Alexander Tong

A Computational Framework for Solving Wasserstein Lagrangian Flows

Rob Brekelmans

Qiang Liu

Alireza Makhzani

The dynamical formulation of the optimal transport can be extended through various choices of the underlying geometry (kinetic energy), and … (see more)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 Schr\"odinger 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.

2024-07-08

Proceedings of the 41st International Conference on Machine Learning (published)

doi.org

arxiv.org

Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold

Lazar Atanackovic

Xi Zhang

Brandon Amos

Leo J Lee

Numerous biological and physical processes can be modeled as systems of interacting samples evolving continuously over time, e.g. the dynami… (see more)cs of communicating cells or physical particles. 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 propose

2024-06-17

ICML.cc/2024/Workshop/GRaM (published)

openreview.net

Iterated Denoising Energy Matching for Sampling from Boltzmann Densities

Cheng-Hao Liu

Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-… (see more)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

2024-05-01

ICML.cc/2024/Conference (poster)

doi.org

openreview.net

Improving and Generalizing Flow-Based Generative Models with Minibatch Optimal Transport

Yanlei Zhang

Continuous normalizing flows (CNFs) are an attractive generative modeling technique, but they have been held back by limitations in their si… (see more)mulation-based maximum likelihood training. We introduce the generalized \textit{conditional flow matching} (CFM) technique, a family of simulation-free training objectives for CNFs. CFM features a stable regression objective like that used to train the stochastic flow in diffusion models but enjoys the efficient inference of deterministic flow models. In contrast to both diffusion models and prior CNF training algorithms, CFM does not require the source distribution to be Gaussian or require evaluation of its density. A variant of our objective is optimal transport CFM (OT-CFM), which creates simpler flows that are more stable to train and lead to faster inference, as evaluated in our experiments. Furthermore, OT-CFM is the first method to compute dynamic OT in a simulation-free way. Training CNFs with CFM improves results on a variety of conditional and unconditional generation tasks, such as inferring single cell dynamics, unsupervised image translation, and Schrödinger bridge inference.

2024-03-11

TMLR (accepted)

openreview.net

Iterated Denoising Energy Matching for Sampling from Boltzmann Densities

Cheng-Hao Liu

Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-… (see more)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

2024-02-09

ArXiv (preprint)

doi.org

arxiv.org

Iterated Denoising Energy Matching for Sampling from Boltzmann Densities

Cheng-Hao Liu

Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-… (see more)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

2024-02-09

ArXiv (preprint)

doi.org

arxiv.org

Iterated Denoising Energy Matching for Sampling from Boltzmann Densities

Cheng-Hao Liu

Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-… (see more)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

2024-02-09

ArXiv (preprint)

doi.org

arxiv.org

Iterated Denoising Energy Matching for Sampling from Boltzmann Densities

Cheng-Hao Liu

Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-… (see more)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

2024-02-09

ArXiv (preprint)

doi.org

arxiv.org

Iterated Denoising Energy Matching for Sampling from Boltzmann Densities

Cheng-Hao Liu

Efficiently generating statistically independent samples from an unnormalized probability distribution, such as equilibrium samples of many-… (see more)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

2024-02-09

ArXiv (preprint)

doi.org

arxiv.org

Assessing Neural Network Representations During Training Using Noise-Resilient Diffusion Spectral Entropy

Danqi Liao

Chen Liu

Benjamin W Christensen

Alexander Tong

Guillaume Huguet

Guy Wolf

Maximilian Nickel

Ian Adelstein

Smita Krishnaswamy

Entropy and mutual information in neural networks provide rich information on the learning process, but they have proven difficult to comput… (see more)e reliably in high dimensions. Indeed, in noisy and high-dimensional data, traditional estimates in ambient dimensions approach a fixed entropy and are prohibitively hard to compute. To address these issues, we leverage data geometry to access the underlying manifold and reliably compute these information-theoretic measures. Specifically, we define diffusion spectral entropy (DSE) in neural representations of a dataset as well as diffusion spectral mutual information (DSMI) between different variables representing data. First, we show that they form noise-resistant measures of intrinsic dimensionality and relationship strength in high-dimensional simulated data that outperform classic Shannon entropy, nonparametric estimation, and mutual information neural estimation (MINE). We then study the evolution of representations in classification networks with supervised learning, self-supervision, or overfitting. We observe that (1) DSE of neural representations increases during training; (2) DSMI with the class label increases during generalizable learning but stays stagnant during overfitting; (3) DSMI with the input signal shows differing trends: on MNIST it increases, while on CIFAR-10 and STL-10 it decreases. Finally, we show that DSE can be used to guide better network initialization and that DSMI can be used to predict downstream classification accuracy across 962 models on ImageNet.

2024-01-01

CISS (published)

doi.org

openreview.net

Learnable Filters for Geometric Scattering Modules

Alexander Tong

Frederik Wenkel

Dhananjay Bhaskar

Kincaid MacDonald

Jackson Grady

Michael Perlmutter

Smita Krishnaswamy

Guy Wolf

2024-01-01

IEEE Transactions on Signal Processing (published)

doi.org

arxiv.org

Metric Flow Matching for Smooth Interpolations on the Data Manifold

Kacper Kapusniak

Peter Potaptchik

Teodora Reu

Leo Zhang

Alexander Tong

Michael M. Bronstein

Joey Bose

Francesco Di Giovanni

Matching objectives underpin the success of modern generative models and rely on constructing conditional paths that transform a source dist… (see more)ribution into a target distribution. Despite being a fundamental building block, conditional paths have been designed principally under the assumption of 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.

2024-01-01

NeurIPS (published)

doi.org

arxiv.org

Opening Conference | Building Safer AI for Youth Mental Health

TRAIL: Responsible AI for Professionals and Leaders

Mila Ventures Founder in Residence

Indigenous Pathfinders in AI

Alexander Tong

Publications

Opening Conference | Building Safer AI for Youth Mental Health

TRAIL: Responsible AI for Professionals and Leaders

Mila Ventures Founder in Residence

Indigenous Pathfinders in AI

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Alexander Tong

Publications