Hubness reduction improves clustering and trajectory inference in single-cell transcriptomic data

Background. Single-cell RNA-seq datasets are characterized by large ambient dimensionality, and their analyses, such as clustering or cell trajectory inference, can be affected by various manifestations of the dimensionality curse. One of these manifestations is the hubness phenomenon, i.e. existence of data points with surprisingly large incoming connectivity degree in the neighbourhood graph. Conventional approach to dampen the unwanted effects of high dimension consists in applying drastic dimensionality reduction, which is critical especially in scRNA-seq. It remains unexplored if this step can be avoided thus retaining more information than contained in the low-dimensional projections, by correcting directly an effect of the high dimension. Results. We investigate the phenomenon of hubness in scRNA-seq data, and its manifestation in spaces of increasing dimensionality. We also link increasing hubness to increased levels of dropout in sequencing data. We show that hub cells do not represent any visible technical or biological bias. The effect of various hubness reduction methods is investigated with respect to the visualization, clustering and trajectory inference tasks in scRNA-seq datasets. We show that applying hubness reduction generates neighbourhood graphs with properties more suitable for applying machine learning methods; and that hubness reduction outperforms other state-of-the-art methods for improving neighbourhood graphs. As a consequence, clustering, trajectory inference and visualisation perform better, especially for datasets characterized by large intrinsic dimensionality. Conclusion. Hubness is an important phenomenon in sequencing data. Reducing hubness can be beneficial for the analysis of scRNA-seq data characterized by large intrinsic dimensionality in which case it can be used as an alternative to drastic dimensionality reduction.

Graphical One-Sided Markets

We study the problem of allocating indivisible objects to a set of rational agents where each agent's final utility depends on the intrinsic valuation of the allocated item as well as the allocation within the agent's local neighbourhood. We specify agents' local neighbourhood in terms of a weighted graph. This extends the model of one-sided markets to incorporate neighbourhood externalities. We consider the solution concept of stability and show that, unlike in the case of one-sided markets, stable allocations may not always exist. When the underlying local neighbourhood graph is symmetric, a 2-stable allocation is guaranteed to exist and any decentralised mechanism where pairs of rational players agree to exchange objects terminates in such an allocation. We show that computing a 2-stable allocation is PLS-complete and further identify subclasses which are tractable. In the case of asymmetric neighbourhood structures, we show that it is NP-complete to check if a 2-stable allocation exists. We then identify structural restrictions where stable allocations always exist and can be computed efficiently. Finally, we study the notion of envy-freeness in this framework.

A HYPERSPECTRAL IMAGE CLASSIFICATION METHOD USING ISOMAP AND RVM

Classification is one of the most significant applications of hyperspectral image processing and even remote sensing. Though various algorithms have been proposed to implement and improve this application, there are still drawbacks in traditional classification methods. Thus further investigations on some aspects, such as dimension reduction, data mining, and rational use of spatial information, should be developed. In this paper, we used a widely utilized global manifold learning approach, isometric feature mapping (ISOMAP), to address the intrinsic nonlinearities of hyperspectral image for dimension reduction. Considering the impropriety of Euclidean distance in spectral measurement, we applied spectral angle (SA) for substitute when constructed the neighbourhood graph. Then, relevance vector machines (RVM) was introduced to implement classification instead of support vector machines (SVM) for simplicity, generalization and sparsity. Therefore, a probability result could be obtained rather than a less convincing binary result. Moreover, taking into account the spatial information of the hyperspectral image, we employ a spatial vector formed by different classes’ ratios around the pixel. At last, we combined the probability results and spatial factors with a criterion to decide the final classification result. To verify the proposed method, we have implemented multiple experiments with standard hyperspectral images compared with some other methods. The results and different evaluation indexes illustrated the effectiveness of our method.

Large Cuts with Local Algorithms on Triangle-Free Graphs

Let $G$ be a $d$-regular triangle-free graph with $m$ edges. We present an algorithm which finds a cut in $G$ with at least $(1/2 + 0.28125/\sqrt{d})m$ edges in expectation, improving upon Shearer's classic result. In particular, this implies that any $d$-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of $G$.Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques. We show that for any fixed $d$, there exists a weighted neighbourhood graph $\mathcal{N}_d$ such that there is a one-to-one correspondence between heavy cuts of $\mathcal{N}_d$ and randomised local algorithms that find large cuts in any $d$-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in $d$-regular graphs: we can compute the optimal cut of $\mathcal{N}_d$ to attain a lower bound on the maximum cut size of any $d$-regular triangle-free graph.

Object Extraction Using Topological Models from Complex Scene Image

Efficient and effective object recognition from a multimedia data are very complex. Automatic object segmentation is usually very hard for natural images; interactive schemes with a few simple markers provide feasible solutions. In this chapter, we propose topological model based region merging. In this work, we will focus on topological models like, Relative Neighbourhood Graph (RNG) and Gabriel graph (GG), etc. From the Initial segmented image, we constructed a neighbourhood graph represented different regions as the node of graph and weight of the edges are the value of dissimilarity measures function for their colour histogram vectors. A method of similarity based region merging mechanism (supervised and unsupervised) is proposed to guide the merging process with the help of markers. The region merging process is adaptive to the image content and it does not need to set the similarity threshold in advance. To the validation of proposed method extensive experiments are performed and the result shows that the proposed method extracts the object contour from the complex background.

An improved Isomap method for manifold learning

Purpose Isometric feature mapping (Isomap) is a very popular manifold learning method and is widely used in dimensionality reduction and data visualization. The most time-consuming step in Isomap is to compute the shortest paths between all pairs of data points based on a neighbourhood graph. The classical Isomap (C-Isomap) is very slow, due to the use of Floyd’s algorithm to compute the shortest paths. The purpose of this paper is to speed up Isomap. Design/methodology/approach Through theoretical analysis, it is found that the neighbourhood graph in Isomap is sparse. In this case, the Dijkstra’s algorithm with Fibonacci heap (Fib-Dij) is faster than Floyd’s algorithm. In this paper, an improved Isomap method based on Fib-Dij is proposed. By using Fib-Dij to replace Floyd’s algorithm, an improved Isomap method is presented in this paper. Findings Using the S-curve, the Swiss-roll, the Frey face database, the mixed national institute of standards and technology database of handwritten digits and a face image database, the performance of the proposed method is compared with C-Isomap, showing the consistency with C-Isomap and marked improvements in terms of the high speed. Simulations also demonstrate that Fib-Dij reduces the computation time of the shortest paths from O(N3) to O(N2lgN). Research limitations/implications Due to the limitations of the computer, the sizes of the data sets in this paper are all smaller than 3,000. Therefore, researchers are encouraged to test the proposed algorithm on larger data sets. Originality/value The new method based on Fib-Dij can greatly improve the speed of Isomap.

Robust L-Isomap with a Novel Landmark Selection Method

Isomap is a widely used nonlinear method for dimensionality reduction. Landmark-Isomap (L-Isomap) has been proposed to improve the scalability of Isomap. In this paper, we focus on two important issues that were not taken into account in L-Isomap, landmark point selection and topological stability. At first, we present a novel landmark point selection method. It first uses a greedy strategy to select some points as landmark candidates and then removes the candidate points that are neighbours of other candidates. The remaining candidate points are the landmark points. The selection method can promote the computation efficiency without sacrificing accuracy. For the topological stability, we define edge density for each edge in the neighbourhood graph. According to the geometrical characteristic of the short-circuit edges, we provide a method to eliminate the short-circuit edge without breaking the data integrity. The approach that integrates L-Isomap with these two improvements is referred to as Robust L-Isomap (RL-Isomap). The effective performance of RL-Isomap is confirmed through several numerical experiments.