Landing Probabilities of Random Walks for Seed-Set Expansion in Hypergraphs Eli Chien Pan Li Olgica Milenkovic Department ECE, UIUC Department ECE, UIUC Department ECE, UIUC Abstract We describe the first known mean-field study of landing probabilities for random walks on hypergraphs. In particular, we examine clique-expansion and tensor methods and evaluate their mean-field characteristics over a class of random hypergraph models for the purpose of seed-set community expansion. We describe parameter regimes in which the two methods outperform each other and propose a hybrid expansion method that uses partial clique-expansion to reduce the projection distortion and low-complexity tensor methods applied directly on the partially expanded hypergraphs. 1 1 Introduction Random walks on graphs are Markov random processes in which given a starting vertex, one moves to a randomly selected neighbor and then repeats the procedure starting from the newly selected vertex [1]. Random walks are used in many graph-based learning algorithms such as PageRank [2] and Label Propagating [3], and they have found a variety of applications in local community detection [4, 5], information retrieval [2] and semi-supervised learning [3]. Random walks are also frequently used to characterize the topological structure of graphs via the hitting time of a vertex from a seed, the commute time between two vertices [6] and the mixing time which also characterizes global graph connectivity [7]. Recently, a new measure of vertex connectivity and similarity, termed a landing probability (LP), was introduced in [8]. A1 Eli Chien and Pan Li contribute equally to this work.Preprint version. LP of a vertex is the probability of a random walk ending at the vertex after making a certain number of steps. Different linear combinations of LPs give rise to different forms of PageRanks (PRs), such as the standard PR [2] and the heat-kernel PR [9], both used for various graph clustering tasks. In particular, Kloumann et al. [8] also initiated the analysis of PRs based on LPs for seed-based community detection. Under the assumption of a generative stochastic block model (SBM) [10] with two blocks, the authors of [8] proved that the empirical average of LPs within the seed community concentrates around a deterministic centroid. Similarly, the empirical averages of LPs outside the seed community also concentrate around another deterministic centroid.

For high-dimensional classification, it is well known that naively performing the Fisher discriminant rule leads to poor results due to diverging spectra and noise accumulation. Therefore, researchers proposed independence rules to circumvent the diverse spectra, and sparse independence rules to mitigate the issue of noise accumulation. However, in biological applications, there are often a group of correlated genes responsible for clinical outcomes, and the use of the covariance information can significantly reduce misclassification rates. The extent of such error rate reductions is unveiled by comparing the misclassification rates of the Fisher discriminant rule and the independence rule. To materialize the gain based on finite samples, a Regularized Optimal Affine Discriminant (ROAD) is proposed based on a covariance penalty. ROAD selects an increasing number of features as the penalization relaxes. Further benefits can be achieved when a screening method is employed to narrow the feature pool before hitting the ROAD. An efficient Constrained Coordinate Descent algorithm (CCD) is also developed to solve the associated optimization problems. Sampling properties of oracle type are established. Simulation studies and real data analysis support our theoretical results and demonstrate the advantages of the new classification procedure under a variety of correlation structures. A delicate result on continuous piecewise linear solution path for the ROAD optimization problem at the population level justifies the linear interpolation of the CCD algorithm.

Time difference of arrival (TDOA) is commonly used to estimate the azimuth of a source in a microphone array. The most common methods to estimate TDOA are based on finding extrema in generalized crosscorrelation waveforms. In this paper we apply microphone array techniques to a manikin head. By considering the entire cross-correlation waveform we achieve azimuth prediction accuracy that exceeds extrema locating methods. We do so by quantizing the azimuthal angle and treating the prediction problem as a multiclass categorization task. We demonstrate the merits of our approach by evaluating the various approaches on Sony's AIBO robot.

Time difference of arrival (TDOA) is commonly used to estimate the azimuth of a source in a microphone array. The most common methods to estimate TDOA are based on finding extrema in generalized crosscorrelation waveforms. In this paper we apply microphone array techniques to a manikin head. By considering the entire cross-correlation waveform we achieve azimuth prediction accuracy that exceeds extrema locating methods. We do so by quantizing the azimuthal angle and treating the prediction problem as a multiclass categorization task. We demonstrate the merits of our approach by evaluating the various approaches on Sony's AIBO robot.

Time difference of arrival (TDOA) is commonly used to estimate the azimuth ofa source in a microphone array. The most common methods to estimate TDOA are based on finding extrema in generalized crosscorrelation waveforms.In this paper we apply microphone array techniques to a manikin head. By considering the entire cross-correlation waveform we achieve azimuth prediction accuracy that exceeds extrema locating methods. We do so by quantizing the azimuthal angle and treating the prediction problem as a multiclass categorization task. We demonstrate the merits of our approach by evaluating the various approaches onSony's AIBO robot.

Driven by the progress in the field of single-trial analysis of EEG, there is a growing interest in brain computer interfaces (BCIs), i.e., systems that enable human subjects to control a computer only by means of their brain signals. In a pseudo-online simulation our BCI detects upcoming finger movements in a natural keyboard typing condition and predicts their laterality. This can be done on average 100-230 ms before the respective key is actually pressed, i.e., long before the onset of EMG. Our approach is appealing for its short response time and high classification accuracy ( 96%) in a binary decision where no human training is involved. We compare discriminative classifiers like Support Vector Machines (SVMs) and different variants of Fisher Discriminant that possess favorable regularization properties for dealing with high noise cases (inter-trial variablity).

Driven by the progress in the field of single-trial analysis of EEG, there is a growing interest in brain computer interfaces (BCIs), i.e., systems that enable human subjects to control a computer only by means of their brain signals. In a pseudo-online simulation our BCI detects upcoming finger movements in a natural keyboard typing condition and predicts their laterality. This can be done on average 100-230 ms before the respective key is actually pressed, i.e., long before the onset of EMG. Our approach is appealing for its short response time and high classification accuracy ( 96%) in a binary decision where no human training is involved. We compare discriminative classifiers like Support Vector Machines (SVMs) and different variants of Fisher Discriminant that possess favorable regularization properties for dealing with high noise cases (inter-trial variablity).