IEEE - Aerospace and Electronic Systems - October 2022 - 45

Bar-Shalom
that I do real research despite working in industry. We then
launched into a spirited discussion about differential geometry
and amazingly good adhoc algorithms such as turbo coding.
Yaakov: Can you give us an idea of your high school
experiences.
Fred: I went to an excellent high school with many bright
students, some of whom became famous, e.g., Paul Auster
and Mark Rudd. My class was about 20% real geniuses who
were very good students and highly competitive. I wasmostly
interested in math and physics and chemistry, but these
courses were never interesting or challenging for me, despite
being called " advanced placement " and " the new math. "
Most ofwhat I learned about physics and math was from the
other smart kids and the library. Otherwise, I loved literature,
playing chess, playing violin in the orchestra, acting in plays,
tennis, football, baseball, Freud, Nietzsche, logical positivism
(A. J. Ayer), and opera. I relished going to the Metropolitan
Opera in New York City every Saturday with my friends to
buy a $2 standing-room ticket for the opera; it was great.
Yaakov: Anything interesting about your college or
grad school experiences?
Fred: The debating team was the single most useful
thing I did as an undergraduate. I got plenty of practice
standing up and giving arguments and rebuttals competing
against teams from other schools in the Northeast of the
United States. The team from Princeton attempted to intimidate
us by bringing a dozen long boxes of3 5 cards, supposedly
with facts and figures to refute our position; this
was an amusing trick that I used years later in a very different
context as we shall see in due course. Once, we debated
the team from Lewisburg Federal Prison where the team
was coached by Jimmy Hoffa. I was a good public speaker
before I started debating, which is why I was recruited to
the team, but intercollegiate debating every week gave me
plenty ofpractice, expert feedback and even more self-confidence.
As a result, I am a fearless public speaker. In addition
to debating, I also learned lots of math and physics; I
took honors math every semester for 4 years. After that I
went to a very expensive and extremely selective graduate
school that shall remain nameless. Most ofmy courses were
in pure math, taught by luminaries like Barry Mazur,
George Mackey, David Mumford, Raoul Bott, and Andy
Gleason. These math courses were edifying and interesting
and fun but a big waste of time otherwise. Maybe they bulletproof
you against being intimidated by any math (or any
person) in the future. Papoulis' textbook on probability is
vastly more useful than measure theory or topology or Hilbert
space or Lie groups or Lie algebras. According to Einstein:
" Mathematicians know many interesting things, but
never quite the ones we physicists want to know " [19]. A
typical working engineer opined: " Most modern mathematical
research brings solutions which nobody understands to
questions that nobody asked " [4]. Carver Mead, a famous
Professor at Caltech, tells us that " mathematics courses, as
they are taught today, are seldom helpful and are often
OCTOBER 2022
downright destructive " [18]. This disparaging attitude about
modern math is widely shared among research physicists
today [20]. But of course, there are exceptions, such as
Gromov's amazing theory ofunderdetermined PDEs, which
is used in my particle flow filter theory [5]. My graduate
school had no classes on artificial intelligence (AI), and
hence I rode my bike down to MIT to take Minsky's and
Papert's course on AI, where we were taught to hate neural
nets (NN). They taught us that single-layer NNs (called
" perceptrons " ) could only distinguish between classes of
images with distinct areas; the proof used very elementary
group theory. When I heard this proof in class I was
shocked, because all the books and papers that I had read
said that perceptrons were a great idea. Also, I had built a
real physical perceptron out ofdiscrete transistors and resistors
as an undergraduate electrical engineer one year before.
My professor gave me an A in the course, because my NN
worked perfectly as designed. But this professor also made
the observation that all of the images that I was able to distinguish
had distinct areas. I had supposed that this was just
a coincidence, but at MIT I had just learned that this was a
mathematical fact. Ofcourse, NNs with multiple layers are
much more powerful, but extremely difficult to train in
those early days. Meanwhile back at my school up the street,
my classmate Paul Werbos was inventing backpropagation,
which is ubiquitous in training multilayer NNs today.
Except that Paul's work was not really appreciated at my
school; it was just the chain rule; it was not mathematically
deep [10]. However, at grad school, I actually learned a few
useful things: Bayesian decision theory, the Fokker-Planck
equation, Ito calculus, nonlinear stochastic optimal control
theory, numerical methods for solving PDEs, mathematical
programming, and how to do symbol manipulation with linear
algebra and multivariable calculus and probability theory
fast and accurately. I had learned about Kalman filters
on my own before I got to grad school by reading a few
chapters ofBryson & Ho's textbook on optimal control theory.
I also learned how to solve a variety ofPDEs that arise
in stochastic nonlinear control theory and physics.
Yaakov: Any big names who had an influence on
you? Gauss?
Fred: Gauss, Einstein, Maxwell, Boltzmann, Hilbert,
Schr€odinger, Feynman, and Gromov.
Yaakov: After grad school, you went to work for a
private company-I assume Raytheon. Do you have stories
about your experience out of school and later when
you became a " luminary " in the field?
Fred: At Raytheon, I had a burning desire to learn
about radars, computers, software, algorithms, and all of
the physics that goes into making radar systems work in
the real world. After about 10 years I tried to leave Raytheon,
but they did not let me; Dennis Picard gave me an
offer that I could not refuse. I was very lucky to work on
so many state-of-the-art phased array radars, such as AN/
TPN-19, Cobra Dane, PAVE PAWS, Cobra Judy,
IEEE A&E SYSTEMS MAGAZINE
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IEEE - Aerospace and Electronic Systems - October 2022

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