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# High Honors for Versatile Engineer/Mathematician

As a recipient of the 2014 National Medal of Science, Thomas Kailath of Stanford University was cited not only for “transformative” scientific contributions to the fields of information and systems science, but also for mentoring activities and entrepreneurial ventures that proved influential in industry.

A longtime (since 1975) member of SIAM, Kailath has written frequently over the years for a variety of SIAM publications. His research monograph Indefinite-Quadratic Estimation and Control: A Unified Approach to $$H^2$$ and $$H^\infty$$ Theories, with Babak Hassibi and Ali H. Sayed, for example, appeared in 1999 as Volume 16 in the SIAM Studies in Applied and Numerical Mathematics series.

Photo courtesy of Thomas Kailath.
SIAM executive director James Crowley fondly recalls a dinner with Kailath at an Indian restaurant in Palo Alto in the early 1990s. Beyond the very pleasant meal and conversation, Crowley remembers Kailath’s description of frequent changes in his research focus––a new emphasis every decade––in the course of his career. “This is certainly borne out by the diversity and richness of Kailath’s research contributions,” Crowley says.

Sayed, Kailath’s frequent co-author and former student, now a member of the faculty at UCLA, points out that throughout Kailath’s work, the interest for the SIAM community lies in his ability to “exploit in insightful and often magical ways the mathematical structure underlying problems in many areas, whether signal processing, control or information theory, or semiconductor manufacturing.” That high-level assessment reflects a dynamic career that has, in fact, been characterized by periodic shifts in focus among different fields.

Especially gratified by the National Medal citation’s reference to “distinctive and sustained mentoring of young scholars,” Kailath commented to SIAM News on the importance of students in his career. In the early years, he said, “almost all my papers were single-authored. But then I realized that if I wanted to address new topical challenges, the best way to do that in a university was to work with groups of brilliant students—and that is what enabled me to change the major focus of my research roughly every decade.”

Kailath’s graduate work at MIT, from 1957 to 1961, on the characterization and identification of random linear time variant channels and on communication via such channels had gained him an international reputation before he joined Stanford as an associate professor in 1963. As director of the Information Systems Laboratory from 1971 to 1981, he was instrumental in building it into a world-class center for communications, control, and signal processing research. He is currently the Hitachi America Professor of Engineering Emeritus in the electrical engineering department at Stanford.

A decade-by-decade perspective on Kailath’s research activities at Stanford begins with the development of an algorithm for exploiting the availability of noiseless feedback and new techniques in the theory of signal detection. Linear Systems, his influential 1980 textbook, resulted from his work during the previous decade using state-space techniques to model and understand the behavior of dynamical systems. That was followed by a decade’s work on multiple-antenna signal processing and the design of VLSI arrays/architectures for signal processing, along with development of the concept of displacement structure.

An overview of his work on the latter can be found in his and Sayed’s extensive 1995 SIAM Review article, in which they relate the development of fast computational algorithms for matrices that have what Kailath named “displacement structure.” An important example is a fast triangularization procedure for such matrices (generalizing a 1917 algorithm of Schur). As they point out in the abstract of their article, “this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra.” Examples include inverse scattering, analytic and unconstrained rational interpolation theory, digital filter design, algebraic coding theory, and adaptive filtering. A collection of review articles on displacement structure can be found in the volume Fast Reliable Algorithms for Matrices with Structure, edited by Kailath and Sayed and published by SIAM in 1999.

Displacement structure was a focus for Kailath and his colleagues for a number of years. The group’s algorithms for solving complex design problems, in which the matrices are very large, Sayed says, are an order of magnitude faster than other algorithms and continue to be widely studied today. And in the process of discovering the algorithms, he says, Kailath’s group discovered fascinating connections with several other areas of mathematics, including interpolation theory and orthogonal polynomials.

In the 1990s, Kailath and colleagues turned their attention to smart antenna technology for wireless communication, as well as to resolution-enhancement techniques for optical lithography in semiconductor manufacturing. The latter work used techniques from signal processing and communications to break a barrier of 100 nm, perceived at that time as the minimum line width achievable by optical lithography. In 1996 Kailath and a group of graduate students formed the company Numerical Technologies, which went public in 2000 and was acquired by Synopsis in 2003.

In the course of his career, Kailath and his students have also made a variety of contributions to probability and statistics. Kailath is an emeritus fellow of the Institute of Mathematical Statistics. He is also a member of the inaugural class of SIAM fellows, cited for his contributions to linear algebra, systems and control, and their applications in engineering.

Another important textbook, Linear Estimation, by Kailath, Sayed, and Hassibi, appeared in 2000. Widely used today as a reference in the broad area of state-space estimation theory, the book offers a unified and motivated window on a range of results in this area that emanated from Kailath’s group.

Students today, Sayed points out, “have the benefit and convenience of rushing to search for articles and references online”; this casts the rewards of his own experience with Kailath as an adviser in a new light. “At a time when students used to visit libraries more often, Kailath’s ability to pinpoint references with precision in archival publications was simply amazing. His immersion in the science of his work was also contagious and lives with me to this day.”