**Race to the bottom** --- the OPTIM@EPFL blog
racetothebottom.xyz/
Blog of OPTIM@EPFL: the chair of continuous optimization at EPFLquarto-1.6.15Mon, 16 Sep 2024 00:00:00 GMTSaddle avoidance and center-stable manifold: a proof from first principles for a general setupAndreea MusatNicolas Boumal
racetothebottom.xyz/posts/saddle-avoidance-general/
]]>nonconvexsaddlesCSMTracetothebottom.xyz/posts/saddle-avoidance-general/Mon, 16 Sep 2024 00:00:00 GMTSaddle avoidance and stable manifold: a proof from first principles for a simple setupNicolas BoumalAndreea Musat
racetothebottom.xyz/posts/saddle-avoidance-simple/
be a iteration map, used to generate sequences as follows: given , let If the sequence converges to a point , then the latter is a fixed point of , that is, . We might converge to any fixed point of (for trivial reasons: just initialize there). It is natural to ask: ]]>nonconvexsaddlesCSMTracetothebottom.xyz/posts/saddle-avoidance-simple/Wed, 21 Aug 2024 00:00:00 GMTForeword to lecture notes by Irène Waldspurger from her Cours PeccotNicolas Boumal
racetothebottom.xyz/posts/foreword-peccot/
the lecture notes for which this foreword is written. ]]>nonconvexrankracetothebottom.xyz/posts/foreword-peccot/Fri, 24 May 2024 00:00:00 GMTBenign landscape of the Brockett cost function: asymmetric caseNicolas Boumal
racetothebottom.xyz/posts/brockett-asymmetric/
be two arbitrary matrices of size (not necessarily symmetric). Some notation: ]]>nonconvexracetothebottom.xyz/posts/brockett-asymmetric/Sun, 28 Apr 2024 00:00:00 GMTPositive semidefinite Laplacians with negative edge weightsNicolas Boumal
racetothebottom.xyz/posts/laplacian-negative-weights/
(symmetric) be the adjacency matrix of an undirected graph, so that is the weight of the edge between nodes and . The degree matrix is diagonal, such that is the (weighted) degree of node . The Laplacian of this graph is the symmetric matrix Simple calculations reveal that encodes a neat quadratic form, namely, where the double sum is for and both ranging from to . ]]>graphsracetothebottom.xyz/posts/laplacian-negative-weights/Sun, 14 Apr 2024 00:00:00 GMTBenign landscape of low-rank approximation: Part IINicolas BoumalChristopher Criscitiello
racetothebottom.xyz/posts/low-rank-approx-corollaries/
Part I, we noted that the optimization problem has a benign landscape: despite the nonconvex constraint, there are no local traps. ]]>nonconvexrankracetothebottom.xyz/posts/low-rank-approx-corollaries/Wed, 13 Dec 2023 00:00:00 GMTBenign landscape of low-rank approximation: Part INicolas BoumalChristopher Criscitiello
racetothebottom.xyz/posts/low-rank-approx/
Eckart–Young–Mirsky theorem states that the best approximation of a matrix by a matrix of rank at most can be computed by truncating an SVD of to dominant components. This is widely known. It is also well known (but rarely proved) that the corresponding optimization problem has a nonconvex yet benign landscape. This post spells out a proof of that fact: ]]>nonconvexrankracetothebottom.xyz/posts/low-rank-approx/Mon, 11 Dec 2023 00:00:00 GMTBenign landscape of the Brockett cost functionNicolas Boumal
racetothebottom.xyz/posts/brockett-symmetric/
be two symmetric matrices of size . Some notation: ]]>nonconvexracetothebottom.xyz/posts/brockett-symmetric/Mon, 04 Dec 2023 00:00:00 GMTOptimality conditions on the orthogonal groupNicolas Boumal
racetothebottom.xyz/posts/optimality-orthogonal/
where is the set of orthogonal matrices and is twice continuously differentiable. ]]>manifoldsracetothebottom.xyz/posts/optimality-orthogonal/Thu, 30 Nov 2023 00:00:00 GMTSimplex, sphere, and Fisher–Rao metricNicolas Boumal
racetothebottom.xyz/posts/fisher-rao-simplex/
]]>manifoldsracetothebottom.xyz/posts/fisher-rao-simplex/Fri, 24 Nov 2023 00:00:00 GMTThe Maximum TheoremNicolas Boumal
racetothebottom.xyz/posts/continuous-maximum/
is a set that depends on , and we consider The Maximum Theorem (Bergé 1963, p116) (see it on google books) is a convenient set of statements about the continuity of the maximal value of that problem as a function of the parameter. In the example, the theorem requires to be compact and to vary continuously “in some sense”. ]]>topologyracetothebottom.xyz/posts/continuous-maximum/Thu, 23 Nov 2023 00:00:00 GMTSmall geodesic trianglesNicolas Boumal
racetothebottom.xyz/posts/small-triangles/
. In a Euclidean space, the vector that brings us from to is the sum of that which brings us from to and that which brings us from to : Curvature breaks the analogous statement on a Riemannian manifold. However, we may still expect that the equality would hold approximately if are near each other. That is indeed the case, as shown below. ]]>manifoldsracetothebottom.xyz/posts/small-triangles/Tue, 21 Nov 2023 00:00:00 GMTRetractions locally preserve distanceNicolas Boumal
racetothebottom.xyz/posts/retraction-distance/
be a Riemannian manifold. For the exponential map, we have the following property: A soft and local version of that extends to retractions, as claimed in (Ring and Wirth 2012, Lem. 6). The statement there is a tad technical (and it also argues a lower-bound). This note aims for a simpler statement. See also (Boumal 2023, Lem. 6.32) for a simpler proof when is compact. ]]>manifoldsracetothebottom.xyz/posts/retraction-distance/Mon, 20 Nov 2023 00:00:00 GMT