initial commit

% This file is part of GpOptimization.

%

% GpOptimization is free software: you can redistribute it and/or modify

% it under the terms of the GNU General Public License as published by

% the Free Software Foundation, version 3 of the License.

%

% GpOptimization is distributed in the hope that it will be useful,

% but WITHOUT ANY WARRANTY; without even the implied warranty of

% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the

% GNU General Public License for more details.

%

% You should have received a copy of the GNU General Public License

% along with GpOptimization. If not, see <http://www.gnu.org/licenses/>.

%

% Copyright (c) by Emile Contal, 2015

function [B] = basis_cst(X)

%%

% Constant mean function

%% Syntax

%   B = basis_cst(X)

%% Arguments

% * _X_ matrix _(n, d)_ where _n_ is the number of data points and _d_ is the dimension

%% Outputs

% * _B_ vector _(n, 1)_ of ones

%% See also

% <basis_none.html basis_none>

B = [ones(size(X,1),1)];

end

% This file is part of GpOptimization.

%

% GpOptimization is free software: you can redistribute it and/or modify

% it under the terms of the GNU General Public License as published by

% the Free Software Foundation, version 3 of the License.

%

% GpOptimization is distributed in the hope that it will be useful,

% but WITHOUT ANY WARRANTY; without even the implied warranty of

% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the

% GNU General Public License for more details.

%

% You should have received a copy of the GNU General Public License

% along with GpOptimization. If not, see <http://www.gnu.org/licenses/>.

%

% Copyright (c) by Emile Contal, 2015

function [B] = basis_none(X)

%%

% Zero mean GP inferance

%% Syntax

%   B = basis_none(X)

%% Arguments

% * _X_ matrix _(n, d)_ where _n_ is the number of data points and _d_ is the dimension

%% See also

% <basis_cst.html basis_cst>

B = [];

end

% This file is part of GpOptimization.

%

% GpOptimization is free software: you can redistribute it and/or modify

% it under the terms of the GNU General Public License as published by

% the Free Software Foundation, version 3 of the License.

%

% GpOptimization is distributed in the hope that it will be useful,

% but WITHOUT ANY WARRANTY; without even the implied warranty of

% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the

% GNU General Public License for more details.

%

% You should have received a copy of the GNU General Public License

% along with GpOptimization. If not, see <http://www.gnu.org/licenses/>.

%

% Copyright (c) by Emile Contal, 2015

function [HP, kernel, noise, fval] = bfgs_search_prior(Xt, Yt, HPini, kfun, basis)

%%

% Optimize prior hyper-parameter with respect to the pseudo-likelihood, using the BFGS algorithm

%% Syntax

%   HP = bfgs_search_prior(Xt, Yt, HPini, kfun, basis)

%% Arguments

% * _Xt_ matrix _(n, d)_ where _n_ is the number of data points and _d_ is the dimension

% * _Yt_ vector _(n, 1)_ of noisy observations

% * _HPini_ vector _(h, 1)_ of initial hyper-parameters formatted as : _[log(sf2) log(sn) log(w1) log(w2)]_

% * _kfun_ kernel function such as _<kernel_se.html kernel_se>_

% * _basis_ basis function such as _<basis_none.html basis_none>_

%% Outputs

% * _HP_ vector _(h, 1)_ of locally optimal log hyper-parameters

% * _kernel_ found kernel function

% * _noise_ found noise standard deviation

%% See also

% <gp_loolik.html gp_loolik>

NelderMeadIters = 50;

Ht = basis(Xt);

f = @(hp) nllcost(Xt, Yt, hp(1), hp(2), hp(3:end), kfun, Ht);

% Starts with few Nelder-Mead iterations

[hpopt, fvalNM] = fminsearch(f, HPini, optimset('maxFunEvals',NelderMeadIters,'display','off'));

% BGFS search

[hpopt, fval] = fminunc(f, hpopt,optimset('display','off','LargeScale','off'));

HP = hpopt;

kernel = @(x,y) kfun(x,y, exp(HP(1)), exp(HP(3:end)));

noise = exp(HP(1)+HP(2));

end

function [nll] = nllcost(Xt, Yt, sf2, rsn, W, kfun, Ht)

Ktt = kfun(Xt, Xt, exp(sf2), exp(W));

noise = exp(sf2+rsn);

BayesInv = gp_inf(Ktt, Yt, noise, Ht);

nll = gp_loolik(Ktt, Yt, noise, BayesInv, Ht);

end

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% This file is part of GpOptimization.

%

% GpOptimization is free software: you can redistribute it and/or modify

% it under the terms of the GNU General Public License as published by

% the Free Software Foundation, version 3 of the License.

%

% GpOptimization is distributed in the hope that it will be useful,

% but WITHOUT ANY WARRANTY; without even the implied warranty of

% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the

% GNU General Public License for more details.

%

% You should have received a copy of the GNU General Public License

% along with GpOptimization. If not, see <http://www.gnu.org/licenses/>.

%

% Copyright (c) by Emile Contal, 2015

function [eNet, PiX, DeltaT, U] = chaining_tree(D2, dmax, step, iMin, iMax, varargin)

%%

% Compute the chaining tree given the canonical distance matrix between point of _X_

%% Syntax

%   [eNet, PiX, DeltaT, U] = chaining_tree(D2, dmax, step, iMin, iMax)

%   [eNet, PiX, DeltaT, U] = chaining_tree(..., 'Name',Value)

%% Arguments

% * _D2_ matrix _(n, n)_ of canonical squared distance

% * _dmax_ scalar of initial diameter of _X_

% * _step_ scalar > 1 for the geometric decay of $\epsilon_i$

% * _iMin_ integer for the first level to consider

% * _iMax_ integer for the last level to consider

%% Name-Value Pair Arguments

% * _a_ scalar > 1 power to use for the geometric decay in the union bound, e.g. 2

% * _lza_ scalar of logarithm of the Riemann zeta of a, e.g. _log(pi^2/6)_

%% Outputs

% * _eNet_ vector _(1, N)_ of indices of points in the final $\epsilon$-net

% * _PiX_ matrix _(h, n)_ of indices of the closest element of the net for all _h=iMax-iMin_ levels

% * _DeltaT_ matrix _(h, N)_ of diameters of the cells of the net for all _h_ levels

% * _U_ vector _(h, 1)_ of negative log probabilities w.r.t the union bounds for all _h_ levels

%% See also

% <gp_dist.html gp_dist> | <enet_greedy.html enet_greedy>

ip = inputParser;

ip.addOptional('a', 2.2);

ip.addOptional('lza', 0.399141);

ip.parse(varargin{:});

opt = ip.Results;

n = size(D2,1);

Ti = [];

nSteps = iMax-iMin;

PiX = zeros(nSteps,n,'uint32');

DeltaT = inf(nSteps,n);

U = inf(nSteps,1);

for ind=1:nSteps

    i = ind+iMin-1;

    ei = dmax*step^-i;

    [ni,Ti] = enet_greedy(D2,ei,Ti);

    U(ind) = log(ni+1) + opt.a*log(i) + opt.lza;

    [TiDist2, TiInd] = min(D2(Ti,:), [], 1);

    PiX(ind,:) = TiInd;

    for j=1:length(Ti)

        DeltaTi_j = sqrt(max(TiDist2(TiInd==j)));

        if ~isempty(DeltaTi_j)

            DeltaT(ind,j) = DeltaTi_j;

        end

    end

end

eNet = Ti;

end

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