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Researcher profile

Justin Solomon

Justin Solomon contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate
Machine LearningGraphicsComputer VisionComputational Geometrycs.CYApplicationsComputational Complexitycond-mat.stat-mechDiscrete Mathematicseess.IVmath.COmath.FANeural and Evolutionary Computingphysics.med-phphysics.soc-phRoboticsSocial and Information Networks

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Published work

19 published item(s)

preprint2026arXiv

Learning Orthonormal Bases for Function Spaces

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.

0 citations0 reviewsNo code linkedOpen access
preprint2026arXiv

Meschers: Geometry Processing of Impossible Objects

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

A Dataset and Benchmark for Mesh Parameterization

UV parameterization is a core task in computer graphics, with applications in mesh texturing, remeshing, mesh repair, mesh editing, and more. It is thus an active area of research, which has led to a wide variety of parameterization methods that excel according to different measures of quality. There is no single metric capturing parameterization quality in practice, since the quality of a parameterization heavily depends on its application; hence, parameterization methods can best be judged by the actual users of the computed result. In this paper, we present a dataset of meshes together with UV maps collected from various sources and intended for real-life use. Our dataset can be used to test parameterization methods in realistic environments. We also introduce a benchmark to compare parameterization methods with artist-provided UV parameterizations using a variety of metrics. This strategy enables us to evaluate the performance of a parameterization method by computing the quality indicators that are valued by the designers of a mesh.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

DeepCurrents: Learning Implicit Representations of Shapes with Boundaries

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Improving Approximate Optimal Transport Distances using Quantization

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.

0 citations0 reviewsNo code linkedOpen access
preprint2022arXiv

Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets

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.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization

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.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Learning Manifold Patch-Based Representations of Man-Made Shapes

Choosing the right representation for geometry is crucial for making 3D models compatible with existing applications. Focusing on piecewise-smooth man-made shapes, we propose a new representation that is usable in conventional CAD modeling pipelines and can also be learned by deep neural networks. We demonstrate its benefits by applying it to the task of sketch-based modeling. Given a raster image, our system infers a set of parametric surfaces that realize the input in 3D. To capture piecewise smooth geometry, we learn a special shape representation: a deformable parametric template composed of Coons patches. Naively training such a system, however, is hampered by non-manifold artifacts in the parametric shapes and by a lack of data. To address this, we introduce loss functions that bias the network to output non-self-intersecting shapes and implement them as part of a fully self-supervised system, automatically generating both shape templates and synthetic training data. We develop a testbed for sketch-based modeling, demonstrate shape interpolation, and provide comparison to related work.

0 citations0 reviewsNo code linkedOpen access
preprint2021arXiv

Synthesis of Frame Field-Aligned Multi-Laminar Structures

In the field of topology optimization, the homogenization approach has been revived as an important alternative to the established, density-based methods because it can represent the microstructural design at a much finer length-scale than the computational grid. The optimal microstructure for a single load case is an orthogonal rank-3 laminate. A rank-3 laminate can be described in terms of frame fields, which are also an important tool for mesh generation in both 2D and 3D. We propose a method for generating multi-laminar structures from frame fields. Rather than relying on integrative approaches that find a parametrization based on the frame field, we find stream surfaces, represented as point clouds aligned with frame vectors, and we solve an optimization problem to find well-spaced collections of such stream surfaces. The stream surface tracing is unaffected by the presence of singularities outside the region of interest. Neither stream surface tracing nor selecting well-spaced surface rely on combed frame fields. In addition to stream surface tracing and selection, we provide two methods for generating structures from stream surface collections. One of these methods produces volumetric solids by summing basis functions associated with each point of the stream surface collection. The other method reinterprets point sampled stream surfaces as a spatial twist continuum and produces a hexahedralization by dualizing a graph representing the structure. We demonstrate our methods on several frame fields produced using the homogenization approach for topology optimization, boundary-aligned, algebraic frame fields, and frame fields computed from closed-form expressions.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Computational Approach to Measuring Vote Elasticity and Competitiveness

The recent wave of attention to partisan gerrymandering has come with a push to refine or replace the laws that govern political redistricting around the country. A common element in several states' reform efforts has been the inclusion of competitiveness metrics, or scores that evaluate a districting plan based on the extent to which district-level outcomes are in play or are likely to be closely contested. In this paper, we examine several classes of competitiveness metrics motivated by recent reform proposals and then evaluate their potential outcomes across large ensembles of districting plans at the Congressional and state Senate levels. This is part of a growing literature using MCMC techniques from applied statistics to situate plans and criteria in the context of valid redistricting alternatives. Our empirical analysis focuses on five states---Utah, Georgia, Wisconsin, Virginia, and Massachusetts---chosen to represent a range of partisan attributes. We highlight situation-specific difficulties in creating good competitiveness metrics and show that optimizing competitiveness can produce unintended consequences on other partisan metrics. These results demonstrate the importance of (1) avoiding writing detailed metric constraints into long-lasting constitutional reform and (2) carrying out careful mathematical modeling on real geo-electoral data in each redistricting cycle.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Algebraic Representations for Volumetric Frame Fields

Field-guided parametrization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, is representation of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog---an octahedral frame field---takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically-sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new odeco frames, so-called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program--based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Automatic phantom test pattern classification through transfer learning with deep neural networks

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Empirical Sampling of Connected Graph Partitions for Redistricting

The space of connected graph partitions underlies statistical models used as evidence in court cases and reform efforts that analyze political districting plans. In response to the demands of redistricting applications, researchers have developed sampling methods that traverse this space, building on techniques developed for statistical physics. In this paper, we study connections between redistricting and statistical physics, and in particular with self-avoiding walks. We exploit knowledge of phase transitions and asymptotic behavior in self avoiding walks to analyze two questions of crucial importance for Markov Chain Monte Carlo analysis of districting plans. First, we examine mixing times of a popular Glauber dynamics based Markov chain and show how the self-avoiding walk phase transitions interact with mixing time. We examine factors new to the redistricting context that complicate the picture, notably the population balance requirements, connectivity requirements, and the irregular graphs used. Second, we analyze the robustness of the qualitative properties of typical districting plans with respect to score functions and a certain lattice-like graph, called the state-dual graph, that is used as a discretization of geographic regions in most districting analysis. This helps us better understand the complex relationship between typical properties of districting plans and the score functions designed by political districting analysts. We conclude with directions for research at the interface of statistical physics, Markov chains, and political districting.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Medial Axis Isoperimetric Profiles

Recently proposed as a stable means of evaluating geometric compactness, the isoperimetric profile of a planar domain measures the minimum perimeter needed to inscribe a shape with prescribed area varying from 0 to the area of the domain. While this profile has proven valuable for evaluating properties of geographic partitions, existing algorithms for its computation rely on aggressive approximations and are still computationally expensive. In this paper, we propose a practical means of approximating the isoperimetric profile and show that for domains satisfying a "thick neck" condition, our approximation is exact. For more general domains, we show that our bound is still exact within a conservative regime and is otherwise an upper bound. Our method is based on a traversal of the medial axis which produces efficient and robust results. We compare our technique with the state-of-the-art approximation to the isoperimetric profile on a variety of domains and show significantly tighter bounds than were previously achievable.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Model Fusion with Kullback--Leibler Divergence

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Octahedral Frames for Feature-Aligned Cross-Fields

We present a method for designing smooth cross fields on surfaces that automatically align to sharp features of an underlying geometry. Our approach introduces a novel class of energies based on a representation of cross fields in the spherical harmonic basis. We provide theoretical analysis of these energies in the smooth setting, showing that they penalize deviations from surface creases while otherwise promoting intrinsically smooth fields. We demonstrate the applicability of our method to quad-meshing and include an extensive benchmark comparing our fields to other automatic approaches for generating feature-aligned cross fields on triangle meshes.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Pillar-based Object Detection for Autonomous Driving

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.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Wasserstein Measure Coresets

The proliferation of large data sets and Bayesian inference techniques motivates demand for better data sparsification. Coresets provide a principled way of summarizing a large dataset via a smaller one that is guaranteed to match the performance of the full data set on specific problems. Classical coresets, however, neglect the underlying data distribution, which is often continuous. We address this oversight by introducing Wasserstein measure coresets, an extension of coresets which by definition takes into account generalization. Our formulation of the problem, which essentially consists in minimizing the Wasserstein distance, is solvable via stochastic gradient descent. This yields an algorithm which simply requires sample access to the data distribution and is able to handle large data streams in an online manner. We validate our construction for inference and clustering.

0 citations0 reviewsNo code linkedOpen access
preprint2019arXiv

Complexity and Geometry of Sampling Connected Graph Partitions

In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use gadgets inspired by their technique to provide families of graphs where the "flip walk" Markov chain used in practice for this sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidence demonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statistical physics of self-avoiding walks, we investigate the sensitivity of certain popular sampling algorithms to the graph topology. Finally, we discuss a few cases where the sampling problem is tractable. Applications to political redistricting have recently brought increased attention to this problem, and we articulate open questions about this application that are highlighted by our results.

0 citations0 reviewsNo code linkedOpen access