# Race to the bottom — the OPTIM@EPFL blog

> Blog of OPTIM@EPFL: the chair of continuous optimization at EPFL

## Pages

- [About](www.racetothebottom.xyz/about.llms.md)
- [Smooth, globally Polyak--Łojasiewicz functions are nonlinear least-squares](www.racetothebottom.xyz/posts/global-polyak-lojasiewicz/index.llms.md)
- [The usual smooth lift of the nuclear norm regularizer enjoys $2 \Rightarrow 1$](www.racetothebottom.xyz/posts/lift-regularizer-nuclear/index.llms.md)
- [Norm of linear combination of unit norm vectors: benign up and down](www.racetothebottom.xyz/posts/sum-sphere-norm/index.llms.md)
- [Differential of the time-one map of a flow at a critical point](www.racetothebottom.xyz/posts/flow-map-differential/index.llms.md)
- [Foreword to lecture notes by Irène Waldspurger from her Cours Peccot](www.racetothebottom.xyz/posts/foreword-peccot/index.llms.md)
- [Benign landscape of low-rank approximation: Part II](www.racetothebottom.xyz/posts/low-rank-approx-corollaries/index.llms.md)
- [The google sheet setup I use to grade assignments](www.racetothebottom.xyz/posts/grading-sheet/index.llms.md)
- [Saddle avoidance and center-stable manifold: a proof from first principles for a general setup](www.racetothebottom.xyz/posts/saddle-avoidance-general/index.llms.md)
- [Benign landscape of low-rank approximation: Part I](www.racetothebottom.xyz/posts/low-rank-approx/index.llms.md)
- [Small geodesic triangles](www.racetothebottom.xyz/posts/small-triangles/index.llms.md)
- [A different way to define embedded submanifolds: Part II](www.racetothebottom.xyz/posts/smooth-retracts-are-manifolds-part2/index.llms.md)
- [Positive semidefinite Laplacians with negative edge weights](www.racetothebottom.xyz/posts/laplacian-negative-weights/index.llms.md)
- [Benign landscape of the Brockett cost function](www.racetothebottom.xyz/posts/brockett-symmetric/index.llms.md)
- [Maximizing the 4-norm over orthogonal matrices has a benign landscape](www.racetothebottom.xyz/posts/four-norm-orthogonal/index.llms.md)
- [The nLab proof that the image of a smooth idempotent map is a smooth manifold](www.racetothebottom.xyz/posts/smooth-retracts-are-manifolds-part1/nlab-proof.llms.md)
- [The sphere-to-simplex lift enjoys $4 \Rightarrow 2$ (and $3$ is not enough)](www.racetothebottom.xyz/posts/sphere-to-simplex/index.llms.md)
- [Pointers for my students on giving talks](www.racetothebottom.xyz/posts/pointers-giving-talks/index.llms.md)
- [A different way to define embedded submanifolds: Part I](www.racetothebottom.xyz/posts/smooth-retracts-are-manifolds-part1/index.llms.md)
- [When the Hessian is a proxy for its Gauss--Newton approximation (and not the other way around)](www.racetothebottom.xyz/posts/Gauss-Newton/index.llms.md)
- [The Sasaki metric for Riemannian submanifolds of Euclidean space](www.racetothebottom.xyz/posts/Sasaki-submanifold/index.llms.md)
- [The zero-set of a nonzero real-analytic function has measure zero](www.racetothebottom.xyz/posts/real-analytic-nullset/index.llms.md)
- [Designing polynomials for Muon's polar factorization: focus on quintics](www.racetothebottom.xyz/posts/polar-poly/index.llms.md)
- [The difference-of-Hadamard lift of the 1-norm regularizer enjoys $2 \Rightarrow 1$](www.racetothebottom.xyz/posts/lift-regularizer-1norm/index.llms.md)
- [Retractions locally preserve distance](www.racetothebottom.xyz/posts/retraction-distance/index.llms.md)
- [Gradient descent can converge to any isolated saddle point](www.racetothebottom.xyz/posts/saddle-convergence-gd/index.llms.md)
- [Simplex, sphere, and Fisher--Rao metric](www.racetothebottom.xyz/posts/fisher-rao-simplex/index.llms.md)
- [Benign landscape of the Brockett cost function: asymmetric case](www.racetothebottom.xyz/posts/brockett-asymmetric/index.llms.md)
- [Optimality conditions on the orthogonal group](www.racetothebottom.xyz/posts/optimality-orthogonal/index.llms.md)
- [An old proof that the Morse Lemma loses only one order of regularity](www.racetothebottom.xyz/posts/Morse-lemma-regularity/index.llms.md)
- [If a smooth function is globally PŁ and coercive, then it has a unique minimizer](www.racetothebottom.xyz/posts/PL-smooth-unique/index.llms.md)
- [Randomization is necessary to escape saddle points in high dimension](www.racetothebottom.xyz/posts/randomized-escape/index.llms.md)
- [The Maximum Theorem](www.racetothebottom.xyz/posts/continuous-maximum/index.llms.md)
- [Saddle avoidance and stable manifold: a proof from first principles for a simple setup](www.racetothebottom.xyz/posts/saddle-avoidance-simple/index.llms.md)
- [Race to the bottom](www.racetothebottom.xyz/index.llms.md)