Joel Andersson

Joel Andersson, Jan-Olov Strömberg

We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an $s$-sparse signal from linear measurements (with high probability) is known to be $m\gtrsim s(\ln s)^3\ln N$. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of $m\times N$-matrices, by considering what we call \emph{low entropy}. We also present an improved condition on the so-called restricted isometry constants, $δ_s$, ensuring sparse recovery via $\ell^1$-minimization. We show that $δ_{2s}<4/\sqrt{41}$ is sufficient and that this can be improved further to almost allow for a sufficient condition of the type $δ_{2s}<2/3$.

Tim Salzmann, Jon Arrizabalaga, Joel Andersson et al.

While real-world problems are often challenging to analyze analytically, deep learning excels in modeling complex processes from data. Existing optimization frameworks like CasADi facilitate seamless usage of solvers but face challenges when integrating learned process models into numerical optimizations. To address this gap, we present the Learning for CasADi (L4CasADi) framework, enabling the seamless integration of PyTorch-learned models with CasADi for efficient and potentially hardware-accelerated numerical optimization. The applicability of L4CasADi is demonstrated with two tutorial examples: First, we optimize a fish's trajectory in a turbulent river for energy efficiency where the turbulent flow is represented by a PyTorch model. Second, we demonstrate how an implicit Neural Radiance Field environment representation can be easily leveraged for optimal control with L4CasADi. L4CasADi, along with examples and documentation, is available under MIT license at https://github.com/Tim-Salzmann/l4casadi