Multi-dimensional Scaling from K-Nearest Neighbourhood Distances

Abstract

Multi-dimensional scaling (MDS) with incomplete distance information represents a significant challenging inverse problem in computational geometry. This technique finds expensive applications in the fields of surface, manifold, and cubicle reconstructions, and is also relevant in the context of social networks. While a majority of existing methodologies tend to provide accurate results primarily when the missing distance indices are chosen randomly or when the omission rate is below 50%, our research proposes an innovative approach. We present a robust MDS framework when distances to the k-nearest neighbors (kNN) are known, even in situations characterized by a high coherence of missing indices. Our proposed strategy starts with a local reconstruction phase based on local correlation. Subsequently, the global reconstruction phase is realized through two distinct models: one based on low-rank semi-definite programming (SDP) and the other rooted in a model utilizing the Frobenius norm. Throughout the global reconstruction, we incorporate the alternating direction method of multipliers (ADMM) and the Riemann gradient descent algorithm (RGrad). Numerical Simulations have demonstrated that for MDS from kNN distances, our proposed model and algorithm outperforms the existed SDP models in terms of the visual effect and error of Gram matrix. We further validate that our approach can reconstruct surfaces from as mere as 1% of kNN distances, which shows that the proposed model is robust to the high coherence of missing indices. Additionally, we propose another MDS model which is applicable from kNN distances with additive noise.