Advances in medical imaging technologies have enabled the collection of longitudinal images, which involve repeated scanning of the same patients over time, to monitor disease progression. However, predictive modeling of such data remains challenging due to high dimensionality, irregular sampling, and data sparsity. To address these issues, we propose ImageFlowNet, a novel model designed to forecast disease trajectories from initial images while preserving spatial details. ImageFlowNet first learns multiscale joint representation spaces across patients and time points, then optimizes deterministic or stochastic flow fields within these spaces using a position-parameterized neural ODE/SDE framework. The model leverages a UNet architecture to create robust multiscale representations and mitigates data scarcity by combining knowledge from all patients. We provide theoretical insights that support our formulation of ODEs, and motivate our regularizations involving high-level visual features, latent space organization, and trajectory smoothness. We validate ImageFlowNet on three longitudinal medical image datasets depicting progression in geographic atrophy, multiple sclerosis, and glioblastoma, demonstrating its ability to effectively forecast disease progression and outperform existing methods. Our contributions include the development of ImageFlowNet, its theoretical underpinnings, and empirical validation on real-world datasets.

Numerous biological and physical processes can be modeled as systems of interacting entities evolving continuously over time, e.g. the dynamics of communicating cells or physical particles. Learning the dynamics of such systems is essential for predicting the temporal evolution of populations across novel samples and unseen environments. Flow-based models allow for learning these dynamics at the population level - they model the evolution of the entire distribution of samples. However, current flow-based models are limited to a single initial population and a set of predefined conditions which describe different dynamics. We argue that multiple processes in natural sciences have to be represented as vector fields on the Wasserstein manifold of probability densities. That is, the change of the population at any moment in time depends on the population itself due to the interactions between samples. In particular, this is crucial for personalized medicine where the development of diseases and their respective treatment response depend on the microenvironment of cells specific to each patient. We propose *Meta Flow Matching* (MFM), a practical approach to integrate along these vector fields on the Wasserstein manifold by amortizing the flow model over the initial populations. Namely, we embed the population of samples using a Graph Neural Network (GNN) and use these embeddings to train a Flow Matching model. This gives MFM the ability to generalize over the initial distributions, unlike previously proposed methods. We demonstrate the ability of MFM to improve the prediction of individual treatment responses on a large-scale multi-patient single-cell drug screen dataset.

Generative modeling of single-cell RNA-seq data is crucial for tasks like trajectory inference, batch effect removal, and simulation of realistic cellular data. However, recent deep generative models simulating synthetic single cells from noise operate on pre-processed continuous gene expression approximations, overlooking the discrete nature of single-cell data, which limits their effectiveness and hinders the incorporation of robust noise models. Additionally, aspects like controllable multi-modal and multi-label generation of cellular data remain underexplored. This work introduces CellFlow for Generation (CFGen), a flow-based conditional generative model that preserves the inherent discreteness of single-cell data. CFGen reliably generates whole-genome, multi-modal, single-cell data, improving the recovery of crucial biological data characteristics while tackling relevant generative tasks such as rare cell type augmentation and batch correction. We also introduce a novel framework for compositional data generation using Flow Matching. By showcasing CFGen on a diverse set of biological datasets and settings, we provide evidence of its value to the fields of computational biology and deep generative models.

Generative modeling of discrete data underlies important applications spanning text-based agents like ChatGPT to the design of the very building blocks of life in protein sequences. However, application domains need to exert control over the generated data by steering the generative process—typically via RLHF—to satisfy a specified property, reward, or affinity metric. In this paper, we study the problem of steering Masked Diffusion Models (MDMs), a recent class of discrete diffusion models that offer a compelling alternative to traditional autoregressive models. We introduce Discrete Denoising Posterior Prediction (DDPP), a novel framework that casts the task of steering pretrained MDMs as a problem of probabilistic inference by learning to sample from a target Bayesian posterior. Our DDPP framework leads to a family of three novel objectives that are all simulation-free, and thus scalable while applying to general non-differentiable reward functions. Empirically, we instantiate DDPP by steering MDMs to perform class-conditional pixel-level image modeling, RLHF-based alignment of MDMs using text based rewards, and finetuning protein language models to generate more diverse secondary structures and shorter proteins. We substantiate our designs via wet-lab validation, where we observe transient expression of reward-optimized protein sequences.

The Cambrian explosion of easily accessible pre-trained diffusion models suggests a demand for methods that combine multiple different pre-trained diffusion models without incurring the significant computational burden of re-training a larger combined model. In this paper, we cast the problem of combining multiple pre-trained diffusion models at the generation stage under a novel proposed framework termed superposition. Theoretically, we derive superposition from rigorous first principles stemming from the celebrated continuity equation and design two novel algorithms tailor-made for combining diffusion models in SuperDiff. SuperDiff leverages a new scalable Itô density estimator for the log likelihood of the diffusion SDE which incurs *no additional overhead* compared to the well-known Hutchinson's estimator needed for divergence calculations. We demonstrate that SuperDiff is scalable to large pre-trained diffusion models as superposition is performed *solely through composition during inference*, and also enjoys painless implementation as it combines different pre-trained vector fields through an automated re-weighting scheme. Notably, we show that SuperDiff is efficient during inference time, and mimics traditional composition operators such as the logical OR and the logical AND. We empirically demonstrate the utility of using SuperDiff for generating more diverse images on CIFAR-10, more faithful prompt conditioned image editing using Stable Diffusion, as well as improved conditional molecule generation and unconditional *de novo* structure design of proteins. https://github.com/necludov/super-diffusion

While score-based generative models are the model of choice across diverse domains, there are limited tools available for controlling inference-time behavior in a principled manner, e.g. for composing multiple pretrained models. Existing classifier-free guidance methods use a simple heuristic to mix conditional and unconditional scores to approximately sample from conditional distributions. However, such methods do not approximate the intermediate distributions, necessitating additional ‘corrector’ steps. In this work, we provide an efficient and principled method for sampling from a sequence of annealed, geometric-averaged, or product distributions derived from pretrained score-based models. We derive a weighted simulation scheme which we call Feynman-Kac Correctors (FKCs) based on the celebrated Feynman-Kac formula by carefully accounting for terms in the appropriate partial differential equations (PDEs). To simulate these PDEs, we propose Sequential Monte Carlo (SMC) resampling algorithms that leverage inference-time scaling to improve sampling quality. We empirically demonstrate the utility of our methods by proposing amortized sampling via inference-time temperature annealing, improving multi-objective molecule generation using pretrained models, and improving classifier-free guidance for text-to-image generation.

Scalable sampling of molecular states in thermodynamic equilibrium is a long-standing challenge in statistical physics. Boltzmann generators tackle this problem by pairing normalizing flows with importance sampling to obtain uncorrelated samples under the target distribution. In this paper, we extend the Boltzmann generator framework with two key contributions, denoting our framework Sequential Boltzmann Generators (SBG). The first is a highly efficient Transformer-based normalizing flow operating directly on all-atom Cartesian coordinates. In contrast to the equivariant continuous flows of prior methods, we leverage exactly invertible non-equivariant architectures which are highly efficient during both sample generation and likelihood evaluation. This efficiency unlocks more sophisticated inference strategies beyond standard importance sampling. In particular, we perform inference-time scaling of flow samples using a continuous-time variant of sequential Monte Carlo, in which flow samples are transported towards the target distribution with annealed Langevin dynamics. SBG achieves state-of-the-art performance w.r.t. all metrics on peptide systems, demonstrating the first equilibrium sampling in Cartesian coordinates of tri-, tetra- and hexa-peptides that were thus far intractable for prior Boltzmann generators.

Sampling efficiently from a target unnormalized probability density remains a core challenge, with relevance across countless high-impact scientific applications. A promising approach towards this challenge is the design of amortized samplers that borrow key ideas, such as probability path design, from state-of-the-art generative diffusion models. However, all existing diffusion-based samplers remain unable to draw samples from distributions at the scale of even simple molecular systems. In this paper, we propose Progressive Inference-Time Annealing (PITA) a novel framework to learn diffusion-based samplers that combines two complementary interpolation techniques: I.) Annealing of the Boltzmann distribution and II.) Diffusion smoothing. PITA trains a sequence of diffusion models from high to low temperatures by sequentially training each model at progressively higher temperatures, leveraging engineered easy access to samples of the temperature-annealed target density. In the subsequent step, PITA enables simulating the trained diffusion model to *procure training samples at a lower temperature* for the next diffusion model through inference-time annealing using a novel Feynman-Kac PDE combined with Sequential Monte Carlo. Empirically, PITA enables, for the first time, equilibrium sampling of $N$-body particle systems, Alanine Dipeptide, and tripeptides in Cartesian coordinates with dramatically lower energy function evaluations.

Modeling the transport dynamics of natural processes from population-level observations is a ubiquitous problem in the natural sciences. Such models rely on key assumptions about the underlying process in order to enable faithful learning of governing dynamics that mimic the actual system behavior. The de facto assumption in current approaches relies on the principle of least action that results in gradient field dynamics and leads to trajectories minimizing an energy functional between two probability measures. However, many real-world systems, such as cell cycles in single-cell RNA, are known to exhibit non-gradient, periodic behavior, which fundamentally cannot be captured by current state-of-the-art methods such as flow and bridge matching. In this paper, we introduce Curly Flow Matching (Curly-FM), a novel approach that is capable of learning non-gradient field dynamics by designing and solving a Schrödinger bridge problem with a non-zero drift reference process---in stark contrast to typical zero-drift reference processes---which is constructed using inferred velocities in addition to population snapshot data. We showcase Curly-FM by solving the trajectory inference problems for single cells, computational fluid dynamics, and ocean currents with approximate velocities. We demonstrate that Curly-FM can learn trajectories that better match both the reference process and population marginals. Curly-FM expands flow matching models beyond the modeling of populations and towards the modeling of known periodic behavior in physical systems. Our code repository is accessible at: https://github.com/kpetrovicc/curly-flow-matching.git

Efficient equilibrium sampling of molecular conformations remains a core challenge in computational chemistry and statistical inference. Classical approaches such as molecular dynamics or Markov chain Monte Carlo inherently lack amortization; the computational cost of sampling must be paid in full for each system of interest. The widespread success of generative models has inspired interest towards overcoming this limitation through learning sampling algorithms. Despite performing competitively with conventional methods when trained on a single system, learned samplers have so far demonstrated limited ability to transfer across systems. We demonstrate that deep learning enables the design of scalable and transferable samplers by introducing Prose, a 285 million parameter all-atom transferable normalizing flow trained on a corpus of peptide molecular dynamics trajectories up to 8 residues in length. Prose draws zero-shot uncorrelated proposal samples for arbitrary peptide systems, achieving the previously intractable transferability across sequence length, whilst retaining the efficient likelihood evaluation of normalizing flows. Through extensive empirical evaluation we demonstrate the efficacy of Prose as a proposal for a variety of sampling algorithms, finding a simple importance sampling-based finetuning procedure to achieve competitive performance to established methods such as sequential Monte Carlo. We open-source the Prose codebase, model weights, and training dataset, to further stimulate research into amortized sampling methods and finetuning objectives.

The computational design of novel protein structures has the potential to impact numerous scientific disciplines greatly. Toward this goal, we introduce \foldflow, a series of novel generative models of increasing modeling power based on the flow-matching paradigm over $3\mathrm{D}$ rigid motions---i.e. the group $\mathrm{SE(3)}$---enabling accurate modeling of protein backbones. We first introduce $\text{FoldFlow-Base}$, a simulation-free approach to learning deterministic continuous-time dynamics and matching invariant target distributions on $\mathrm{SE(3)}$. We next accelerate training by incorporating Riemannian optimal transport to create $\text{FoldFlow-OT}$, leading to the construction of both more simple and stable flows. Finally, we design \foldflowsfm, coupling both Riemannian OT and simulation-free training to learn stochastic continuous-time dynamics over $\mathrm{SE(3)}$. Our family of $\text{FoldFlow}$, generative models offers several key advantages over previous approaches to the generative modeling of proteins: they are more stable and faster to train than diffusion-based approaches, and our models enjoy the ability to map any invariant source distribution to any invariant target distribution over $\mathrm{SE(3)}$. Empirically, we validate $\text{FoldFlow}$, on protein backbone generation of up to $300$ amino acids leading to high-quality designable, diverse, and novel samples.

Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-body systems, is a foundational problem in science. In this paper, we propose Iterated Denoising Energy Matching (iDEM), an iterative algorithm that uses a novel stochastic score matching objective leveraging solely the energy function and its gradient---and no data samples---to train a diffusion-based sampler. Specifically, iDEM alternates between (I) sampling regions of high model density from a diffusion-based sampler and (II) using these samples in our stochastic matching objective to further improve the sampler. iDEM is scalable to high dimensions as the inner matching objective, is *simulation-free*, and requires no MCMC samples. Moreover, by leveraging the fast mode mixing behavior of diffusion, iDEM smooths out the energy landscape enabling efficient exploration and learning of an amortized sampler. We evaluate iDEM on a suite of tasks ranging from standard synthetic energy functions to invariant $n$-body particle systems. We show that the proposed approach achieves state-of-the-art performance on all metrics and trains $2-5\times$ faster, which allows it to be the first method to train using energy on the challenging $55$-particle Lennard-Jones system.

The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry (*kinetic energy*), and the regularization of density paths (*potential energy*). These combinations yield different variational problems (*Lagrangians*), encompassing many variations of the optimal transport problem such as the Schrödinger bridge, unbalanced optimal transport, and optimal transport with physical constraints, among others. In general, the optimal density path is unknown, and solving these variational problems can be computationally challenging. We propose a novel deep learning based framework approaching all of these problems from a unified perspective. Leveraging the dual formulation of the Lagrangians, our method does not require simulating or backpropagating through the trajectories of the learned dynamics, and does not need access to optimal couplings. We showcase the versatility of the proposed framework by outperforming previous approaches for the single-cell trajectory inference, where incorporating prior knowledge into the dynamics is crucial for correct predictions.

Proteins are essential for almost all biological processes and derive their diverse functions from complex $3 \rm D$ structures, which are in turn determined by their amino acid sequences. In this paper, we exploit the rich biological inductive bias of amino acid sequences and introduce FoldFlow++, a novel sequence-conditioned $\text{SE}(3)$-equivariant flow matching model for protein structure generation. FoldFlow++ presents substantial new architectural features over the previous FoldFlow family of models including a protein large language model to encode sequence, a new multi-modal fusion trunk that combines structure and sequence representations, and a geometric transformer based decoder. To increase diversity and novelty of generated samples -- crucial for de-novo drug design -- we train FoldFlow++ at scale on a new dataset that is an order of magnitude larger than PDB datasets of prior works, containing both known proteins in PDB and high-quality synthetic structures achieved through filtering. We further demonstrate the ability to align FoldFlow++ to arbitrary rewards, e.g. increasing secondary structures diversity, by introducing a Reinforced Finetuning (ReFT) objective. We empirically observe that FoldFlow++ outperforms previous state-of-the-art protein structure-based generative models, improving over RFDiffusion in terms of unconditional generation across all metrics including designability, diversity, and novelty across all protein lengths, as well as exhibiting generalization on the task of equilibrium conformation sampling. Finally, we demonstrate that a fine-tuned FoldFlow++ makes progress on challenging conditional design tasks such as designing scaffolds for the VHH nanobody.

Matching objectives underpin the success of modern generative models and rely on constructing conditional paths that transform a source distribution into a target distribution. Despite being a fundamental building block, conditional paths have been designed principally under the assumption of $\textit{Euclidean geometry}$, resulting in straight interpolations. However, this can be particularly restrictive for tasks such as trajectory inference, where straight paths might lie outside the data manifold, thus failing to capture the underlying dynamics giving rise to the observed marginals. In this paper, we propose Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching where interpolants are approximate geodesics learned by minimizing the kinetic energy of a data-induced Riemannian metric. This way, the generative model matches vector fields on the data manifold, which corresponds to lower uncertainty and more meaningful interpolations. We prescribe general metrics to instantiate MFM, independent of the task, and test it on a suite of challenging problems including LiDAR navigation, unpaired image translation, and modeling cellular dynamics. We observe that MFM outperforms the Euclidean baselines, particularly achieving SOTA on single-cell trajectory prediction.

Modeling stochastic and irregularly sampled time series is a challenging problem found in a wide range of applications, especially in medicine. Neural stochastic differential equations (Neural SDEs) are an attractive modeling technique for this problem, which parameterize the drift and diffusion terms of an SDE with neural networks. However, current algorithms for training Neural SDEs require backpropagation through the SDE dynamics, greatly limiting their scalability and stability. To address this, we propose **Trajectory Flow Matching** (TFM), which trains a Neural SDE in a *simulation-free* manner, bypassing backpropagation through the dynamics. TFM leverages the flow matching technique from generative modeling to model time series. In this work we first establish necessary conditions for TFM to learn time series data. Next, we present a reparameterization trick which improves training stability. Finally, we adapt TFM to the clinical time series setting, demonstrating improved performance on four clinical time series datasets both in terms of absolute performance and uncertainty prediction, a crucial parameter in this setting.

Although data diffusion embeddings are ubiquitous in unsupervised learning and have proven to be a viable technique for uncovering the underlying intrinsic geometry of data, diffusion embeddings are inherently limited due to their discrete nature. To this end, we propose neural FIM, a method for computing the Fisher information metric (FIM) from point cloud data - allowing for a continuous manifold model for the data. Neural FIM creates an extensible metric space from discrete point cloud data such that information from the metric can inform us of manifold characteristics such as volume and geodesics. We demonstrate Neural FIM’s utility in selecting parameters for the PHATE visualization method as well as its ability to obtain information pertaining to local volume illuminating branching points and cluster centers embeddings of a toy dataset and two single-cell datasets of IPSC reprogramming and PBMCs (immune cells).

One of the grand challenges of cell biology is inferring the gene regulatory network (GRN) which describes interactions between genes and their products that control gene expression and cellular function. We can treat this as a causal discovery problem but with two non-standard challenges: (1) regulatory networks are inherently cyclic so we should not model a GRN as a directed acyclic graph (DAG), and (2) observations have significant measurement noise so for typical sample sizes, there will always be a large equivalence class of graphs that are likely given the data, and we want methods that capture this uncertainty. Existing methods either focus on challenge (1), identifying cyclic structure from dynamics, or on challenge (2) learning complex Bayesian posteriors over directed acyclic graphs, but not both. In this paper we leverage the fact that it is possible to estimate the ``velocity'' of the expression of a gene with RNA velocity techniques to develop an approach that addresses both challenges. Because we have access to velocity information, we can treat the Bayesian structure learning problem as a problem of sparse identification of a dynamical system, capturing cyclic feedback loops through time. We leverage Generative Flow Networks (GFlowNets) to estimate the posterior distribution over the combinatorial space of possible sparse dependencies. Our results indicate that our method learns posteriors that better encapsulate the distributions of cyclic structures compared to counterpart state-of-the-art Bayesian structure learning approaches.

Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology and physics. While it is thought that these methods preserve underlying manifold structure of data by learning a proxy for geodesic distances, no specific theoretical links have been established. Here, we establish such a link via results in Riemannian geometry explicitly connecting heat diffusion to manifold distances. In this process, we also formulate a more general heat kernel based manifold embedding method that we call heat geodesic embeddings. This novel perspective makes clearer the choices available in manifold learning and denoising. Results show that our method outperforms existing state of the art in preserving ground truth manifold distances, and preserving cluster structure in toy datasets. We also showcase our method on single cell RNA-sequencing datasets with both continuum and cluster structure, where our method enables interpolation of withheld timepoints of data. Finally, we show that parameters of our more general method can be configured to give results similar to PHATE (a state-of-the-art diffusion based manifold learning method) as well as SNE (an attraction/repulsion neighborhood based method that forms the basis of t-SNE).

In modern relational machine learning it is common to encounter large graphs that arise via interactions or similarities between observations in many domains. Further, in many cases the target entities for analysis are actually signals on such graphs. We propose to compare and organize such datasets of graph signals by using an earth mover’s distance (EMD) with a geodesic cost over the underlying graph. Typically, EMD is computed by optimizing over the cost of transporting one probability distribution to another over an underlying metric space. However, this is inefficient when computing the EMD between many signals. Here, we propose an unbalanced graph EMD that efficiently embeds the unbalanced EMD on an underlying graph into an L1space, whose metric we call unbalanced diffusion earth mover’s distance (UDEMD). Next, we show how this gives distances between graph signals that are robust to noise. Finally, we apply this to organizing patients based on clinical notes, embedding cells modeled as signals on a gene graph, and organizing genes modeled as signals over a large cell graph. In each case, we show that UDEMD-based embeddings find accurate distances that are highly efficient compared to other methods.

We present a method called Manifold Interpolating Optimal-Transport Flow (MIOFlow) that learns stochastic, continuous population dynamics from static snapshot samples taken at sporadic timepoints. MIOFlow combines dynamic models, manifold learning, and optimal transport by training neural ordinary differential equations (Neural ODE) to interpolate between static population snapshots as penalized by optimal transport with manifold ground distance. Further, we ensure that the flow follows the geometry by operating in the latent space of an autoencoder that we call a geodesic autoencoder (GAE). In GAE the latent space distance between points is regularized to match a novel multiscale geodesic distance on the data manifold that we define. We show that this method is superior to normalizing flows, Schr\"odinger bridges and other generative models that are designed to flow from noise to data in terms of interpolating between populations. Theoretically, we link these trajectories with dynamic optimal transport. We evaluate our method on simulated data with bifurcations and merges, as well as scRNA-seq data from embryoid body differentiation, and acute myeloid leukemia treatment.

A major challenge in embedding or visualizing clinical patient data is the heterogeneity of variable types including continuous lab values, categorical diagnostic codes, as well as missing or incomplete data. In particular, in EHR data, some variables are missing not at random (MNAR) but deliberately not collected and thus are a source of information. For example, lab tests may be deemed necessary for some patients on the basis of suspected diagnosis, but not for others. Here we present the MURAL forest – an unsupervised random forest for representing data with disparate variable types (e.g., categorical, continuous, MNAR). MURAL forests consist of a set of decision trees where node-splitting variables are chosen at random, such that the marginal entropy of all other variables is minimized by the split. This allows us to also split on MNAR variables and discrete variables in a way that is consistent with the continuous variables. The end goal is to learn the MURAL embedding of patients using average tree distances between those patients. These distances can be fed to nonlinear dimensionality reduction method like PHATE to derive visualizable embeddings. While such methods are ubiquitous in continuous-valued datasets (like single cell RNA-sequencing) they have not been used extensively in mixed variable data. We showcase the use of our method on one artificial and two clinical datasets. We show that using our approach, we can visualize and classify data more accurately than competing approaches. Finally, we show that MURAL can also be used to compare cohorts of patients via the recently proposed tree-sliced Wasserstein distances.

We propose a new fast method of measuring distances between large numbers of related high dimensional datasets called the Diffusion Earth Mover’s Distance (EMD). We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data. In such cases where the graph is a discretization of an underlying Riemannian closed manifold, we prove that Diffusion EMD is topologically equivalent to the standard EMD with a geodesic ground distance. Diffusion EMD can be computed in {Õ}(n) time and is more accurate than similarly fast algorithms such as tree-based EMDs. We also show Diffusion EMD is fully differentiable, making it amenable to future uses in gradient-descent frameworks such as deep neural networks. Finally, we demonstrate an application of Diffusion EMD to single cell data collected from 210 COVID-19 patient samples at Yale New Haven Hospital. Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods. This distance matrix between patients can be embedded into a higher level patient manifold which uncovers structure and heterogeneity in patients. More generally, Diffusion EMD is applicable to all datasets that are massively collected in parallel in many medical and biological systems.

Optimal transport has recently been reintroduced to the machine learning community thanks in part to novel efficient optimization procedures allowing for medium to large scale applications. We propose a Python toolbox that implements several key optimal transport ideas for the machine learning community. The toolbox contains implementations of a number of founding works of OT for machine learning such as Sinkhorn algorithm and Wasserstein barycenters, but also provides generic solvers that can be used for conducting novel fundamental research. This toolbox, named POT for Python Optimal Transport, is open source with an MIT license.

Biomolecular graph analysis has recently gained much attention in the emerging field of geometric deep learning. Here we focus on organizing biomolecular graphs in ways that expose meaningful relations and variations between them. We propose a geometric scattering autoencoder (GSAE) network for learning such graph embeddings. Our embedding network first extracts rich graph features using the recently proposed geometric scattering transform. Then, it leverages a semi-supervised variational autoencoder to extract a low-dimensional embedding that retains the information in these features that enable prediction of molecular properties as well as characterize graphs. We show that GSAE organizes RNA graphs both by structure and energy, accurately reflecting b istable R NA s tructures. A lso, the model is generative and can sample new folding trajectories.

It is increasingly common to encounter data in the form of cross-sectional population measurements over time, particularly in biomedical settings. Recent attempts to model individual trajectories from this data use optimal transport to create pairwise matchings between time points. However, these methods cannot model non-linear paths common in many underlying dynamic systems. We establish a link between continuous normalizing flows and dynamic optimal transport to model the expected paths of points over time. Continuous normalizing flows are generally under constrained, as they are allowed to take an arbitrary path from the source to the target distribution. We present \emph{TrajectoryNet}, which controls the continuous paths taken between distributions. We show how this is particularly applicable for studying cellular dynamics in data from single-cell RNA sequencing (scRNA-seq) technologies, and that TrajectoryNet improves upon recently proposed static optimal transport-based models that can be used for interpolating cellular distributions.

While neural networks are powerful approximators used to classify or embed data into lower dimensional spaces, they are often regarded as black boxes with uninterpretable features. Here we propose Graph Spectral Regularization for making hidden layers more interpretable without significantly impacting performance on the primary task. Taking inspiration from spatial organization and localization of neuron activations in biological networks, we use a graph Laplacian penalty to structure the activations within a layer. This penalty encourages activations to be smooth either on a predetermined graph or on a feature-space graph learned from the data via co-activations of a hidden layer of the neural network. We show numerous uses for this additional structure including cluster indication and visualization in biological and image data sets.

Archetypal analysis is a data decomposition method that describes each observation in a dataset as a convex combination of ”pure types” or archetypes. These archetypes represent extrema of a data space in which there is a trade-off between features, such as in biology where different combinations of traits provide optimal fitness for different environments. Existing methods for archetypal analysis work well when a linear relationship exists between the feature space and the archetypal space. However, such methods are not applicable to systems where the feature space is generated non-linearly from the combination of archetypes, such as in biological systems or image transformations. Here, we propose a reformulation of the problem such that the goal is to learn a non-linear transformation of the data into a latent archetypal space. To solve this problem, we introduce Archetypal Analysis network (AAnet), which is a deep neural network framework for learning and generating from a latent archetypal representation of data. We demonstrate state-of-the-art recovery of ground-truth archetypes in non-linear data domains, show AAnet can generate from data geometry rather than from data density, and use AAnet to identify biologically meaningful archetypes in single-cell gene expression data.

Many fundamental problems in shared-memory distributed computing, including mutual exclusion [James E. Burns and Nancy A. Lynch, 1993], consensus [Leqi Zhu, 2016], and implementations of many sequential objects [Prasad Jayanti et al., 2000], are known to require linear space in the worst case. However, these lower bounds all work by constructing particular executions for any given algorithm that may be both very long and very improbable. The significance of these bounds is justified by an assumption that any space that is used in some execution must be allocated for all executions. This assumption is not consistent with the storage allocation mechanisms of actual practical systems. We consider the consequences of adopting a per-execution approach to space complexity, where an object only counts toward the space complexity of an execution if it is used in that execution. This allows us to show that many known randomized algorithms for fundamental problems in shared-memory distributed computing have expected space complexity much lower than the worst-case lower bounds, and that many algorithms that are adaptive in time complexity can also be made adaptive in space complexity. For the specific problem of mutual exclusion, we develop a new algorithm that illustrates an apparent trade-off between low expected space complexity and low expected RMR complexity. Whether this trade-off is necessary is an open problem. For some applications, it may be helpful to pay only for objects that are updated, as opposed to those that are merely read. We give a data structure that requires no space to represent objects that are not updated at the cost of a small overhead on those that are.