We study the theoretical limits of the $\ell_0$ (quasi) norm based optimization algorithms when employed for solving classical compressed sensing or sparse regression problems. Considering standard contexts with deterministic signals and statistical systems, we utilize \emph{Fully lifted random duality theory} (Fl RDT) and develop a generic analytical program for studying performance of the \emph{maximum-likelihood} (ML) decoding. The key ML performance parameter, the residual \emph{root mean square error} ($\textbf{RMSE}$), is uncovered to exhibit the so-called \emph{phase-transition} (PT) phenomenon. The associated aPT curve, which separates the regions of systems dimensions where \emph{an} $\ell_0$ based algorithm succeeds or fails in achieving small (comparable to the noise) ML optimal $\textbf{RMSE}$ is precisely determined as well. In parallel, we uncover the existence of another dPT curve which does the same separation but for practically feasible \emph{descending} $\ell_0$ ($d\ell_0$) algorithms. Concrete implementation and practical relevance of the Fl RDT typically rely on the ability to conduct a sizeable set of the underlying numerical evaluations which reveal that for the ML decoding the Fl RDT converges astonishingly fast with corrections in the estimated quantities not exceeding $\sim 0.1\%$ already on the third level of lifting. Analytical results are supplemented by a sizeable set of numerical experiments where we implement a simple variant of $d\ell_0$ and demonstrate that its practical performance very accurately matches the theoretical predictions. Completely surprisingly, a remarkably precise agreement between the simulations and the theory is observed for fairly small dimensions of the order of 100.

Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with representative datasets. Recently, an augmented framework has been developed to overcome some limitations that emerged in the application of the original framework. In this paper, we propose a new class of continuous-depth neural networks with delay, named Neural Delay Differential Equations (NDDEs). To compute the corresponding gradients, we use the adjoint sensitivity method to obtain the delayed dynamics of the adjoint. Differential equations with delays are typically seen as dynamical systems of infinite dimension that possess more fruitful dynamics. Compared to NODEs, NDDEs have a stronger capacity of nonlinear representations. We use several illustrative examples to demonstrate this outstanding capacity. Firstly, we successfully model the delayed dynamics where the trajectories in the lower-dimensional phase space could be mutually intersected and even chaotic in a model-free or model-based manner. Traditional NODEs, without any argumentation, are not directly applicable for such modeling. Secondly, we achieve lower loss and higher accuracy not only for the data produced synthetically by complex models but also for the CIFAR10, a well-known image dataset. Our results on the NDDEs demonstrate that appropriately articulating the elements of dynamical systems into the network design is truly beneficial in promoting network performance.

I am about to submit this new paper on using simulations as a way to incorporate expert knowledge into machine learning, via a similarity kernel. Has anyone seen this idea before (instead of feature data - machine learning training, feature data- simulation - kernel capturing similarity between all samples - machine learning training.? I'd like to see the work and cite it if appropriate.