This repository contains the code used in the Master Thesis `Randomized low-rank approximation of matrices and tensors' by Davide Pradovera ([email protected]).

The main folder contains the following files:

• LICENSE
• README.md
• startup.m (file to add library directories to MATLAB path)
• PM.m (power method for general matrices)
• gen_matrix_decay.m (algorithm to generate matrices with prescribed decay)
• gen_TT_tensor_decay.m (algorithm to generate TT-tensors whose unfoldings have a prescribed decay)
• recompress.m (method to recompress a matrix in low-rank format)
• round_nonortho.m (rounding of a TT-tensor without orthogonalization step)
• truncate_nonortho.m (truncation of a TT-tensor without orthogonalization step)

The MAT folder contains the files regarding the compression of matrices (Chapter 2 in the Thesis):

• round_randomized.m (rounding of a matrix using Algorithm 3 in the Thesis)
• round_randomized_2norm.m (rounding of a matrix using Algorithm 5 in the Thesis)
• round_randomized_2norm_PM.m (rounding of a matrix using Algorithm 4 in the Thesis)
• round_randomized_normA.m (rounding of a matrix using Algorithm 2 in the Thesis)
• truncate_randomized.m (truncation of a matrix using Algorithm 1 in the Thesis with Gaussian random vectors)
• truncate_randomized_rank1.m (truncation of a matrix using Algorithm 1 in the Thesis with rank-1 random vectors)
• truncate_randomized_uniform.m (truncation of a matrix using Algorithm 1 in the Thesis with uniform random vectors)
• example.m (example of compression of matrix generated using ../gen_matrix_decay.m)

The MAT_HADA folder contains the files regarding the compression of Hadamard products of matrices (Section 4.1 in the Thesis):

• round_hada_randomized.m (rounding of a Hadamard product of matrices using Algorithm 13 in the Thesis)
• truncate_hada_randomized.m (truncation of a Hadamard product of matrices using a modification of Algorithm 1 in the Thesis)
• example.m (example of compression of the Hadamard product of matrices generated using ../gen_matrix_decay.m)

The TT folder contains the files regarding the compression of TT-tensors (Chapter 3 in the Thesis):

• find_range_unfolding_TT.m (single step within Algorithm 11 in the Thesis)
• round_TT_randomized.m (rounding of a TT-tensor using Algorithm 12 in the Thesis)
• truncate_TT_randomized.m (truncation of a TT-tensor using Algorithm 11 in the Thesis)
• update_stored_range_TT.m (algorithm for the extraction of new samples within Algorithms 11 and 12 in the Thesis)
• example.m (example of compression of TT-tensors generated using ../gen_TT_tensor_decay.m)

The TT_HADA folder contains the files regarding the compression of Hadamard products of TT-tensors (Section 4.3 in the Thesis):

• find_range_hada_unfolding_TT.m (single step within Algorithm 14 in the Thesis)
• left_project_hada_TT.m (computation of line 10 within Algorithm 14 in the Thesis)
• multiply_hada_TT.m (computation of line 6 within Algorithm 14 in the Thesis)
• round_hada_TT_randomized.m (rounding of a Hadamard product of TT-tensors using Algorithm 15 in the Thesis)
• truncate_hada_TT_randomized.m (truncation of a Hadamard product of TT-tensors using Algorithm 14 in the Thesis)
• update_stored_range_hada_TT.m (algorithm for the extraction of new samples within Algorithms 14 and 15 in the Thesis)
• example.m (example of compression of the Hadamard product of TT-tensors generated using ../gen_TT_tensor_decay.m)