SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Variational distributions To investigate the basic effectiveness of GeoPhy algorithm, we employ simple constructions for the variational distributions  $Q_{\theta}(z)$ ,  $Q_{\phi}(B_{\tau}|\tau)$ , and  $R_{\psi}(z|\tau)$ . We use an independent distribution for each tip node coordinate, i.e.  $Q_{\theta}(z) = \prod_{i=1}^{N} Q_{\theta_i}(z_i)$ , where we use a  $d$ -dimensional normal or wrapped normal distribution for the coordinates of each tip node  $z_i$ . For the conditional distribution of branch lengths given tree topology,  $Q_{\phi}(B_{\tau}|\tau)$ , we use the diagonal lognormal distribution whose location and scale parameters are given as a function of the LTFs of the topology  $\tau$ , which is comprised of GNNs as proposed in [40]. For the model of  $R_{\psi}(z|\tau)$ , we also employ an independent distribution:  $R_{\psi}(z|\tau) = \prod_{i=1}^{N} R_{\psi_i}(z_i|\tau)$ , where, we use the same type of distribution as  $Q_{\theta_i}(z_i)$ , independent of  $\tau$ .
$\widehat{g}_{\chi}$  is an unbiased estimator of  $\nabla \left\langle \mathbb{V}_{Q(z)}[(\widehat{g}_{\theta})]\right\rangle$  as shown in [7].
Differentiability for discrete optimization Discrete optimization problems often suffer from the lack of informative gradients of the objective functions. To address this issue, continuous relaxation for discrete optimization has been actively studied, such as a widely-used reparameterization trick with the Gumbel-softmax distribution [10, 20]. Beyond categorical variables, recent approaches have further advanced the continuous relaxation techniques to more complex discrete objects, including spanning trees [33]. However, it is still nontrivial to extend such techniques to the case of binary tree topologies. As outlined in equation (4), we have introduced a distribution over binary tree topologies  $\mathcal{T}$  derived from continuous distributions  $Q(z)$ . This method facilitates a gradient-based optimization further aided by variance reduction techniques.
Gradient-based algorithms for tree optimization For the hierarchical clustering (HC), which reconstructs a tree relationship based on the distance measures between samples, gradient-based algorithms [23, 2, 3] have been proposed based on Dasgupta's cost function [4]. In particular, Chami et al. [2] proposed to decode tree topology from hyperbolic coordinates while the optimization is performed for a relaxed cost function, which is differentiable with respect to the coordinates. However, these approaches are not readily applicable to more general problems, including phylogenetic inference, as their formulations depend on the specific form of the cost functions.
Phylogenetic analysis in hyperbolic space The approach of embedding phylogenetic trees into hyperbolic spaces has been explored for visualization and an interpretation of novel samples with existing phylogeny [21, 11]. For the inference task, a maximum-likelihood approach was proposed in [35], which however assumed a simplified likelihood function of pairwise distances. A recent study [19] proposed an MCMC-based algorithm for sampling  $(\tau, B_{\tau})$ , which were linked from coordinates  $z$  using the NJ algorithm [31]. However, there remained the issue of an unevaluated Jacobian determinant, which posed a challenge in evaluating inference objectives. Given that we only use topology  $\tau$  as described in equation (4), the variational lower bound for the inference can be unbiasedly evaluated through sampling, as shown in Proposition 1.
We applied GeoPhy to perform phylogenetic inference on biological sequence datasets of 27 to 64 species compiled in Lakner et al. [17].
Models and training As a prior distribution of  $P(\tau)$  and  $P(B_{\tau}|\tau)$ , we assumed a uniform distribution over all topologies, and an exponential distribution  $\mathrm{Exp}(10)$  independent for all branches, respectively, as commonly used in the literature [42, 16]. For the neural network used in the parameterization of  $Q_{\phi}(B_{\tau}|\tau)$ , we employed edge convolutional operation (EDGE), which was well-performed architecture in [40]. For the stochastic gradient optimizations, we used the Adam optimizer [13] with a learning rate of 0.001. We trained for GeoPhy until one million Monte Carlo tree samples were consumed for the gradient estimation of our loss function. This number equals the number of likelihood evaluations (NLEs) and is used for a standardized comparison of experimental runs [34, 42]. The marginal log-likelihood (MLL) value is estimated with 1000 MC samples. More details of the experimental setup are found in Appendix C.
Initialization of coordinates We initialized the mean parameters of the tip coordinate distribution  $Q_{\theta}(z)$  with the multi-dimensional scaling (MDS) algorithm when  $Q_{\theta}$  was given as normal distributions. For  $Q_{\theta}$  comprised of wrapped normal distributions, we used the hyperbolic MDS algorithm (hMDS) proposed in [32] for the initialization. For a distance matrix used for MDS and hMDS, we used the Hamming distance between each pair of the input sequences  $Y$  as similar to [19]. For the scale parameters, we used 0.1 for all experiments. For  $R_{\psi}(z)$ , we used the same mean parameters as  $Q_{\theta}(z)$  and 1.0 for the scale parameters.
5.1 Exploration of stable learning conditions
To achieve a stable and fast convergence in the posterior approximations, we compared several control variates (CVs) for the variance reduction of the gradients by using DS1 dataset [8]. We
Figure 2: Comparison of marginal log-likelihood (MLL) estimates for DS1 dataset with different control variates. For a variational distribution,  $Q(z)$ , an independent two-dimensional Normal distribution with a diagonal (diag) or full covariance matrix was used for each tip node.  $K$  stands for the number of Monte Carlo samples used for gradient estimation. For reference, we show the mean MLL value (gray dashed lines;  $-7108.42 \pm 0.18$ ) estimated with the MrBayes stepping-stone (SS) method in [42]. We also show MLL estimates obtained with VBPI-GNN (red dashed lines) using VIMCO estimator ( $K = 10$ ). The iterations are counted as the number of likelihood evaluations (NLEs). Legend: NoCV stands for no control variates, LOO for leave-one-out CV, and IW for the case of using importance-weighted ELBO as the learning objective.
demonstrate that the choice of control variates is crucial for optimization (Fig. 2). In particular, adaptive control variates (LAX) for individual Monte Carlo samples and the leave-one-out (LOO) CVs for multiple Monte Carlo samples (with  $K = 3$ ) significantly accelerate the convergence of the marginal log-likelihood (MLL) estimates, yielding promising results comparable to VBPI-GNN. Although IW-ELBO was effective when  $Q(z)$  was comprised of diagonal Normal distributions, no advantages were observed for the case with full covariance matrices. Similar tendencies in the MLL estimates were observed for the other dimension ( $d = 3, 4$ ) and wrapped normal distributions (Appendix D.1).
5.2 Comparison of different topological distributions
We investigated effective choices of tip coordinate distributions  $Q(z)$ , which yielded tree topologies, by comparing different combinations of distribution types (normal  $\mathcal{N}$  or wrapped normal  $\mathcal{W}\mathcal{N}$ ), space dimensions  $(d = 2,3,4)$ , and covariance matrix types (diagonal or full), across selected CVs (LAX, LOO, and LOO+LAX). While overall performance was stable and comparable for the range of configurations, the most flexible ones: the normal and wrapped normal distributions with  $d = 4$  and a full covariance matrix, indicated relatively high MLL estimates within their respective groups (Table 2 in Appendix D), implying the importance of flexibility in the variational distributions.
5.3 Performance evaluation across eight benchmark datasets
To demonstrate the inference performance of GeoPhy, we compared the marginal log-likelihood (MLL) estimates for the eight real datasets (DS1-8) [8, 6, 37, 9, 17, 43, 38, 30]. The gold-standard values were obtained using the stepping-stone (SS) algorithm [36] in MrBayes [29], wherein each evaluation involved ten independent runs, each with four chains of 10,000,000 iterations, as reported in [42]. In Table 1, we summarize the MLL estimates of GeoPhy and other approximate Bayesian inference approaches for the eight datasets. While VBPI-GNN [40] employs a preselected set of tree topologies as its support set before execution, it is known to provide reasonable MLL estimates near the reference values. The other approaches including CSMC [34], VCSMC [24],  $\phi$ -CSMC [16] and GeoPhy (ours) tackle the more challenging general problem of model optimization by considering all candidate topologies without preselection. We compared the GeoPhy for two configurations of  $Q(z)$ : a wrapped normal distribution  $\mathcal{W}\mathcal{N}$  with a 4-dimensional full covariance matrix, and a 2-dimensional diagonal matrix, with three choices of CVs for optimization. Results for other settings, including those of the normal distributions in Euclidean spaces, are provided in Table 3 in Appendix D. There, we noted a slight advantage of wrapped normal distributions over their Euclidean counterparts. In
Table 1, we observe that the superior performance of the larger model  $\mathcal{W}\mathcal{N}(\mathrm{full},4)$  over  $\mathcal{W}\mathcal{N}(\mathrm{diag},2)$ , observed in Table 2, is consistently reproduced across all datasets. Moreover, although  $\mathcal{W}\mathcal{N}(\mathrm{full},2)$  is relatively less performant in our comparison, almost all the estimates outperformed the other CSMC-based methods, implying the stability and efficiency of our approach.
Table 1: Comparison of the marginal log-likelihood (MLL) estimates with different approaches in eight benchmark datasets. The MLL values for MrBayes SS and VBPI-GNN, which employs the VIMCO estimator of 10 samples and EDGE GNN, are sourced from [40]. The values of CSMC, VCSMC, and  $\phi$ -CSMSC are referenced from [16]. We adopt the nomenclature from the literature [42] and refer to the dataset named DS7 in [16] as DS8. The MLL values for our approach (GeoPhy) are shown for two different  $Q(z)$  configurations: a wrapped normal distribution  $\mathcal{W}\mathcal{N}$  with 4-dimensional full covariance matrix, and 2-dimensional diagonal matrix. In each training, we used three different sets of CVs: LAX with  $K = 1$ , LOO with  $K = 3$  which we labeled LOO(3), and a combination of LOO and LAX, denoted as  $\mathrm{LOO}(3)+$ . The bold figures are the best (highest) values obtained with GeoPhy and the tree CSMC-based methods, all of which perform an approximate Bayesian inference without the preselection of topologies. We underlined GeoPhy's MLL estimates that outperformed the other CSMC-based methods, demonstrating superior performance across various configurations.
Dataset	DS1	DS2	DS3	DS4	DS5	DS6	DS7	DS8
#Taxa (N)	27	29	36	41	50	50	59	64
#Sites (M)	1949	2520	1812	1137	378	1133	1824	1008
MrBayes SS	-7108.42	-26367.57	-33735.44	-13330.06	-8214.51	-6724.07	-37332.76	-8649.88
[29, 42]	(0.18)	(0.48)	(0.5)	(0.54)	(0.28)	(0.86)	(2.42)	(1.75)
VBPI-GNN	-7108.41	-26367.73	-33735.12	-13329.94	-8214.64	-6724.37	-37332.04	-8650.65
[40]	(0.14)	(0.07)	(0.09)	(0.19)	(0.38)	(0.4)	(0.26)	(0.45)
CSMC	-8306.76	-27884.37	-35381.01	-15019.21	-8940.62	-8029.51	-11013.57
[34, 16]	(166.27)	(226.6)	(218.18)	(100.61)	(46.44)	(83.67)	(113.49)
VCSMC	-9180.34	-28700.7	-37211.2	-17106.1	-9449.65	-9296.66
[24, 16]	(170.27)	(4892.67)	(397.97)	(362.74)	(2578.58)	(2046.7)
φ-CSMC	-7290.36	-30568.49	-33798.06	-13582.24	-8367.51	-7013.83	-9209.18
[16]	(7.23)	(31.34)	(6.62)	(35.08)	(8.87)	(16.99)	(18.03)
WNN(full,4)	-7111.55	-26379.48	-33757.79	-13342.71	-8240.87	-6735.14	-37377.86	-8663.51
(0.07)	(11.60)	(8.07)	(1.61)	(9.80)	(2.64)	(29.48)	(6.85)
LOO(3)	-7119.77	-26368.44	-33736.01	-13339.26	-8234.06	-6733.91	-37350.77	-8671.32
(11.80)	(0.13)	(0.03)	(3.19)	(7.53)	(0.57)	(11.74)	(5.99)
LOO(3)+	-7116.09	-26368.54	-33735.85	-13337.42	-8233.89	-6735.90	-37358.96	-8660.48
(10.67)	(0.12)	(0.12)	(1.32)	(6.63)	(1.13)	(13.06)	(0.78)
WNN(diag,2)	-7126.89	-26444.84	-33823.74	-13358.16	-8251.45	-6745.60	-37516.88	-8719.44
(10.06)	(27.91)	(15.62)	(9.79)	(9.72)	(8.36)	(69.88)	(60.54)
LOO(3)	-7130.67	-26380.41	-33737.75	-13346.94	-8239.36	-6741.63	-37382.28	-8690.41
(10.67)	(14.40)	(2.48)	(4.25)	(4.62)	(3.23)	(31.96)	(15.92)
LOO(3)+	-7128.40	-26375.28	-33736.91	-13347.32	-8235.41	-6742.40	-37411.28	-8683.22
(9.78)	(11.78)	(1.91)	(4.42)	(5.70)	(1.94)	(56.74)	(13.13)
Figure 3: Superimposed probability densities of topological distributions  $\sum_{i=1}^{N} Q(z_i)$  up to 100,000, 200,000, and 500,000 steps (MC samples) from the left. For  $Q(z)$ , we employed a wrapped normal distribution with a two-dimensional full covariance matrix. The experiments used the DS3 dataset ( $N = 36$ ). The majority-rule consensus phylogenetic tree obtained with MrBayes and each step of GeoPhy are shown in blue and red lines, respectively. The center area is magnified by transforming the radius  $r$  of the Poincaré coordinates into  $\tanh 2.1r$ .
We further analyze the accuracy and expressivity of tree topology distribution  $Q(\tau)$  in comparison with MCMC tree samples. As the training progresses, the majority consensus tree from GeoPhy
Figure 4: First to Third: Marginal log-likelihood (MLL) estimates and Robinson-Foulds (RF) distance between the consensus trees obtained from topology distribution  $Q(\tau)$  and MrBayes (MCMC) for datasets DS1, DS3, and DS7, respectively. Fourth: Comparison of bipartition frequencies derived from the posterior distribution of tree topologies for MrBayes, VBPI-GNN, and GeoPhy (ours).
progressively aligns with those obtained via MCMC (Fig. 3). Furthermore, we also observed that the Robinson-Foulds (RF) distance between the consensus trees derived from the topological distribution  $Q(\tau)$  and MCMC (MrBayes) are highly aligned with the MLL estimates (Fig. 4 First to Third), underscoring the validity of MLL-based evaluations. While GeoPhy provides a tree topology mode akin to MCMCs with a close MLL value, it may require more expressivity in  $Q(z)$  to fully represent tree topology distributions. In Fig. 4 Fourth, we illustrate the room for improvement in the expressivity of  $Q(\tau)$  in representing the bipartition frequency of tree samples, where VBPI-GNN aligns more closely with MrBayes. Further analysis is found in Appendix D.3.
In terms of execution time, we compared GeoPhy with VBPI-GNN and MrBayes, obtaining results that are promising and comparable to VBPI-GNN (Fig. 10 Left). Though we have not included CSMC-based methods in the comparison due to deviations in MLL values, they tend to capitalize on parallelism for faster execution as demonstrated in [16].
We developed a novel differential phylogenetic inference framework named GeoPhy, which optimized a variational distribution of tree topology and branch lengths without the preselection of candidate topologies. We also proposed a practical implementation for stable model optimization through choices of distribution models and control variates. In experiments conducted with real sequence datasets, GeoPhy consistently outperformed other approximate Bayesian methods that considered whole topologies.
7 Limitations and Future work
Although GeoPhy exhibits remarkable performance on standard benchmarks without preselecting topologies, it may be necessary to use more expressive distributions for  $Q(z)$  than independent parametric distributions to reach a level of performance comparable to state-of-the-art VBPI or gold-standard MCMC evaluations. Another limitation of our study lies in its exclusive focus on efficiency, measured in terms of the number of likelihood evaluations (NLEs), without paying significant attention to optimizing the computational cost per iteration. The general design of our variational distribution allows us to replace the tree link function such as UPGMA as presented in Fig. 9 Right. Future work should explore alternative design choices in continuous tree topology representations and the link functions to address these limitations. Extending our framework to include complex models remains crucial. Given the potential of scalable phylogenetic methods to enhance our understanding of viral and bacterial evolution, further exploration could substantially impact public health and disease control.
We appreciate the valuable comments on our study provided by Dr. Tsukasa Fukunaga of Waseda University and the anonymous reviewers. TM was partially supported by JSPS KAKENHI JP20H04239 and JST ACT-X JPMJAX22AI. MH was partially supported by JST CREST JPMJCR21F1 and AMED JP22ama121055, JP21ae0121049 and JP21gm0010008.
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A Details of Background
A.1 Summary of calculations for wrapped normal distributions
The wrapped normal distribution, as proposed in [25], is a probability distribution defined on hyperbolic spaces, which is easy to sample from and evaluate its probability density at arbitrary coordinates in the hyperbolic spaces. In the following, we provide a detailed summary of the calculation involved in applying the wrapped normal distributions within the context of the Lorentz model of hyperbolic spaces.
The Lorentz model, denoted as  $\mathbb{H}^d$ , represents the  $d$ -dimensional hyperbolic space as a submanifold of a  $d + 1$  dimensional Euclidean space. Given  $u, v \in \mathbb{R}^{d + 1}$ , we can define the pseudo-inner product and pseudo-norm as follows:
$$
\langle u, v \rangle_ {L} := - u _ {0} v _ {0} + \sum_ {j = 1} ^ {d} u _ {j} v _ {j}, \| u \| _ {L} := \sqrt {\langle u , u \rangle_ {L}}. \tag {17}
$$
The distance between hyperbolic coordinates  $\nu, \mu \in \mathbb{H}^d$  is defined as follows:
$$
\mathrm {d} (\nu , \mu) := \cosh^ {- 1} (- \langle \nu , \mu \rangle_ {L}), \tag {18}
$$
where  $\cosh^{-1}$  denotes the inverse function of the hyperbolic cosine function. Consider hyperbolic coordinates  $\nu, \mu \in \mathbb{H}^d$  and tangent vectors  $u \in T_{\mu} \mathbb{H}^d$  and  $v \in T_{\nu} \mathbb{H}^d$ . An exponential map  $\exp_{\mu}(u) \in \mathbb{H}^d$ , a logarithm map  $\log_{\mu}(\nu) \in T_{\mu} \mathbb{H}^d$ , and a parallel transport map  $\mathrm{PT}_{\nu \to \mu}(v) \in T_{\mu} \mathbb{H}^d$ , can be calculated as follows:
$$
\exp_ {\mu} (u) = \cosh (\| u \| _ {L}) \mu + \sinh (\| u \| _ {L}) \frac {u}{\| u \| _ {L}}, \tag {19}
$$
$$
\log_ {\mu} (\nu) = \frac {\cosh^ {- 1} (\alpha)}{\sqrt {\alpha^ {2} - 1}} \left(\nu - \alpha \mu\right), \tag {20}
$$
$$
\mathrm {P T} _ {\nu \rightarrow \mu} (v) = v + \frac {\left\langle \mu - \alpha \nu , v \right\rangle_ {L}}{\alpha + 1} (\nu + \mu), \tag {21}
$$
where we denote  $\alpha = -\langle \nu, \mu \rangle_L$ , and  $\cosh^{-1}$  represents the inverse function of  $\cosh$ .