Periodic Reporting for period 4 - PATHWISE (Pathwise methods and stochastic calculus in the path towards understanding high-dimensional phenomena)
Período documentado: 2023-07-01 hasta 2024-06-30
The project concerns the behavior of objects in a high-dimensional setting, in search of phenomena that arise as the number of degrees of freedom of a system tends to infinity. The theory underlying these objects has recently seen a boost in applications, in accordance with the explosion of interest in data science and machine learning, two fields of which this theory is a cornerstone. The project puts emphasis on expanding the theoretical foundations relevant, for example, to the understanding of interacting particle systems in statistical mechanics or neural networks or reinforcement learning problems with a large decision-space.
The intuition derived from low-dimensional examples in various fields such as topology and partial differential equations may suggest that an attempt to understand the behavior of high-dimensional objects is futile, as a system's behavior quickly becomes complex and intractable as the dimension increases. This is manifested in a meta-phenomenon referred to as the "curse of dimensionality" which, simply put, refers to the exponential growth of the number of configurations of a system with respect to dimension.
Surprisingly, in many cases of interest which often include ones relevant to real-life applications, high-dimensional systems turn out to be well-behaved and tractable and sometimes it is even the case that the behavior becomes more regular as the dimension, or the number of degrees of freedom, increases. An exemplary mathematical result which illustrates this is the central limit theorem, which shows that when we average a growing number of independent random quantities, the Gaussian (normal) distribution emerges. Broadly speaking, this is the type of phenomenon that the project aims to reveal.
The project is concerned in particular with one facet of this theory we refer to as "dimension free" behavior, which alludes to the fact that many quantities of interest and many central inequalities and bounds related to high dimensional objects have no explicit dependence on the dimension. This phenomenon is observed in several important classes of distributions. In particular we focus on the Gaussian measure and on measures with a convex potential (called "log concave" measures). To this end, the project aims to make progress on several conjectures which predict the dimension-free behavior of high-dimensional objects. One example of such conjecture is the Kannan-Lovasz-Simonovitz conjecture which asserts that for high-dimensional measures with convex potentials, half-spaces (hence, sets which only depend on one direction) will be approximate minimizers.
The project revolves around an emerging method, which is based on a connection with the theory of stochastic calculus, the theory that describes the motion of diffusing particles. The "pathwise method" attempts to analyse a high dimensional system by associating with it a certain stochsatic evolution driven by a Brownian motion, in a way that quantities of interest of the system such as entropy or variance can be expressed in terms of the stochastic process.
The intuition derived from low-dimensional examples in various fields such as topology and partial differential equations may suggest that an attempt to understand the behavior of high-dimensional objects is futile, as a system's behavior quickly becomes complex and intractable as the dimension increases. This is manifested in a meta-phenomenon referred to as the "curse of dimensionality" which, simply put, refers to the exponential growth of the number of configurations of a system with respect to dimension.
Surprisingly, in many cases of interest which often include ones relevant to real-life applications, high-dimensional systems turn out to be well-behaved and tractable and sometimes it is even the case that the behavior becomes more regular as the dimension, or the number of degrees of freedom, increases. An exemplary mathematical result which illustrates this is the central limit theorem, which shows that when we average a growing number of independent random quantities, the Gaussian (normal) distribution emerges. Broadly speaking, this is the type of phenomenon that the project aims to reveal.
The project is concerned in particular with one facet of this theory we refer to as "dimension free" behavior, which alludes to the fact that many quantities of interest and many central inequalities and bounds related to high dimensional objects have no explicit dependence on the dimension. This phenomenon is observed in several important classes of distributions. In particular we focus on the Gaussian measure and on measures with a convex potential (called "log concave" measures). To this end, the project aims to make progress on several conjectures which predict the dimension-free behavior of high-dimensional objects. One example of such conjecture is the Kannan-Lovasz-Simonovitz conjecture which asserts that for high-dimensional measures with convex potentials, half-spaces (hence, sets which only depend on one direction) will be approximate minimizers.
The project revolves around an emerging method, which is based on a connection with the theory of stochastic calculus, the theory that describes the motion of diffusing particles. The "pathwise method" attempts to analyse a high dimensional system by associating with it a certain stochsatic evolution driven by a Brownian motion, in a way that quantities of interest of the system such as entropy or variance can be expressed in terms of the stochastic process.
**Project Overview and Key Results**
Since the start of the project, significant progress has been made in both establishing new inequalities related to the "concentration of mass" phenomenon and advancing the dimension-free analysis of high-dimensional objects. We have also extended the "pathwise analysis" technique in new and impactful directions. Below, we summarize some key outcomes and findings:
1. **Pathwise Methods in Boolean Function Analysis**
Boolean functions are foundational in theoretical computer science, and our research has focused on adapting pathwise techniques to the Boolean hypercube. This has led to the discovery of new concentration bounds and provided alternative proofs for existing ones. Notably, we resolved a 25-year-old conjecture of Talagrand by proving an inequality that generalizes two central results in this field—the Kahn-Kalai-Linial theorem and Talagrand's influence inequality. Moreover, our approach has yielded robust versions of several related inequalities and offered a new pathway for proving quantitative versions of the FKG inequality.
2. **Decomposition and Concentration for Measures on the Discrete Hypercube**
We explored natural conditions under which measures on the Boolean hypercube exhibit "well-behaved" behavior. Specifically, we sought measures that are either highly concentrated or can be decomposed into a small mixture of such concentrated measures. Through pathwise methods, we derived both concentration and decomposition bounds. These results have significant implications for models in statistical physics, improving the analysis of sampling algorithm complexity and supporting the accuracy of "mean field" approximations.
3. **Robust Functional Inequalities in Gaussian Space**
Functional inequalities like the logarithmic Sobolev inequality and the Shannon-Stam inequality play essential roles in statistics and information theory. We developed robust versions of these inequalities in Gaussian space by characterizing functions or measures that approximately saturate them. Our work yielded robust forms of the log-Sobolev, Shannon-Stam, and Talagrand's transportation-entropy inequalities, extending the utility of these results across several domains.
4. **A General Framework for Markov Chain Concentration Bounds**
In collaboration with Yuansi Chen, we proposed a novel framework combining spectral independence and pathwise analysis to both construct and analyze Markov chains in discrete and continuous settings. This approach resulted in the first optimal mixing time for the hardcore model, among other models. Additionally, our follow-up work improved several bounds for sampling uniform measures on convex sets, showing that the hit-and-run random walk has sample complexity equivalent to the ball walk.
5. **Concentration Inequalities and Social Choice**
In recent work with Avi Wigderson and Pei Wu, we provided optimal estimates for the "It ain't over till it's over" theorem in social choice theory. Collaborating with Mikulincer and Raghavendra, we are also finalizing a manuscript that improves the "majority is stablest" theorem, achieving polynomial rather than exponential dependence on influences. This work potentially paves the way towards resolving a conjecture of Courtade and Kumar. Additionally, in collaboration with Kindler, Minzer, and Lifshitz, we have proposed simplified proofs of concentration inequalities surrounding the Kahn-Kalai-Linial bound and strengthened Friedgut's theorem. All of these contributions are rooted in extensions of the pathwise analysis technique.
Since the start of the project, significant progress has been made in both establishing new inequalities related to the "concentration of mass" phenomenon and advancing the dimension-free analysis of high-dimensional objects. We have also extended the "pathwise analysis" technique in new and impactful directions. Below, we summarize some key outcomes and findings:
1. **Pathwise Methods in Boolean Function Analysis**
Boolean functions are foundational in theoretical computer science, and our research has focused on adapting pathwise techniques to the Boolean hypercube. This has led to the discovery of new concentration bounds and provided alternative proofs for existing ones. Notably, we resolved a 25-year-old conjecture of Talagrand by proving an inequality that generalizes two central results in this field—the Kahn-Kalai-Linial theorem and Talagrand's influence inequality. Moreover, our approach has yielded robust versions of several related inequalities and offered a new pathway for proving quantitative versions of the FKG inequality.
2. **Decomposition and Concentration for Measures on the Discrete Hypercube**
We explored natural conditions under which measures on the Boolean hypercube exhibit "well-behaved" behavior. Specifically, we sought measures that are either highly concentrated or can be decomposed into a small mixture of such concentrated measures. Through pathwise methods, we derived both concentration and decomposition bounds. These results have significant implications for models in statistical physics, improving the analysis of sampling algorithm complexity and supporting the accuracy of "mean field" approximations.
3. **Robust Functional Inequalities in Gaussian Space**
Functional inequalities like the logarithmic Sobolev inequality and the Shannon-Stam inequality play essential roles in statistics and information theory. We developed robust versions of these inequalities in Gaussian space by characterizing functions or measures that approximately saturate them. Our work yielded robust forms of the log-Sobolev, Shannon-Stam, and Talagrand's transportation-entropy inequalities, extending the utility of these results across several domains.
4. **A General Framework for Markov Chain Concentration Bounds**
In collaboration with Yuansi Chen, we proposed a novel framework combining spectral independence and pathwise analysis to both construct and analyze Markov chains in discrete and continuous settings. This approach resulted in the first optimal mixing time for the hardcore model, among other models. Additionally, our follow-up work improved several bounds for sampling uniform measures on convex sets, showing that the hit-and-run random walk has sample complexity equivalent to the ball walk.
5. **Concentration Inequalities and Social Choice**
In recent work with Avi Wigderson and Pei Wu, we provided optimal estimates for the "It ain't over till it's over" theorem in social choice theory. Collaborating with Mikulincer and Raghavendra, we are also finalizing a manuscript that improves the "majority is stablest" theorem, achieving polynomial rather than exponential dependence on influences. This work potentially paves the way towards resolving a conjecture of Courtade and Kumar. Additionally, in collaboration with Kindler, Minzer, and Lifshitz, we have proposed simplified proofs of concentration inequalities surrounding the Kahn-Kalai-Linial bound and strengthened Friedgut's theorem. All of these contributions are rooted in extensions of the pathwise analysis technique.
One should mention the recent breakthrough by Y. Chen ["An almost constant lower bound for the KLS conjecture", Geom. Funct. Anal.] where very substantial progress was made in regards to the Kannan-Lovasz-Simonovitz conjecture, and several follow up works, including the one by Klartag, which comes very close to proving these conjectures. These works build on the very methods proposed by project - pathwise analysis and in particular "stochastic localization" process (which were partially developed by the PI).