Multidimensional scaling for large genomic data sets

<p>Abstract</p> <p>Background</p> <p>Multi-dimensional scaling (MDS) is aimed to represent high dimensional data in a low dimensional space with preservation of the similarities between data points. This reduction in dimensionality is crucial for analyzing and revealing the genuine structure hidden in the data. For noisy data, dimension reduction can effectively reduce the effect of noise on the embedded structure. For large data set, dimension reduction can effectively reduce information retrieval complexity. Thus, MDS techniques are used in many applications of data mining and gene network research. However, although there have been a number of studies that applied MDS techniques to genomics research, the number of analyzed data points was restricted by the high computational complexity of MDS. In general, a non-metric MDS method is faster than a metric MDS, but it does not preserve the true relationships. The computational complexity of most metric MDS methods is over <it>O(N</it>²<it>)</it>, so that it is difficult to process a data set of a large number of genes <it>N</it>, such as in the case of whole genome microarray data.</p> <p>Results</p> <p>We developed a new rapid metric MDS method with a low computational complexity, making metric MDS applicable for large data sets. Computer simulation showed that the new method of split-and-combine MDS (SC-MDS) is fast, accurate and efficient. Our empirical studies using microarray data on the yeast cell cycle showed that the performance of K-means in the reduced dimensional space is similar to or slightly better than that of K-means in the original space, but about three times faster to obtain the clustering results. Our clustering results using SC-MDS are more stable than those in the original space. Hence, the proposed SC-MDS is useful for analyzing whole genome data.</p> <p>Conclusion</p> <p>Our new method reduces the computational complexity from <it>O</it>(<it>N</it>³) to <it>O</it>(<it>N</it>) when the dimension of the feature space is far less than the number of genes <it>N</it>, and it successfully reconstructs the low dimensional representation as does the classical MDS. Its performance depends on the grouping method and the minimal number of the intersection points between groups. Feasible methods for grouping methods are suggested; each group must contain both neighboring and far apart data points. Our method can represent high dimensional large data set in a low dimensional space not only efficiently but also effectively.</p

Re-Encryption Method Designed by Row Complete Matrix

With the prevalence of Internet access, document storage has become a fundamental web service in recent years. One important topic is how to design a secure channel for efficiently sharing documents with another receiver. In this paper, we demonstrate a re-encryption method that is designed with row complete matrices. With this new method, the document owner can share a ciphertext in the cloud with another receiver by sending a serial number to the server and giving the receiver a corresponding key at the same time. This method ensures that the server cannot obtain the information about key and plaintext and that the receiver cannot obtain the original key of the owner either. Only the owner has the knowledge of all the information. Using this re-encryption system, the cloud server can provide a secure file-sharing service without worrying about the shared key management problem. Moreover, the cost of re-encryption will not increase even when the encryption is strengthened with longer encryption keys

An Asymmetric Subspace Watermarking Method for Copyright Protection

We present an asymmetric watermarking method for copyright protection that uses different matrix operations to embed & extract a watermark. It allows for the public release of all information, except the secret key.We investigate the conditions for a high detection probability, a low false positive probability, & the possibility of unauthorized users successfully hacking into our system. The robustness of our method is demonstrated by the simulation of various attacks

Split-and-Combine Singular Value Decomposition for Large-Scale Matrix

The singular value decomposition (SVD) is a fundamental matrix decomposition in linear algebra. It is widely applied in many modern techniques, for example, high- dimensional data visualization, dimension reduction, data mining, latent semantic analysis, and so forth. Although the SVD plays an essential role in these fields, its apparent weakness is the order three computational cost. This order three computational cost makes many modern applications infeasible, especially when the scale of the data is huge and growing. Therefore, it is imperative to develop a fast SVD method in modern era. If the rank of matrix is much smaller than the matrix size, there are already some fast SVD approaches. In this paper, we focus on this case but with the additional condition that the data is considerably huge to be stored as a matrix form. We will demonstrate that this fast SVD result is sufficiently accurate, and most importantly it can be derived immediately. Using this fast method, many infeasible modern techniques based on the SVD will become viable

Subspace Watermarking Method for Copyright Protection:Symmetric and Asymmetric

對於保護數位影像資料，數位影像浮水印技術是一般公認有效的 保護方法。我們提供了一套浮水印技術，這套技術針對每一張受 保護的圖片計算最適合加入浮水印的空間。在此空間上我們分別 設計出對稱與非對稱的加密方式來保護影像。不論是對稱或是非 對稱的方法，保護影像的能力與錯誤判別的機率皆相似。我們提 供的方法可以提昇現有浮水印技術的保護能力。The watermarking method has emerged as an important tool for content tracing, authentication, and data hiding in multimedia applications. We propose a watermarking strategy in which the watermark of a host is selected from the robust features of the estimated forged images of the host. We introduce the concept of watermark space which is obtained from Monte Carlo simulations of potential pirate attacks on the host image. The subspace approach can be used for symmetric and asymmetric watermarking scheme for copyright protection. Both of them reveal the phenomena of a high detection probability and low false positive probability. In either scheme, significant efforts have been made to explore the probability of successful hacking into our system. The result shows our schemes sustain most known attacks.1 Introduction 1 2 Watermark Space 5 3 Symmetric Subspace Watermarking Method 16 4 Conventional Asymmetric Watermarking Schemes 36 5 Asymmetric Subspace Watermarking Method 43 6 Conclusion 62 Reference 6

The Leveraged Wavelets and Galerkin-Wavelets Methods

. We present a scheme that leverage orthonormal or biorthogonal wavelets to a new system of biorthogonal wavelets. The leveraged biorthogonal wavelets will have some nice properties. If we start with orthonormal wavelets, the leveraged scaling functions and wavelets are compactly supported and are differentiable. The derivatives of the leveraged wavelets are orthogonal to their translations; the derivatives of the leveraged scaling functions are nearly orthogonal to their translations; and the derivatives of the leveraged scaling functions and wavelets are orthogonal to each other. This feature may be valuable for the numerical solution of differential equations. If we start with B-splines and cooperating with the lifting scheme of Sweldens, our leverage scheme can reproduce all of those biorthogonal wavelets by Cohen, Daubechies and Feauveau. There is a simple algorithm to calculate new filter coefficients from the old filter coefficients. In the end of this article we test the newly ..