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. In biology and other natural science settings we often have the problem that data are measured from the same system but with different sensors or in different days where sensors are calibrated differently. This is often termed batch effect in biology and can include, for example, drastic variations between subjects, experimental settings, or even times of day when an experiment is conducted. In such settings it is important to globally and locally align the datasets such that they can be combined for effective further analysis. Otherwise, measurement artifacts may dominate downstream analysis. For instance, clustering the data will group samples by measurement time or sensor used rather than by biological or meaningful differences between datapoints.Recent works regard the two datasets as views of the same system and construct a multiview diffusion geometry but all of them require at least partial bijection, if not full one, between views BID11 BID12 BID24 Tuia & Camps-Valls, 2016) . Other work directly attempt to match data points directly in ambient space, or by local data geometry and these can be very sensitive to differences in sampling density rather than data geometry BID10 . Here, we present a principled approach called harmonic alignment to for correct this type of effect based on the manifold assumption.The manifold assumption holds that high dimensional data originates from an intrinsically low dimensional smoothly varying space that is mapped via nonlinear functions to observable high dimensional measurements. Thus, we assume that the datasets are from transformed versions of the same low dimensional manifold. We learn the manifolds separately from the two datasets using diffusion geometric approaches and then find an isometric transformation to map from one manifold to the other. Note that we are not aligning points to points. Indeed there may be sampling differences and density differences in the data. However, our manifold learning approach uses an anisotropic kernel that detects the geometry of the data to align rather than point-by-point matching which is done in other methods.Our method involves first embedding each dataset separately into diffusion components, and then finding an isometric transformation that aligns these diffusion representations. To find such transformation, we utilize the duality between diffusion coordinates and geometric harmonics that act as generalized Fourier harmonics in the graph space. The diffusion components are eigenvectors of a Markov-normalized data diffusion operator, whose eigenvalues indicate frequency of the eigenvector. We attempt to find a transformation from one set of eigenvectors to another, via feature correspondences in the data.While datapoint correspondences may be difficult or impossible to obtain since many biological measurements are destructive, feature correspondences are often available. For instance single-cell measurements of cells from the same device, thus containing counts for the same genes, albeit affected by batch differences. Thus when corresponding features are transformed via the graph Fourier transform (GFT) into diffusion coordinates, the representations should be similar, with potentially small frequency-proximal perturbations. For instance, slowly varying features across the manifold should be load to low-frequency eigenvectors of the Markov matrix. This insight allows us to create a correlation matrix between the eigenvectors of one dataset to another based on correlation between feature loadings to the eigenvectors. However, since we know that eigenvectors represent frequency harmonics, we need not compute the entire correlation matrix but rather only the near-diagonal values. This implies that the two manifolds must be perturbed such that low and high frequency eigenvector space are similar. We then find a linear transformation between that maps the eigenvectors of one space into those of the other that maximizes these correlations by orthogonalizing this matrix. This transformation allows us to align the two datasets with each other. Finally, given an aligned representation, we build a robust unified diffusion geometry that is invariant batch effects and sample-specific artifacts and low-pass filter this geometry to denoise the unified manifold. Thus, in addition to aligning the manifolds our method denoises the manifolds as well.We demonstrate the results of our method on artificial manifolds created from rotated MNIST digits, corrupted MNIST digits, as well as on single-cell biological data measuring peripheral blood cells. In each case our method successfully aligns the manifolds such that they have appropriate neighbors within and across the two datasets. We show an application to transfer learning where a lazy classifier trained on one dataset is applied to the other dataset after alignment. Further, comparisons with recently developed methods such as the MNN-based method of BID10 show significant improvements in performance and denoising ability. We presented a novel method for aligning or batch-normalizing two datasets that involves learning and aligning their intrinsic manifold dimensions. Our method leverages the fact that common or corresponding features in the two datasets should have similar harmonics on the graph of the data. Our harmonic alignment method finds an isometric transformation that maximizes the similarity of frequency harmonics of common features. Results show that our method successfully aligns artifi- cially misaligned as well as biological data containing batch effect. Our method has the advantages that it aligns manifold geometry and not density (and thus is insensitive to sampling differences in data) and further our method denoises the datasets to obtain alignments of significant manifold dimensions rather than noise. Future applications of harmonic alignment can include integration of data from different measurement types performed on the same system, where features have known correlations.
We propose a method for aligning the latent features learned from different datasets using harmonic correlations. Proposes using feature correspondences to preform manifold alignment between batches of data from the same samples to avoid the collection of noisy measurements.