Hessian approx. fnctn handle based on finite differences of the gradient. function hessfun = approxhessianFD(problem) function hessfun = approxhessianFD(problem, options) Input: A Manopt problem structure (already containing the manifold and enough information to compute the cost gradient) and an options structure (optional), containing one option: options.stepsize (positive double; default: 2^-14). If the gradient cannot be computed or approximated on 'problem', a warning is issued. Output: Returns a function handle, encapsulating a generic finite difference approximation of the Hessian of the problem cost. The finite difference is based on computations of the gradient. The returned hessfun has this calling pattern: function hessfd = hessfun(x, xdot) function hessfd = hessfun(x, xdot, storedb) function hessfd = hessfun(x, xdot, storedb, key) x is a point on the manifold problem.M, xdot is a tangent vector to that manifold at x, storedb is a StoreDB object, and key is the StoreDB key to point x. Usage: Typically, the user will set problem.M and other fields to define the cost and the gradient (typically, problem.cost and problem.grad or problem.egrad). Then, to use this generic purpose Hessian approximation: problem.approxhess = approxhessianFD(problem, options); See also: trustregions

- StoreDB
- canGetApproxGradient Checks whether an approximate gradient can be computed for this problem.
- canGetGradient Checks whether the gradient can be computed for a problem structure.
- getGradient Computes the gradient of the cost function at x.
- mergeOptions Merges two options structures with one having precedence over the other.

- positive_definite_karcher_mean Computes a Karcher mean of a collection of positive definite matrices.
- arc Adaptive regularization by cubics (ARC) minimization algorithm for Manopt
- preconhessiansolve Preconditioner based on the inverse Hessian, by solving linear systems.
- trustregions Riemannian trust-regions solver for optimization on manifolds.

- function hessfd = funhandle(x, xdot, storedb, key)
- function hessfd = hessianFD(stepsize, problem, x, xdot, storedb, key)

0001 function hessfun = approxhessianFD(problem, options) 0002 % Hessian approx. fnctn handle based on finite differences of the gradient. 0003 % 0004 % function hessfun = approxhessianFD(problem) 0005 % function hessfun = approxhessianFD(problem, options) 0006 % 0007 % Input: 0008 % 0009 % A Manopt problem structure (already containing the manifold and enough 0010 % information to compute the cost gradient) and an options structure 0011 % (optional), containing one option: 0012 % options.stepsize (positive double; default: 2^-14). 0013 % 0014 % If the gradient cannot be computed or approximated on 'problem', 0015 % a warning is issued. 0016 % 0017 % Output: 0018 % 0019 % Returns a function handle, encapsulating a generic finite difference 0020 % approximation of the Hessian of the problem cost. The finite difference 0021 % is based on computations of the gradient. 0022 % 0023 % The returned hessfun has this calling pattern: 0024 % 0025 % function hessfd = hessfun(x, xdot) 0026 % function hessfd = hessfun(x, xdot, storedb) 0027 % function hessfd = hessfun(x, xdot, storedb, key) 0028 % 0029 % x is a point on the manifold problem.M, xdot is a tangent vector to that 0030 % manifold at x, storedb is a StoreDB object, and key is the StoreDB key to 0031 % point x. 0032 % 0033 % Usage: 0034 % 0035 % Typically, the user will set problem.M and other fields to define the 0036 % cost and the gradient (typically, problem.cost and problem.grad or 0037 % problem.egrad). Then, to use this generic purpose Hessian approximation: 0038 % 0039 % problem.approxhess = approxhessianFD(problem, options); 0040 % 0041 % See also: trustregions 0042 0043 % The Riemannian Trust-Region method, used in combination with the present 0044 % Hessian approximation, is called RTR-FD. Some convergence theory for it 0045 % is available in this paper: 0046 % 0047 % @incollection{boumal2015rtrfd 0048 % author={Boumal, N.}, 0049 % title={Riemannian trust regions with finite-difference Hessian approximations are globally convergent}, 0050 % year={2015}, 0051 % booktitle={Geometric Science of Information} 0052 % } 0053 0054 % This file is part of Manopt: www.manopt.org. 0055 % Original author: Nicolas Boumal, April 8, 2015. 0056 % Contributors: 0057 % Change log: 0058 % 0059 % Feb. 19, 2015 (NB): 0060 % It is sufficient to ensure positive radial linearity to guarantee 0061 % (together with other assumptions) that this approximation of the 0062 % Hessian will confer global convergence to the trust-regions method. 0063 % Formerly, in-code comments referred to the necessity of having 0064 % complete radial linearity, and that this was harder to achieve. 0065 % This appears not to be necessary after all, which simplifies the 0066 % code. 0067 % 0068 % April 3, 2015 (NB): 0069 % Works with the new StoreDB class system. 0070 % 0071 % April 8, 2015 (NB): 0072 % Changed to approxhessianFD, which now returns a function handle 0073 % that encapsulates the getHessianFD functionality. Will be better 0074 % aligned with the other Hessian approximations to come (which may 0075 % want to use storedb.internal), and now allows specifying the step 0076 % size. 0077 % 0078 % March 17, 2020 (NB): 0079 % Following a bug report by Marco Sutti, added the instruction 0080 % storedb.remove(key1); 0081 % to avoid memory usage ramping up when many inner iterations are 0082 % needed inside of tCG for trustregions. The deciding factor is that 0083 % there is no need to cache the gradient at the artificially produced 0084 % point used here for finite differencing, as this point is not 0085 % visible outside of this function: there is no reason we would visit 0086 % it again. 0087 0088 % This Hessian approximation is based on the gradient: 0089 % check availability. 0090 if ~canGetGradient(problem) && ~canGetApproxGradient(problem) 0091 warning('manopt:approxhessianFD:nogradient', ... 0092 'approxhessianFD requires the gradient to be computable.'); 0093 end 0094 0095 % Set local defaults here, and merge with user options, if any. 0096 localdefaults.stepsize = 2^-14; 0097 if ~exist('options', 'var') || isempty(options) 0098 options = struct(); 0099 end 0100 options = mergeOptions(localdefaults, options); 0101 0102 % Finite-difference parameter: how far do we look? 0103 stepsize = options.stepsize; 0104 0105 % Build and return the function handle here. This extra construct via 0106 % funhandle makes it possible to make storedb and key optional. 0107 hessfun = @funhandle; 0108 function hessfd = funhandle(x, xdot, storedb, key) 0109 % Allow omission of the key, and even of storedb. 0110 if ~exist('key', 'var') 0111 if ~exist('storedb', 'var') 0112 storedb = StoreDB(); 0113 end 0114 key = storedb.getNewKey(); 0115 end 0116 hessfd = hessianFD(stepsize, problem, x, xdot, storedb, key); 0117 end 0118 0119 end 0120 0121 0122 function hessfd = hessianFD(stepsize, problem, x, xdot, storedb, key) 0123 % This function does the actual work. 0124 % 0125 % Original code: Dec. 30, 2012 (NB). 0126 0127 % Extract the input vector norm. 0128 norm_xdot = problem.M.norm(x, xdot); 0129 0130 % First, check whether the step xdot is not too small. 0131 if norm_xdot < eps 0132 hessfd = problem.M.zerovec(x); 0133 return; 0134 end 0135 0136 % Determine how far to retract xdot, so that the point reached does not 0137 % depend on the norm of xdot. This is what ensures radial linearity of 0138 % this present Hessian approximation. 0139 c = stepsize / norm_xdot; 0140 0141 % Compute the gradient at the current point. 0142 grad = getGradient(problem, x, storedb, key); 0143 0144 % Compute a point a little further along xdot, and the gradient there. 0145 % Since this is a new point, we need a new key for it, for storedb. 0146 x1 = problem.M.retr(x, xdot, c); 0147 key1 = storedb.getNewKey(); 0148 grad1 = getGradient(problem, x1, storedb, key1); 0149 0150 % Remove any caching associated to that new point, since there is no 0151 % reason we would be visiting it again. 0152 storedb.remove(key1); 0153 0154 % Transport grad1 back from x1 to x. 0155 grad1 = problem.M.transp(x1, x, grad1); 0156 0157 % Return the finite difference of them: (grad1 - grad)/c. 0158 hessfd = problem.M.lincomb(x, 1/c, grad1, -1/c, grad); 0159 0160 end

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