Convex Formulations for Training Two-Layer ReLU Neural Networks

Solving non-convex, NP-hard optimization problems is crucial for training machine learning models, including neural networks. However, non-convexity often leads to black-box machine learning models with unclear inner workings. While convex formulations have been used for verifying neural network robustness, their application to training neural networks remains less explored. In response to this challenge, we reformulate the problem of training infinite-width two-layer ReLU networks as a convex completely positive program in a finite-dimensional (lifted) space. Despite the convexity, solving this problem remains NP-hard due to the complete positivity constraint. To overcome this challenge, we introduce a semidefinite relaxation that can be solved in polynomial time. We then experimentally evaluate the tightness of this relaxation, demonstrating its competitive performance in test accuracy across a range of classification tasks.

Alignist: CAD-Informed Orientation Distribution Estimation by Fusing Shape and Correspondences

Object pose distribution estimation is crucial in robotics for better path planning and handling of symmetric objects. Recent distribution estimation approaches employ contrastive learning-based approaches by maximizing the likelihood of a single pose estimate in the absence of a CAD model. We propose a pose distribution estimation method leveraging symmetry respecting correspondence distributions and shape information obtained using a CAD model. Contrastive learning-based approaches require an exhaustive amount of training images from different viewpoints to learn the distribution properly, which is not possible in realistic scenarios. Instead, we propose a pipeline that can leverage correspondence distributions and shape information from the CAD model, which are later used to learn pose distributions. Besides, having access to pose distribution based on correspondences before learning pose distributions conditioned on images, can help formulate the loss between distributions. The prior knowledge of distribution also helps the network to focus on getting sharper modes instead. With the CAD prior, our approach converges much faster and learns distribution better by focusing on learning sharper distribution near all the valid modes, unlike contrastive approaches, which focus on a single mode at a time. We achieve benchmark results on SYMSOL-I and T-Less datasets.

UV-free Texture Generation with Denoising and Geodesic Heat Diffusions

Seams, distortions, wasted UV space, vertex-duplication, and varying resolution over the surface are the most prominent issues of the standard UV-based texturing of meshes. These issues are particularly acute when automatic UV-unwrapping techniques are used. For this reason, instead of generating textures in automatically generated UV-planes like most state-of-the-art methods, we propose to represent textures as coloured point-clouds whose colours are generated by a denoising diffusion probabilistic model constrained to operate on the surface of 3D objects. Our sampling and resolution agnostic generative model heavily relies on heat diffusion over the surface of the meshes for spatial communication between points. To enable processing of arbitrarily sampled point-cloud textures and ensure long-distance texture consistency we introduce a fast re-sampling of the mesh spectral properties used during the heat diffusion and introduce a novel heat-diffusion-based self-attention mechanism. Our code and pre-trained models are available at github.com/simofoti/UV3-TeD.

Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms

We present a novel set of rigorous and computationally efficient topology-based complexity notions that exhibit a strong correlation with the generalization gap in modern deep neural networks (DNNs). DNNs show remarkable generalization properties, yet the source of these capabilities remains elusive, defying the established statistical learning theory. Recent studies have revealed that properties of training trajectories can be indicative of generalization. Building on this insight, state-of-the-art methods have leveraged the topology of these trajectories, particularly their fractal dimension, to quantify generalization. Most existing works compute this quantity by assuming continuous- or infinite-time training dynamics, complicating the development of practical estimators capable of accurately predicting generalization without access to test data. In this paper, we respect the discrete-time nature of training trajectories and investigate the underlying topological quantities that can be amenable to topological data analysis tools. This leads to a new family of reliable topological complexity measures that provably bound the generalization error, eliminating the need for restrictive geometric assumptions. These measures are computationally friendly, enabling us to propose simple yet effective algorithms for computing generalization indices. Moreover, our flexible framework can be extended to different domains, tasks, and architectures. Our experimental results demonstrate that our new complexity measures correlate highly with generalization error in industry-standards architectures such as transformers and deep graph networks. Our approach consistently outperforms existing topological bounds across a wide range of datasets, models, and optimizers, highlighting the practical relevance and effectiveness of our complexity measures.

NeRF-Feat: 6D Object Pose Estimation using Feature Rendering

Object Pose Estimation is a crucial component in robotic grasping and augmented reality. Learning based approaches typically require training data from a highly accurate CAD model or labeled training data acquired using a complex setup. We address this by learning to estimate pose from weakly labeled data without a known CAD model. We propose to use a NeRF to learn object shape implicitly which is later used to learn view-invariant features in conjunction with CNN using a contrastive loss. While NeRF helps in learning features that are view-consistent, CNN ensures that the learned features respect symmetry. During inference, CNN is used to predict view-invariant features which can be used to establish correspondences with the implicit 3d model in NeRF. The correspondences are then used to estimate the pose in the reference frame of NeRF. Our approach can also handle symmetric objects unlike other approaches using a similar training setup. Specifically, we learn viewpoint invariant, discriminative features using NeRF which are later used for pose estimation. We evaluated our approach on LM, LM-Occlusion, and T-Less dataset and achieved benchmark accuracy despite using weakly labeled data.

Attending to Topological Spaces: The Cellular Transformer

Topological Deep Learning seeks to enhance the predictive performance of neural network models by harnessing topological structures in input data. Topological neural networks operate on spaces such as cell complexes and hypergraphs, that can be seen as generalizations of graphs. In this work, we introduce the Cellular Transformer (CT), a novel architecture that generalizes graph-based transformers to cell complexes. First, we propose a new formulation of the usual self- and cross-attention mechanisms, tailored to leverage incidence relations in cell complexes, e.g., edge-face and node-edge relations. Additionally, we propose a set of topological positional encodings specifically designed for cell complexes. By transforming three graph datasets into cell complex datasets, our experiments reveal that CT not only achieves state-of-the-art performance, but it does so without the need for more complex enhancements such as virtual nodes, in-domain structural encodings, or graph rewiring.

NRDF: Neural Riemannian Distance Fields for Learning Articulated Pose Priors

Faithfully modeling the space of articulations is a crucial task that allows recovery and generation of realistic poses, and remains a notorious challenge. To this end, we introduce Neural Riemannian Distance Fields (NRDFs), data-driven priors modeling the space of plausible articulations, represented as the zero-level-set of a neural field in a high-dimensional product-quaternion space. To train NRDFs only on positive examples, we introduce a new sampling algorithm, ensuring that the geodesic distances follow a desired distribution, yielding a principled distance field learning paradigm. We then devise a projection algorithm to map any random pose onto the level-set by an adaptive-step Riemannian optimizer, adhering to the product manifold of joint rotations at all times. NRDFs can compute the Riemannian gradient via backpropagation and by mathematical analogy, are related to Riemannian flow matching, a recent generative model. We conduct a comprehensive evaluation of NRDF against other pose priors in various downstream tasks, i.e., pose generation, image-based pose estimation, and solving inverse kinematics, highlighting NRDF's superior performance. Besides humans, NRDF's versatility extends to hand and animal poses, as it can effectively represent any articulation.

HyperSDFusion: Bridging Hierarchical Structures in Language and Geometry for Enhanced 3D Text2Shape Generation

3D shape generation from text is a fundamental task in 3D representation learning. The text-shape pairs exhibit a hierarchical structure, where a general text like "chair" covers all 3D shapes of the chair, while more detailed prompts refer to more specific shapes. Furthermore, both text and 3D shapes are inherently hierarchical structures. However, existing Text2Shape methods, such as SDFusion, do not exploit that. In this work, we propose HyperSDFusion, a dual-branch diffusion model that generates 3D shapes from a given text. Since hyperbolic space is suitable for handling hierarchical data, we propose to learn the hierarchical representations of text and 3D shapes in hyperbolic space. First, we introduce a hyperbolic text-image encoder to learn the sequential and multi-modal hierarchical features of text in hyperbolic space. In addition, we design a hyperbolic text-graph convolution module to learn the hierarchical features of text in hyperbolic space. In order to fully utilize these text features, we introduce a dual-branch structure to embed text features in 3D feature space. At last, to endow the generated 3D shapes with a hierarchical structure, we devise a hyperbolic hierarchical loss. Our method is the first to explore the hyperbolic hierarchical representation for text-to-shape generation. Experimental results on the existing text-to-shape paired dataset, Text2Shape, achieved state-of-the-art results.

Position Paper: Challenges and Opportunities in Topological Deep Learning

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

TopoX: A Suite of Python Packages for Machine Learning on Topological Domains

We introduce topox, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. topox consists of three packages: toponetx facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; topoembedx provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; topomodelx is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of topox is available under MIT license at https://github.com/pyt-team.