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Spectral clustering partitions data into similar groups in the eigenspace of the affinity matrix. The accuracy of the spectral clustering algorithm is affected by the affine equivariance realized in the translation of distance to similarity relationship. The similarity value computed as a Gaussian of the distance between data objects is sensitive to the scale factor $\sigma$. The value of $\sigma$, a control parameter of drop in affinity value, is generally a fixed constant or determined by manual tuning. In this research work, $\sigma$ is determined automatically from the distance values i.e. the similarity relationship that exists in the real data space. The affinity value of a data pair is determined as a location estimate of the spread of distance values of the data points with the other points. The scale factor ${\sigma }_i$ corresponding to a data point $x_i$ is computed as the trimean of its distance vector and used in fixing the scale to compute the affinity matrix. Our proposed automatic scale parameter for spectral clustering resulted in a robust similarity matrix which is affine equivariant with the distance distribution and also eliminates the overhead of manual tuning to find the best $\sigma$ value. The performance of spectral clustering using such affinity matrices was analyzed using UCI data sets and image databases. The obtained scores for NMI, ARI, Purity and F-score were observed to be equivalent to those of existing works and better for most of the data sets. The proposed scale factor was used in various state-of-the-art spectral clustering algorithms and it proves to perform well irrespective of the normalization operations applied in the algorithms. A comparison of clustering error rates obtained for various data sets across the algorithms shows that the proposed automatic scale factor is successful in clustering the data sets equivalent to that obtained using manually tuned best $\sigma$ value. Thus the automatic scale factor proposed in this research work eliminates the need for exhaustive grid search for the best scale parameter that results in best clustering performance.

A new method of projective synchronization of unidentical chaotic systems is proposed in this letter. This method is based on the stability criterion of linear systems. The response of two unidentical chaotic systems can synchronize up to any desired scaling factor by a suitable separation of the systems. The new method of projective synchronization is suitable not only for the three-dimensional coupled partially linear systems, but also for higher dimensional even hyperchaotic systems. The simplicity and effectiveness of the proposed method are illustrated by the Lorenz system, the four-dimensional partially linear system, the four-dimensional hyperchaotic Rösser system and Chua's circuit system as four numerical examples.

Conjugate gradient methods constitute excellent neural network training methods which are characterized by their simplicity and their very low memory requirements. In this paper, we propose a new spectral conjugate gradient method which guarantees sufficient descent using any line search, avoiding thereby the usually inefficient restarts. Moreover, we establish the global convergence of our proposed method under some assumptions. Experimental results provide evidence that our proposed method is preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness.

An Expert Vision motion analysis system with five analog video cameras was used to evaluate the mouth movement during two facial animations (smile and puffy face). Sixteen skin markers were adhered to the face of subject according to the anatomic landmarks to represent the functional movement of the facial muscles. The trajectory of the four or eight peri-oral skin markers was simultaneously evaluated by Challis technique, instead of individual movement of single marker. The 4-marker method is a little less accurate than 8-marker method. But the 4-marker method can be incorporated as a part of our modality of facial motion analysis, including two/three dimensional displacement of individual marker and absolute/relative displacement of paired markers. It was much easy for data acqusition and no extra marker was needed in the whole modality of our facial motion analysis. Physicians can use this Challis technique to evaluate grouped movement of facial markers, as a whole, in different animation.