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nengo_spa.algebras¶
The following items are reexported by nengo_spa.algebras
:
Abstract base class for algebras. 

Definition of constants for common properties of vectors in an algebra. 


The side in a binary operation for which a special element’s properties hold. 
Holographic Reduced Representations (HRRs) algebra. 

Vector properties supported by the 

Vectorderived Transformation Binding (VTB) algebra. 

Vector properties supported by the 

Transposed Vectorderived Transformation Binding (TVTB) algebra. 

Vector properties supported by the 
Base classes¶
Abstract base class for algebras. 

Abstract base class for implementing signs for an algebra. 

Definition of constants for common properties of vectors in an algebra. 


The side in a binary operation for which a special element’s properties hold. 

A generic sign implementation. 

class
nengo_spa.algebras.base.
ElementSidedness
(value)[source]¶ The side in a binary operation for which a special element’s properties hold.

LEFT
= 'left'¶

RIGHT
= 'right'¶

TWO_SIDED
= 'twosided'¶


class
nengo_spa.algebras.base.
AbstractAlgebra
[source]¶ Abstract base class for algebras.
Custom algebras can be defined by implementing the interface of this abstract base class.

is_valid_dimensionality
(d)[source]¶ Checks whether d is a valid vector dimensionality.
 Parameters
d (int) – Dimensionality
 Returns
True, if d is a valid vector dimensionality for the use with the algebra.
 Return type
bool

create_vector
(d, properties, *, rng=None)[source]¶ Create a vector fulfilling given properties in the algebra.
Valid properties and combinations thereof depend on the concrete algebra. It is suggested that the properties is either a set of str (if order does not matter) or a list of str (if order does matter). Use the constants defined in
CommonProperties
where appropriate. Parameters
d (int) – Vector dimensionality
properties – Definition of properties for the vector to fulfill. Type and specification format depend on the concrete algbra, but it is suggested to use either a set or list of str (depending on whether order of properties matters) utilizing the constants defined in
CommonProperties
where applicable.rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.
 Returns
Random vector with desired properties.
 Return type
ndarray

make_unitary
(v)[source]¶ Returns a unitary vector based on the vector v.
A unitary vector does not change the length of a vector it is bound to.
 Parameters
v ((d,) ndarray) – Vector to base unitary vector on.
 Returns
Unitary vector.
 Return type
ndarray

superpose
(a, b)[source]¶ Returns the superposition of a and b.
This is commonly elementwise addition.
 Parameters
a ((d,) ndarray) – Left operand in superposition.
b ((d,) ndarray) – Right operand in superposition.
 Returns
Superposed vector.
 Return type
(d,) ndarray

bind
(a, b)[source]¶ Returns the binding of a and b.
The resulting vector should in most cases be dissimilar to both inputs.
 Parameters
a ((d,) ndarray) – Left operand in binding.
b ((d,) ndarray) – Right operand in binding.
 Returns
Bound vector.
 Return type
(d,) ndarray

binding_power
(v, exponent)[source]¶ Returns the binding power of v using the exponent.
For a positive exponent, the binding power is defined as binding (exponent1) times bindings of v to itself. For a negative exponent, the binding power is the approximate inverse bound to itself according to the prior definition. Depending on the algebra, fractional exponents might be valid or return a ValueError, if not. Usually, a fractional binding power will require that v has a positive sign.
Note the following special exponents:
an exponent of 1 will return the approximate inverse,
an exponent of 0 will return the identity vector,
and an exponent of 1 will return v itself.
The default implementation supports integer exponents only and will apply the
bind
method multiple times. It requires the algebra to have a left identity. Parameters
v ((d,) ndarray) – Vector to bind repeatedly to itself.
exponent (int or float) – Exponent of the binding power.
 Returns
Binding power of v.
 Return type
(d,) ndarray
See also

invert
(v, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Invert vector v.
A vector bound to its inverse will result in the identity vector.
Some algebras might not have an inverse only on specific sides. In that case a NotImplementedError may be raised for nonexisting inverses.
 Parameters
v ((d,) ndarray) – Vector to invert.
sidedness (ElementSidedness, optional) – Side in the binding operation on which the returned value acts as inverse.
 Returns
Inverted vector.
 Return type
(d,) ndarray

get_binding_matrix
(v, swap_inputs=False)[source]¶ Returns the transformation matrix for binding with a fixed vector.
 Parameters
v ((d,) ndarray) – Fixed vector to derive binding matrix for.
swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.
 Returns
Transformation matrix to perform binding with v.
 Return type
(d, d) ndarray

get_inversion_matrix
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Returns the transformation matrix for inverting a vector.
Some algebras might not have an inverse only on specific sides. In that case a NotImplementedError may be raised for nonexisting inverses.
 Parameters
d (int) – Vector dimensionality (determines the matrix size).
sidedness (ElementSidedness, optional) – Side in the binding operation on which a transformed vectors acts as inverse.
 Returns
Transformation matrix to invert a vector.
 Return type
(d, d) ndarray

implement_superposition
(n_neurons_per_d, d, n)[source]¶ Implement neural network for superposing vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
n (int) – Number of vectors to superpose in the network.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of length n of inputs to the network, and output the network output. Return type
tuple

implement_binding
(n_neurons_per_d, d, unbind_left, unbind_right)[source]¶ Implement neural network for binding vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
unbind_left (bool) – Whether the left input should be unbound from the right input.
unbind_right (bool) – Whether the right input should be unbound from the left input.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of the left and the right input in that order, and output the network output. Return type
tuple

sign
(v)[source]¶ Returns the sign of v defined by the algebra.
The exact definition of the sign depends on the concrete algebra, but should be analogous to the sign of a (complex) number in so far that binding two vectors with the same sign produces a “positive” vector. There might, however, be multiple types of negative signs, where binding vectors with different types of negative signs will produce another “negative” vector.
Furthermore, if the algebra supports fractional binding powers, it should do so for all “nonnegative” vectors, but not “negative” vectors.
If an algebra does not have the notion of a sign, it may raise a
NotImplementedError
. Parameters
v ((d,) ndarray) – Vector to determine sign of.
 Returns
The sign of the input vector.
 Return type
See also

abs
(v)[source]¶ Returns the absolute vector of v defined by the algebra.
The exact definition of “absolute vector” may depend on the concrete algebra. It should be a “positive” vector (see
sign
) that relates to the input vector.The default implementation requires that the possible signs of the algebra correspond to actual vectors within the algebra. It will bind the sign vector (from the left side) to the vector v.
If an algebra does not have the notion of a sign or absolute vector, it may raise a
NotImplementedError
. Parameters
v ((d,) ndarray) – Vector to obtain the absolute vector of.
 Returns
The absolute vector relating to the input vector.
 Return type
(d,) ndarray

absorbing_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the standard absorbing element of dimensionality d.
An absorbing element will produce a scaled version of itself when bound to another vector. The standard absorbing element is the absorbing element with norm 1.
Some algebras might not have an absorbing element other than the zero vector. In that case a NotImplementedError may be raised.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – Side in the binding operation on which the element absorbs.
 Returns
Standard absorbing element.
 Return type
(d,) ndarray

identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the identity element of dimensionality d.
The identity does not change the vector it is bound to.
Some algebras might not have an identity element. In that case a NotImplementedError may be raised.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – Side in the binding operation on which the element acts as identity.
 Returns
Identity element.
 Return type
(d,) ndarray

negative_identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Returns the negative identity element of dimensionality d.
The negative identity only changes the sign of the vector it is bound to.
Some algebras might not have a negative identity element (or even the notion of a sign). In that case a :py:class`NotImplementedError` may be raised.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – Side in the binding operation on which the element acts as negative identity.
 Returns
Negative identity element.
 Return type
(d,) ndarray

zero_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the zero element of dimensionality d.
The zero element produces itself when bound to a different vector. Usually this will be the zero vector.
Some algebras might not have a zero element. In that case a NotImplementedError may be raised.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – Side in the binding operation on which the element acts as zero.
 Returns
Zero element.
 Return type
(d,) ndarray


class
nengo_spa.algebras.base.
AbstractSign
[source]¶ Abstract base class for implementing signs for an algebra.

class
nengo_spa.algebras.base.
GenericSign
(sign)[source]¶ A generic sign implementation.
 Parameters
sign (1, 0, 1, None) – The represented sign. None is used for an indefinite sign.

class
nengo_spa.algebras.base.
CommonProperties
[source]¶ Definition of constants for common properties of vectors in an algebra.
Use these for best interoperability between algebras.

UNITARY
= 'unitary'¶ A unitary vector does not change the length of a vector it is bound to.

POSITIVE
= 'positive'¶ A positive vector does not change the sign of a vector it is bound to.
A positive vector allows for fractional binding powers.

Holographic reduced representations (HRR)¶
Holographic Reduced Representations (HRRs) algebra. 

Vector properties supported by the 


Represents a sign in the 

class
nengo_spa.algebras.hrr_algebra.
HrrAlgebra
[source]¶ Holographic Reduced Representations (HRRs) algebra.
Uses elementwise addition for superposition, circular convolution for binding with an approximate inverse.
The circular convolution \(c\) of vectors \(a\) and \(b\) is given by
\[c[i] = \sum_j a[j] b[i  j]\]where negative indices on \(b\) wrap around to the end of the vector.
This computation can also be done in the Fourier domain,
\[c = DFT^{1} ( DFT(a) \odot DFT(b) )\]where \(DFT\) is the Discrete Fourier Transform operator, and \(DFT^{1}\) is its inverse.
Circular convolution as a binding operation is associative, commutative, distributive.
More information on circular convolution as a binding operation can be found in [plate2003].
 plate2003
Plate, Tony A. Holographic Reduced Representation: Distributed Representation for Cognitive Structures. Stanford, CA: CSLI Publications, 2003.

is_valid_dimensionality
(d)[source]¶ Checks whether d is a valid vector dimensionality.
For circular convolution all positive numbers are valid dimensionalities.
 Parameters
d (int) – Dimensionality
 Returns
True, if d is a valid vector dimensionality for the use with the algebra.
 Return type
bool

create_vector
(d, properties, *, rng=None)[source]¶ Create a vector fulfilling given properties in the algebra.
 Parameters
d (int) – Vector dimensionality
properties (set of str) – Definition of properties for the vector to fulfill. Valid set elements are constants defined in
HrrProperties
.rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.
 Returns
Random vector with desired properties.
 Return type
ndarray

make_unitary
(v)[source]¶ Returns a unitary vector based on the vector v.
A unitary vector does not change the length of a vector it is bound to.
 Parameters
v ((d,) ndarray) – Vector to base unitary vector on.
 Returns
Unitary vector.
 Return type
ndarray

superpose
(a, b)[source]¶ Returns the superposition of a and b.
This is commonly elementwise addition.
 Parameters
a ((d,) ndarray) – Left operand in superposition.
b ((d,) ndarray) – Right operand in superposition.
 Returns
Superposed vector.
 Return type
(d,) ndarray

bind
(a, b)[source]¶ Returns the binding of a and b.
The resulting vector should in most cases be dissimilar to both inputs.
 Parameters
a ((d,) ndarray) – Left operand in binding.
b ((d,) ndarray) – Right operand in binding.
 Returns
Bound vector.
 Return type
(d,) ndarray

binding_power
(v, exponent)[source]¶ Returns the binding power of v using the exponent.
The binding power is defined as binding (exponent1) times bindings of v to itself. Fractional binding powers are supported.
Note the following special exponents:
an exponent of 1 will return the approximate inverse,
an exponent of 0 will return the identity vector,
and an exponent of w1cne will return v itself.
The following relations hold for integer exponents, and for unitary vectors:
\(v^a \circledast v^b = v^{a+b}\),
\((v^a)^b = v^{ab}\).
If \(a \geq 0\) and \(b \geq 0\), then the first relation holds also for nonunitary vectors with real exponents.
 Parameters
v ((d,) ndarray) – Vector to bind repeatedly to itself.
exponent (int or float) – Exponent of the binding power.
 Returns
Binding power of v.
 Return type
(d,) ndarray
See also

invert
(v, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Invert vector v.
This turns circular convolution into circular correlation, meaning that
A*B*~B
is approximatelyA
.Examples
For the vector
[1, 2, 3, 4, 5]
, the inverse is[1, 5, 4, 3, 2]
. Parameters
v ((d,) ndarray) – Vector to invert.
sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the inverse is twosided.
 Returns
Inverted vector.
 Return type
(d,) ndarray

get_binding_matrix
(v, swap_inputs=False)[source]¶ Returns the transformation matrix for binding with a fixed vector.
 Parameters
v ((d,) ndarray) – Fixed vector to derive binding matrix for.
swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.
 Returns
Transformation matrix to perform binding with v.
 Return type
(d, d) ndarray

get_inversion_matrix
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Returns the transformation matrix for inverting a vector.
 Parameters
d (int) – Vector dimensionality (determines the matrix size).
sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the inverse is twosided.
 Returns
Transformation matrix to invert a vector.
 Return type
(d, d) ndarray

implement_superposition
(n_neurons_per_d, d, n)[source]¶ Implement neural network for superposing vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
n (int) – Number of vectors to superpose in the network.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of length n of inputs to the network, and output the network output. Return type
tuple

implement_binding
(n_neurons_per_d, d, unbind_left, unbind_right)[source]¶ Implement neural network for binding vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
unbind_left (bool) – Whether the left input should be unbound from the right input.
unbind_right (bool) – Whether the right input should be unbound from the left input.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of the left and the right input in that order, and output the network output. Return type
tuple

sign
(v)[source]¶ Returns the HRR sign of v.
See
AbstractAlgebra.sign
for general information on the notion of a sign for algbras, andHrrSign
for details specific to HRRs. Parameters
v ((d,) ndarray) – Vector to determine sign of.
 Returns
The sign of the input vector.
 Return type

absorbing_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the standard absorbing element of dimensionality d.
An absorbing element will produce a scaled version of itself when bound to another vector. The standard absorbing element is the absorbing element with norm 1.
The absorbing element for circular convolution is the vector \((1, 1, \dots, 1)^{\top} / \sqrt{d}\).
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the standard absorbing element is twosided.
 Returns
Standard absorbing element.
 Return type
(d,) ndarray

identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the identity element of dimensionality d.
The identity does not change the vector it is bound to.
The identity element for circular convolution is the vector \((1, 0, \dots, 0)^{\top}\).
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the identity is twosided.
 Returns
Identity element.
 Return type
(d,) ndarray

negative_identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the negative identity element of dimensionality d.
The negative identity element for circular convolution is the vector \((1, 0, \dots, 0)^{\top}\).
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the identity is twosided.
 Returns
Negative identity element.
 Return type
(d,) ndarray

zero_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the zero element of dimensionality d.
The zero element produces itself when bound to a different vector. For circular convolution this is the zero vector.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – This argument has no effect because the HRR algebra is commutative and the zero element is twosided.
 Returns
Zero element.
 Return type
(d,) ndarray

class
nengo_spa.algebras.hrr_algebra.
HrrSign
(dc_sign, nyquist_sign)[source]¶ Represents a sign in the
HrrAlgebra
.For odd dimensionalities, the sign is equal to the sign of the DC component of the Fourier representation of the vector. For even dimensionalities the sign is constituted out of the signs of the DC component and Nyquist frequency. Thus, for even dimensionalities, there is a total of four subsigns excluding zero. The overall sign is considered positive if the DC component is positive and the Nyquist component is nonnegative; the sign is considered negative if either component is negative; and the sign is considered zero if both are zero. Binding two Semantic Pointers with the same subsign will yield a positive Semantic Pointer. See the table below for details.
¶ Sign (DC, Nyquist)
+ (+1, ≥0)
− (+1, <0)
− (1, ≥0)
− (−1, <0)
(0, 0)
+ (+1, ≥0)
+ (+1, ≥0)
− (+1, <0)
− (−1, ≥0)
− (−1, <0)
(0, 0)
− (+1, <0)
+ (1, ≥0)
− (−1, <0)
− (−1, ≥0)
(0, 0)
− (−1, ≥0)
+ (1, ≥0)
− (+1, <0)
(0, 0)
− (−1, <0)
+ (1, ≥0)
(0, 0)
(0, 0)
(0, 0)
 Parameters
dc_sign (int) – Sign of the DC component.
nyquist_sign (int) – Sign of the Nyquist frequency component. Set to zero for odd dimensionalities by definition.

dc_sign
¶

nyquist_sign
¶

to_vector
(d)[source]¶ Return the vector in the algebra corresponding to the sign.
DC sign
Nyquist sign
Vector
1
>=0
[ 1, 0, 0, …] (identity)
1
<0
[ 0, 1, 0, 0, …]
1
>=0
[ 0, 1, 0, …]
1
<0
[1, 0, 0, 0, …] (negative identity)
0
0
[ 0, 0, 0, …] (zero)
 Parameters
d (int) – Vector dimensionality.
 Returns
Vector corresponding to the sign.
 Return type
(d,) ndarray

class
nengo_spa.algebras.hrr_algebra.
HrrProperties
[source]¶ Vector properties supported by the
HrrAlgebra
.
UNITARY
= 'unitary'¶ A unitary vector does not change the length of a vector it is bound to.

POSITIVE
= 'positive'¶ A positive vector does not change the sign of a vector it is bound to.
A positive vector allows for fractional binding powers.

Vectorderived transformation binding (VTB)¶
Vectorderived Transformation Binding (VTB) algebra. 

Vector properties supported by the 


Represents a sign in the 

class
nengo_spa.algebras.vtb_algebra.
VtbAlgebra
[source]¶ Vectorderived Transformation Binding (VTB) algebra.
VTB uses elementwise addition for superposition. The binding operation \(\mathcal{B}(x, y)\) is defined as
\[\begin{split}\mathcal{B}(x, y) := V_y x = \left[\begin{array}{ccc} V_y' & 0 & 0 \\ 0 & V_y' & 0 \\ 0 & 0 & V_y' \end{array}\right] x\end{split}\]with
\[\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d  d' + 1} & y_{d  d' + 2} & \dots & y_d \end{array}\right]\end{split}\]and
\[d'^2 = d.\]The approximate inverse \(y^+\) for \(y\) is permuting the elements such that \(V_{y^+} = V_y^T\).
Note that VTB requires the vector dimensionality to be square.
The VTB binding operation is neither associative nor commutative. Furthermore, there are right inverses and identities only. By transposing the \(V_y\) matrix, the closely related
TvtbAlgebra
(Transposed VTB) algebra is obtained which does have twosided identities and inverses.Additional information about VTB can be found in
See also

is_valid_dimensionality
(d)[source]¶ Checks whether d is a valid vector dimensionality.
For VTB all square numbers are valid dimensionalities.
 Parameters
d (int) – Dimensionality
 Returns
True, if d is a valid vector dimensionality for the use with the algebra.
 Return type
bool

create_vector
(d, properties, *, rng=None)[source]¶ Create a vector fulfilling given properties in the algebra.
Creating positive vectors requires SciPy.
 Parameters
d (int) – Vector dimensionality
properties (set of str) – Definition of properties for the vector to fulfill. Valid set elements are constants defined in
VtbProperties
.rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.
 Returns
Random vector with desired properties.
 Return type
ndarray

make_unitary
(v)[source]¶ Returns a unitary vector based on the vector v.
A unitary vector does not change the length of a vector it is bound to.
 Parameters
v ((d,) ndarray) – Vector to base unitary vector on.
 Returns
Unitary vector.
 Return type
ndarray

superpose
(a, b)[source]¶ Returns the superposition of a and b.
This is commonly elementwise addition.
 Parameters
a ((d,) ndarray) – Left operand in superposition.
b ((d,) ndarray) – Right operand in superposition.
 Returns
Superposed vector.
 Return type
(d,) ndarray

bind
(a, b)[source]¶ Returns the binding of a and b.
The resulting vector should in most cases be dissimilar to both inputs.
 Parameters
a ((d,) ndarray) – Left operand in binding.
b ((d,) ndarray) – Right operand in binding.
 Returns
Bound vector.
 Return type
(d,) ndarray

invert
(v, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Invert vector v.
A vector bound to its inverse will result in the identity vector.
VTB has a right inverse only.
Deprecated since version 1.2.0: Calling this method with the default
sidedness=ElementSidedness.TWO_SIDED
returns the right inverse for backwards compatibility, but has been deprecated and will be removed in the next major release. Parameters
v ((d,) ndarray) – Vector to invert.
sidedness (ElementSidedness) – Must be set to
ElementSidedness.RIGHT
.
 Returns
Right inverse of vector.
 Return type
(d,) ndarray

binding_power
(v, exponent)[source]¶ Returns the binding power of v using the exponent.
The binding power is defined as binding (exponent1) times bindings of v to itself.
Fractional binding powers are supported for “positive” vectors if SciPy is available.
Note the following special exponents:
an exponent of 1 will return the inverse,
an exponent of 0 will return the identity vector,
and an exponent of 1 will return v itself.
Be aware that the binding power for the VTB algebra does not satisfy the usual properties of exponentiation:
\(\mathcal{B}(v^a, v^b) = v^{a+b}\) does not hold,
\((v^a)^b = v^{ab}\) does not hold.
 Parameters
v ((d,) ndarray) – Vector to bind repeatedly to itself.
exponent (int or float) – Exponent of the binding power.
 Returns
Binding power of v.
 Return type
(d,) ndarray
See also

get_binding_matrix
(v, swap_inputs=False)[source]¶ Returns the transformation matrix for binding with a fixed vector.
 Parameters
v ((d,) ndarray) – Fixed vector to derive binding matrix for.
swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.
 Returns
Transformation matrix to perform binding with v.
 Return type
(d, d) ndarray

get_swapping_matrix
(d)[source]¶ Get matrix to swap operands in bound state.
 Parameters
d (int) – Dimensionality of vector.
 Returns
Matrix to multiply with a vector to switch left and right operand in bound state.
 Return type
(d, d) ndarry

get_inversion_matrix
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Returns the transformation matrix for inverting a vector.
VTB has a right inverse only.
Deprecated since version 1.2.0: Calling this method with the default
sidedness=ElementSidedness.TWO_SIDED
returns the right transformation matrix for the right inverse for backwards compatibility, but has been deprecated and will be removed in the next major release. Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness) – Must be set to
ElementSidedness.RIGHT
.
 Returns
Transformation matrix to invert a vector.
 Return type
(d, d) ndarray

implement_superposition
(n_neurons_per_d, d, n)[source]¶ Implement neural network for superposing vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
n (int) – Number of vectors to superpose in the network.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of length n of inputs to the network, and output the network output. Return type
tuple

implement_binding
(n_neurons_per_d, d, unbind_left, unbind_right)[source]¶ Implement neural network for binding vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
unbind_left (bool) – Whether the left input should be unbound from the right input.
unbind_right (bool) – Whether the right input should be unbound from the left input.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of the left and the right input in that order, and output the network output. Return type
tuple

sign
(v)[source]¶ Returns the sign of v defined by the algebra.
The exact definition of the sign depends on the concrete algebra, but should be analogous to the sign of a (complex) number in so far that binding two vectors with the same sign produces a “positive” vector. There might, however, be multiple types of negative signs, where binding vectors with different types of negative signs will produce another “negative” vector.
Furthermore, if the algebra supports fractional binding powers, it should do so for all “nonnegative” vectors, but not “negative” vectors.
If an algebra does not have the notion of a sign, it may raise a
NotImplementedError
. Parameters
v ((d,) ndarray) – Vector to determine sign of.
 Returns
The sign of the input vector.
 Return type
See also
AbstractAlgebra.abs

abs
(v)[source]¶ Returns the absolute vector of v defined by the algebra.
The exact definition of “absolute vector” may depend on the concrete algebra. It should be a “positive” vector (see
sign
) that relates to the input vector.The default implementation requires that the possible signs of the algebra correspond to actual vectors within the algebra. It will bind the sign vector (from the left side) to the vector v.
If an algebra does not have the notion of a sign or absolute vector, it may raise a
NotImplementedError
. Parameters
v ((d,) ndarray) – Vector to obtain the absolute vector of.
 Returns
The absolute vector relating to the input vector.
 Return type
(d,) ndarray

absorbing_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ VTB has no absorbing element except the zero vector.
Always raises a
NotImplementedError
.

identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the identity element of dimensionality d.
VTB has a right identity only.
Deprecated since version 1.2.0: Calling this method with the default
sidedness=ElementSidedness.TWO_SIDED
returns the right identity for backwards compatibility, but has been deprecated and will be removed in the next major release. Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness) – Must be set to
ElementSidedness.RIGHT
.
 Returns
Right identity element.
 Return type
(d,) ndarray

negative_identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the negative identity element of dimensionality d.
VTB has a right negative identity only.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – Must be set to
ElementSidedness.RIGHT
.
 Returns
Negative identity element.
 Return type
(d,) ndarray

zero_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the zero element of dimensionality d.
The zero element produces itself when bound to a different vector. For VTB this is the zero vector.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – This argument has no effect because the zero element of the VTB algebra is twosided.
 Returns
Zero element.
 Return type
(d,) ndarray


class
nengo_spa.algebras.vtb_algebra.
VtbSign
(sign)[source]¶ Represents a sign in the
VtbAlgebra
.

class
nengo_spa.algebras.vtb_algebra.
VtbProperties
[source]¶ Vector properties supported by the
VtbAlgebra
.
UNITARY
= 'unitary'¶ A unitary vector does not change the length of a vector it is bound to.

POSITIVE
= 'positive'¶ A positive vector does not change the sign of a vector it is bound to.
A positive vector allows for fractional binding powers.

Transposed vectorderived transformation binding (TVTB)¶
Transposed Vectorderived Transformation Binding (TVTB) algebra. 

Vector properties supported by the 


Represents a sign in the 

class
nengo_spa.algebras.tvtb_algebra.
TvtbAlgebra
[source]¶ Transposed Vectorderived Transformation Binding (TVTB) algebra.
TVTB uses elementwise addition for superposition. The binding operation \(\mathcal{B}(x, y)\) is defined as
\[\begin{split}\mathcal{B}(x, y) := V_y^T x = \left[\begin{array}{ccc} V_y'^T & 0 & 0 \\ 0 & V_y'^T & 0 \\ 0 & 0 & V_y'^T \end{array}\right] x\end{split}\]with
\[\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d  d' + 1} & y_{d  d' + 2} & \dots & y_d \end{array}\right]\end{split}\]and
\[d'^2 = d.\]The approximate inverse \(y^+\) for \(y\) is permuting the elements such that \(V_{y^+} = V_y^T\).
Note that TVTB requires the vector dimensionality to be square.
The TVTB binding operation is neither associative nor commutative. In contrast to VTB, however, TVTB has twosided identities and inverses. Other properties are equivalent to VTB.
See also

is_valid_dimensionality
(d)[source]¶ Checks whether d is a valid vector dimensionality.
For TVTB all square numbers are valid dimensionalities.
 Parameters
d (int) – Dimensionality
 Returns
True, if d is a valid vector dimensionality for the use with the algebra.
 Return type
bool

create_vector
(d, properties, *, rng=None)[source]¶ Create a vector fulfilling given properties in the algebra.
Creating positive vectors requires SciPy.
 Parameters
d (int) – Vector dimensionality
properties (set of str) – Definition of properties for the vector to fulfill. Valid set elements are constants defined in
TvtbProperties
.rng (numpy.random.RandomState, optional) – The random number generator to use to create the vector.
 Returns
Random vector with desired properties.
 Return type
ndarray

make_unitary
(v)[source]¶ Returns a unitary vector based on the vector v.
A unitary vector does not change the length of a vector it is bound to.
 Parameters
v ((d,) ndarray) – Vector to base unitary vector on.
 Returns
Unitary vector.
 Return type
ndarray

superpose
(a, b)[source]¶ Returns the superposition of a and b.
This is commonly elementwise addition.
 Parameters
a ((d,) ndarray) – Left operand in superposition.
b ((d,) ndarray) – Right operand in superposition.
 Returns
Superposed vector.
 Return type
(d,) ndarray

bind
(a, b)[source]¶ Returns the binding of a and b.
The resulting vector should in most cases be dissimilar to both inputs.
 Parameters
a ((d,) ndarray) – Left operand in binding.
b ((d,) ndarray) – Right operand in binding.
 Returns
Bound vector.
 Return type
(d,) ndarray

binding_power
(v, exponent)[source]¶ Returns the binding power of v using the exponent.
The binding power is defined as binding (exponent1) times bindings of v to itself.
Fractional binding powers are supported for “positive” vectors if SciPy is available.
Note the following special exponents:
an exponent of 1 will return the inverse,
an exponent of 0 will return the identity vector,
and an exponent of 1 will return v itself.
The following relations hold for integer exponents:
\(\mathcal{B}(v^a, v^b) = v^{a+b}\),
\((v^a)^b = v^{ab}\).
(Technically, these relations also hold for positive unitary vectors, but the only such vector is the identity vector.)
 Parameters
v ((d,) ndarray) – Vector to bind repeatedly to itself.
exponent (int or float) – Exponent of the binding power.
 Returns
Binding power of v.
 Return type
(d,) ndarray
See also

invert
(v, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Invert vector v.
A vector bound to its inverse will result in the identity vector.
 Parameters
v ((d,) ndarray) – Vector to invert.
sidedness (ElementSidedness) – This argument has no effect because the inverse of the TVTB algebra is twosided.
 Returns
Inverse of vector.
 Return type
(d,) ndarray

get_binding_matrix
(v, swap_inputs=False)[source]¶ Returns the transformation matrix for binding with a fixed vector.
 Parameters
v ((d,) ndarray) – Fixed vector to derive binding matrix for.
swap_inputs (bool, optional) – By default the matrix will be such that v becomes the right operand in the binding. By setting swap_inputs, the matrix will be such that v becomes the left operand. For binding operations that are commutative (such as circular convolution), this has no effect.
 Returns
Transformation matrix to perform binding with v.
 Return type
(d, d) ndarray

get_inversion_matrix
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Returns the transformation matrix for inverting a vector.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness) – This argument has no effect because the inverse of the TVTB algebra is twosided.
 Returns
Transformation matrix to invert a vector.
 Return type
(d, d) ndarray

implement_superposition
(n_neurons_per_d, d, n)[source]¶ Implement neural network for superposing vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
n (int) – Number of vectors to superpose in the network.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of length n of inputs to the network, and output the network output. Return type
tuple

implement_binding
(n_neurons_per_d, d, unbind_left, unbind_right)[source]¶ Implement neural network for binding vectors.
 Parameters
n_neurons_per_d (int) – Neurons to use per dimension.
d (int) – Dimensionality of the vectors.
unbind_left (bool) – Whether the left input should be unbound from the right input.
unbind_right (bool) – Whether the right input should be unbound from the left input.
 Returns
Tuple (net, inputs, output) where net is the implemented
nengo.Network
, inputs a sequence of the left and the right input in that order, and output the network output. Return type
tuple

sign
(v)[source]¶ Returns the sign of v defined by the algebra.
The exact definition of the sign depends on the concrete algebra, but should be analogous to the sign of a (complex) number in so far that binding two vectors with the same sign produces a “positive” vector. There might, however, be multiple types of negative signs, where binding vectors with different types of negative signs will produce another “negative” vector.
Furthermore, if the algebra supports fractional binding powers, it should do so for all “nonnegative” vectors, but not “negative” vectors.
If an algebra does not have the notion of a sign, it may raise a
NotImplementedError
. Parameters
v ((d,) ndarray) – Vector to determine sign of.
 Returns
The sign of the input vector.
 Return type
See also
AbstractAlgebra.abs

absorbing_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ TVTB has no absorbing element except the zero vector.
Always raises a
NotImplementedError
.

identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the identity element of dimensionality d.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness) – This argument has no effect because the identity of the TVTB algebra is twosided.
 Returns
Identity element.
 Return type
(d,) ndarray

negative_identity_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the negative identity element of dimensionality d.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – This argument has no effect because the negative identity of the TVTB algebra is twosided.
 Returns
Negative identity element.
 Return type
(d,) ndarray

zero_element
(d, sidedness=<ElementSidedness.TWO_SIDED: 'twosided'>)[source]¶ Return the zero element of dimensionality d.
The zero element produces itself when bound to a different vector. For VTB this is the zero vector.
 Parameters
d (int) – Vector dimensionality.
sidedness (ElementSidedness, optional) – This argument has no effect because the zero element of the VTB algebra is twosided.
 Returns
Zero element.
 Return type
(d,) ndarray


class
nengo_spa.algebras.tvtb_algebra.
TvtbSign
(sign)[source]¶ Represents a sign in the
TvtbAlgebra
.

class
nengo_spa.algebras.tvtb_algebra.
TvtbProperties
[source]¶ Vector properties supported by the
TvtbAlgebra
.
UNITARY
= 'unitary'¶ A unitary vector does not change the length of a vector it is bound to.

POSITIVE
= 'positive'¶ A positive vector does not change the sign of a vector it is bound to.
A positive vector allows for fractional binding powers.
