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diff --git a/topology.tex b/topology.tex index 56af0c6..32f1a6a 100644 --- a/topology.tex +++ b/topology.tex @@ -171,6 +171,20 @@ studied properties of the cost function \mathscr C(\mathbf x)=\frac12\sum_{k=1}^M\big[V_k(\mathbf x)-V_0\big]^2 \end{equation} which achieves zero only for configurations that satisfy all the constraints. +Introduced in Ref.~\cite{Fyodorov_2019_A}, the existence of solutions and the +geometric structure of the cost function were studied for the linear problem in +a series of papers \cite{Fyodorov_2019_A, Fyodorov_2020_Counting, +Fyodorov_2022_Optimization} and later reviewed \cite{Vivo_2024_Random}. Some +work on the equilibrium measure of the cost function in the nonlinear problem +was made in Ref.~\cite{Tublin_2022_A}, and the problem was solved in +Ref.~\cite{Urbani_2023_A}. Subsequent work has studied varied dynamics applied +to the cost function, including gradient descent, Hessian descent, Langevin, +stochastic gradient descent, and approximate message passing +\cite{Kamali_2023_Dynamical, Kamali_2023_Stochastic, Montanari_2023_Solving, +Montanari_2024_On}. Finally, some progress has been made on aspects of the +geometric structure of the cost function in the nonlinear problem +\cite{Kent-Dobias_2024_Conditioning, Kent-Dobias_2024_Algorithm-independent}. + From the perspective of the cost function, the set of solutions looks like a network of flat canyons at the bottom of the cost landscape. Here we dispense with the cost function and study the set of solutions directly. This set can be written as |