Module act_many: operations with a set of TT-tensors

Package teneva, module act_many: operations with a set of TT-tensors.

This module contains the basic operations with a set of multiple TT-tensors (Y1, Y2, …), including “add_many” and “outer_many”.

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teneva.act_many.add_many(Y_many, e=1e-10, r=1000000000000.0, trunc_freq=15)

Compute Y1 + Y2 + … + Ym in the TT-format.

Parameters:

• Y_many (list) – the list of TT-tensors (some of them may be int/float).

• e (float) – desired approximation accuracy. The result will be truncated to this accuracy.

• r (int) – maximum rank of the result.

• trunc_freq (int) – frequency of intermediate summation result truncation.

Returns:

TT-tensor, which represents the element wise sum of all given tensors. If all the tensors are numbers, then result will be int/float.

Return type:

list

Examples:

# 10 random TT-tensors with TT-rank 2:
Y_all = [teneva.rand([5]*10, 2) for _ in range(10)]

# Compute the sum:
Y = teneva.add_many(Y_all, e=1.E-4, r=50, trunc_freq=2)

# Show the result:
teneva.show(Y)

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

# TT-tensor    10D : |5| |5|  |5|  |5|  |5|  |5|  |5|  |5|  |5| |5|
# <rank>  =   17.9 :   \5/ \20/ \20/ \20/ \20/ \20/ \20/ \20/ \5/
#

This function also supports float arguments:

Y_all = [
    42.,
    teneva.rand([5]*10, 2),
    33.,
    teneva.rand([5]*10, 4)
]
Y = teneva.add_many(Y_all, e=1.E-4, r=50, trunc_freq=2)
teneva.show(Y)

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

# TT-tensor    10D : |5| |5| |5| |5| |5| |5| |5| |5| |5| |5|
# <rank>  =    6.7 :   \5/ \7/ \7/ \7/ \7/ \7/ \7/ \7/ \5/
#

If all arguments are numbers, then function returns the sum of numbers:

Y_all = [10., 20., 2., 10.]
Y = teneva.add_many(Y_all, e=1.E-4, r=50, trunc_freq=2)
print(Y)

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

# 42.0
#

teneva.act_many.outer_many(Y_many)

Compute outer product of given TT-tensors.

Parameters:

Y_many (list) – list of TT-tensors.

Returns:

TT-tensor, which is the outer product of given TT-tensors.

Return type:

list

Note

See also “outer” function, which computes outer kronecker product of two given TT-tensors.

Examples:

Y1 = teneva.rand([4]*5, 2) # 5-dim random TT-tensor with TT-rank 2
Y2 = teneva.rand([3]*5, 3) # 5-dim random TT-tensor with TT-rank 3
Y3 = teneva.rand([2]*5, 4) # 5-dim random TT-tensor with TT-rank 4

Y = teneva.outer_many([Y1, Y2, Y3]) # Compute the outer product
teneva.show(Y)                      # Print the resulting TT-tensor

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

# TT-tensor    15D : |4| |4| |4| |4| |4| |3| |3| |3| |3| |3| |2| |2| |2| |2| |2|
# <rank>  =    2.6 :   \2/ \2/ \2/ \2/ \1/ \3/ \3/ \3/ \3/ \1/ \4/ \4/ \4/ \4/
#

Y1_full = teneva.full(Y1) # Compute tensors in the full format
Y2_full = teneva.full(Y2) # to check the result
Y3_full = teneva.full(Y3)
Y_full = teneva.full(Y)

Z_full = np.tensordot(Y1_full, Y2_full, 0)
Z_full = np.tensordot(Z_full, Y3_full, 0)

e = np.linalg.norm(Y_full - Z_full) # Compute error for TT-tensor vs full tensor
e /= np.linalg.norm(Z_full)         #

print(f'Error     : {e:-8.2e}')     # Rel. error for TT-tensor vs full tensor

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

# Error     : 3.61e-16
#