ICML 2023 Topological Deep Learning Challenge : Design and Results

This paper presents the computational challenge on topological deep learning that was hosted within the ICML 2023 Workshop on Topology and Geometry in Machine Learning. The competition asked participants to provide open-source implementations of topological neural networks from the literature by contributing to the python packages TopoNetX (data processing) and TopoModelX (deep learning). The challenge attracted twenty-eight qualifying submissions in its two-month duration. This paper describes the design of the challenge and summarizes its main findings.

VidStyleODE: Disentangled Video Editing via StyleGAN and NeuralODEs

We propose $\textbf{VidStyleODE}$, a spatiotemporally continuous disentangled $\textbf{Vid}$eo representation based upon $\textbf{Style}$GAN and Neural-$\textbf{ODE}$s. Effective traversal of the latent space learned by Generative Adversarial Networks (GANs) has been the basis for recent breakthroughs in image editing. However, the applicability of such advancements to the video domain has been hindered by the difficulty of representing and controlling videos in the latent space of GANs. In particular, videos are composed of content (i.e., appearance) and complex motion components that require a special mechanism to disentangle and control. To achieve this, VidStyleODE encodes the video content in a pre-trained StyleGAN $\mathcal{W}_+$ space and benefits from a latent ODE component to summarize the spatiotemporal dynamics of the input video. Our novel continuous video generation process then combines the two to generate high-quality and temporally consistent videos with varying frame rates. We show that our proposed method enables a variety of applications on real videos: text-guided appearance manipulation, motion manipulation, image animation, and video interpolation and extrapolation. Project website: https://cyberiada.github.io/VidStyleODE

Chordal Averaging on Flag Manifolds and Its Applications

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix groups, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis.

Disentangling Content and Motion for Text-Based Neural Video Manipulation

Giving machines the ability to imagine possible new objects or scenes from linguistic descriptions and produce their realistic renderings is arguably one of the most challenging problems in computer vision. Recent advances in deep generative models have led to new approaches that give promising results towards this goal. In this paper, we introduce a new method called DiCoMoGAN for manipulating videos with natural language, aiming to perform local and semantic edits on a video clip to alter the appearances of an object of interest. Our GAN architecture allows for better utilization of multiple observations by disentangling content and motion to enable controllable semantic edits. To this end, we introduce two tightly coupled networks: (i) a representation network for constructing a concise understanding of motion dynamics and temporally invariant content, and (ii) a translation network that exploits the extracted latent content representation to actuate the manipulation according to the target description. Our qualitative and quantitative evaluations demonstrate that DiCoMoGAN significantly outperforms existing frame-based methods, producing temporally coherent and semantically more meaningful results.

6D Camera Relocalization in Visually Ambiguous Extreme Environments

We propose a novel method to reliably estimate the pose of a camera given a sequence of images acquired in extreme environments such as deep seas or extraterrestrial terrains. Data acquired under these challenging conditions are corrupted by textureless surfaces, image degradation, and presence of repetitive and highly ambiguous structures. When naively deployed, the state-of-the-art methods can fail in those scenarios as confirmed by our empirical analysis. In this paper, we attempt to make camera relocalization work in these extreme situations. To this end, we propose: (i) a hierarchical localization system, where we leverage temporal information and (ii) a novel environment-aware image enhancement method to boost the robustness and accuracy. Our extensive experimental results demonstrate superior performance in favor of our method under two extreme settings: localizing an autonomous underwater vehicle and localizing a planetary rover in a Mars-like desert. In addition, our method achieves comparable performance with state-of-the-art methods on the indoor benchmark (7-Scenes dataset) using only 20% training data.

Q-FW: A Hybrid Classical-Quantum Frank-Wolfe for Quadratic Binary Optimization

We present a hybrid classical-quantum framework based on the Frank-Wolfe algorithm, Q-FW, for solving quadratic, linearly-constrained, binary optimization problems on quantum annealers (QA). The computational premise of quantum computers has cultivated the re-design of various existing vision problems into quantum-friendly forms. Experimental QA realizations can solve a particular non-convex problem known as the quadratic unconstrained binary optimization (QUBO). Yet a naive-QUBO cannot take into account the restrictions on the parameters. To introduce additional structure in the parameter space, researchers have crafted ad-hoc solutions incorporating (linear) constraints in the form of regularizers. However, this comes at the expense of a hyper-parameter, balancing the impact of regularization. To date, a true constrained solver of quadratic binary optimization (QBO) problems has lacked. Q-FW first reformulates constrained-QBO as a copositive program (CP), then employs Frank-Wolfe iterations to solve CP while satisfying linear (in)equality constraints. This procedure unrolls the original constrained-QBO into a set of unconstrained QUBOs all of which are solved, in a sequel, on a QA. We use D-Wave Advantage QA to conduct synthetic and real experiments on two important computer vision problems, graph matching and permutation synchronization, which demonstrate that our approach is effective in alleviating the need for an explicit regularization coefficient.

Point2Cyl: Reverse Engineering 3D Objects from Point Clouds to Extrusion Cylinders

We propose Point2Cyl, a supervised network transforming a raw 3D point cloud to a set of extrusion cylinders. Reverse engineering from a raw geometry to a CAD model is an essential task to enable manipulation of the 3D data in shape editing software and thus expand their usages in many downstream applications. Particularly, the form of CAD models having a sequence of extrusion cylinders -- a 2D sketch plus an extrusion axis and range -- and their boolean combinations is not only widely used in the CAD community/software but also has great expressivity of shapes, compared to having limited types of primitives (e.g., planes, spheres, and cylinders). In this work, we introduce a neural network that solves the extrusion cylinder decomposition problem in a geometry-grounded way by first learning underlying geometric proxies. Precisely, our approach first predicts per-point segmentation, base/barrel labels and normals, then estimates for the underlying extrusion parameters in differentiable and closed-form formulations. Our experiments show that our approach demonstrates the best performance on two recent CAD datasets, Fusion Gallery and DeepCAD, and we further showcase our approach on reverse engineering and editing.

Riemannian Functional Map Synchronization for Probabilistic Partial Correspondence in Shape Networks

Functional maps are efficient representations of shape correspondences, that provide matching of real-valued functions between pairs of shapes. Functional maps can be modelled as elements of the Lie group $SO(n)$ for nearly isometric shapes. Synchronization can subsequently be employed to enforce cycle consistency between functional maps computed on a set of shapes, hereby enhancing the accuracy of the individual maps. There is an interest in developing synchronization methods that respect the geometric structure of $SO(n)$, while introducing a probabilistic framework to quantify the uncertainty associated with the synchronization results. This paper introduces a Bayesian probabilistic inference framework on $SO(n)$ for Riemannian synchronization of functional maps, performs a maximum-a-posteriori estimation of functional maps through synchronization and further deploys a Riemannian Markov-Chain Monte Carlo sampler for uncertainty quantification. Our experiments demonstrate that constraining the synchronization on the Riemannian manifold $SO(n)$ improves the estimation of the functional maps, while our Riemannian MCMC sampler provides for the first time an uncertainty quantification of the results.

Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks

Disobeying the classical wisdom of statistical learning theory, modern deep neural networks generalize well even though they typically contain millions of parameters. Recently, it has been shown that the trajectories of iterative optimization algorithms can possess fractal structures, and their generalization error can be formally linked to the complexity of such fractals. This complexity is measured by the fractal's intrinsic dimension, a quantity usually much smaller than the number of parameters in the network. Even though this perspective provides an explanation for why overparametrized networks would not overfit, computing the intrinsic dimension (e.g., for monitoring generalization during training) is a notoriously difficult task, where existing methods typically fail even in moderate ambient dimensions. In this study, we consider this problem from the lens of topological data analysis (TDA) and develop a generic computational tool that is built on rigorous mathematical foundations. By making a novel connection between learning theory and TDA, we first illustrate that the generalization error can be equivalently bounded in terms of a notion called the 'persistent homology dimension' (PHD), where, compared with prior work, our approach does not require any additional geometrical or statistical assumptions on the training dynamics. Then, by utilizing recently established theoretical results and TDA tools, we develop an efficient algorithm to estimate PHD in the scale of modern deep neural networks and further provide visualization tools to help understand generalization in deep learning. Our experiments show that the proposed approach can efficiently compute a network's intrinsic dimension in a variety of settings, which is predictive of the generalization error.

Multiway Non-rigid Point Cloud Registration via Learned Functional Map Synchronization

We present SyNoRiM, a novel way to jointly register multiple non-rigid shapes by synchronizing the maps relating learned functions defined on the point clouds. Even though the ability to process non-rigid shapes is critical in various applications ranging from computer animation to 3D digitization, the literature still lacks a robust and flexible framework to match and align a collection of real, noisy scans observed under occlusions. Given a set of such point clouds, our method first computes the pairwise correspondences parameterized via functional maps. We simultaneously learn potentially non-orthogonal basis functions to effectively regularize the deformations, while handling the occlusions in an elegant way. To maximally benefit from the multi-way information provided by the inferred pairwise deformation fields, we synchronize the pairwise functional maps into a cycle-consistent whole thanks to our novel and principled optimization formulation. We demonstrate via extensive experiments that our method achieves a state-of-the-art performance in registration accuracy, while being flexible and efficient as we handle both non-rigid and multi-body cases in a unified framework and avoid the costly optimization over point-wise permutations by the use of basis function maps.