SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $L(\cdot;0,\Sigma)$  denotes some likelihood model with zero mean and covariance  $\Sigma$ , and the mean function  $u$  can be given a prior either Besov or Q-EP. Because of the definition (1) in terms of expanded series, there is no explicit formula for the posterior prediction using Besov prior. By contrast, a tractable formula exists for the posterior predictive distribution for (15) with Q-EP prior  $\mu_0 = \mathbf{q} - \mathcal{E}\mathcal{P}(0,\mathcal{C})$  when the likelihood happens to be  $L(\cdot;0,C) = \mathrm{q} - \mathrm{ED}(\mathbf{0},\mathbf{C})$ , as stated in the following theorem.
Theorem 3.5 (Posterior Prediction). Given covariates  $\mathbf{x} = \{x_{i}\}_{i = 1}^{N}$  and observations  $\mathbf{y} = \{y_{i}\}_{i = 1}^{N}$  following q-ED in the model (15) with q-  $\mathcal{E}\mathcal{P}$  prior for the same  $q > 0$ , we have the following posterior predictive distribution for  $u(x_{*})$  at (a) new point(s)  $x_{*}$ :
$$
u \left(x _ {*}\right) | \mathbf {y}, \mathbf {x}, x _ {*} \sim \mathrm {q} - \operatorname {E D} \left(\boldsymbol {\mu} ^ {*}, \mathbf {C} ^ {*}\right), \quad \boldsymbol {\mu} ^ {*} = \mathbf {C} _ {*} ^ {\top} (\mathbf {C} + \Sigma) ^ {- 1} \mathbf {y}, \quad \mathbf {C} ^ {*} = \mathbf {C} _ {* *} - \mathbf {C} _ {*} ^ {\top} (\mathbf {C} + \Sigma) ^ {- 1} \mathbf {C} _ {*} \tag {16}
$$
where  $\mathbf{C} = \mathcal{C}(\mathbf{x},\mathbf{x})$ $\mathbf{C}_{*} = \mathcal{C}(\mathbf{x},x_{*})$  and  $\mathbf{C}_{**} = \mathcal{C}(x_{*},x_{*})$
Proof. See Appendix A.6.
Remark 4. From Theorem 3.5, we know that  $Q$ -EP has same predictive mean as  $GP$ . But their predictive covariances differ by a constant  $\left(\frac{2}{d}\right)^{\frac{2}{q}}\frac{\Gamma\left(\frac{d}{2} + \frac{2}{q}\right)}{\Gamma\left(\frac{d}{2}\right)}$  (asymptotically 1) based on Proposition 3.1.
When the likelihood  $L$  is not Q-EP, e.g. multinomial, such conjugacy is absent. Then we refer to the following sampling method for the posterior inference.
3.3.1 Inference by White-Noise MCMC
We follow [13] to consider the pushforward  $(T)$  of Gaussian white noise  $\nu_{0}$  for non-Gaussian measures  $\mu_0 = T^\# \nu_0$ . More specifically, we construct a measurable transformation  $T:\mathbf{z}\to \mathbf{u}$  that maps standard Gaussian random variables to  $q$ -exponential random variables. The transformation based on the stochastic representation (6) is more straightforward than that for Besov based on series expansions proposed by [13].
Recall the stochastic representation (6) of  $\mathbf{u} \sim \mathrm{q - ED}_d(\mathbf{0},\mathbf{C})$ :  $\mathbf{u} = R\mathbf{L}S$  with  $R^q \sim \chi_d^2$  and  $S \sim \mathrm{Unif}(\mathcal{S}^{d + 1})$ . We can rewrite  $S \stackrel{d}{=} \frac{\mathbf{z}}{\|\mathbf{z}\|_2}$ ,  $R^q \stackrel{d}{=} \| \mathbf{z}\|_2^2$ , for  $\mathbf{z} \sim \mathcal{N}_d(\mathbf{0},\mathbf{I}_d)$ . Therefore, we have the pushforward mapping  $(T)$  and its inverse  $(T^{-1})$  as
$$
\mathbf {u} = T (\mathbf {z}) = \mathbf {L} \mathbf {z} \| \mathbf {z} \| ^ {\frac {2}{q} - 1}, \quad \mathbf {z} = T ^ {- 1} (\mathbf {u}) = \mathbf {L} ^ {- 1} \mathbf {u} \| \mathbf {L} ^ {- 1} \mathbf {u} \| ^ {\frac {q}{2} - 1} \tag {17}
$$
Figure B.1 illustrates that sampling with the white-noise representation (17) is indistinguishable from sampling by the stochastic representation (6). Then we can apply such white-noise representation to dimension-independent MCMC algorithms including preconditioned Crank-Nicolson (pCN) [14], infinite-dimensional Metropolis adjusted Langevin algorithm ( $\infty$ -MALA) [10], infinite-dimensional Hamiltonian Monte Carlo ( $\infty$ -HMC) [7], and infinite-dimensional manifold MALA ( $\infty$ -mMALA) [8] and HMC ( $\infty$ -mHMC) [9]. See Algorithm 1 for an implementation on pCN, hence named white-noise  $pCN$  ( $wn-pCN$ ).
3.3.2 Hyper-parameter Tuning
As in GP, there are hyper-parameters in the covariance function of Q-EP, e.g. variance magnitude  $(\sigma^2)$  and correlation length  $(\rho)$ , that require careful adjustment and fine tuning. If the data process  $y(x)$  and its mean  $u(x)$  are both Q-EP with the same  $q$ , then we can have the marginal likelihood [38] as another Q-EP (c.f. Theorem 3.5). In general, when there is no such tractability, hyper-parameter tuning by optimizing the marginal likelihood is unavailable. However, we could impose conjugate hyper-priors on some parameters or even marginalize them to facilitate the inference of them (See Appendix A.7 for a proposition on such conditional conjugacy for the variance magnitude  $\sigma^2$ ).
To tune the correlation length  $(\rho)$ , we could impose a hyper-prior for  $\rho$  and sample from  $p(\rho|\mathbf{u})$ . We then alternate updating  $\mathbf{u}$  and hyper-parameters in a Gibbs scheme. In general, one could also use Bayesian optimization methods [33, 18, 4] for the hyper-parameter tuning.
4 NUMERICAL EXPERIMENTS
In this section, we compare GP, Besov and Q-EP by modeling time series (temporal), reconstructing images (spatial) from computed tomography and solving a (spatiotemporal) inverse problem (Appendix C.4). These numerical experiments demonstrate that our proposed Q-EP enables faster convergence in obtaining a better maximum a posteriori (MAP) estimate. What is more, white-noise MCMC based inference provides appropriate uncertainty quantification (UQ) (by the posterior standard deviation). More numerical results can be found in the supplementary materials which also contain some demonstration codes. All the computer codes are publicly available at https://github.com/lanzithinking/Q-EXP.
4.1 Time Series Modeling
We first consider two simulated time series, one with step jumps and the other with sharp turnings, whose true trajectories are as follows:
$$
u _ {J} (t) = 1, \quad t \in [ 0, 1 ]; \quad 0. 5, \quad t \in (1, 1. 5 ]; \quad 2, \quad t \in (1. 5, 2 ]; \quad 0, \quad o t h e r w i s e
$$
$$
u _ {\mathrm {T}} (t) = 1. 5 t, \quad t \in [ 0, 1 ]; \quad 3. 5 - 2 t, \quad t \in (1, 1. 5 ]; \quad 3 t - 4, \quad t \in (1. 5, 2 ]; \quad 0, \quad o t h e r w i s e
$$
We generate the time series  $\{y_i\}$  by adding Gaussian noises to the true trajectories evaluated at  $N = 200$  evenly spaced points  $\{t_i\}$  in [0,2], that is,  $y_{i}^{*} = u_{*}(t_{i}) + \varepsilon_{i}$ ,  $\varepsilon_{i}\stackrel {ind}{\sim}N(0,\sigma_{*}^{2}(t_{i}))$ ,  $i = 1,\dots ,N$ ,  $*, = \mathrm{J},\mathrm{T}$ . Let  $\sigma_{\mathrm{J}} / \| u_{\mathrm{J}}\| = 0.015$  for  $t_i\in [0,2]$  and  $\sigma_{\mathrm{T}} / \| u_{\mathrm{T}}\| = 0.01$  if  $t_i\in [0,1]$ ; 0.07 if  $t_i\in (1,2]$ . In addition, we also consider two real data sets of Tesla and Google stock prices in 2022. See Figures 4 (and Figures C.2) for the true trajectories (blue lines) and realizations (orange dots) respectively.
(a) Time series with sharp turnings (model fitting).
(c) Tesla stock prices in 2022 (model fitting).
Figure 4: (a)(c) MAP estimates by GP (left), Besov (middle) and Q-EP (right) models. (b)(d) Predictions by GP (left) and Q-EP (right) models. Orange dots are actual realizations (data points). Blue solid lines are true trajectories. Black ticks indicate the training data points. Red dashed lines are MAP estimates. Red dot-dashed lines are predictions with shaded region being credible bands.
(b) Time series with turnings (prediction).
(d) Tesla stock prices in 2022 (prediction).
We use the above likelihood and test three priors: GP, Besov and Q-EP. For Besov, we choose the Fourier basis  $\phi_0(t) = \sqrt{2}$ ,  $\phi_{\ell}(t) = \cos (\pi \ell t)$ ,  $\ell \in \mathbb{N}$  (results with other wavelet bases including Haar, Shannon, Meyer and Mexican Hat are worse hence omitted). For both GP and Q-EP, we adopt the Matérn kernel with  $\nu = \frac{1}{2}$ ,  $\sigma^2 = 1$ ,  $\rho = 0.5$  and  $s = 1$ :  $C(t,t') = \sigma^{2}\frac{2^{1 - \nu}}{\Gamma(\nu)} w^{\nu}K_{\nu}(w)$ ,  $w = \sqrt{2\nu} (\| t - t'\| /\rho)^s$ . In both Besov and Q-EP, we set  $q = 1$ . Figures 4a and 4c (and Figures C.2a and C.2c) compare the MAP estimates (red dashed lines). We can see that Q-EP yields the best estimates closest to the true trajectories in the simulation and the best fit to the Tesla/Google stock prices. We also investigate the negative posterior densities and relative errors,  $\| \hat{u}_* - u_*\| /\| u_*\|$ , as functions of iterations in Figure C.1. Though incomparable in the absolute values, the negative posterior densities indicate faster convergence in both GP and Q-EP models than in Besov model. The error reducing plots on the right panels of subplots in Figure C.1 indicate that Q-EP prior model can achieve the smallest errors. Table 1 compares them in terms of root mean of squared error (RMSE) and log-likelihood (LL).
Table 1: Time series modeling: root mean of squared errors (RMSE) and log-likelihood (LL) values at MAP estimates by GP, Besov and Q-EP prior models.
root mean squared errors (RMSE)	log-likelihood (LL)
Data Sets	GP	Besov	Q-EP	GP	Besov	Q-EP
simulation (jumps)	1.2702	2.1603	1.1083	-31.4582	-89.8549	-74.0590
simulation (turnings)	1.4270	2.4556	0.9987	-39.8234	-56.7874	-87.3124
Tesla stocks	180.3769	136.8769	51.2236	-488.6458	-281.3796	-39.4070
Google stocks	44.4236	39.4809	36.8686	-386.1546	-305.0058	-265.9790
Then we consider the prediction problem. In the simulations, the last 1/8 portion and every other of the last but 3/8 part of the data points are selected for testing. The models with GP and Q-EP priors are trained on the rest of the data, as indicated by short " ticks" in Figures 4b and 4d (and Figures C.2b and C.2d). For the Tesla/google stocks, we select every other day in the first half year, every 4 days in the 3rd quarter and every 8 days in the last quarter for training and test on the rest. They pose challenges on both interpolation (among observations) and extrapolation (at no-observation region) tasks. As we can see in those figures, uncertainty grows as the data become scarce. Nevertheless, the Q-EP yields smaller errors than GP. Note, such prediction is not immediately available for models with Besov prior.
Figure 5: Shepp-Logan phantom: true image, observation (sinogram), and MAP estimates by GP, Besov and Q-EP models with relative errors  $68.10\%$ ,  $70.27\%$  and  $40.87\%$  respectively.
4.2 Computed Tomography Imaging
Computed tomography (CT) is a medical imaging technique used to obtain detailed internal images of human body. CT scanners use a rotating X-ray tube and a row of detectors to measure X-ray attenuations by different tissues inside the body from different angles. Denote the true imaging as a function  $u(x)$  on the square unit  $D = [0,1]^2$  taking values as the pixels. The observed data,  $\mathbf{y}$ , (a.k.a. sinogram) are results of Radon transformation (A) of the discretized  $n \times n$  field  $\mathbf{u}$  with  $n_{\theta}$  angles and  $n_s$  sensors, contaminated by noise  $\pmb{\varepsilon}$  [6]:
$$
\mathbf {y} = \mathbf {A} \mathbf {u} + \boldsymbol {\varepsilon}, \quad \boldsymbol {\varepsilon} \sim \mathcal {N} (\mathbf {0}, \sigma_ {\varepsilon} ^ {2} \mathbf {I}), \quad \mathbf {y} \in \mathbb {R} ^ {n _ {\theta} n _ {s}}, \quad \mathbf {A} \in \mathbb {R} ^ {n _ {\theta} n _ {s} \times n ^ {2}}, \quad \mathbf {u} \in \mathbb {R} ^ {n ^ {2}}
$$
In general  $n_{\theta}n_{s} \ll d = n^{2}$  so the linear inverse problem is under-determined. Bayesian approach could fill useful prior information (e.g. edges) in the sparse data.
We first consider the Shepp-Logan phantom, a standard test image created by Shepp and Logan in [42] to model a human head and to test image reconstruction algorithms. In this simulation, we create the true image  $u^{\dagger}$  for a resolution of  $n^2 = 128 \times 128$  and project it at  $n_\theta = 90$  angles with  $n_s = 100$  equally spaced sensors. The generated sinogram is then added by noise with signal noise ratio  $\mathrm{SNR} = \| \mathbf{A}\mathbf{u}^{\dagger} \| / \| \pmb{\varepsilon} \| = 100$ . The first two panels of Figure 5 show the truth and the observation.
Table 2: Posterior estimates of Shepp-Logan phantom by GP, Besov and Q-EP prior models: relative error, RLE :=  $\| \hat{u} - u^{\dagger} \| / \| u^{\dagger} \|$ , of MAP ( $\hat{u} = u^{*}$ ) and posterior mean ( $\hat{u} = \overline{u}$ ) respectively, log-likelihood (LL), PSNR, SSIM and HarrPSI. Numbers in the bracket are standard deviations obtained repeating the experiments for 10 times with different random seeds.
MAP	Posterior Mean
GP	Besov	Q-EP	GP	Besov	Q-EP
RLE	0.6810	0.7027	0.4087	0.4917(6.16e-7)	0.4894(3.53e-5)	0.4890(4.79e-5)
LL	-1.55e+6	-1.54e+6	-1.57e+5	-5.21e+5(8.47)	-4.80e+5(196.34)	-4.56e+5(307.97)
PSNR	15.5531	15.2806	19.9887	18.3826(1.09e-5)	18.4226(6.27e-4)	18.4303(8.51e-4)
SSIM	0.4028	0.3703	0.5967	0.5561(3.92e-7)	0.5535(2.38e-4)	0.5403(5.26e-4)
HaarPSI	0.0961	0.0870	0.3105	0.3126(1.52e-8)	0.3126(3.36e-4)	0.3122(3.06e-4)
Note, the computation involving a full sized  $(d\times d)$  kernel matrix  $\mathbf{C}$  for GP and Q-EP is prohibitive. Therefore, we consider its Mercer's expansion (12) with Fourier basis for a truncation at the first  $L = 2000$  items. Figure 5 shows that while GP and Besov models reconstruct very blurry phantom images, the Q-EP prior model produces MAP estimate of the highest quality. For each of the three models, we also apply wn-pCN to generate 10000 posterior samples (after discarding 5000) and use them to reconstruct  $u$  (posterior mean or median) and quantify uncertainty (posterior standard deviation).
Table 2 summarizes the errors relative to MAP  $(u^{*})$  and posterior mean  $(\overline{u})$  respectively,  $\| \hat{u} - u^{\dagger} \| / \| u^{\dagger} \|$  (with  $\hat{u}$  being  $u^{*}$  or  $\overline{u}$ ), log-likelihood (LL), and several quality metrics in imaging analysis including the peak signal-to-noise ratio (PSNR) [20], the structured similarity index (SSIM) [46], and the Haar wavelet-based perceptual similarity index (HaarPSI) [39]. Q-EP attains the lowest error and highest quality scores in most cases. In Figure C.3, we compare the uncertainty by these models. It seems that GP has uncertainty filed with more recognizable shape than the other two. However, the posterior standard deviation by GP is much smaller (about  $1\%$  of that with Q-EP) compared with the other two. Therefore, this raises a red flag that GP could be over-confident about a less accurate estimate.
Figure 6: CT of human head (upper) and torso (lower): true image, observation (sinogram), and MAP estimates by GP, Besov and Q-EP models with relative errors  $29.99\%$ ,  $22.41\%$  and  $22.24\%$  (for head) and  $26.11\%$ ,  $21.77\%$  and  $21.53\%$  (for torso) respectively.
Finally, we apply these methods to CT scans of a human cadaver and torso from the Visible Human Project [1]. These images contain  $n^2 = 512 \times 512$  pixels and the sinograms are obtained with  $n_\theta = 200$  angles and  $n_s = 512$  sensors. The first two panels of each row in Figure 6 show a highly calibrated CT reconstruction (treated as "truth") and the observed sinogram. The rest three panels illustrate that both Besov and Q-EP models outperform GP in reconstructions, as verified in the quantitative summaries in Table C.2. Figure C.4 indicates that GP tends to underestimate the uncertainty.
In these CT reconstruction examples, we observe larger discrepancy of performance between Besov and Q-EP in the low-dimensional data-sparse application (Shepp-Logan phantom at resolution  $n^2 = 128 \times 128$  with  $n_\theta = 90$  angles and  $n_s = 100$  sensors) compared with the high-dimensional data-intensive applications (two human body CTs at resolution  $n^2 = 512 \times 512$  with  $n_\theta = 200$  angles and  $n_s = 512$  sensors). This may be due to the truncation in Mercer's kernel representation (12) and different rates of posterior contraction [21, 22, 2]. We will explore them in another journal paper.
5 CONCLUSION
In this paper, we propose the  $q$ -exponential process (Q-EP) as a prior on  $L^q$  functions with a flexible parameter  $q > 0$  to control the degree of regularization. Usually,  $q = 1$  is adopted to capture abrupt changes or sharp contrast in data such as edges in the image as the Besov prior has recently gained popularity for. Compared with GP, Q-EP can impose sharper regularization through  $q$ . Compared with Besov, Q-EP enjoys the explicit formula with more control on the correlation structure as GP. The numerical experiments in time series modeling, image reconstruction and Bayesian inverse problems demonstrate our proposed Q-EP is superior in Bayesian functional data modeling.
In the numerical experiments of current work, we manually grid-search for the optimal hyperparameters. The reported results are not sensitive to some of these hyper-parameters such as the variance magnitude  $(\sigma^2)$  and the correlation length  $(\rho)$  but may change drastically to others like the regularity parameter  $(\nu)$  and the smoothness parameter  $(s)$ . In future, we will incorporate hyper-priors for some of those parameters and adopt a hierarchical scheme to overcome such shortcoming. We plan to study the properties such as regularity of function draws of Q-EP and the posterior contraction, and compare the contraction rates among GP, Besov and Q-EP priors [21, 22, 2]. Future work will also consider operator based kernels such as graph Laplacian [16, 17, 29].
SL is supported by NSF grant DMS-2134256.
[45] Umberto Villa, Noemi Petra, and Omar Ghattas. hIPPYlib: An extensible software framework for large-scale inverse problems governed by PDEs; part i: Deterministic Inversion and Linearized Bayesian Inference. ACM Transactions on Mathematical Software, 47(2):1-34, jun 2021.
[46] Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 13(4):600-612, 2004.
[47] MIKE WEST. On scale mixtures of normal distributions. Biometrika, 74(3):646-648, 09 1987.
Supplement Document for "Bayesian Learning via Q-Exponential Process"
A PROOFS
A.1 Proof of Theorem 3.3
Proof. First we prove the exchangeability of  $\mathrm{q - ED}_d(\pmb {\mu},\mathbf{C})$  with general (non-identity) covariance matrix  $\mathbf{C} = [\mathcal{C}(t_i,t_j)]_{d\times d}$  for some kernel function  $\mathcal{C}$ . It actually holds for all elliptic distributions including MVN. Their densities contain the essential quadratic form  $r(\mathbf{u}) = \mathbf{u}^{\top}\mathbf{C}^{-1}\mathbf{u}$  which is invariant under any permutation of coordinates.
Denote  $\mathbf{u} = [u_{t_1},\dots ,u_{t_i},\dots ,u_{t_j},\dots ,u_{t_d}]^T$  . Without loss of generality, we only need to show  $r(\mathbf{u})$  is invariant by switching two coordinates, say,  $t_i\leftrightarrow t_j$  . Denote  $\mathbf{u}' = [u_{t_1},\dots ,u_{t_j},\dots ,u_{t_i},\dots ,u_{t_d}]^T$  - Switching  $t_i$  and  $t_j$  leads to a different covariance matrix  $\mathbf{C}'$  obtained by switching both  $i$  -th and  $j$  -th rows and columns simultaneously in  $\mathbf{C}$  . If we denote the elementary matrix  $E_{ij}$  as derived from switching  $i$  -th and  $j$  -th rows of the identity matrix I. Then we have
$$
\mathbf {u} ^ {\prime} = E _ {i j} \mathbf {u}, \quad \mathbf {C} ^ {\prime} = E _ {i j} \mathbf {C} E _ {i j}
$$
Note  $E_{ij}$  is idempotent, i.e.  $E_{ij} = E_{ij}^{-1}$ . Therefore
$$
\left(\mathbf {u} ^ {\prime}\right) ^ {\top} \left(\mathbf {C} ^ {\prime}\right) ^ {- 1} \mathbf {u} ^ {\prime} = \mathbf {u} ^ {\top} E _ {i j} E _ {i j} \mathbf {C} ^ {- 1} E _ {i j} E _ {i j} \mathbf {u} = \mathbf {u} ^ {\top} \mathbf {C} ^ {- 1} \mathbf {u}
$$
Next, the consistency directly follows from Kano's consistency Theorem 3.2 with our choice of  $g(r)$ . The proof is hence completed.
A.2 Theorem of Q-EP as a mixture of Gaussians
Theorem A.1. Suppose  $\mathbf{u} \sim \mathrm{q - ED}_d(0,\mathbf{C})$  for  $0 < q < 2$ , then there exist an random variable  $V > 0$  and a standard normal random vector  $\mathbf{Z} \sim \mathcal{N}_d(\mathbf{0},\mathbf{I})$  independent of each other such that  $\mathbf{u} \stackrel{d}{=} \mathbf{Z} / V$ .
Proof. Based on [3], it suffices to show  $(- \frac{d}{dr})^k g(r) \geq 0$  for all  $k \in \mathbb{N}$ . Observe that  $g'(r) = \left[\left(\frac{q}{2} - 1\right)\frac{d}{2} r^{\left(\frac{q}{2} - 1\right)\frac{d}{2} - 1} - \frac{q}{4} r^{\left(\frac{q}{2} - 1\right)\left(\frac{d}{2} + 1\right)}\right] \exp\left\{-\frac{r^{\frac{q}{2}}}{2}\right\} \leq 0$  when  $q \leq 2$ . Denote  $(- \frac{d}{dr})^k g(r) := p_k(r^{\left(\frac{q}{2} - 1\right)/2}, r^{-1}) \exp\left\{-\frac{r^{\frac{q}{2}}}{2}\right\}$  where the coefficients of polynomial  $p_k$  are all non-negative. Then we have
$$
\left(- \frac {d}{d r}\right) ^ {k + 1} g (r) = \left[ - \frac {d}{d r} p _ {k} \left(r ^ {\left(\frac {q}{2} - 1\right) / 2}, r ^ {- 1}\right) + \frac {q}{4} r ^ {\left(\frac {q}{2} - 1\right)} p _ {k} \left(r ^ {\left(\frac {q}{2} - 1\right) / 2}, r ^ {- 1}\right) \right] \exp \{- \frac {r ^ {\frac {q}{2}}}{2} \}
$$
where  $p_{k+1}(r^{\left(\frac{q}{2}-1\right)/2}, r^{-1})$  being the term in the square bracket has all positive coefficients because the powers  $\left(\frac{q}{2}-1\right)/2$  and  $-1$  appear as coefficients in  $\frac{d}{dr} p_k(r^{\left(\frac{q}{2}-1\right)/2}, r^{-1})$  and are both negative. The proof is completed by induction.
A.3 Proposition of distribution of  $r(\mathbf{u})$
The following proposition determines the distribution of  $R = \sqrt{r(\mathbf{u})}$  as  $q$ -root of a gamma (also chi-squared) distribution thus gives a complete recipe for generating random vector  $\mathbf{u} \sim \mathrm{q - ED}_d(0,\mathbf{C})$  based on the stochastic representation (6).
Proposition A.1. If  $\mathbf{u} \sim \mathrm{q - ED}_d(0,\mathbf{C})$ , then we have
$$
R ^ {q} = r ^ {\frac {q}{2}} \sim \Gamma \left(\alpha = \frac {d}{2}, \beta = \frac {1}{2}\right) = \chi_ {d} ^ {2}, \quad a n d \quad \mathrm {E} [ R ^ {k} ] = 2 ^ {\frac {k}{q}} \frac {\Gamma \left(\frac {d}{2} + \frac {k}{q}\right)}{\Gamma \left(\frac {d}{2}\right)} \stackrel {{\prime}} {{\sim}} d ^ {\frac {k}{q}}, a s d \rightarrow \infty , \forall k \in \mathbb {N} \tag {18}
$$
Proof. With out chosen  $g(r)$ , the density of  $r$  becomes
$$
f (r) \propto r ^ {\frac {d}{2} - 1} r ^ {\left(\frac {q}{2} - 1\right) \frac {d}{2}} \exp \left\{- \frac {r ^ {\frac {q}{2}}}{2} \right\} = r ^ {\frac {q}{2} \cdot \frac {d}{2} - 1} \exp \left\{- \frac {r ^ {\frac {q}{2}}}{2} \right\}
$$
A change of variable  $r \to r^{\frac{q}{2}}$  yields the density of  $R^{q} = r^{\frac{q}{2}}$  that can be recognized as the density of  $\chi_{d}^{2}$ .
On the other hand, since  $\nu := R^q \sim \Gamma\left(\alpha = \frac{d}{2}, \beta = \frac{1}{2}\right)$ , we have:
$$
\begin{array}{l} \mathrm {E} [ R ^ {k} ] = \int_ {0} ^ {\infty} v ^ {\frac {k}{q}} f (v) d v = \frac {1}{\Gamma (\frac {d}{2})} \left(\frac {1}{2}\right) ^ {\frac {d}{2}} \int_ {0} ^ {\infty} v ^ {\frac {k}{q} + \frac {d}{2} - 1} \exp \left\{- \frac {1}{2} v \right\} d v \\ = 2 ^ {\frac {k}{q}} \frac {\Gamma (\frac {d}{2} + \frac {k}{q})}{\Gamma (\frac {d}{2})} \dot {\sim} 2 ^ {\frac {k}{q}} \left(\frac {d}{2}\right) ^ {\frac {k}{q}} = d ^ {\frac {k}{q}} \\ \end{array}
$$
where we use  $\Gamma(x + \alpha) \dot{\sim} \Gamma(x)x^{\alpha}$  as  $x \to \infty$  with  $x = \frac{d}{2}$  and  $\alpha = \frac{k}{q}$  when  $d \to \infty$ .
A.4 Proof of Proposition 3.1
Proof. By Theorem 2.6.4 in [19] for  $\mathrm{q - ED}_d(\pmb {\mu},\mathbf{C}) = \mathrm{EC}_d(\pmb {\mu},\mathbf{C},g)$  with our chosen  $g$ , we know  $\operatorname {E}[\mathbf{u}] = \pmb{\mu}$  and  $\operatorname {Cov}(\mathbf{u}) = (\operatorname {E}[R^2 ] / \operatorname {rank}(\mathbf{C}))\mathbf{C}$ . It follows by letting  $k = 2$  in Proposition A.1 and using the similar asymptotic analysis.
A.5 Proof of Theorem 3.4