nengo_spa.algebras¶
Algebras define the specific superposition and (un)binding operations.
Abstract base class for algebras. |
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Holographic Reduced Representations (HRRs) algebra. |
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Vector-derived Transformation Binding (VTB) algebra. |
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class
nengo_spa.algebras.AbstractAlgebra[source]¶ Abstract base class for algebras.
Custom algebras can be defined by implementing the interface of this abstract base class.
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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
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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
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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
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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
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invert(v)[source]¶ Invert vector v.
A vector bound to its inverse will result in the identity vector.
- Parameters
v ((d,) ndarray) – Vector to invert.
- Returns
Inverted vector.
- Return type
(d,) ndarray
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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
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get_inversion_matrix(d)[source]¶ Returns the transformation matrix for inverting a vector.
- Parameters
d (int) – Vector dimensionality (determines the matrix size).
- Returns
Transformation matrix to invert a vector.
- Return type
(d, d) ndarray
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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
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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
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absorbing_element(d)[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.
- Returns
Standard absorbing element.
- Return type
(d,) ndarray
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class
nengo_spa.algebras.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.
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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
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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
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invert(v)[source]¶ Invert vector v.
A vector bound to its inverse will result in the identity vector.
- Parameters
v ((d,) ndarray) – Vector to invert.
- Returns
Inverted vector.
- Return type
(d,) ndarray
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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
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get_inversion_matrix(d)[source]¶ Returns the transformation matrix for inverting a vector.
- Parameters
d (int) – Vector dimensionality (determines the matrix size).
- Returns
Transformation matrix to invert a vector.
- Return type
(d, d) ndarray
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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
-
absorbing_element(d)[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.
- Returns
Standard absorbing element.
- Return type
(d,) ndarray
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identity_element(d)[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.
- Returns
Identity element.
- Return type
(d,) ndarray
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class
nengo_spa.algebras.VtbAlgebra[source]¶ Vector-derived Transformation Binding (VTB) algebra.
VTB uses elementwise addition for superposition. The binding operation \(\mathcal{B}(x, y)\) is defined as
\[\begin{split}\mathcal{B}(x, y) := V_y x = \left[\begin{array}{ccc} V_y' & 0 & 0 \\ 0 & V_y' & 0 \\ 0 & 0 & V_y' \end{array}\right] x\end{split}\]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 \(V_{y^+} = V_y\).
Note that VTB requires the vector dimensionality to be square.
The VTB binding operation is neither associative nor commutative.
Publications with further information are forthcoming.
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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
-
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)[source]¶ Invert vector v.
A vector bound to its inverse will result in the identity vector.
- Parameters
v ((d,) ndarray) – Vector to invert.
- 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
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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
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get_inversion_matrix(d)[source]¶ Returns the transformation matrix for inverting a vector.
- Parameters
d (int) – Vector dimensionality (determines the matrix size).
- Returns
Transformation matrix to invert a vector.
- Return type
(d, d) ndarray
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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
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