Choice of Preconditioner for Riemannian Trust Region Method












1












$begingroup$


I'm using MANOPT's (MATLAB toolbox for Riemannian optimization) implementation of the Riemannian trust region method from



Absil, P.-A.; Mahony, R.; Sepulchre, R., Optimization algorithms on matrix manifolds., Princeton, NJ: Princeton University Press (ISBN 978-0-691-13298-3/hbk). xv, 224 p. (2008). ZBL1147.65043.



to minimize a smooth, partially separable cost function
$$f(mathbf{x}_1,ldots,mathbf{x}_n)=sum_{i,j} f_{i,j}(mathbf{x}_i,mathbf{x}_j)$$
on the product manifold $mathcal{N}=mathcal{M}^n$ where $mathcal{M}subsetmathbb{R}^m$ is a regular submanifold. In other words: each $mathbf{x}_i$ is an element of $mathcal{M}$. In general, the matrix representation (in the ambient space) of the Riemannian Hessian of the cost must be assumed to be indefinite. Due to the partially separable structure it is a sparse block matrix.



Unfortunately the performance of the algorithm in terms of execution time is very unsatisfactory. It performs lots of inner iterations (Steihaug-Toint CG), often exceeding the maximum number of inner iterations allowed. Checking the eigenvalues of the Hessian shows that they are scattered over a wide range and some are quite close to zero, which would suggest using a preconditioner. The MANOPT tutorial states, that the preconditioner should be a symmetric positive definite operator on the tangent space at the current (outer) iterate.



I've implemented a preconditioner $P$ based on an incomplete Cholesky factorization of the Hessian
$$ H+alpha I = LL^top+R$$
with a diagonal shift $alpha$ s.t. $H+alpha I>0$ (i.e. Cholesky works). Currently $alpha$ is selected by trial and error, starting with an estimate of the absolute value of the most negative eigenvalue and gradually increasing the parameter until the decomposition succeeds. The preconditioner itself is given by:
$$P=P_kLL^top P_k$$
where $P_k$ is the orthogonal projection onto the tangent space at the k-th (outer) iterate. With this preconditioner I'm indeed observing a reduced inner iteration count. However, given the same initial iterate, the optimization is now converging to a different solution.



Now my question is, whether this is normal/can be expected to happen and if yes, why? Could this be an issue of several local minima (the cost definitely has several local minima)? Moreover I'm wondering whether a different approach to the preconditioning issue might give better results. I've found few publications dealing with general-purpose preconditioners for the Riemannian trust region method and none for the indefinite Hessian case. Any input would be appreciated!










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I'm using MANOPT's (MATLAB toolbox for Riemannian optimization) implementation of the Riemannian trust region method from



    Absil, P.-A.; Mahony, R.; Sepulchre, R., Optimization algorithms on matrix manifolds., Princeton, NJ: Princeton University Press (ISBN 978-0-691-13298-3/hbk). xv, 224 p. (2008). ZBL1147.65043.



    to minimize a smooth, partially separable cost function
    $$f(mathbf{x}_1,ldots,mathbf{x}_n)=sum_{i,j} f_{i,j}(mathbf{x}_i,mathbf{x}_j)$$
    on the product manifold $mathcal{N}=mathcal{M}^n$ where $mathcal{M}subsetmathbb{R}^m$ is a regular submanifold. In other words: each $mathbf{x}_i$ is an element of $mathcal{M}$. In general, the matrix representation (in the ambient space) of the Riemannian Hessian of the cost must be assumed to be indefinite. Due to the partially separable structure it is a sparse block matrix.



    Unfortunately the performance of the algorithm in terms of execution time is very unsatisfactory. It performs lots of inner iterations (Steihaug-Toint CG), often exceeding the maximum number of inner iterations allowed. Checking the eigenvalues of the Hessian shows that they are scattered over a wide range and some are quite close to zero, which would suggest using a preconditioner. The MANOPT tutorial states, that the preconditioner should be a symmetric positive definite operator on the tangent space at the current (outer) iterate.



    I've implemented a preconditioner $P$ based on an incomplete Cholesky factorization of the Hessian
    $$ H+alpha I = LL^top+R$$
    with a diagonal shift $alpha$ s.t. $H+alpha I>0$ (i.e. Cholesky works). Currently $alpha$ is selected by trial and error, starting with an estimate of the absolute value of the most negative eigenvalue and gradually increasing the parameter until the decomposition succeeds. The preconditioner itself is given by:
    $$P=P_kLL^top P_k$$
    where $P_k$ is the orthogonal projection onto the tangent space at the k-th (outer) iterate. With this preconditioner I'm indeed observing a reduced inner iteration count. However, given the same initial iterate, the optimization is now converging to a different solution.



    Now my question is, whether this is normal/can be expected to happen and if yes, why? Could this be an issue of several local minima (the cost definitely has several local minima)? Moreover I'm wondering whether a different approach to the preconditioning issue might give better results. I've found few publications dealing with general-purpose preconditioners for the Riemannian trust region method and none for the indefinite Hessian case. Any input would be appreciated!










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      I'm using MANOPT's (MATLAB toolbox for Riemannian optimization) implementation of the Riemannian trust region method from



      Absil, P.-A.; Mahony, R.; Sepulchre, R., Optimization algorithms on matrix manifolds., Princeton, NJ: Princeton University Press (ISBN 978-0-691-13298-3/hbk). xv, 224 p. (2008). ZBL1147.65043.



      to minimize a smooth, partially separable cost function
      $$f(mathbf{x}_1,ldots,mathbf{x}_n)=sum_{i,j} f_{i,j}(mathbf{x}_i,mathbf{x}_j)$$
      on the product manifold $mathcal{N}=mathcal{M}^n$ where $mathcal{M}subsetmathbb{R}^m$ is a regular submanifold. In other words: each $mathbf{x}_i$ is an element of $mathcal{M}$. In general, the matrix representation (in the ambient space) of the Riemannian Hessian of the cost must be assumed to be indefinite. Due to the partially separable structure it is a sparse block matrix.



      Unfortunately the performance of the algorithm in terms of execution time is very unsatisfactory. It performs lots of inner iterations (Steihaug-Toint CG), often exceeding the maximum number of inner iterations allowed. Checking the eigenvalues of the Hessian shows that they are scattered over a wide range and some are quite close to zero, which would suggest using a preconditioner. The MANOPT tutorial states, that the preconditioner should be a symmetric positive definite operator on the tangent space at the current (outer) iterate.



      I've implemented a preconditioner $P$ based on an incomplete Cholesky factorization of the Hessian
      $$ H+alpha I = LL^top+R$$
      with a diagonal shift $alpha$ s.t. $H+alpha I>0$ (i.e. Cholesky works). Currently $alpha$ is selected by trial and error, starting with an estimate of the absolute value of the most negative eigenvalue and gradually increasing the parameter until the decomposition succeeds. The preconditioner itself is given by:
      $$P=P_kLL^top P_k$$
      where $P_k$ is the orthogonal projection onto the tangent space at the k-th (outer) iterate. With this preconditioner I'm indeed observing a reduced inner iteration count. However, given the same initial iterate, the optimization is now converging to a different solution.



      Now my question is, whether this is normal/can be expected to happen and if yes, why? Could this be an issue of several local minima (the cost definitely has several local minima)? Moreover I'm wondering whether a different approach to the preconditioning issue might give better results. I've found few publications dealing with general-purpose preconditioners for the Riemannian trust region method and none for the indefinite Hessian case. Any input would be appreciated!










      share|cite|improve this question









      $endgroup$




      I'm using MANOPT's (MATLAB toolbox for Riemannian optimization) implementation of the Riemannian trust region method from



      Absil, P.-A.; Mahony, R.; Sepulchre, R., Optimization algorithms on matrix manifolds., Princeton, NJ: Princeton University Press (ISBN 978-0-691-13298-3/hbk). xv, 224 p. (2008). ZBL1147.65043.



      to minimize a smooth, partially separable cost function
      $$f(mathbf{x}_1,ldots,mathbf{x}_n)=sum_{i,j} f_{i,j}(mathbf{x}_i,mathbf{x}_j)$$
      on the product manifold $mathcal{N}=mathcal{M}^n$ where $mathcal{M}subsetmathbb{R}^m$ is a regular submanifold. In other words: each $mathbf{x}_i$ is an element of $mathcal{M}$. In general, the matrix representation (in the ambient space) of the Riemannian Hessian of the cost must be assumed to be indefinite. Due to the partially separable structure it is a sparse block matrix.



      Unfortunately the performance of the algorithm in terms of execution time is very unsatisfactory. It performs lots of inner iterations (Steihaug-Toint CG), often exceeding the maximum number of inner iterations allowed. Checking the eigenvalues of the Hessian shows that they are scattered over a wide range and some are quite close to zero, which would suggest using a preconditioner. The MANOPT tutorial states, that the preconditioner should be a symmetric positive definite operator on the tangent space at the current (outer) iterate.



      I've implemented a preconditioner $P$ based on an incomplete Cholesky factorization of the Hessian
      $$ H+alpha I = LL^top+R$$
      with a diagonal shift $alpha$ s.t. $H+alpha I>0$ (i.e. Cholesky works). Currently $alpha$ is selected by trial and error, starting with an estimate of the absolute value of the most negative eigenvalue and gradually increasing the parameter until the decomposition succeeds. The preconditioner itself is given by:
      $$P=P_kLL^top P_k$$
      where $P_k$ is the orthogonal projection onto the tangent space at the k-th (outer) iterate. With this preconditioner I'm indeed observing a reduced inner iteration count. However, given the same initial iterate, the optimization is now converging to a different solution.



      Now my question is, whether this is normal/can be expected to happen and if yes, why? Could this be an issue of several local minima (the cost definitely has several local minima)? Moreover I'm wondering whether a different approach to the preconditioning issue might give better results. I've found few publications dealing with general-purpose preconditioners for the Riemannian trust region method and none for the indefinite Hessian case. Any input would be appreciated!







      riemannian-geometry nonlinear-optimization






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 7 at 15:13









      chrisfchrisf

      62




      62






















          0






          active

          oldest

          votes












          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3065101%2fchoice-of-preconditioner-for-riemannian-trust-region-method%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          0






          active

          oldest

          votes








          0






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3065101%2fchoice-of-preconditioner-for-riemannian-trust-region-method%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Probability when a professor distributes a quiz and homework assignment to a class of n students.

          Aardman Animations

          Are they similar matrix