# Neural Radiance Fields with Hash-Low-Rank Decomposition
## 1 Introduction  
High-quality rendering at novel viewpoints remains a pivotal challenge in both computer vision and computer graphics. Traditional explicit 3D representations, such as point clouds [1; 2; 3], voxels [4; 5; 6; 7], and meshes [8; 9], are widely used in industry due to their convenience. However, these representations struggle to balance information quality and storage size. In addition, constructing high-quality data with these methods requires substantial human effort, which conflicts with the efficiency-driven demands of modern production. In recent years, implicit neural representation [7; 10] (INR) has demonstrated immense research value and application potential in computer graphics and vision. One of the most representative works, neural radiance field [11] (NeRF) revolutionized the paradigm for novel view synthesis shortly after its introduction. NeRF stores volume density and color in a 3D field and trains a multi-layer perceptron (MLP) to learn the function of this field. By designing a coarse-to-fine training framework and using positional encoding, NeRF achieves compact 3D scene representation and photo-realistic rendering with relatively small memory usage, showcasing its superiority over traditional 3D representations. Compared to purely explicit methods, implicit methods like NeRF can simultaneously achieve high-quality geometry and appearance, whereas explicit learning methods often struggle to effectively decouple the two. However, despite NeRF's effective balance between storage size and information quality, its forward rendering process is slow due to numerous deep MLP queries. During the rendering process, each ray requires a considerable number of points to query the MLP, and the total number of rays often reaches thousands or more. Meanwhile, the compact MLP implies highly compressed scene information, necessitating longer training times and more iterations for gradient descent. These factors result in traditional NeRF requiring a day or even longer to complete training. To accelerate NeRF's rendering speed, recent approaches have opted to explicitly store part of the scene information using learnable parameters to store scenes in voxels [4], point clouds [2; 12], or other 3D representations [5; 6] while using a small implicit MLP tointegrate and decode the information. This combination of explicit and implicit methods achieves faster rendering speeds. Due to the increase in learnable parameters and the powerful fitting capability of MLPs, these methods outperform pure MLP-based NeRF in both fitting speed and rendering quality. These methods can be categorized into dense voxel grids [4], sparse voxel grids [13], octrees [5; 6], and hash tables [14]. Among them, Instant-NGP [14] employs multi-level hash encoding and specific spatial hash functions, significantly improving training speed and rendering quality with fewer parameters than NeRF, making it one of the most representative NeRF improvements. TensoRF [13], on the other hand, uses tensor low-rank decomposition on dense voxel grids, decomposing the 3D feature grid into multiple low-rank or rank-one components, and it uses matrix multiplication to fuse features extracted from each component, achieving compact scene reconstruction. However, both methods still exhibit some feature wastage and contention. In multi-hashing methods, there is no clear strategy for resolving hash collisions. This can cause multiple high-frequency signal regions to share the same hash table entries, leading to them competing for features during the gradient descent process. As a result, these signals may not be effectively captured; in TensoRF, the VM decomposition for high-quality rendering may result in high-rank component information redundancy, such as parameters being occupied by blank regions at the edges of the scene.