## Abstract

Optimal Transport (OT) has proven useful to infer single-cell trajectories of developing biological systems by aligning distributions across time points. Recently, Parameterized Monge Maps (PMM) were introduced to learn the optimal map between two distributions. Here, we apply PMM to model single-cell dynamics and show that PMM fails to account for asymmetric shifts in cell state distributions. To alleviate this limitation, we propose Unbalanced Parameterised Monge Maps (UPMM). We first describe the novel formulation and show on synthetic data how our method extends discrete unbalanced OT to the continuous domain. Then, we demonstrate that UPMM outperforms well-established trajectory inference methods on real-world developmental single-cell data.

## 1 Introduction

Single-cell RNA sequencing (scRNA-seq) data allows to study cellular development at an unprecedented resolution. To recover these developmental landscapes, various trajectory inference (TI) methods have been proposed [Sae+19]. RNA velocity-based methods ([La +18], [Ber+20]) derive the developmental dynamics by modeling spliced and unspliced gene expression rates with ordinary differential equations, yielding velocity vectors in gene space for every single cell. Optimal Transport (OT, [PC+19]) has also proven useful for inferring developmental dynamics in temporal scRNA-seq data[Sch+19]. However, this approach is limited by its computational complexity and only yields couplings between cell populations as opposed to generating vectors in gene expression space. In this work, we alleviate these limitations by suggesting a TI algorithm based on Optimal Transport via Input Convex Neural Networks (ICNNs, [AXK17]), which we refer to as Parameterized Monge Maps (PMM) [Mak+20]). PMM has been successfully applied to modeling perturbation responses in scRNA-seq data [Bun+21; BKC22]. However, one major disadvantage of applying PMM to scRNA-seq data is the lack of an unbalanced formulation, allowing the model to adapt the source and target distribution during training. This concept is needed in many applications, e.g. to account for different proliferation rates of cells.

We adapt PMM in two different ways to overcome this limitation. First, we show how to adapt the distribution based on prior biological knowledge to allow for different proliferation rates, which has been successfully used in the discrete OT case [Sch+19]. Second, we propose Unbalanced Parameterized Monge Maps, a novel algorithm based on PMM. On simulated data, we demonstrate that it mimics the behavior of discrete unbalanced OT. Finally, we apply our algorithms to a scRNAseq dataset of pancreatic endocrinogenesis [Bas+19] and disentangle fine-grained cell state trajectories. We show that our proposed methods outperform the established trajectory inference methods scVelo [Ber+20], Waddington OT [Sch+19], and TrajectoryNet [Ton+20].

## 2 Methods

Let *P* and *Q* be two probability distributions defined on ℝ^{d}. Let *X* and *Y* be random variables such that *X* ~ *P* and *Y* ~ *Q*, respectively. The Monge Map is defined as

If distances in the space are measured in the squared Euclidean distance, this optimization problem can be rewritten to ([Vil03], Theorem 1.3)
where *𝒞𝒱𝒳* is the set of integrable convex functions and *f* ^{*} denotes the convex conjugate of *f* defined by *f **(*y*)= sup_{x}⟨*x, y*⟩− *f*(*x*). Moreover, the Monge Map *T* : *Q* → *P* can be obtained as the gradient of the convex function *f* ^{*}:

Thus, Makkuva et al. propose to learn *f* and *g* with Input Convex Neural Networks (ICNNs, [AXK17]) leading to the optimization problem

Thus, we obtain a fully parameterized vector field by taking the gradient of the potential *g* since *T* (*y*)= ∇ *g*(*y*). See Appendix A.1 for a detailed derivation.

### 2.1 PMM with transformed distributions

For modeling scRNA-seq data capturing different developmental stages, a notion of unbalancedness is crucial as some cells might proliferate faster than others. Schiebinger et al. propose using proliferation and apoptosis genes to model cell growth and death, respectively, in the discrete OT case. Similarly, we transform the source distribution *Q* with a function γ : *Q* → *Q* depending on proliferation and apoptosis marker genes. Hence, we solve (1) with *X* ~ γ(*Q*) (Appendix A.12).

If prior knowledge is not available, γ cannot be defined. Consequently, PMM cannot account for unbalanced distribution shifts (e.g. across time points). To alleviate this limitation, we propose Unbalanced Parameterized Monge Maps (UPMM).

### 2.2 Unbalanced Parameterized Monge Maps

The goal of UPMM is to mimic the behavior of discrete unbalanced optimal transport (Appendix A.2) in the continuous domain. Therefore, we define the objective function of UPMM as

Here, (*X, y*)*dy* and , where denotes the Wasserstein-2 optimal coupling of *X* and *Y*. We estimate batch-wise with discrete regularised OT [Cut13]. Analogous to the discrete unbalanced OT formulation, decreasing τ_{a} and τ_{b} increases unbalancedness in source and target distribution, respectively^{3}. If and , hence we recover PMM as defined in (4). Algorithm A.3 shows the full training procedure, with training details and hyperparameters described in Appendix A.4.

For simulated data (Appendix A.5.1), Figure 1 visually confirms that UPMM with τ = τ_{a} = τ_{b} =1 yields the same results as PMM. By gradually decreasing τ, we arrive at a map with similar behavior as the one obtained from the discrete unbalanced case. The corresponding potentials are shown in Appendix A.11.1.

### 2.3 Benchmarks and metrics

To assess the performance of PMM, UPMM, and PMM with growth rate prior (Appendix A.12), which we refer to as PMM+GR, we apply our three newly proposed TI methods to a pancreatic endocrinogenesis scRNA-seq dataset (Appendix A.5.2). We compare our methods to the wellestablished TI methods scVelo, Waddington OT, and TrajectoryNet (Appendix A.6). To evaluate how accurately different methods recover biological ground truth, we rely on CellRank [Lan+22] to quantify cell fate probabilities.

The performance of all methods is assessed with three metrics. First, we consider aggregated *cell type transitions* based on biological knowledge. We consider cell type transitions in three classes, the endocrine branch (EB), Ngn3 EP, and non-endocrine branch (NEB) (Appendix A.8.1), which we refer to as lineages in the following. Here, we report the mean of correct cell transitions for each lineage, while full results can be found in Appendix A.8. Second, *cell type redundancy* determines to which extent terminal cell populations are recovered with CellRank (Appendix A.9). This quantifies how well terminal cell states can be recovered from the learned dynamics. Third, we analyze *velocity consistency* across transcriptomically similar cells (Appendix A.10) justified by the fact that cells that are close in gene expression space should have a similar cell trajectory.

## 3 Results

Table 1 shows that, overall, our newly proposed TI methods PMM+GR and UPMM recover correct cell type transitions best. None of the competing methods yield comparable results across all considered sets of cell types. TrajectoryNet maps most Ngn3 high endocrine progenitors (Ngn3 EP) incorrectly. Appendix A.13 reveals that most of these cells are mapped to Acinar cells. While scVelo performs well on Ngn3 EP cells it fails on various other cell types, for example having a large bias to map EP cells to Beta cells. WOT accurately recovers the NEB lineage but fails to achieve similar performance on the other two lineages. Table 1 shows that PMM yields promising results in the EB lineage but its performance on Ngn3 EP is very poor. This can be explained by the asymmetric distribution shift of cell states between the two time points. In effect, Acinar cells proliferate much faster than cells in the EB lineage as can be seen in Appendix A.5.2. Hence, Ngn3 EP is mainly mapped to the NEB branch. In contrast, PMM+GR can compensate for that by considering proliferation and apoptosis genes. Hence, almost all Ngn3 (EP) cells are mapped correctly. Similarly, UPMM explicitly models unbalancedness, thus ranking second overall. While table 1 reports aggregated results, table 2 in Appendix A.8.2 shows more detailed results.

Figure 2 complements these quantitative results by projecting the high dimensional velocities onto a two-dimensional UMAP [MHM18] representation (Appendix A.7). For example, the velocity flow estimated by PMM contradicts the biological ground truth for the Ngn3 EP cells, whose velocity vectors should point towards the upper left of the UMAP in (2c). While UPMM accounts considerably well for the unbalancedness (2d), PMM+GR yields optimal directionalities of Ngn3 EP cells (2e). Plots of the potential values of the UMAP-embedded cells can be found in Appendix A.11.2.

The superior performance of PMM+GR and UPMM is confirmed by the cell type redundancy metric in Table 1. They both yield optimal matches between recovered and known terminal states. Table 1 also shows that velocity vectors of similar cells are significantly higher correlated for PMM, UPMM, PMM+GR, and TrajectoryNet than for scVelo. WOT does not yield velocity vectors and hence neither can the velocity consistency metric be computed nor the velocity stream embedding be plotted (Appendix A.10). Overall, our proposed methods outperform well-established trajectory inference methods. While PMM is a promising approach to model cell dynamics we showed that incorporating unbalancedness is crucial for the accurate identification of cell fate maps.

## 4 Discussion

By extending PMM to the unbalanced setting we propose a new, powerful trajectory inference algorithm that is able to recover fine-grained cell fates while being robust to unbalanced population growth of cells. Given its ability to learn a generative continuous map of cellular differentiation, we believe PMM+GR and UPMM would also show competitive performance for interpolating cellular states to unseen cell types during training. Our methods are particularly valuable for development cell atlas building ([Han+21]) due to their performance, scalability and applicability to unseen data during training. Moreover, our algorithm naturally extends to multiple modalities, e.g. allowing for a fully parameterized vector field in chromatin or protein space. Instead of seeing RNA velocity and UPMM as competing approaches, a promising approach would be to superpose the velocity vector fields to combine strengths from both methods.