SlowGuess/NeurIPS_2023_OCR huggingface dataset:
VAE with Riemannian manifold latent space. The latent space of VAE reflects the geometrical property of the representations of the data. The efficacy of setting the latent space to be hyperbolic space (Mathieu et al., 2019; Nagano et al., 2019; Cho et al., 2022) or elliptic space (Xu & Durrett, 2018; Davidson et al., 2018) has been verified for various datasets. Skopek et al. (2019) further extend the approach to enable the latent space to be the product of Riemannian manifolds with different learnable curvatures. On top of these, we explore the method of setting the latent space to be the diagonal Gaussian manifold, which can be viewed as the product of hyperbolic spaces, and provide a novel viewpoint on prior work with information geometry.
Distributions over hyperbolic space. Defining a tractable distribution over hyperbolic space is challenging. Nagano et al. (2019) suggest hyperbolic wrapped normal distribution (HWN) from the observation that the tangent space is Euclidean space. Leveraging operations defined on the tangent spaces, e.g., parallel transport, enables an easy sampling algorithm. Mathieu et al. (2019) propose a rejection sampling method for the Riemannian normal distribution defined on the Poincaré disk model, namely Poincaré normal distribution. This method rejects the pathological samples and enables accurate sampling from the distribution in exchange for high computational complexity.
Although these distributions are widely adopted in many applications (Skopek et al., 2019; Mathieu & Nickel, 2020; Cho et al., 2022), one can barely adopt the full covariance matrix due to the difficulties
Algorithm 1 Decoder
Input Parameter  $(\alpha, \beta) \in \mathcal{G}_c, \gamma$ , Decoding layers  $\mathrm{Dec}(\cdot)$
Output Reconstruction  $\mathbf{x}'$
1: Sample  $\mu \sim \mathcal{N}(\alpha, \beta^2\gamma^2)$
2: Sample log  $\sigma^2\sim$  Gamma  $\left(\frac{1}{4c\gamma^2} +1,\frac{1}{4c\beta^2\gamma^2}\right)$
3:  $\mathbf{x}' = \mathrm{Dec}([\mu, \log \sigma^2])$
4: return x'
Table 1: Density estimation on real-world datasets.  $d$  denotes the latent dimension. We report the negative test log-likelihoods of average 10 runs for Breakout, CUB, Food101, and Oxford102 with  $95\%$  confidence interval. N/A in the log-likelihood indicates that the results are not available due to the failure of all runs, and N/A in the standard deviation indicates the results are not available due to failures of some runs. The best results are bolded.
d	E-VAE	L-VAE	P-VAE	GM-VAE (c=1)	GM-VAE (c=1/2)	GM-VAE (c=3/2)
Breakout	2	124.74±0.86	122.58N/A	270.05±2.84	121.52±1.00	122.64±1.13	122.47±1.98
4	66.39±0.76	66.70±0.32	271.73±42.95	65.83±0.49	66.39±0.50	65.80±0.49
8	44.97±0.37	45.25±0.27	81.55±64.61	45.14±0.30	45.31±0.36	45.36±0.49
CUB	50	992.05±1.38	993.03±1.64	990.49±2.26	985.46±3.82	986.27±3.81	979.14±3.70
60	969.99±3.13	968.79±3.70	964.02±3.55	958.00±3.25	960.88±3.46	956.77±2.53
70	949.13±2.72	948.88±3.19	944.24±4.40	939.08±3.12	942.34±3.44	937.15±2.76
Food101	50	1297.81±4.51	1298.45±6.32	1293.26±7.14	1286.30±6.19	1299.58±7.02	1290.57±8.23
60	1224.03±8.31	1227.16±5.18	1218.09±3.88	1213.31±3.88	1216.63±4.56	1207.30±5.12
70	1164.95±3.80	1165.39±5.54	1165.91±4.91	1152.80±3.35	1160.97±4.18	1149.56±3.41
Oxford102	50	1297.41±2.69	1296.41±1.56	1294.12±1.80	1292.90±3.43	1289.43±2.46	1289.99±1.72
60	1253.80±2.57	1256.52±2.99	1251.77±1.82	1245.49±2.18	1248.72±1.62	1247.47±2.51
70	1231.52±3.18	1229.38±3.44	1219.75±1.72	1215.07±2.52	1218.54±3.85	1214.85±2.56
in Monte-Carlo based KL approximation. The number of samples to approximate the KL divergence increases exponentially when the full covariance matrix is used (Cho et al., 2022), so it is common to use isotropic or diagonal covariance instead. Especially in the Poincaré normal, the computation of KL divergence is slow due to the expensive rejection sampling.
RL with hyperbolic space. The hierarchical relationship between the states lying on the trajectories earned from RL agents has been gaining attention recently. Nagano et al. (2019) have studied that the hierarchical structure of Atari2600 Breakout game states can be well-captured with hyperbolic VAEs. We compare the same task, where GM-VAE outperforms the previous work. Cetin et al. (2022) suggest using hyperbolic space as the geometric prior for representation learning in model-free RL agent, showing improvements in generalization performances. Here, we focus on model-based RL, especially the method of using the world model (Ha & Schmidhuber, 2018; Hafner et al., 2020), and open the possibility of applying hyperbolic space to broader domains of RL by solving the bottleneck of the numerical stability.
In this section, we demonstrate the performances of GM-VAE on two tasks: density estimation of image datasets and model-based RL. We remark on the practical properties of GM-VAE shown in the experiments with additional analyses.
5.1 Density estimation on image datasets
We conduct density estimation on image datasets to measure the effectiveness of hyperbolic latent space against Euclidean space with the proposed GM-VAE. We use three datasets: the images from Atari2600 Breakout with binarization (Breakout) (Nagano et al., 2019), Oxford 102 Flower (Oxford102) (Nilsback & Zisserman, 2008), Food101 (Bossard et al., 2014), and Caltech-UCSD Birds200-2011 (CUB) (Wah et al., 2011). The datasets are chosen with the four lowest  $\delta$ -hyperbolicity ( $\delta$ -H), a metric that measures how the given images are well-embed in hyperbolic space. Low  $\delta$ -H implies that the dataset is likely to embed in hyperbolic space. The details about  $\delta$ -H are available in Appendix D. The values of  $\delta$ -H for the four datasets and other candidate datasets are in Appendix D.2. Several studies show that the images from the chosen datasets have an implicit hierarchical structure (Nagano et al., 2019; Li et al., 2019; Bossard et al., 2014; Kerdels & Peters, 2015).
We compare GM-VAE with the three baseline models: VAE with Euclidean latent space  $(\mathcal{E}$  -VAE), and hyperbolic VAE equipped with HWN (L-VAE) and Poincaré normal  $(\mathcal{P}$  -VAE). We use the product
latent space for both  $\mathcal{L}$ -VAE and  $\mathcal{P}$ -VAE, and set the curvature value to  $-1$ . The other details on the implementation and experimental setups are described in Appendix E.1.
The results are reported at Table 1. GM-VAE outperforms the baselines in all the settings, except one case of Breakout. Especially in CUB and Oxford102, GM-VAE outperforms the baselines regardless of the curvature value. In Breakout,  $\mathcal{P}$ -VAE shows inferior performance due to unstable training, and  $\mathcal{L}$ -VAE fails in some of the runs with small latent dimension. The results of  $\mathcal{P}$ -VAE and  $\mathcal{L}$ -VAE with non-product latent space, a common choice in previous work, are also present in Appendix F.
5.2 State representation learning in model-based RL
We focus on the model-based RL task to verify the utility of GM-VAE on various tasks. Specifically, we apply GM-VAE to a world model, which aims to learn the representation of the environments (Ha & Schmidhuber, 2018; Hafner et al., 2019a,b, 2020). We use DreamerV2 (Hafner et al., 2020) as the baseline model to evaluate the performance of GM-VAE in modeling environments. DreamerV2 is composed of a recurrent state space model (RSSM) (Hafner et al., 2019b) and three predictors for the image  $p_{\phi}(x_t|h_t,z_t)$ , the reward  $p_{\phi}(r_t|h_t,z_t)$ , and the discount factor  $p_{\phi}(\gamma_t|h_t,z_t)$ , where  $x_{t}$  is the observation which the format is the image,  $r_t$  is the reward,  $\gamma_{t}$  is the discounting factor,  $h_t$  is the deterministic recurrent state, and  $z_{t}$  is the stochastic state. The model is trained by maximizing the likelihood of  $p(\mathbf{x},\mathbf{r},\boldsymbol {\gamma}\mid \mathbf{a})$  given observations  $\mathbf{x}$ , rewards  $\mathbf{r}$ , and discount factors  $\gamma$  earned from the sequence of actions  $\mathbf{a}$  of an agent. By deriving the evidence lower bound of  $p(\mathbf{x},\mathbf{r},\boldsymbol {\gamma}\mid \mathbf{a})$ , the world model is learned to optimize the likelihood with the variational distribution  $q_{\theta}(z_{t}\mid h_{t},x_{t})$  with the following objective  $\mathcal{L}(\phi ,\theta)$  as:
$$
\mathcal {L} (\phi , \theta) = \mathbb {E} \left[ \sum_ {t = 1} ^ {T} \left(- \log p _ {\phi} \left(x _ {t}, r _ {t}, \gamma_ {t} \mid h _ {t}, z _ {t}\right) + \beta \mathrm {D} _ {\mathrm {K L}} \left[ q _ {\theta} \left(z _ {t} \mid h _ {t}, x _ {t}\right) \| p _ {\phi} \left(z _ {t} \mid h _ {t}\right) \right]\right) \right],
$$
where  $\beta$  is KL loss scaling factor,  $T$  is the length of input sequence,  $p_{\phi}$  is the prior, and  $q_{\phi}$  is the approximated posterior. GM-VAE is employed by replacing the space of  $z_{t}$  with the Gaussian manifold and two components in RSSM, the representation model  $q_{\theta}(z_t|h_t,x_t)$  and transition predictor  $p_{\phi}(z_t|h_t)$ , with PGM normal.
We compare evaluation scores between different types of latent space on world model learning over the Atari2600 environments. The agents are trained with 100M environment steps. We select games having the  $\delta$ -H values of the four lowest and the two highest among 60 popular Atari2600 games. The other details on the implementation and experimental setups are described in Appendix E.2 with the  $\delta$ -H for all 60 games in Appendix D.3.
With a commonly-used hyperbolic distribution, i.e., HWN, we observe that training the world model fails due to the numerical stability issue. On the other hand, GM-VAE shows competitive results with the baselines in Euclidean and discrete latent space in all the games we test. The results are reported in Figure 2b. We note that the reproduced Euclidean baseline results by using the official code are better than those reported in Hafner et al. (2020).
5.3 Remark on GM-VAE
Numerical stability. One notable property of GM-VAE is the numerical stability during training compared to  $\mathcal{L}$ -VAE and  $\mathcal{P}$ -VAE. During the experiments,  $\mathcal{L}$ -VAE and  $\mathcal{P}$ -VAE fail to run in some of the Breakout image density estimations and all the seeds of model-based RL due to the numerical instability. Similar observations are also reported in several previous works (Mathieu et al., 2019; Chen et al., 2021; Skopek et al., 2019). The sampling from a hyperbolic distribution is a major cause of the numerical instability. Consequently, the sampling-based KL divergence computation can be unstable.
We first show that sampling from PGM normal can be stabilized via a simple reparameterization trick. To train GM-VAE, one needs to obtain sample  $\mu$  and  $\sigma$  from PGN normal  $\mathcal{K}_c(\alpha, \beta, \gamma)$ . Sampling  $\mu$  can be done from  $\mathcal{N}(\alpha, \beta^2\gamma^2)$  without numerical issues. Sampling  $\sigma$  can be done from  $\operatorname{Gamma}(a, b)$  where  $a = 1/4c\gamma^2 + 1$ ,  $b = 1/4c\beta^2\gamma^2$  as shown in Appendix C.2. However, due to machine precision error, often,  $\beta$  violates the manifold constraints, i.e.,  $\beta = 0$ . Eventually, direct sampling of  $\sigma$  can cause the numerical instability. To avoid  $\beta$  being zero, we use the output of the VAE encoder as  $\log \beta^2$  whose value ranges over the entire real numbers and is more stable even when  $\beta$  is close to
(a) Latent space analysis
Figure 2: The results of model-based RL experiment. (a) The dots from yellow to purple represent the latent states from the world model in the Atari2600 Breakout with decreasing rewards. Along the red geodesic dashed line passing, we sample for images to visualize the learned representations. As the sample shows, we can observe a hierarchical structure at different stages of the game along the geodesic. (b) We compare the methods of using Euclidean, discrete, and hyperbolic latent space. We report averaged rewards over four runs and bold the best reward.
Latent space	Euc.	Disc.	Hyp.	δ-H
Breakout	329.0	256.8	319.3	0.12
Alien	3412.5	3120.0	3485.0	0.14
Zaxxon	34275	38825	38950	0.14
Ice Hockey	25.50	11.80	20.75	0.14
Freeway	32.8	33.0	33.0	0.38
Krull	53290	36135	66185	0.38
zero. With  $\log \beta^2$ , instead of sampling  $\sigma^2$ , we sample  $\log \sigma^2 = \log \epsilon + \log b$ , where  $\epsilon$  is sampled from  $\mathrm{Gamma}(a, 1)$ , through the reparameterization of the Gamma distribution, where  $\log b$  can be directly computed from  $\log \beta^2$ .
We can show that the KL divergence between an arbitrary PGM normal and prior distribution  $\mathcal{K}_c(\mathbf{0},I,I)$  has a closed-form solution without any sampling. The KL divergence of PGM normal is the sum of the KL divergences between two Gaussian distributions and between two Gamma distributions, as shown in Appendix C.3. First, the KL divergence between a univariate Gaussian distribution  $\mathcal{N}(\mu ,\sigma^2)$  and the prior distribution can be obtained with log  $\sigma^2$  as shown in Equation 4. Second, the KL divergence between the two Gamma distributions,  $\mathrm{Gamma}(a_1,b_1)$  and  $\mathrm{Gamma}(a_2,b_2)$ , written as:
$$
D _ {\mathrm {K L}} (\operatorname {G a m m a} (a _ {1}, b _ {1}) \parallel \operatorname {G a m m a} (a _ {2}, b _ {2})) = a _ {2} \log \frac {b _ {1}}{b _ {2}} - \ln \frac {\Gamma (a _ {1})}{\Gamma (a _ {2})} + (a _ {1} - a _ {2}) \psi (a _ {1}) - \left(1 - \frac {b _ {2}}{b _ {1}}\right) a _ {1},
$$
where  $\psi$  is the digamma function, can be computed using  $\log b$ .
Training time comparison. Common bottle-necks of the mode hyperbolic VAEs arise from the complex manifold constraints and the difficulty of sampling from the hyperbolic distributions. For example, the Poincaré disk model of  $\mathcal{P}$ -VAE and the Lorentz model of  $\mathcal{L}$ -VAE requires the samples to be inside of a unit disk and to be on a hyperboloid with constraint  $\{\mathbf{x} \in \mathbb{R}^{n+1} \mid -x_0^2 + \sum_{i=1}^n x_i^2\}$ , respectively. Such manifolds need complex transformations, e.g., clipping, projection, or geometric transformations using the Riemannian operations, to
Table 2: The training time of the VAEs in density estimation of the Breakout image dataset. We report the training time of the VAEs in seconds per epoch. GM-VAE is  $1.93\mathrm{x}$  faster than  $\mathcal{P}$ -VAE and  $1.41\mathrm{x}$  faster than  $\mathcal{L}$ -VAE in the experiments held on a single A100 40GB PCI GPU.
(b) Results
E-VAE	L-VAE	P-VAE	GM-VAE
24.5	35.9	49.2	25.5
match the manifold constraint so making the training of the hyperbolic VAEs slower. The Gaussian manifold, on the other hand, has a much simple manifold constraint and even does not require any transformations if we utilize the log space of  $\sigma$ .
We report the time consumptions of the VAEs with the latent dimension of 8 per epoch in the density estimation of Breakout at Table 2. The results demonstrate that the algorithmic distinctions enable GM-VAE to be trained much faster than the baseline hyperbolic VAEs and even similar to  $\mathcal{E}$ -VAE.
Latent space analysis. We present a plot of the learned representation in the hyperbolic space at Figure 2a for qualitative analysis. We take the world model with GM-VAE trained for Breakout and illustrate the geodesic starting from the origin in the figure with four generated samples along with the geodesic. We also provide the scatter plot of game states with their cumulative rewards represented in different colors. The brighter the color, the higher the cumulative reward. The scatter
plot reveals that the states with high cumulative rewards are distributed near the origin. Together with the samples from the geodesic, we can observe that the hierarchical structure of Breakout is well captured in the latent space.
Note that in the Poincaré disk model, the depth of the hierarchy is expected to be shown as the distance from the origin (Nickel & Kiela, 2017). When the root node is placed near the origin, the leaf nodes are likely to be placed near the boundary of the open disk. The geodesic lines starting from the origin to the boundary of the Poincaré disk model are identical to the geodesics of the Gaussian manifold starting from  $(0,1)$ , i.e., the origin of the Gaussian manifold. The connection implies that the data hierarchy should be aligned along the geodesic curves if the hierarchy is well captured.
To quantitatively measure the correlation between the cumulative rewards and the states, we measure the Pearson correlation between the cumulative reward and the norm of the states. We obtain a correlation coefficient of 0.46 from the hyperbolic latent space, whereas the correlation coefficient of the Euclidean latent space is 0.40, showing the hyperbolic space better captures the hierarchy along the increasing norm. More experimental details are explained in Appendix G.2.
6 Conclusion & Future Work
In this work, we propose a novel method of representation learning with GM-VAE, utilizing the Gaussian manifold for the latent space. With the newly-proposed PGM normal defined over the Gaussian manifold, which shows better stability and ease of sampling compared to the commonly-used ones, we verify the efficacy of our method on several tasks. Our method achieves outperforming results on density estimation with image datasets and competitive results on model-based RL compared to the baselines. We explain the behavior of GM-VAE in terms of solving the frequent numerical issue of commonly-used hyperbolic VAEs. The analysis of latent space exhibits that the hierarchy lying in the dataset can be preserved by using GM-VAE.
We suggest that the numerical stability of our method can be helpful for scaling the generative models, e.g., very deep VAE (Child, 2021), endowed with hyperbolic geometrical priors. As GM-VAE is beneficial for capturing hierarchy with promising results in modeling RL environment, another potential future work can be extending the use of hyperbolic space, such as learning a skill tree for solving complex long-horizon tasks (Shi et al., 2022). We believe that the connection between the statistical manifold and hyperbolic space provides new insight to the research community and hope to see more interesting connections and analyses in the future.
This work was partly supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-01906, Artificial Intelligence Graduate School Program (POSTECH)) and National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. RS-2023-00217286) and National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2021R1C1C1011375)
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A Machine Precision Error Analysis on Hyperbolic Space
Figure 3: The machine precision errors of the hyperbolic models. For the Poincare half-plane model, we report the upper bound of the machine precision error derived by Yu & Sa (2021). For the Lorentz model, we report the error between the Lorentzian product of the point and -1, which is the manifold constraint of the Lorentz model. For the Poincare disk model, we report the threshold that the norm of the point becomes one. The precision is set to 32 bits. The analysis reveals that all the three models suffer from numerical stability issue with points which the distance between the origin is farther than 20.
B Gaussian Manifold
B.1 Curvature of the Gaussian manifold
We construct a Riemannian manifold  $\{(\mu, \sigma) \mid \mu \in \mathbb{R}, \sigma \in \mathbb{R}_{>0}\}$  with a positive constant  $c$  and the metric tensor  $\sigma^{-2} \mathrm{diag}(1, 1/c)$ , which we will name Gaussian manifold. We need to show the value of the curvature.
First, we need to compute the Christoeffel symbols of the Gaussian manifold defined as:
$$
\Gamma_ {i j} ^ {k} = \frac {1}{2} g ^ {k l} \left(\frac {\partial g _ {j l}}{\partial g _ {i}} + \frac {\partial g _ {i l}}{\partial g _ {j}} - \frac {\partial g _ {i j}}{\partial g _ {l}}\right),
$$
where  $g_{ij}$  is the  $(i,j)$  element of the metric tensor and  $g^{ij}$  is the  $(i,j)$  element of the inverse of the metric tensor.
The Christoeffel symbols of the Gaussian manifold are:
$$
\Gamma_ {i j} ^ {1} = \left( \begin{array}{c c} 0 & - \frac {1}{\sigma} \\ - \frac {1}{\sigma} & 0 \end{array} \right)
$$
$$
\Gamma_ {i j} ^ {2} = \left( \begin{array}{c c} \frac {c}{\sigma} & 0 \\ 0 & - \frac {1}{\sigma} \end{array} \right).
$$
Then, the sectional curvature of the space  $\kappa_{g}$  with given tangent vectors  $du, dv$  is computed as:
$$
\begin{array}{l} \kappa_ {g} = \frac {\langle R (d \mu , d \sigma) d \sigma , d \mu \rangle}{\det g} \\ = \frac {1}{\det  g} \cdot g _ {1 m} \left(\frac {\partial \Gamma_ {2 2} ^ {m}}{\partial \mu} - \frac {\partial \Gamma_ {1 2} ^ {m}}{\partial \sigma} + \Gamma_ {2 2} ^ {p} \Gamma_ {1 p} ^ {m} - \Gamma_ {1 2} ^ {p} \Gamma_ {2 p} ^ {m}\right) \\ = \frac {- \frac {1}{\sigma^ {4}}}{\frac {1}{c \sigma^ {4}}} \\ = - c. \\ \end{array}
$$