Cocktail party math
Mathematicians bring new life to an old party
When you attend upcoming holiday parties, pay special attention to the noise around you. With all the voices, all the clinking glasses and plates, and all the cheery holiday tunes, you might find it tough to focus on just one conversation.
MU mathematicians Peter Casazza and Dan Edidin have spent plenty of time pondering this problem. No, they’re not trying to eavesdrop on your party chatter. But others — law enforcement surveillance crews, homeland security agents and 911 operators, for example — have plenty of interest in finding the most effective way to pick out individual voices among crowds.
Engineers have been trying to solve this so-called “Cocktail Party Problem” since the 1950s. Now Casazza and Edidin, working with Siemens Corporate Research, have helped move a solution one step closer to reality.
What is done and what remains
The Cocktail Party Problem is complex, but at its most basic it comes down to a question: Can you separate individual voices out of a crowd and retain their distinct vocal characteristics?
The most common method to do this is called “signal reconstruction with noisy phase.” This process allows a recording to retain the characteristics of a person’s voice, but it also tends to include some distracting background noise.
“Engineers have conjectured for 40 or 50 years that you should be able to do signal reconstruction without noisy phase,” Casazza says. “If you could do signal reconstruction without noisy phase, then the noise wouldn’t bother you.” Casazza and Edidin now have proved, through a mathematical solution, that the engineers were right.
Casazza is quick to point out that he and Edidin have not solved the Cocktail Party Problem. They have proved part of it and helped pave the way for a practical solution. But in Casazza’s mind, it isn’t truly solved until engineers can develop a product that accurately and consistently does the job. That’s why he is trying to develop an algorithm that works with Siemens software so that engineers there can take the work further.
“It’s an engineering problem; engineers have to solve it,” Casazza says. “All we can do is provide mathematics for them to solve it.”
The problematic tie that binds
Casazza and Edidin’s work has made headlines, but Casazza thinks the bigger news came as an offshoot of that work. In working on the Cocktail Party Problem, he came across an unexpected nemesis: the Kadison-Singer Problem.
Kadison-Singer is an open (unsolved) problem that was officially posed in 1959. It is a less public-friendly problem because of its sheer complexity, but it is legendary in math and science circles.
When Casazza spotted Kadison-Singer in his own work, it set off a chain reaction. Casazza and his wife, fellow MU mathematician Janet Tremain, have since shown that Kadison-Singer is equal to several other unsolved problems in 12 different fields — from pure mathematics to applied mathematics and engineering. In other words, the same problem has been puzzling bright minds in 12 different forms for nearly 50 years.
The problem remains unsolved, but Casazza and Tremain’s work has so much significance that it earned publication in the Proceedings of the National Academy of Sciences, one of the toughest places for mathematicians to get published, Casazza says. It’s the first such publication for an MU mathematician since 1949.
The work also has brought together top researchers from varied fields. Most of these researchers — usually segmented by academic disciplines — had not met each other before. Many had left their own version of Kadison-Singer for dead or unsolvable. At a September 2006 conference sponsored by the American Institute of Mathematics, one attendee summed up the result of Casazza and Tremain’s discovery: “Pete has opened the coffin. Now we have to decide what we are going to do about it.”