From 9f5601e65fd952e1588977686e4bfb1d05a0c28a Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 2 Sep 2022 17:46:20 +0200 Subject: Added first draft popular summary. --- popular summary.txt | 5 +++++ 1 file changed, 5 insertions(+) create mode 100644 popular summary.txt diff --git a/popular summary.txt b/popular summary.txt new file mode 100644 index 0000000..8829dcf --- /dev/null +++ b/popular summary.txt @@ -0,0 +1,5 @@ +Understanding the stationary points of a function can tell you a lot about the function. From these points (where the function's derivative vanishes) you can infer topological and geometric properties and, when the function is an energy landscape, these inferences can provide physical insight. For complex energy landscapes in high dimensions, physical insight is often sorely lacking. Our work extends existing methods to count stationary points to a broader class of complex landscapes. Namely, this encompasses landscapes whose equilibrium properties are described by the replica symmetry breaking (RSB) theory of Giorgio Parisi that won the Nobel Prize in physics last year. + +In the paper, we derive an expression for the typical number of stationary points in a class of complex random landscapes. In order to find the typical number rather than the mean number, which is biased by outliers, we use the replica method. We find a form for the solution inspired by Parisi's equilibrium solution, and show it is consistent with known properties at the very lowest energies. We then take the solution and apply it to two specific models with novel RSB structure in their energy landscapes. + +A correct accounting of stationary points for these and other complex landscapes promises to yield important insight into physics in many disciplines where such landscapes appear, from the condensed matter of glasses to the performance of machine learning algorithms. -- cgit v1.2.3-70-g09d2 From 35f22e49b8a62b3a1148751f19e504366de3343f Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Mon, 5 Sep 2022 13:22:58 +0000 Subject: Update on Overleaf. --- popular summary.txt | 10 ++++++++-- 1 file changed, 8 insertions(+), 2 deletions(-) diff --git a/popular summary.txt b/popular summary.txt index 8829dcf..1bf4887 100644 --- a/popular summary.txt +++ b/popular summary.txt @@ -1,5 +1,11 @@ -Understanding the stationary points of a function can tell you a lot about the function. From these points (where the function's derivative vanishes) you can infer topological and geometric properties and, when the function is an energy landscape, these inferences can provide physical insight. For complex energy landscapes in high dimensions, physical insight is often sorely lacking. Our work extends existing methods to count stationary points to a broader class of complex landscapes. Namely, this encompasses landscapes whose equilibrium properties are described by the replica symmetry breaking (RSB) theory of Giorgio Parisi that won the Nobel Prize in physics last year. +Understanding the stationary points of a function tells you a lot about the function. +From these points (where the function's derivative vanishes) you can infer topological and geometric properties and, when the function is an energy landscape, these inferences can provide physical insight. -In the paper, we derive an expression for the typical number of stationary points in a class of complex random landscapes. In order to find the typical number rather than the mean number, which is biased by outliers, we use the replica method. We find a form for the solution inspired by Parisi's equilibrium solution, and show it is consistent with known properties at the very lowest energies. We then take the solution and apply it to two specific models with novel RSB structure in their energy landscapes. +For complex or rugged energy landscapes of many variables, general insights are often sorely lacking, hence the very intense interdisciplinary +activity on the subject over the last three decades. +Our work extends existing methods to count stationary points to a substantially broader class of complex landscapes. Namely, this encompasses *all complex* landscapes whose equilibrium properties are described by the replica symmetry breaking (RSB) theory of Giorgio Parisi recognized by the Nobel Prize in physics last year. + +In the paper, we derive an expression for the typical number of stationary points in these landscapes. In order to find the typical number +one is forced to use the replica method. We find a form for the solution inspired by Parisi's equilibrium solution, and show it is consistent with known properties at the very lowest energies. We then take the solution and apply it to two specific models with novel RSB structure in their energy landscapes. A correct accounting of stationary points for these and other complex landscapes promises to yield important insight into physics in many disciplines where such landscapes appear, from the condensed matter of glasses to the performance of machine learning algorithms. -- cgit v1.2.3-70-g09d2 From 8b741d55f0412f05e072d6c8c83108ebf6ead0fd Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 5 Sep 2022 16:03:52 +0200 Subject: Last submission details. --- defense.txt | 1 + popular summary.txt | 10 ++-------- 2 files changed, 3 insertions(+), 8 deletions(-) create mode 100644 defense.txt diff --git a/defense.txt b/defense.txt new file mode 100644 index 0000000..8720f3a --- /dev/null +++ b/defense.txt @@ -0,0 +1 @@ +Our work advances significantly the state-of-the-art in the study of complexity, allowing the analysis of the geometric properties of many model energies that do not belong to the narrow class for which quenched averages and annealed averages are equal. This advances path-breaking work by Ros et al. in PRX 9 011003 (2019) by extending the methods they pioneered. Providing a conjecture for the quenched complexity also pushes forward the mathematics of complexity, which has been interested in these models as in Ben Arous et al., Comm. Pure Appl. Math., 72 2282. diff --git a/popular summary.txt b/popular summary.txt index 1bf4887..48ac919 100644 --- a/popular summary.txt +++ b/popular summary.txt @@ -1,11 +1,5 @@ -Understanding the stationary points of a function tells you a lot about the function. -From these points (where the function's derivative vanishes) you can infer topological and geometric properties and, when the function is an energy landscape, these inferences can provide physical insight. +Understanding the stationary points of a function tells you a lot about the function. From these points (where the function's derivative vanishes) you can infer topological and geometric properties and, when the function is an energy landscape, these inferences can provide physical insight. For complex or rugged energy landscapes of many variables, insight is often sorely lacking, hence the very intense interdisciplinary activity on the subject over the last three decades. Our work extends existing methods to count stationary points to a substantially broader class of complex landscapes. Namely, this encompasses all complex landscapes whose equilibrium properties are described by the replica symmetry breaking (RSB) theory of Giorgio Parisi recognized by the Nobel Prize in physics last year. -For complex or rugged energy landscapes of many variables, general insights are often sorely lacking, hence the very intense interdisciplinary -activity on the subject over the last three decades. -Our work extends existing methods to count stationary points to a substantially broader class of complex landscapes. Namely, this encompasses *all complex* landscapes whose equilibrium properties are described by the replica symmetry breaking (RSB) theory of Giorgio Parisi recognized by the Nobel Prize in physics last year. - -In the paper, we derive an expression for the typical number of stationary points in these landscapes. In order to find the typical number -one is forced to use the replica method. We find a form for the solution inspired by Parisi's equilibrium solution, and show it is consistent with known properties at the very lowest energies. We then take the solution and apply it to two specific models with novel RSB structure in their energy landscapes. +In the paper, we derive an expression for the typical number of stationary points in these landscapes. In order to find the typical number, one must use the replica method. We find a form for the solution inspired by Parisi's equilibrium solution, and show it is consistent with known properties at the very lowest energies. We then take the solution and apply it to two specific models with novel RSB structure in their energy landscapes. A correct accounting of stationary points for these and other complex landscapes promises to yield important insight into physics in many disciplines where such landscapes appear, from the condensed matter of glasses to the performance of machine learning algorithms. -- cgit v1.2.3-70-g09d2