In many natural science settings, and in particular in biology, an often encountered problem is that while data are measured from the same system, is is collected by different equipment 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 to analyze them [e.g., 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. However, most of these require at least partial bijection, if not full one, between views. Other approaches attempt to directly match data points, either in their ambient space or by local data geometry, and these can be very sensitive to differences in sampling density rather than data geometry . Here, we present a principled approach called harmonic alignment for correcting this type of effect based on the popular 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 for alignment purposes, rather than working by point-by-point matching as done in previous methods. Our method involves first embedding each dataset separately into harmonic 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 graph (or manifold) spaces. These harmonic diffusion components are eigenvectors of a Markov-normalized data diffusion operator, whose corresponding eigenvalues encode frequencies of these eigenvectors. We attempt to find a transformation from one set of eigenvectors to another, via feature correspondences in the data.
While data point correspondences may be difficult or impossible to obtain, since many biological measurements are destructive, feature correspondences are often available. For instance, single-cell experiments performed on the same device will have counts from the same genes as their features, even though the measurements are typically affected by batch differences. When these corresponding features are transformed via the graph Fourier transform (GFT) into diffusion components, their representations should be similar, with potentially small frequency-proximal perturbations. For example, the GFT of slowly varying features across the manifold should be relatively localized to the low-frequency harmonics. This insight allows us to create a correlation matrix between the graph harmonics of one dataset to another based on correlation between the GFT of features. Since we are only interested in finding frequency-proximal correlations (i.e., we do not want to correlate high frequency harmonics with low frequency harmonics between datasets), we need not compute the entire correlation matrix but rather only its near-diagonal values. We then find the closest orthogonal approximation to our correlation matrix, yielding an isometric linear transformation that maps the diffusion representation of one space into those of the other. 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 to batch effects and sample-specific artifacts, and further low-pass filter this geometry to denoise the unified manifold. Thus, in addition to aligning data manifolds, our method also denoises them along the way.
We demonstrate the results of our method on artificial manifolds created from rotated MNIST digits, corrupted MNIST digits, and single-cell biological data measuring peripheral blood cells. In each case, our method successfully aligns the data manifolds such that they have appropriate neighbors both within and across the two datasets. Further, we show an application of our approach in transfer learning by applying a lazy classifier to one unlabeled dataset based on labels provided by another dataset (with batch effects between them), and compare the classification accuracy before and after alignment. Finally, comparisons with recently developed methods such as the MNN-based method fromHaghverdi et al.  show significant improvements in performance and denoising ability are achieved by our harmonic alignment methods.
2 Harmonic alignment
A typical and effective assumption in machine learning is that high dimensional data originates from an intrinsic low dimensional manifold that is mapped via nonlinear functions to observable high dimensional measurements; this is commonly referred to as the manifold assumption. Formally, letbe a hidden dimensional manifold that is only observable via a collection of nonlinear functions that enable its immersion in a high dimensional ambient space as from which data is collected. Conversely, given a dataset of high dimensional observations, manifold learning methods assume its data points originate from a sampling of the underlying manifold via , , and aim to learn a low dimensional intrinsic representation that approximates the manifold geometry of [see, for example, 12, 13, 14, 15, and references therein].
A popular approach towards this manifold learning task is to construct diffusion geometry from data , and then embed data points into diffusion coordinates that provide a natural global coordinate system derived from Laplace operator over manifold geometries, as explained in Section 2.1. However, this approach, as well as other manifold learning ones, implicitly assumes that the feature functions represent data collection technologies (e.g., sensors or markers) that operate in a consistent manner on all samples in . While this assumption may be valid in some fields, biological data collection (and more generally, data collection in natural sciences) is often highly affected by a family of phenomena known as batch effects
, which introduce nonnegligible variance between different data batches due to various uncontrollable factors, as explained in Section1. These include, for example, drastic variations between subjects, experimental settings, or even times of day when an experiment is conducted.
Therefore, in such settings, one should consider a collection of samples , each originating from feature functions that aim to measure the same quantities in the data, but are also affected by sample-dependent artifacts. While each sample can be analyzed to find its intrinsic structure, their union into a single dataset often yields an incoherent geometry biased by batch effects, where neither the relations between samples or within each sample can be clearly seen. To address such artifacts, and construct a unified geometry of multiple batches (i.e., samples or datasets) together, we propose to first embed each batch separately in diffusion coordinates, and then find an isometric transformation that aligns these diffusion representations.
In order to find such transformation, we utilize the duality between diffusion coordinates and geometric harmonics that act as generalized Fourier harmonics, as shown in graph signal processing . As explained in Section 2.2, this duality allows us to capture cross-batch correlations between diffusion coordinates, and orthogonalize the resulting correlation matrix to provide a map between batch-specific diffusion representations. Finally, given an aligned representation, we build a robust unified diffusion geometry that is invariant to both batch effects and batch-specific artifacts. While our approach generalizes naturally to any number of batches, for simplicity, we focus our formulation here on the case of two batches. A succinct summary of these steps is presented in Algorithm 1.
2.1 Diffusion geometry & graph harmonics
The first step in our approach is to capture the intrinsic geometry of each batch using the diffusion maps method from Coifman and Lafon , which non-linearly embeds the data in a new coordinate system (i.e., diffusion coordinates) that is often considered as representing a data manifold, or more generally a diffusion geometry, over the data. We note that as we are only considering an individual batch in this section, we may drop the superscript batch index notation from time to time, for simplicity, when its existence is clearly implied from context.
The diffusion maps construction starts by considering local similarities defined via a kernel , that capture local neighborhoods in the data. We note that a popular choice for is the Gaussian kernel , where
is interpreted as a user-configurable neighborhood size. Next, these similarities are normalized to defined transition probabilitiesthat are organized in an that describes a Markovian diffusion process over the intrinsic geometry of the data. Finally, a diffusion map is defined by taking the eigenvalues and (corresponding) eignevectors of , and mapping each data point to an
dimensional vector, where represents a diffusion-time (i.e., number of transitions considered in the diffusion process). In this work, we denote the diffusion map for the entire batch as . We refer the reader to Coifman and Lafon  for further details and mathematical derivation, but note that in general, as increases, most of the eigenvalue weights , , become numerically negligible, and thus truncated diffusion map coordinates (i.e., using only nonnegligible ones) can be used for dimensionality reduction purposes.
Much work has been done in various fields on applications of diffusion maps as a whole, as well as individual diffusion coordinates (i.e., eigenvectors of ), in data analysis [e.g., 18, 19, 20, 21, 22]. In particular, as discussed in Coifman and Lafon  and Nadler et al. , the diffusion coordinates are closely related to eigenvectors of Laplace operators on manifolds, as well as their discretizations as eigenvectors of graph Laplacians, which were studied previously, for example, in Belkin and Niyogi . Indeed, the similarities measured in can be considered as determining edge weights of a graph structure defined over the data. Formally, we define this graph by considering every data point in as a vertex on the graph, and then defining weighted edges between them via an adjacency matrix with , . Then, the (normalized) graph Laplacian is defined as with being a diagonal degree matrix (i.e., with ). Finally, it is easy to see that , and thus it can be verified that the eigenvectors of can be written as , with corresponding eigenvalues . It should be noted that if data is uniformly sampled from a manifold [as considered in 24]
, these two sets of eigenvectors coincide and the diffusion coordinates can be considered as Laplacian eigenvectors (or eigenfunctions, in continuous settings).
A central tenet of graph signal processing is that the Laplacian eigenfunctions can be regarded as generalized Fourier harmonics [17, 25], i.e., graph harmonics. Indeed, a classic result in spectral graph theory shows that the discrete Fourier basis can be derived as Laplacian eigenvectors of the ring graphs [see, e.g., 26, Proposition 10]. Based on this interpretation, a graph Fourier transform (GFT) is defined on graph signals (i.e., functions over the vertices of the graph) as , , similar to the definition of the classic discrete Fourier transform (DFT). Further, we can also write the GFT in terms of the diffusion coordinates as , given their relation to graph harmonics. Therefore, up to appropriate weighting, the diffusion coordinates can conceptually be interpreted as intrinsic harmonics of the data, and conversely, the graph harmonics can be considered (conceptually) as intrinsic coordinates of data manifolds. In Section 2.2, we leverage this duality between coordinates and harmonics in order to capture relations between data manifolds of individual batches, which are then used in Section 2.3 to align these intrinsic coordinates and construct a unified data manifold over them.
2.2 Cross-graph harmonic correlation
We now turn our attention to considering the relation between two batches via their their intrinsic data manifold structure, as it is captured by diffusion coordinates or, equivalently, graph harmonics. We note that, as discussed extensively in Coifman and Lafon , a naïve construction of an intrinsic data graph with a Gaussian kernel (as described, for simplicity, in Section 2.1) may be severely distorted by density variations in the data. Such distortion would detrimental in our case, as the resulting diffusion geometry and its harmonic structure would not longer reflect a stable (i.e., batch-invariant) intrinsic “shape” of the data. Therefore, we follow the normalization suggested in there to separate data geometry from density, and define a graph structure (i.e., adjacency matrix) over each batch via an anistotropic kernel given by
where is the previously defined Gaussian kernel. Further, we note that, in practice, we chose to remove self-edges (i.e., set ), in order to conform with standard settings in graph signal processing, but this has no effect on the derivation of the proposed method in this section. The resulting graph structure is then used, as previously described, to construct the intrinsic harmonic structure given by on each batch.
While the intrinsic geometry constructed by our graph structures should describe similar “shapes” for the two batches, there is no guarantee that their computed intrinsic coordinates will match. Indeed, it is easy to see how various permutations of these coordinates can be obtained if some eigenvalues have multiplicities greater than one (i.e., their monotonic order is no longer deterministic), but even beyond that, in practical settings batch effects often result in various misalignments (e.g., rotations and other affine transformations) between derived intrinsic coordinates. Therefore, to properly recover relations between multiple batches, we aim to quantify relations between their coordinates, or more accurately, between their graph harmonics.
We note that if we had even a partial overlap between data points in the two batches, this task would be trivially enabled by taking correlations between these harmonics. However, given that here we assume a setting without such predetermined overlap, we have to rely on other properties that are independent of individual data points. To this end, we now consider the feature functions and our initial assumption that corresponding functions aim to measure equivalent quantities in the batches (or datasets). Therefore, while they may differ in the original raw form, we expect their expression over the intrinsic structure of the data (e.g., as captured by GFT coefficients) to correlate, at least partially, between batches. Therefore, we use this property to compute cross-batch correlations between graph harmonics based on the GFT of corresponding data features. To formulate this, it is convenient to extend the definition of the GFT from functions (or vectors) to matrices, by slight abuse of the inner product notation, as , where consists of data features as columns and has graph harmonics as columns (both with rows representing data points).
Notice that the resulting Fourier matrix , for each batch, no longer depends on individual data points, and instead it expresses the graph harmonics in terms of data features. Therefore, we can now use this matrix to formulate cross-batch harmonic correlations by considering inner products between rows of these matrices. Further, we need not consider all the correlations between graph harmonics, since we also have access to their corresponding frequency information, expressed via the associated Laplacian eigenvalues . Therefore, instead of computing correlations between every pair of harmonics across batches, we only consider them within local frequency bands, defined via appropriate graph filters, as explained in the following.
Let be a smooth window defined on the interval as . Then, by translating this window along the along the real line, we obtain equally spaced wavelet windows that can be applied to the eigenvalues in order to smoothly partition the spectrum of each data graph. This construction is known as the itersine filter bank, which can be shown to be a tight frame . The resulting windows are centered at frequencies . The generating function for these wavelets ensures that each halfway overlaps with . This property implies that there are smooth transitions between the weights of consecutive frequency bands. Furthermore, as a tight frame, this filter bank has the property that for any Laplacian eigenvalue . This, in turn, ensures that any filtering we do using the filter bank will behave uniformly across the spectrum. Together, these two properties imply that cross-batch correlations between harmonics within and between bands across the respective batch spectra will be robust. To obtain such band-limited correlations we construct the following filter bank tensor
Each of this matrix corresponds to the Fourier matrix with rows scaled by . Then, we use these filter-bank tensors to compute band-limited correlations via , and finally merge these to generate a combined matrix , which we refer to as the harmonic (cross-batch) correlation matrix. This step, when combined with the half-overlaps discussed above, allows flexibility in aligning harmonics across bands, which is demonstrated in practice in Section 3.
2.3 Isometric alignment
Given the harmonic correlation matrix , we now define an isometric transformation between the intrinsic coordinate systems of the two data manifolds representing and . Such transformation ensures our alignment fits the two coordinate systems together without breaking the rigid structure of each batch, thus preserving their intrinsic structure. To formulate such transformation, we recall that isometric transformations are given by orthogonal matrices, and thus we can rephrase our task as finding the best approximation of
by an orthogonal matrix. Such approximation is a well studied problem, dating back toSchönemann 
, which showed that it can be obtained directly from the singular value decompositionby taking .
Finally, given the isometric transformation defined by , we can now align of the data manifolds of two batches, and define a unified intrinsic coordinate system for the entire data in . While such alignment could equivalently be phrased in terms of either diffusion coordinates or harmonic coordinates , we opt here for the latter, as they relates more directly to the computed harmonic correlations. Therefore, we construct the transformed embedding matrix as
where we drop the superscript for (as they are clear from context), are diagonal matrices that consists of the nonzero Laplacian eigenvalues of each view, and consist of the corresponding eigenvectors (i.e., harmonics) as its columns. We note that the truncated zero-valued eigenvalues correspond to zero frequencies (i.e., flat constant harmonics), and therefore they only encode global shifts that we anyway aim to remove in the alignment process. Accordingly, this truncation is also applied to the harmonic correlation matrix prior to its orthogonalization, in order to ensure the orthogonality of as a transformation on truncated harmonic coordinates. Finally, we note that this construction in terms of Laplacian eigenpairs is equivalent to a diffusion map, albeit using a slightly different derivation of a discretized heat kernel [popular, for example, in graph signal processing works such as 25], with the parameter again serving an analogous purpose to diffusion time.
3 Empirical results
3.1 Rotational Alignment
As a first proof of concept, we first demonstrate harmonic alignment of two circular manifolds. To generate these manifolds, we rotated two different MNIST examples of the digit ‘3’ in a full circle (i.e., consisting of degrees), and sampled a point for each degree (see Figure (a)a). As we noted in Section 2, the manifold coordinates obtained by diffusion maps capture intrinisic structure that should be invariant to the phase of this rotation (e.g., the starting angle). However, as seen in this example, it is clear that the two generated manifolds are out of phase with one another, even in their respective diffusion coordinates.
Figure (b)b demonstrates the simple rotation that is learned by harmonic alignment between the two embeddings. On the left side, we see the out-of-phase embeddings. Taking nearest neighbors in this space illustrates the disconnection between the embeddings: nearest neighbors are only formed for within-sample points. After alignment, however, we see that the embeddings are in phase with each other because nearest neighbors in the aligned space span both samples and are in the same orientation with each other.
3.2 Feature Corruption
Next, we assess the ability of harmonic alignment in recovering -nearest neighborhoods after random feature corruption (see Figure 6). To do this, we drew random samples from MNIST to get batch datasets and , both of size of data points (i.e., digit images). Then, for each trial in this experiment we drew
samples from a unit-variance Normal distribution to create arandom matrix. We orthogonalized this matrix to yield the corruption matrix . To vary the amount of feature corruption, we produced partial corruption matrices (for several values of ) by randomly substituting (i.e., ) of the columns in with columns of the identity matrix. Right multiplication of by these matrices yields corrupted images with only preserved pixels (see Figure (b)b, ‘corrupted’). To assess the recovery of -nearest neighborhoods, we then performed a lazy classification on data points (i.e., rows) in by only using the labels of neighbors from . The results of this experiment, performed for are reported in Figure (a)a. For robustness, at each we sampled three different non-overlapping pairs , and for each pair we sampled three matrices, each with random identity columns, for a total of nine trials per .
In general, both unaligned mutual nearest neighbors (MNN) and harmonic alignment with any filter set cannot recover -nearest neighborhoods under total corruption; note that in our case, it clearly violates our (partial) feature correspondence assumption. Therefore, we start our analysis with overlap, which achieves 10% accuracy, essentially giving the baseline (random chance) accuracy given that MNIST has ten classes. However, when using sufficiently many bandlimited filters (see Section 2.2), our harmonic alignment quickly recovers over accuracy and consistently outperforms both MNN and unaligned classifications, except under under high correspondence (i.e., when ).
Next we examined the ability of harmonic alignment to reconstruct corrupted data (see Figure (b)b). We performed the same corruption procedure as before with and selected ten examples of each digit in MNIST. We show the ground truth from and the corrupted result in Figure (b)b. Then, reconstruction was performed by setting each pixel in a new image to the dominant class average of the nearest neighbors from . In the unaligned case, we see that most examples turn into smeared fives or ones; this is likely a random intersection formed by and (e.g., accounting for the baseline random chance classification accuracy). On the other hand, the reconstructions produced by harmonic alignment resemble their original input examples.
In Figure 10, we compare the runtime, k-nn accuracy, and transfer learning capabilities of our method with two other contemporary alignment methods. First, we examine the unsupervised algorithm proposed by Wang and Mahadevan  for generating weight matrices between two different samples. The algorithm first creates a local distance matrix of size around each point and its four nearest neighbors. Then, it computes an optimal match between k-nn distance matrices of each pair of points in and by comparing all permutations of the k-nn matrices and computing the minimal Frobenius norm between such permuted matrices. We report runtime results for , as failed to complete after running for eight hours. Because the method in Wang and Mahadevan  merely gives a weight matrix that can be used with separate algorithms for computing the final features, we report accuracy results using their implementation. Regardless of input size, we were unable to recover k-nearest neighborhoods for datasets with 35% uncorrupted columns (Figure (b)b), in spite of the computational cost of the method (Figure (a)a).
A more scalable approach to manifold alignment [i.e., 11] has recently emerged in computational biology literature. This approach uses mutual nearest neighbor (MNN) matching to compute mapping between datasets based on the assumption that if two points are truly neighbors they will resemble each other in both datasets. Because this approach amounts to building a k-nearest neighbors graph for each dataset and then choosing a set of neighbors between each dataset, MNN scales comparably to our method (see Figure (a)a). Additionally, MNN is able to recover 20-30% of k-nearest neighborhoods when only 35% of features match (see Figure (b)b). However, while this is an improvement over Wang and Mahadevan , its results are substantially lower than those of the proposed harmonic alignment method. We note tha, additionally, the performance of harmonic alignment monotonically correlates (i.e., improves) with input size, whereas MNN did not improve with more points in our tests.
3.4 Transfer Learning
An interesting use of manifold alignment algorithms is transfer learning. In these setting, an algorithm is trained to perform well on a small (e.g., preliminarly collected) dataset, and the goal is to extend the algorithm to a new larger dataset (e.g., as more data is being collected) after alignment. In Figure (c)c we explore the utility of harmonic alignment in transfer learning and compare it to MNN and the method proposed by Wang and Mahadevan . In this experiment, we first randomly selected uncorrupted examples of MNIST digits, and constructed their diffusion map to use as our training set. Next, we took 65% corrupted unlabeled points (see Section 3.2) in batches of , , , and , as a test set to perform lazy classification on using the labels from the uncorrupted examples. As shown in (c)c, as the testing set increases up to a ratio of one training sample to eight testing samples, harmonic alignment consistently outperformed both competing methods, and remains above 60% post alignment lazy-classification accuracy even with ratio. In addition to showing the use of manifold alignment in transfer learning, this demonstrates the robustness of our algorithm to imbalance between batches.
3.5 Biological data
To illustrate the need for robust manifold alignment in computational biology, we turn to a simple real-world example obtained from Amodio et al.  (Figure 14). This dataset was collected by mass cytometry (CyTOF) of peripheral blood mononuclear cells (PBMC) from patients who contracted dengue fever. Subsequently, the Montgomery lab at Yale University experimentally introduced these PBMCs to Zika virus strains.
The canonical response to dengue infection is upregulation of interferon gamma (IFN), as discussed in Chesler and Reiss , Chakravarti and Kumaria , and Braga et al. . During early immune response, IFN works in tandem with acute phase cytokines such as tumor necrosis factor to induce febrile response and inhibit viral replication . In the PBMC dataset, we thus expect to see upregulation of these two cytokines together, which we explore in Figure 14.
In Figure (a)a, we show the relationship between IFN and TNF without denoising. Note that there is a substantial difference between the IFN distributions of sample 1 and sample 2 (”Earth Mover’s Distance” (EMD) = 2.699). In order to identify meaningful relationships in CyTOF data, it is common to denoise it first . We used a graph low-pass filter proposed in van Dijk et al.  to denoise the cytokine data. The results of this denoising are shown in Figure (b)b. This procedure introduced more technical artifacts by enhancing the difference between batches, as seen by the increased EMD (3.127) between the IFN distributions of both patients. This is likely due to a substantial connectivity difference between the two batch subgraphs in the total graph of the combined dataset.
Next, we performed harmonic alignment of the two patient profiles. We show the results of this in Figure (c)c. Harmonic alignment corrected the difference between IFN distributions and restored the canonical correlation of IFN and TNF. This example illustrates the utility of harmonic alignment for biological data, where it can be used for integrated analysis of data collected across different experiments, patients, and time points.
We presented a novel method for aligning or batch-normalizing datasets, which involves learning and aligning their intrinsic manifold dimensions. Our method leverages the fact that common or corresponding features across datasets should have similar harmonics on the intrinsic geometry of the data, represented by as a graph. Ourharmonic alignment method finds an isometric transformation that maximizes the similarity of graph-based frequency harmonics of common features. Results show that our method successfully aligns artificially misaligned samples, as well as biological data containing batch effects. Our method has the advantages that it aligns manifold geometry rather than density (and thus is insensitive to sampling differences in data). Further, our method inherently denoises the datasets to obtain alignments of significant manifold dimensions rather than noise. We expect future applications of harmonic alignment to include, for example, integration of data from different measurement types performed on the same system, where only a subset of features have known correlations.
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