Testing Small Set Expansion in General Graphs

We consider the problem of testing small set expansion for general graphs. A graph $G$ is a $(k,\phi)$-expander if every subset of volume at most $k$ has conductance at least $\phi$. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes. We give testers with two-sided error and one-sided error in the adjacency list model that allows degree and neighbor queries to the oracle of the input graph. The testers take as input an $n$-vertex graph $G$, a volume bound $k$, an expansion bound $\phi$ and a distance parameter $\varepsilon>0$. For the two-sided error tester, with probability at least $2/3$, it accepts the graph if it is a $(k,\phi)$-expander and rejects the graph if it is $\varepsilon$-far from any $(k^*,\phi^*)$-expander, where $k^*=\Theta(k\varepsilon)$ and $\phi^*=\Theta(\frac{\phi^4}{\min\{\log(4m/k),\log n\}\cdot(\ln k)})$. The query complexity and running time of the tester are $\widetilde{O}(\sqrt{m}\phi^{-4}\varepsilon^{-2})$, where $m$ is the number of edges of the graph. For the one-sided error tester, it accepts every $(k,\phi)$-expander, and with probability at least $2/3$, rejects every graph that is $\varepsilon$-far from $(k^*,\phi^*)$-expander, where $k^*=O(k^{1-\xi})$ and $\phi^*=O(\xi\phi^2)$ for any $0<\xi<1$. The query complexity and running time of this tester are $\widetilde{O}(\sqrt{\frac{n}{\varepsilon^3}}+\frac{k}{\varepsilon \phi^4})$. We also give a two-sided error tester with smaller gap between $\phi^*$ and $\phi$ in the rotation map model that allows (neighbor, index) queries and degree queries.Comment: 23 pages; STACS 201

Characterization of the neuroendocrine pancreatic tumors nature by MDCT enhancement pattern: a radio-pathological correlation

Introduction Pre-operative suspicion of neuroendocrine pancreatic lesions nature arises both from clinical (presence and the type of secreted hormone) and imaging findings. However, imaging suggestion of lesion nature is based quite only on nodular dimension and on the presence of local and distant spreading. Aim of the study was to determine the nature of neuroendocrine pancreatic lesions by analysing lesions enhancement pattern at MDCT and by comparing it with histological findings, including the MVD. Materials and Methods We included 45 patients submitted to surgical resection for pancreatic neuroendocrine tumor. All preoperative CT examinations were performed by a multidetector CT. Post-contrastographic study included 4 phases: early arterial (delay 15-20”), pancreatic (delay 35”), venous (delay 70”) and late phases (delay 180”). Two different patterns of enhancement were defined: pattern A, including lesions showing early enhancement (during early arterial or pancreatic phase) and a rapid wash-out; pattern B, including lesions with wash-in in the early arterial or pancreatic phase with no wash-out nor in the late phase (pattern B1), and lesions showing enhancement only in the venous and/or late phases (pattern B2). Results 66 lesions were detected (30 pattern A, 26 B1 and 10 B2). At pathology 28 lesions were adenomas, 14 borderline and 24 carcinomas: 24/30 lesions showing pattern A were benign, 5 borderline and 1 carcinoma; 23/36 lesions showing pattern B were carcinomas, 9 borderline and 4 adenomas. Among the 26 B1 lesions, 13 were carcinomas, 9 borderline and 4 adenomas, while all 10 B2 lesions were malignant. Pattern A showed PPV of benignancy of 80%, and pattern B NPV of benignancy of 89%. MVD was evaluated in 22 lesions obtaining significant differences among the 3 histological and the 3 enhancement pattern. Significant differences between B1 and B2 malignant lesions existed by considering metastases (only B2 lesions) and fibrosis (all B2 lesions). Conclusion The enhancement pattern at CT is related to MVD and the histological type, thus representing a further criterium for suggesting nature of neuroendocrine lesions. The low MVD of B2 lesions, associated with the presence of fibrosis, may justify the delayed enhancement of these lesions

Consensus graph and spectral representation for one-step multi-view kernel based clustering

Recently, multi-view clustering has received much attention in the fields of machine learning and pattern recognition. Spectral clustering for single and multiple views has been the common solution. Despite its good clustering performance, it has a major limitation: it requires an extra step of clustering. This extra step, which could be the famous k-means clustering, depends heavily on initialization, which may affect the quality of the clustering result. To overcome this problem, a new method called Multiview Clustering via Consensus Graph Learning and Nonnegative Embedding (MVCGE) is presented in this paper. In the proposed approach, the consensus affinity matrix (graph matrix), consensus representation and cluster index matrix (nonnegative embedding) are learned simultaneously in a unified framework. Our proposed method takes as input the different kernel matrices corresponding to the different views. The proposed learning model integrates two interesting constraints: (i) the cluster indices should be as smooth as possible over the consensus graph and (ii) the cluster indices are set to be as close as possible to the graph convolution of the consensus representation. In this approach, no post-processing such as k-means or spectral rotation is required. Our approach is tested with real and synthetic datasets. The experiments performed show that the proposed method performs well compared to many state-of-the-art approaches

Trees with Given Stability Number and Minimum Number of Stable Sets

We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of size $\lceil \frac{n-1}{n-\alpha} \rceil$ or $\lfloor \frac{n-1}{n-\alpha}\rfloor$, so that every vertex is included in at most two distinct stars, and the centers of these stars form a stable set of the tree.Comment: v2: Referees' comments incorporate

GSLAM: Initialization-robust Monocular Visual SLAM via Global Structure-from-Motion

Many monocular visual SLAM algorithms are derived from incremental structure-from-motion (SfM) methods. This work proposes a novel monocular SLAM method which integrates recent advances made in global SfM. In particular, we present two main contributions to visual SLAM. First, we solve the visual odometry problem by a novel rank-1 matrix factorization technique which is more robust to the errors in map initialization. Second, we adopt a recent global SfM method for the pose-graph optimization, which leads to a multi-stage linear formulation and enables L1 optimization for better robustness to false loops. The combination of these two approaches generates more robust reconstruction and is significantly faster (4X) than recent state-of-the-art SLAM systems. We also present a new dataset recorded with ground truth camera motion in a Vicon motion capture room, and compare our method to prior systems on it and established benchmark datasets.Comment: 3DV 2017 Project Page: https://frobelbest.github.io/gsla