Accurate forecasting of multivariate time series data is important in many engineering and scientific applications. Recent state-of-the-art works ignore the inter-relations between variates, using their model on each variate independently. This raises several research questions related to proper modeling of multivariate data. In this work, we propose to view multivariate forecasting as a multi-task learning problem, facilitating the analysis of forecasting by considering the angle between task gradients and their balance. To do so, we analyze linear models to characterize the behavior of tasks. Our analysis suggests that tasks can be defined by grouping similar variates together, which we achieve via a simple clustering that depends on correlation-based similarities. Moreover, to balance tasks, we scale gradients with respect to their prediction error. Then, each task is solved with a linear model within our MTLinear framework. We evaluate our approach on challenging benchmarks in comparison to strong baselines, and we show it obtains on-par or better results on multivariate forecasting problems.

Deep learning models with a large number of parameters, often referred to as over-parameterized models, have achieved exceptional performance across various tasks. Despite concerns about overfitting, these models frequently generalize well to unseen data, thanks to effective regularization techniques, with data augmentation being among the most widely used. While data augmentation has shown great success in classification tasks using label-preserving transformations, its application in regression problems has received less attention. Recently, a novel manifold learning approach for generating synthetic data was proposed, utilizing a first-order approximation of the data manifold. Building on this foundation, we present a theoretical framework and practical tools for approximating and sampling general data manifolds. Furthermore, we introduce the Curvature-Enhanced Manifold Sampling (CEMS) method for regression tasks. CEMS leverages a second-order representation of the data manifold to enable efficient sampling and reconstruction of new data points. Extensive evaluations across multiple datasets and comparisons with state-of-the-art methods demonstrate that CEMS delivers superior performance in both in-distribution and out-of-distribution scenarios, while introducing only minimal computational overhead. Code is available at https://github.com/azencot-group/CEMS.

Learning disentangled representations in sequential data is a key goal in deep learning, with broad applications in vision, audio, and time series. While real-world data involves multiple interacting semantic factors over time, prior work has mostly focused on simpler two-factor static and dynamic settings, primarily because such settings make data collection easier, thereby overlooking the inherently multi-factor nature of real-world data. We introduce the first standardized benchmark for evaluating multi-factor sequential disentanglement across six diverse datasets spanning video, audio, and time series. Our benchmark includes modular tools for dataset integration, model development, and evaluation metrics tailored to multi-factor analysis. We additionally propose a post-hoc Latent Exploration Stage to automatically align latent dimensions with semantic factors, and introduce a Koopman-inspired model that achieves state-of-the-art results. Moreover, we show that Vision-Language Models can automate dataset annotation and serve as zero-shot disentanglement evaluators, removing the need for manual labels and human intervention. Together, these contributions provide a robust and scalable foundation for advancing multi-factor sequential disentanglement. Our code is available on GitHub, and the datasets and trained models are available on Hugging Face.

Recent advances in generative modeling have positioned diffusion models as state-of-the-art tools for sampling from complex data distributions. While these models have shown remarkable success across single-modality domains such as images and audio, extending their capabilities to *Modality Translation (MT)*, translating information across different sensory modalities, remains an open challenge. Existing approaches often rely on restrictive assumptions, including shared dimensionality, Gaussian source priors, and modality-specific architectures, which limit their generality and theoretical grounding. In this work, we propose the Latent Denoising Diffusion Bridge Model (LDDBM), a general-purpose framework for modality translation based on a latent-variable extension of Denoising Diffusion Bridge Models. By operating in a shared latent space, our method learns a bridge between arbitrary modalities without requiring aligned dimensions. We introduce a contrastive alignment loss to enforce semantic consistency between paired samples and design a domain-agnostic encoder-decoder architecture tailored for noise prediction in latent space. Additionally, we propose a predictive loss to guide training toward accurate cross-domain translation and explore several training strategies to improve stability. Our approach supports arbitrary modality pairs and performs strongly on diverse MT tasks, including multi-view to 3D shape generation, image super-resolution, and multi-view scene synthesis. Comprehensive experiments and ablations validate the effectiveness of our framework, establishing a new strong baseline in general modality translation. For more information, see our project page: https://sites.google.com/view/lddbm/home.

Diffusion-based generative models have demonstrated exceptional performance, yet their iterative sampling procedures remain computationally expensive. A prominent strategy to mitigate this cost is *distillation*, with *offline distillation* offering particular advantages in terms of efficiency, modularity, and flexibility. In this work, we identify two key observations that motivate a principled distillation framework: (1) while diffusion models have been viewed through the lens of dynamical systems theory, powerful and underexplored tools can be further leveraged; and (2) diffusion models inherently impose structured, semantically coherent trajectories in latent space. Building on these observations, we introduce the *Koopman Distillation Model* (KDM), a novel offline distillation approach grounded in Koopman theory - a classical framework for representing nonlinear dynamics linearly in a transformed space. KDM encodes noisy inputs into an embedded space where a learned linear operator propagates them forward, followed by a decoder that reconstructs clean samples. This enables single-step generation while preserving semantic fidelity. We provide theoretical justification for our approach: (1) under mild assumptions, the learned diffusion dynamics admit a finite-dimensional Koopman representation; and (2) proximity in the Koopman latent space correlates with semantic similarity in the generated outputs, allowing for effective trajectory alignment. Empirically, KDM achieves state-of-the-art performance across standard *offline distillation* benchmarks - improving FID scores by up to 40% in a single generation step.

Generating realistic time series data is critical for applications in healthcare, finance, and climate science. However, irregular sampling and missing values present significant challenges. While prior methods address these irregularities, they often yield suboptimal results and incur high computational costs. Recent advances in regular time series generation, such as the diffusion-based ImagenTime model, demonstrate strong, fast, and scalable generative capabilities by transforming time series into image representations, making them a promising solution. However, extending ImagenTime to irregular sequences using simple masking introduces ``unnatural'' neighborhoods, where missing values replaced by zeros disrupt the learning process. To overcome this, we propose a novel two-step framework: first, a Time Series Transformer completes irregular sequences, creating natural neighborhoods; second, a vision-based diffusion model with masking minimizes dependence on the completed values. This hybrid approach leverages the strengths of both completion and masking, enabling robust and efficient generation of realistic time series. Our method achieves state-of-the-art performance across benchmarks, delivering a relative improvement in discriminative score by 70% and in computational cost by 85%.

Generative modeling of time series is a central challenge in time series analysis, particularly under data-scarce conditions. Despite recent advances in generative modeling, a comprehensive understanding of how state-of-the-art generative models perform under limited supervision remains lacking. In this work, we conduct the first large-scale study evaluating leading generative models in data-scarce settings, revealing a substantial performance gap between full-data and data-scarce regimes. To close this gap, we propose a unified diffusion-based generative framework that can synthesize high-fidelity time series across diverse domains using just a few examples. Our model is pretrained on a large, heterogeneous collection of time series datasets, enabling it to learn generalizable temporal representations. It further incorporates architectural innovations such as dynamic convolutional layers for flexible channel adaptation and dataset token conditioning for domain-aware generation. Without requiring abundant supervision, our unified model achieves state-of-the-art performance in few-shot settings—outperforming domain-specific baselines across a wide range of subset sizes. Remarkably, it also surpasses all baselines even when tested on full datasets benchmarks, highlighting the strength of pretraining and cross-domain generalization. We hope this work encourages the community to revisit few-shot generative modeling as a key problem in time series research and pursue unified solutions that scale efficiently across domains. Code is available at https://github.com/azencot-group/ImagenFew.

Generating realistic time series data is important for many engineering and scientific applications. Existing work tackles this problem using generative adversarial networks (GANs). However, GANs are unstable during training, and they can suffer from mode collapse. While variational autoencoders (VAEs) are known to be more robust to the these issues, they are (surprisingly) less considered for time series generation. In this work, we introduce Koopman VAE (KoVAE), a new generative framework that is based on a novel design for the model prior, and that can be optimized for either regular and irregular training data. Inspired by Koopman theory, we represent the latent conditional prior dynamics using a linear map. Our approach enhances generative modeling with two desired features: (i) incorporating domain knowledge can be achieved by leveraging spectral tools that prescribe constraints on the eigenvalues of the linear map; and (ii) studying the qualitative behavior and stability of the system can be performed using tools from dynamical systems theory. Our results show that KoVAE outperforms state-of-the-art GAN and VAE methods across several challenging synthetic and real-world time series generation benchmarks. Whether trained on regular or irregular data, KoVAE generates time series that improve both discriminative and predictive metrics. We also present visual evidence suggesting that KoVAE learns probability density functions that better approximate the empirical ground truth distribution.

One of the fundamental representation learning tasks is unsupervised sequential disentanglement, where latent codes of inputs are decomposed to a single static factor and a sequence of dynamic factors. To extract this latent information, existing methods condition the static and dynamic codes on the entire input sequence. Unfortunately, these models often suffer from information leakage, i.e., the dynamic vectors encode both static and dynamic information, or vice versa, leading to a non-disentangled representation. Attempts to alleviate this problem via reducing the dynamic dimension and auxiliary loss terms gain only partial success. Instead, we propose a novel and simple architecture that mitigates information leakage by offering a simple and effective subtraction inductive bias while conditioning on a single sample. Remarkably, the resulting variational framework is simpler in terms of required loss terms, hyper-parameters, and data augmentation. We evaluate our method on multiple data-modality benchmarks including general time series, video, and audio, and we show beyond state-of-the-art results on generation and prediction tasks in comparison to several strong baselines.

Data augmentation (DA) methods tailored to specific domains generate synthetic samples by applying transformations that are appropriate for the characteristics of the underlying data domain, such as rotations on images and time warping on time series data. In contrast, domain-independent approaches, e.g. mixup, are applicable to various data modalities, and as such they are general and versatile. While regularizing classification tasks via DA is a well-explored research topic, the effect of DA on regression problems received less attention. To bridge this gap, we study the problem of domain-independent augmentation for regression, and we introduce FOMA: a new data-driven domain-independent data augmentation method. Essentially, our approach samples new examples from the tangent planes of the train distribution. Augmenting data in this way aligns with the network tendency towards capturing the dominant features of its input signals. We evaluate FOMA on in-distribution generalization and out-of-distribution robustness benchmarks, and we show that it improves the generalization of several neural architectures. We also find that strong baselines based on mixup are less effective in comparison to our approach. Our code is publicly available at https://github.com/azencot-group/FOMA

Lately, there has been a surge in interest surrounding generative modeling of time series data. Most existing approaches are designed either to process short sequences or to handle long-range sequences. This dichotomy can be attributed to gradient issues with recurrent networks, computational costs associated with transformers, and limited expressiveness of state space models. Towards a unified generative model for varying-length time series, we propose in this work to transform sequences into images. By employing invertible transforms such as the delay embedding and the short-time Fourier transform, we unlock three main advantages: i) We can exploit advanced diffusion vision models; ii) We can remarkably process short- and long-range inputs within the same framework; and iii) We can harness recent and established tools proposed in the time series to image literature. We validate the effectiveness of our method through a comprehensive evaluation across multiple tasks, including unconditional generation, interpolation, and extrapolation. We show that our approach achieves consistently state-of-the-art results against strong baselines. In the unconditional generation tasks, we show remarkable mean improvements of $58.17$% over previous diffusion models in the short discriminative score and $132.61$% in the (ultra-)long classification scores. Code is at https://github.com/azencot-group/ImagenTime.

Analyzing the inner mechanisms of deep neural networks is a fundamental task in machine learning. Existing work provides limited analysis or it depends on local theories, such as fixed-point analysis. In contrast, we propose to analyze trained neural networks using an operator theoretic approach which is rooted in Koopman theory, the Koopman Analysis of Neural Networks (KANN). Key to our method is the Koopman operator, which is a linear object that globally represents the dominant behavior of the network dynamics. The linearity of the Koopman operator facilitates analysis via its eigenvectors and eigenvalues. Our method reveals that the latter eigendecomposition holds semantic information related to the neural network inner workings. For instance, the eigenvectors highlight positive and negative n-grams in the sentiments analysis task; similarly, the eigenvectors capture the salient features of healthy heart beat signals in the ECG classification problem.

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation from being flat in its principal directions. We find that state-of-the-art trained convolutional neural networks for image classification have a characteristic curvature profile along layers: an initial steep increase, followed by a long phase of a plateau, and followed by another increase. In contrast, this behavior does not appear in untrained networks in which the curvature flattens. We also show that the curvature gap between the last two layers has a strong correlation with the generalization capability of the network. Moreover, we find that the intrinsic dimension of latent codes is not necessarily indicative of curvature. Finally, we observe that common regularization methods such as mixup yield flatter representations when compared to other methods. Our experiments show consistent results over a variety of deep learning architectures and multiple data sets.

Unsupervised disentanglement is a long-standing challenge in representation learning. Recently, self-supervised techniques achieved impressive results in the sequential setting, where data is time-dependent. However, the latter methods employ modality-based data augmentations and random sampling or solve auxiliary tasks. In this work, we propose to avoid that by generating, sampling, and comparing empirical distributions from the underlying variational model. Unlike existing work, we introduce a self-supervised sequential disentanglement framework based on contrastive estimation with no external signals, while using common batch sizes and samples from the latent space itself. In practice, we propose a unified, efficient, and easy-to-code sampling strategy for semantically similar and dissimilar views of the data. We evaluate our approach on video, audio, and time series benchmarks. Our method presents state-of-the-art results in comparison to existing techniques. The code is available at https://github.com/azencot-group/SPYL.

Abstract We propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace–Beltrami operator and thus favor low‐frequency eigenfunctions, we aim for a basis that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem allowing to better exploit non‐smooth descriptors. To effectively search in this high‐dimensional space of solutions, we incorporate into our minimization state‐of‐the‐art regularizers. We solve the resulting highly non‐linear and non‐convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester‐type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.

Finding latent structures in data is drawing increasing attention in diverse fields such as image and signal processing, fluid dynamics, and machine learning. In this work we examine the problem of finding the main modes of gradient flows. Gradient descent is a fundamental process in optimization where its stochastic version is prominent in training of neural networks. Here our aim is to establish a consistent theory for gradient flows $\boldsymbol{\psi}_t = P(\boldsymbol{\psi})$, where $P$ is a nonlinear homogeneous operator. Our proposed framework stems from analytic solutions of homogeneous flows, previously formalized by Cohen and Gilboa, where the initial condition $\boldmath{\psi}_0$ admits the nonlinear eigenvalue problem $P(\boldsymbol{\psi}_0)=\lambda \boldsymbol{\psi}_0 $. We first present an analytic solution for dynamic mode decomposition (DMD) in such cases. We show an inherent flaw of DMD, which is unable to recover the essential dynamics of the flow. It is evident that DMD is best suited for homogeneous flows of degree one. We propose an adaptive time sampling scheme and show its dynamics are analogue to homogeneous flows of degree one with a fixed step size. Moreover, we adapt DMD to yield a real spectrum, using symmetric matrices. Our analytic solution of the proposed scheme recovers the dynamics perfectly and yields zero error. We then proceed to show the relation between the orthogonal modes $\{\phi_i\}$ and their decay profiles under the gradient flow. We formulate orthogonal nonlinear spectral decomposition (OrthoNS), which recovers the essential latent structures of the gradient descent process. Definitions for spectrum and filtering are given, and a Parseval-type identity is shown. Experimental results on images show the resemblance to direct computations of nonlinear spectral decomposition. A significant speedup (by about two orders of magnitude) is achieved for this application using the proposed method.

Recurrent neural networks are widely used on time series data, yet such models often ignore the underlying physical structures in such sequences. A new class of physics-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems. In this work, we propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics. Key to our approach is a new analysis which explores the interplay between consistent dynamics and their associated Koopman operators. Our network is directly related to the derived analysis, and its computational requirements are comparable to other baselines. We evaluate our method on a wide range of high-dimensional and short-term dependent problems, and it achieves accurate estimates for significant prediction horizons, while also being robust to noise.

Abstract We propose a new method for computing accurate point‐to‐point mappings between a pair of triangle meshes given imperfect initial correspondences. Unlike the majority of existing techniques, we optimize for a map while leveraging information from the inverse map, yielding results which are highly consistent with respect to composition of mappings. Remarkably, our method considers only a linear number of candidate points on the target shape, allowing us to work directly with high resolution meshes, and to avoid a delicate and possibly error‐prone up‐sampling procedure. Key to this dimensionality reduction is a novel candidate selection process, where the mapped points drift over the target shape, finalizing their location based on intrinsic distortion measures. Overall, we arrive at an iterative scheme where at each step we optimize for the map and its inverse by solving two relaxed Quadratic Assignment Problems using off‐the‐shelf optimization tools. We provide quantitative and qualitative comparison of our method with several existing techniques, and show that it provides a powerful matching tool when accurate and consistent correspondences are required.

Abstract We propose a novel approach for shape matching between triangular meshes that, in contrast to existing methods, can match crease features. Our approach is based on a hybrid optimization scheme, that solves simultaneously for an elastic deformation of the source and its projection on the target. The elastic energy we minimize is invariant to rigid body motions, and its non‐linear membrane energy component favors locally injective maps. Symmetrizing this model enables feature aligned correspondences even for non‐isometric meshes. We demonstrate the advantage of our approach over state of the art methods on isometric and non‐isometric datasets, where we improve the geodesic distance from the ground truth, the conformal and area distortions, and the mismatch of the mean curvature functions. Finally, we show that our computed maps are applicable for surface interpolation, consistent cross‐field computation, and consistent quadrangular remeshing of a set of shapes.

Abstract We present a new structure‐preserving numerical scheme for solving the Euler‐Poincaré Differential (EPDiff) equation on arbitrary triangle meshes. Unlike existing techniques, our method solves the difficult non‐linear EPDiff equation by constructing energy preserving, yet fully explicit, update rules. Our approach uses standard differential operators on triangle meshes, allowing for a simple and efficient implementation. Key to the structure‐preserving features that our method exhibits is a novel numerical splitting scheme. Namely, we break the integration into three steps which rely on linear solves with a fixed sparse matrix that is independent of the simulation and thus can be pre‐factored. We test our method in the context of simulating concentrated reconnecting wavefronts on flat and curved domains. In particular, EPDiff is known to generate geometrical fronts which exhibit wave‐like behavior when they interact with each other. In addition, we also show that at a small additional cost, we can produce globally‐supported periodic waves by using our simulated fronts with wavefronts tracking techniques. We provide quantitative graphs showing that our method exactly preserves the energy in practice. In addition, we demonstrate various interesting results including annihilation and recreation of a circular front, a wave splitting and merging when hitting an obstacle and two separate fronts propagating and bending due to the curvature of the domain.

We propose a novel technique for computing consistent cross fields on a pair of triangle meshes given an input correspondence, which we use as guiding fields for approximately consistent quadrangulations. Unlike the majority of existing methods our approach does not assume that the meshes share the same connectivity or even have the same number of vertices, and furthermore does not place any restrictions on the topology (genus) of the shapes. Importantly, our method is robust with respect to small perturbations of the given correspondence, as it only relies on the transportation of real-valued functions and thus avoids the costly and error-prone estimation of the map differential. Key to this robustness is a novel formulation, which relies on the previously-proposed notion of power vectors , and we show how consistency can be enforced without pre-alignment of local basis frames, in which these power vectors are computed. We demonstrate that using the same formulation we can both compute a quadrangulation that would respect a given symmetry on the same shape or a map across a pair of shapes. We provide quantitative and qualitative comparison of our method with several baselines and show that it both provides more accurate results and allows to handle more general cases than existing techniques.

The motion of a thin viscous film of fluid on a curved surface exhibits many intricate visual phenomena, which are challenging to simulate using existing techniques. A possible alternative is to use a reduced model, involving only the temporal evolution of the mass density of the film on the surface. However, in this model, the motion is governed by a fourth-order nonlinear PDE, which involves geometric quantities such as the curvature of the underlying surface, and is therefore difficult to discretize. Inspired by a recent variational formulation for this problem on smooth surfaces, we present a corresponding model for triangle meshes. We provide a discretization for the curvature and advection operators which leads to an efficient and stable numerical scheme, requires a single sparse linear solve per time step, and exactly preserves the total volume of the fluid. We validate our method by qualitatively comparing to known results from the literature, and demonstrate various intricate effects achievable by our method, such as droplet formation, evaporation, droplets interaction and viscous fingering. Finally, we extend our method to incorporate non-linear van der Waals forcing terms which stabilize the motion of the film and allow additional effects such as pearling.

Abstract A tangent vector field on a surface is the generator of a smooth family of maps from the surface to itself, known as the flow. Given a scalar function on the surface, it can be transported, or advected, by composing it with a vector field's flow. Such transport is exhibited by many physical phenomena, e.g., in fluid dynamics. In this paper, we are interested in the inverse problem: given source and target functions, compute a vector field whose flow advects the source to the target. We propose a method for addressing this problem, by minimizing an energy given by the advection constraint together with a regularizing term for the vector field. Our approach is inspired by a similar method in computational anatomy, known as LDDMM, yet leverages the recent framework of functional vector fields for discretizing the advection and the flow as operators on scalar functions. The latter allows us to efficiently generalize LDDMM to curved surfaces, without explicitly computing the flow lines of the vector field we are optimizing for. We show two approaches for the solution: using linear advection with multiple vector fields, and using non‐linear advection with a single vector field. We additionally derive an approximated gradient of the corresponding energy, which is based on a novel vector field transport operator. Finally, we demonstrate applications of our machinery to intrinsic symmetry analysis, function interpolation and map improvement.

Vector fields on surfaces are fundamental in various applications in computer graphics and geometry processing. In many cases, in addition to representing vector fields, the need arises to compute their derivatives , for example, for solving partial differential equations on surfaces or for designing vector fields with prescribed smoothness properties. In this work, we consider the problem of computing the Levi-Civita covariant derivative , that is, the tangential component of the standard directional derivative, on triangle meshes. This problem is challenging since, formally, tangent vector fields on polygonal meshes are often viewed as being discontinuous, hence it is not obvious what a good derivative formulation would be. We leverage the relationship between the Levi-Civita covariant derivative of a vector field and the directional derivative of its component functions to provide a simple, easy-to-implement discretization for which we demonstrate experimental convergence. In addition, we introduce two linear which provide access to additional constructs in Riemannian geometry that are not easy to discretize otherwise, including the parallel transport operator which can be seen simply as a certain matrix exponential. Finally, we show the applicability of our operator to various tasks, such as fluid simulation on curved surfaces and vector field design, by posing algebraic constraints on the covariant derivative operator.

Abstract Fluid simulation plays a key role in various domains of science including computer graphics. While most existing work addresses fluids on bounded Euclidean domains, we consider the problem of simulating the behavior of an incompressible fluid on a curved surface represented as an unstructured triangle mesh. Unlike the commonly used Eulerian description of the fluid using its time‐varying velocity field, we propose to model fluids using their vorticity, i.e., by a (time varying) scalar function on the surface. During each time step, we advance scalar vorticity along two consecutive, stationary velocity fields. This approach leads to a variational integrator in the space continuous setting. In addition, using this approach, the update rule amounts to manipulating functions on the surface using linear operators, which can be discretized efficiently using the recently introduced functional approach to vector fields. Combining these time and space discretizations leads to a conceptually and algorithmically simple approach, which is efficient, time‐reversible and conserves vorticity by construction. We further demonstrate that our method exhibits no numerical dissipation and is able to reproduce intricate phenomena such as vortex shedding from boundaries.

Abstract In this paper, we introduce a novel coordinate‐free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace‐Beltrami and symmetry operators, as well as to a straight‐forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.

We develop a novel formulation for the notion of shape differences, aimed at providing detailed information about the location and nature of the differences or distortions between the two shapes being compared. Our difference operator, derived from a shape map, is much more informative than just a scalar global shape similarity score, rendering it useful in a variety of applications where more refined shape comparisons are necessary. The approach is intrinsic and is based on a linear algebraic framework, allowing the use of many common linear algebra tools (e.g, SVD, PCA) for studying a matrix representation of the operator. Remarkably, the formulation allows us not only to localize shape differences on the shapes involved, but also to compare shape differences across pairs of shapes, and to analyze the variability in entire shape collections based on the differences between the shapes. Moreover, while we use a map or correspondence to define each shape difference, consistent correspondences between the shapes are not necessary for comparing shape differences, although they can be exploited if available. We give a number of applications of shape differences, including parameterizing the intrinsic variability in a shape collection, exploring shape collections using local variability at different scales, performing shape analogies, and aligning shape collections.