Algorithm 1.

Orientable Diffusion Map (ODM)

Require: Data points x1, x2, …, xn ∈ ℝp. Apply local PCA for each xi and its N ≪ n nearest neighbors to find a p × d matrix Oi whose columns form an orthonormal basis to a subspace that approximates the tangent space Txiℳ. For nearby points xi and xj define $O_{\mathit{\text{ij}}}={\text{argmin}}_{O\in O(d)}\Vert O-O_i^T{O}_j{\Vert }_F$ given by $O_{\mathit{\text{ij}}}={\mathit{\text{UV}}}^T$ via the SVD $O_i^T{O}_j=U\mathrm{\Sigma }{V}^T$. Define a n × n matrix Z whose entries are zij = det Oij for nearby points, and zij = 0 otherwise. Define 𝒵 = D−1Z, where D is diagonal with $D_{\mathit{\text{ii}}}={\displaystyle {\sum }_{j=1}^n|z_{\mathit{\text{ij}}}|}$. Compute the top eigenvector ${\upsilon }_1^{𝒵}$ of 𝒵. Examine the histogram of ${\upsilon }_1^{𝒵}$ to decide the orientability of the manifold. If the manifold is orientable, estimate the reflections as ${\widehat{z}}_i=\text{sign }{\upsilon }_1^{𝒵}(i)$. Compute the other eigenvectors υ2, υ3, …, υk of 𝒵 corresponding to the largest eigenvalues and embed the data points in ℝk−1 using $$x_i\mapsto {\widehat{z}}_i({\lambda }_2^t{\upsilon }_2^{𝒵}(i),{\lambda }_3^t{\upsilon }_3^{𝒵}(i),\dots ,{\lambda }_k^t{\upsilon }_k^{𝒵}(i)),\text{ with }t>0.$$