Cocktail party problem
The Cocktail Party Problem refers to the challenge of isolating a single audio signal from a complex mix of sounds, akin to focusing on one conversation amidst the noise of a busy social gathering. This concept highlights the difficulty digital systems have in performing this task, which humans typically manage with relative ease. The problem has broad applications in data analysis and signal processing, affecting fields such as telecommunications, surveillance, and cognitive science. Pioneering work by Colin Cherry and subsequent advancements have sought to develop techniques that enhance the separation of voices in recordings using mathematical methods, particularly through Fourier transforms. Despite progress, current technologies still struggle to achieve perfect clarity, leaving some background noise. Additionally, the metaphor of the cocktail party extends into various mathematical realms, including combinatorics and graph theory, addressing questions about group dynamics and partner selection in games. Researchers continue to explore these mathematical challenges, yielding insights applicable across both pure and applied mathematics.
Cocktail party problem
Summary: A metaphorical cocktail party is the setting for a source separation problem and other challenges.
The eponymous Cocktail Party Problem is a source separation problem in digital signal processing, wherein digital systems have difficulty separating out one signal among many—the metaphorical conversation in a noisy cocktail party, which is comparatively easily handled by the human brain. More broadly, distinguishing signal from noise is a data analysis challenge with many specific applications. The metaphor of the cocktail party also lends itself to a number of other problems in combinatorics, graph theory, probability, and functional analysis.
Conversations and Background Noise
With all the noise at a party, it can be difficult to focus on one conversation, although many people are able to do so. Telecommunication professor Colin Cherry conducted experiments in this area, and he is considered by some to be a pioneer in cognitive science. Many people can even recognize the sound of their name from across a noisy room. However, this is not as easy when heard on a recording. One cocktail party problem arises from concerns about separating each individual’s voice characteristics in a recording from the other voices and background noise. People in surveillance and intelligence are inherently interested in such a problem, and scientists and engineers have worked on solutions since at least the 1950s. One common method is mathematical signal processing. Mathematicians and engineers digitize a signal using a Fourier transform, named for Joseph Fourier. They process it using a variety of methods to remove noise and other extraneous information, and then reconstruct the signal using the inverse transform.
While the process may result in an improved recording of one person’s voice, early twenty-first-century technology and methods do not provide perfect separation, so the recording still includes at least some distracting background noise. However, engineers have conjectured that the signal should be able to be reconstructed without the noisy phase. Mathematicians Radu Balan, Peter Casazza, and Dan Edidin made progress on the problem in 2006, when they showed—using a neural net—that it is mathematically possible to retain the voice characteristics without the noise. Scientists continue to work on developing algorithms for practical use. Casazza made another fundamental mathematical discovery during his work on the cocktail party problem. He and his wife, Janet Tremain, also a mathematician, showed that the Kadison–Singer problem, named for mathematicians Richard Kadison and Isadore “Iz” Singer, is equivalent to other unsolved problems in areas of pure and applied mathematics and engineering, such as operator theory, harmonic analysis, and signal processing.
Mathematicians also investigate other party problems, like the probability that when people at a party are chosen to be partners for a card game—like bridge—no randomly chosen partners will contain spouses or members of the same family. The solutions require finding specific combinations or permutations of the guests. Under certain constraints, the maximal probability for some problems may be bounded at less than certainty as the number of people at the party grows. There are also connections between this question and the card game War, as well as with a related set of problems that focus on orders and arrangements of guests around a single dinner table or in various groupings, with applications in areas like queuing theory and assignment problems. The classic dining philosophers problem is yet another variation that has applications in resource sharing and task allocation in computer science.
Another party problem asks how many people must be present at a party in order to ensure that there will be a group of three people who share the characteristic of being acquaintances or strangers. There is no guarantee that three people will all know each other or will be strangers in parties of five or less people since counterexamples exist. The Java game HEXI, named so because the game is played on the vertices of a hexagon, is modeled on this question. The six vertices are connected by edges and each player takes a turn coloring an edge his or her color. One color represents acquaintances, and the other represents strangers. The goal of the game is to avoid making a triangle of the same color. Mathematicians model this question using graph theory, and show that in any group of at least six people, it is possible to find a group of three people satisfying one of the mutually exclusive relationships. Hence HEXI will always have a loser. Instead of people at a party or vertices of a polygon, one could explore other objects like nations embroiled in a conflict, sequences of randomly generated numbers, or stars. Mathematicians investigate problems like these concerning the existence of regular patterns in sets of objects in Ramsey theory, named for Frank Ramsey.
Bibliography
Albertson, Michael. “People Who Know People.” Mathematics Magazine 67, no. 4 (1994).
Brodie, Marc. “Avoiding Your Spouse at a Party Leads to War.” Mathematics Magazine 75, no. 3 (2002).
Casazza, Peter, and Janet Tremain. “The Kadison–Singer Problem in Mathematics and Engineering.” Proceedings of the National Academy of Sciences 103, no. 7 (2006).
“Mathematicians Solve the ‘Cocktail Party Problem.’” PHYSorg.com. August 22, 2006. http://www.physorg.com/news75477497.html.