Module props: various simple properties of TT-tensors

Package teneva, module props: various properties of TT-tensors.

This module contains the basic properties of TT-tensors, including “erank”, “ranks”, “shape”, etc.

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teneva.props.erank(Y)

Compute effective TT-rank of the given TT-tensor.

Effective TT-rank r of a TT-tensor Y with shape [n_1, n_2, …, n_d] and TT-ranks r_0, r_1, …, r_d (r_0 = r_d = 1) is a solution of equation n_1 r + sum_{lpha=2}^{d-1} n_lpha r^2 + n_d r = sum_{lpha=1}^{d} n_lpha r_{lpha-1} r_{lpha}.

The representation with a constant TT-rank r (r_0 = 1, r_1 = r_2 = … = r_{d-1} = r, r_d = 1) yields the same total number of parameters as in the original decomposition of the tensor Y.

Parameters:

Y (list) – TT-tensor.

Returns:

effective TT-rank.

Return type:

float

Examples:

# 10-dim random TT-tensor with TT-rank 2:
Y = teneva.rand([5]*10, 2)

# The effective TT-rank:
teneva.erank(Y)

# >>> ----------------------------------------
# >>> Output:

# 2.0
#

Note that it also works for 2-dimensional arrays (i.e., matrices):

# 2-dim random TT-tensor (matrix) with TT-rank 20:
Y = teneva.rand([5]*2, 20)

# The effective TT-rank:
teneva.erank(Y)

# >>> ----------------------------------------
# >>> Output:

# 20
#

teneva.props.ranks(Y)

Return the TT-ranks of the given TT-tensor.

Parameters:

Y (list) – TT-tensor.

Returns:

TT-ranks in form of the 1D array of ints of the length d+1, where d is a number of tensor dimensions (the first and last elements are equal 1).

Return type:

np.ndarray

Examples:

# 5-dim random TT-tensor with TT-rank 2:
Y = teneva.rand([10, 12, 8, 8, 30], 2)

# TT-ranks of the TT-tensor:
teneva.ranks(Y)

# >>> ----------------------------------------
# >>> Output:

# array([1, 2, 2, 2, 2, 1])
#

teneva.props.shape(Y)

Return the shape of the given TT-tensor.

Parameters:

Y (list) – TT-tensor.

Returns:

shape of the tensor in form of the 1D array of ints of the length d, where d is a number of tensor dimensions.

Return type:

np.ndarray

Examples:

# 5-dim random TT-tensor with TT-rank 2:
Y = teneva.rand([10, 12, 8, 8, 30], 2)

# Shape of the TT-tensor:
teneva.shape(Y)

# >>> ----------------------------------------
# >>> Output:

# array([10, 12,  8,  8, 30])
#

teneva.props.size(Y)

Return the size (number of parameters) of the given TT-tensor.

Parameters:

Y (list) – TT-tensor.

Returns:

total number of parameters in the TT-representation (it is a sum of sizes of all TT-cores).

Return type:

int

Examples:

# 5-dim random TT-tensor with TT-rank 2:
Y = teneva.rand([10, 12, 8, 8, 30], 2)

# Size of the TT-tensor:
teneva.size(Y)

# >>> ----------------------------------------
# >>> Output:

# 192
#