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.

Neural operators, such as Fourier Neural Operators (FNO), form a principled approach for learning solution operators for partial differential equations (PDE) and other mappings between function spaces. However, many real-world problems require high-resolution training data, and the training time and limited GPU memory pose big barriers. One solution is to train neural operators in mixed precision to reduce the memory requirement and increase training speed. However, existing mixed-precision training techniques are designed for standard neural networks, and we find that their direct application to FNO leads to numerical overflow and poor memory efficiency. Further, at first glance, it may appear that mixed precision in FNO will lead to drastic accuracy degradation since reducing the precision of the Fourier transform yields poor results in classical numerical solvers. We show that this is not the case; in fact, we prove that reducing the precision in FNO still guarantees a good approximation bound, when done in a targeted manner. Specifically, we build on the intuition that neural operator learning inherently induces an approximation error, arising from discretizing the infinite-dimensional ground-truth input function, implying that training in full precision is not needed. We formalize this intuition by rigorously characterizing the approximation and precision errors of FNO and bounding these errors for general input functions. We prove that the precision error is asymptotically comparable to the approximation error. Based on this, we design a simple method to optimize the memory-intensive half-precision tensor contractions by greedily finding the optimal contraction order. Through extensive experiments on different state-of-the-art neural operators, datasets, and GPUs, we demonstrate that our approach reduces GPU memory usage by up to 50% and improves throughput by 58% with little or no reduction in accuracy.

Modeling the complex three-dimensional (3D) dynamics of relational systems is an important problem in the natural sciences, with applications ranging from molecular simulations to particle mechanics. Machine learning methods have achieved good success by learning graph neural networks to model spatial interactions. However, these approaches do not faithfully capture temporal correlations since they only model next-step predictions. In this work, we propose Equivariant Graph Neural Operator (EGNO), a novel and principled method that directly models dynamics as trajectories instead of just next-step prediction. Different from existing methods, EGNO explicitly learns the temporal evolution of 3D dynamics where we formulate the dynamics as a function over time and learn neural operators to approximate it. To capture the temporal correlations while keeping the intrinsic SE(3)-equivariance, we develop equivariant temporal convolutions parameterized in the Fourier space and build EGNO by stacking the Fourier layers over equivariant networks. EGNO is the first operator learning framework that is capable of modeling solution dynamics functions over time while retaining 3D equivariance. Comprehensive experiments in multiple domains, including particle simulations, human motion capture, and molecular dynamics, demonstrate the significantly superior performance of EGNO against existing methods, thanks to the equivariant temporal modeling. Our code is available at https://github.com/MinkaiXu/egno.

Existing neural operator architectures face challenges when solving multiphysics problems with coupled partial differential equations (PDEs) due to complex geometries, interactions between physical variables, and the limited amounts of high-resolution training data. To address these issues, we propose *Codomain Attention Neural Operator* (CoDA-NO), which tokenizes functions along the codomain or channel space, enabling self-supervised learning or pretraining of multiple PDE systems. Specifically, we extend positional encoding, self-attention, and normalization layers to function spaces. CoDA-NO can learn representations of different PDE systems with a single model. We evaluate CoDA-NO's potential as a backbone for learning multiphysics PDEs over multiple systems by considering few-shot learning settings. On complex downstream tasks with limited data, such as fluid flow simulations, fluid-structure interactions, and Rayleigh-Bénard convection, we found CoDA-NO to outperform existing methods by over 36%.

We propose the geometry-informed neural operator (GINO), a highly efficient approach for learning the solution operator of large-scale partial differential equations with varying geometries. GINO uses a signed distance function (SDF) representation of the input shape and neural operators based on graph and Fourier architectures to learn the solution operator. The graph neural operator handles irregular grids and transforms them into and from regular latent grids on which Fourier neural operator can be efficiently applied. We provide an efficient implementation of GINO using an optimized hashing approach, which allows efficient learning in a shared, compressed latent space with reduced computation and memory costs. GINO is discretization-invariant, meaning the trained model can be applied to arbitrary discretizations of the continuous domain and applies to any shape or resolution. To empirically validate the performance of our method on large-scale simulation, we generate the industry-standard aerodynamics dataset of 3D vehicle geometries with Reynolds numbers as high as five million. For this large-scale 3D fluid simulation, numerical methods are expensive to compute surface pressure. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points. The cost-accuracy experiments show a 26,000x speed-up compared to optimized GPU-based computational fluid dynamics (CFD) simulators on computing the drag coefficient. When tested on new combinations of geometries and boundary conditions (inlet velocities), GINO obtains a one-fourth reduction in error rate compared to deep neural network approaches.

Reinforcement Learning in large action spaces is a challenging problem. This is especially true for cooperative multi-agent reinforcement learning (MARL), which often requires tractable learning while respecting various constraints like communication budget and information about other agents. In this work, we focus on the fundamental hurdle affecting both value-based and policy-gradient approaches: an exponential blowup of the action space with the number of agents. For value-based methods, it poses challenges in accurately representing the optimal value function for value-based methods, thus inducing suboptimality. For policy gradient methods, it renders the critic ineffective and exacerbates the problem of the lagging critic. We show that from a learning theory perspective, both problems can be addressed by accurately representing the associated action-value function with a low-complexity hypothesis class. This requires accurately modelling the agent interactions in a sample efficient way. To this end, we propose a novel tensorised formulation of the Bellman equation. This gives rise to our method Tesseract, which utilises the view of Q-function seen as a tensor where the modes correspond to action spaces of different agents. Algorithms derived from Tesseract decompose the Q-tensor across the agents and utilise low-rank tensor approximations to model the agent interactions relevant to the task. We provide PAC analysis for Tesseract based algorithms and highlight their relevance to the class of rich observation MDPs. Empirical results in different domains confirm the gains in sample efficiency using Tesseract as supported by the theory.

Recent generative adversarial networks (GANs) are able to generate impressive photo-realistic images. However, controllable generation with GANs remains an open research problem. Achieving controllable generation requires semantically interpretable and disentangled factors of variation. It is challenging to achieve this goal using simple fixed distributions such as Gaussian distribution. Instead, we propose an unsupervised framework to learn a distribution of latent codes that control the generator through self-training. Self-training provides an iterative feedback in the GAN training, from the discriminator to the generator, and progressively improves the proposal of the latent codes as training proceeds. The latent codes are sampled from a latent variable model that is learned in the feature space of the discriminator. We consider a normalized independent component analysis model and learn its parameters through tensor factorization of the higher-order moments. Our framework exhibits better disentanglement compared to other variants such as the variational autoencoder, and is able to discover semantically meaningful latent codes without any supervision. We empirically demonstrate on both cars and faces datasets that each group of elements in the learned code controls a mode of variation with a semantic meaning, e.g. pose or background change. We also demonstrate with quantitative metrics that our method generates better results compared to other approaches.

Data augmentation is a simple yet effective way to improve the robustness of deep neural networks (DNNs). Diversity and hardness are two complementary dimensions of data augmentation to achieve robustness. For example, AugMix explores random compositions of a diverse set of augmentations to enhance broader coverage, while adversarial training generates adversarially hard samples to spot the weakness. Motivated by this, we propose a data augmentation framework, termed AugMax, to unify the two aspects of diversity and hardness. AugMax first randomly samples multiple augmentation operators and then learns an adversarial mixture of the selected operators. Being a stronger form of data augmentation, AugMax leads to a significantly augmented input distribution which makes model training more challenging. To solve this problem, we further design a disentangled normalization module, termed DuBIN (Dual-Batch-and-Instance Normalization), that disentangles the instance-wise feature heterogeneity arising from AugMax. Experiments show that AugMax-DuBIN leads to significantly improved out-of-distribution robustness, outperforming prior arts by 3.03%, 3.49%, 1.82% and 0.71% on CIFAR10-C, CIFAR100-C, Tiny ImageNet-C and ImageNet-C. Codes and pretrained models are available: https://github.com/VITA-Group/AugMax.

Natural human-computer interaction and audio-visual human behaviour sensing systems, which would achieve robust performance in-the-wild are more needed than ever as digital devices are increasingly becoming an indispensable part of our life. Accurately annotated real-world data are the crux in devising such systems. However, existing databases usually consider controlled settings, low demographic variability, and a single task. In this paper, we introduce the SEWA database of more than 2,000 minutes of audio-visual data of 398 people coming from six cultures, 50 percent female, and uniformly spanning the age range of 18 to 65 years old. Subjects were recorded in two different contexts: while watching adverts and while discussing adverts in a video chat. The database includes rich annotations of the recordings in terms of facial landmarks, facial action units (FAU), various vocalisations, mirroring, and continuously valued valence, arousal, liking, agreement, and prototypic examples of (dis)liking. This database aims to be an extremely valuable resource for researchers in affective computing and automatic human sensing and is expected to push forward the research in human behaviour analysis, including cultural studies. Along with the database, we provide extensive baseline experiments for automatic FAU detection and automatic valence, arousal, and (dis)liking intensity estimation.

Tensors, or multidimensional arrays, are data structures that can naturally represent visual data of multiple dimensions. Inherently able to efficiently capture structured, latent semantic spaces and high-order interactions, tensors have a long history of applications in a wide span of computer vision problems. With the advent of the deep learning paradigm shift in computer vision, tensors have become even more fundamental. Indeed, essential ingredients in modern deep learning architectures, such as convolutions and attention mechanisms, can readily be considered as tensor mappings. In effect, tensor methods are increasingly finding significant applications in deep learning, including the design of memory and compute efficient network architectures, improving robustness to random noise and adversarial attacks, and aiding the theoretical understanding of deep networks. This article provides an in-depth and practical review of tensors and tensor methods in the context of representation learning and deep learning, with a particular focus on visual data analysis and computer vision applications. Concretely, besides fundamental work in tensor-based visual data analysis methods, we focus on recent developments that have brought on a gradual increase in tensor methods, especially in deep learning architectures and their implications in computer vision applications. To further enable the newcomer to grasp such concepts quickly, we provide companion Python notebooks, covering key aspects of this article and implementing them, step-by-step with TensorLy.

The prominence of deep learning, large amount of annotated data and increasingly powerful hardware made it possible to reach remarkable performance for supervised classification tasks, in many cases saturating the training sets. However the resulting models are specialized to a single very specific task and domain. Adapting the learned classification to new domains is a hard problem due to at least three reasons: (1) the new domains and the tasks might be drastically different; (2) there might be very limited amount of annotated data on the new domain and (3) full training of a new model for each new task is prohibitive in terms of computation and memory, due to the sheer number of parameters of deep CNNs. In this paper, we present a method to learn new-domains and tasks incrementally, building on prior knowledge from already learned tasks and without catastrophic forgetting. We do so by jointly parametrizing weights across layers using low-rank Tucker structure. The core is task agnostic while a set of task specific factors are learnt on each new domain. We show that leveraging tensor structure enables better performance than simply using matrix operations. Joint tensor modelling also naturally leverages correlations across different layers. Compared with previous methods which have focused on adapting each layer separately, our approach results in more compact representations for each new task/domain. We apply the proposed method to the 10 datasets of the Visual Decathlon Challenge and show that our method offers on average about 7.5× reduction in number of parameters and competitive performance in terms of both classification accuracy and Decathlon score.

Training deep neural networks with spatio-temporal (i.e., 3D) or multidimensional convolutions of higher-order is computationally challenging due to millions of unknown parameters across dozens of layers. To alleviate this, one approach is to apply low-rank tensor decompositions to convolution kernels in order to compress the network and reduce its number of parameters. Alternatively, new convolutional blocks, such as MobileNet, can be directly designed for efficiency. In this paper, we unify these two approaches by proposing a tensor factorization framework for efficient multidimensional (separable) convolutions of higher-order. Interestingly, the proposed framework enables a novel higher-order transduction, allowing to train a network on a given domain (e.g., 2D images or N-dimensional data in general) and using transduction to generalize to higher-order data such as videos (or (N+K)--dimensional data in general), capturing for instance temporal dynamics while preserving the learnt spatial information. We apply the proposed methodology, coined CP-Higher-Order Convolution (HO-CPConv), to spatio-temporal facial emotion analysis. Most existing facial affect models focus on static imagery and discard all temporal information. This is due to the above-mentioned burden of training 3D convolutional nets and the lack of large bodies of video data annotated by experts. We address both issues with our proposed framework. Initial training is first done on static imagery before using transduction to generalize to the temporal domain. We demonstrate superior performance on three challenging large scale affect estimation datasets, AffectNet, SEWA, and AFEW-VA.

This paper is on highly accurate and highly efficient human pose estimation. Recent works based on Fully Convolutional Networks (FCNs) have demonstrated excellent results for this difficult problem. While residual connections within FCNs have proved to be quintessential for achieving high accuracy, we re-analyze this design choice in the context of improving both the accuracy and the efficiency over the state-of- the-art. In particular, we make the following contributions: (a) We propose gated skip connections with per-channel learnable parameters to control the data flow for each channel within the module within the macro-module. (b) We introduce a hybrid network that combines the HourGlass and U-Net architectures which minimizes the number of identity connections within the network and increases the performance for the same parameter budget. Our model achieves state-of-the-art results on the MPII and LSP datasets. In addition, with a reduction of 3× in model size and complexity, we show no decrease in performance when compared to the original HourGlass network.

Speech-driven facial animation involves using a speech signal to generate realistic videos of talking faces. Recent deep learning approaches to facial synthesis rely on extracting low-dimensional representations and concatenating them, followed by a decoding step of the concatenated vector. This accounts for only first-order interactions of the features and ignores higher-order interactions. In this paper we propose a polynomial fusion layer that models the joint representation of the encodings by a higher-order polynomial, with the parameters modelled by a tensor decomposition. We demonstrate the suitability of this approach through experiments on generated videos evaluated on a range of metrics on video quality, audiovisual synchronisation and generation of blinks.

Learning from spatio-temporal data has numerous applications such as human-behavior analysis, object tracking, video compression, and physics simulation. However, existing methods still perform poorly on challenging video tasks such as long-term forecasting. This is because these kinds of challenging tasks require learning long-term spatio-temporal correlations in the video sequence. In this paper, we propose a higher-order convolutional LSTM model that can efficiently learn these correlations, along with a succinct representations of the history. This is accomplished through a novel tensor train module that performs prediction by combining convolutional features across time. To make this feasible in terms of computation and memory requirements, we propose a novel convolutional tensor-train decomposition of the higher-order model. This decomposition reduces the model complexity by jointly approximating a sequence of convolutional kernels as a low-rank tensor-train factorization. As a result, our model outperforms existing approaches, but uses only a fraction of parameters, including the baseline models. Our results achieve state-of-the-art performance in a wide range of applications and datasets, including the multi-steps video prediction on the Moving-MNIST-2 and KTH action datasets as well as early activity recognition on the Something-Something V2 dataset.

Abstract Conditional image generation lies at the heart of computer vision and conditional generative adversarial networks (cGAN) have recently become the method of choice for this task, owing to their superior performance. The focus so far has largely been on performance improvement, with little effort in making cGANs more robust to noise. However, the regression (of the generator) might lead to arbitrarily large errors in the output, which makes cGANs unreliable for real-world applications. In this work, we introduce a novel conditional GAN model, calledRoCGAN, which leverages structure in the target space of the model to address the issue. Specifically, we augment the generator with an unsupervised pathway, which promotes the outputs of the generator to span the target manifold, even in the presence of intense noise. We prove that RoCGAN share similar theoretical properties as GAN and establish with both synthetic and real data the merits of our model. We perform a thorough experimental validation on large scale datasets for natural scenes and faces and observe that our model outperforms existing cGAN architectures by a large margin. We also empirically demonstrate the performance of our approach in the face of two types of noise (adversarial and Bernoulli).

Convolutional neural networks typically consist of many convolutional layers followed by one or more fully connected layers. While convolutional layers map between high-order activation tensors, the fully connected layers operate on flattened activation vectors. Despite empirical success, this approach has notable drawbacks. Flattening followed by fully connected layers discards multilinear structure in the activations and requires many parameters. We address these problems by incorporating tensor algebraic operations that preserve multilinear structure at every layer. First, we introduce Tensor Contraction Layers (TCLs) that reduce the dimensionality of their input while preserving their multilinear structure using tensor contraction. Next, we introduce Tensor Regression Layers (TRLs), which express outputs through a low-rank multilinear mapping from a high-order activation tensor to an output tensor of arbitrary order. We learn the contraction and regression factors end-to-end, and produce accurate nets with fewer parameters. Additionally, our layers regularize networks by imposing low-rank constraints on the activations (TCL) and regression weights (TRL). Experiments on ImageNet show that, applied to VGG and ResNet architectures, TCLs and TRLs reduce the number of parameters compared to fully connected layers by more than 65% while maintaining or increasing accuracy. In addition to the space savings, our approach's ability to leverage topological structure can be crucial for structured data such as MRI. In particular, we demonstrate significant performance improvements over comparable architectures on three tasks associated with the UK Biobank dataset.

Recent findings indicate that over-parametrization, while crucial for successfully training deep neural networks, also introduces large amounts of redundancy. Tensor methods have the potential to efficiently parametrize over-complete representations by leveraging this redundancy. In this paper, we propose to fully parametrize Convolutional Neural Networks (CNNs) with a single high-order, low-rank tensor. Previous works on network tensorization have focused on parametrizing individual layers (convolutional or fully connected) only, and perform the tensorization layer-by-layer separately. In contrast, we propose to jointly capture the full structure of a neural network by parametrizing it with a single high-order tensor, the modes of which represent each of the architectural design parameters of the network (e.g. number of convolutional blocks, depth, number of stacks, input features, etc). This parametrization allows to regularize the whole network and drastically reduce the number of parameters. Our model is end-to-end trainable and the low-rank structure imposed on the weight tensor acts as an implicit regularization. We study the case of networks with rich structure, namely Fully Convolutional Networks (FCNs), which we propose to parametrize with a single 8th-order tensor. We show that our approach can achieve superior performance with small compression rates, and attain high compression rates with negligible drop in accuracy for the challenging task of human pose estimation.

While it is relatively easy and natural for humans to detect and interpret non-verbal cues, it is a hard task for computer systems. Automatic recognition of emotions has been the subject of extensive studies in the past decade, but despite the various methods that have been implemented, the problem remains challenging. In particular, most existing works focus on predicting a set discrete stereotypical emotion categories. We are instead interested in predicting continuous values of valence and arousal, which are able to model accurately a broader range of spontaneous emotions. Moreover, as opposed to much of the prior work that focused on controlled (laboratory) conditions, we are interested in analysis in naturalist (in-the-wild) conditions. To do so, we propose to leverage the structure in the data using tensor methods. In addition to preserving the structure, these have the potential to also reduce the total number of parameters in the models, thus improving the computational performance. We first consider a model with analytic solution in the form of a Tucker Tensor Regression. We then investigate a deep, gradient based method, namely Tensor Regression Networks. We perform thorough experiments on two publicly available databases, AFEW-VA and SEWA, for facial affect estimation in-the-wild, in terms of valence and arousal levels. Experimental results demonstrate that tensor-based methods successfully leverage the structure in the data and on average outperform baseline methods.

Deep generative models have significantly advanced image generation, enabling generation of visually pleasing images with realistic texture. Apart from the texture, it is the shape geometry of objects that strongly dictates their appearance. However, currently available generative models do not incorporate geometric information into the image generation process. This often yields visual objects of degenerated quality. In this work, we propose a regularized Geometry-Aware Generative Adversarial Network (GAGAN) which disentangles appearance and shape in the latent space. This regularized GAGAN enables the generation of images with both realistic texture and shape. Specifically, we condition the generator on a statistical shape prior. The prior is enforced through mapping the generated images onto a canonical coordinate frame using a differentiable geometric transformation. In addition to incorporating geometric information, this constrains the search space and increases the model’s robustness. We show that our approach is versatile, able to generalise across domains (faces, sketches, hands and cats) and sample sizes (from as little as $$\sim \, 200{-}30{,}000$$ to more than 200, 000). We demonstrate superior performance through extensive quantitative and qualitative experiments in a variety of tasks and settings. Finally, we leverage our model to automatically and accurately detect errors or drifting in facial landmarks detection and tracking in-the-wild.

Tensors are higher-order extensions of matrices. While matrix methods form the cornerstone of traditional machine learning and data analysis, tensor methods have been gaining increasing traction. However, software support for tensor operations is not on the same footing. In order to bridge this gap, we have developed TensorLy, a Python library that provides a high-level API for tensor methods and deep tensorized neural networks. TensorLy aims to follow the same standards adopted by the main projects of the Python scientific community, and to seamlessly integrate with them. Its BSD license makes it suitable for both academic and commercial applications. TensorLy's backend system allows users to perform computations with several libraries such as NumPy or PyTorch to name but a few. They can be scaled on multiple CPU or GPU machines. In addition, using the deep-learning frameworks as backend allows to easily design and train deep tensorized neural networks. TensorLy is available at https://github.com/tensorly/tensorly

Deep generative models learned through adversarial training have become increasingly popular for their ability to generate naturalistic image textures. However, aside from their texture, the visual appearance of objects is significantly influenced by their shape geometry; information which is not taken into account by existing generative models. This paper introduces the Geometry-Aware Generative Adversarial Networks (GAGAN) for incorporating geometric information into the image generation process. Specifically, in GAGAN the generator samples latent variables from the probability space of a statistical shape model. By mapping the output of the generator to a canonical coordinate frame through a differentiable geometric transformation, we enforce the geometry of the objects and add an implicit connection from the prior to the generated object. Experimental results on face generation indicate that the GAGAN can generate realistic images of faces with arbitrary facial attributes such as facial expression, pose, and morphology, that are of better quality than current GAN-based methods. Our method can be used to augment any existing GAN architecture and improve the quality of the images generated.

Active appearance models (AAMs) are generative models of shape and appearance that have proven very attractive for their ability to handle wide changes in illumination, pose, and occlusion when trained in the wild, while not requiring large training data set like regression-based or deep learning methods. The problem of fitting an AAM is usually formulated as a non-linear least squares one and the main way of solving it is a standard Gauss-Newton algorithm. In this paper, we extend AAMs in two ways: we first extend the Gauss-Newton framework by formulating a bidirectional fitting method that deforms both the image and the template to fit a new instance. We then formulate a second order method by deriving an efficient Newton method for AAMs fitting. We derive both methods in a unified framework for two types of AAMs, holistic and part-based, and additionally show how to exploit the structure in the problem to derive fast yet exact solutions. We perform a thorough evaluation of all algorithms on three challenging and recently annotated in-the-wild data sets, and investigate fitting accuracy, convergence properties, and the influence of noise in the initialization. We compare our proposed methods to other algorithms and show that they yield state-of-the-art results, out-performing other methods while having superior convergence properties.

Finding landmarks on objects like faces is a challenging computer vision problem, especially in real life conditions (or in-the-wild) and Active Appearance Models have been widely used to solve it. State-of-the-art algorithms for fitting an AAM to a new image are based on Gauss-Newton (GN) optimization. Recently fast GN algorithms have been proposed for both forward additive and inverse compositional fitting frameworks. In this paper, we propose a fast and exact bi-directional (Fast-Bd) approach to AAM fitting by combining both approaches. Although such a method might appear to increase computational burden, we show that by capitalizing on results from optimization theory, an exact solution, as computationally efficient as the original forward or inverse formulation, can be derived. Our proposed bi-directional approach achieves state-of-the-art performance and superior convergence properties. These findings are validated on two challenging, in-the-wild data sets, LFPW and Helen, and comparison is provided to the state-of-the art methods for Active Appearance Models fitting.

Active Appearance Models (AAMs) are statistical models of shape and appearance widely used in computer vision to detect landmarks on objects like faces. Fitting an AAM to a new image can be formulated as a non-linear least-squares problem which is typically solved using iterative methods. Owing to its efficiency, Gauss-Newton optimization has been the standard choice over more sophisticated approaches like Newton. In this paper, we show that the AAM problem has structure which can be used to solve efficiently the original Newton problem without any approximations. We then make connections to the original Gauss-Newton algorithm and study experimentally the effect of the additional terms introduced by the Newton formulation on both fitting accuracy and convergence. Based on our derivations, we also propose a combined Newton and Gauss-Newton method which achieves promising fitting and convergence performance. Our findings are validated on two challenging in-the-wild data sets.