Maximilian Nickel

ICML Conference 2025 Conference Paper

Representative Ranking for Deliberation in the Public Sphere

• Manon Revel
• Smitha Milli
• Tyler Lu
• Jamelle Watson-Daniels
• Maximilian Nickel

Online comment sections, such as those on news sites or social media, have the potential to foster informal public deliberation, However, this potential is often undermined by the frequency of toxic or low-quality exchanges that occur in these settings. To combat this, platforms increasingly leverage algorithmic ranking to facilitate higher-quality discussions, e. g. , by using civility classifiers or forms of prosocial ranking. Yet, these interventions may also inadvertently reduce the visibility of legitimate viewpoints, undermining another key aspect of deliberation: representation of diverse views. We seek to remedy this problem by introducing guarantees of representation into these methods. In particular, we adopt the notion of justified representation (JR) from the social choice literature and incorporate a JR constraint into the comment ranking setting. We find that enforcing JR leads to greater inclusion of diverse viewpoints while still being compatible with optimizing for user engagement or other measures of conversational quality.

ICLR Conference 2024 Conference Paper

Generalized Schrödinger Bridge Matching

• Guan-Horng Liu
• Yaron Lipman
• Maximilian Nickel
• Brian Karrer
• Evangelos A. Theodorou
• Ricky T. Q. Chen

Modern distribution matching algorithms for training diffusion or flow models directly prescribe the time evolution of the marginal distributions between two boundary distributions. In this work, we consider a generalized distribution matching setup, where these marginals are only implicitly described as a solution to some task-specific objective function. The problem setup, known as the Generalized Schrödinger Bridge (GSB), appears prevalently in many scientific areas both within and without machine learning. We propose Generalized Schödinger Bridge Matching (GSBM), a new matching algorithm inspired by recent advances, generalizing them beyond kinetic energy minimization and to account for nonlinear state costs. We show that such a generalization can be cast as solving conditional stochastic optimal control, for which efficient variational approximations can be used, and further debiased with the aid of path integral theory. Compared to prior methods for solving GSB problems, our GSBM algorithm always preserves a feasible transport map between the boundary distributions throughout training, thereby enabling stable convergence and significantly improved scalability. We empirically validate our claims on an extensive suite of experimental setups, including crowd navigation, opinion depolarization, LiDAR manifolds, and image domain transfer. Our work brings new algorithmic opportunities for training diffusion models enhanced with task-specific optimality structures.

NeurIPS Conference 2024 Conference Paper

No Free Delivery Service: Epistemic limits of passive data collection in complex social systems

• Maximilian Nickel

Rapid model validation via the train-test paradigm has been a key driver for the breathtaking progress in machine learning and AI. However, modern AI systems often depend on a combination of tasks and data collection practices that violate all assumptions ensuring test validity. Yet, without rigorous model validation we cannot ensure the intended outcomes of deployed AI systems, including positive social impact, nor continue to advance AI research in a scientifically sound way. In this paper, I will show that for widely considered inference settings in complex social systems the train-test paradigm does not only lack a justification but is indeed invalid for any risk estimator, including counterfactual and causal estimators, with high probability. These formal impossibility results highlight a fundamental epistemic issue, i. e. , that for key tasks in modern AI we cannot know whether models are valid under current data collection practices. Importantly, this includes variants of both recommender systems and reasoning via large language models, and neither naïve scaling nor limited benchmarks are suited to address this issue. I am illustrating these results via the widely used MovieLens benchmark and conclude by discussing the implications of these results for AI in social systems, including possible remedies such as participatory data curation and open science.

ICLR Conference 2023 Conference Paper

Flow Matching for Generative Modeling

• Yaron Lipman
• Ricky T. Q. Chen
• Heli Ben-Hamu
• Maximilian Nickel
• Matthew Le 0001

We introduce a new paradigm for generative modeling built on Continuous Normalizing Flows (CNFs), allowing us to train CNFs at unprecedented scale. Specifically, we present the notion of Flow Matching (FM), a simulation-free approach for training CNFs based on regressing vector fields of fixed conditional probability paths. Flow Matching is compatible with a general family of Gaussian probability paths for transforming between noise and data samples---which subsumes existing diffusion paths as specific instances. Interestingly, we find that employing FM with diffusion paths results in a more robust and stable alternative for training diffusion models. Furthermore, Flow Matching opens the door to training CNFs with other, non-diffusion probability paths. An instance of particular interest is using Optimal Transport (OT) displacement interpolation to define the conditional probability paths. These paths are more efficient than diffusion paths, provide faster training and sampling, and result in better generalization. Training CNFs using Flow Matching on ImageNet leads to consistently better performance than alternative diffusion-based methods in terms of both likelihood and sample quality, and allows fast and reliable sample generation using off-the-shelf numerical ODE solvers.

ICML Conference 2023 Conference Paper

Hyperbolic Image-text Representations

• Karan Desai
• Maximilian Nickel
• Tanmay Rajpurohit
• Justin Johnson 0001
• Ramakrishna Vedantam

Visual and linguistic concepts naturally organize themselves in a hierarchy, where a textual concept "dog" entails all images that contain dogs. Despite being intuitive, current large-scale vision and language models such as CLIP do not explicitly capture such hierarchy. We propose MERU, a contrastive model that yields hyperbolic representations of images and text. Hyperbolic spaces have suitable geometric properties to embed tree-like data, so MERU can better capture the underlying hierarchy in image-text datasets. Our results show that MERU learns a highly interpretable and structured representation space while being competitive with CLIP’s performance on standard multi-modal tasks like image classification and image-text retrieval.

ICML Conference 2023 Conference Paper

Neural FIM for learning Fisher information metrics from point cloud data

• Oluwadamilola Fasina
• Guillaume Huguet
• Alexander Tong 0001
• Yanlei Zhang
• Guy Wolf
• Maximilian Nickel
• Ian Adelstein
• Smita Krishnaswamy

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).

ICML Conference 2023 Conference Paper

On Kinetic Optimal Probability Paths for Generative Models

• Neta Shaul
• Ricky T. Q. Chen
• Maximilian Nickel
• Matthew Le 0001
• Yaron Lipman

Recent successful generative models are trained by fitting a neural network to an a-priori defined tractable probability density path taking noise to training examples. In this paper we investigate the space of Gaussian probability paths, which includes diffusion paths as an instance, and look for an optimal member in some useful sense. In particular, minimizing the Kinetic Energy (KE) of a path is known to make particles’ trajectories simple, hence easier to sample, and empirically improve performance in terms of likelihood of unseen data and sample generation quality. We investigate Kinetic Optimal (KO) Gaussian paths and offer the following observations: (i) We show the KE takes a simplified form on the space of Gaussian paths, where the data is incorporated only through a single, one dimensional scalar function, called the data separation function. (ii) We characterize the KO solutions with a one dimensional ODE. (iii) We approximate data-dependent KO paths by approximating the data separation function and minimizing the KE. (iv) We prove that the data separation function converges to $1$ in the general case of arbitrary normalized dataset consisting of $n$ samples in $d$ dimension as $n/\sqrt{d}\rightarrow 0$. A consequence of this result is that the Conditional Optimal Transport (Cond-OT) path becomes kinetic optimal as $n/\sqrt{d}\rightarrow 0$. We further support this theory with empirical experiments on ImageNet.

ICML Conference 2022 Conference Paper

Matching Normalizing Flows and Probability Paths on Manifolds

• Heli Ben-Hamu
• Samuel Cohen
• Avishek Joey Bose
• Brandon Amos
• Maximilian Nickel
• Aditya Grover
• Ricky T. Q. Chen
• Yaron Lipman

Continuous Normalizing Flows (CNFs) are a class of generative models that transform a prior distribution to a model distribution by solving an ordinary differential equation (ODE). We propose to train CNFs on manifolds by minimizing probability path divergence (PPD), a novel family of divergences between the probability density path generated by the CNF and a target probability density path. PPD is formulated using a logarithmic mass conservation formula which is a linear first order partial differential equation relating the log target probabilities and the CNF’s defining vector field. PPD has several key benefits over existing methods: it sidesteps the need to solve an ODE per iteration, readily applies to manifold data, scales to high dimensions, and is compatible with a large family of target paths interpolating pure noise and data in finite time. Theoretically, PPD is shown to bound classical probability divergences. Empirically, we show that CNFs learned by minimizing PPD achieve state-of-the-art results in likelihoods and sample quality on existing low-dimensional manifold benchmarks, and is the first example of a generative model to scale to moderately high dimensional manifolds.

NeurIPS Conference 2022 Conference Paper

Semi-Discrete Normalizing Flows through Differentiable Tessellation

• Ricky T. Q. Chen
• Brandon Amos
• Maximilian Nickel

Mapping between discrete and continuous distributions is a difficult task and many have had to resort to heuristical approaches. We propose a tessellation-based approach that directly learns quantization boundaries in a continuous space, complete with exact likelihood evaluations. This is done through constructing normalizing flows on convex polytopes parameterized using a simple homeomorphism with an efficient log determinant Jacobian. We explore this approach in two application settings, mapping from discrete to continuous and vice versa. Firstly, a Voronoi dequantization allows automatically learning quantization boundaries in a multidimensional space. The location of boundaries and distances between regions can encode useful structural relations between the quantized discrete values. Secondly, a Voronoi mixture model has near-constant computation cost for likelihood evaluation regardless of the number of mixture components. Empirically, we show improvements over existing methods across a range of structured data modalities.

ICML Conference 2021 Conference Paper

CURI: A Benchmark for Productive Concept Learning Under Uncertainty

• Ramakrishna Vedantam
• Arthur Szlam
• Maximilian Nickel
• Ari S. Morcos
• Brenden M. Lake

Humans can learn and reason under substantial uncertainty in a space of infinitely many compositional, productive concepts. For example, if a scene with two blue spheres qualifies as “daxy, ” one can reason that the underlying concept may require scenes to have “only blue spheres” or “only spheres” or “only two objects. ” In contrast, standard benchmarks for compositional reasoning do not explicitly capture a notion of reasoning under uncertainty or evaluate compositional concept acquisition. We introduce a new benchmark, Compositional Reasoning Under Uncertainty (CURI) that instantiates a series of few-shot, meta-learning tasks in a productive concept space to evaluate different aspects of systematic generalization under uncertainty, including splits that test abstract understandings of disentangling, productive generalization, learning boolean operations, variable binding, etc. Importantly, we also contribute a model-independent “compositionality gap” to evaluate the difficulty of generalizing out-of-distribution along each of these axes, allowing objective comparison of the difficulty of each compositional split. Evaluations across a range of modeling choices and splits reveal substantial room for improvement on the proposed benchmark.

ICLR Conference 2021 Conference Paper

Learning Neural Event Functions for Ordinary Differential Equations

• Ricky T. Q. Chen
• Brandon Amos
• Maximilian Nickel

The existing Neural ODE formulation relies on an explicit knowledge of the termination time. We extend Neural ODEs to implicitly defined termination criteria modeled by neural event functions, which can be chained together and differentiated through. Neural Event ODEs are capable of modeling discrete and instantaneous changes in a continuous-time system, without prior knowledge of when these changes should occur or how many such changes should exist. We test our approach in modeling hybrid discrete- and continuous- systems such as switching dynamical systems and collision in multi-body systems, and we propose simulation-based training of point processes with applications in discrete control.

NeurIPS Conference 2021 Conference Paper

Moser Flow: Divergence-based Generative Modeling on Manifolds

• Noam Rozen
• Aditya Grover
• Maximilian Nickel
• Yaron Lipman

We are interested in learning generative models for complex geometries described via manifolds, such as spheres, tori, and other implicit surfaces. Current extensions of existing (Euclidean) generative models are restricted to specific geometries and typically suffer from high computational costs. We introduce Moser Flow (MF), a new class of generative models within the family of continuous normalizing flows (CNF). MF also produces a CNF via a solution to the change-of-variable formula, however differently from other CNF methods, its model (learned) density is parameterized as the source (prior) density minus the divergence of a neural network (NN). The divergence is a local, linear differential operator, easy to approximate and calculate on manifolds. Therefore, unlike other CNFs, MF does not require invoking or backpropagating through an ODE solver during training. Furthermore, representing the model density explicitly as the divergence of a NN rather than as a solution of an ODE facilitates learning high fidelity densities. Theoretically, we prove that MF constitutes a universal density approximator under suitable assumptions. Empirically, we demonstrate for the first time the use of flow models for sampling from general curved surfaces and achieve significant improvements in density estimation, sample quality, and training complexity over existing CNFs on challenging synthetic geometries and real-world benchmarks from the earth and climate sciences.

ICLR Conference 2021 Conference Paper

Neural Spatio-Temporal Point Processes

• Ricky T. Q. Chen
• Brandon Amos
• Maximilian Nickel

We propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and enable flexible, high-fidelity models of discrete events that are localized in continuous time and space. Central to our approach is a combination of continuous-time neural networks with two novel neural architectures, \ie, Jump and Attentive Continuous-time Normalizing Flows. This approach allows us to learn complex distributions for both the spatial and temporal domain and to condition non-trivially on the observed event history. We validate our models on data sets from a wide variety of contexts such as seismology, epidemiology, urban mobility, and neuroscience.

NeurIPS Conference 2020 Conference Paper

Riemannian Continuous Normalizing Flows

• Emile Mathieu
• Maximilian Nickel

Normalizing flows have shown great promise for modelling flexible probability distributions in a computationally tractable way. However, whilst data is often naturally described on Riemannian manifolds such as spheres, torii, and hyperbolic spaces, most normalizing flows implicitly assume a flat geometry, making them either misspecified or ill-suited in these situations. To overcome this problem, we introduce Riemannian continuous normalizing flows, a model which admits the parametrization of flexible probability measures on smooth manifolds by defining flows as the solution to ordinary differential equations. We show that this approach can lead to substantial improvements on both synthetic and real-world data when compared to standard flows or previously introduced projected flows.

NeurIPS Conference 2019 Conference Paper

Hyperbolic Graph Neural Networks

• Qi Liu
• Maximilian Nickel
• Douwe Kiela

Learning from graph-structured data is an important task in machine learning and artificial intelligence, for which Graph Neural Networks (GNNs) have shown great promise. Motivated by recent advances in geometric representation learning, we propose a novel GNN architecture for learning representations on Riemannian manifolds with differentiable exponential and logarithmic maps. We develop a scalable algorithm for modeling the structural properties of graphs, comparing Euclidean and hyperbolic geometry. In our experiments, we show that hyperbolic GNNs can lead to substantial improvements on various benchmark datasets.

ICML Conference 2018 Conference Paper

Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry

• Maximilian Nickel
• Douwe Kiela

We are concerned with the discovery of hierarchical relationships from large-scale unstructured similarity scores. For this purpose, we study different models of hyperbolic space and find that learning embeddings in the Lorentz model is substantially more efficient than in the Poincar{é}-ball model. We show that the proposed approach allows us to learn high-quality embeddings of large taxonomies which yield improvements over Poincar{é} embeddings, especially in low dimensions. Lastly, we apply our model to discover hierarchies in two real-world datasets: we show that an embedding in hyperbolic space can reveal important aspects of a company’s organizational structure as well as reveal historical relationships between language families.

AAAI Conference 2016 Conference Paper

Holographic Embeddings of Knowledge Graphs

• Maximilian Nickel
• Lorenzo Rosasco
• Tomaso Poggio

Learning embeddings of entities and relations is an efficient and versatile method to perform machine learning on relational data such as knowledge graphs. In this work, we propose holographic embeddings (HOLE) to learn compositional vector space representations of entire knowledge graphs. The proposed method is related to holographic models of associative memory in that it employs circular correlation to create compositional representations. By using correlation as the compositional operator, HOLE can capture rich interactions but simultaneously remains efficient to compute, easy to train, and scalable to very large datasets. Experimentally, we show that holographic embeddings are able to outperform state-ofthe-art methods for link prediction on knowledge graphs and relational learning benchmark datasets.

NeurIPS Conference 2014 Conference Paper

Reducing the Rank in Relational Factorization Models by Including Observable Patterns

• Maximilian Nickel
• Xueyan Jiang
• Volker Tresp

Tensor factorizations have become popular methods for learning from multi-relational data. In this context, the rank of a factorization is an important parameter that determines runtime as well as generalization ability. To determine conditions under which factorization is an efficient approach for learning from relational data, we derive upper and lower bounds on the rank required to recover adjacency tensors. Based on our findings, we propose a novel additive tensor factorization model for learning from latent and observable patterns in multi-relational data and present a scalable algorithm for computing the factorization. Experimentally, we show that the proposed approach does not only improve the predictive performance over pure latent variable methods but that it also reduces the required rank --- and therefore runtime and memory complexity --- significantly.

ICML Conference 2011 Conference Paper

A Three-Way Model for Collective Learning on Multi-Relational Data

• Maximilian Nickel
• Volker Tresp
• Hans-Peter Kriegel