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Christopher Beckham

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9 papers

2 author rows

JMLR Journal 2025 Journal Article

Score-Based Diffusion Models in Function Space

Jae Hyun Lim
Nikola B. Kovachki
Ricardo Baptista
Christopher Beckham
Kamyar Azizzadenesheli
Jean Kossaifi
Vikram Voleti
Jiaming Song

Diffusion models have recently emerged as a powerful framework for generative modeling. They consist of a forward process that perturbs input data with Gaussian white noise and a reverse process that learns a score function to generate samples by denoising. Despite their tremendous success, they are mostly formulated on finite-dimensional spaces, e.g., Euclidean, limiting their applications to many domains where the data has a functional form, such as in scientific computing and 3D geometric data analysis. This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. In DDOs, the forward process perturbs input functions gradually using a Gaussian process. The generative process is formulated by a function-valued annealed Langevin dynamic. Our approach requires an appropriate notion of the score for the perturbed data distribution, which we obtain by generalizing denoising score matching to function spaces that can be infinite-dimensional. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution. We theoretically and numerically verify the applicability of our approach on a set of function-valued problems, including generating solutions to the Navier-Stokes equation viewed as the push-forward distribution of forcings from a Gaussian Random Field (GRF), as well as volcano InSAR and MNIST-SDF. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2025. ( edit, beta )

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TMLR Journal 2024 Journal Article

Exploring validation metrics for offline model-based optimisation with diffusion models

Christopher Beckham
Alexandre Piché
David Vazquez
Christopher Pal

In model-based optimisation (MBO) we are interested in using machine learning to design candidates that maximise some measure of reward with respect to a black box function called the (ground truth) oracle, which is expensive to compute since it involves executing a real world process. In offline MBO we wish to do so without assuming access to such an oracle during training or validation, with makes evaluation non-straightforward. While an approximation to the ground oracle can be trained and used in place of it during model validation to measure the mean reward over generated candidates, the evaluation is approximate and vulnerable to adversarial examples. Measuring the mean reward of generated candidates over this approximation is one such `validation metric', whereas we are interested in a more fundamental question which is finding which validation metrics correlate the most with the ground truth. This involves proposing validation metrics and quantifying them over many datasets for which the ground truth is known, for instance simulated environments. This is encapsulated under our proposed evaluation framework which is also designed to measure extrapolation, which is the ultimate goal behind leveraging generative models for MBO. While our evaluation framework is model agnostic we specifically evaluate denoising diffusion models due to their state-of-the-art performance, as well as derive interesting insights such as ranking the most effective validation metrics as well as discussing important hyperparameters.

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TMLR Journal 2024 Journal Article

Robust Guided Diffusion for Offline Black-Box Optimization

Can Chen
Christopher Beckham
Zixuan Liu
Xue Liu
Christopher Pal

Offline black-box optimization aims to maximize a black-box function using an offline dataset of designs and their measured properties. Two main approaches have emerged: the forward approach, which learns a mapping from input to its value, thereby acting as a proxy to guide optimization, and the inverse approach, which learns a mapping from value to input for conditional generation. (a) Although proxy-free~(classifier-free) diffusion shows promise in robustly modeling the inverse mapping, it lacks explicit guidance from proxies, essential for generating high-performance samples beyond the training distribution. Therefore, we propose \textit{proxy-enhanced sampling} which utilizes the explicit guidance from a trained proxy to bolster proxy-free diffusion with enhanced sampling control. (b) Yet, the trained proxy is susceptible to out-of-distribution issues. To address this, we devise the module \textit{diffusion-based proxy refinement}, which seamlessly integrates insights from proxy-free diffusion back into the proxy for refinement. To sum up, we propose \textit{\textbf{R}obust \textbf{G}uided \textbf{D}iffusion for Offline Black-box Optimization}~(\textbf{RGD}), combining the advantages of proxy~(explicit guidance) and proxy-free diffusion~(robustness) for effective conditional generation. RGD achieves state-of-the-art results on various design-bench tasks, underscoring its efficacy. Our code is \href{https://github.com/GGchen1997/RGD}{here}.

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NeurIPS Conference 2023 Conference Paper

Parallel-mentoring for Offline Model-based Optimization

Can (Sam) Chen
Christopher Beckham
Zixuan Liu
Xue (Steve) Liu
Chris Pal

We study offline model-based optimization to maximize a black-box objective function with a static dataset of designs and scores. These designs encompass a variety of domains, including materials, robots, DNA sequences, and proteins. A common approach trains a proxy on the static dataset and performs gradient ascent to obtain new designs. However, this often results in poor designs due to the proxy inaccuracies for out-of-distribution designs. Recent studies indicate that (a) gradient ascent with a mean ensemble of proxies generally outperforms simple gradient ascent, and (b) a trained proxy provides weak ranking supervision signals for design selection. Motivated by (a) and (b), we propose $\textit{parallel-mentoring}$ as an effective and novel method that facilitates mentoring among proxies, creating a more robust ensemble to mitigate the out-of-distribution issue. We focus on the three-proxy case in the main paper and our method consists of two modules. The first module, $\textit{voting-based pairwise supervision}$, operates on three parallel proxies and captures their ranking supervision signals as pairwise comparison labels. These labels are combined through majority voting to generate consensus labels, which incorporates ranking supervision signals from all proxies and enables mutual mentoring. Yet, label noise arises due to possible incorrect consensus. To alleviate this, we introduce an $\textit{adaptive soft-labeling}$ module with soft-labels initialized as consensus labels. Based on bi-level optimization, this module fine-tunes proxies in the inner level and learns more accurate labels in the outer level to adaptively mentor proxies, resulting in a more robust ensemble. Experiments validate the effectiveness of our method. Our code is available here.

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ICML Conference 2019 Conference Paper

Manifold Mixup: Better Representations by Interpolating Hidden States

Vikas Verma
Alex Lamb
Christopher Beckham
Amir Najafi 0002
Ioannis Mitliagkas
David Lopez-Paz
Yoshua Bengio

Deep neural networks excel at learning the training data, but often provide incorrect and confident predictions when evaluated on slightly different test examples. This includes distribution shifts, outliers, and adversarial examples. To address these issues, we propose \manifoldmixup{}, a simple regularizer that encourages neural networks to predict less confidently on interpolations of hidden representations. \manifoldmixup{} leverages semantic interpolations as additional training signal, obtaining neural networks with smoother decision boundaries at multiple levels of representation. As a result, neural networks trained with \manifoldmixup{} learn flatter class-representations, that is, with fewer directions of variance. We prove theory on why this flattening happens under ideal conditions, validate it empirically on practical situations, and connect it to the previous works on information theory and generalization. In spite of incurring no significant computation and being implemented in a few lines of code, \manifoldmixup{} improves strong baselines in supervised learning, robustness to single-step adversarial attacks, and test log-likelihood.

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NeurIPS Conference 2019 Conference Paper

On Adversarial Mixup Resynthesis

Christopher Beckham
Sina Honari
Vikas Verma
Alex Lamb
Farnoosh Ghadiri
R Devon Hjelm
Yoshua Bengio
Chris Pal

In this paper, we explore new approaches to combining information encoded within the learned representations of auto-encoders. We explore models that are capable of combining the attributes of multiple inputs such that a resynthesised output is trained to fool an adversarial discriminator for real versus synthesised data. Furthermore, we explore the use of such an architecture in the context of semi-supervised learning, where we learn a mixing function whose objective is to produce interpolations of hidden states, or masked combinations of latent representations that are consistent with a conditioned class label. We show quantitative and qualitative evidence that such a formulation is an interesting avenue of research.

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NeurIPS Conference 2018 Conference Paper

Unsupervised Depth Estimation, 3D Face Rotation and Replacement

Joel Ruben Antony Moniz
Christopher Beckham
Simon Rajotte
Sina Honari
Chris Pal

We present an unsupervised approach for learning to estimate three dimensional (3D) facial structure from a single image while also predicting 3D viewpoint transformations that match a desired pose and facial geometry. We achieve this by inferring the depth of facial keypoints of an input image in an unsupervised manner, without using any form of ground-truth depth information. We show how it is possible to use these depths as intermediate computations within a new backpropable loss to predict the parameters of a 3D affine transformation matrix that maps inferred 3D keypoints of an input face to the corresponding 2D keypoints on a desired target facial geometry or pose. Our resulting approach, called DepthNets, can therefore be used to infer plausible 3D transformations from one face pose to another, allowing faces to be frontalized, transformed into 3D models or even warped to another pose and facial geometry. Lastly, we identify certain shortcomings with our formulation, and explore adversarial image translation techniques as a post-processing step to re-synthesize complete head shots for faces re-targeted to different poses or identities.

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NeurIPS Conference 2017 Conference Paper

ExtremeWeather: A large-scale climate dataset for semi-supervised detection, localization, and understanding of extreme weather events

Evan Racah
Christopher Beckham
Tegan Maharaj
Samira Ebrahimi Kahou
Mr. Prabhat
Chris Pal

Then detection and identification of extreme weather events in large-scale climate simulations is an important problem for risk management, informing governmental policy decisions and advancing our basic understanding of the climate system. Recent work has shown that fully supervised convolutional neural networks (CNNs) can yield acceptable accuracy for classifying well-known types of extreme weather events when large amounts of labeled data are available. However, many different types of spatially localized climate patterns are of interest including hurricanes, extra-tropical cyclones, weather fronts, and blocking events among others. Existing labeled data for these patterns can be incomplete in various ways, such as covering only certain years or geographic areas and having false negatives. This type of climate data therefore poses a number of interesting machine learning challenges. We present a multichannel spatiotemporal CNN architecture for semi-supervised bounding box prediction and exploratory data analysis. We demonstrate that our approach is able to leverage temporal information and unlabeled data to improve the localization of extreme weather events. Further, we explore the representations learned by our model in order to better understand this important data. We present a dataset, ExtremeWeather, to encourage machine learning research in this area and to help facilitate further work in understanding and mitigating the effects of climate change. The dataset is available at extremeweatherdataset. github. io and the code is available at https: //github. com/eracah/hur-detect.

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ICML Conference 2017 Conference Paper

Unimodal Probability Distributions for Deep Ordinal Classification

Christopher Beckham
Christopher Pal

Probability distributions produced by the cross-entropy loss for ordinal classification problems can possess undesired properties. We propose a straightforward technique to constrain discrete ordinal probability distributions to be unimodal via the use of the Poisson and binomial probability distributions. We evaluate this approach in the context of deep learning on two large ordinal image datasets, obtaining promising results.

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