Periodic Reporting for period 3 - HARMONIC (Discrete harmonic analysis for computer science)
Berichtszeitraum: 2022-03-01 bis 2023-08-31
This project aims to develop Discrete Harmonic Analysis, a mathematical field at the intersection of several different areas, and apply it to computer science.
The idea is to build on the success of Boolean Function Analysis, a fundamental tool in theoretical computer science, which studies a notion of Fourier transform which is well-suited to problems in computer science.
Boolean Function Analysis is a special case of Discrete Harmonic Analysis, in which the objects being studied are of a specific type (functions accepting bits as inputs).
Yet in some applications, more complicated objects are called for, and this necessitates replacing Boolean Function Analysis with the more general Discrete Harmonic Analysis.
While theoretical computer science does not concern itself with immediate applications, in the past few years it has given rise to several start-ups with applications to cryptocurrencies and beyond.
The backbone of these start-ups is a special type of protocol, Probabilistically Checkable Proofs, whose construction is based on Boolean Function Analysis.
Probabilistically Checkable Proofs are notorious for their supposed impracticality, although recent progress has called this belief into question.
Using Discrete Harmonic Analysis, it might be possible to further optimize this type of protocols, making them applicable in much wider settings.
The idea is to build on the success of Boolean Function Analysis, a fundamental tool in theoretical computer science, which studies a notion of Fourier transform which is well-suited to problems in computer science.
Boolean Function Analysis is a special case of Discrete Harmonic Analysis, in which the objects being studied are of a specific type (functions accepting bits as inputs).
Yet in some applications, more complicated objects are called for, and this necessitates replacing Boolean Function Analysis with the more general Discrete Harmonic Analysis.
While theoretical computer science does not concern itself with immediate applications, in the past few years it has given rise to several start-ups with applications to cryptocurrencies and beyond.
The backbone of these start-ups is a special type of protocol, Probabilistically Checkable Proofs, whose construction is based on Boolean Function Analysis.
Probabilistically Checkable Proofs are notorious for their supposed impracticality, although recent progress has called this belief into question.
Using Discrete Harmonic Analysis, it might be possible to further optimize this type of protocols, making them applicable in much wider settings.
Most of the research so far has concentrated on studying Discrete Harmonic Analysis on the symmetric group, a fundamental object of study in several areas of mathematics.
The research has successfully generalized many fundamental results of Boolean Function Analysis to this challenging setting.
In particular, the theory of complexity measures of functions has been generalized; the important technical tool of hypercontractivity has been ported to this domain; and a sharp version of the Friedgut–Kalai–Naor theorem was proved.
A different tract of research studies High-Dimensional Expanders, which are hard-to-construct algebraic objects with several wondrous properties.
The research has generalized the basic setup of Boolean Function Analysis to this setting, which is the first step toward developing full-fledged Discrete Harmonic Analysis on this domain; these results have already been used in subsequent work by other researchers.
These results were then used to construct "hard instances" for the Sum-of-Squares meta-algorithm, paving the way to future use of High-Dimensional Expanders in the construction of better Probabilistically Checkable Proofs.
Another area of study is approximate polymorphisms, a problem arising from classical impossibility results in Social Choice Theory such as Arrow's theorem and the Gibbard-Satterthwaite theorem.
These two results state that in some forms of elections, the only social choice functions satisfying certain natural properties are those in which one agent (the "dictator") determines the result.
Further work showed that even if we relax the properties, all we are left with is social choice functions which are very close to dictators.
Our work extends these results to another impossible result in Social Choice Theory, the Doctrinal Paradox.
The research has successfully generalized many fundamental results of Boolean Function Analysis to this challenging setting.
In particular, the theory of complexity measures of functions has been generalized; the important technical tool of hypercontractivity has been ported to this domain; and a sharp version of the Friedgut–Kalai–Naor theorem was proved.
A different tract of research studies High-Dimensional Expanders, which are hard-to-construct algebraic objects with several wondrous properties.
The research has generalized the basic setup of Boolean Function Analysis to this setting, which is the first step toward developing full-fledged Discrete Harmonic Analysis on this domain; these results have already been used in subsequent work by other researchers.
These results were then used to construct "hard instances" for the Sum-of-Squares meta-algorithm, paving the way to future use of High-Dimensional Expanders in the construction of better Probabilistically Checkable Proofs.
Another area of study is approximate polymorphisms, a problem arising from classical impossibility results in Social Choice Theory such as Arrow's theorem and the Gibbard-Satterthwaite theorem.
These two results state that in some forms of elections, the only social choice functions satisfying certain natural properties are those in which one agent (the "dictator") determines the result.
Further work showed that even if we relax the properties, all we are left with is social choice functions which are very close to dictators.
Our work extends these results to another impossible result in Social Choice Theory, the Doctrinal Paradox.
We expect the development of Discrete Harmonic Analysis on the symmetric group to continue.
Following on our success on generalizing the FKN theorem, we hope to prove the more general Kindler–Safra theorem on the symmetric group.
Moreover, we intend to explore applications of this progress to other areas of theoretical computer science, specifically to the area of hardness of approximation, which studies the limit of algorithms for approximately solving optimization problems.
Another work in progress on the symmetric group studies the analog of the Fourier expansion for functions on the symmetric group.
Such an analog is known for several domains, but not for the symmetric group.
Our work suggests such an analog, and proves several interesting properties about it.
This work should interest not only computer scientists but also mathematicians specializing in representation theory and in algebraic combinatorics.
Our work on the symmetric group suggested a sharp form of the Kindler-Safra theorem on the biased Boolean cube, which is work-in-progress.
Another work-in-progress is the characterization of the junta threshold for low degree Boolean functions on the slice.
Using similar ideas, we hope to develop Discrete Harmonic Analysis on another important domain, the Grassmann scheme, which was recently used to prove a major conjecture in theoretical computer science, the 2-to-2 conjecture.
In particular, we plan to use insights from our work on the symmetric group to this domain.
Finally, we plan to continue working on approximate polymorphisms.
Recent work has characterized approximate polymorphisms of Boolean functions, and we plan to extend this to predicates and to general alphabets.
These results should interest computer scientists as well as social choice theorists and mathematicians specializing in universal algebra.
Following on our success on generalizing the FKN theorem, we hope to prove the more general Kindler–Safra theorem on the symmetric group.
Moreover, we intend to explore applications of this progress to other areas of theoretical computer science, specifically to the area of hardness of approximation, which studies the limit of algorithms for approximately solving optimization problems.
Another work in progress on the symmetric group studies the analog of the Fourier expansion for functions on the symmetric group.
Such an analog is known for several domains, but not for the symmetric group.
Our work suggests such an analog, and proves several interesting properties about it.
This work should interest not only computer scientists but also mathematicians specializing in representation theory and in algebraic combinatorics.
Our work on the symmetric group suggested a sharp form of the Kindler-Safra theorem on the biased Boolean cube, which is work-in-progress.
Another work-in-progress is the characterization of the junta threshold for low degree Boolean functions on the slice.
Using similar ideas, we hope to develop Discrete Harmonic Analysis on another important domain, the Grassmann scheme, which was recently used to prove a major conjecture in theoretical computer science, the 2-to-2 conjecture.
In particular, we plan to use insights from our work on the symmetric group to this domain.
Finally, we plan to continue working on approximate polymorphisms.
Recent work has characterized approximate polymorphisms of Boolean functions, and we plan to extend this to predicates and to general alphabets.
These results should interest computer scientists as well as social choice theorists and mathematicians specializing in universal algebra.