Fast Frechet Distance Between Curves With Long Edges

Computing the Fr\'echet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fr\'echet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fr\'echet distance between them: (1) a linear-time algorithm for deciding the Fr\'echet distance between two curves, (2) an algorithm that computes the Fr\'echet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve

Reducing Computational and Statistical Complexity in Machine Learning Through Cardinality Sparsity

High-dimensional data has become ubiquitous across the sciences but causes computational and statistical challenges. A common approach for dealing with these challenges is sparsity. In this paper, we introduce a new concept of sparsity, called cardinality sparsity. Broadly speaking, we call a tensor sparse if it contains only a small number of unique values. We show that cardinality sparsity can improve deep learning and tensor regression both statistically and computationally. On the way, we generalize recent statistical theories in those fields

Unfolding simple chains inside circles

It is an open problem to determined whether a polygonal chain can be straightened inside a confi ning region if its links are not allowed to cross. In this paper we propose a special case: whether a polygonal chain can be straightened inside a circle without allowing its links to cross. We prove that this is possible if the straightened confi guration can fi t within circle. Then we show that these simple chains have just one equivalence class of confi gurations

Semi-supervised Vector-Quantization in Visual SLAM using HGCN

In this paper, two semi-supervised appearance based loop closure detection technique, HGCN-FABMAP and HGCN-BoW are introduced. Furthermore an extension to the current state of the art localization SLAM algorithm, ORB-SLAM, is presented. The proposed HGCN-FABMAP method is implemented in an off-line manner incorporating Bayesian probabilistic schema for loop detection decision making. Specifically, we let a Hyperbolic Graph Convolutional Neural Network (HGCN) to operate over the SURF features graph space, and perform vector quantization part of the SLAM procedure. This part previously was performed in an unsupervised manner using algorithms like HKmeans, kmeans++,..etc. The main Advantage of using HGCN, is that it scales linearly in number of graph edges. Experimental results shows that HGCN-FABMAP algorithm needs far more cluster centroids than HGCN-ORB, otherwise it fails to detect loop closures. Therefore we consider HGCN-ORB to be more efficient in terms of memory consumption, also we conclude the superiority of HGCN-BoW and HGCN-FABMAP with respect to other algorithms