ArrayXXd CMT::BlobNonlinearity::gradient(const ArrayXXd& inputs) const {
if(inputs.rows() != 1)
throw Exception("Data has to be stored in one row.");
ArrayXXd diff = ArrayXXd::Zero(mNumComponents, inputs.cols());
diff.rowwise() += inputs.row(0);
diff.colwise() -= mMeans;
ArrayXXd diffSq = diff.square();
ArrayXd precisions = mLogPrecisions.exp();
ArrayXd weights = mLogWeights.exp();
ArrayXXd negEnergy = diffSq.colwise() * (-precisions / 2.);
ArrayXXd negEnergyExp = negEnergy.exp();
ArrayXXd gradient(3 * mNumComponents, inputs.cols());
// gradient of mean
gradient.topRows(mNumComponents) = (diff * negEnergyExp).colwise() * (weights * precisions);
// gradient of log-precisions
gradient.middleRows(mNumComponents, mNumComponents) = (diffSq / 2. * negEnergyExp).colwise() * (-weights * precisions);
// gradient of log-weights
gradient.bottomRows(mNumComponents) = negEnergyExp.colwise() * weights;
return gradient;
}
ArrayXXd CMT::BlobNonlinearity::operator()(const ArrayXXd& inputs) const {
if(inputs.rows() != 1)
throw Exception("Data has to be stored in one row.");
ArrayXXd diff = ArrayXXd::Zero(mNumComponents, inputs.cols());
diff.rowwise() += inputs.row(0);
diff.colwise() -= mMeans;
ArrayXXd negEnergy = diff.square().colwise() * (-mLogPrecisions.exp() / 2.);
return (mLogWeights.exp().transpose().matrix() * negEnergy.exp().matrix()).array() + mEpsilon;
}
ArrayXXd CMT::BlobNonlinearity::derivative(const ArrayXXd& inputs) const {
if(inputs.rows() != 1)
throw Exception("Data has to be stored in one row.");
ArrayXXd diff = ArrayXXd::Zero(mNumComponents, inputs.cols());
diff.rowwise() -= inputs.row(0);
diff.colwise() += mMeans;
ArrayXd precisions = mLogPrecisions.exp();
ArrayXXd negEnergy = diff.square().colwise() * (-precisions / 2.);
return (mLogWeights.exp() * precisions).transpose().matrix() * (diff * negEnergy.exp()).matrix();
}
bool CMT::Mixture::train(
const MatrixXd& data,
const MatrixXd& dataValid,
const Parameters& parameters,
const Component::Parameters& componentParameters)
{
if(parameters.initialize && !initialized())
initialize(data, parameters, componentParameters);
ArrayXXd logJoint(numComponents(), data.cols());
Array<double, Dynamic, 1> postSum;
Array<double, 1, Dynamic> logLik;
ArrayXXd post;
ArrayXXd weights;
// training and validation log-loss for checking convergence
double avgLogLoss = numeric_limits<double>::infinity();
double avgLogLossNew;
double avgLogLossValid = evaluate(dataValid);
double avgLogLossValidNew = avgLogLossValid;
int counter = 0;
// backup model parameters
VectorXd priors = mPriors;
vector<Component*> components;
for(int k = 0; k < numComponents(); ++k)
components.push_back(mComponents[k]->copy());
for(int i = 0; i < parameters.maxIter; ++i) {
// compute joint probability of data and assignments (E)
#pragma omp parallel for
for(int k = 0; k < numComponents(); ++k)
logJoint.row(k) = mComponents[k]->logLikelihood(data) + log(mPriors[k]);
// compute normalized posterior (E)
logLik = logSumExp(logJoint);
// average negative log-likelihood in bits per component
avgLogLossNew = -logLik.mean() / log(2.) / dim();
if(parameters.verbosity > 0) {
if(i % parameters.valIter == 0) {
// print training and validation error
cout << setw(6) << i;
cout << setw(14) << setprecision(7) << avgLogLossNew;
cout << setw(14) << setprecision(7) << avgLogLossValidNew << endl;
} else {
// print training error
cout << setw(6) << i << setw(14) << setprecision(7) << avgLogLossNew << endl;
}
}
// test for convergence
if(avgLogLoss - avgLogLossNew < parameters.threshold)
return true;
avgLogLoss = avgLogLossNew;
// compute normalized posterior (E)
post = (logJoint.rowwise() - logLik).exp();
postSum = post.rowwise().sum();
weights = post.colwise() / postSum;
// optimize prior weights (M)
if(parameters.trainPriors) {
mPriors = postSum / data.cols() + parameters.regularizePriors;
mPriors /= mPriors.sum();
}
// optimize components (M)
if(parameters.trainComponents) {
#pragma omp parallel for
for(int k = 0; k < numComponents(); ++k)
mComponents[k]->train(data, weights.row(k), componentParameters);
} else {
return true;
}
if((i + 1) % parameters.valIter == 0) {
// check validation error
avgLogLossValidNew = evaluate(dataValid);
if(avgLogLossValidNew < avgLogLossValid) {
// backup new found model parameters
priors = mPriors;
for(int k = 0; k < numComponents(); ++k)
*components[k] = *mComponents[k];
avgLogLossValid = avgLogLossValidNew;
} else {
counter++;
if(parameters.valLookAhead > 0 && counter >= parameters.valLookAhead) {
// set parameters to best parameters found during training
mPriors = priors;
for(int k = 0; k < numComponents(); ++k) {
*mComponents[k] = *components[k];
delete components[k];
}
return true;
}
}
}
}
if(parameters.verbosity > 0)
cout << setw(6) << parameters.maxIter << setw(11) << setprecision(5) << evaluate(data) << endl;
return false;
}
bool CMT::Mixture::train(
const MatrixXd& data,
const Parameters& parameters,
const Component::Parameters& componentParameters)
{
if(data.rows() != dim())
throw Exception("Data has wrong dimensionality.");
if(parameters.initialize && !initialized())
initialize(data, parameters, componentParameters);
ArrayXXd logJoint(numComponents(), data.cols());
Array<double, Dynamic, 1> postSum;
Array<double, 1, Dynamic> logLik;
ArrayXXd post;
ArrayXXd weights;
double avgLogLoss = numeric_limits<double>::infinity();
double avgLogLossNew;
for(int i = 0; i < parameters.maxIter; ++i) {
// compute joint probability of data and assignments (E)
#pragma omp parallel for
for(int k = 0; k < numComponents(); ++k)
logJoint.row(k) = mComponents[k]->logLikelihood(data) + log(mPriors[k]);
// compute normalized posterior (E)
logLik = logSumExp(logJoint);
// average negative log-likelihood in bits per component
avgLogLossNew = -logLik.mean() / log(2.) / dim();
if(parameters.verbosity > 0)
cout << setw(6) << i << setw(14) << setprecision(7) << avgLogLossNew << endl;
// test for convergence
if(avgLogLoss - avgLogLossNew < parameters.threshold)
return true;
avgLogLoss = avgLogLossNew;
// compute normalized posterior (E)
post = (logJoint.rowwise() - logLik).exp();
postSum = post.rowwise().sum();
weights = post.colwise() / postSum;
// optimize prior weights (M)
if(parameters.trainPriors) {
mPriors = postSum / data.cols() + parameters.regularizePriors;
mPriors /= mPriors.sum();
}
// optimize components (M)
if(parameters.trainComponents) {
#pragma omp parallel for
for(int k = 0; k < numComponents(); ++k)
mComponents[k]->train(data, weights.row(k), componentParameters);
} else {
return true;
}
}
if(parameters.verbosity > 0)
cout << setw(6) << parameters.maxIter << setw(14) << setprecision(7) << evaluate(data) << endl;
return false;
}
Array<double, 1, Dynamic> CMT::logMeanExp(const ArrayXXd& array) {
Array<double, 1, Dynamic> arrayMax = array.colwise().maxCoeff() - 1.;
return arrayMax + (array.rowwise() - arrayMax).exp().colwise().mean().log();
}
pair<pair<ArrayXXd, ArrayXXd>, Array<double, 1, Dynamic> > CMT::STM::computeDataGradient(
const MatrixXd& input,
const MatrixXd& output) const
{
// make sure nonlinearity is differentiable
DifferentiableNonlinearity* nonlinearity =
dynamic_cast<DifferentiableNonlinearity*>(mNonlinearity);
if(!nonlinearity)
throw Exception("Nonlinearity has to be differentiable.");
if(input.rows() != dimIn())
throw Exception("Input has wrong dimensionality.");
if(output.rows() != 1)
throw Exception("Output has wrong dimensionality.");
if(input.cols() != output.cols())
throw Exception("Number of inputs and outputs should be the same.");
if(dimInNonlinear() && !dimInLinear()) {
Array<double, 1, Dynamic> responses;
ArrayXXd jointEnergy;
if(numFeatures() > 0)
jointEnergy = mWeights * (mFeatures.transpose() * input).array().square().matrix()
+ mPredictors * input;
else
jointEnergy = mPredictors * input;
jointEnergy.colwise() += mBiases.array();
jointEnergy *= mSharpness;
responses = logSumExp(jointEnergy);
// posterior over components for each input
MatrixXd posterior = (jointEnergy.rowwise() - responses).array().exp();
responses /= mSharpness;
Array<double, 1, Dynamic> tmp0 = (*mNonlinearity)(responses);
Array<double, 1, Dynamic> tmp1 = -mDistribution->gradient(output, tmp0);
Array<double, 1, Dynamic> tmp2 = nonlinearity->derivative(responses);
ArrayXXd avgPredictor = mPredictors.transpose() * posterior;
ArrayXXd tmp3;
if(numFeatures() > 0) {
ArrayXXd avgWeights = (2. * mWeights).transpose() * posterior;
tmp3 = mFeatures * (avgWeights * (mFeatures.transpose() * input).array()).matrix();
} else {
tmp3 = ArrayXXd::Zero(avgPredictor.rows(), avgPredictor.cols());
}
return make_pair(
make_pair(
(tmp3 + avgPredictor).rowwise() * (tmp1 * tmp2),
ArrayXXd::Zero(output.rows(), output.cols())),
mDistribution->logLikelihood(output, tmp0));
} else if(dimInNonlinear() && dimInLinear()) {
// split inputs into linear and nonlinear components
MatrixXd inputNonlinear = input.topRows(dimInNonlinear());
MatrixXd inputLinear = input.bottomRows(dimInLinear());
Array<double, 1, Dynamic> responses;
ArrayXXd jointEnergy;
if(numFeatures() > 0)
jointEnergy = mWeights * (mFeatures.transpose() * inputNonlinear).array().square().matrix()
+ mPredictors * input;
else
jointEnergy = mPredictors * inputNonlinear;
jointEnergy.colwise() += mBiases.array();
jointEnergy *= mSharpness;
responses = logSumExp(jointEnergy);
// posterior over components for each input
MatrixXd posterior = (jointEnergy.rowwise() - responses).array().exp();
responses /= mSharpness;
responses += (mLinearPredictor.transpose() * inputLinear).array();
Array<double, 1, Dynamic> tmp0 = (*mNonlinearity)(responses);
Array<double, 1, Dynamic> tmp1 = -mDistribution->gradient(output, tmp0);
Array<double, 1, Dynamic> tmp2 = nonlinearity->derivative(responses);
ArrayXXd avgPredictor = mPredictors.transpose() * posterior;
ArrayXXd tmp3;
if(numFeatures() > 0) {
ArrayXXd avgWeights = (2. * mWeights).transpose() * posterior;
tmp3 = mFeatures * (avgWeights * (mFeatures.transpose() * inputNonlinear).array()).matrix();
} else {
tmp3 = ArrayXXd::Zero(avgPredictor.rows(), avgPredictor.cols());
}
// concatenate gradients of nonlinear and linear component
ArrayXXd inputGradient(dimIn(), input.cols());
inputGradient <<
(tmp3 + avgPredictor).rowwise() * (tmp1 * tmp2),
mLinearPredictor * (tmp1 * tmp2).matrix();
return make_pair(
make_pair(
inputGradient,
ArrayXXd::Zero(output.rows(), output.cols())),
mDistribution->logLikelihood(output, tmp0));
} else if(dimInLinear()) {
double avgBias = logSumExp(mSharpness * mBiases)(0, 0) / mSharpness;
Array<double, 1, Dynamic> responses = (mLinearPredictor.transpose() * input).array() + avgBias;
Array<double, 1, Dynamic> tmp0 = (*mNonlinearity)(responses);
Array<double, 1, Dynamic> tmp1 = -mDistribution->gradient(output, tmp0);
Array<double, 1, Dynamic> tmp2 = nonlinearity->derivative(responses);
return make_pair(
make_pair(
mLinearPredictor * (tmp1 * tmp2).matrix(),
ArrayXXd::Zero(output.rows(), output.cols())),
mDistribution->logLikelihood(output, tmp0));
}
return make_pair(
make_pair(
ArrayXXd::Zero(input.rows(), input.cols()),
ArrayXXd::Zero(output.rows(), output.cols())),
logLikelihood(input, output));
}
