We propose a novel framework for combining datasets via alignment of their associated intrinsic dimensions. Our approach assumes that the two datasets are sampled from a common latent space, i.e., they measure equivalent systems. Thus, we expect there to exist a natural (albeit unknown) alignment of the data manifolds associated with the intrinsic geometry of these datasets, which are perturbed by measurement artifacts in the sampling process. Importantly, we do not assume any individual correspondence (partial or complete) between data points. Instead, we rely on our assumption that a subset of data features have correspondence across datasets. We leverage this assumption to estimate relations between intrinsic manifold dimensions, which are given by diffusion map coordinates over each of the datasets. We compute a correlation matrix between diffusion coordinates of the datasets by considering graph (or manifold) Fourier coefficients of corresponding data features. We then orthogonalize this correlation matrix to form an isometric transformation between the diffusion maps of the datasets. Finally, we apply this transformation to the diffusion coordinates and construct a unified diffusion geometry of the datasets together. We show that this approach successfully corrects misalignment artifacts, and allows for integrated data.
The authors pointed out that the measurements in biology and natural science suffer from batch effects such as the variations between batches of data measured at different times or by different sensors. In order to analyze different batches of data, an alignment or a calibration is frequently needed. The authors propose to use that though there is variation among different batches, these batches all share an underlying intrinsic manifold structure, which may admit a set of alignable coordinates.
Technically, the authors propose to choose the diffusion kernel method, which is one of the spectral methods, to extract the harmonic like eigenfunctions defined on the manifold, for each of the batches of data. Using these harmonic-like coordinates, the authors assume there exists an isometric rotation in the between each pair of batches such that their coordinates can be aligned under this orthogonal rotation.
Comments:
Overall I think the problems pointed out do exist and this is an interesting proposal to use the manifold structure to align the data. But there are some weak points in this proposal:
1. It's well known that spectral methods are frequently sensitive to perturbations of the datasets. At the beginning of section 2.2 the authors propose to use a normalization to construct the kernel, however, I don't quite understand how this would solve the instability to perturbations. 
2. In my opinion, the equation (1) is the most interesting construction in this paper. This motivation for this tensors construction is not strong enough and I would suggest put more detail into this construction. My understanding is the window functions g introduced here serve for an invariance purpose such that when the frequency slightly shift (or rotate), the correlation computed should be stable. But the tradeoff of choosing a proper window should be discussed carefully, potentially with different dataset since different dataset may have a different sensitivity to perturbations across different batches. 
3. In section 3.1, the first motivating example is quite confusing. The authors demonstrated the alignment of two rotated MNIST digits, 3. For each digit, the underlying manifold is S1. S1 is diffeomorphic to its rotation. So I'm not so sure what's the underlying manifold geometry used to align them.  My understanding is that this alignment doesn't come from the S1 manifold but comes from some additional structure in the image signal. Fig1(b) is also a little confusing that I couldn't figure out what's drawn there.
Overall, to use the underlying manifold structure to align data batches is an interesting and straightforward proposal, but I hope the authors can address these question carefully and make the argument stronger.