Implicit Neural Representations (INRs) rely heavily on architectural choices for good generalization. Developing theoretically grounded approaches for architecture design remains an active area of research. Via theoretical analysis of the infinite-width limit, we establish a methodology that characterizes INR’s generalization by means of kernel alignment. We first formulate the optimal kernel that minimizes pointwise expected squared error, then demonstrate that the Neural Tangent Kernel of the composed function (INR with input encoding) can approximate any positive semidefinite dot-product kernels through input feature mapping adjustments. Building upon these insights, we propose a Kernel Alignment Regularizer (KAR) that naturally integrates with existing INR systems to enhance kernel alignment. We further develop Plug-in Encoding for Aligned Kernels (PEAK) to refine INR models with KAR using learnable input encoding. This work contributes to the ongoing research efforts in bridging theory and practice for principled INR architecture design. Code is available at https://github.com/lizhemin15/KAR.

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can solely represent multi-dimensional discrete data on meshgrid, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR) parameterized by multilayer perceptrons (MLPs), which can continuously represent data beyond meshgrid with powerful representation abilities. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization, and utilize MLPs to paramterize factor functions of the tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.

This paper proposes a regularizer called Implicit Neural Representation Regularizer (INRR) to improve the generalization ability of the Implicit Neural Representation (INR). The INR is a fully connected network that can represent signals with details not restricted by grid resolution. However, its generalization ability could be improved, especially with nonuniformly sampled data. The proposed INRR is based on learned Dirichlet Energy (DE) that measures similarities between rows/columns of the matrix. The smoothness of the Laplacian matrix is further integrated by parameterizing DE with a tiny INR. INRR improves the generalization of INR in signal representation by perfectly integrating the signal's self-similarity with the smoothness of the Laplacian matrix. Through well-designed numerical experiments, the paper also reveals a series of properties derived from INRR, including momentum methods like convergence trajectory and multi-scale similarity. Moreover, the proposed method could improve the performance of other signal representation methods.

Abstract It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low‐rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorisation component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. In this work, a regularised deep matrix factorised (RDMF) model for image restoration is proposed, which utilises the implicit bias of the low rank of deep neural networks and the explicit bias of total variation. RDMF is a powerful and flexible framework for inverse problems in image processing while the combination of implicit and explicit regularisation represents the intrinsic characteristics of a natural image. The effectiveness of the RDMF model with extensive experiments are demonstrated, in which the method surpasses the recently proposed models in common examples, especially for the restoration from very few observations. This work sheds light on a more general framework for solving other inverse problems by combining the implicit bias of deep learning with explicit regularisation.

The explicit low-rank regularization, e.g., nuclear norm regularization, has been widely used in imaging sciences. However, it has been found that implicit regularization outperforms explicit ones in various image processing tasks. Another issue is that the fixed explicit regularization limits the applicability to broad images since different images favor different features captured by different explicit regularizations. As such, this paper proposes a new adaptive and implicit low-rank regularization that captures the low-rank prior dynamically from the training data. The core of our new adaptive and implicit low-rank regularization is parameterizing the Laplacian matrix in the Dirichlet energy-based regularization, which we call the adaptive and implicit regularization (AIR). Theoretically, we show that the adaptive regularization of AIR enhances the implicit regularization and vanishes at the end of training. We validate AIR's effectiveness on various benchmark tasks, indicating that the AIR is particularly favorable for the scenarios when the missing entries are nonuniform. The code can be found at https://github.com/lizhemin15/AIR-Net.

User-generated content (UGC) such as blogs and twitters are exploding in modern Internet services. In such systems, recommender systems are needed to help people filter vast amount of UGC generated by other users. However, traditional rec-ommendation models do not use user authorship of items. In this paper, we show that with this additional information, we can significantly improve the performance of recommendations. A generative model that combines hierarchical topic modeling and matrix factorization is proposed. Empirical results show that our model outperforms other state-of-the-art models, and can provide interpretable topic structures for users and items. Furthermore, since user interests can be inferred from their productions, rec-ommendations can be made for users that do not have any ratings to solve the cold-start problem.