Rob Brekelmans

Xianghao Kong, Rob Brekelmans, Greg Ver Steeg

Denoising diffusion models have spurred significant gains in density modeling and image generation, precipitating an industrial revolution in text-guided AI art generation. We introduce a new mathematical foundation for diffusion models inspired by classic results in information theory that connect Information with Minimum Mean Square Error regression, the so-called I-MMSE relations. We generalize the I-MMSE relations to exactly relate the data distribution to an optimal denoising regression problem, leading to an elegant refinement of existing diffusion bounds. This new insight leads to several improvements for probability distribution estimation, including theoretical justification for diffusion model ensembling. Remarkably, our framework shows how continuous and discrete probabilities can be learned with the same regression objective, avoiding domain-specific generative models used in variational methods. Code to reproduce experiments is provided at http://github.com/kxh001/ITdiffusion and simplified demonstration code is at http://github.com/gregversteeg/InfoDiffusionSimple.

Kirill Neklyudov, Rob Brekelmans, Daniel Severo et al. · meta-ai, utoronto

Learning the continuous dynamics of a system from snapshots of its temporal marginals is a problem which appears throughout natural sciences and machine learning, including in quantum systems, single-cell biological data, and generative modeling. In these settings, we assume access to cross-sectional samples that are uncorrelated over time, rather than full trajectories of samples. In order to better understand the systems under observation, we would like to learn a model of the underlying process that allows us to propagate samples in time and thereby simulate entire individual trajectories. In this work, we propose Action Matching, a method for learning a rich family of dynamics using only independent samples from its time evolution. We derive a tractable training objective, which does not rely on explicit assumptions about the underlying dynamics and does not require back-propagation through differential equations or optimal transport solvers. Inspired by connections with optimal transport, we derive extensions of Action Matching to learn stochastic differential equations and dynamics involving creation and destruction of probability mass. Finally, we showcase applications of Action Matching by achieving competitive performance in a diverse set of experiments from biology, physics, and generative modeling.

Rob Brekelmans, Sicong Huang, Marzyeh Ghassemi et al. · utoronto

Mutual information (MI) is a fundamental quantity in information theory and machine learning. However, direct estimation of MI is intractable, even if the true joint probability density for the variables of interest is known, as it involves estimating a potentially high-dimensional log partition function. In this work, we present a unifying view of existing MI bounds from the perspective of importance sampling, and propose three novel bounds based on this approach. Since accurate estimation of MI without density information requires a sample size exponential in the true MI, we assume either a single marginal or the full joint density information is known. In settings where the full joint density is available, we propose Multi-Sample Annealed Importance Sampling (AIS) bounds on MI, which we demonstrate can tightly estimate large values of MI in our experiments. In settings where only a single marginal distribution is known, we propose Generalized IWAE (GIWAE) and MINE-AIS bounds. Our GIWAE bound unifies variational and contrastive bounds in a single framework that generalizes InfoNCE, IWAE, and Barber-Agakov bounds. Our MINE-AIS method improves upon existing energy-based methods such as MINE-DV and MINE-F by directly optimizing a tighter lower bound on MI. MINE-AIS uses MCMC sampling to estimate gradients for training and Multi-Sample AIS for evaluating the bound. Our methods are particularly suitable for evaluating MI in deep generative models, since explicit forms of the marginal or joint densities are often available. We evaluate our bounds on estimating the MI of VAEs and GANs trained on the MNIST and CIFAR datasets, and showcase significant gains over existing bounds in these challenging settings with high ground truth MI.

Kirill Neklyudov, Rob Brekelmans, Alexander Tong et al. · utoronto

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.

Longxuan Yu, Yunshu Wu, Yu Fu et al.

Discrete Masked diffusion language models generate text by iterative parallel decoding, but few-step decoding suffers from a tradeoff between length and quality: with a fixed step budget, standard methods can generate a short, high-quality output, or they can produce long but repetitive text. Continuous denoising can sidestep this tradeoff by evolving all positions jointly in embedding space, but building such a model from scratch at scale remains an open problem. We show that a pretrained masked DLM can instead be lightly adapted to support continuous embedding-space denoising. Starting from LLaDA-8B-Instruct, we continue-pretrain for only 1,000 steps with Discrete Stochastic Localization (DSL), replacing binary masking with continuous per-token Gaussian noise as a soft mask. The adapted model supports continuous inference that evolves all positions jointly in embedding space and defers hard token commitment to the final step. On zero-shot summarization at low step budgets (<=16 forward passes), DSL-LLaDA-SDE achieves the best ROUGE-1 on all four benchmarks and largely avoids the premature-termination / repetition tradeoff of iterative unmasking. The same adaptation also yields selective noisy-state robustness: the model corrects corrupted tokens while preserving clean ones. Control experiments using standard masked diffusion training with the same compute demonstrate neither behavior.

Rob Brekelmans, Tim Genewein, Jordi Grau-Moya et al.

Policy regularization methods such as maximum entropy regularization are widely used in reinforcement learning to improve the robustness of a learned policy. In this paper, we show how this robustness arises from hedging against worst-case perturbations of the reward function, which are chosen from a limited set by an imagined adversary. Using convex duality, we characterize this robust set of adversarial reward perturbations under KL and alpha-divergence regularization, which includes Shannon and Tsallis entropy regularization as special cases. Importantly, generalization guarantees can be given within this robust set. We provide detailed discussion of the worst-case reward perturbations, and present intuitive empirical examples to illustrate this robustness and its relationship with generalization. Finally, we discuss how our analysis complements and extends previous results on adversarial reward robustness and path consistency optimality conditions.

Yunshu Wu, Jiayi Cheng, Partha Thakuria et al.

Non-autoregressive (NAR) generation reduces decoding latency by predicting many tokens in parallel, but iterative refinement often suffers from error accumulation and distribution shift under self-generated drafts. Masked diffusion language models (MDLMs) and their remasking samplers (e.g., ReMDM) can be viewed as modern NAR iterative refinement, where generation repeatedly revises a partially observed draft. In this work we show that \emph{training alone} can substantially improve the step-efficiency of MDLM/ReMDM sampling. We propose \textsc{DSL} (Discrete Stochastic Localization), which trains a single SNR-invariant denoiser across a continuum of corruption levels, bridging intermediate draft noise and mask-style endpoint corruption within one Diffusion Transformer. On OpenWebText, \textsc{DSL} fine-tuning yields large MAUVE gains at low step budgets, surpassing the MDLM+ReMDM baseline with \(\sim\)4$\times$ fewer denoiser evaluations, and matches autoregressive quality at high budgets. Analyses show improved self-correction and uncertainty calibration, making remasking markedly more compute-efficient.

Shaorong Zhang, Longxuan Yu, Rob Brekelmans et al.

Masked Diffusion Models (MDMs) significantly accelerate inference by trading off sequential determinism. However, the theoretical mechanisms governing generation order and the risks inherent in parallelization remain under-explored. In this work, we provide a unified information-theoretic framework to decouple and analyze two fundamental sources of failure: order sensitivity and parallelization bias. Our analysis yields three key insights: (1) The benefits of Easy-First decoding (prioritizing low-entropy tokens) are magnified as model error increases; (2) factorized parallel decoding introduces intrinsic sampling errors that can lead to arbitrary large Reverse KL divergence, capturing "incoherence" failures that standard Forward KL metrics overlook; and (3) while verification can eliminate sampling error, it incurs an exponential cost governed by the total correlation within a block. Conversely, heuristics like remasking, though computationally efficient, cannot guarantee distributional correctness. Experiments on a controlled Block-HMM and large-scale MDMs (LLaDA) for arithmetic reasoning validate our theoretical framework.

Rob Brekelmans, Frank Nielsen

Markov Chain Monte Carlo methods for sampling from complex distributions and estimating normalization constants often simulate samples from a sequence of intermediate distributions along an annealing path, which bridges between a tractable initial distribution and a target density of interest. Prior works have constructed annealing paths using quasi-arithmetic means, and interpreted the resulting intermediate densities as minimizing an expected divergence to the endpoints. To analyze these variational representations of annealing paths, we extend known results showing that the arithmetic mean over arguments minimizes the expected Bregman divergence to a single representative point. In particular, we obtain an analogous result for quasi-arithmetic means, when the inputs to the Bregman divergence are transformed under a monotonic embedding function. Our analysis highlights the interplay between quasi-arithmetic means, parametric families, and divergence functionals using the rho-tau representational Bregman divergence framework, and associates common divergence functionals with intermediate densities along an annealing path.

Marta Skreta, Tara Akhound-Sadegh, Viktor Ohanesian et al.

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. Our code is available at https://github.com/martaskrt/fkc-diffusion.

Yunshu Wu, Jiayi Cheng, Longxuan Yu et al.

Continuous diffusion is a natural framework for non-autoregressive generation but has generally lagged behind masked discrete diffusion models (MDMs) on discrete sequence generation. We argue that the bottleneck is not continuity itself, but a representation in which denoising depends on timestep-indexed noise regimes. We introduce \emph{Discrete Stochastic Localization} (DSL), a continuous-state framework with unit-sphere token embeddings whose Bayes-optimal denoiser is invariant to the nominal signal-to-noise ratio (SNR) under the localization channel. One trained network then supports an entire family of per-token SNR paths, with endpoint masked-diffusion paths as a special case. Fine-tuning a pretrained MDLM checkpoint with DSL substantially improves distributional faithfulness (MAUVE) on OpenWebText across all step budgets from $T{=}128$ to $T{=}1024$, and the same checkpoint supports random-order autoregressive sampling, as well as a hybrid continuous-then-discrete sampler using as few as T=48 total steps -- without distillation or retraining.

Nima Shoghi, Yuxuan Liu, Yuning Shen et al.

Molecular dynamics (MD) simulations remain the gold standard for studying protein dynamics, but their computational cost limits access to biologically relevant timescales. Recent generative models have shown promise in accelerating simulations, yet they struggle with long-horizon generation due to architectural constraints, error accumulation, and inadequate modeling of spatio-temporal dynamics. We present STAR-MD (Spatio-Temporal Autoregressive Rollout for Molecular Dynamics), a scalable SE(3)-equivariant diffusion model that generates physically plausible protein trajectories over microsecond timescales. Our key innovation is a causal diffusion transformer with joint spatio-temporal attention that efficiently captures complex space-time dependencies while avoiding the memory bottlenecks of existing methods. On the standard ATLAS benchmark, STAR-MD achieves state-of-the-art performance across all metrics--substantially improving conformational coverage, structural validity, and dynamic fidelity compared to previous methods. STAR-MD successfully extrapolates to generate stable microsecond-scale trajectories where baseline methods fail catastrophically, maintaining high structural quality throughout the extended rollout. Our comprehensive evaluation reveals severe limitations in current models for long-horizon generation, while demonstrating that STAR-MD's joint spatio-temporal modeling enables robust dynamics simulation at biologically relevant timescales, paving the way for accelerated exploration of protein function.

Stephen Zhao, Rob Brekelmans, Alireza Makhzani et al.

Numerous capability and safety techniques of Large Language Models (LLMs), including RLHF, automated red-teaming, prompt engineering, and infilling, can be cast as sampling from an unnormalized target distribution defined by a given reward or potential function over the full sequence. In this work, we leverage the rich toolkit of Sequential Monte Carlo (SMC) for these probabilistic inference problems. In particular, we use learned twist functions to estimate the expected future value of the potential at each timestep, which enables us to focus inference-time computation on promising partial sequences. We propose a novel contrastive method for learning the twist functions, and establish connections with the rich literature of soft reinforcement learning. As a complementary application of our twisted SMC framework, we present methods for evaluating the accuracy of language model inference techniques using novel bidirectional SMC bounds on the log partition function. These bounds can be used to estimate the KL divergence between the inference and target distributions in both directions. We apply our inference evaluation techniques to show that twisted SMC is effective for sampling undesirable outputs from a pretrained model (a useful component of harmlessness training and automated red-teaming), generating reviews with varied sentiment, and performing infilling tasks.

Haibo Wang, Yuxuan Qiu, Yanze Wang et al.

Simulating transition dynamics between metastable states is a fundamental challenge in dynamical systems and stochastic processes with wide real-world applications in understanding protein folding, chemical reactions and neural activities. However, the computational challenge often lies on sampling exponentially many paths in which only a small fraction ends in the target metastable state due to existence of high energy barriers. To amortize the cost, we propose a data-driven approach to warm-up the simulation by learning nonlinear interpolations from local dynamics. Specifically, we infer a potential energy function from local dynamics data. To find plausible paths between two metastable states, we formulate a generalized flow matching framework that learns a vector field to sample propable paths between the two marginal densities under the learned energy function. Furthermore, we iteratively refine the model by assigning importance weights to the sampled paths and buffering more likely paths for training. We validate the effectiveness of the proposed method to sample probable paths on both synthetic and real-world molecular systems.

Stephen Zhao, Aidan Li, Rob Brekelmans et al.

Reinforcement learning (RL) has become a predominant technique to align language models (LMs) with human preferences or promote outputs which are deemed to be desirable by a given reward function. Standard RL approaches optimize average reward, while methods explicitly focused on reducing the probability of undesired outputs typically come at a cost to average-case performance. To improve this tradeoff, we introduce RePULSe, a new training method that augments the standard RL loss with an additional loss that uses learned proposals to guide sampling low-reward outputs, and then reduces those outputs' probability. We run experiments demonstrating that RePULSe produces a better tradeoff of expected reward versus the probability of undesired outputs and is more adversarially robust, compared to standard RL alignment approaches and alternatives.

Shaorong Zhang, Rob Brekelmans, Greg Ver Steeg

Diffusion Posterior Sampling (DPS) provides a principled Bayesian approach to inverse problems by sampling from $p(x_0 \mid y)$. However, in practice, the goal of inverse problem solving is not to cover the posterior but to recover the most accurate reconstruction, where optimization-based diffusion solvers often excel despite lacking a clear probabilistic foundation. We introduce Local MAP Sampling (LMAPS), a new inference framework that iteratively solving local MAP subproblems along the diffusion trajectory. This perspective clarifies their connection to global MAP estimation and DPS, offering a unified probabilistic interpretation for optimization-based methods. Building on this foundation, we develop practical algorithms with a probabilistically interpretable covariance approximation, a reformulated objective for stability and interpretability, and a gradient approximation for non-differentiable operators. Across a broad set of image restoration and scientific tasks, LMAPS achieves state-of-the-art performance, including $\geq 2$ dB gains on motion deblurring, JPEG restoration, and quantization, and $>1.5$ dB improvements on inverse scattering benchmarks.

Shaorong Zhang, Rob Brekelmans, Yunshu Wu et al.

Diffusion models provide a powerful way to incorporate complex prior information for solving inverse problems. However, existing methods struggle to correctly incorporate guidance from conflicting signals in the prior and measurement, and often failed to maximizing the consistency to the measurement, especially in the challenging setting of non-Gaussian or unknown noise. To address these issues, we propose Measurement-Aligned Sampling (MAS), a novel framework for linear inverse problem solving that flexibly balances prior and measurement information. MAS unifies and extends existing approaches such as DDNM, TMPD, while generalizing to handle both known Gaussian noise and unknown or non-Gaussian noise types. Extensive experiments demonstrate that MAS consistently outperforms state-of-the-art methods across a variety of tasks, while maintaining relatively low computational cost.

Grégoire Delétang, Jordi Grau-Moya, Markus Kunesch et al.

We extend temporal-difference (TD) learning in order to obtain risk-sensitive, model-free reinforcement learning algorithms. This extension can be regarded as modification of the Rescorla-Wagner rule, where the (sigmoidal) stimulus is taken to be either the event of over- or underestimating the TD target. As a result, one obtains a stochastic approximation rule for estimating the free energy from i.i.d. samples generated by a Gaussian distribution with unknown mean and variance. Since the Gaussian free energy is known to be a certainty-equivalent sensitive to the mean and the variance, the learning rule has applications in risk-sensitive decision-making.

Vaden Masrani, Rob Brekelmans, Thang Bui et al.

Many common machine learning methods involve the geometric annealing path, a sequence of intermediate densities between two distributions of interest constructed using the geometric average. While alternatives such as the moment-averaging path have demonstrated performance gains in some settings, their practical applicability remains limited by exponential family endpoint assumptions and a lack of closed form energy function. In this work, we introduce $q$-paths, a family of paths which is derived from a generalized notion of the mean, includes the geometric and arithmetic mixtures as special cases, and admits a simple closed form involving the deformed logarithm function from nonextensive thermodynamics. Following previous analysis of the geometric path, we interpret our $q$-paths as corresponding to a $q$-exponential family of distributions, and provide a variational representation of intermediate densities as minimizing a mixture of $α$-divergences to the endpoints. We show that small deviations away from the geometric path yield empirical gains for Bayesian inference using Sequential Monte Carlo and generative model evaluation using Annealed Importance Sampling.

Rob Brekelmans, Frank Nielsen, Alireza Makhzani et al.

The exponential family is well known in machine learning and statistical physics as the maximum entropy distribution subject to a set of observed constraints, while the geometric mixture path is common in MCMC methods such as annealed importance sampling. Linking these two ideas, recent work has interpreted the geometric mixture path as an exponential family of distributions to analyze the thermodynamic variational objective (TVO). We extend these likelihood ratio exponential families to include solutions to rate-distortion (RD) optimization, the information bottleneck (IB) method, and recent rate-distortion-classification approaches which combine RD and IB. This provides a common mathematical framework for understanding these methods via the conjugate duality of exponential families and hypothesis testing. Further, we collect existing results to provide a variational representation of intermediate RD or TVO distributions as a minimizing an expectation of KL divergences. This solution also corresponds to a size-power tradeoff using the likelihood ratio test and the Neyman Pearson lemma. In thermodynamic integration bounds such as the TVO, we identify the intermediate distribution whose expected sufficient statistics match the log partition function.

Rob Brekelmans, Vaden Masrani, Thang Bui et al.

Annealed importance sampling (AIS) is the gold standard for estimating partition functions or marginal likelihoods, corresponding to importance sampling over a path of distributions between a tractable base and an unnormalized target. While AIS yields an unbiased estimator for any path, existing literature has been primarily limited to the geometric mixture or moment-averaged paths associated with the exponential family and KL divergence. We explore AIS using $q$-paths, which include the geometric path as a special case and are related to the homogeneous power mean, deformed exponential family, and $α$-divergence.

Vu Nguyen, Vaden Masrani, Rob Brekelmans et al.

Achieving the full promise of the Thermodynamic Variational Objective (TVO), a recently proposed variational lower bound on the log evidence involving a one-dimensional Riemann integral approximation, requires choosing a "schedule" of sorted discretization points. This paper introduces a bespoke Gaussian process bandit optimization method for automatically choosing these points. Our approach not only automates their one-time selection, but also dynamically adapts their positions over the course of optimization, leading to improved model learning and inference. We provide theoretical guarantees that our bandit optimization converges to the regret-minimizing choice of integration points. Empirical validation of our algorithm is provided in terms of improved learning and inference in Variational Autoencoders and Sigmoid Belief Networks.

Rob Brekelmans, Vaden Masrani, Frank Wood et al.

The recently proposed Thermodynamic Variational Objective (TVO) leverages thermodynamic integration to provide a family of variational inference objectives, which both tighten and generalize the ubiquitous Evidence Lower Bound (ELBO). However, the tightness of TVO bounds was not previously known, an expensive grid search was used to choose a "schedule" of intermediate distributions, and model learning suffered with ostensibly tighter bounds. In this work, we propose an exponential family interpretation of the geometric mixture curve underlying the TVO and various path sampling methods, which allows us to characterize the gap in TVO likelihood bounds as a sum of KL divergences. We propose to choose intermediate distributions using equal spacing in the moment parameters of our exponential family, which matches grid search performance and allows the schedule to adaptively update over the course of training. Finally, we derive a doubly reparameterized gradient estimator which improves model learning and allows the TVO to benefit from more refined bounds. To further contextualize our contributions, we provide a unified framework for understanding thermodynamic integration and the TVO using Taylor series remainders.

Ayush Jaiswal, Rob Brekelmans, Daniel Moyer et al.

Supervised machine learning models often associate irrelevant nuisance factors with the prediction target, which hurts generalization. We propose a framework for training robust neural networks that induces invariance to nuisances through learning to discover and separate predictive and nuisance factors of data. We present an information theoretic formulation of our approach, from which we derive training objectives and its connections with previous methods. Empirical results on a wide array of datasets show that the proposed framework achieves state-of-the-art performance, without requiring nuisance annotations during training.

Rob Brekelmans, Daniel Moyer, Aram Galstyan et al.

Compression is at the heart of effective representation learning. However, lossy compression is typically achieved through simple parametric models like Gaussian noise to preserve analytic tractability, and the limitations this imposes on learning are largely unexplored. Further, the Gaussian prior assumptions in models such as variational autoencoders (VAEs) provide only an upper bound on the compression rate in general. We introduce a new noise channel, \emph{Echo noise}, that admits a simple, exact expression for mutual information for arbitrary input distributions. The noise is constructed in a data-driven fashion that does not require restrictive distributional assumptions. With its complex encoding mechanism and exact rate regularization, Echo leads to improved bounds on log-likelihood and dominates $β$-VAEs across the achievable range of rate-distortion trade-offs. Further, we show that Echo noise can outperform flow-based methods without the need to train additional distributional transformations.

Daniel Moyer, Shuyang Gao, Rob Brekelmans et al.

Representations of data that are invariant to changes in specified factors are useful for a wide range of problems: removing potential biases in prediction problems, controlling the effects of covariates, and disentangling meaningful factors of variation. Unfortunately, learning representations that exhibit invariance to arbitrary nuisance factors yet remain useful for other tasks is challenging. Existing approaches cast the trade-off between task performance and invariance in an adversarial way, using an iterative minimax optimization. We show that adversarial training is unnecessary and sometimes counter-productive; we instead cast invariant representation learning as a single information-theoretic objective that can be directly optimized. We demonstrate that this approach matches or exceeds performance of state-of-the-art adversarial approaches for learning fair representations and for generative modeling with controllable transformations.

Shuyang Gao, Rob Brekelmans, Greg Ver Steeg et al.

Advances in unsupervised learning enable reconstruction and generation of samples from complex distributions, but this success is marred by the inscrutability of the representations learned. We propose an information-theoretic approach to characterizing disentanglement and dependence in representation learning using multivariate mutual information, also called total correlation. The principle of total Cor-relation Ex-planation (CorEx) has motivated successful unsupervised learning applications across a variety of domains, but under some restrictive assumptions. Here we relax those restrictions by introducing a flexible variational lower bound to CorEx. Surprisingly, we find that this lower bound is equivalent to the one in variational autoencoders (VAE) under certain conditions. This information-theoretic view of VAE deepens our understanding of hierarchical VAE and motivates a new algorithm, AnchorVAE, that makes latent codes more interpretable through information maximization and enables generation of richer and more realistic samples.

Greg Ver Steeg, Rob Brekelmans, Hrayr Harutyunyan et al.

Scientists often seek simplified representations of complex systems to facilitate prediction and understanding. If the factors comprising a representation allow us to make accurate predictions about our system, but obscuring any subset of the factors destroys our ability to make predictions, we say that the representation exhibits informational synergy. We argue that synergy is an undesirable feature in learned representations and that explicitly minimizing synergy can help disentangle the true factors of variation underlying data. We explore different ways of quantifying synergy, deriving new closed-form expressions in some cases, and then show how to modify learning to produce representations that are minimally synergistic. We introduce a benchmark task to disentangle separate characters from images of words. We demonstrate that Minimally Synergistic (MinSyn) representations correctly disentangle characters while methods relying on statistical independence fail.