Daniel@0: function [h, hdata] = glmhess(net, x, t, hdata)
Daniel@0: %GLMHESS Evaluate the Hessian matrix for a generalised linear model.
Daniel@0: %
Daniel@0: %	Description
Daniel@0: %	H = GLMHESS(NET, X, T) takes a GLM network data structure NET,   a
Daniel@0: %	matrix X of input values, and a matrix T of target values and returns
Daniel@0: %	the full Hessian matrix H corresponding to the second derivatives of
Daniel@0: %	the negative log posterior distribution, evaluated for the current
Daniel@0: %	weight and bias values as defined by NET. Note that the target data
Daniel@0: %	is not required in the calculation, but is included to make the
Daniel@0: %	interface uniform with NETHESS.  For linear and logistic outputs, the
Daniel@0: %	computation is very simple and is  done (in effect) in one line in
Daniel@0: %	GLMTRAIN.
Daniel@0: %
Daniel@0: %	[H, HDATA] = GLMHESS(NET, X, T) returns both the Hessian matrix H and
Daniel@0: %	the contribution HDATA arising from the data dependent term in the
Daniel@0: %	Hessian.
Daniel@0: %
Daniel@0: %	H = GLMHESS(NET, X, T, HDATA) takes a network data structure NET, a
Daniel@0: %	matrix X of input values, and a matrix T of  target values, together
Daniel@0: %	with the contribution HDATA arising from the data dependent term in
Daniel@0: %	the Hessian, and returns the full Hessian matrix H corresponding to
Daniel@0: %	the second derivatives of the negative log posterior distribution.
Daniel@0: %	This version saves computation time if HDATA has already been
Daniel@0: %	evaluated for the current weight and bias values.
Daniel@0: %
Daniel@0: %	See also
Daniel@0: %	GLM, GLMTRAIN, HESSCHEK, NETHESS
Daniel@0: %
Daniel@0: 
Daniel@0: %	Copyright (c) Ian T Nabney (1996-2001)
Daniel@0: 
Daniel@0: % Check arguments for consistency
Daniel@0: errstring = consist(net, 'glm', x, t);
Daniel@0: if ~isempty(errstring);
Daniel@0:   error(errstring);
Daniel@0: end
Daniel@0: 
Daniel@0: ndata = size(x, 1);
Daniel@0: nparams = net.nwts;
Daniel@0: nout = net.nout;
Daniel@0: p = glmfwd(net, x);
Daniel@0: inputs = [x ones(ndata, 1)];
Daniel@0: 
Daniel@0: if nargin == 3
Daniel@0:    hdata = zeros(nparams);	% Full Hessian matrix
Daniel@0:    % Calculate data component of Hessian
Daniel@0:    switch net.outfn
Daniel@0: 
Daniel@0:    case 'linear'
Daniel@0:       % No weighting function here
Daniel@0:       out_hess = [x ones(ndata, 1)]'*[x ones(ndata, 1)];
Daniel@0:       for j = 1:nout
Daniel@0:          hdata = rearrange_hess(net, j, out_hess, hdata);
Daniel@0:       end
Daniel@0:    case 'logistic'
Daniel@0:       % Each output is independent
Daniel@0:       e = ones(1, net.nin+1);
Daniel@0:       link_deriv = p.*(1-p);
Daniel@0:       out_hess = zeros(net.nin+1);
Daniel@0:       for j = 1:nout
Daniel@0:          inputs = [x ones(ndata, 1)].*(sqrt(link_deriv(:,j))*e);
Daniel@0:          out_hess = inputs'*inputs;   % Hessian for this output
Daniel@0:          hdata = rearrange_hess(net, j, out_hess, hdata);
Daniel@0:       end
Daniel@0:       
Daniel@0:    case 'softmax'
Daniel@0:       bb_start = nparams - nout + 1;	% Start of bias weights block
Daniel@0:       ex_hess = zeros(nparams);	% Contribution to Hessian from single example
Daniel@0:       for m = 1:ndata
Daniel@0:          X = x(m,:)'*x(m,:);
Daniel@0:          a = diag(p(m,:))-((p(m,:)')*p(m,:));
Daniel@0:          ex_hess(1:nparams-nout,1:nparams-nout) = kron(a, X);
Daniel@0:          ex_hess(bb_start:nparams, bb_start:nparams) = a.*ones(net.nout, net.nout);
Daniel@0:          temp = kron(a, x(m,:));
Daniel@0:          ex_hess(bb_start:nparams, 1:nparams-nout) = temp;
Daniel@0:          ex_hess(1:nparams-nout, bb_start:nparams) = temp';
Daniel@0:          hdata = hdata + ex_hess;
Daniel@0:       end
Daniel@0:     otherwise
Daniel@0:       error(['Unknown activation function ', net.outfn]);
Daniel@0:     end
Daniel@0: end
Daniel@0: 
Daniel@0: [h, hdata] = hbayes(net, hdata);
Daniel@0: 
Daniel@0: function hdata = rearrange_hess(net, j, out_hess, hdata)
Daniel@0: 
Daniel@0: % Because all the biases come after all the input weights,
Daniel@0: % we have to rearrange the blocks that make up the network Hessian.
Daniel@0: % This function assumes that we are on the jth output and that all outputs
Daniel@0: % are independent.
Daniel@0: 
Daniel@0: bb_start = net.nwts - net.nout + 1;	% Start of bias weights block
Daniel@0: ob_start = 1+(j-1)*net.nin; 	% Start of weight block for jth output
Daniel@0: ob_end = j*net.nin;         	% End of weight block for jth output
Daniel@0: b_index = bb_start+(j-1);   	% Index of bias weight
Daniel@0: % Put input weight block in right place
Daniel@0: hdata(ob_start:ob_end, ob_start:ob_end) = out_hess(1:net.nin, 1:net.nin);
Daniel@0: % Put second derivative of bias weight in right place
Daniel@0: hdata(b_index, b_index) = out_hess(net.nin+1, net.nin+1);
Daniel@0: % Put cross terms (input weight v bias weight) in right place
Daniel@0: hdata(b_index, ob_start:ob_end) = out_hess(net.nin+1,1:net.nin);
Daniel@0: hdata(ob_start:ob_end, b_index) = out_hess(1:net.nin, net.nin+1);
Daniel@0: 
Daniel@0: return