A Tutorial on Spectral Clustering

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed

Analyzing and clustering neural data

This thesis aims to analyze neural data in an overall effort by the Charles Stark Draper Laboratory to determine an underlying pattern in brain activity in healthy individuals versus patients with a brain degenerative disorder. The neural data comes from ECoG (electrocorticography) applied to either humans or primates. Each ECoG array has electrodes that measure voltage variations which neuroscientists claim correlates to neurons transmitting signals to one another. ECoG differs from the less invasive technique of EEG (electroencephalography) in that EEG electrodes are placed above a patients scalp while ECoG involves drilling small holes in the skull to allow electrodes to be closer to the brain. Because of this ECoG boasts an exceptionally high signal-to-noise ratio and less susceptibility to artifacts than EEG [6]. While wearing the ECoG caps, the patients are asked to perform a range of different tasks. The tasks performed by patients are partitioned into different levels of mental stress i.e. how much concentration is presumably required. The specific dataset used in this thesis is derived from cognitive behavior experiments performed on primates at MGH (Massachusetts General Hospital). The content of this thesis can be thought of as a pipelined process. First the data is collected from the ECoG electrodes, then the data is pre-processed via signal processing techniques and finally the data is clustered via unsupervised learning techniques. For both the pre-processing and the clustering steps, different techniques are applied and then compared against one another. The focus of this thesis is to evaluate clustering techniques when applied to neural data. For the pre-processing step, two types of bandpass filters, a Butterworth Filter and a Chebyshev Filter were applied. For the clustering step three techniques were applied to the data, K-means Clustering, Spectral Clustering and Self-Tuning Spectral Clustering. We conclude that for pre-processing the results from both filters are very similar and thus either filter is sufficient. For clustering we conclude that K- means has the lowest amount of overlap between clusters. K-means is also the most time-efficient of the three techniques and is thus the ideal choice for this application.2016-10-27T00:00:00

Identification of Invariant Sensorimotor Structures as a Prerequisite for the Discovery of Objects

Perceiving the surrounding environment in terms of objects is useful for any general purpose intelligent agent. In this paper, we investigate a fundamental mechanism making object perception possible, namely the identification of spatio-temporally invariant structures in the sensorimotor experience of an agent. We take inspiration from the Sensorimotor Contingencies Theory to define a computational model of this mechanism through a sensorimotor, unsupervised and predictive approach. Our model is based on processing the unsupervised interaction of an artificial agent with its environment. We show how spatio-temporally invariant structures in the environment induce regularities in the sensorimotor experience of an agent, and how this agent, while building a predictive model of its sensorimotor experience, can capture them as densely connected subgraphs in a graph of sensory states connected by motor commands. Our approach is focused on elementary mechanisms, and is illustrated with a set of simple experiments in which an agent interacts with an environment. We show how the agent can build an internal model of moving but spatio-temporally invariant structures by performing a Spectral Clustering of the graph modeling its overall sensorimotor experiences. We systematically examine properties of the model, shedding light more globally on the specificities of the paradigm with respect to methods based on the supervised processing of collections of static images.Comment: 24 pages, 10 figures, published in Frontiers Robotics and A

Image Segmentation by Making Use of Spectral Clustering

Import 05/08/2014Cílem bakalářské práce je ověřit možnosti segmentace obrazu s využitím spektrální dekompozice Laplaceovy matice sítě, kterou obraz vytváří, s následným shlukováním. V diplomové práci proveďte: 1) Seznamte se s problematikou spektrálního shlukování (vhodným materiálem je např.: Ulrike von Luxburg, A Tutorial on Spectral Clustering). 2) Metodu spektrálního shlukování implementujte a experimentálně ověřte pro segmentaci obrazu. Pozornost věnujte efektivnímu postupu výpočtu vlastních čísel a vektorů Laplaceovy matice (lze použít např. knihovnu ARPACK), jakož i výběru vhodné shlukovací metody. 3) Metodu implementovanou dle předchozího odstavce rozšiřte na tzv. difúzní spektrální shlukování (Gaura, Sojka: Diffusion Spectral Clustering for Image Segmentation). 4) Chování metod otestujte a shrňte dosažené výsledky. Programátorské práce proveďte v C/C++.The goal of this bachelor work is to verify the possibilities of image segmentation by making use of the Laplacian matrix of the graph (grid) that is created by image, which is followed by subsequent clustering. In the bachelor work, do the following: 1) Get acquainted with the technique of spectral clustering (e.g. from the paper "A tutorial on Spectral Clustering" by Ulrike von Luxburg). 2) Verify the usefulness of spectral clustering in the area of image segmentation. Pay attention to the selection of efficient method of computing eigenvalues and eigenvectors (e.g. the ARPACK library can be used). Also, pay attention to the selection of the clustering method. 3) Improve the method from the previous paragraph into the method of diffusion spectral clustering (Gaura, Sojka: Diffusion Spectral Clustering for Image Segmentation). 4) Both the methods should be tested. The results should be summarised. The software should be created in C/C++.470 - Katedra aplikované matematikyvýborn