L1 Tucker Decomposition

This repository contains alternating optimization algorithms for L1-norm tucker decomposition

Dependencies: Tensor Toolbox (https://tensortoolbox.com/)

Algorithms:

L1HOSVD

• Usage: [U, G, Xhat, L1met, stats]=L1HOSVD(X, Ks, Uin, 'tol', tol, 'maxit', maxit, 'X_clean', X_clean, 'Un_true', Un_true)
• Inputs:
  • X: I-way Input Tensor of size (D = [D1 D2 ... DI])
  • Ks: Low-rank tensor core size (Ks = [d1 d2 ... dI])
  • Uin: Initial Factor matrices (cell array of length I, where Uin{i} is a matrix of size Di by di)
  • (Optional) tol: optimization tolerance (default value: 1e-4)
  • (Optional) maxit: maximum number of algorithm iterations (default value: 50)
  • (Optional) X_clean: Nominal input tensor (input tensor without noises or outliers) [If given, stats.RERR returns reconstruction error vs. basis updates]
  • (Optional) Un_true: Nominal factor matrices (If given, algorithm returns subspace error vs. basis update indexes, in stats.SERR)
• Outputs:
  • U: Factor matrices
  • G: Tensor core (tensor of size Ks)
  • Xhat: Reconstruction of X
  • L1met: The value of the objective function (L1-metric of tensor core) at the end of computation
  • stats: struct containing some information about algorithm execution
    • stats.update_types : list of the indexes of basis (0 for updating B), in the order they are updated in algorithm execution
    • stats.RERR: list of reconstruction errors after each basis update
    • stats.SERR: list of subspace errors after each basis update

L1HOOI function:

• Usage: [U,G,Xhat,L1met_end,stats,funcname,stats_T1] = L1HOOI(X, Ks, Uin, 'tol', tol, 'maxit', maxit, 'X_clean', X_clean, 'Un_true', Un_true, 'T',inf, 'selection', selection)
• Inputs:
  • X: I-way Input Tensor of size (D = [D1 D2 ... DI])
  • Ks: Low-rank tensor core size (Ks = [d1 d2 ... dI])
  • Uin: Initial Factor matrices (cell array of length I, where Uin{i} is a matrix of size Di by di)
  • (Optional) proj: 'L2'(default) for L1HOOI with L2-projection, 'L1' for L1HOOI with L1-projection
  • (Optional) tol: optimization tolerance (default value: 1e-8)
  • (Optional) maxit: maximum number of algorithm iterations (default value: 1000)
  • (Optional) X_clean: Nominal input tensor (input tensor without noises or outliers) [If given, stats.RERR returns reconstruction error vs. basis updates]
  • (Optional) Un_true: Nominal factor matrices (If given, algorithm returns subspace error vs. basis update indexes, in stats.SERR)
  • (Optional) T: If set to a number, it controls the maximum iterations for the outer loop( e.g. set T = 1 for one round of U matrix updates)
  • (Optional) selection: input options are 'default' (for increasing order), 'random' (update in a random order of basis matrices randomly picked at each outer iteration), or a list specifying the order of basis updates (e.g. [3, 1, 2])
• Outputs:
  • U: Factor matrices
  • G: Tensor core (tensor of size Ks)
  • Xhat: Reconstruction of X
  • **L1met_end: The value of the objective function (L1-metric of tensor core) at the end of computation
  • stats: struct containing some information about algorithm execution
    • stats.update_types : list of the indexes of basis (0 for updating B), in the order they are updated in algorithm execution
    • stats.RERR: list of reconstruction errors after each basis update
    • stats.SERR: list of subspace errors after each basis update
  • funcname name of the function configuration (e.g. 'L1HOOI/L2proj': is returned for above function call if T is not given)
  • stats_T1: struct containing some information about algorithm execution, until the end of round 1

L1HOOI Configurations:

• L1-HOOI/L2 projection (Increasing Order): [U,G,Xhat] = L1HOOI(X, Ks, Uin)
• L1-HOOI/L1 projection (Increasing Order): [U,G,Xhat] = L1HOOI(X, Ks, Uin, 'proj', 'L1')
• L1-HOOI/L2 projection (random permutation Order): [U,G,Xhat] = L1HOOI(X, Ks, Uin, 'selection', 'random')
• L1-HOOI/L1 projection (random permutation Order): [U,G,Xhat] = L1HOOI(X, Ks, Uin, 'proj', 'L1', 'selection', 'random')
• L1-HOOI/L2 projection (T=1): [U, G, Xhat] = L1HOOI(X, Ks, Uin, 'T', 1) (or use [~ ,~ ,~ ,~ ,~ ,~ , stats_T1] = L1HOOI(X, Ks, Uin) for concurrent computation of L1-HOOI/L2 projection and L1-HOOI/L2 projection(T=1))
• L1-HOOI/L1 projection (T=1): [U, G, Xhat] = L1HOOI(X, Ks, Uin, 'proj', 'L1', 'T', 1)