Franklin T. Luk

Principal component analysis for distributed data sets with updating

Identifying the patterns of large data sets is a key requirement in data mining. A powerful technique for this purpose is the principal component analysis (PCA). PCA-based clustering algorithms are effective when the data sets are found in the same location. In applications where the large data sets are physically far apart, moving huge amounts of data to a single location can become an impractical, or even impossible, task. A way around this problem was proposed in [10], where truncated singular value decompositions (SVDs) are computed locally and used to reduce the communication costs. Unfortunately, truncated SVDs introduce local approximation errors that could add up and would adversely affect the accuracy of the final PCA. In this paper, we introduce a new method to compute the PCA without incurring local approximation errors. In addition, we consider the situation of updating the PCA when new data arrive at the various locations.

Algorithmic fault tolerance using the Lanczos method

We consider the problem of algorithm-based fault tolerance, and make two major contributions. First, we show how very general sequences of polynomials can be used to generate the checksums, so as to reduce the chance of numerical overflows. Second, we show how the Lanczos process can be applied in the error location and correction steps, so as to save on the amount of work and to facilitate actual hardware implementation. 1. Background. Many important signal processing and control problems require computational solution in real time. Much research has gone into the development of special purpose algorithms and associated hardware. The latter are usually called systolic arrays in academia, and application specific integrated circuits (ASICs) in industry. In many critical situations, so much depends on the ability of the combined software/hardware system to deliver reliable and accurate numerical results that fault tolerance is indispensable. Often, weight constraints forbid the use of multiple modular redundancy and one must resort to a software technique to handle errors. A top choice is Algorithm-Based Fault Tolerance (ABFT), originally developed by Abraham and students [9, 10], to provide a low-cost error protection for basic matrix operations. Their work was extended by Luk et al. [11, 13, 14] to applications that include matrix equation solvers, triangular decompositions, and recursive least squares. A theoretical framework for error correction was developed for the cases of one error [10], two errors [1], and multiple errors [7]. Interestingly, the model in [7] turns out to be the Reed-Solomon code [17]. However, the

Fast singular value algorithm for Hankel matrices

We present an O(n2logn) algorithm for finding all the singular values of an n-by-n complex Hankel matrix.

A fast eigenvalue algorithm for Hankel matrices

Abstract We present an algorithm that can find all the eigenvalues of an n×n complex Hankel matrix in O (n 2 log n) operations. Our scheme consists of an O (n 2 log n) Lanczos-type tridiagonalization procedure and an O (n) QR-type diagonalization method.

A new matrix decomposition for signal processing

We extend the generalized singular value decomposition to a new decomposition that can be updated at a low cost. In addition, we show how a forgetting factor can be incorporated in our decomposition.

Reducing boundary distortion in image restoration

The ill-conditioned nature of the inverse problem of image restoration produces dramatic consequences when the boundaries of the image are incorrectly modeled. Solutions to this problem are difficult to find in the literature. In this paper, we develop and analyze two sets of new approaches that are effective with algebraic restoration procedures.

Using complex-orthogonal transformations to diagonalize a complex symmetric matrix

In this paper, we propose the use of complex-orthogonal transformations for finding the eigenvalues of a complex symmetric matrix. Using these special transformations can significantly reduce computational costs because the tridiagonal structure of a complex symmetric matrix is maintained.

A fast method to diagonalize a Hankel matrix

Abstract We consider a Vandermonde factorization of a Hankel matrix, and propose a new approach to compute the full decomposition in O( n 2 ) operations. The method is based on the use of a variant of the Lanczos method to compute a tridiagonal matrix whose eigenvalues are the modes generating the entries in the Hankel matrix. By adapting existing methods to solve for these eigenvalues and then for the coefficients, one arrives at a method to compute the entire decomposition in O( n 2 ) operations. The method is illustrated with a simple numerical example.

Fixed-point arithmetic for mobile devices: a fingerprinting verification case study

Mobile devices use embedded processors with low computing capabilities to reduce power consumption. Since floating-point arithmetic units are power hungry, computationally intensive jobs must be accomplished with either digital signal processors or hardware co-processors. In this paper, we propose to perform fixed-point arithmetic on an integer hardware unit. We illustrate the advantages of our approach by implementing fingerprint verification on mobile devices.

A symmetric rank-revealing Toeplitz matrix decomposition

In signal and image processing, regularization often requires a rank-revealing decomposition of a symmetric Toeplitz matrix with a small rank deficiency. In this paper, we present an efficient factorization method that exploits symmetry as well as the rank and Toeplitz properties of the given matrix.