2 3 4 5 6 7 8 MLE (m= 5) 9.249 9.927 12.261 13.437 17.958 21.655 26.092 40.034 MLE (m= 20) 8.086 8.645 11.309 12.728 16.174 21.155 25.298 38.033 Local PCA 10.702 12.064 15.368 16.653 17.331 17.523 16.725 17.889 PPCA 4357 3638 N/A N/A N/A N/A N/A N/A fact that the classical methods tend to underestimate the iD of high dimensional data [ 20 ]. In addition, the diffusion iD estimates are also much larger compared to those reported for natural image datasets in previous work [15, 6]. This could be due to the inherently noisy nature of radio astronomy data. Figure 3 shows the relationship between the signal-to-noise ratio (SNR) and the iD of each galaxy, along with their FR morphology. Overall, there is no strong relationship between SNR and iD. However, at low iD values ( <100 ), FR II sources tend to exhibit higher SNRs, suggesting that galaxies with lower iD are less affected by noise. On the other hand, if a galaxy has a high SNR, it does not necessarily have a low iD. In future work, we will extend this approach to estimate the iD using the features learned by selfsupervised learning based models trained on the RGZ dataset ([ 21 ]). We will use the results to determine what degree of compression can be achieved for different subsets of the data. Acknowledgments and Disclosure of Funding This work was supported by the Engineering and Physical Sciences Research Council EP/Y030826/1. 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The dataset was constructed using the sample selection and classification described in [ 23 ], who made use of the parent galaxy sample from [ 24 ]. Optical data from data release 7 of Sloan Digital Sky Survey [SDSS DR7; 25 ] was cross-matched with NRAO VLA Sky Survey [NVSS; 26 ] and Faint Images of the Radio Sky at Twenty-Centimeters [FIRST; 27 ] radio surveys. The galaxies are labelled using the FRI and FRII morphological types based on the definition of [ 28 ] and further divided into their subtypes. In addition to labelling the sources as FRI, FRII and their subtypes, each source is also flagged as ‘Confident’ or ‘Uncertain’ to indicate the human classifiers’ confidence while labelling the dataset. In this work we use the MiraBest Confident subset and consider only the binary FRI/FRII classification. The training and validation sets are created by splitting the predefined training data into a ratio of 80:20. The final split consists of 584 training samples, 145 validation samples, and 104 withheld test samples. B Score Spectra Figure 5 shows the score spectra for 100 randomly selected samples for each label. 8 (a) (b) Figure 5: Score spectrum plots for up to 100 randomly selected RGZ sources from different (a) mean and (b) standard deviation intervals of the energy distribution. Here, the x-axis shows the singular values which go up to the ambient dimension (72x72 for RGZ images) and the y-axis shows the magnitude of each singular value. The point at which there is a sharp drop in singular values indicates the normal dimension. This can be subtracted from the ambient dimension to calculate the intrinsic dimension. 9