Existing losses such as the eikonal loss is to ensure that the implicit function indeed outputs cannot guarantee the recovered implicit function to be a the signed distance. A standard regularization loss used is distance function, even when the implicit function satisfies the eikonal equation: it constrains the norm of the gradient the eikonal equation almost everywhere. Furthermore, the of an implicit function to be 1 almost everywhere. If eikonal loss suffers from stability issues in optimization and the implicit function is a signed distance function, then it the remedies that introduce area or divergence minimization satisfies the eikonal equation. However, the converse is not can lead to oversmoothing. We address these challenges true. Figure 1 shows an example: on the left, we optimize by designing a loss function that when minimized can an implicit function to satisfy the eikonal equation, while converge to the true distance function, is stable, and naturally it successfully does so, it converges to a solution that is far penalize large surface area. We provide theoretical from the actual distance [5, 6].

Cooperative co-evolution (CC) algorithms, based on the divide-and-conquer strategy, have emerged as the predominant approach to solving large-scale global optimization (LSGO) problems. The efficiency and accuracy of the grouping stage significantly impact the performance of the optimization process. While the general separability grouping (GSG) method has overcome the limitation of previous differential grouping (DG) methods by enabling the decomposition of non-additively separable functions, it suffers from high computational complexity. To address this challenge, this article proposes a composite separability grouping (CSG) method, seamlessly integrating DG and GSG into a problem decomposition framework to utilize the strengths of both approaches. CSG introduces a step-by-step decomposition framework that accurately decomposes various problem types using fewer computational resources. By sequentially identifying additively, multiplicatively and generally separable variables, CSG progressively groups non-separable variables by recursively considering the interactions between each non-separable variable and the formed non-separable groups. Furthermore, to enhance the efficiency and accuracy of CSG, we introduce two innovative methods: a multiplicatively separable variable detection method and a non-separable variable grouping method. These two methods are designed to effectively detect multiplicatively separable variables and efficiently group non-separable variables, respectively. Extensive experimental results demonstrate that CSG achieves more accurate variable grouping with lower computational complexity compared to GSG and state-of-the-art DG series designs.

We present new insights and a novel paradigm (StEik) for learning implicit neural representations (INR) of shapes. In particular, we shed light on the popular eikonal loss used for imposing a signed distance function constraint in INR. We show analytically that as the representation power of the network increases, the optimization approaches a partial differential equation (PDE) in the continuum limit that is unstable. We show that this instability can manifest in existing network optimization, leading to irregularities in the reconstructed surface and/or convergence to sub-optimal local minima, and thus fails to capture fine geometric and topological structure. We show analytically how other terms added to the loss, currently used in the literature for other purposes, can actually eliminate these instabilities. However, such terms can over-regularize the surface, preventing the representation of fine shape detail. Based on a similar PDE theory for the continuum limit, we introduce a new regularization term that still counteracts the eikonal instability but without over-regularizing. Furthermore, since stability is now guaranteed in the continuum limit, this stabilization also allows for considering new network structures that are able to represent finer shape detail. We introduce such a structure based on quadratic layers. Experiments on multiple benchmark data sets show that our new regularization and network are able to capture more precise shape details and more accurate topology than existing state-of-the-art.

Subspace clustering is a class of extensively studied clustering methods and the spectral-type approaches are its important subclass whose key first step is to learn a coefficient matrix with block diagonal structure. To realize this step, sparse subspace clustering (SSC), low rank representation (LRR) and block diagonal representation (BDR) were successively proposed and have become the state-of-the-arts (SOTAs). Among them, the former two minimize their convex objectives by imposing sparsity and low rankness on the coefficient matrix respectively, but so-desired block diagonality cannot neccesarily be guaranteed practically while the latter designs a block diagonal matrix induced regularizer but sacrifices convexity. For solving this dilemma, inspired by Convex Biclustering, in this paper, we propose a simple yet efficient spectral-type subspace clustering method named Adaptive Block Diagonal Representation (ABDR) which strives to pursue so-desired block diagonality as BDR by coercively fusing the columns/rows of the coefficient matrix via a specially designed convex regularizer, consequently, ABDR naturally enjoys their merits and can adaptively form more desired block diagonality than the SOTAs without needing to prefix the number of blocks as done in BDR. Finally, experimental results on synthetic and real benchmarks demonstrate the superiority of ABDR.