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# nengo_spa.algebras¶

The following items are re-exported 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 HrrAlgebra. Vector-derived Transformation Binding (VTB) algebra. Vector properties supported by the VtbAlgebra. Transposed Vector-derived Transformation Binding (TVTB) algebra. Vector properties supported by the TvtbAlgebra.

## 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. ElementSidedness(value) The side in a binary operation for which a special element’s properties hold. GenericSign(sign) A generic sign implementation.
class nengo_spa.algebras.base.ElementSidedness(value)[source]

Bases: enum.Enum

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

LEFT = 'left'
RIGHT = 'right'
TWO_SIDED = 'two-sided'
class nengo_spa.algebras.base.AbstractAlgebra[source]

Bases: object

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
dint

Dimensionality

Returns
bool

True, if d is a valid vector dimensionality for the use with the algebra.

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
dint

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.

rngnumpy.random.RandomState, optional

The random number generator to use to create the vector.

Returns
ndarray

Random vector with desired properties.

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
ndarray

Unitary vector.

superpose(a, b)[source]

Returns the superposition of a and b.

Parameters
a(d,) ndarray

Left operand in superposition.

b(d,) ndarray

Right operand in superposition.

Returns
(d,) ndarray

Superposed vector.

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
(d,) ndarray

Bound vector.

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 (exponent-1) 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.

exponentint or float

Exponent of the binding power.

Returns
(d,) ndarray

Binding power of v.

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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 non-existing inverses.

Parameters
v(d,) ndarray

Vector to invert.

sidednessElementSidedness, optional

Side in the binding operation on which the returned value acts as inverse.

Returns
(d,) ndarray

Inverted vector.

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_inputsbool, 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
(d, d) ndarray

Transformation matrix to perform binding with v.

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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 non-existing inverses.

Parameters
dint

Vector dimensionality (determines the matrix size).

sidednessElementSidedness, optional

Side in the binding operation on which a transformed vectors acts as inverse.

Returns
(d, d) ndarray

Transformation matrix to invert a vector.

implement_superposition(n_neurons_per_d, d, n)[source]

Implement neural network for superposing vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

nint

Number of vectors to superpose in the network.

Returns
tuple

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.

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)[source]

Implement neural network for binding vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

unbind_leftbool

Whether the left input should be unbound from the right input.

unbind_rightbool

Whether the right input should be unbound from the left input.

Returns
tuple

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.

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 “non-negative” 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
AbstractSign

The sign of the input vector.

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 inverse of 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
(d,) ndarray

The absolute vector relating to the input vector.

absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

Side in the binding operation on which the element absorbs.

Returns
(d,) ndarray

Standard absorbing element.

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

Side in the binding operation on which the element acts as identity.

Returns
(d,) ndarray

Identity element.

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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:classNotImplementedError may be raised.

Parameters
dint

Vector dimensionality.

sidednessElementSidedness, optional

Side in the binding operation on which the element acts as negative identity.

Returns
(d,) ndarray

Negative identity element.

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

Side in the binding operation on which the element acts as zero.

Returns
(d,) ndarray

Zero element.

class nengo_spa.algebras.base.AbstractSign[source]

Bases: abc.ABC

Abstract base class for implementing signs for an algebra.

is_positive()[source]

Return whether the sign is positive.

is_negative()[source]

Return whether the sign is negative.

is_zero()[source]

Return whether the sign neither positive nor negative (i.e. zero), but definite.

is_indefinite()[source]

Return whether the sign is neither positive nor negative nor zero.

to_vector(d)[source]

Return the vector in the algebra corresponding to the sign.

Parameters
dint

Vector dimensionality.

Returns
(d,) ndarray

Vector corresponding to the sign.

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.

is_positive()[source]

Return whether the sign is positive.

is_negative()[source]

Return whether the sign is negative.

is_zero()[source]

Return whether the sign neither positive nor negative (i.e. zero), but definite.

is_indefinite()[source]

Return whether the sign is neither positive nor negative nor zero.

class nengo_spa.algebras.base.CommonProperties[source]

Bases: object

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 HrrAlgebra. HrrSign(dc_sign, nyquist_sign) Represents a sign in the HrrAlgebra.
class nengo_spa.algebras.hrr_algebra.HrrAlgebra[source]

Holographic Reduced Representations (HRRs) algebra.

Uses element-wise 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
dint

Dimensionality

Returns
bool

True, if d is a valid vector dimensionality for the use with the algebra.

create_vector(d, properties, *, rng=None)[source]

Create a vector fulfilling given properties in the algebra.

Parameters
dint

Vector dimensionality

propertiesset of str

Definition of properties for the vector to fulfill. Valid set elements are constants defined in HrrProperties.

rngnumpy.random.RandomState, optional

The random number generator to use to create the vector.

Returns
ndarray

Random vector with desired properties.

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
ndarray

Unitary vector.

superpose(a, b)[source]

Returns the superposition of a and b.

Parameters
a(d,) ndarray

Left operand in superposition.

b(d,) ndarray

Right operand in superposition.

Returns
(d,) ndarray

Superposed vector.

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
(d,) ndarray

Bound vector.

binding_power(v, exponent)[source]

Returns the binding power of v using the exponent.

The binding power is defined as binding (exponent-1) 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 non-unitary vectors with real exponents.

Parameters
v(d,) ndarray

Vector to bind repeatedly to itself.

exponentint or float

Exponent of the binding power.

Returns
(d,) ndarray

Binding power of v.

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

Invert vector v.

This turns circular convolution into circular correlation, meaning that A*B*~B is approximately A.

Parameters
v(d,) ndarray

Vector to invert.

sidednessElementSidedness, optional

This argument has no effect because the HRR algebra is commutative and the inverse is two-sided.

Returns
(d,) ndarray

Inverted vector.

Examples

For the vector [1, 2, 3, 4, 5], the inverse is [1, 5, 4, 3, 2].

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_inputsbool, 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
(d, d) ndarray

Transformation matrix to perform binding with v.

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

Returns the transformation matrix for inverting a vector.

Parameters
dint

Vector dimensionality (determines the matrix size).

sidednessElementSidedness, optional

This argument has no effect because the HRR algebra is commutative and the inverse is two-sided.

Returns
(d, d) ndarray

Transformation matrix to invert a vector.

implement_superposition(n_neurons_per_d, d, n)[source]

Implement neural network for superposing vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

nint

Number of vectors to superpose in the network.

Returns
tuple

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.

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)[source]

Implement neural network for binding vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

unbind_leftbool

Whether the left input should be unbound from the right input.

unbind_rightbool

Whether the right input should be unbound from the left input.

Returns
tuple

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.

sign(v)[source]

Returns the HRR sign of v.

See AbstractAlgebra.sign for general information on the notion of a sign for algbras, and HrrSign for details specific to HRRs.

Parameters
v(d,) ndarray

Vector to determine sign of.

Returns
HrrSign

The sign of the input vector.

absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

This argument has no effect because the HRR algebra is commutative and the standard absorbing element is two-sided.

Returns
(d,) ndarray

Standard absorbing element.

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

This argument has no effect because the HRR algebra is commutative and the identity is two-sided.

Returns
(d,) ndarray

Identity element.

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

This argument has no effect because the HRR algebra is commutative and the identity is two-sided.

Returns
(d,) ndarray

Negative identity element.

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

This argument has no effect because the HRR algebra is commutative and the zero element is two-sided.

Returns
(d,) ndarray

Zero element.

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 sub-signs excluding zero. The overall sign is considered positive if the DC component is positive and the Nyquist component is non-negative; 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 sub-sign will yield a positive Semantic Pointer. See the table below for details.

Resulting Semantic Pointer signs from HRR binding two Semantic Pointers. (Only the upper triangle is given as the matrix is symmetric.)

Sign (DC, Nyquist)

+ (+1, +1)

− (+1, -1)

− (-1, +1)

− (−1, -1)

(0, 0)

+ (+1, +1)

+ (+1, +1)

− (+1, -1)

− (−1, +1)

− (−1, -1)

(0, 0)

− (+1, -1)

+ (1, +1)

− (−1, -1)

− (−1, +1)

(0, 0)

− (−1, +1)

+ (1, +1)

− (+1, -1)

(0, 0)

− (−1, -1)

+ (1, +1)

(0, 0)

(0, 0)

(0, 0)

Parameters
dc_signint

Sign of the DC component.

nyquist_signint

Sign of the Nyquist frequency component. Will be set to the dc_sign if zero.

dc_sign
nyquist_sign
is_positive()[source]

Return whether the sign is positive.

is_negative()[source]

Return whether the sign is negative.

is_indefinite()[source]

Return whether the sign is neither positive nor negative nor zero.

to_vector(d)[source]

Return the vector in the algebra corresponding to the sign.

DC sign

Nyquist sign

Vector

1

1

[ 1, 0, 0, …] (identity)

1

-1

[ 0, 1, 0, 0, …]

-1

1

[ 0, -1, 0, …]

-1

-1

[-1, 0, 0, 0, …] (negative identity)

0

0

[ 0, 0, 0, …] (zero)

Parameters
dint

Vector dimensionality.

Returns
(d,) ndarray

Vector corresponding to the sign.

class nengo_spa.algebras.hrr_algebra.HrrProperties[source]

Bases: object

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.

## Vector-derived transformation binding (VTB)¶

 Vector-derived Transformation Binding (VTB) algebra. Vector properties supported by the VtbAlgebra. VtbSign(sign) Represents a sign in the VtbAlgebra.
class nengo_spa.algebras.vtb_algebra.VtbAlgebra[source]

Vector-derived Transformation Binding (VTB) algebra.

VTB uses elementwise addition for superposition. The binding operation $$\mathcal{B}(x, y)$$ is defined as

$\begin{split}\mathcal{B}(x, y) := V_y x = \left[\begin{array}{ccc} V_y' & 0 & 0 \\ 0 & V_y' & 0 \\ 0 & 0 & \ddots \end{array}\right] x\end{split}$

with $$d'$$ blocks

where

$\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d - d' + 1} & y_{d - d' + 2} & \dots & y_d \end{array}\right]\end{split}$

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 two-sided identities and inverses.

is_valid_dimensionality(d)[source]

Checks whether d is a valid vector dimensionality.

For VTB all square numbers are valid dimensionalities.

Parameters
dint

Dimensionality

Returns
bool

True, if d is a valid vector dimensionality for the use with the algebra.

create_vector(d, properties, *, rng=None)[source]

Create a vector fulfilling given properties in the algebra.

Creating positive vectors requires SciPy.

Parameters
dint

Vector dimensionality

propertiesset of str

Definition of properties for the vector to fulfill. Valid set elements are constants defined in VtbProperties.

rngnumpy.random.RandomState, optional

The random number generator to use to create the vector.

Returns
ndarray

Random vector with desired properties.

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
ndarray

Unitary vector.

superpose(a, b)[source]

Returns the superposition of a and b.

Parameters
a(d,) ndarray

Left operand in superposition.

b(d,) ndarray

Right operand in superposition.

Returns
(d,) ndarray

Superposed vector.

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
(d,) ndarray

Bound vector.

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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.

sidednessElementSidedness

Must be set to ElementSidedness.RIGHT.

Returns
(d,) ndarray

Right inverse of vector.

binding_power(v, exponent)[source]

Returns the binding power of v using the exponent.

The binding power is defined as binding (exponent-1) 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.

exponentint or float

Exponent of the binding power.

Returns
(d,) ndarray

Binding power of v.

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_inputsbool, 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
(d, d) ndarray

Transformation matrix to perform binding with v.

get_swapping_matrix(d)[source]

Get matrix to swap operands in bound state.

Parameters
dint

Dimensionality of vector.

Returns
(d, d) ndarry

Matrix to multiply with a vector to switch left and right operand in bound state.

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness

Must be set to ElementSidedness.RIGHT.

Returns
(d, d) ndarray

Transformation matrix to invert a vector.

implement_superposition(n_neurons_per_d, d, n)[source]

Implement neural network for superposing vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

nint

Number of vectors to superpose in the network.

Returns
tuple

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.

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)[source]

Implement neural network for binding vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

unbind_leftbool

Whether the left input should be unbound from the right input.

unbind_rightbool

Whether the right input should be unbound from the left input.

Returns
tuple

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.

sign(v)[source]
abs(v)[source]
absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

VTB has no absorbing element except the zero vector.

Always raises a NotImplementedError.

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness

Must be set to ElementSidedness.RIGHT.

Returns
(d,) ndarray

Right identity element.

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

Return the negative identity element of dimensionality d.

VTB has a right negative identity only.

Parameters
dint

Vector dimensionality.

sidednessElementSidedness, optional

Must be set to ElementSidedness.RIGHT.

Returns
(d,) ndarray

Negative identity element.

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

This argument has no effect because the zero element of the VTB algebra is two-sided.

Returns
(d,) ndarray

Zero element.

class nengo_spa.algebras.vtb_algebra.VtbSign(sign)[source]

Represents a sign in the VtbAlgebra.

The sign depends on the symmetry and positive/negative definiteness of the binding matrix derived from the vector. For all non-symmetric matrices, the sign is indefinite. It is also indefinite, if the matrices’ eigenvalues have different signs. A symmetric, positive (negative) definite binding matrix corresponds to a positive (negative) sign (equivalent to all eigenvalues being greater than 0, respectively lower than 0). If all eigenvalues are equal to 0, the sign is also 0.

to_vector(d)[source]

Return the vector in the algebra corresponding to the sign.

Parameters
dint

Vector dimensionality.

Returns
(d,) ndarray

Vector corresponding to the sign.

class nengo_spa.algebras.vtb_algebra.VtbProperties[source]

Bases: object

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 vector-derived transformation binding (TVTB)¶

 Transposed Vector-derived Transformation Binding (TVTB) algebra. Vector properties supported by the TvtbAlgebra. TvtbSign(sign) Represents a sign in the TvtbAlgebra.
class nengo_spa.algebras.tvtb_algebra.TvtbAlgebra[source]

Transposed Vector-derived Transformation Binding (TVTB) algebra.

TVTB uses elementwise addition for superposition. The binding operation $$\mathcal{B}(x, y)$$ is defined as

$\begin{split}\mathcal{B}(x, y) := V_y^T x = \left[\begin{array}{ccc} V_y'^T & 0 & 0 \\ 0 & V_y'^T & 0 \\ 0 & 0 & \ddots \end{array}\right] x\end{split}$

with $$d'$$ blocks

where

$\begin{split}V_y' = d^{\frac{1}{4}} \left[\begin{array}{cccc} y_1 & y_2 & \dots & y_{d'} \\ y_{d' + 1} & y_{d' + 2} & \dots & y_{2d'} \\ \vdots & \vdots & \ddots & \vdots \\ y_{d - d' + 1} & y_{d - d' + 2} & \dots & y_d \end{array}\right]\end{split}$

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 two-sided identities and inverses. Other properties are equivalent to VTB.

is_valid_dimensionality(d)[source]

Checks whether d is a valid vector dimensionality.

For TVTB all square numbers are valid dimensionalities.

Parameters
dint

Dimensionality

Returns
bool

True, if d is a valid vector dimensionality for the use with the algebra.

create_vector(d, properties, *, rng=None)[source]

Create a vector fulfilling given properties in the algebra.

Creating positive vectors requires SciPy.

Parameters
dint

Vector dimensionality

propertiesset of str

Definition of properties for the vector to fulfill. Valid set elements are constants defined in TvtbProperties.

rngnumpy.random.RandomState, optional

The random number generator to use to create the vector.

Returns
ndarray

Random vector with desired properties.

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
ndarray

Unitary vector.

superpose(a, b)[source]

Returns the superposition of a and b.

Parameters
a(d,) ndarray

Left operand in superposition.

b(d,) ndarray

Right operand in superposition.

Returns
(d,) ndarray

Superposed vector.

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
(d,) ndarray

Bound vector.

binding_power(v, exponent)[source]

Returns the binding power of v using the exponent.

The binding power is defined as binding (exponent-1) 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.

exponentint or float

Exponent of the binding power.

Returns
(d,) ndarray

Binding power of v.

invert(v, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

Invert vector v.

A vector bound to its inverse will result in the identity vector.

Parameters
v(d,) ndarray

Vector to invert.

sidednessElementSidedness

This argument has no effect because the inverse of the TVTB algebra is two-sided.

Returns
(d,) ndarray

Inverse of vector.

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_inputsbool, 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
(d, d) ndarray

Transformation matrix to perform binding with v.

get_inversion_matrix(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

Returns the transformation matrix for inverting a vector.

Parameters
dint

Vector dimensionality.

sidednessElementSidedness

This argument has no effect because the inverse of the TVTB algebra is two-sided.

Returns
(d, d) ndarray

Transformation matrix to invert a vector.

implement_superposition(n_neurons_per_d, d, n)[source]

Implement neural network for superposing vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

nint

Number of vectors to superpose in the network.

Returns
tuple

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.

implement_binding(n_neurons_per_d, d, unbind_left, unbind_right)[source]

Implement neural network for binding vectors.

Parameters
n_neurons_per_dint

Neurons to use per dimension.

dint

Dimensionality of the vectors.

unbind_leftbool

Whether the left input should be unbound from the right input.

unbind_rightbool

Whether the right input should be unbound from the left input.

Returns
tuple

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.

sign(v)[source]
absorbing_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

TVTB has no absorbing element except the zero vector.

Always raises a NotImplementedError.

identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

Return the identity element of dimensionality d.

Parameters
dint

Vector dimensionality.

sidednessElementSidedness

This argument has no effect because the identity of the TVTB algebra is two-sided.

Returns
(d,) ndarray

Identity element.

negative_identity_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[source]

Return the negative identity element of dimensionality d.

Parameters
dint

Vector dimensionality.

sidednessElementSidedness, optional

This argument has no effect because the negative identity of the TVTB algebra is two-sided.

Returns
(d,) ndarray

Negative identity element.

zero_element(d, sidedness=<ElementSidedness.TWO_SIDED: 'two-sided'>)[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
dint

Vector dimensionality.

sidednessElementSidedness, optional

This argument has no effect because the zero element of the VTB algebra is two-sided.

Returns
(d,) ndarray

Zero element.

class nengo_spa.algebras.tvtb_algebra.TvtbSign(sign)[source]

Represents a sign in the TvtbAlgebra.

The sign depends on the symmetry and positive/negative definiteness of the binding matrix derived from the vector. For all non-symmetric matrices, the sign is indefinite. It is also indefinite, if the matrices’ eigenvalues have different signs. A symmetric, positive (negative) definite binding matrix corresponds to a positive (negative) sign (equivalent to all eigenvalues being greater than 0, respectively lower than 0). If all eigenvalues are equal to 0, the sign is also 0.

to_vector(d)[source]

Return the vector in the algebra corresponding to the sign.

Parameters
dint

Vector dimensionality.

Returns
(d,) ndarray

Vector corresponding to the sign.

class nengo_spa.algebras.tvtb_algebra.TvtbProperties[source]

Bases: object

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.