Method and apparatus for constructing symmetry-aware neural network

Abstract

Disclosed is a method for constructing a symmetry-aware neural network. The method comprises receiving input data and a list of symmetries associated with the input data; computing a first linear space of symmetry-equivariant maps; determining a first equivariant basis for the list of symmetries; and constructing at least one equivariant layer of the neural network using the first equivariant basis. The first equivariant basis is a basis of the first linear space of symmetry-equivariant maps. When receiving one or more new symmetries, the method comprises computing a second linear space of symmetry-equivariant maps for the one or more new symmetries; computing an intersection of the second linear space with the first linear space; determining a second equivariant basis as a basis of the computed intersection; and updating the at least one equivariant layer using the second equivariant basis. There is no need to change the architecture of the neural network.

Description

[0001]METHOD AND APPARATUS FOR CONSTRUCTING SYMMETRY-AWARE NEURAL NETWORKTECHNICAL FIELD The present disclosure relates generally to the field of artificial intelligence (AI). For instance, the present disclosure relates to a method and an apparatus for generating symmetry-aware neural networks. BACKGROUND ARTIn semantic communications, one goal is to develop devices and machines that can communicate with one another optimallyfrom a semantic perspective. To achieve this goal, those devices and machines need to understand the semantics and concepts behind the observed and / or communicated data. One way to look at the problem is by utilizing a discovered structure or symmetries among the observed and / or communicated data. The main goal would be to use the minimum possible amount of data to achieve a required percentage of success in a given task (going beyond Shannon information theoretic bounds).One major research line in that perspective is to have deep neural networks (DNNs) that learn and / or be embodied with thesemantics of any dataset and to be able to generalize well to unseen situations.Thus, DNNs need to learn by concept rather than learning by approximation (interpolation), otherwise the accuracy of theresults obtained under training might vary a lot from the ones obtained under testing / validation.SUMMARYFor these neural networks to learn by concept, two main tasks could be described: 1) discovering a list of symmetries, and 2)building symmetry-aware neural networks (according to the discovered list of symmetries). The present disclosure focuses onthe second task with the assumption that the first task is already achieved and a list of given symmetries (not necessarily an all- inclusive list) is always available.For building symmetry-aware neural networks, some mathematical knowledge on analyzing a list of given symmetries withmeans of group-equivariance are discussed. Mathematically, if a dataset has some relations or symmetries (for examplesymmetries of a cube) between its data elements, then one could identify some group (for example group of symmetries of the cube) that acts on the data representation in a way that would transform some data samples to other ones according to the existing relations (for example a 90 degrees rotation along an axis going through the centers of two opposite faces on a given data sample would transform one face of the cube to an adjacent one). In general, the group does not need to be known, but only the list of given symmetries is needed. Then mathematically, this list will generate a group (that we do not need to compute) where the elements of the list are the generators of the group. Thus, the list of these symmetries is all that is needed.In a more specific manner relating to machine learning, when an available dataset has a list of one or more discoveredsymmetries between its data elements and where this list generates a group acting on the data representation, then it would be desirable to train the neural network only on a few data samples and enable it to generalize to all other data samples with some given knowledge about the underlying list of symmetries. However, most of existing DNNs are unable to generalize well and efficiently when given few abstract / semantic data samples that inherits some sort of symmetry and / or correlation that can be represented mathematically in a representation of a finite group. Classical neural networks (without equivariance) usually achieve low test / validation accuracy thus poor generalization (extrapolation) when only a small portion of the real dataset is available for training. Advanced DNNs (with certain types of equivariance), such as Transformers, need heavy computations (long running time, high energy consumption, waste of system resources and slow convergence) and they still do not generalize well when dealing with a small abstract / semantic dataset. One way to enable the network to be aware of any symmetry or relation between the input data with the minimum needed data samples is to implement equivariance with respect to the group generated by the list of discovered symmetries. Here, group equivariance means that applying an action of the group on the initial data representation followed by some function is equivalent to applying the function first and then the action of the group on the new data representation. In machine learning, the function could be a linear one (such as multiplying by the weights) or a non-linear one (such as applying the activation functions).In a machine learning context, it is desirable to have an equivariant neural network. A neural network is usually composed ofstacked layers of neurons. The collection of real valued activities of neurons of a given layer is organized as a vector and thus the set of all possible collections of activities of neurons of a given layer is seen as a real vector space. Each layer receives an input activation vector and transforms it into an output activation vector. A simple linear layer is mainly composed of a weight matrix that transforms the input activation vector into an output activation vector by means of linear matrix multiplication. A given layer can contain a linear component and a non-linear component, usually the latter referred to as a non-linear activation function. When talking about an equivariant layer, it is required that all components of the layer (overall transformations) respect the equivariance properties. An equivariant neural network (or sub-network) is achieved when all its layers respect the desired equivariance. Note that, in theory, one can always construct a pure linear neural network. However, non-linear activations have been shown to be very crucial for the learning convergence in a neural network. Thus, it is important to consider these non-linearities when implementing an equivariant neural network which could impose some limitations to be handled. When devising group-equivariant transformations within a layer, actions of a fixed group on the vector spaces of input andoutput activities of the layer are usually given. Since the point-wise non-linear functions are effective in the diverse applicationsof machine learning, one could restrict the non-linear scope to only point-wise non-linearities. With those non-linearities being applied separately at each neuron, it is possible to treat those actions in a linear manner (and hence more computable manner) if the actions of the group of symmetries consist only in permuting the coordinates of the activities’ vectors. In other words, the latter condition means that the actions of the group of symmetries could be seen as a collection of permutation matrices (closed under multiplication and inversion) which act on the vector spaces of activation vectors. Those actions are called“permutation representations” in the mathematical theory of representation of finite groups.The idea of group-equivariance has been studied many times where equivariance is applied to the raw input data (for example images or sentences) such that: a) only permutation equivariance is considered, b) some specific transformations (such astranslation, reflections, and rotations) are considered, or c) in a more generic manner for larger set of groups such as compactgroups, arbitrary matrix groups, Clifford groups, and in special cases of permutation groups when represented as direct sum, direct product, or wreath product of two other permutation subgroups. In those studies, equivariance is ensured through the more or less complete exploitation of the decomposition of representations into irreducible sub-representations, approximations (by different methods) of the projectors on those irreducible sub-representations (and hence computation and approximation of appropriated passage matrices) and more or less complete exploitation of the structure of endomorphisms of irreducible representations. Since the different methods considered for implementing equivariance were applied on the raw input level, many inefficiencies and difficulties could be faced: (i) finding the group acting on those raw data elements would be a challenging task, and decomposing therepresentations of such groups (which could be infinite), and where dimensions on the input level are usually much higher than on the semantic one, would be even more challenging and very costly in terms of complexity, running time, and energy consumption; and (ii) restricting to famous transformations, such as rotations, translations, and reflections on the raw level, would lead tomissing very essential groups acting on the semantic level (for example permuting the values of the features / attributes rather than the raw data itself), and thus would lead to being unable to learn concepts and generalize well with minimal training data. In addition, the traditional methods, even with the most generic forms of equivariance, assumes knowledge of the group of transformations acting on the input data and do not utilize well the structure and parameters of the representation decomposition. Therefore, they lead to having repeated / redundant computations and high complexity in implementing group-equivariant layers. Even in the case where the representation decomposition parameters are utilized, the problem still relies in the fact thatcomputing this decomposition (if possible, e.g., with free algebraic software like GAP) could be very costly in terms of timeand complexity. The disadvantages could be summarized as follows: i) Raw-input equivariance (missing groups acting on semantic level),ii) Costly and / or complicated computations of equivariance parameters, andii) Limited scope of representations / groups is achieved.Therefore, a main problem is that existing neural networks are unable to generalize well and efficiently when given fewabstract / semantic data samples that inherits some sort of symmetry and / or correlation (a list of discovered symmetries in thedata representation is given). Thus, a method is provided to build symmetry-aware neural networks (equivariance with respectto all possible combinations of the symmetries elements) to deal with the case when the representation decomposition parameters (irreducible multiplicities and / or passage matrices) are unavailable or are very costly to be used or computed. For building a symmetry-aware neural network, a computer-implemented method is provided. The method comprises the following steps: -obtaining input data ^ = {^^ , ^^ , … ∈ ℝ^∗^ and a list of symmetries ^(^) ≔ {^^ , ^^ , … , ^^} associatedwith the input data of a layer, in which n denotes a multiplicity of the input data, d denotes a representation dimension,and l denotes a quantity of the symmetries;- computing an equivariant basis ^^(^) = {^^ , ^^, … ∈ ℝ^∗^∗^that is with respect to the list of symmetries, wherein the equivariant basis ^^(^) is a basis of a linear space ℰ(^) of equivariant maps, and b denotesa quantity of the equivariant maps of the equivariant basis; -multiplying the whole input data ^ by each of the equivariant basis elements in the equivariant basis ^^(^), toobtain a list of transformed input data ^ = {^^ , ^^, … , ^^} ∈ ℝ^∗^∗^;- constructing weight matrices of the layer and determining a multiplicity m of an output of the layer based oninput and output dimensions of the layer; -optimizing the weight matrices to obtain learned weight matrices ^ = {^^ , ^^ , … , ^^} ∈ -multiplying each learned weight matrix ^^ ∈ ^ by a corresponding transformed input data ^^ in the list,respectively, to obtain a plurality of output matrices ^ = {^^ , ^^ , … , ^^} ∈ -computing a sum over all the b matrices of the output matrices O to obtain a two-dimensional array of m vectorseach of dimension d; and -applying a point-wise activation function to a flattened array of the m vectors, to obtain an output of the layer.In this way, the symmetry-aware neural network built accordingly enables the implementation of equivariance with respect toa list of symmetries without necessarily knowing the entire group. The neural network may learn on a semantic / abstract levelrather than on a raw input. In this way, the neural network may have a simple architecture (comprising one or more equivariantlayers) and may still generalize very well even when trained with a small portion of a complete dataset. The learning andconvergence phase can be much quicker. Extremely higher simplicity in the network may be achieved with few parameters tooptimize. As used herein, an artificial neural network may refer to a branch of machine learning models that loosely models the structureof the neurons in a biological brain and "learns" to perform tasks by considering examples. Typically, neurons are aggregatedinto layers. Signals propagate from a first layer (the input layer) via intermediate layers, if any, to a last layer (the output layer). An edge between a neuron of a given layer to a neuron of a next layer typically has a weight that adjusts as learning proceeds, for example using (error) backpropagation. It is noted that the list of symmetries associated with the input data is always known in this disclosure. How to determine / generate the list of symmetries is not relevant to the present disclosure.As used herein, a canonical basis of a vector space ℝ^ for some integers ^ may refer to a standard basis of said vector spaceℝ^. As used herein, an activation function may refer to a function of a neuron of an artificial neural network that calculates an output of the neuron based on a sum of its inputs and multiplied by their individual weights.Optionally, if two or more of the plurality of layers share a same list of symmetries, the equivariant basis ^^(^) is computedonly once for the two or more layers. In a possible implementation form, the step of computing the equivariant basis that is with respect to the list of symmetries may comprise the following steps: -computing a linear space of equivariant maps ℇ(^^) with respect to each symmetry ^^; and- computing an intersection of the computed linear spaces (ℇ(^) ≔ ⋂^^∈^(^) ℇ(^^) ), to obtain a list of linearmaps as the equivariant basis ^^(^) that is the basis of the intersection ℰ(^).In a possible implementation form, a cyclic equivariance may be used for computing the linear space ℇ(^^) of equivariant mapswith respect to each symmetry ^^.In a possible implementation form, the weight matrices may be optimized through a stochastic gradient descent or Adamalgorithm. In a possible implementation form, one or more invariant biases that are invariant with respect to the list of symmetries maybe added to obtain the array of vectors.In a possible implementation form, the method may further comprises learning the one or more invariant biases.In a possible implementation form, the activation function may be a non-linear activation function.In a possible implementation form, the activation function may be a rectified linear unit (ReLU) or Tanh activation function.In a possible implementation form, the list of transformed input data may comprise b transformed input data of dimension n byd. The weight matrices may comprise b weight matrices each of dimension m by n. The plurality of output matrices maycomprise b output matrices each of dimension m by d. n, d, b, m, and n are positive integers.Accordingly, an apparatus is provided. The apparatus comprises a neural network comprising a plurality of layers. The apparatus is configured to: -obtain input data ^ = {^^ , ^^ , … , ^^} ∈ ℝ^∗^ and a list of symmetries ^(^) ≔ {^^ , ^^ , … , ^^} associated withthe input data of a layer, wherein n denotes a multiplicity of the input data, d denotes a representation dimension, andl denotes a quantity of the symmetries;- compute an equivariant basis ^^(^) = {^^, ^^ , … , ^^} ∈ ℝ^∗^∗^that is with respect to the list of symmetries, wherein the equivariant basis ^^(^) is a basis of a linear space ℰ(^) of equivariant maps, and b denotes a quantity ofthe equivariant maps of the equivariant basis; -multiply the whole input data ^ by each of the equivariant basis elements in the equivariant basis ^^(^), toobtain a list of transformed input data ^ = {^^ , ^^, … , ^^} ∈ ℝ^∗^∗^; -construct weight matrices of the layer and determine a multiplicity m of an output of the layer based on inputand output dimensions of the layer; -optimize the weight matrices to obtain learned weight matrices ^ = {^^, ^^ , … , ^^} ∈ -multiply each learned weight matrix ^^ ∈ ^ by a corresponding transformed input data ^^ in the list,respectively, to obtain a plurality of output matrices ^ = {^^ , ^^ , … , ^^} ∈ -compute a sum over all the b matrices of the output matrices O to obtain a two-dimensional array of m vectorseach of dimension d; and -apply a point-wise activation function to a flattened array of the m vectors, to obtain an output of the layer.Optionally, the apparatus may be any device that can be adapted to perform and / or accelerate AI computing (e.g., as an AI accelerator). For instance, the apparatus may be (but not limited to): a central processing unit (CPU), graphics processing unit(GPU), Neural Processing Unit (NPU), or any device comprising the CPU and / or GPU and / or NPU.In a possible implementation form, if two or more of the plurality of layers share a same list of symmetries, the apparatus isconfigured to compute the equivariant basis ^^(^) only once for the two or more layers.In a possible implementation form, for computing the equivariant basis that is with respect to the list of symmetries, the apparatus may be configured to: -compute a linear space of equivariant maps ℇ(^^) with respect to each symmetry ^^; and- compute an intersection of the computed linear spaces (ℇ(^) ≔ ), to obtain a list of linear mapsas the equivariant basis ^^(^) that is the basis of the intersection ℰ(^).In a possible implementation form, a cyclic equivariance may be used for computing the linear space ℇ(^^) of equivariant mapswith respect to each symmetry ^^.In a possible implementation form, the weight matrices may be optimized through a stochastic gradient descent or Adamalgorithm. In a possible implementation form, one or more invariant biases that are invariant with respect to the list of symmetries may be added to obtain the array of vectors. In a possible implementation form, the apparatus may be further configured to learn the one or more invariant biases. In a possible implementation form, the activation function may be a non-linear activation function. In a possible implementation form, the activation function may be a rectified linear unit (ReLU) or Tanh activation function.In a possible implementation form, the list of transformed input data may comprise b transformed input data of dimension n byd. The weight matrices may comprise b weight matrices each of dimension m by n. The plurality of output matrices maycomprise b output matrices each of dimension m by d. n, d, b, m, and n are positive integers.The above-mentioned method and apparatus provide a generic solution for constructing a symmetry-aware neural network, which involves computing an equivariant basis for an equivariant layer implementation with respect to a given (fixed) list of symmetries at a given instant of time.In some cases, symmetries are not available all at once. The symmetry-aware neural networks are dependent on symmetriesdiscovery tasks and are inflexible with newly discovered symmetries. Thus, it is an objective of this disclosure to build flexiblesymmetry-aware neural networks when symmetries are given, received, or discovered in different time instances and not all atonce (e.g., in an evolving environment).A first aspect of this disclosure provides a computer-implemented method for constructing (and / or updating) a symmetry-awareneural network. The method comprises the following steps: -receiving input data and a list of symmetries associated with the input data;- computing a first linear space of symmetry-equivariant maps and determining a first equivariant basis for thelist of symmetries, wherein the first equivariant basis is a basis of the first linear space of equivariant maps; -constructing at least one equivariant layer of the neural network using the first equivariant basis.In response to receiving one or more new symmetries, the method comprises the following steps:- computing a second linear space of symmetry-equivariant maps for the one or more new symmetries;- computing an intersection of the second linear space with the first linear space;- determining a second equivariant basis as a basis of the computed intersection; and- updating the at least one equivariant layer using the second equivariant basis.In this way, the neural network can be incrementally updated without a need to be built from scratch in response to receivingthe one or more new symmetries. Optionally, receiving the input data and a list of symmetries associated with the input data may comprise: obtaining the input data ^ = {^^ , ^^ , … , ^^} ∈ ℝ^∗^ and the list of symmetries ^^(^) = {^^ , ^^ , … , ^^} associated withthe input data of a layer. n denotes a multiplicity of the input data, d denotes a representation dimension, and l denotesa quantity of the symmetries.Optionally, computing the first linear space of symmetry-equivariant maps comprises: computing an intersection of linearspaces of equivariant maps ℇ(^^) with respect to each symmetry ^^ in the list ^^ as the first linear space. It can be similarlyapplied for computing the second linear space ℇ^, which is an intersection of linear spaces of symmetry-equivariant mapscorresponding to the one or more new symmetries.Optionally, determining the first equivariant basis for the list of symmetries may comprise: computing an intersection of thecomputed linear spaces (ℇ^ ≔ ⋂^^∈^^ ℇ(^^) ), to obtain a list of linear maps as the first equivariant basis ^^^ that is the basisof the intersection ℰ^. It can be similarly applied for determining the second equivariant basis ^ℰ^ ∩ℇ^ , which is a basis of anintersection between the two linear spaces ℇ^and ℇ^.Optionally, constructing the at least one equivariant layer of the neural network using the first equivariant basis may comprise:- multiplying the whole input data ^ by each of the equivariant basis elements in the first equivariant basis ^^^,to obtain a list of transformed input data ^ = {^^, ^^ , … , ^^} ∈ ℝ^∗^∗^; -constructing weight matrices of the equivariant layer and determining a multiplicity m of an output of theequivariant layer based on input and output dimensions of the equivariant layer; -optimizing the weight matrices to obtain learned weight matrices ^ = {^^ , ^^ , … , ^^} ∈ ℝ^∗^∗^.It can be similarly applied for updating the at least one equivariant layer using the second equivariant basis ^ℰ^ ∩ℇ^ , which isan updated equivariant basis in view of the one or more new symmetries. The learned weight matrices may be used as follows: -multiplying each learned weight matrix ^^ ∈ ^ by a corresponding transformed input data ^^ in the list,respectively, to obtain a plurality of output matrices ^ = {^^ , ^^ , … , ^^} ∈ -computing a sum over all the b matrices of the output matrices O to obtain a two-dimensional array of m vectorseach of dimension d; and -applying a point-wise activation function to a flattened array of the m vectors, to obtain an output of theequivariant layer.It is noted that the list of transformed input data comprises b transformed input data of dimension n by d, the weight matricescomprises b weight matrices each of dimension m by n, the plurality of output matrices comprises b output matrices each ofdimension m by d, wherein n, d, b, m, and n are positive integers. These dimensions are kept unchanged even after receivingthe one or more new symmetries. Thus, there is no need to change the architecture of the neural network. Optionally, a cyclic equivariance may be used for computing the first linear space and the second linear space. Optionally, the weight matrices may be optimized through a stochastic gradient descent or Adam algorithm. Optionally, one or more invariant biases that are invariant with respect to the list of symmetries and with respect to the one or more new symmetries may be added to obtain the array of vectors. Optionally, the method may comprise learning the one or more invariant biases.Optionally, the activation function may be a non-linear activation function, for instance, a rectified linear unit (ReLU), anexponential linear unit (ELU), or a Tanh activation function. In an implementation form of the first aspect, the update of the at least one equivariant layer is performed without a need for changing an architecture of the neural network. In a further implementation form of the first aspect, for updating of the at least one equivariant layer, an input dimension and output dimension of the at least one equivariant layer are kept unchanged. In a further implementation form of the first aspect, the method further comprises training the neural network with the at least one updated equivariant layer. In a further implementation form of the first aspect, for training the neural network with the at least one updated equivariant layer, the method comprises the following steps: -keeping weights obtained before receiving the one or more new symmetries unchanged; and- resuming training the neural network with the at least one updated equivariant layer until convergence or afurther new symmetry is received. In a further implementation form of the first aspect, for training the neural network with the at least one updated equivariant layer, the method comprises the following steps: -initializing the neural network with the at least one updated equivariant layer with random weights, to obtain avanilla neural network; and -training the vanilla neural network until convergence or a further new symmetry is received.A second aspect of this disclosure provides an apparatus for constructing (and / or updating) a symmetry-aware neural network. The apparatus is configured to: -receive an input data and a list of symmetries associated with the input data;- compute a first linear space of symmetry-equivariant maps and determining a first equivariant basis for the listof symmetries, wherein the first equivariant basis is a basis of the first linear space of equivariant maps; and -construct at least one equivariant layer of the neural network using the first equivariant basis.In response to receiving one or more new symmetries, the apparatus is configured to: -compute a second linear space of symmetry-equivariant maps for the one or more new symmetries;- compute an intersection of the second linear space with the first linear space;- determine a second equivariant basis as a basis of the computed intersection; and- update the at least one equivariant layer using the second equivariant basis.In an implementation form of the second aspect, the update of the at least one equivariant layer is performed by the apparatus without a need for changing an architecture of the neural network. In a further implementation form of the second aspect, for updating of the at least one equivariant layer, the apparatus is configured to keep an input dimension and output dimension of the at least one equivariant layer unchanged. In a further implementation form of the second aspect, the apparatus is further configured to train the neural network with the at least one updated equivariant layer. In a further implementation form of the second aspect, for training the neural network with the at least one updated equivariant layer, the apparatus is configured to: -keep weights obtained before receiving the one or more new symmetries unchanged; and- resume training the neural network with the at least one updated equivariant layer until convergence or a furthernew symmetry is received. In a further implementation form of the second aspect, for training the neural network with the at least one updated equivariant layer, the apparatus is configured to: -initialize the neural network with at least one updated equivariant layer with random weights, to obtain a vanillaneural network; and -train the vanilla neural network until convergence or a further new symmetry is received.The device of the second aspect may share the same optional features and advantages with the method of the first aspect. A third aspect of this disclosure provides a system. The system comprises at least one apparatus according to the second aspect or any implementation form thereof. A fourth aspect of this disclosure provides a computer program comprising instructions which, when the program is executed by a computer, cause the computer to carry out the method according to the first aspect or any implementation form thereof.A fifth aspect of this disclosure provides a computer-readable medium comprising instructions which, when executed by acomputer, cause the computer to carry out the method according to the first aspect or any implementation form thereof.A sixth aspect of this disclosure provides an equivariant neural network layer (also referred to as a symmetry-aware neuralnetwork layer). The equivariant neural network layer is adapted (e.g., through a processor associated therewith) to perform themethod according to the first aspect or any implementation form thereof.A seventh aspect of this disclosure provides a neural network comprising one or more equivariant layers each according to thesixth aspect. BRIEF DESCRIPTION OF DRAWINGS The above-described aspects and implementations will now be explained with reference to the accompanying drawings, in which the same or similar reference numerals designate the same or similar elements. The drawings are to be regarded as being schematic representations, and elements illustrated in the drawings are not necessarily shown to scale. Rather, the various elements are represented such that their function and general purpose become apparent to those skilled in the art.FIG. 1 illustrates a method for constructing a symmetry-aware neural network;FIG. 2 illustrates the method of FIG. 1 in more detail;FIG. 3 illustrates an example of an equivariant linear layer;FIG. 4 illustrates an example of computing an equivariant basis;FIG. 5 illustrates an example of weight matrices construction and optimization;FIGs 6A-6C illustrate an exemplary multinary classification problem;FIG. 7A shows an example of SET card game dataset;FIG.7B shows simulation result using the SET dataset in FIG.7A;FIG. 8 shows a method for constructing a symmetry-aware neural network according to this disclosure;FIG. 9 shows a diagram of updating equivariant layers of a symmetry-aware neural network with an updated equivariantbasis during training; FIG.10 shows an example of a neural network architecture; andFIGs 11A-11C show simulation results.DETAILED DESCRIPTIONS OF DRAWINGS In the following description, reference is made to the accompanying drawings, which form part of the disclosure, and which show, by way of illustration, specific aspects of implementations of the present disclosure or specific aspects in which implementations of the present disclosure may be used. It is understood that implementations of the present disclosure may be used in other aspects and comprise structural or logical changes not depicted in the figures. The following detailed description, therefore, is not to be taken in a limiting sense, and the scope of the present disclosure is defined by the appended claims. For instance, it is understood that a disclosure in connection with a described method may also hold true for a corresponding apparatus or system configured to perform the method and vice versa. For example, if one or a plurality of specific method steps are described, a corresponding device may include one or a plurality of units, e.g. functional units, to perform the describedone or plurality of method steps (e.g. one unit performing the one or plurality of steps, or a plurality of units each performingone or more of the plurality of steps), even if such one or more units are not explicitly described or illustrated in the figures.On the other hand, for example, if a specific apparatus is described based on one or a plurality of units, e.g. functional units, acorresponding method may include one step to perform the functionality of the one or plurality of units (e.g. one step performing the functionality of the one or plurality of units, or a plurality of steps each performing the functionality of one or more of the plurality of units), even if such one or plurality of steps are not explicitly described or illustrated in the figures. Further, it is understood that the features of the various exemplary implementations and / or aspects described herein may be combined with each other, unless specifically noted otherwise. FIG.1 illustrates a method 10 in accordance with the present disclosure in its broadest implementation. The method 10 implements an equivariant neural layer of an artificial neural network. In general, the method 10 computes a basis of an intersection for all the symmetries in a list of linear spaces of maps equivariant with respect to the action of a symmetry in this list and build an equivariant neural layer using the computed equivariant basis. The method 10 is applied to any layer and comprises the following steps. Then the layer as an equivariant layer is built in multiple steps, starting by step 13. Step 14: constructing weight matrices whose dimensions are determined by the input (given) and output (chosen) multiplicitiesof the data representation within the built equivariant layer. Accordingly, a multiplicity m of an output of the layer is determinedbased on input and output dimensions of the layer. Step 17: computing a sum over all the b matrices of the output matrices O to obtain a two-dimensional array of m vectors eachof dimension d.Step 18: applying a point-wise activation function to a flattened array of the m vectors, to obtain an output of the layer.It is noted that the two-dimensional array obtained in step 17 is flattened into a one-dimensional array in step 18, such that thepoint-wise activation function can be applied to the flattened array of vectors. Optionally, any pointwise non-linear activationfunction may be applied in step 18 on the transformed input by the described equivariant basis given the new data representation is just a multiple of the initial input data representation.FIG. 2 illustrates the method 10 of FIG. 1 in more detail.The equivariant basis ^^(^) that is with respect to the list of symmetries in step 12 may be computed as follows:- Step 121: for each symmetry, a linear space of group-equivariant maps is generated, in which the group here isgenerated by the selected symmetry. That is, a linear space of equivariant maps ℇ(^^) with respect to each symmetry^^is computed. Optionally, cyclic equivariance may be applied. -Step 122: an intersection of all linear spaces given in previous step is computed, which may be a list of linearmaps which forms an equivariant basis of the intersection between all the computed spaces. That is, an intersection of the computed linear spaces is computed, which may be represented as ℇ(^) ≔ As a result, a list oflinear maps as the equivariant basis ^^(^) that is the basis of the intersection ℰ(^) is obtained.FIG.3 illustrates an equivariant linear layer 31. As illustrated in FIG. 3, a neural network 300 usually comprises several layers (e.g., 301, 302, 303, 304) in a feed-forward structure. Each layer receives an input activation vector and transforms it into an output activation vector through linear (weight matrices) and / or non-linear (non-linear activation functions) components in the layer. The present disclosure is mainly considered to deal with any neural network architecture where the main goal is to let the neural network (or a system comprising the neural network) to be aware of existing symmetries (that is provided as a list of symmetries in addition to the input activation vector) and thus to optimize the learning process. Thus, the present disclosure is applicable to any system or scenario which involves neural networks where a list of symmetries in the given data representation is available. This disclosure provides an equivariant linear layer 31 that may be applied to any layer of a neural network. That is, any oneor more of the layers 301, 302, 303, 304 of the neural network 300 may be the equivariant linear layer 31. In general, asymmetry-aware neural network may comprise one or more equivariant linear layers 31 and one or more fully-connected layers.Referring back to FIG. 1, steps 12, 13, 14, 15, 16, and 17 are all implemented within the equivariant linear layer 31 as follows:starting with step 11, based on the obtained input representation and the list of symmetries (referred to as generator (Gen.)),step 12 is performed. For instance, a space of symmetry-equivariant maps corresponding to each symmetry in the list iscomputed (e.g., as in step 121). Then, a basis of the intersection between all the computed spaces in the previous step 121 iscomputed (e.g., as in step 122). Then, a symmetries-equivariant basis ^^(^) of the space (^) is computed as a list of ^equivariant maps of dimension ^ ∗ ^ where ^ is the given representation dimension.The free weights in the step of “applying free weights” in FIG.3 may be understood that, for each element (b elements in total)of the computed equivariant basis ^^(^), one could assign a full weight matrix (without any structure or constraints) whosesize is given byinput (given) and output (arbitrary) multiplicities of the data representation at the specific layer in action, denotedas n and m respectively. This full weight matrix may be understood as the free weights. Any optimization algorithm may beused in the learning process for optimizing these free weight matrices as in step 15. For the “applying” part of the step of “applying free weights”, each input vector is transformed into the equivariant basis by computing the product of the inputvector with each equivariant linear map in ^^(^) as in step 13. As a result, a list of b 2D-arrays each of dimension n*d isobtained. Then, as in step 16, the learnt weights may be applied on these 2D-arrays by computing their corresponding products(each array is multiplied by its corresponding weight matrix) leading to b matrices each of dimension m*d. Then, as in step 17,the sum over all these b matrices is computed leading to one 2D-array of m vectors each of dimension d (when flattened givesone output vector of dimension m*d). Optionally, invariant biases that are invariant with respect to the list of symmetries may be learnt and added to the output.Then, a point-wise (with respect to each integer position ^ ∈ [1, ^ ∗ ^] of the flattened output vector) non-linear activationfunction (such as ReLU or Tanh) may be applied as in step 18.FIG. 4 illustrates an example of computing an equivariant basis, which is based on step 121 and 122 of FIG. 1. As illustratedin FIG.4, cyclic equivariance may be used for the sake of efficiency in step 121.FIG. 5 illustrates an example of weight matrices construction and optimization. The representations in FIG. 5 are in accordancewith the annotations introduced in FIG. 1-FIG. 4.As clearly depicted in FIG.5, the present disclosure results in the following number of parameters to be optimized per layer: ^× ^ × ^ (^ = {^^, ^^ , … , ^^} ∈ ℝ^∗^∗^). In comparison to the case of classical fully connected layers with similar inputand output dimensions to the above described layer, the following number of parameters to be optimized per layer: ^2 × ^×^. As long as b is much smaller than d2 (which is almost the case in application scenarios, e.g., as shown in FIGs. 6-7), thepresent disclosure leads to a symmetry-aware neural network that has much less parameters to optimize.FIGs 6A-6C shows an exemplary multinary classification problem.This disclosure has been tested on a multinary classification problem into six possible category types where objects are imageswith three main features and where each feature takes two possible values. It is a framework of a Shepard and al. experiment(or simply, Shepard experiment) (see Shepard, R. N., Hovland, C. I., & Jenkins, H. M. (1961). Learning and memorization of classifications. Psychological Monographs: General and Applied, 75(13), 1–42. https: / / doi.org / 10.1037 / h0093825).As illustrated in FIG. 6A, an example of a simple dataset used for the Shepard experiment with features being shape (triangleor square), size (large or small) and color (filled or not).For category Type-I, knowledge of one feature of the stimuli (for example the color of the objects in the frames) is enough to distinguish between the two sets of stimuli. For category Type-II, knowledge of two features (for example color and size) is needed to have an economical rule for distinguishing between the two sets of stimuli. For categories Type-III, IV, and V, knowledge of three dimensions (with some exceptions) is needed for the distinguishing but also one can observe an abstract geometric structure underlying each of these types: snake, corner, and isolated point. For category Type-VI, knowledge on all three dimensions without any exceptions is needed (geometric structure: opposed diagonals).FIG. 6B shows geometric structures of 70 representatives of all the six possible category types in Shepard experiment.There are three independent features each taking two values so each single image in FIG. 6A could be seen as a vertex on acube. Therefore, a quadruple of images like in FIG. 6A could be seen as a collection of four vertices on a cube. The six differenttypes of FIG. 6B then correspond to the different geometric structures of those quadruples of vertices on the cube. Each typerealized like this is invariant by the group of the symmetries of the cube. The list of given symmetries in this disclosure in this example may be a list of discovered permutations that transform an abstract representation of a given input set of four images (seen as four vertices of the cube) into a new abstract representation within the same category type (for example if the initial input could be seen abstractly as a face of the cube, then the transformed set of vertices should be also a face of the cube). The list of given symmetries may vary in size, but in general its elements are adapted to generate at least a subgroup of the group of symmetries of the cube. FIG. 6C shows generalization capabilities of a classical fully connected neural network (on the left-hand side of FIG. 6C) compared to the generalization capabilities of an artificial neural network in accordance with the present disclosure (on the right-hand side of FIG.6C) given the multinary classification problem in previous FIG.6A and FIG.6B. The fully connected (which is non-symmetry-aware) neural network serves as a reference for comparison.The training dataset comprises nine representatives (one for each category and three for generators of the group of symmetries of the cube) out of the total of 70 representatives of all the six possible category types. The number of epochs amounts to 10, the batch size is 256, the number of batches per epoch totals 64, a learning rate is set to 0.0015, Tanh serves as non-linear activation function, and a bias can be learned.The left-hand side of FIG. 6C shows a limited generalization capability of the classical fully connected neural network. Inparticular, the accuracy of validation almost corresponds to random guessing.However, the right-hand side of FIG. 6C depicts an improved generalization capabilities of the deep neural network inaccordance with the present disclosure respecting the list of given symmetries, here specifically equivariance with respect to the group of cube symmetries.In addition to the improved generalization capabilities, a significant reduction in the number of parameters to be optimized inthe weight matrix and thus in complexity of the network (running time, memory and energy consumption) is achieved. In thisexample, the number of optimized parameters in a single symmetry-aware linear layer is given by: ^ × ^ × ^ = 756 ^^,where m and n are the output and input multiplicities, respectively, of the input representation in a given layer. When comparedto the number of parameters in a fully connected layer given by ^2 × ^ × ^ = 482 ^^,the following reduction of parameters in each of the three symmetry-aware layers may be obtained:(482^^ − 756 ^^) / 482 ^^×100% = ^^. ^^ %.FIG.7A shows an example of SET (card game) dataset. FIG.7A shows 81 cards. Each card has four features with three possible values for each feature. The four features and their corresponding values are: Color (Violet, Green, Red), Number (One, Two, Three), Shape (Diamond, Oval, Squiggle), and Shading (Solid, Striped, Open). Normally in the SET game, the goal is to identify triplets of cards that form a SET (the specific rules are not of interest in this application). However here it is focused on the different possible types given by any tuple of three different cards. In total,there could be 85320 possible combinations (with each combination being a different representative) of three cards tuples (witheach card being an object). These representatives could be grouped / classified into 20 possible category types based on their structure. As an additional input, we are also given a list of symmetries relating the given dataset. Recall that whenever any data symmetry is applied on a specific representative in a given type, then its transformation (including identity) output should stay within the same type. Here it is noted that the group of all symmetries behind the SET game dataset could be identified as a Wreath Product (denoted as≀) of two symmetric groups given by features and values as follows: . In fact, the above remark is also applicable to the previous embodiment on Shepard experiment, and it can be generalized to any case where the objects have a given set of features and where all the features can take the same number of possible values. Then mathematically, in this case, the types are just the orbits of the wreath product, where there is in total 31104 group elements (possible symmetries) in this example. In addition to the improved generalization capabilities with a hugely reduced training dataset (e.g., only 3% of the whole dataset) in the above example, the number of parameters in each symmetry-aware layer compared to a classical fully connected layer is also reduced, in this example by: .The disclosure of FIGs 1 – 7 provides a method to build a symmetry-aware neural network that requires only to know the datarepresentation and the list of symmetries without the need to know the parameters of the representation decomposition intoirreducibles. In this way, running time for computing the equivariant basis is much quicker and the complexity is much lower.Further, manual tasks such as transforming the passage matrix from complex domain to real domain can be avoided. The symmetry-aware neural network built according to the present disclosure enables the implementation of equivariance withrespect to a list of symmetries (so not necessarily knowing the group) on a semantic / abstract level rather than on a raw inputand for any representation of any finite group. In this way, a neural network with a simple architecture (comprising one or moreequivariant layers) can generalize very well given very small portion of the complete dataset. The learning and convergencephase can be much quicker. Extremely higher simplicity in the network may be achieved with few parameters to optimize. Thesymmetry-aware neural network has a better generalization capability even trained with a small portion of a training dataset.According to this disclosure, a list of equivariant linear maps is computed, which directly transforms the input into the desiredequivariant linear space without imposing any conditions or custom structure on the weights’ matrix. Known optimizationalgorithms and the existing tools may be utilized without any need for custom or specific adaptations. Building such symmetry-aware neural networks are dependent on symmetries discovery task and are inflexible with newly discovered symmetries. This disclosure further provides a method to build flexible symmetry-aware neural networks when symmetries are given or discovered in different instances of time and not all at once (could be in an evolving environment also). A general idea is to build a symmetry-aware neural network layer in an incremental way by updating the equivariant basis oncea new symmetry is added to a known list of symmetries. The updated equivariant basis is given by the basis of the intersectionof the previous linear space of equivariant maps (before adding the new symmetry) with a new linear space of a new equivariantmaps based on the new symmetry.FIG. 8 shows a method 80 for constructing (and / or updating) a symmetry-aware neural network according to this disclosure.The method 80 is computer-implemented and comprises the following steps 81 - 88.Step 81: receiving input data and a list of symmetries associated with the input data. Step 82: computing a first linear space of symmetry-equivariant maps. In this disclosure, the symmetry-equivariant maps may also be simply referred to as equivariant maps. Step 83: determining a first equivariant basis for the list of symmetries. The first equivariant basis is a basis of the first linear space of equivariant maps. Step 84: constructing at least one equivariant layer of the neural network using the first equivariant basis. Steps 81-84 may share the same features of the method for building a symmetry-aware neural network introduced above withrespect to FIG. 1 – 7. For instance, Step 81 may be based on Step 11. Step 82 may be based on Step 121. Step 83 may be basedon Step 122. Step 84 may be based on Steps 14 – 18.In this disclosure, in response to receiving one or more new symmetries, the method 80 comprises the following steps 85 – 88.It is noted that how a new symmetry is discovered is not relevant to this disclosure. Any means commonly known in the field for symmetry discovering may be used. Step 85: computing a second linear space of symmetry-equivariant maps for the one or more new symmetries. Step 86: computing an intersection of the second linear space with the first linear space. Step 87: determining a second equivariant basis as a basis of the computed intersection.Step 88: updating the at least one equivariant layer using the second equivariant basis.Step 85 may be based on Step 121, in which case the received one or more new symmetries is used. Step 87 may be based onStep 122, in which case the computed intersection is used. Step 88 may be based on Step 14 – 18, in which case a resumedtraining or a new training of the at least one equivariant layer is performed based on the second equivariant basis.According to this disclosure, the update of the at least one equivariant layer is performed without a need for changing anarchitecture of the neural network. For updating of the at least one equivariant layer, an input dimension and output dimensionof the at least one equivariant layer are kept unchanged. Optionally, the method 80 may further comprise training the neural network with the at least one updated equivariant layer. Optionally, for training the neural network with the at least one updated equivariant layer, the method comprises the following steps: -keeping weights obtained before receiving the one or more new symmetries unchanged; and- resuming training the neural network with the at least one updated equivariant layer until convergence or afurther new symmetry is received.Alternatively, for training the neural network with the at least one updated equivariant layer, the method comprises thefollowing steps: -initializing the neural network with the at least one updated equivariant layer with random weights, to obtain avanilla neural network; and -training the vanilla neural network until convergence or a further new symmetry is received.The method 80 may be executed by any apparatus or system thereof that is suitable for constructing or updating the symmetry- aware neural network.It is noted that, optionally, the symmetry-aware neural network of this disclosure may be used for image processing (e.g., objectrecognition / classification). The input data may comprise image data comprising one or more objects. The symmetries may comprise information on internal relevant properties of the one or more objects to be recognized / classified. Making the networksymmetry-aware can allow the training to be much more efficient and can allow this network to recognize or classify correctlyon situations which were never seen during training. However, it shall be understood that this disclosure is not limited to image processing. FIG.9 shows a diagram of updating equivariant layers of a symmetry-aware neural network with an updated equivariant basis during training.At time ^ = ^0 during a training phase of the symmetry-aware neural network, an initial linear space (corresponding to the firstlinear space in FIG. 8) of symmetry-equivariant maps (and its equivariant basis) for an initial list of symmetries ^I is computed.Then, one or more equivariant layers is built using the computed equivariant basis (corresponding to the first equivariant basisin FIG. 8). Further, the neural network is trained until convergence or a new symmetry is given / discovered.At time ^ = ^n, a new symmetry (or a list of new symmetries) ^n is discovered / obtained. Instead of re-computing an equivariantbasis for the complete symmetries from scratch, a new linear space (corresponding to the second linear space in FIG. 8) ofsymmetry-equivariant maps (and its equivariant basis) for the new symmetry (or the list of new symmetries) ^n is computed.Then, an intersection of the new linear space at ^ = ^n with the initial linear space ^ = ^0 is computed, and an updated equivariantbasis (corresponding to the second equivariant basis in FIG.8) of the computed intersection is determined. Then, one or more equivariant layers are updated with respect to the updated equivariant basis but without the need to change the architecture (dependent only on representation dimension and arbitrary multiplicities). The neural network is trained untilconvergence, or a further new symmetry is discovered (and the solution of ^ = ^n is iterated). Optionally, the training of theneural network may be resumed, or restarted from a vanilla neural network of which parameters are re-initialized.The solution of this disclosure can adapt to slowly discovered symmetries. It is suitable for evolving environment. There is no need to redo everything from scratch each time a new symmetry is found. The equivariant basis obtained previously can beupdated. When symmetries are unknown and it is not sure if the already discovered symmetries are enough, then the incrementalimplementation of equivariance can guide the task of symmetry discovery. This disclosure directly transforms the related input data (may be raw or semantic) into the desired equivariant linear space without imposing any conditions or custom structure on the weights matrix, making the network very flexible and adaptable to newly discovered symmetries. This disclosure can adapt quickly to new symmetries without restarting from scratch for building the equivariant network. Since running time for computing the updated group-equivariant basis is much quicker. This disclosure also gains advantage of incremental training and knowledge transfer. The learning and convergence phase can be distributed over time and does not suffer from a bottleneck. The network can have better generalization capabilities even with a small amount (e.g., 1%) of a training dataset. This disclosure may also contribute to the symmetry discovery task in a feedback-like manner. This disclosure can be applied to any neural network system architecture where the main goal is to let the system (more specifically the neural network) to be equivariant with respect to an evolving (with time) list of symmetries in an incremental manner step by step over time. Thus, this disclosure is applicable to any system or scenario which involves neural networks where a list of symmetries in the given data representation could be available over different instances of time. This disclosurecan also be applied in an example scenario where a final list of symmetries is already given / known initially at time 0. However,for training a neural network, the final list is divided into sub-parts and making the symmetries artificially evolve over time.This may be beneficial to the generalization objective. FIG.10 shows an example of a neural network architecture. This disclosure can be used to obtain an equivariant neural networkwith respect to an evolving list of discovered symmetries. As depicted in FIG. 10, an exemplary network model may comprisefour blocks: a one-layer pair-wise relational network followed by a two-layers two-head self-attention transformer, then two symmetry-aware (or equivariant) linear layers with their non-linear activation function, and finally two fully connected layers (FCL) separated by one non-linear activation function, with ELU being the non-linear activation function for all the concerned layers.The goal is to make the whole network, until reaching the FCLs, equivariant with respect to an evolving list of symmetries.The FCLs are used to insure invariance after the equivariant network for the sake of classification tasks. In FIG. 10, ELL refersto Equivariant Linear Layer, and Att. refers to “Attention”. The neural network architecture shown in FIG.10 may be trained with the SET game dataset shown in FIG.7A. Each card of the SET game data set has four features with three possible values for each feature. The four features and their correspondingvalues are: Color (Violet, Green, Red), Number (One, Two, Three), Shape (Diamond, Oval, Squiggle), and Shading (Solid,Striped, Open). Normally in the SET game, the goal is to identify triplets of cards that form a SET (the specific rules are not of interest in this disclosure). Here it is focused on the different possible types given by any tuple of three different cards. In total, there are 85320 possible combinations (with each combination being a different representative) of three cards tuples (with each card being an object).These representatives could be grouped or classified into 20 possible category types based on their structure. As an additionalinput, a list of symmetries is given relating the given dataset. Recall that whenever any data symmetry is applied on a specificrepresentative in a given type, then its transformation (including identity) output should stay within the same type.FIG. 11A shows a training simulation result according to this disclosure where symmetries are given or discovered indifferent time instances.The simulation results are based on that each of the different 81 cards (called stimuli) is represented in ℝ12 (the dimensionof the data representation as faces), and the dataset is built from random tuples of three cards represented in ℝ3∗12 (inputmultiplicity n=3 at the first layer). FIG. 11B shows a training simulation result of a non-equivariant neural network. FIG. 11C shows a training simulation result of a symmetry-aware neural network using a list of fully discovered symmetries all at once.When applying the solution according to this disclosure, only a small amount (e.g., 1%) of all the possible representatives(85320 possible combinations of tuples of three cards) are selected randomly for the training phase. The symmetry-awarenetwork is able to generalize with a relative good accuracy to the other 99% even before discovering the final list ofsymmetries (being four generators of the symmetry group in this example). The simulation results also highlight the advantage of the incremental equivariance according to this disclosure compared to the case when full equivariance is applied once the full list of symmetries is discovered: the convergence with accurate generalization is much faster (there is no need to wait till the full list of symmetries is available). According to this disclosure,the final generalization accuracy is better thanks to learning by parts in a step-by-step manner. The simulation results shown inFIG. 11A-11C are based on the following learning parameters: number of epochs = 200, batch size = 300, number of batchesper epoch = 64, learning rate = 0.0002, and a learnable Bias is enabled.The method of this disclosure can be performed by any device that is suitable for performing and / or accelerating AI computing.For instance, the device may be (but not limited to): a CPU, a GPU, a NPU, an AI accelerator, or a chipset thereof, or anycomputing module comprising the CPU and / or GPU and / or NPU and / or the AI accelerator.The present disclosure has been described in conjunction with various implementations as examples. However, other variationscan be understood and effected by those persons skilled in the art and practicing the claimed matter, from the studies of thedrawings, this disclosure and the independent claims. In the claims as well as in the description, the word “comprising” doesnot exclude other elements or steps and the indefinite article “a” or “an” does not exclude a plurality. A single element or other unit may fulfill the functions of several entities or items recited in the claims. The mere fact that certain measures are recited in the mutual different dependent claims does not indicate that a combination of these measures cannot be used in an advantageous implementation. A computer program may be stored / distributed on a suitable medium, such as an optical storage medium or a solid-state medium supplied together with or as part of other hardware, but may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems.

Claims

CLAIMS1. A computer-implemented method for constructing a symmetry-aware neural network, wherein the method comprises:receiving (81) input data and a list of symmetries associated with the input data; computing (82) a first linear space of symmetry-equivariant maps and determining (83) a first equivariant basis for the list of symmetries, wherein the first equivariant basis is a basis of the first linear space of symmetry-equivariant maps; constructing (84) at least one equivariant layer of the neural network using the first equivariant basis; and in response to receiving one or more new symmetries, further comprising: computing (85) a second linear space of symmetry-equivariant maps for the one or more new symmetries; computing (86) an intersection of the second linear space with the first linear space; determining (87) a second equivariant basis as a basis of the computed intersection; and updating (88) the at least one equivariant layer using the second equivariant basis.

2. The method according to claim 1, wherein the update (88) of the at least one equivariant layer is performed without aneed for changing an architecture of the neural network.

3. The method according to claim 2, wherein for updating (88) of the at least one equivariant layer, an input dimensionand output dimension of the at least one equivariant layer are kept unchanged.

4. The method according to any one of claims 1 to 3, further comprising training the neural network with the at least oneupdated equivariant layer.

5. The method according to claim 4, wherein for training the neural network with the at least one updated equivariantlayer, the method comprises: keeping weights obtained before receiving the one or more new symmetries unchanged; and resuming training the neural network with the at least one updated equivariant layer until convergence or a further new symmetry is received.

6. The method according to claim 4, wherein for training the neural network with the at least one updated equivariantlayer, the method comprises: initializing the neural network with the at least one updated equivariant layer with random weights, to obtain a vanillaneural network; and training the vanilla neural network until convergence or a further new symmetry is received.

7. An apparatus for constructing a symmetry-aware neural network, wherein the apparatus is configured to:receive an input data and a list of symmetries associated with the input data; compute a first linear space of symmetry-equivariant maps and determining a first equivariant basis for the list of symmetries, wherein the first equivariant basis is a basis of the first linear space of symmetry-equivariant maps; and construct at least one equivariant layer of the neural network using the first equivariant basis; wherein in response to receiving one or more new symmetries, the apparatus is configured to: compute a second linear space of symmetry-equivariant maps for the one or more new symmetries; compute an intersection of the second linear space with the first linear space; determine a second equivariant basis as a basis of the computed intersection; and update the at least one equivariant layer using the second equivariant basis.

8. The apparatus according to claim 7, wherein the update of the at least one equivariant layer is performed by theapparatus without a need for changing an architecture of the neural network.

9. The apparatus according to claim 8, wherein for updating of the at least one equivariant layer, the apparatus isconfigured to keep an input dimension and output dimension of the at least one equivariant layer unchanged.

10. The apparatus according to any one of claims 6 to 9, wherein the apparatus is further configured to train the neuralnetwork with the at least one updated equivariant layer.

11. The apparatus according to claim 10, wherein for training the neural network with the at least one updated equivariantlayer, the apparatus is configured to: keep weights obtained before receiving the one or more new symmetries unchanged; and resume training the neural network with the at least one updated equivariant layer until convergence or a further new symmetry is received.

12. The apparatus according to claim 10, wherein for training the neural network with the at least one updated equivariantlayer, the apparatus is configured to: initialize the neural network with the at least one updated equivariant layer with random weights, to obtain a vanilla neural network; and train the vanilla neural network until convergence or a further new symmetry is received.

13. A computer program comprising instructions which, when the program is executed by a computer, cause the computerto carry out the method according to any one of claims 1 to 6.

14. A computer-implemented method (10) for constructing a symmetry-aware neural network, wherein the neural networkcomprises a plurality of layers, and the method comprises: obtaining (11) input data ^ = {^^ , ^^ , … , ^^} ∈ ℝ^∗^ and a list of symmetries ^(^) ≔ {^^, ^^ , … , ^^} associated withthe input data of a layer, wherein n denotes a multiplicity of the input data, d denotes a representation dimension, andl denotes a quantity of the symmetries;computing (12) an equivariant basis ^^(^) = {^^, ^^ , … , ^^} ∈ ℝ^∗^∗^that is with respect to the list of symmetries, wherein the equivariant basis ^^(^) is a basis of a linear space ℰ(^) of equivariant maps, and b denotes a quantity ofthe equivariant maps of the equivariant basis; multiplying (13) the whole input data ^ by each of the equivariant basis elements in the equivariant basis ^^(^), toobtain a list of transformed input data ^ = {^^ , ^^, … , ^^} ∈ ℝ^∗^∗^; constructing (14) weight matrices of the layer and determining a multiplicity m of an output of the layer based on inputand output dimensions of the layer; optimizing (15) the weight matrices to obtain learned weight matrices ^ = {^^, ^^ , … , ^^} ∈multiplying (16) each learned weight matrix ^^ ∈ ^ by a corresponding transformed input data ^^ in the list,respectively, to obtain a plurality of output matrices ^ = {^^ , ^^ , … , ^^} ∈computing (17) a sum over all the b matrices of the output matrices O to obtain a two-dimensional array of m vectorseach of dimension d; and applying (18) a point-wise activation function to a flattened array of the m vectors, to obtain an output of the layer.

15. The method (10) according to claim 14, wherein the step of computing (12) the equivariant basis ^^(^) that is withrespect to the list of symmetries comprises: computing (121) a linear space of equivariant maps ℇ(^^) with respect to each symmetry ^^; andcomputing (122) an intersection of the computed linear spaces (ℇ(^) ≔ℇ(^^) ), to obtain a list of linear mapsas the equivariant basis ^^(^) that is the basis of the intersection ℰ(^).

16. The method (10) according to claim 15, wherein a cyclic equivariance is used for computing the linear space ℇ(^^) ofequivariant maps with respect to each symmetry ^^.

17. The method (10) according to any one of claims 14 to 16, wherein the weight matrices are optimized through astochastic gradient descent or Adam algorithm.

18. The method (10) according to any one of claims 14 to 17, wherein one or more invariant biases that are invariant withrespect to the list of symmetries are added to obtain the array of vectors.

19. The method (10) according to claim 18, further comprising learning the one or more invariant biases.

20. The method (10) according to any one of claims 14 to 19, wherein the activation function is a non-linear activationfunction.

21. The method (10) according to any one of claims 14 to 20, wherein the activation function is a rectified linear unit,ReLU, or Tanh activation function.

22. The method (10) according to any one of claims 14 to 21, wherein the list of transformed input data comprises btransformed input data of dimension n by d, the weight matrices comprise b weight matrices each of dimension m by n, theplurality of output matrices comprises b output matrices each of dimension m by d, wherein n, d, b, m, and n are positiveintegers.

23. An apparatus comprising a symmetry-aware neural network, wherein the neural network comprises a plurality of layers,and the apparatus is configured to: obtain (11) input data ^ = {^^ , ^^ ,∈ ℝ^∗^ and a list of symmetries ^(^) ≔ {^^ , ^^ , … , ^^} associated with theinput data of a layer, wherein n denotes a multiplicity of the input data, d denotes a representation dimension, and ldenotes a quantity of the symmetries; compute (12) an equivariant basis ^^(^) = {^^ , ^^ , … , ^^} ∈ ℝ^∗^∗^that is with respect to the list of symmetries, wherein the equivariant basis ^^(^) is a basis of a linear space ℰ(^) of equivariant maps, and b denotes a quantity ofthe equivariant maps of the equivariant basis; multiply (13) the whole input data ^ by each of the equivariant basis elements in the equivariant basis ^^(^), to obtaina list of transformed input data ^ = {^^ , ^^, … , ^^} ∈ ℝ^∗^∗^; construct (14) weight matrices of the layer and determine a multiplicity m of an output of the layer based on input andoutput dimensions of the layer; optimize (15) the weight matrices to obtain learned weight matrices ^ = {^^, ^^ , … , ^^} ∈ ℝ^∗^∗^; multiply (16) each learned weight matrix ^^ ∈ ^ by a corresponding transformed input data ^^ in the list, respectively,to obtain a plurality of output matrices ^ = {^^ , ^^, … , ^^} ∈ ℝ^∗^∗^;compute (17) a sum over all the b matrices of the output matrices O to obtain a two-dimensional array of m vectorseach of dimension d; and apply (18) a point-wise activation function to a flattened array of the m vectors, to obtain an output of the layer.

24. The apparatus according to claim 23, wherein if two or more of the plurality of layers share a same list of symmetries,the apparatus is configured to compute the equivariant basis ^^(^) only once for the two or more layers.

25. The apparatus according to claim 23 or 24, wherein for computing the equivariant basis ^^(^) that is with respect tothe list of symmetries, the apparatus is configured to: compute a linear space of equivariant maps ℇ(^^) with respect to each symmetry ^^; andcompute an intersection of the computed linear spaces (ℇ(^) ≔ℇ(^^) ), to obtain a list of linear maps as theequivariant basis ^^(^) that is the basis of the intersection ℰ(^).

26. The apparatus according to claim 25, wherein a cyclic equivariance is used for computing the linear space ℇ(^^) ofequivariant maps with respect to each symmetry ^^.

27. The apparatus according to any one of claims 23 to 26, wherein one or more invariant biases that are invariant withrespect to the list of symmetries are added to obtain the array of vectors.

28. A computer program comprising instructions which, when the program is executed by a computer, cause the computerto carry out the method according to any one of claims 14 to 22.

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