Module vectors: collection of explicit useful QTT-vectors

Package teneva, module vectors: various useful QTT-vectors.

This module contains the collection of functions for explicit construction of various useful QTT-vectors (delta function and others).

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teneva.vectors.vector_delta(q, i, v=1.0)

Build QTT-vector that is zero everywhere except for a given index.

Construct a QTT-vector of the length 2^q with only one nonzero element in position i, that is equal to a given value v.

Parameters:

• q (int) – quantization level. The resulting vector will have the total length 2^q.

• i (int) – the index for nonzero element (< 2^q). Note that “negative index notation” is supported.

• v (float) – the value of the vector at index i.

Returns:

TT-tensor representing the QTT-vector.

Return type:

list

Examples:

q = 5                            # Quantization level (the size is 2^q)
i = 2                            # The index for nonzero element
v = 42.                          # The value of the vector at index "i"
Y = teneva.vector_delta(q, i, v) # Build QTT-vector

teneva.show(Y)

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

# TT-tensor     5D : |2| |2| |2| |2| |2|
# <rank>  =    1.0 :   \1/ \1/ \1/ \1/
#

Let check the result:

Y_full = teneva.full(Y)          # Transform QTT-vector to full format
Y_full = Y_full.flatten('F')
i_max = np.argmax(Y_full)        # Find index and value for max
y_max = Y_full[i_max]

# Find number of nonzero vector items:
s = len([y for y in Y_full if abs(y) > 1.E-10])

print(f'The max value index      :', i_max)
print(f'The max value            :', y_max)
print(f'Number of nonzero items  :', s)

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

# The max value index      : 2
# The max value            : 42.0
# Number of nonzero items  : 1
#

We can also build some big QTT-vector by “delta” function and check the norm of the result:

q = 100                          # Quantization level (the size is 2^q)
i = 99                           # The index for nonzero element
v = 42.                          # The value of the vector at index "i"
Y = teneva.vector_delta(q, i, v) # Build QTT-vector

teneva.norm(Y)

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

# 42.0
#