Neural 3D Morphable Models: Spiral Convolutional Networks for 3D Shape Representation Learning and Generation

Generative models for 3D geometric data arise in many important applications in 3D computer vision and graphics. In this paper, we focus on 3D deformable shapes that share a common topological structure, such as human faces and bodies. Morphable Models and their variants, despite their linear formulation, have been widely used for shape representation, while most of the recently proposed nonlinear approaches resort to intermediate representations, such as 3D voxel grids or 2D views. In this work, we introduce a novel graph convolutional operator, acting directly on the 3D mesh, that explicitly models the inductive bias of the fixed underlying graph. This is achieved by enforcing consistent local orderings of the vertices of the graph, through the spiral operator, thus breaking the permutation invariance property that is adopted by all the prior work on Graph Neural Networks. Our operator comes by construction with desirable properties (anisotropic, topology-aware, lightweight, easy-to-optimise), and by using it as a building block for traditional deep generative architectures, we demonstrate state-of-the-art results on a variety of 3D shape datasets compared to the linear Morphable Model and other graph convolutional operators.Comment: to appear at ICCV 201

Accelerating Generic Graph Neural Networks via Architecture, Compiler, Partition Method Co-Design

Graph neural networks (GNNs) have shown significant accuracy improvements in a variety of graph learning domains, sparking considerable research interest. To translate these accuracy improvements into practical applications, it is essential to develop high-performance and efficient hardware acceleration for GNN models. However, designing GNN accelerators faces two fundamental challenges: the high bandwidth requirement of GNN models and the diversity of GNN models. Previous works have addressed the first challenge by using more expensive memory interfaces to achieve higher bandwidth. For the second challenge, existing works either support specific GNN models or have generic designs with poor hardware utilization. In this work, we tackle both challenges simultaneously. First, we identify a new type of partition-level operator fusion, which we utilize to internally reduce the high bandwidth requirement of GNNs. Next, we introduce partition-level multi-threading to schedule the concurrent processing of graph partitions, utilizing different hardware resources. To further reduce the extra on-chip memory required by multi-threading, we propose fine-grained graph partitioning to generate denser graph partitions. Importantly, these three methods make no assumptions about the targeted GNN models, addressing the challenge of model variety. We implement these methods in a framework called SwitchBlade, consisting of a compiler, a graph partitioner, and a hardware accelerator. Our evaluation demonstrates that SwitchBlade achieves an average speedup of $1.85\times$ and energy savings of $19.03\times$ compared to the NVIDIA V100 GPU. Additionally, SwitchBlade delivers performance comparable to state-of-the-art specialized accelerators

The 1/N1/N expansion of tensor models with two symmetric tensors

It is well known that tensor models for a tensor with no symmetry admit a $1/N$ expansion dominated by melonic graphs. This result relies crucially on identifying \emph{jackets} which are globally defined ribbon graphs embedded in the tensor graph. In contrast, no result of this kind has so far been established for symmetric tensors because global jackets do not exist. In this paper we introduce a new approach to the $1/N$ expansion in tensor models adapted to symmetric tensors. In particular we do not use any global structure like the jackets. We prove that, for any rank $D$, a tensor model with two symmetric tensors and interactions the complete graph $K_{D+1}$ admits a $1/N$ expansion dominated by melonic graphs.Comment: misprints corrected, references adde

Asymptotic expansion of the multi-orientable random tensor model

Three-dimensional random tensor models are a natural generalization of the celebrated matrix models. The associated tensor graphs, or 3D maps, can be classified with respect to a particular integer or half-integer, the degree of the respective graph. In this paper we analyze the general term of the asymptotic expansion in N, the size of the tensor, of a particular random tensor model, the multi-orientable tensor model. We perform their enumeration and we establish which are the dominant configurations of a given degree.Comment: 27 pages, 24 figures, several minor modifications have been made, one figure has been added; accepted for publication in "Electronic Journal of Combinatorics

Dimer Models from Mirror Symmetry and Quivering Amoebae

Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T^2 subspace of the T^3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume.Comment: 55 pages, 27 figures, LaTeX2