SlowGuess/NeurIPS_2023_OCR huggingface dataset:
data	Noise level	Method	PearsonR	SpearmanR	Norm Fro N2	Norm inf N2
Swiss roll	0.1	Diffusion Map	0.974 ± 0.01	0.983 ± 0.007	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Heat-Geo	0.995 ± 0.003	0.996 ± 0.002	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Heat-PHATE	0.99 ± 0.002	0.997 ± 0.001	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	PHATE	0.357 ± 0.007	0.357 ± 0.007	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Shortest Path	0.998 ± 0.0	0.997 ± 0.0	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	TSNE	0.951 ± 0.014	0.952 ± 0.01	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	UMAP	0.765 ± 0.059	0.737 ± 0.058	0.015 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Euclidean	0.379 ± 0.002	0.42 ± 0.002	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Diffusion Map	0.983 ± 0.01	0.983 ± 0.007	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Heat-Geo	0.995 ± 0.003	0.996 ± 0.002	0.018 ± 0.0	0.026 ±
Swiss roll	0.1	Heat-PHATE	0.999 ± 0.001	0.997 ± 0.001	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	PHATE	0.357 ± 0.007	0.357 ± 0.007	0.018 ± 0.0	0.026 ± 0.0
SWISS ROLL	0.1	Shortest Path	0.998 ± 0.0	0.997 ± 0.0	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	TSNE	0.951 ± 0.014	0.952 ± 0.01	0.018 ± 0,0	0.026 ± 0.0
Swiss roll	0.1	UMAP	0.765 ± 0.059	0.737 ± 0.058	0.015 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Euclidean	0.379 ± 0.002
Swiss roll	0.1	Diffusion Map	0.983 ± 0.01	0.983 ± 0.007	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Heat-Geo	0.995 ± 0.003	0.996 ± 0.0	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Heat-PHATE	0.999 ± 0.001	0.997 ± 0.0	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	PHATE	0.357 ± 0.007	0.357 ± 0.007	0.018 ± 0.0	0.026 ± 0.0
SWISS ROLL	0.1	Shortest Path	1.335 ± 0.014	1.335 ± 0.014	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	TSNE	0.951 ± 0.014	0.952 ± 0.01	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	UMAP	0.785 ± 0.024	0.785 ± 0.024	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Euclidean	0.379 ± 0.002
Swiss roll	0.1	Diffusion Map	0.983 ± 0.01	0.983 ± 0.007	0.018 ± 0.0	0.018 ± 0.0
Swiss roll	0.1	Heat-Geo	0.995 ± 0.0	0.997 ± 0.0	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Heat-PHATE	0.999 ± 0.0	0.997 ± 0.0	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	PHATE	0.357 ± 0.007	0.357 ± 0.007	0.018 ± 0.0	0.026 ± 0.0
SWISS	0.1	Shortest Path	1.335 ± 0.014	1.335 ± 0.014	0.018 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	TSNE	0.951 ± 0.014	0.952 ± 0.01	0.012 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	UMAP	0.765 ± 0.059	0.737 ± 0.058	0.015 ± 0.0	0.026 ± 0.0
Swiss roll	0.1	Euclidean	0.379 ± 0,002	0.42 ± 0.002	0.018 ± 0.0	0.026 ± 0.0
Table 8: Comparison of the estimated distance matrices with the ground truth geodesic distance matrices on the Swiss roll dataset, using a two-dimensional embedding. Best models on average are bolded (not necessarily significant).
Table 9: Comparison of the estimated distance matrices with the ground truth geodesic distance matrices on the Tree dataset, using a two-dimensional embedding. Best models on average are bolded (not necessarily significant).
data	Noise level	Method	Homogeneity	Adjusted Rand Score	Adjusted Mutual Info Score
Swiss roll	0.1	Heat-Geo	0.82 ± 0.008	0.668 ± 0.034	0.74 ± 0.018
Swiss roll	0.1	Phate	0.731 ± 0.035	0.546 ± 0.044	0.652 ± 0.046
Swiss roll	0.1	TSNE	0.748 ± 0.067	0.537 ± 0.1	0.668 ± 0.068
Swiss roll	0.1	UMAP	0.81 ± 0.036	0.611 ± 0.039	0.726 ± 0.045
Swiss roll	0.5	Heat-Geo	0.813 ± 0.026	0.656 ± 0.049	0.733 ± 0.022
Swiss roll	0.5	Phate	0.735 ± 0.048	0.543 ± 0.064	0.656 ± 0.053
Swiss roll	0.5	TSNE	0.764 ± 0.07	0.564 ± 0.097	0.684 ± 0.065
Swiss roll	0.5	UMAP	0.826 ± 0.019	0.664 ± 0.073	0.744 ± 0.032
Swiss roll	1.0	Heat-Geo	0.722 ± 0.051	0.548 ± 0.091	0.652 ± 0.056
Swiss roll	1.0	Phate	0.482 ± 0.014	0.317 ± 0.031	0.428 ± 0.021
Swiss roll	1.0	TSNE	0.757 ± 0.037	0.562 ± 0.058	0.679 ± 0.042
Swiss roll	1.0	UMAP	0.726 ± 0.041	0.51 ± 0.077	0.65 ± 0.05
Swiss roll high	0.1	Heat-Geo	0.82 ± 0.015	0.666 ± 0.033	0.739 ± 0.019
Swiss roll high	0.1	Phate	0.705 ± 0.03	0.518 ± 0.048	0.628 ± 0.04
Swiss roll high	0.1	TSNE	0.757 ± 0.078	0.558 ± 0.115	0.677 ± 0.08
Swiss roll high	0.1	UMAP	0.796 ± 0.03	0.624 ± 0.048	0.714 ± 0.037
Swiss roll high	0.5	Heat-Geo	0.805 ± 0.021	0.655 ± 0.047	0.725 ± 0.035
Swiss roll high	0.5	Phate	0.745 ± 0.04	0.562 ± 0.061	0.664 ± 0.047
Swiss roll high	0.5	TSNE	0.747 ± 0.075	0.538 ± 0.11	0.668 ± 0.075
Swiss roll high	0.5	UMAP	0.787 ± 0.041	0.573 ± 0.067	0.703 ± 0.032
Swiss roll high	1.0	Heat-Geo	0.7 ± 0.045	0.534 ± 0.057	0.644 ± 0.032
Swiss roll high	1.0	Phate	0.552 ± 0.047	0.386 ± 0.056	0.496 ± 0.04
Swiss roll high	1.0	TSNE	0.754 ± 0.034	0.548 ± 0.068	0.675 ± 0.036
Swiss roll high	1.0	UMAP	0.76 ± 0.041	0.56 ± 0.077	0.68 ± 0.05
Swiss roll very high	0.1	Heat-Geo	0.818 ± 0.033	0.668 ± 0.074	0.738 ± 0.039
Swiss roll very high	0.1	Phate	0.688 ± 0.043	0.497 ± 0.053	0.614 ± 0.053
Swiss roll very high	0.1	TSNE	0.741 ± 0.07	0.544 ± 0.101	0.662 ± 0.075
Swiss roll very high	0.1	UMAP	0.816 ± 0.042	0.65 ± 0.069	0.733 ± 0.054
Swiss roll very high	0.5	Heat-Geo	0.73 ± 0.045	0.605 ± 0.093	0.701 ± 0.028
Swiss roll very high	0.5	Phate	0.758 ± 0.034	0.55 ± 0.037	0.676 ± 0.014
Swiss roll very high	0.5	TSNE	0.77 ± 0.054	0.557 ± 0.093	0.708 ± 0.031
Swiss roll very high	0.5	UMAP	0.789 ± 0.052	0.574 ± 0.101	0.707 ± 0.061
Swiss roll very high	1.0	Heat-Geo	0.592 ± 0.033	0.427 ± 0.063	0.545 ± 0.031
Swiss roll very high	1.0	Phate	0.531 ± 0.042	0.377 ± 0.046	0.486 ± 0.045
Swiss roll very high	1.0	TSNE	0.738 ± 0.019	0.551 ± 0.039	0.662 ± 0.025
Swiss roll very high	1.0	UMAP	0.736 ± 0.057	0.542 ± 0.102	0.66 ± 0.061
Tree	0.1	Heat-Geo	0.784 ± 0.051	0.734 ± 0.07	0.786 ± 0.051
Tree	0.1	Phate	0.55 ± 0.042	0.409 ± 0.064	0.555 ± 0.042
Tree	0.1	TSNE	0.706 ± 0.054	0.61 ± 0.075	0.712 ± 0.055
Tree	0.1	UMAP	0.678 ± 0.086	0.584 ± 0.12	0.681 ± 0.086
Tree	0.5	Heat-Geo	0.545 ± 0.121	0.411 ± 0.154	0.577 ± 0.094
Tree	0.5	Phate	0.529 ± 0.111	0.404 ± 0.151	0.555 ± 0.095
Tree	0.5	TSNE	0.647 ± 0.049	0.591 ± 0.065	0.65 ± 0.048
Tree	0.5	UMAP	0.645 ± 0.051	0.565 ± 0.058	0.652 ± 0.05
Tree	1.0	Heat-Geo	0.398 ± 0.07	0.3 ± 0.077	0.42 ± 0.07
Tree	1.0	Phate	0.418 ± 0.08	0.337 ± 0.093	0.43 ± 0.075
Tree	1.0	TSNE	0.405 ± 0.077	0.378 ± 0.074	0.405 ± 0.077
Tree	1.0	UMAP	0.432 ± 0.086	0.395 ± 0.098	0.432 ± 0.085
Table 10: Clustering results on swiss roll (with distribution) and tree. Best models on average are bolded (not necessarily significant).
Embeddings of HeatGeo for different hyperparameters
Harnack Regularization
Diffusion Time
Number of neighbours
Order of Heat Kernel Approximation (Euler)
Figure 8: Embeddings of Heat Geodesic Embedding for different choices of hyperparameters on the EB dataset. We evaluate the impact of the Harnack regularization, the diffusion time, the number of neighbours in the kNN, and the order of the approximation for Euler and Chebyshev approximations.
Order of Heat Kernel Approximation (Chebyshev)
Figure 9: Impact of diffusion time on the Pearson correlation between the estimated distance matrix and ground truth distance matrix for different methods on the Swiss roll dataset.
Pearson-  $\rho$  in function of Order of the heat kernel approximation on Swiss Roll data
Figure 10: Impact of Checbshev approximation order on the Pearson correlation between the estimated distance matrix and ground truth distance matrix for different methods on the Swiss roll dataset.
Figure 11: Impact of number of neighbours on the Pearson correlation between the estimated distance matrix and ground truth distance matrix for different methods on the Swiss roll dataset.
Figure 12: Impact of Harnack regularization on the Pearson correlation between the estimated distance matrix and ground truth distance matrix for HeatGeo on the Swiss roll dataset.
Pearson  $\rho$  in function of Graph type on Swiss Roll 10D data
Figure 13: Pearson correlation between estimated and ground truth distance matrices for the 10-dimensional Swiss roll dataset for various graph constructions. Standard deviations are computed over the 5 test folds.
Table 11: Clustering quality metrics for different methods. We report the homogeneity and the adjusted mutual information (aMI). Best models on average are bolded (higher is better).
data	Noise level	Method	Homogeneity	Adjusted Rand Score	Adjusted Mutual Info Score
Mnist	0	Diff-Map	0.556 ± 0.002	0.347 ± 0.002	0.622 ± 0.002
Mnist	0	Heat-Geo	0.785 ± 0.0	0.695 ± 0.0	0.829 ± 0.001
Mnist	0	Phate	0.822 ± 0.01	0.72 ± 0.017	0.835 ± 0.011
Mnist	0	TSNE	0.903 ± 0.003	0.871 ± 0.002	0.902 ± 0.003
Mnist	0	UMAP	0.851 ± 0.016	0.846 ± 0.005	0.86 ± 0.015
Coil	0	Diff-Map	0.21 ± 0.036	0.041 ± 0.015	0.142 ± 0.024
Coil	0	Heat-Geo	0.849 ± 0.016	0.67 ± 0.029	0.806 ± 0.022
Coil	0	Phate	0.804 ± 0.017	0.615 ± 0.028	0.735 ± 0.021
Coil	0	TSNE	0.907 ± 0.014	0.79 ± 0.03	0.88 ± 0.02
Coil	0	UMAP	0.871 ± 0.009	0.725 ± 0.019	0.826 ± 0.012
Pbmc	0	Diff-Map	0.026 ± 0.001	0.011 ± 0.0	0.038 ± 0.001
Pbmc	0	Heat-Geo	0.734 ± 0.009	0.724 ± 0.019	0.768 ± 0.017
Pbmc	0	Phate	0.798 ± 0.012	0.818 ± 0.009	0.785 ± 0.01
Pbmc	0	TSNE	0.605 ± 0.019	0.437 ± 0.032	0.544 ± 0.022
Pbmc	0	UMAP	0.177 ± 0.037	0.097 ± 0.033	0.148 ± 0.035
Table 12: 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.
Method	Time (s)
UMAP	4.21 ± 0.68
t-SNE	80.15 ± 62.44
Isomap	2.58 ± 0.03
PHATE	3.30 ± 0.05
Diff Map	8.62 ± 0.23
HeatGeo (Backward Euler)	2.60 ± 0.05
HeatGeo (Chebyshev)	2.11 ± 0.05