Justin Solomon

Ahmed H. Mahmoud, Rahul Goel, Jonathan Ragan-Kelley et al.

We present a GPU-based system for automatic differentiation (AD) of functions defined on triangle meshes, designed to exploit the locality and sparsity in mesh-based computation. Our system evaluates derivatives using per-element forward-mode AD, confining all computation to registers and shared memory and assembling global gradients, sparse Jacobians, and sparse Hessians directly on the GPU. By avoiding global computation graphs, intermediate buffers, and device-host synchronization, our approach minimizes memory traffic and enables efficient differentiation under both static and dynamically changing sparsity. Our programming model lets users express energy terms over mesh neighborhoods, while our system automatically manages parallel execution, derivative propagation, sparse assembly, and matrix-free operations such as Hessian-vector products. Our system supports both scalar- and vector-valued objectives, dynamic interaction-driven sparsity updates, and seamless integration with external GPU sparse linear solvers. We evaluate our system on applications including elastic and cloth simulation, surface parameterization, mesh smoothing, frame field design, ARAP deformation, and spherical manifold optimization. Across these tasks, our system consistently outperforms state-of-the-art differentiation frameworks, including PyTorch, JAX, Warp, DrJIT, EnzymeAD, and Thallo. We demonstrate speedups across a range of solver types, from Newton and Gauss-Newton for nonlinear least squares to L-BFGS and gradient descent, and across different derivative usage modes, including Hessian-vector products as well as full sparse Hessian and Jacobian construction. Our system is available as open source at https://github.com/owensgroup/RXMesh.

Behrooz Zarebavami, Ahmed H. Mahmoud, Ana Dodik et al.

We present a fast sparse matrix permutation algorithm tailored to linear systems arising from triangle meshes. Our approach produces nested-dissection-style permutations while significantly reducing permutation runtime overhead. Rather than enforcing strict balance and separator optimality, the algorithm deliberately relaxes these design decisions to favor fast partitioning and efficient elimination-tree construction. Our method decomposes permutation into patch-level local orderings and a compact quotient-graph ordering of separators, preserving the essential structure required by sparse Cholesky factorization while avoiding its most expensive components. We integrate our algorithm into vendor-maintained sparse Cholesky solvers on both CPUs and GPUs. Across a range of graphics applications, including single factorizations and repeated factorizations, our method reduces permutation time and improves the sparse Cholesky solve performance by up to 6.27x. Our code is available at https://github.com/BehroozZare/fast-permute.

Tal Shnitzer, Anthony Ou, Mírian Silva et al.

There is a rapidly growing number of open-source Large Language Models (LLMs) and benchmark datasets to compare them. While some models dominate these benchmarks, no single model typically achieves the best accuracy in all tasks and use cases. In this work, we address the challenge of selecting the best LLM out of a collection of models for new tasks. We propose a new formulation for the problem, in which benchmark datasets are repurposed to learn a "router" model for this LLM selection, and we show that this problem can be reduced to a collection of binary classification tasks. We demonstrate the utility and limitations of learning model routers from various benchmark datasets, where we consistently improve performance upon using any single model for all tasks.

Arman Maesumi, Tanish Makadia, Aruna Anderson et al.

Intrinsic methods fill the default toolbox for geometry processing on meshes. Intrinsic operators, in particular the Laplacian, underlie methods that require invariance to isometry and have hence been employed in many algorithms for shape analysis, learning, and editing. However, intrinsic methods are predicated on assumptions that quickly become brittle when working with in-the-wild geometry, where (i) mesh quality is not guaranteed, and (ii) many meshes are modeled with multiple connected components. In such settings, volumetric constructions are better-defined, since restrictions on surface topology can be relaxed. This paper presents a Monte Carlo method for estimating the Dirichlet-to-Neumann (DtN) operator -- a boundary-to-boundary volumetric operator -- and its associated Steklov eigenmodes. We build on recent developments in Monte Carlo geometry processing by casting this boundary operator itself as the subject of estimation. The DtN operator, defined through a volumetric stochastic process, is then generalized to the exterior domain, where it couples disconnected components through the surrounding ambient space. We show that our method is orders of magnitude faster than existing boundary-element approaches for computing Steklov spectra while remaining robust to poor triangulations, high-resolution meshes, and multi-component geometry. To demonstrate this scalability, we compute interior and exterior Steklov eigenspectra for approximately 450,000 shapes from the uncurated Objaverse dataset. We incorporate these operators into Steklov-CLIP, a mesh-based neural network that uses volumetric spectral operators for large-scale contrastive 3D representation learning. The resulting network learns semantically meaningful global and dense shape representations, illustrating that geometrically-principled volumetric operators can be made practical at the scale of modern 3D datasets.

Lingxiao Li, Samuel Hurault, Justin Solomon

We present a discretization-free scalable framework for solving a large class of mass-conserving partial differential equations (PDEs), including the time-dependent Fokker-Planck equation and the Wasserstein gradient flow. The main observation is that the time-varying velocity field of the PDE solution needs to be self-consistent: it must satisfy a fixed-point equation involving the probability flow characterized by the same velocity field. Instead of directly minimizing the residual of the fixed-point equation with neural parameterization, we use an iterative formulation with a biased gradient estimator that bypasses significant computational obstacles with strong empirical performance. Compared to existing approaches, our method does not suffer from temporal or spatial discretization, covers a wider range of PDEs, and scales to high dimensions. Experimentally, our method recovers analytical solutions accurately when they are available and achieves superior performance in high dimensions with less training time compared to alternatives.

Lingxiao Li, Qiang Liu, Anna Korba et al.

Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.

Christopher Scarvelis, Justin Solomon

We introduce an optimal transport-based model for learning a metric tensor from cross-sectional samples of evolving probability measures on a common Riemannian manifold. We neurally parametrize the metric as a spatially-varying matrix field and efficiently optimize our model's objective using a simple alternating scheme. Using this learned metric, we can nonlinearly interpolate between probability measures and compute geodesics on the manifold. We show that metrics learned using our method improve the quality of trajectory inference on scRNA and bird migration data at the cost of little additional cross-sectional data.

Yuchen Zeng, Kristjan Greenewald, Kangwook Lee et al.

Traditional machine learning models focus on achieving good performance on the overall training distribution, but they often underperform on minority groups. Existing methods can improve the worst-group performance, but they can have several limitations: (i) they require group annotations, which are often expensive and sometimes infeasible to obtain, and/or (ii) they are sensitive to outliers. Most related works fail to solve these two issues simultaneously as they focus on conflicting perspectives of minority groups and outliers. We address the problem of learning group annotations in the presence of outliers by clustering the data in the space of gradients of the model parameters. We show that data in the gradient space has a simpler structure while preserving information about minority groups and outliers, making it suitable for standard clustering methods like DBSCAN. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art both in terms of group identification and downstream worst-group performance.

Dustin Klebe, Tal Shnitzer, Mikhail Yurochkin et al.

Pretrained unimodal encoders incorporate rich semantic information into embedding space structures. To be similarly informative, multi-modal encoders typically require massive amounts of paired data for alignment and training. We introduce a semi-supervised Geometrically Regularized Alignment (GeRA) method to align the embedding spaces of pretrained unimodal encoders in a label-efficient way. Our method leverages the manifold geometry of unpaired (unlabeled) data to improve alignment performance. To prevent distortions to local geometry during the alignment process, potentially disrupting semantic neighborhood structures and causing misalignment of unobserved pairs, we introduce a geometric loss term. This term is built upon a diffusion operator that captures the local manifold geometry of the unimodal pretrained encoders. GeRA is modality-agnostic and thus can be used to align pretrained encoders from any data modalities. We provide empirical evidence to the effectiveness of our method in the domains of speech-text and image-text alignment. Our experiments demonstrate significant improvement in alignment quality compared to a variaty of leading baselines, especially with a small amount of paired data, using our proposed geometric regularization.

Ana Dodik, Oded Stein, Vincent Sitzmann et al.

We propose a variational technique to optimize for generalized barycentric coordinates that offers additional control compared to existing models. Prior work represents barycentric coordinates using meshes or closed-form formulae, in practice limiting the choice of objective function. In contrast, we directly parameterize the continuous function that maps any coordinate in a polytope's interior to its barycentric coordinates using a neural field. This formulation is enabled by our theoretical characterization of barycentric coordinates, which allows us to construct neural fields that parameterize the entire function class of valid coordinates. We demonstrate the flexibility of our model using a variety of objective functions, including multiple smoothness and deformation-aware energies; as a side contribution, we also present mathematically-justified means of measuring and minimizing objectives like total variation on discontinuous neural fields. We offer a practical acceleration strategy, present a thorough validation of our algorithm, and demonstrate several applications.

Christopher Scarvelis, Haitz Sáez de Ocáriz Borde, Justin Solomon

Score-based generative models (SGMs) sample from a target distribution by iteratively transforming noise using the score function of the perturbed target. For any finite training set, this score function can be evaluated in closed form, but the resulting SGM memorizes its training data and does not generate novel samples. In practice, one approximates the score by training a neural network via score-matching. The error in this approximation promotes generalization, but neural SGMs are costly to train and sample, and the effective regularization this error provides is not well-understood theoretically. In this work, we instead explicitly smooth the closed-form score to obtain an SGM that generates novel samples without training. We analyze our model and propose an efficient nearest-neighbor-based estimator of its score function. Using this estimator, our method achieves competitive sampling times while running on consumer-grade CPUs.

Hamidreza Kamkari, Mohammad Sina Nabizadeh, Justin Solomon

Infinite-dimensional orthonormal basis expansions play a central role in representing and computing with function spaces due to their favorable linear algebraic properties. However, common bases such as Fourier or wavelets are fixed and do not adapt to the structure of a given problem or dataset. In this paper, we aim to represent these bases with neural networks and optimize them. Our key idea is that any target infinite-dimensional orthonormal basis can be viewed either as a point on the Lie manifold of the orthogonal group, or equivalently, as the endpoint of a continuous path on that manifold that connects a reference basis, e.g. Fourier, to that target. Paths on the Lie manifold satisfy ordinary differential equations (ODEs) governed by skew-adjoint integral operators. Using neural networks to define finite-rank generators of such ODEs allows us to parameterize and optimize orthonormal bases in function space. While relying on finite-rank generators to model infinite operators might seem restrictive, we prove a universality result: even with a rank-2 generator, the integrated solutions of the ODE are dense in the orthogonal group under the appropriate operator topology. In other words, for any target orthonormal basis, there exists a path originating from a reference basis and driven by finite-rank generators that gets arbitrarily close to that target basis. We demonstrate the flexibility of our framework by transforming the Fourier basis into the principal components of a functional dataset, eigenfunctions of linear operators, or dynamic modes of energy-preserving physical simulations.

Alice Petrov, Mohammad Sina Nabizadeh, Ana Dodik et al.

Many geometry processing pipelines implicitly assume their input data is a manifold, or is sampled from one, with a unique tangent plane at every point. Geometric data, however, routinely contains sharp features like edges, corners, self-intersections, branching junctions, and other singularities, rendering standard methods ill-defined at these points. To bring geometry processing to these and other singular spaces, we introduce the ``tangent blow-up,'' a representation inspired by algebraic geometry that restores structure at singularities by lifting to the product of the ambient space and the Grassmannian of tangent planes. After iterating this construction, points that coincide in position but differ in tangent direction, curvature, or higher-order contact become well-separated. We equip the tangent blow-up with a product metric and define discretized differential operators, such as the gradient, divergence, and Laplacian, directly in the lifted domain. We demonstrate our framework across geodesic computation, segmentation, surface parameterization, and curvature estimation.

Ana Dodik, Isabella Yu, Kartik Chandra et al.

Impossible objects, geometric constructions that humans can perceive but that cannot exist in real life, have been a topic of intrigue in visual arts, perception, and graphics, yet no satisfying computer representation of such objects exists. Previous work embeds impossible objects in 3D, cutting them or twisting/bending them in the depth axis. Cutting an impossible object changes its local geometry at the cut, which can hamper downstream graphics applications, such as smoothing, while bending makes it difficult to relight the object. Both of these can invalidate geometry operations, such as distance computation. As an alternative, we introduce Meschers, meshes capable of representing impossible constructions akin to those found in M.C. Escher's woodcuts. Our representation has a theoretical foundation in discrete exterior calculus and supports the use-cases above, as we demonstrate in a number of example applications. Moreover, because we can do discrete geometry processing on our representation, we can inverse-render impossible objects. We also compare our representation to cut and bend representations of impossible objects.

Rickard Brüel-Gabrielsson, Tongzhou Wang, Manel Baradad et al.

Despite dropout's ubiquity in machine learning, its effectiveness as a form of data augmentation remains under-explored. We address two key questions: (i) When is dropout effective as an augmentation strategy? (ii) Is dropout uniquely effective under these conditions? To explore these questions, we propose Deep Augmentation, a network- and modality-agnostic method that applies dropout or PCA transformations to targeted layers in neural networks. Through extensive experiments on contrastive learning tasks in NLP, computer vision, and graph learning, we find that uniformly applying dropout across layers does not consistently improve performance. Instead, dropout proves most beneficial in deeper layers and can be matched by alternative augmentations (e.g., PCA). We also show that a stop-gradient operation is critical for ensuring dropout functions effectively as an augmentation, and that performance trends invert when moving from contrastive tasks to supervised tasks. Our analysis suggests that Deep Augmentation helps mitigate inter-layer co-adaptation -- a notable issue in self-supervised learning due to the absence of labeled data. Drawing on these insights, we outline a procedure for selecting the optimal augmentation layer and demonstrate that Deep Augmentation can outperform traditional input-level augmentations. This simple yet powerful approach can be seamlessly integrated into a wide range of architectures and modalities, yielding notable gains in both performance and generalization.

S. Mazdak Abulnaga, Esra Abaci Turk, Mikhail Bessmeltsev et al.

We present a volumetric mesh-based algorithm for parameterizing the placenta to a flattened template to enable effective visualization of local anatomy and function. MRI shows potential as a research tool as it provides signals directly related to placental function. However, due to the curved and highly variable in vivo shape of the placenta, interpreting and visualizing these images is difficult. We address interpretation challenges by mapping the placenta so that it resembles the familiar ex vivo shape. We formulate the parameterization as an optimization problem for mapping the placental shape represented by a volumetric mesh to a flattened template. We employ the symmetric Dirichlet energy to control local distortion throughout the volume. Local injectivity in the mapping is enforced by a constrained line search during the gradient descent optimization. We validate our method using a research study of 111 placental shapes extracted from BOLD MRI images. Our mapping achieves sub-voxel accuracy in matching the template while maintaining low distortion throughout the volume. We demonstrate how the resulting flattening of the placenta improves visualization of anatomy and function. Our code is freely available at https://github.com/mabulnaga/placenta-flattening .

Nicolas Girard, Dmitriy Smirnov, Justin Solomon et al.

While state of the art image segmentation models typically output segmentations in raster format, applications in geographic information systems often require vector polygons. To help bridge the gap between deep network output and the format used in downstream tasks, we add a frame field output to a deep segmentation model for extracting buildings from remote sensing images. We train a deep neural network that aligns a predicted frame field to ground truth contours. This additional objective improves segmentation quality by leveraging multi-task learning and provides structural information that later facilitates polygonization; we also introduce a polygonization algorithm that utilizes the frame field along with the raster segmentation. Our code is available at https://github.com/Lydorn/Polygonization-by-Frame-Field-Learning.

Trevor Henderson, Justin Solomon

This paper proposes a new method to interpolate between two audio signals. As an interpolation parameter is changed, the pitches in one signal slide to the pitches in the other, producing a portamento, or musical glide. The assignment of pitches in one sound to pitches in the other is accomplished by solving a 1-dimensional optimal transport problem. In addition, we introduce several techniques that preserve the audio fidelity over this highly non-linear transformation. A portamento is a natural way for a musician to transition between notes, but traditionally it has only been possible for instruments with a continuously variable pitch like the human voice or the violin. Audio transport extends the portamento to any instrument, even polyphonic ones. Moreover, the effect can be used to transition between different instruments, groups of instruments, or really any transient-less audio signals. The audio transport effect operates in real-time; we open-source implementation is provided. In experiments with sinusoidal inputs, the interpolating effect is indistinguishable from ideal sine sweeps. In general, the effect produces clear, musical results for a wide variety of inputs.

S. Mazdak Abulnaga, Esra Abaci Turk, Mikhail Bessmeltsev et al.

We present a volumetric mesh-based algorithm for flattening the placenta to a canonical template to enable effective visualization of local anatomy and function. Monitoring placental function in vivo promises to support pregnancy assessment and to improve care outcomes. We aim to alleviate visualization and interpretation challenges presented by the shape of the placenta when it is attached to the curved uterine wall. To do so, we flatten the volumetric mesh that captures placental shape to resemble the well-studied ex vivo shape. We formulate our method as a map from the in vivo shape to a flattened template that minimizes the symmetric Dirichlet energy to control distortion throughout the volume. Local injectivity is enforced via constrained line search during gradient descent. We evaluate the proposed method on 28 placenta shapes extracted from MRI images in a clinical study of placental function. We achieve sub-voxel accuracy in mapping the boundary of the placenta to the template while successfully controlling distortion throughout the volume. We illustrate how the resulting mapping of the placenta enhances visualization of placental anatomy and function. Our code is freely available at https://github.com/mabulnaga/placenta-flattening .

Jiacheng Zhu, Kristjan Greenewald, Kimia Nadjahi et al.

Parameter-efficient fine-tuning optimizes large, pre-trained foundation models by updating a subset of parameters; in this class, Low-Rank Adaptation (LoRA) is particularly effective. Inspired by an effort to investigate the different roles of LoRA matrices during fine-tuning, this paper characterizes and leverages unexpected asymmetry in the importance of low-rank adapter matrices. Specifically, when updating the parameter matrices of a neural network by adding a product $BA$, we observe that the $B$ and $A$ matrices have distinct functions: $A$ extracts features from the input, while $B$ uses these features to create the desired output. Based on this observation, we demonstrate that fine-tuning $B$ is inherently more effective than fine-tuning $A$, and that a random untrained $A$ should perform nearly as well as a fine-tuned one. Using an information-theoretic lens, we also bound the generalization of low-rank adapters, showing that the parameter savings of exclusively training $B$ improves the bound. We support our conclusions with experiments on RoBERTa, BART-Large, LLaMA-2, and ViTs.

Artem Lukoianov, Haitz Sáez de Ocáriz Borde, Kristjan Greenewald et al.

While 2D diffusion models generate realistic, high-detail images, 3D shape generation methods like Score Distillation Sampling (SDS) built on these 2D diffusion models produce cartoon-like, over-smoothed shapes. To help explain this discrepancy, we show that the image guidance used in Score Distillation can be understood as the velocity field of a 2D denoising generative process, up to the choice of a noise term. In particular, after a change of variables, SDS resembles a high-variance version of Denoising Diffusion Implicit Models (DDIM) with a differently-sampled noise term: SDS introduces noise i.i.d. randomly at each step, while DDIM infers it from the previous noise predictions. This excessive variance can lead to over-smoothing and unrealistic outputs. We show that a better noise approximation can be recovered by inverting DDIM in each SDS update step. This modification makes SDS's generative process for 2D images almost identical to DDIM. In 3D, it removes over-smoothing, preserves higher-frequency detail, and brings the generation quality closer to that of 2D samplers. Experimentally, our method achieves better or similar 3D generation quality compared to other state-of-the-art Score Distillation methods, all without training additional neural networks or multi-view supervision, and providing useful insights into relationship between 2D and 3D asset generation with diffusion models.

Ana Dodik, Ahmed H. Mahmoud, Justin Solomon

We propose a system for differentiating through solutions to geometry processing problems. Our system differentiates a broad class of geometric algorithms, exploiting existing fast problem-specific schemes common to geometry processing, including local-global and ADMM solvers. It is compatible with machine learning frameworks, opening doors to new classes of inverse geometry processing applications. We marry the scatter-gather approach to mesh processing with tensor-based workflows and rely on the adjoint method applied to user-specified imperative code to generate an efficient backward pass behind the scenes. We demonstrate our approach by differentiating through mean curvature flow, spectral conformal parameterization, geodesic distance computation, and as-rigid-as-possible deformation, examining usability and performance on these applications. Our system allows practitioners to differentiate through existing geometry processing algorithms without needing to reformulate them, resulting in low implementation effort, fast runtimes, and lower memory requirements than differentiable optimization tools not tailored to geometry processing.

Artem Lukoianov, Chenyang Yuan, Justin Solomon et al.

Recent work has shown that the generalization ability of image diffusion models arises from the locality properties of the trained neural network. In particular, when denoising a particular pixel, the model relies on a limited neighborhood of the input image around that pixel, which, according to the previous work, is tightly related to the ability of these models to produce novel images. Since locality is central to generalization, it is crucial to understand why diffusion models learn local behavior in the first place, as well as the factors that govern the properties of locality patterns. In this work, we present evidence that the locality in deep diffusion models emerges as a statistical property of the image dataset and is not due to the inductive bias of convolutional neural networks, as suggested in previous work. Specifically, we demonstrate that an optimal parametric linear denoiser exhibits similar locality properties to deep neural denoisers. We show, both theoretically and experimentally, that this locality arises directly from pixel correlations present in the image datasets. Moreover, locality patterns are drastically different on specialized datasets, approximating principal components of the data's covariance. We use these insights to craft an analytical denoiser that better matches scores predicted by a deep diffusion model than prior expert-crafted alternatives. Our key takeaway is that while neural network architectures influence generation quality, their primary role is to capture locality patterns inherent in the data.

Christopher Scarvelis, Justin Solomon

Penalizing the nuclear norm of a function's Jacobian encourages it to locally behave like a low-rank linear map. Such functions vary locally along only a handful of directions, making the Jacobian nuclear norm a natural regularizer for machine learning problems. However, this regularizer is intractable for high-dimensional problems, as it requires computing a large Jacobian matrix and taking its singular value decomposition. We show how to efficiently penalize the Jacobian nuclear norm using techniques tailor-made for deep learning. We prove that for functions parametrized as compositions $f = g \circ h$, one may equivalently penalize the average squared Frobenius norm of $Jg$ and $Jh$. We then propose a denoising-style approximation that avoids the Jacobian computations altogether. Our method is simple, efficient, and accurate, enabling Jacobian nuclear norm regularization to scale to high-dimensional deep learning problems. We complement our theory with an empirical study of our regularizer's performance and investigate applications to denoising and representation learning.

Christopher Scarvelis, Justin Solomon

Training a diffusion model approximates a map from a data distribution $ρ$ to the optimal score function $s_t$ for that distribution. Can we differentiate this map? If we could, then we could predict how the score, and ultimately the model's samples, would change under small perturbations to the training set before committing to costly retraining. We give a closed-form procedure for computing this map's directional derivatives, relying only on black-box access to a pre-trained score model and its derivatives with respect to its inputs. We extend this result to estimate the sensitivity of a diffusion model's samples to additive perturbations of its target measure, with runtime comparable to sampling from a diffusion model and computing log-likelihoods along the sample path. Our method is robust to numerical and approximation error, and the resulting sensitivities correlate with changes in an image diffusion model's samples after retraining and fine-tuning.

Anh Truong, Ahmed H. Mahmoud, Mina Konaković Luković et al.

Processing visual data often involves small adjustments or sequences of changes, e.g., image filtering, surface smoothing, and animation. While established graphics techniques like normal mapping and video compression exploit redundancy to encode such small changes efficiently, the problem of encoding small changes to neural fields -- neural network parameterizations of visual or physical functions -- has received less attention. We propose a parameter-efficient strategy for updating neural fields using low-rank adaptations (LoRA). LoRA, a method from the parameter-efficient fine-tuning LLM community, encodes small updates to pre-trained models with minimal computational overhead. We adapt LoRA for instance-specific neural fields, avoiding the need for large pre-trained models and yielding lightweight updates. We validate our approach with experiments in image filtering, geometry editing, video compression, and energy-based editing, demonstrating its effectiveness and versatility for representing neural field updates.

Rickard Brüel-Gabrielsson, Jiacheng Zhu, Onkar Bhardwaj et al.

Fine-tuning large language models (LLMs) with low-rank adaptations (LoRAs) has become common practice, often yielding numerous copies of the same LLM differing only in their LoRA updates. This paradigm presents challenges for systems that serve real-time responses to queries that each involve a different LoRA. Prior works optimize the design of such systems but still require continuous loading and offloading of LoRAs, as it is infeasible to store thousands of LoRAs in GPU memory. To mitigate this issue, we investigate the efficacy of compression when serving LoRAs. We propose a method for the joint compression of LoRAs into a shared basis paired with LoRA-specific scaling matrices. We extend our algorithm to learn clusters of LoRAs that are amenable to joint compression, allowing it to scale gracefully to large LoRA collections. Our experiments with up to 1000 LoRAs demonstrate that compressed LoRAs preserve performance while offering major throughput gains in realistic serving scenarios with over a thousand LoRAs, maintaining 80% of the throughput of serving a single LoRA.

Kimia Nadjahi, Kristjan Greenewald, Rickard Brüel Gabrielsson et al.

The ability of machine learning (ML) algorithms to generalize well to unseen data has been studied through the lens of information theory, by bounding the generalization error with the input-output mutual information (MI), i.e., the MI between the training data and the learned hypothesis. Yet, these bounds have limited practicality for modern ML applications (e.g., deep learning), due to the difficulty of evaluating MI in high dimensions. Motivated by recent findings on the compressibility of neural networks, we consider algorithms that operate by slicing the parameter space, i.e., trained on random lower-dimensional subspaces. We introduce new, tighter information-theoretic generalization bounds tailored for such algorithms, demonstrating that slicing improves generalization. Our bounds offer significant computational and statistical advantages over standard MI bounds, as they rely on scalable alternative measures of dependence, i.e., disintegrated mutual information and $k$-sliced mutual information. Then, we extend our analysis to algorithms whose parameters do not need to exactly lie on random subspaces, by leveraging rate-distortion theory. This strategy yields generalization bounds that incorporate a distortion term measuring model compressibility under slicing, thereby tightening existing bounds without compromising performance or requiring model compression. Building on this, we propose a regularization scheme enabling practitioners to control generalization through compressibility. Finally, we empirically validate our results and achieve the computation of non-vacuous information-theoretic generalization bounds for neural networks, a task that was previously out of reach.

Ana Dodik, Vincent Sitzmann, Justin Solomon et al.

Skinning is a popular way to rig and deform characters for animation, to compute reduced-order simulations, and to define features for geometry processing. Methods built on skinning rely on weight functions that distribute the influence of each degree of freedom across the mesh. Automatic skinning methods generate these weight functions with minimal user input, usually by solving a variational problem on a mesh whose boundary is the skinned surface. This formulation necessitates tetrahedralizing the volume bounded by the surface, which brings with it meshing artifacts, the possibility of tetrahedralization failure, and the impossibility of generating weights for surfaces that are not closed. We introduce a mesh-free and robust automatic skinning method that generates high-quality skinning weights comparable to the current state of the art without volumetric meshes. Our method reliably works even on open surfaces and triangle soups where current methods fail. We achieve this through the use of a Lagrangian representation for skinning weights, which circumvents the need for finite elements while optimizing the biharmonic energy.

Nicholas Sharp, Cristian Romero, Alec Jacobson et al.

Physical systems ranging from elastic bodies to kinematic linkages are defined on high-dimensional configuration spaces, yet their typical low-energy configurations are concentrated on much lower-dimensional subspaces. This work addresses the challenge of identifying such subspaces automatically: given as input an energy function for a high-dimensional system, we produce a low-dimensional map whose image parameterizes a diverse yet low-energy submanifold of configurations. The only additional input needed is a single seed configuration for the system to initialize our procedure; no dataset of trajectories is required. We represent subspaces as neural networks that map a low-dimensional latent vector to the full configuration space, and propose a training scheme to fit network parameters to any system of interest. This formulation is effective across a very general range of physical systems; our experiments demonstrate not only nonlinear and very low-dimensional elastic body and cloth subspaces, but also more general systems like colliding rigid bodies and linkages. We briefly explore applications built on this formulation, including manipulation, latent interpolation, and sampling.

S. Mazdak Abulnaga, Oded Stein, Polina Golland et al.

Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence--a natural extension relevant to shapes extracted from simulation, medical imaging, and volume rendering--presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align closely to the boundary.

Tal Shnitzer, Mikhail Yurochkin, Kristjan Greenewald et al.

The need for efficiently comparing and representing datasets with unknown alignment spans various fields, from model analysis and comparison in machine learning to trend discovery in collections of medical datasets. We use manifold learning to compare the intrinsic geometric structures of different datasets by comparing their diffusion operators, symmetric positive-definite (SPD) matrices that relate to approximations of the continuous Laplace-Beltrami operator from discrete samples. Existing methods typically assume known data alignment and compare such operators in a pointwise manner. Instead, we exploit the Riemannian geometry of SPD matrices to compare these operators and define a new theoretically-motivated distance based on a lower bound of the log-Euclidean metric. Our framework facilitates comparison of data manifolds expressed in datasets with different sizes, numbers of features, and measurement modalities. Our log-Euclidean signature (LES) distance recovers meaningful structural differences, outperforming competing methods in various application domains.

Rickard Brüel-Gabrielsson, Mikhail Yurochkin, Justin Solomon

Several recent works use positional encodings to extend the receptive fields of graph neural network (GNN) layers equipped with attention mechanisms. These techniques, however, extend receptive fields to the complete graph, at substantial computational cost and risking a change in the inductive biases of conventional GNNs, or require complex architecture adjustments. As a conservative alternative, we use positional encodings to expand receptive fields to $r$-hop neighborhoods. More specifically, our method augments the input graph with additional nodes/edges and uses positional encodings as node and/or edge features. We thus modify graphs before inputting them to a downstream GNN model, instead of modifying the model itself. This makes our method model-agnostic, i.e., compatible with any of the existing GNN architectures. We also provide examples of positional encodings that are lossless with a one-to-one map between the original and the modified graphs. We demonstrate that extending receptive fields via positional encodings and a virtual fully-connected node significantly improves GNN performance and alleviates over-squashing using small $r$. We obtain improvements on a variety of models and datasets and reach competitive performance using traditional GNNs or graph Transformers.

Lingxiao Li, Noam Aigerman, Vladimir G. Kim et al.

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Most past algorithms either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of found solutions using ad hoc heuristics. We present an end-to-end method to learn the proximal operator of a family of training problems so that multiple local minima can be quickly obtained from initial guesses by iterating the learned operator, emulating the proximal-point algorithm that has fast convergence. The learned proximal operator can be further generalized to recover multiple optima for unseen problems at test time, enabling applications such as object detection. The key ingredient in our formulation is a proximal regularization term, which elevates the convexity of our training loss: by applying recent theoretical results, we show that for weakly-convex objectives with Lipschitz gradients, training of the proximal operator converges globally with a practical degree of over-parameterization. We further present an exhaustive benchmark for multi-solution optimization to demonstrate the effectiveness of our method.

Paul Zhang, Dmitriy Smirnov, Justin Solomon

Much of computer-generated animation is created by manipulating meshes with rigs. While this approach works well for animating articulated objects like animals, it has limited flexibility for animating less structured free-form objects. We introduce Wassersplines, a novel trajectory inference method for animating unstructured densities based on recent advances in continuous normalizing flows and optimal transport. The key idea is to train a neurally-parameterized velocity field that represents the motion between keyframes. Trajectories are then computed by advecting keyframes through the velocity field. We solve an additional Wasserstein barycenter interpolation problem to guarantee strict adherence to keyframes. Our tool can stylize trajectories through a variety of PDE-based regularizers to create different visual effects. We demonstrate our tool on various keyframe interpolation problems to produce temporally-coherent animations without meshing or rigging.

David Palmer, Dmitriy Smirnov, Stephanie Wang et al.

Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such as signed distance fields) parameterized by deep neural networks. Many of these methods, however, learn only closed surfaces and are unable to reconstruct shapes with boundary curves. We propose a hybrid shape representation that combines explicit boundary curves with implicit learned interiors. Using machinery from geometric measure theory, we parameterize currents using deep networks and use stochastic gradient descent to solve a minimal surface problem. By modifying the metric according to target geometry coming, e.g., from a mesh or point cloud, we can use this approach to represent arbitrary surfaces, learning implicitly defined shapes with explicitly defined boundary curves. We further demonstrate learning families of shapes jointly parameterized by boundary curves and latent codes.

Yue Wang, Justin Solomon

3D object detection often involves complicated training and testing pipelines, which require substantial domain knowledge about individual datasets. Inspired by recent non-maximum suppression-free 2D object detection models, we propose a 3D object detection architecture on point clouds. Our method models 3D object detection as message passing on a dynamic graph, generalizing the DGCNN framework to predict a set of objects. In our construction, we remove the necessity of post-processing via object confidence aggregation or non-maximum suppression. To facilitate object detection from sparse point clouds, we also propose a set-to-set distillation approach customized to 3D detection. This approach aligns the outputs of the teacher model and the student model in a permutation-invariant fashion, significantly simplifying knowledge distillation for the 3D detection task. Our method achieves state-of-the-art performance on autonomous driving benchmarks. We also provide abundant analysis of the detection model and distillation framework.

Yue Wang, Vitor Guizilini, Tianyuan Zhang et al.

We introduce a framework for multi-camera 3D object detection. In contrast to existing works, which estimate 3D bounding boxes directly from monocular images or use depth prediction networks to generate input for 3D object detection from 2D information, our method manipulates predictions directly in 3D space. Our architecture extracts 2D features from multiple camera images and then uses a sparse set of 3D object queries to index into these 2D features, linking 3D positions to multi-view images using camera transformation matrices. Finally, our model makes a bounding box prediction per object query, using a set-to-set loss to measure the discrepancy between the ground-truth and the prediction. This top-down approach outperforms its bottom-up counterpart in which object bounding box prediction follows per-pixel depth estimation, since it does not suffer from the compounding error introduced by a depth prediction model. Moreover, our method does not require post-processing such as non-maximum suppression, dramatically improving inference speed. We achieve state-of-the-art performance on the nuScenes autonomous driving benchmark.

Kristjan Greenewald, Anming Gu, Mikhail Yurochkin et al.

Mixup is a popular regularization technique for training deep neural networks that improves generalization and increases robustness to certain distribution shifts. It perturbs input training data in the direction of other randomly-chosen instances in the training set. To better leverage the structure of the data, we extend mixup in a simple, broadly applicable way to \emph{$k$-mixup}, which perturbs $k$-batches of training points in the direction of other $k$-batches. The perturbation is done with displacement interpolation, i.e. interpolation under the Wasserstein metric. We demonstrate theoretically and in simulations that $k$-mixup preserves cluster and manifold structures, and we extend theory studying the efficacy of standard mixup to the $k$-mixup case. Our empirical results show that training with $k$-mixup further improves generalization and robustness across several network architectures and benchmark datasets of differing modalities. For the wide variety of real datasets considered, the performance gains of $k$-mixup over standard mixup are similar to or larger than the gains of mixup itself over standard ERM after hyperparameter optimization. In several instances, in fact, $k$-mixup achieves gains in settings where standard mixup has negligible to zero improvement over ERM.

Alexander Korotin, Lingxiao Li, Aude Genevay et al.

Despite the recent popularity of neural network-based solvers for optimal transport (OT), there is no standard quantitative way to evaluate their performance. In this paper, we address this issue for quadratic-cost transport -- specifically, computation of the Wasserstein-2 distance, a commonly-used formulation of optimal transport in machine learning. To overcome the challenge of computing ground truth transport maps between continuous measures needed to assess these solvers, we use input-convex neural networks (ICNN) to construct pairs of measures whose ground truth OT maps can be obtained analytically. This strategy yields pairs of continuous benchmark measures in high-dimensional spaces such as spaces of images. We thoroughly evaluate existing optimal transport solvers using these benchmark measures. Even though these solvers perform well in downstream tasks, many do not faithfully recover optimal transport maps. To investigate the cause of this discrepancy, we further test the solvers in a setting of image generation. Our study reveals crucial limitations of existing solvers and shows that increased OT accuracy does not necessarily correlate to better results downstream.

Petr Mokrov, Alexander Korotin, Lingxiao Li et al.

Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the so-called JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated to each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications. Our approach relies on input-convex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Unlike previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate our algorithm's performance by computing diffusions following the Fokker-Planck equation and apply it to unnormalized density sampling as well as nonlinear filtering.

Dmitriy Smirnov, Michael Gharbi, Matthew Fisher et al.

Artists and video game designers often construct 2D animations using libraries of sprites -- textured patches of objects and characters. We propose a deep learning approach that decomposes sprite-based video animations into a disentangled representation of recurring graphic elements in a self-supervised manner. By jointly learning a dictionary of possibly transparent patches and training a network that places them onto a canvas, we deconstruct sprite-based content into a sparse, consistent, and explicit representation that can be easily used in downstream tasks, like editing or analysis. Our framework offers a promising approach for discovering recurring visual patterns in image collections without supervision.

Gaspard Beugnot, Aude Genevay, Kristjan Greenewald et al.

Optimal transport (OT) is a popular tool in machine learning to compare probability measures geometrically, but it comes with substantial computational burden. Linear programming algorithms for computing OT distances scale cubically in the size of the input, making OT impractical in the large-sample regime. We introduce a practical algorithm, which relies on a quantization step, to estimate OT distances between measures given cheap sample access. We also provide a variant of our algorithm to improve the performance of approximate solvers, focusing on those for entropy-regularized transport. We give theoretical guarantees on the benefits of this quantization step and display experiments showing that it behaves well in practice, providing a practical approximation algorithm that can be used as a drop-in replacement for existing OT estimators.

Alexander Korotin, Lingxiao Li, Justin Solomon et al.

Wasserstein barycenters provide a geometric notion of the weighted average of probability measures based on optimal transport. In this paper, we present a scalable algorithm to compute Wasserstein-2 barycenters given sample access to the input measures, which are not restricted to being discrete. While past approaches rely on entropic or quadratic regularization, we employ input convex neural networks and cycle-consistency regularization to avoid introducing bias. As a result, our approach does not resort to minimax optimization. We provide theoretical analysis on error bounds as well as empirical evidence of the effectiveness of the proposed approach in low-dimensional qualitative scenarios and high-dimensional quantitative experiments.

Justin Solomon, Kristjan Greenewald, Haikady N. Nagaraja

We introduce $k$-variance, a generalization of variance built on the machinery of random bipartite matchings. $K$-variance measures the expected cost of matching two sets of $k$ samples from a distribution to each other, capturing local rather than global information about a measure as $k$ increases; it is easily approximated stochastically using sampling and linear programming. In addition to defining $k$-variance and proving its basic properties, we provide in-depth analysis of this quantity in several key cases, including one-dimensional measures, clustered measures, and measures concentrated on low-dimensional subsets of $\mathbb R^n$. We conclude with experiments and open problems motivated by this new way to summarize distributional shape.

Yue Wang, Alireza Fathi, Jiajun Wu et al.

A common dilemma in 3D object detection for autonomous driving is that high-quality, dense point clouds are only available during training, but not testing. We use knowledge distillation to bridge the gap between a model trained on high-quality inputs at training time and another tested on low-quality inputs at inference time. In particular, we design a two-stage training pipeline for point cloud object detection. First, we train an object detection model on dense point clouds, which are generated from multiple frames using extra information only available at training time. Then, we train the model's identical counterpart on sparse single-frame point clouds with consistency regularization on features from both models. We show that this procedure improves performance on low-quality data during testing, without additional overhead.

Lingxiao Li, Aude Genevay, Mikhail Yurochkin et al.

Wasserstein barycenters provide a geometrically meaningful way to aggregate probability distributions, built on the theory of optimal transport. They are difficult to compute in practice, however, leading previous work to restrict their supports to finite sets of points. Leveraging a new dual formulation for the regularized Wasserstein barycenter problem, we introduce a stochastic algorithm that constructs a continuous approximation of the barycenter. We establish strong duality and use the corresponding primal-dual relationship to parametrize the barycenter implicitly using the dual potentials of regularized transport problems. The resulting problem can be solved with stochastic gradient descent, which yields an efficient online algorithm to approximate the barycenter of continuous distributions given sample access. We demonstrate the effectiveness of our approach and compare against previous work on synthetic examples and real-world applications.

Yue Wang, Alireza Fathi, Abhijit Kundu et al.

We present a simple and flexible object detection framework optimized for autonomous driving. Building on the observation that point clouds in this application are extremely sparse, we propose a practical pillar-based approach to fix the imbalance issue caused by anchors. In particular, our algorithm incorporates a cylindrical projection into multi-view feature learning, predicts bounding box parameters per pillar rather than per point or per anchor, and includes an aligned pillar-to-point projection module to improve the final prediction. Our anchor-free approach avoids hyperparameter search associated with past methods, simplifying 3D object detection while significantly improving upon state-of-the-art.

Sebastian Claici, Mikhail Yurochkin, Soumya Ghosh et al.

We propose a method to fuse posterior distributions learned from heterogeneous datasets. Our algorithm relies on a mean field assumption for both the fused model and the individual dataset posteriors and proceeds using a simple assign-and-average approach. The components of the dataset posteriors are assigned to the proposed global model components by solving a regularized variant of the assignment problem. The global components are then updated based on these assignments by their mean under a KL divergence. For exponential family variational distributions, our formulation leads to an efficient non-parametric algorithm for computing the fused model. Our algorithm is easy to describe and implement, efficient, and competitive with state-of-the-art on motion capture analysis, topic modeling, and federated learning of Bayesian neural networks.

Rafael B. Fricks, Justin Solomon, Ehsan Samei

Imaging phantoms are test patterns used to measure image quality in computer tomography (CT) systems. A new phantom platform (Mercury Phantom, Gammex) provides test patterns for estimating the task transfer function (TTF) or noise power spectrum (NPF) and simulates different patient sizes. Determining which image slices are suitable for analysis currently requires manual annotation of these patterns by an expert, as subtle defects may make an image unsuitable for measurement. We propose a method of automatically classifying these test patterns in a series of phantom images using deep learning techniques. By adapting a convolutional neural network based on the VGG19 architecture with weights trained on ImageNet, we use transfer learning to produce a classifier for this domain. The classifier is trained and evaluated with over 3,500 phantom images acquired at a university medical center. Input channels for color images are successfully adapted to convey contextual information for phantom images. A series of ablation studies are employed to verify design aspects of the classifier and evaluate its performance under varying training conditions. Our solution makes extensive use of image augmentation to produce a classifier that accurately classifies typical phantom images with 98% accuracy, while maintaining as much as 86% accuracy when the phantom is improperly imaged.