tensor.cpp

// Xerus - A General Purpose Tensor Library
// Copyright (C) 2014-2017 Benjamin Huber and Sebastian Wolf. 
// 
// Xerus is free software: you can redistribute it and/or modify
// it under the terms of the GNU Affero General Public License as published
// by the Free Software Foundation, either version 3 of the License,
// or (at your option) any later version.
// 
// Xerus 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 Affero General Public License for more details.
// 
// You should have received a copy of the GNU Affero General Public License
// along with Xerus. If not, see <http://www.gnu.org/licenses/>.
//
// For further information on Xerus visit https://libXerus.org 
// or contact us at contact@libXerus.org.

/**
* @file
* @brief Implementation of the Tensor class.
*/

#include <xerus/tensor.h>
#include <xerus/misc/check.h>

#include <xerus/misc/containerSupport.h>
#include <xerus/misc/basicArraySupport.h>
#include <xerus/misc/math.h>
#include <xerus/misc/internal.h>

#include <xerus/blasLapackWrapper.h>
#include <xerus/cholmod_wrapper.h>
#include <xerus/sparseTimesFullContraction.h>
#include <xerus/index.h>
#include <xerus/tensorNetwork.h>

#include <xerus/cholmod_wrapper.h>

#include <fstream>
#include <iomanip>

namespace xerus {
	size_t Tensor::sparsityFactor = 4;
	
	/*- - - - - - - - - - - - - - - - - - - - - - - - - - Constructors - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -*/
	
	Tensor::Tensor(const Representation _representation) : Tensor(DimensionTuple({}), _representation) { } 
	
	
	Tensor::Tensor(DimensionTuple _dimensions, const Representation _representation, const Initialisation _init) 
		: dimensions(std::move(_dimensions)), size(misc::product(dimensions)), representation(_representation)
	{
		REQUIRE(size != 0, "May not create tensors with an dimension == 0.");
		
		if(representation == Representation::Dense) {
			denseData.reset(new value_t[size], internal::array_deleter_vt);
			if(_init == Initialisation::Zero) {
				misc::set_zero(denseData.get(), size);
			}
		} else {
			sparseData.reset(new std::map<size_t, value_t>());
		}
	}
	
	
	Tensor::Tensor(DimensionTuple _dimensions, std::unique_ptr<value_t[]>&& _data)
	: dimensions(std::move(_dimensions)), size(misc::product(dimensions)), representation(Representation::Dense), denseData(_data.release()) {
		REQUIRE(size != 0, "May not create tensors with an dimension == 0.");
	}
	
	
	Tensor::Tensor(DimensionTuple _dimensions, const std::function<value_t()>& _f) : Tensor(std::move(_dimensions), Representation::Dense, Initialisation::None) {
		value_t* const realData = denseData.get();
		for (size_t i=0; i < size; ++i) {
			realData[i] = _f();
		}
	}
	
	
	Tensor::Tensor(DimensionTuple _dimensions, const std::function<value_t(const size_t)>& _f) : Tensor(std::move(_dimensions), Representation::Dense, Initialisation::None) {
		value_t* const realData = denseData.get();
		for (size_t i=0; i < size; ++i) {
			realData[i] = _f(i);
		}
	}
	
	
	Tensor::Tensor(DimensionTuple _dimensions, const std::function<value_t(const MultiIndex&)>& _f) : Tensor(std::move(_dimensions), Representation::Dense, Initialisation::None) {
		value_t* const realData = denseData.get();
		MultiIndex multIdx(degree(), 0);
		size_t idx = 0;
		while (true) {
			realData[idx] = _f(multIdx);
			// Increasing indices
			idx++;
			size_t changingIndex = degree()-1;
			multIdx[changingIndex]++;
			while(multIdx[changingIndex] == dimensions[changingIndex]) {
				multIdx[changingIndex] = 0;
				changingIndex--;
				// Return on overflow 
				if(changingIndex >= degree()) { return; }
				multIdx[changingIndex]++;
			}
		}
	}
	
	
	Tensor::Tensor(DimensionTuple _dimensions, const size_t _N, const std::function<std::pair<size_t, value_t>(size_t, size_t)>& _f) : Tensor(std::move(_dimensions), Representation::Sparse, Initialisation::Zero) {
		REQUIRE(_N <= size, "Cannot create more non zero entries that the dimension of the Tensor.");
		for (size_t i = 0; i < _N; ++i) {
			std::pair<size_t, value_t> entry = _f(i, size);
			REQUIRE(entry.first < size, "Postion is out of bounds " << entry.first);
			REQUIRE(!misc::contains(*sparseData, entry.first), "Allready contained " << entry.first);
			sparseData->insert(std::move(entry));
		}
	}
	
	
	Tensor Tensor::ones(DimensionTuple _dimensions) {
		Tensor ret(std::move(_dimensions), Representation::Dense, Initialisation::None);
		value_t* const data = ret.get_dense_data();
		for(size_t i = 0; i < ret.size; ++i) {
			data[i] = 1.0;
		}
		return ret;
	}
		
	Tensor Tensor::identity(DimensionTuple _dimensions) {
		REQUIRE(_dimensions.size()%2 == 0, "Identity tensor must have even degree, here: " << _dimensions.size());
		const size_t d = _dimensions.size();
		
		Tensor ret(std::move(_dimensions), Representation::Sparse);
		MultiIndex position(d, 0);
		
		if(d == 0) {
			ret[position] = 1.0;
		} else {
			bool notMultiLevelBreak = true;
			while(notMultiLevelBreak) {
				ret[position] = 1.0;
				
				position[0]++; position[d/2]++;
				size_t node = 0;
				while(position[node] == std::min(ret.dimensions[node], ret.dimensions[d/2+node])) {
					position[node] = position[d/2+node] = 0;
					node++;
					if(node == d/2) {notMultiLevelBreak = false; break;}
					position[node]++; position[d/2+node]++;
				}
			}
		}
		
		return ret;
	}
	
	
	Tensor Tensor::kronecker(DimensionTuple _dimensions) {
		Tensor ret(std::move(_dimensions), Representation::Sparse);
		if(ret.degree() == 0) {
			ret[{}] = 1.0;
		} else {
			for(size_t i = 0; i < misc::min(ret.dimensions); ++i) {
				ret[MultiIndex(ret.degree(), i)] = 1.0;
			}
		}
		return ret;
	}
	
	
	Tensor Tensor::dirac(DimensionTuple _dimensions, const MultiIndex& _position) {
		Tensor ret(std::move(_dimensions), Representation::Sparse);
		ret[_position] = 1.0;
		return ret;
	}
	
	
	Tensor Tensor::dirac(DimensionTuple _dimensions, const size_t _position) {
		Tensor ret(std::move(_dimensions), Representation::Sparse);
		ret[_position] = 1.0;
		return ret;
	}
	
	
	Tensor Tensor::dense_copy() const {
		Tensor ret(*this);
		ret.use_dense_representation();
		return ret;
	}
	
	
	Tensor Tensor::sparse_copy() const {
		Tensor ret(*this);
		ret.use_sparse_representation();
		return ret;
	}
	
	Tensor& Tensor::operator=(const TensorNetwork& _network) {
		return operator=(Tensor(_network));
	}
	
	
	/*- - - - - - - - - - - - - - - - - - - - - - - - - - Information - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -*/
	size_t Tensor::degree() const {
		return dimensions.size();
	}
	
	
	bool Tensor::has_factor() const {
// 		return std::abs(1.0-factor) > std::numeric_limits<double>::epsilon();
		#pragma GCC diagnostic push
			#pragma GCC diagnostic ignored "-Wfloat-equal"
			return (factor != 1.0);
		#pragma GCC diagnostic pop
	}
	
	
	bool Tensor::is_dense() const {
		INTERNAL_CHECK((representation == Representation::Dense && denseData && !sparseData) || (representation == Representation::Sparse && sparseData && !denseData), "Internal Error: " << bool(representation) << bool(denseData) << bool(sparseData));
		return representation == Representation::Dense;
	}
	
	
	bool Tensor::is_sparse() const {
		INTERNAL_CHECK((representation == Representation::Dense && denseData && !sparseData) || (representation == Representation::Sparse && sparseData && !denseData), "Internal Error: " << bool(representation) << bool(denseData) << bool(sparseData));
		return representation == Representation::Sparse;
	}
	
	
	size_t Tensor::sparsity() const {
		if(is_sparse()) {
			return sparseData->size();
		}
		return size;
	}
	
	
	size_t Tensor::count_non_zero_entries(const value_t _eps) const {
		if(is_dense()) {
			size_t count = 0;
			for(size_t i = 0; i < size; ++i) {
				if(std::abs(denseData.get()[i]) > _eps ) { count++; }
			}
			return count;
		}
		size_t count = 0;
		for(const auto& entry : *sparseData) {
			if(std::abs(entry.second) > _eps) { count++; } 
		}
		return count;
	}
	
	
	bool Tensor::all_entries_valid() const {
		if(is_dense()) {
			for(size_t i = 0; i < size; ++i) {
				if(!std::isfinite(denseData.get()[i])) { return false; } 
			}
		} else {
			for(const auto& entry : *sparseData) {
				if(!std::isfinite(entry.second)) {return false; } 
			}
		}
		return true;
	}
	
	
	size_t Tensor::reorder_cost() const {
		return is_sparse() ? 10*sparsity() : size;
	}
	
	
	value_t Tensor::one_norm() const {
		if(is_dense()) {
			return std::abs(factor)*blasWrapper::one_norm(denseData.get(), size);
		} 
		value_t norm = 0;
		for(const auto& entry : *sparseData) {
			norm += std::abs(entry.second);
		}
		return std::abs(factor)*norm;
	}
	
	value_t Tensor::frob_norm() const {
		if(is_dense()) {
			return std::abs(factor)*blasWrapper::two_norm(denseData.get(), size);
		} 
		value_t norm = 0;
		for(const auto& entry : *sparseData) {
			norm += misc::sqr(entry.second);
		}
		return std::abs(factor)*sqrt(norm);
	}
	
	
	/*- - - - - - - - - - - - - - - - - - - - - - - - - - Basic arithmetics - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -*/
	Tensor& Tensor::operator+=(const Tensor& _other) {
		plus_minus_equal<1>(*this, _other);
		return *this;
	}
	
	
	Tensor& Tensor::operator-=(const Tensor& _other) {
		plus_minus_equal<-1>(*this, _other);
		return *this;
	}
	
	
	Tensor& Tensor::operator*=(const value_t _factor) {
		factor *= _factor;
		return *this;
	}
	
	
	Tensor& Tensor::operator/=(const value_t _divisor) {
		factor /= _divisor;
		return *this;
	}
	
	
	/*- - - - - - - - - - - - - - - - - - - - - - - - - - Access - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -*/
	value_t& Tensor::operator[](const size_t _position) {
		REQUIRE(_position < size, "Position " << _position << " does not exist in Tensor of dimensions " << dimensions);
		
		ensure_own_data_and_apply_factor();
		
		if(is_dense()) {
			return denseData.get()[_position];
		}
		value_t& result = (*sparseData)[_position];
		use_dense_representation_if_desirable();
		if (is_dense()) {
			return denseData.get()[_position];
		}
		return result;
	}
	

	value_t Tensor::operator[](const size_t _position) const {
		REQUIRE(_position < size, "Position " << _position << " does not exist in Tensor of dimensions " << dimensions);
		
		if(is_dense()) {
			return factor*denseData.get()[_position];
		} 
		const std::map<size_t, value_t>::const_iterator entry = sparseData->find(_position);
		if(entry == sparseData->end()) {
			return 0.0;
		}
		return factor*entry->second;
	}
	
	
	value_t& Tensor::operator[](const MultiIndex& _positions) {
		return operator[](multiIndex_to_position(_positions, dimensions));
	}
	
	
	value_t Tensor::operator[](const MultiIndex& _positions) const {
		return operator[](multiIndex_to_position(_positions, dimensions));
	}
	
	
	value_t& Tensor::at(const size_t _position) {
		REQUIRE(_position < size, "Position " << _position << " does not exist in Tensor of dimensions " << dimensions);
		REQUIRE(!has_factor(), "at() must not be called if there is a factor.");
		REQUIRE((is_dense() && denseData.unique()) || (is_sparse() && sparseData.unique()) , "Data must be unique to call at().");
		
		if(is_dense()) {
			return denseData.get()[_position];
		} 
			value_t& result = (*sparseData)[_position];
			use_dense_representation_if_desirable();
			if (is_dense()) {
				return denseData.get()[_position];
			} 
				return result;
			
		
	}
	
	
	value_t Tensor::cat(const size_t _position) const {
		REQUIRE(_position < size, "Position " << _position << " does not exist in Tensor of dimensions " << dimensions);
		REQUIRE(!has_factor(), "at() must not be called if there is a factor.");
		
		if(is_dense()) {
			return denseData.get()[_position];
		} 
			const std::map<size_t, value_t>::const_iterator entry = sparseData->find(_position);
			if(entry == sparseData->end()) {
				return 0.0;
			} 
				return entry->second;
			
		
	}
	
	
	value_t* Tensor::get_dense_data() {
		use_dense_representation();
		ensure_own_data_and_apply_factor();
		return denseData.get();
	}
	
	
	value_t* Tensor::get_unsanitized_dense_data() {
		REQUIRE(is_dense(), "Unsanitized dense data requested, but representation is not dense!");
		return denseData.get();
	}
	
	
	const value_t* Tensor::get_unsanitized_dense_data() const  {
		REQUIRE(is_dense(), "Unsanitized dense data requested, but representation is not dense!");
		return denseData.get();
	}
	
	
	value_t* Tensor::override_dense_data()  {
		factor = 1.0;
		if(!denseData.unique()) {
			sparseData.reset();
			denseData.reset(new value_t[size], internal::array_deleter_vt);
			representation = Representation::Dense;
		}
		return denseData.get();
	}
	
	
	const std::shared_ptr<value_t>& Tensor::get_internal_dense_data() {
		REQUIRE(is_dense(), "Internal dense data requested, but representation is not dense!");
		return denseData;
	}
	
	
	std::map<size_t, value_t>& Tensor::get_sparse_data() {
		CHECK(is_sparse(), warning, "Request for sparse data although the Tensor is not sparse.");
		use_sparse_representation();
		ensure_own_data_and_apply_factor();
		return *sparseData.get();
	}
	
	
	std::map<size_t, value_t>& Tensor::get_unsanitized_sparse_data() {
		REQUIRE(is_sparse(), "Unsanitized sparse data requested, but representation is not sparse!");
		return *sparseData.get();
	}
	
	
	const std::map<size_t, value_t>& Tensor::get_unsanitized_sparse_data() const  {
		REQUIRE(is_sparse(), "Unsanitized sparse data requested, but representation is not sparse!");
		return *sparseData.get();
	}
	
	
	std::map<size_t, value_t>& Tensor::override_sparse_data() {
		factor = 1.0;
		if(sparseData.unique()) {
			INTERNAL_CHECK(is_sparse(), "Internal Error");
			sparseData->clear();
		} else {
			denseData.reset();
			sparseData.reset(new std::map<size_t, value_t>());
			representation = Representation::Sparse;
		}
		
		return *sparseData.get();
	}
	
	
	const std::shared_ptr<std::map<size_t, value_t>>& Tensor::get_internal_sparse_data() {
		REQUIRE(is_sparse(), "Internal sparse data requested, but representation is not sparse!");
		return sparseData;
	}
	
	
	/*- - - - - - - - - - - - - - - - - - - - - - - - - - Indexing - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -*/
	internal::IndexedTensor<Tensor> Tensor::operator()(const std::vector<Index>&  _indices) {
		return internal::IndexedTensor<Tensor>(this, _indices, false);
	}
	
	
	internal::IndexedTensor<Tensor> Tensor::operator()(      std::vector<Index>&& _indices) {
		return internal::IndexedTensor<Tensor>(this, std::move(_indices), false);
	}
	
	
	internal::IndexedTensorReadOnly<Tensor> Tensor::operator()(const std::vector<Index>&  _indices) const {
		return internal::IndexedTensorReadOnly<Tensor>(this, _indices);
	}
	
	
	internal::IndexedTensorReadOnly<Tensor> Tensor::operator()(      std::vector<Index>&& _indices) const {
		return internal::IndexedTensorReadOnly<Tensor>(this, std::move(_indices));
	}
	
	
	/*- - - - - - - - - - - - - - - - - - - - - - - - - - Modififiers - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -*/
	void Tensor::reset(DimensionTuple _newDim, const Representation _representation, const Initialisation _init) {
		const size_t oldDataSize = size;
		dimensions = std::move(_newDim);
		size = misc::product(dimensions);
		factor = 1.0;
		
		if(_representation == Representation::Dense) {
			if(representation == Representation::Dense) {
				if(oldDataSize != size || !denseData.unique()) {
					denseData.reset(new value_t[size], internal::array_deleter_vt);
				}
			} else {
				sparseData.reset();
				denseData.reset(new value_t[size], internal::array_deleter_vt);
				representation = _representation;
			}
			
			if(_init == Initialisation::Zero) {
				memset(denseData.get(), 0, size*sizeof(value_t));
			}
		} else {
			if(representation == Representation::Sparse) {
				if(sparseData.unique()) {
					sparseData->clear();
				} else {
					sparseData.reset(new std::map<size_t, value_t>());
				}
			} else {
				denseData.reset();
				sparseData.reset(new std::map<size_t, value_t>());
				representation = _representation;
			}
		}
	}
	
	
	void Tensor::reset(DimensionTuple _newDim, const Initialisation _init) {
		const size_t oldDataSize = size;
		dimensions = std::move(_newDim);
		size = misc::product(dimensions);
		factor = 1.0;
		
		if(representation == Representation::Dense) {
			if(oldDataSize != size || !denseData.unique()) {
				denseData.reset(new value_t[size], internal::array_deleter_vt);
			}
			
			if(_init == Initialisation::Zero) {
				memset(denseData.get(), 0, size*sizeof(value_t));
			}
		} else {
			if(sparseData.unique()) {
				sparseData->clear();
			} else {
				sparseData.reset(new std::map<size_t, value_t>());
			}
		}
	}
	
	void Tensor::reset() {
		dimensions.clear();
		factor = 1.0;
		
		if(representation == Representation::Dense) {
			if(size != 1 || !denseData.unique()) {
				denseData.reset(new value_t[1], internal::array_deleter_vt);
			}
			*denseData.get() = 0.0;
		} else {
			if(sparseData.unique()) {
				sparseData->clear();
			} else {
				sparseData.reset(new std::map<size_t, value_t>());
			}
		}
		size = 1;
	}
	
	
	void Tensor::reset(DimensionTuple _newDim, const std::shared_ptr<value_t>& _newData) {
		dimensions = std::move(_newDim);
		size = misc::product(dimensions);
		factor = 1.0;
		
		if(representation == Representation::Sparse) {
			sparseData.reset();
			representation = Representation::Dense;
		}
		
		denseData = _newData;
	}
	
	
	void Tensor::reset(DimensionTuple _newDim, std::unique_ptr<value_t[]>&& _newData) {
		dimensions = std::move(_newDim);
		size = misc::product(dimensions);
		factor = 1.0;
		
		if(representation == Representation::Sparse) {
			sparseData.reset();
			representation = Representation::Dense;
		}
		
		denseData.reset(_newData.release());
	}
	
	void Tensor::reset(DimensionTuple _newDim, std::map<size_t, value_t>&& _newData) {
		dimensions = std::move(_newDim);
		size = misc::product(dimensions);
		factor = 1.0;
		
		if(representation == Representation::Dense) {
			denseData.reset();
			representation = Representation::Sparse;
		}
		
		std::shared_ptr<std::map<size_t, value_t>> newD(new std::map<size_t, value_t>(std::move(_newData)));
		sparseData = std::move(newD);
	}
		
	void Tensor::reinterpret_dimensions(DimensionTuple _newDimensions) {
		REQUIRE(misc::product(_newDimensions) == size, "New dimensions must not change the size of the tensor in reinterpretation: " << misc::product(_newDimensions) << " != " << size);
		dimensions = std::move(_newDimensions); // NOTE pass-by-value
	}
	
	
	void Tensor::resize_mode(const size_t _mode, const size_t _newDim, size_t _cutPos) {
		REQUIRE(_mode < degree(), "Can't resize mode " << _mode << " as the tensor is only order " << degree());

		if (dimensions[_mode] == _newDim) { return; }  // Trivial case: Nothing to do

		const size_t oldDim = dimensions[_mode];
		_cutPos = std::min(_cutPos, oldDim);
		
		REQUIRE(_newDim > 0, "Dimension must be larger than 0! Is " << _newDim);
		REQUIRE(_newDim > oldDim || _cutPos >= oldDim -_newDim, "Cannot remove " << oldDim -_newDim << " slates starting (exclusivly) at position " << _cutPos);
		
		const size_t dimStepSize = misc::product(dimensions, _mode+1, degree());
		const size_t oldStepSize = oldDim*dimStepSize;
		const size_t newStepSize = _newDim*dimStepSize;
		const size_t blockCount = size / oldStepSize; //  == misc::product(dimensions, 0, _n);
		const size_t newsize = blockCount*newStepSize;
		
		if(is_dense()) {
			std::unique_ptr<value_t[]> tmpData(new value_t[newsize]);
			
			if (_newDim > oldDim) { // Add new slates
				const size_t insertBlockSize = (_newDim-oldDim)*dimStepSize;
				if(_cutPos == 0) {
					for ( size_t i = 0; i < blockCount; ++i ) {
						value_t* const currData = tmpData.get()+i*newStepSize;
						misc::set_zero(currData, insertBlockSize);
						misc::copy(currData+insertBlockSize, denseData.get()+i*oldStepSize, oldStepSize);
					}
				} else if(_cutPos == oldDim) {
					for ( size_t i = 0; i < blockCount; ++i ) {
						value_t* const currData = tmpData.get()+i*newStepSize;
						misc::copy(currData, denseData.get()+i*oldStepSize, oldStepSize);
						misc::set_zero(currData+oldStepSize, insertBlockSize);
					}
				} else {
					const size_t preBlockSize = _cutPos*dimStepSize;
					const size_t postBlockSize = (oldDim-_cutPos)*dimStepSize;
					for (size_t i = 0; i < blockCount; ++i) {
						value_t* const currData = tmpData.get()+i*newStepSize;
						misc::copy(currData, denseData.get()+i*oldStepSize, preBlockSize);
						misc::set_zero(currData+preBlockSize, insertBlockSize);
						misc::copy(currData+preBlockSize+insertBlockSize, denseData.get()+i*oldStepSize+preBlockSize, postBlockSize);
					}
				}
			} else { // Remove slates
				if (_cutPos < oldDim) {
					const size_t removedSlates = (oldDim-_newDim);
					const size_t removedBlockSize = removedSlates*dimStepSize;
					const size_t preBlockSize = (_cutPos-removedSlates)*dimStepSize;
					const size_t postBlockSize = (oldDim-_cutPos)*dimStepSize;
					
					INTERNAL_CHECK(removedBlockSize+preBlockSize+postBlockSize == oldStepSize && preBlockSize+postBlockSize == newStepSize, "IE");
					
					for (size_t i = 0; i < blockCount; ++i) {
						value_t* const currData = tmpData.get()+i*newStepSize;
						misc::copy(currData, denseData.get()+i*oldStepSize, preBlockSize);
						misc::copy(currData+preBlockSize, denseData.get()+i*oldStepSize+preBlockSize+removedBlockSize, postBlockSize);
					}
				} else {
					for (size_t i = 0; i < blockCount; ++i) {
						misc::copy(tmpData.get()+i*newStepSize, denseData.get()+i*oldStepSize, newStepSize);
					}
				}
			}
			denseData.reset(tmpData.release(), internal::array_deleter_vt);
		
		} else {
			std::unique_ptr<std::map<size_t, value_t>> tmpData(new std::map<size_t, value_t>());
			
			if (_newDim > oldDim) { // Add new slates
				const size_t slatesAdded = _newDim-oldDim;
				for(const auto& entry : *sparseData.get()) {
					// Decode the position as i*oldStepSize + j*DimStepSize + k
					const size_t k = entry.first%dimStepSize;
					const size_t j = (entry.first%oldStepSize)/dimStepSize;
					const size_t i = entry.first/oldStepSize;
					
					if(j < _cutPos) { // Entry remains at current position j
						tmpData->emplace(i*newStepSize+j*dimStepSize+k, entry.second);
					} else { // Entry moves to position j+slatesAdded
						tmpData->emplace(i*newStepSize+(j+slatesAdded)*dimStepSize+k, entry.second);
					}
				}
			} else { // Remove slates
				const size_t slatesRemoved = oldDim-_newDim;
				for(const auto& entry : *sparseData.get()) {
					// Decode the position as i*oldStepSize + j*DimStepSize + k
					const size_t k = entry.first%dimStepSize;
					const size_t j = (entry.first%oldStepSize)/dimStepSize;
					const size_t i = entry.first/oldStepSize;
					
					if(j < _cutPos-slatesRemoved) { // Entry remains at current position j
						tmpData->emplace(i*newStepSize+j*dimStepSize+k, entry.second);
					} else if(j >= _cutPos) { // Entry advances in position j
						tmpData->emplace(i*newStepSize+(j-slatesRemoved)*dimStepSize+k, entry.second);
					}
				}
			}
			sparseData.reset(tmpData.release());
		}
		
		dimensions[_mode] = _newDim;
		size = newsize;
	}
	
	
	void Tensor::fix_mode(const size_t _mode, const size_t _slatePosition) {
		REQUIRE(_slatePosition < dimensions[_mode], "The given slatePosition must be smaller than the corresponding dimension. Here " << _slatePosition << " >= " << dimensions[_mode] << ", dim = " << dimensions << "=" << size << " mode " << _mode);
		
		const size_t stepCount = misc::product(dimensions, 0, _mode);
		const size_t blockSize = misc::product(dimensions, _mode+1, degree());
		const size_t stepSize = dimensions[_mode]*blockSize;
		const size_t slateOffset = _slatePosition*blockSize;
		
		if(is_dense()) {
			std::unique_ptr<value_t[]> tmpData(new value_t[stepCount*blockSize]);
			
			// Copy data
			for(size_t i = 0; i < stepCount; ++i) {
				misc::copy(tmpData.get()+i*blockSize, denseData.get()+i*stepSize+slateOffset, blockSize);
			}
			
			denseData.reset(tmpData.release(), &internal::array_deleter_vt);
		} else {
			std::unique_ptr<std::map<size_t, value_t>> tmpData(new std::map<size_t, value_t>());
			
			for(const auto& entry : *sparseData.get()) {
				// Decode the position as i*stepSize + j*blockSize + k
				const size_t k = entry.first%blockSize;
				const size_t j = (entry.first%stepSize)/blockSize;
				const size_t i = entry.first/stepSize;
				
				if(j == _slatePosition) {
					tmpData->emplace(i*blockSize+k, entry.second);
				}
			}
			
			sparseData.reset(tmpData.release());
		}
		
		// Adjust dimensions
		dimensions.erase(dimensions.begin()+long(_mode));
		size = stepCount*blockSize;
	}
	
	
	void Tensor::remove_slate(const size_t _mode, const size_t _pos) {
		REQUIRE(_mode < degree(), "");
		REQUIRE(_pos < dimensions[_mode], _pos << " " << dimensions[_mode]);
		REQUIRE(dimensions[_mode] > 1, "");
		
		resize_mode(_mode, dimensions[_mode]-1, _pos+1);
	}
	
	
	void Tensor::perform_trace(size_t _firstMode, size_t _secondMode) {
		REQUIRE(_firstMode != _secondMode, "Given indices must not coincide");
		REQUIRE(dimensions[_firstMode] == dimensions[_secondMode], "Dimensions of trace indices must coincide.");
		
		if(_firstMode > _secondMode) { std::swap(_firstMode, _secondMode); }
		
		
		
		const size_t front = misc::product(dimensions, 0, _firstMode);
		const size_t mid = misc::product(dimensions, _firstMode+1, _secondMode);
		const size_t back = misc::product(dimensions, _secondMode+1, degree());
		const size_t traceDim = dimensions[_firstMode];
		const size_t frontStepSize = traceDim*mid*traceDim*back;
		const size_t traceStepSize = mid*traceDim*back+back;
		const size_t midStepSize = traceDim*back;
		
		size = front*mid*back;
		
		if(is_dense()) {
			std::unique_ptr<value_t[]> newData(new value_t[size]);
			misc::set_zero(newData.get(), size);
			
			for(size_t f = 0; f < front; ++f) {
				for(size_t t = 0; t < traceDim; ++t) {
					for(size_t m = 0; m < mid; ++m) {
						misc::add_scaled(newData.get()+(f*mid+m)*back, factor, denseData.get()+f*frontStepSize+t*traceStepSize+m*midStepSize, back);
					}
				}
			}
			
			denseData.reset(newData.release(), internal::array_deleter_vt);
		} else {
			std::unique_ptr<std::map<size_t, value_t>> newData( new std::map<size_t, value_t>());
			
			for(const auto& entry : *sparseData) {
				size_t pos = entry.first;
				const size_t backIdx = pos%back;
				pos /= back;
				const size_t traceBackIdx = pos%traceDim;
				pos /= traceDim;
				const size_t midIdx = pos%mid;
				pos /= mid;
				const size_t traceFrontIdx = pos%traceDim;
				pos /= traceDim;
				const size_t frontIdx = pos;
				
				if(traceFrontIdx == traceBackIdx) {
					(*newData)[(frontIdx*mid + midIdx)*back + backIdx] += factor*entry.second;
				}
			}
			
			sparseData.reset(newData.release());
		}
		
		dimensions.erase(dimensions.begin()+_secondMode);
		dimensions.erase(dimensions.begin()+_firstMode);
		factor = 1.0;
	}
	
	
	void Tensor::modify_diagonal_entries(const std::function<void(value_t&)>& _f) {
		ensure_own_data_and_apply_factor();
		
		if(degree() == 0) {
			_f(at(0)); 
		} else {
			size_t stepSize = 1;
			for(size_t i = 1; i < degree(); ++i) {
				stepSize *= dimensions[i];
				stepSize += 1;
			}
			
			const size_t numDiags = misc::min(dimensions);
			
			for(size_t i = 0; i < numDiags; ++i){
				_f(at(i*stepSize));
			}
		}
	}
	
	
	void Tensor::modify_diagonal_entries(const std::function<void(value_t&, const size_t)>& _f) {
		ensure_own_data_and_apply_factor();
		
		if(degree() == 0) {
			_f(at(0), 0); 
		} else {
			size_t stepSize = 1;
			for(size_t i = 1; i < degree(); ++i) {
				stepSize *= dimensions[i];
				stepSize += 1;
			}
			
			const size_t numDiags = misc::min(dimensions);
			
			for(size_t i = 0; i < numDiags; ++i){
				_f(at(i*stepSize), i);
			}
		}
	}
	
	
	void Tensor::modify_entries(const std::function<void(value_t&)>& _f) {
		ensure_own_data_and_apply_factor();
		if(is_dense()) {
			for(size_t i = 0; i < size; ++i) { _f(at(i)); }
		} else {
			for(size_t i = 0; i < size; ++i) {
				value_t val = cat(i);
				const value_t oldVal = val;
				_f(val);
				if(misc::hard_not_equal(val, 0.0)) {
					at(i) = val;
				} else if( misc::hard_not_equal(oldVal, 0.0)) {
					IF_CHECK( size_t succ =) sparseData->erase(i);
					INTERNAL_CHECK(succ == 1, "Internal Error");
				}
			}
		}
	}
	

	void Tensor::modify_entries(const std::function<void(value_t&, const size_t)>& _f) {
		ensure_own_data_and_apply_factor();
		if(is_dense()) {
			for(size_t i = 0; i < size; ++i) { _f(at(i), i); }
		} else {
			for(size_t i = 0; i < size; ++i) {
				value_t val = cat(i);
				const value_t oldVal = val;
				_f(val, i);
				if(misc::hard_not_equal(val, 0.0)) {
					at(i) = val;
				} else if( misc::hard_not_equal(oldVal, 0.0)) {
					IF_CHECK( size_t succ =) sparseData->erase(i);
					INTERNAL_CHECK(succ == 1, "Internal Error");
				}
			}
		}
	}
	
	
	void Tensor::modify_entries(const std::function<void(value_t&, const MultiIndex&)>& _f) {
		ensure_own_data_and_apply_factor();
		
		MultiIndex multIdx(degree(), 0);
		size_t idx = 0;
		while (true) {
			if(is_dense()) {
				_f(at(idx), multIdx);
			} else {
				value_t val = cat(idx);
				const value_t oldVal = val;
				_f(val, multIdx);