SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\frac {\widetilde {D _ {\mathrm {T}}} (x _ {i} , x _ {j}) ^ {2}}{D _ {\mathrm {T}} (x _ {i} , x _ {j}) ^ {2}} = \frac {\sum_ {k \neq i , j} \left(d _ {t} (x _ {i} , x _ {k}) - d _ {t} (x _ {j} , x _ {k})\right) ^ {2} + 2 (d _ {t} (x _ {i} , x _ {j}) + \epsilon) ^ {2}}{D _ {\mathrm {T}} (x _ {i} , x _ {j}) ^ {2}} = 1 + \frac {4 \epsilon d (x _ {i} , x _ {j}) + 2 \epsilon^ {2}}{D _ {\mathrm {T}} (x _ {i} , x _ {j}) ^ {2}},
$$
$$
\frac {4 \epsilon d (x _ {i} , x _ {j}) + 2 \epsilon^ {2}}{D _ {T} (x _ {i} , x _ {j}) ^ {2}} \leq \frac {2 \epsilon d _ {t} (x _ {i} , x _ {j}) + \epsilon^ {2}}{d _ {t} (x _ {i} , x _ {j}) ^ {2}}
$$
For  $\epsilon > 0$ , this gives
$$
\epsilon \geq \frac {4 d _ {t} (x _ {i} , x _ {j}) ^ {3} - 2 d _ {t} (x _ {i} , x _ {j}) D _ {T} (x _ {i} , x _ {j}) ^ {2}}{D _ {t} (x _ {i} , x _ {j}) ^ {2} - 2 d _ {t} (x _ {i} , x _ {j}) ^ {2}} = - 2 d _ {t} (x _ {i}, x _ {j}).
$$
For  $\epsilon < 0$ , we have
$$
\epsilon \leq \frac {4 d _ {t} (x _ {i} , x _ {j}) ^ {3} - 2 d _ {t} (x _ {i} , x _ {j}) D _ {T} (x _ {i} , x _ {j}) ^ {2}}{D _ {t} (x _ {i} , x _ {j}) ^ {2} - 2 d _ {t} (x _ {i} , x _ {j}) ^ {2}} = - 2 d _ {t} (x _ {i}, x _ {j}).
$$
Thus  $\epsilon \in (-\infty, -2d_t(x_i, x_j)) \cup (0, \infty)$ . As we require the perturbation factor  $\epsilon < d_t(x_i, x_j)$ , hence we choose  $\epsilon \in (0, \infty)$ .
Proof of Lemma 5.1. Since the eigendecomposition of  $\mathbf{H}_t$  form an orthonormal basis of  $\mathbb{R}^n$ , and since its first eigenvector is constant, we can write the diffusion distance  $\| \delta_i \mathbf{H}_t - \delta_i \mathbf{H}_t \|_2^2 = \sum_{k \geq 0} e^{-2t\lambda_k} (\psi_{ki} - \psi_{kj})^2 = \sum_{k \geq 1} e^{-2t\lambda_k} (\psi_{ki} - \psi_{kj})^2$ . In particular, this defines the  $k$  dimensional embedding  $x \mapsto (e^{-t\lambda_1} \psi_1(x), \ldots, e^{-t\lambda_k} \psi_k(x))$ .
$$
P H _ {\boldsymbol {H} _ {t}} = (1 / 4 t) ^ {2} \left\| d _ {t} (x _ {i}, \cdot) - d _ {t} (x _ {j}, \cdot) \right\| _ {2} ^ {2},
$$
and it is equivalent to a multiscale random walk distance with kernel  $\sum_{k > 0}m_t(k)\pmb{P}^k$ , where  $m_t(k)\coloneqq t^k e^{-t} / k!$ .
Proof of Proposition 5.2. For  $\sigma = 0$ , we have  $d_{t}(x_{i},x_{j}) = -4t\log (H_{t})_{ij}$ , the relation between the PHATE potential and the heat-geodesic follows from the definition
$$
\begin{array}{l} P H _ {\boldsymbol {H} _ {t}} = \sum_ {k} \left(- \log \boldsymbol {H} _ {t} (x _ {i}, x _ {k}) + \log \boldsymbol {H} _ {t} (x _ {j}, x _ {k})\right) ^ {2} \\ = (1 / 4 t) ^ {2} \left\| d _ {t} \left(x _ {i}, \cdot\right) - d _ {t} \left(x _ {j}, \cdot\right) \right\| _ {2} ^ {2}. \\ \end{array}
$$
Using the heat kernel  $H_{t}$  with the random walk Laplacian  $L_{rw} = Q^{-1}L = I_{n} - Q^{-1}W$  corresponds to a multiscale random walk kernel. Recall that we can write  $L_{rw}$  in terms of the symmetric Laplacian  $L_{rw} = Q^{-1 / 2}L_sQ^{1 / 2}$ , meaning that the two matrices are similar, hence admit the same eigenvalues  $\Lambda$ . We also know that  $L_{s}$  is diagonalizable, since we can write  $L_{s} = Q^{-1 / 2}LQ^{-1 / 2} = Q^{-1 / 2}\Psi \Lambda \Psi^{T}Q^{-1 / 2}$ . In particular, we have  $L_{rw} = S\Lambda S^{-1}$ , where  $S\coloneqq Q^{-1 / 2}\Psi$ . The random walk matrix can be written as  $P = I_n - R_{rw}$ , hence its eigenvalues are  $(I_n - \Lambda)$ , and we can write  $P^t = S(I_n - \Lambda)^t S^{-1}$ . Similarly, the heat kernel with the random walk Laplacian can be written as  $H_{t} = Se^{-t\Lambda}S^{-1}$ . Interestingly, we can relate the eigenvalues of  $H_{t}$  and  $P$  with the Poisson distribution. Note the probability mass function of a Poisson as  $m_t(k)\coloneqq t^k e^{-t} / k!$ , for  $t\geq 0$ , we have
$$
e ^ {- t (1 - \mu)} = e ^ {- t} \sum_ {k \geq 0} \frac {(t \mu) ^ {k}}{k !} = \sum_ {k \geq 0} m _ {t} (k) \mu^ {k}. \tag {7}
$$
We note that (7) is the probability generating function of a Poisson distribution with parameter  $t$ , i.e.  $\mathbb{E}[\mu^X]$ , where  $X \sim \mathrm{Poisson}(t)$ . With this relationship, we can express  $\pmb{H}_t$  as a linear combination of  $\pmb{P}^t$  weighted by the Poisson distribution. Indeed, substituting  $\lambda = 1 - \mu$  in (7) links the eigenvalues of  $\pmb{H}_t$  and  $\pmb{P}$ . We write the heat kernel as a linear combination of random walks weighted by the Poisson distribution, we have
Proof of Proposition 5.5. The squared MDS weighted by the heat kernel corresponds to
$$
\begin{array}{l} \sum_ {i j} h _ {t} (x _ {i}, x _ {j}) (d _ {i j} ^ {2} - \| y _ {i} - y _ {j} \| ^ {2}) ^ {2} = \sum_ {i j} h _ {t} (x _ {i}, x _ {j}) (- t \log h _ {t} (x _ {i}, x _ {j}) - \| y _ {i} - y _ {j} \| ^ {2}) ^ {2} \\ = \sum_ {i j} h _ {t} (x _ {i}, x _ {j}) t ^ {2} (\log h _ {t} (x _ {i}, x _ {j}) - \log \exp (- \| y _ {i} - y _ {j} \| ^ {2} / t) ^ {2}. \\ \end{array}
$$
If there exists an embedding that attain a zero loss, then it is the same as  $\sum_{ij}h_t(x_i,x_j)(\log h_t(x_i,x_j) - \log \exp (-\| y_i - y_j\|^2 /t) = D_{\mathrm{KL}}(h_t\| q).$
A.3 Algorithm details
We present a detailed version of the Heat Geodesic Embedding algorithm in Alg.2.
For the knee-point detection we use the Kneedle algorithm [26]. It identifies a knee-point as a point where the curvature decreases maximally between points (using finite differences). We summarize the four main steps of the algorithm for a function  $f(x)$ , and we refer to [26] for additional details.
Algorithm 2 Heat Geodesic Embedding
1: Input:  $N \times d$  dataset matrix  $\mathbf{X}$ , denoising parameter  $\rho \in [0,1]$ , Harnack regularization  $\sigma > 0$ , output dimension  $k$ .  
2: Returns:  $N \times e$  embedding matrix  $\mathbf{E}$ .  
3:  $\triangleright$  1. Calculate Heat Operator  $\mathbf{H}_t$   
4: if  $t$  is "auto" then  
5:  $\lfloor t \leftarrow$  Kneedle  $\{H(H_t)\}_t$   
6:  $\mathbf{W} \leftarrow$  kernel  $(\mathbf{X})$   
7:  $\mathbf{L} \leftarrow Q - \mathbf{W}$   
8: if Exact then  
9:  $\mathbf{H}_t \leftarrow \Psi e^{-t\Lambda} \Psi^T$   
10: else  
11:  $\mathbf{H}_t \leftarrow p_K(\mathbf{L}, t)$   
12:  $\triangleright$  2. Calculate Pairwise Distances  $D$   
13:  $\mathbf{D} \leftarrow -4t \log \mathbf{H}_t$   
14:  $\mathbf{D} \leftarrow (1 - \rho) \mathbf{D} + \rho D_T$   
15: Return  $\mathbf{E} \leftarrow$  MetricMDS  $(\mathbf{D}, \| \cdot \|_2, k)$
1. Smoothing with a spline to preserve the shape of the function.
2. Normalize the values, so the algorithm does not depend on the magnitude of the observations.
3. Computing the set of finite differences for  $x$  and  $y \coloneqq f(x)$ , e.g.  $y_{d_i} \coloneqq f(x_i) - x_i$ .
4. Evaluating local maxima of the difference curve  $y_{d_i}$ , and select the knee-point using a threshold based on the average difference between consecutive  $x$ .
B Experiments and datasets details
Our experiments compare our approach with multiple state-of-the-art baselines for synthetic datasets (for which the true geodesic distance is known) and real-world datasets. For all models, we perform sample splitting with a 50/50 validation-test split. The validation and test sets each consist of 5 repetitions with different random initializations. The hyper-parameters are selected according to the performance on the validation set. We always report the results on the test set, along with the standard deviations computed over the five repetitions. We use the following state-of-the-art methods in our experiments: our Heat Geodesic Embedding, diffusion maps[5], PHATE [22], Heat-PHATE (a variation of PHATE using the Heat Kernel), Rand-Geo (a variation of Heat Geodesic Embedding where we use the random walk kernel), Shortest-path which estimates the geodesic distance by computing the shortest path between two nodes in a graph built on the point clouds,  $t$ -SNE[32], and UMAP[19].
B.1 Datasets
We consider two synthetic datasets, the well known Swiss roll and the tree datasets. The exact geodesic distance can be computed for these datasets. We additionally consider real-world datasets: PBMC, IPSC [22], EB [22], and two from the from the 2022 NeurIPS multimodal single-cell integration challenge<sup>4</sup>.
B.1.1 Swiss Roll
The Swiss roll dataset consists of data points samples on a smooth manifold inspired by shape of the famous alpine pastry. In its simplest form, it is a 2-dimensional surface embedded in  $\mathbb{R}^3$  given by
$$
\begin{array}{l} x = t \cdot \cos (t) \\ y = h \\ z = t \cdot \sin (t) \\ \end{array}
$$
<sup>4</sup>https://www.kaggle.com/competitions/open-problems-multimodal/
where  $t \in [T_0, T_1]$  and  $h \in [0, W]$ . In our experiments we used  $T_0 = \frac{3}{2}\pi$ ,  $T_1 = \frac{9}{2}\pi$ , and  $W = 5$ . We use two sampling mechanisms for generating the data points: uniformly and clustered. In the first, we sample points uniformly at random in the  $[T_0, T_1] \times [0, W]$  plane. In the second, we sample according to a mixture of isotropic multivariate Gaussian distributions in the same plane with equal weights, means  $[(7, W/2), (12, W/2)]$ , and standard deviations [1, 1]. In the clustered case, data samples are given a label  $y$  according to the Gaussian mixture component from which they were sampled.
We consider variations of the Swiss roll by projecting the data samples in higher dimension using a random rotation matrix sampled from the Haar distribution. We use three different ambient dimensions: 3, 10, and 50.
Finally, we add isotropic Gaussian noise to the data points in the ambient space with a standard deviation  $\sigma$ .
B.1.2 Tree
The tree dataset is created by generating  $K$  branches from a  $D$ -dimensional Brownian motion that are eventually glued together. Each branch is sampled from a multidimensional Brownian motion  $d\mathbf{X}_{\mathbf{k}} = 2d\mathbf{W}(t)$  at times  $t = 0,1,2,\dots,L - 1$  for  $k\in [K]$ . The first branch is taken as the main branch and the remaining branches are glued to the main branch by setting  $X_{k} = X_{k} + X_{0}[i_{k}]$  where  $i_k$  is a random index of the main branch vector. The total number of samples is thus  $L\cdot K$
In our experiments, we used  $L = 500$ ,  $K = 5$ , and  $D = 5,10$  (i.e., two versions with different dimensions of the ambient space).
We compare the performance of the different methods according to several metrics. For synthetic datasets, where ground truth geodesic distance is available, we directly compare the estimated distance matrices and ground truth geodesic distance matrices. For real-world datasets, we use clustering quality and continuous interpolation as evaluation metrics.
B.2.1 Distance matrix evaluation
The following methods use an explicit distance matrix: diffusion maps, Heat Geodesic Embedding, Heat-Phate, Phate, Rand-Geo and Shortest Path. For these methods, we compare their ability their ability to recover the ground truth distance matrix several metrics. Letting  $D$  and  $\hat{D}$  the ground truth and inferred distance matrices respectively, and  $N$  the number of points in the dataset, we use the following metrics.
Pearson  $\rho$  We compute the average Pearson correlation between the rows of the distance matrices,  $\frac{1}{N}\sum_{i = 1}^{N}r_{D_i,\hat{D}_i}$ , where  $r_{x,y}$  is the Pearson correlation coefficient between vectors  $x$  and  $y$ .  $D_{i}$  stands for the  $i$ -th row of  $D$ .
Spearman  $\rho$  We compute the average Spearman correlation between the rows of the distance matrices,  $\frac{1}{N}\sum_{i=1}^{N}r_{D_i,\hat{D}_i}$ , where  $r_{x,y}$  is the Spearman correlation coefficient between vectors  $x$  and  $y$ .  $D_i$  stands for the  $i$ -th row of  $D$ .
Frobenius Norm We use  $\| D - \hat{D}\| _F$ , where  $\| A\| _F = \sqrt{\sum_{i = 1}^N\sum_{j = 1}^N|A_{i,j}|^2}$
Maximum Norm We use  $\| D - \hat{D}\|_{\infty}$ , where  $\| A\|_{\infty} = \max_{i,j}|A_{i,j}|$
B.2.2 Embedding evaluation
Some methods produce low-dimensional embeddings without using an explicit distance matrix for the data points. This is the case for UMAP and t-SNE. To compare against these methods, we use the distance matrix obtained by considering Euclidean distance between the low-dimensional embeddings. We used 2-dimensional embeddings in our experiments. For diffusion maps, we obtain
these embeddings by using the first two eigenvectors of the diffusion operator only. For Heat Geodesic Embedding, Heat-PHATE, PHATE, Rand-GEO and Shortest Path, we use multidimensional scaling (MDS) on the originally inferred distance matrix.
**Clustering** We evaluate the ability of Heat Geodesic Embedding to create meaningful embeddings when clusters are present in the data. To this end, we run a k-means clustering on the two dimensional embeddings obtained with each method and compare them against the ground truth labels. For the Tree dataset, we use the branches as clusters. For the Swiss roll dataset, we sample data points on the manifold according to a mixture of Gaussians and use the mixture component as the ground truth cluster label.
Interpolation To quantitatively evaluate the quality of the continuous embeddings, we first embed the entire dataset and obfuscate all samples from a particular time point  $(e.g., t = 2)$ . We then estimate the distribution of the missing time point by using displacement interpolation [35] between the adjacent time points  $(e.g., t = 1$  and  $t = 3)$ . We report the Earth Mover Distance (EMD) between the predicted distribution and true distribution. A low EMD suggests that the obfuscated embeddings are naturally located between the previous and later time points, and that the generated embedding captures the temporal evolution of the data adequately.
B.3 Hyperparameters
In Table 5, we report the values of hyperparameters used to compute the different embeddings.
Hyperparameter	Description	Values
Heat Geodesic Embedding
k	Number of neighbours in k-NN graph	5,10,15
order	order of the approximation	30
t	Diffusion time	0.1,1,10,50,auto
Approximation method	Approximation method for Heat Kernel	Euler, Chebyshev
Laplacian	Type of laplacian	Combinatorial
Harnack ρ	Harnack Regularization	0,0.25,0.5,0.75,1,1.5
PHATE
n-PCA	Number of PCA components	50,100
t	Diffusion time	1,5,10,20,auto
k	Number of neighbours	10
Diffusion Maps
k	Number of neighbours in k-NN graph	5,10,15
t	Diffusion time	1,5,10,20
Shortest Path
k	Number of neighbours in k-NN graph	5,10,15
UMAP
k	Number of neighbours	5,10,15
min-dist	Minimum distance	0.1,0.5,0.99
t-SNE
p	Perplexity	10,30,100
early exaggeration	Early exaggeration parameter	12
Table 5: Hyperparameters used in our experiments
B.4 Hardware
The experiments were performed on a compute node with 16 Intel Xeon Platinum 8358 Processors and 64GB RAM.
C Additional results
C.1 HeatGeo weighted
Following Sec. 5, we know that weighting the MDS loss by the heat kernel corresponds to a specific parametrization of SNE, and thus promote the identification of cluster. In Fig. 5, we show the embeddings of four Gaussian distributions in 10 dimensions (top), and the PBMC dataset (bottom). The reference embedding is using t-SNE, as it models as it also minimizes the KL between the ambient and embedded distributions. We see that HeatGeo weighted form cluster that are shaped like a Gaussian. This is expected as Prop. 5.5, indicates that this is equivalent to minimizing the  $D_{\mathrm{KL}}$  between the heat kernel and a Gaussian affinity kernel.
Figure 5: Embeddings of four Gaussian distributions in 10 dimensions (top), and the PBMC dataset (bottom). HeatGeo with weight is equivalent to minimizing the  $D_{\mathrm{KL}}$  between the heat kernel and a Gaussian affinity kernel, hence produces clusters shaped similar to a Gaussian.
C.2 Truncated distance
In Fig.6, we discretize the interval [0, 51] in 51 nodes, and we compute the heat-geodesic distance of the midpoint with respect to the other points, effectively approximating the Euclidean distance. Using Chebyshev polynomials of degree of 20, we see that the impact of the truncation is greater as the diffusion time increases. The backward Euler methods does not result in a truncated distance.
Figure 6: Approximation of the squared Euclidean distance with the Heat-geodesic for the exact computation, Backward Euler approximation, and Chebyshev polynomials. For larger diffusion time, the Chebyshev approximation results in a thresholded distance. The Harnack regularization ensures  $d_{t}(x,x) = 0$ .
Figure 7: Impact of the Checbshev approximation order on the embedding of HeatGeo for the PBMC dataset.
C.3 Harnack inequality
For complete Riemannian manifolds that satisfy the parabolic Harnack inequality (PHI) we have  $h_t(x,y) \simeq V^{-1}(x,\sqrt{t}) e^{-d(x,y)^2 / t}$  so that  $-t \log h_t(x,y) \simeq t \log V(x,\sqrt{t}) + d^2(x,y)$  [25].
$$
h _ {t} (x, x) = \frac {1}{V (x , \sqrt {t})} \tag {8}
$$
$$
V (x, \sqrt {t}) = h _ {t} (x, x) ^ {- 1} \tag {9}
$$
We then have,
$$
d ^ {2} (x, y) \simeq - t \log h _ {t} (x, y) - t \log V (x, \sqrt {t})
$$
$$
d ^ {2} (x, y) \simeq - t \log h _ {t} (x, y) - t \log h _ {t} (x, x) ^ {- 1}
$$
$$
d ^ {2} (x, y) \simeq - t \log h _ {t} (x, y) + t \log h _ {t} (x, x)
$$
C.3.1 Case studies for specific manifolds
The circle -  $\mathbb{S}_1$  We now show that our expression for the Heat Geodesic Embedding-distance is monotonically increasing with respect to the ground truth geodesic distance  $d\in \mathbb{R}^{+}$  for a fixed diffusion time  $t$  and for any Harnack regularization in  $\mathbb{S}_1$ .
Our expression for the Heat Geodesic Embedding-distance is
$$
\hat {d} = \sqrt {- 4 t \log (h _ {t} (d)) + 4 t \log (h _ {t} (0))}
$$
As the square-root is monotonic, and  $4t\log h_t(0)$  is constant with respect to  $d$ , we need to show that  $f(d) = -\log (h_t(d))$  is monotonically increasing.
For  $\mathbb{S}_1$ , we have
$$
h _ {t} (d) = \sum_ {m \in \mathbb {Z}} \frac {1}{\sqrt {4 \pi t}} e ^ {- \frac {(d + 2 \pi m) ^ {2}}{4 t}}
$$
As log is monotonically increasing, it suffices to show that  $\sum_{m\in \mathbb{Z}}e^{-\frac{(d + 2\pi m)^2}{4t}}$  is monotonically decreasing, which is the case as for any  $d^{\prime} > d,\forall m\in \mathbb{Z}$ , we have
$$
e ^ {- \frac {(d + 2 \pi m) ^ {2}}{4 t}} > e ^ {- \frac {(d ^ {\prime} + 2 \pi m) ^ {2}}{4 t}}.
$$
In general, one can see that (1) the heat kernel depending only on the geodesic distance and (2) the heat kernel being monotonically decreasing with respect to the geodesic distance are sufficient conditions for preserving ordering of pair-wise distances with Heat Geodesic Embedding.
The sphere  $-\mathbb{S}_n$  The above result can be applied to the higher-dimensional sphere  $\mathbb{S}_n$ . It is known that the heat kernel on manifold of constant curvatures is a function of the geodesic distance  $(d)$  and time only. For  $\mathbb{S}_n$  the heat kernel is given by
$$
h _ {t} (x, y) = \sum_ {l = 0} ^ {\infty} e ^ {- l (l + n) - 2 t} \frac {2 l + n - 2}{n - 2} C _ {l} ^ {\frac {n}{2} - 1} (\cos (d))
$$
with  $I$  the regularized incomplete beta function and  $C$  the Gegenbauer polynomials.
Furthermore, Nowak et al. [24] showed that the heat kernel of the sphere is monotonically decreasing. The distance inferred from Heat Geodesic Embedding thus preserves ordering of the pair-wise distances.
Euclidean  $(\mathbb{R}^3)$  For the euclidean space, we have for the volume of  $\sqrt{t}$ -geodesic ball and for the heat kernel:
$$
V _ {\sqrt {t}} = \frac {4}{3} \pi t ^ {3 / 2}
$$
$$
h _ {t} (x, y) = \frac {1}{(4 \pi t) ^ {3 / 2}} e ^ {- \frac {\rho^ {2}}{4 t}}.
$$
Recalling Harnack inequality,
$$
\frac {c _ {1}}{V (x , \sqrt {t})} e ^ {- \frac {d (x , y) ^ {2}}{c _ {2} t}} \leq h _ {t} (x, y) \leq \frac {c _ {3}}{V (x , \sqrt {t})} e ^ {- \frac {d (x , y) ^ {2}}{c _ {4} t}}
$$
With  $c_{2} = c_{4} = 4$ , we have
$$
\frac {c _ {1}}{V (x , \sqrt {t})} \leq \frac {1}{(4 \pi t) ^ {3 / 2}} \leq \frac {c _ {3}}{V (x , \sqrt {t})}
$$
In this case, the bound can be made tight, by setting
$$
\begin{array}{l} c _ {1} = c _ {3} = \frac {V (x , \sqrt {t})}{(4 \pi t) ^ {3 / 2}} \\ = \frac {\frac {4}{3} \pi t ^ {3 / 2}}{(4 \pi t) ^ {3 / 2}} \\ = \frac {1}{3 \sqrt {4 \pi}} = \frac {1}{6 \sqrt {\pi}}, \\ \end{array}
$$
we recover the exact geodesic distance.
C.4 Quantitative results
C.4.1 Distance matrix evaluation
We report the performance of the different methods in terms of the ground truth geodesic matrix reconstruction in Table. 6 for the Swiss roll dataset and in Table. 7, for the Tree dataset.
C.4.2 Distance matrix evaluation via two-dimensional embeddings
We report the performance of the different methods in terms of the ground truth geodesic matrix reconstruction in Table 8 for the Swiss roll dataset and in Table 9, for the Tree dataset.
C.4.3 Clustering quality evaluation
On Tables 10, we report the performance on clustering quality for the synthetic datasets with different noise level.
C.5 Impact of the different hyperparameters
We investigate the impact of the different hyperparameters on the quality of the embeddings. In Figure 8, we show the embeddings of HeatGeo for different values of diffusion time, number of neighbours, order, and Harnack regularization.
In Figures 9, 10, 11, and 12, we show the impact of different hyperparameters on the Pearson correlation between the estimated distance matrix and ground truth distance matrix for different methods on the Swiss roll dataset.
C.6 Graph construction
We compare the embeddings of the heat-geodesic distance for different graph construction. Throughout the paper we used the graph construction from PHATE [22]. In the following we present additional results depending on the choice of kernel to construct the graph. Specifically, we use a simple nearest neighbor (kNN) graph implemented in [7], the graph from UMAP [19], and the implementation in the package Scany [36] for single-cell analysis. In figure, we present the embeddings 2500 points of a tree with five branches in 10 dimensions, where the observations are perturbed with a standard Gaussian noise. All methods used five nearest neighbors and a diffusion time of 20. In Figure 13, we show the evolution of the Pearson correlation between estimated and ground truth distance matrices for the 10-dimensional Swiss roll dataset for various graph constructions. We note that the results are stable across different graph construction strategies.
Table 6: Comparison of the estimated distance matrices with the ground truth geodesic distance matrices on the Swiss roll dataset. Best models on average are bolded (not necessarily significant).
Table 7: Comparison of the estimated distance matrices with the ground truth geodesic distance matrices on the Tree roll dataset. Best models on average are bolded (not necessarily significant).
C.7 Time Complexity
In Table 12, we present the average computing time for creating embeddings and corresponding distance matrix for the different methods. All methods are applied on the Swiss roll dataset in three dimension with 2000 samples. We present empirical averages and standard deviations over ten repetitions. The experiments were run on a Apple M2 Pro chip with 16G RAM.