Recurrent Kernel Networks

06/07/2019 ∙ by Dexiong Chen, et al. ∙ University Claude Bernard Lyon 1 Inria 9

Substring kernels are classical tools for representing biological sequences or text. However, when large amounts of annotated data is available, models that allow end-to-end training such as neural networks are often prefered. Links between recurrent neural networks (RNNs) and substring kernels have recently been drawn, by formally showing that RNNs with specific activation functions were points in a reproducing kernel Hilbert space (RKHS). In this paper, we revisit this link by generalizing convolutional kernel networks---originally related to a relaxation of the mismatch kernel---to model gaps in sequences. It results in a new type of recurrent neural network which can be trained end-to-end with backpropagation, or without supervision by using kernel approximation techniques. We experimentally show that our approach is well suited to biological sequences, where it outperforms existing methods for protein classification tasks.

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1 Introduction

Learning from biological sequences is important for a variety of scientific fields such as evolution [8] or human health [15]

. In order to use classical statistical models, a first step is often to map sequences to vectors of fixed size, while retaining relevant features for the considered learning task. For a long time, such features were extracted from sequence alignment, either against a reference or among each others 

[3]. The resulting features are appropriate for sequences that are similar enough, but they become ill-defined when sequences are not suited to alignment. This includes important cases such as microbial genomes, distant species, or human diseases, and calls for alternative representations [7].

String kernels provide generic representations for biological sequences, most of which do not require global alignment  [33]. In particular, a classical approach maps sequences to a huge-dimensional feature space by enumerating statistics about all occuring subsequences. These subsequences may be simple classical -mers leading to the spectrum kernel [20], -mers up to mismatches [21], or gap-allowing subsequences [23]. Other approaches involve kernels based on a generative model [16, 34], or based on local alignments between sequences [35] inspired by convolution kernels [11, 36].

The goal of kernel design is then to encode prior knowledge in the learning process. For instance, modeling gaps in biological sequences is important since it allows taking into account short insertion and deletion events, a common source of genetic variation. However, even though kernel methods are good at encoding prior knowledge, they provide fixed task-independent representations. When large amounts of data are available, approaches that optimize the data representation for the prediction task are now often prefered. For instance, convolutional neural networks 

[18] are commonly used for DNA sequence modeling [1, 2, 40]

, and have been successful for natural language processing 

[17]

. While convolution filters learned over images are interpreted as image patches, those learned over sequences are viewed as sequence motifs. RNNs such as long short-term memory networks (LSTMs) 

[13] are also commonly used in both biological [14] and natural language processing contexts [5, 25].

Motivated by the regularization mechanisms of kernel methods, which are useful when the amount of data is small and are yet imperfect in neural networks, hybrid approaches have been developed between the kernel and neural networks paradigms [6, 26, 39]. Closely related to our work, the convolutional kernel network (CKN) model originally developed for images [24] was successfully adapted to biological sequences in [4]. CKNs for sequences consist in a continuous relaxation of the mismatch kernel: while the latter represents a sequence by its content in -mers up to a few discrete errors, the former considers a continuous relaxation, leading to an infinite-dimensional sequence representation. Finally, a kernel approximation relying on the Nyström method [37]

projects the mapped sequences to a linear subspace of the RKHS, spanned by a finite number of motifs. When these motifs are learned end-to-end with backpropagation, learning with CKNs can also be thought of as performing feature selection in the—infinite dimensional—RKHS.

In this paper, we generalize CKNs for sequences by allowing gaps in motifs, motivated by genomics applications. The kernel map retains the convolutional structure of CKNs but the kernel approximation that we introduce can be computed using a recurrent network, which we call recurrent kernel network (RKN). This RNN arises from the dynamic programming structure used to compute efficiently the substring kernel of [23], a link already exploited by [19] to derive their sequence neural network. Both our kernels rely on the RNN to build a representation of an input sequence by computing a string kernel between this sequence and a set of learnable filters. Yet, our model exhibits major differences with [19], who use the regular substring kernel of [23] and compose this representation with another non-linear map—by applying an activation function to the output of the RNN. By contrast, we obtain our RKHS by relaxing the substring kernel to allow inexact matching at the compared positions. The resulting feature space can be interpreted as a continuous neighborhood around all substrings (with gaps) of the described sequence. Furthermore, our RNN provides a finite-dimensional approximation of the relaxed kernel, relying on the Nyström approximation method [37]. As a consequence, RKNs may be learned in an unsupervised manner (in such a case, the goal is to approximate the kernel map), and with supervision, which may be interpreted as performing feature selection in the RKHS.

Contributions.

In this paper, we make the following contributions:
  We generalize convolutional kernel networks for sequences [4] to allow gaps, an important option for biological data. As in [4], we observe that the kernel formulation brings practical benefits over traditional CNNs or RNNs [14] when the amount of labeled data is small or moderate.
 

 We provide a kernel point of view on recurrent neural networks, with new unsupervised and supervised learning algorithms. The resulting RKHS can be interpreted in terms of gappy motifs, and end-to-end learning amounts to performing feature selection in this RKHS.


  Based on [27]

, we propose a new way to simulate max pooling in RKHSs, thus solving a classical discrepancy between theory and practice in the literature of string kernels, where sums are often replaced by a maximum operator that does not ensure positive definiteness 

[35].

2 Background on Kernel Methods and String Kernels

Kernel methods consist in mapping data points living in a set to a possibly infinite-dimensional Hilbert space , through a mapping function , before learning a simple predictive model in  [32]. The so-called kernel trick allows to perform learning without explicitly computing this mapping, as long as the inner-product  between two points  can be efficiently computed. Whereas kernel methods traditionally lack scalability since they require computing an Gram matrix, where is the amount of training data, recent approaches based on approximations have managed to make kernel methods work at large scale in many cases [29, 37].

For sequences in , which is the set of sequences of any possible length over an alphabet , the mapping often enumerates subsequence content. For instance, the spectrum kernel maps sequences to a fixed-length vector , where is the set of -mers—length- sequence of characters in for some in , and if contains an exact occurence of and otherwise [20]. The mismatch kernel [21] operates similarly, but if contains an occurence of up to a few letters, which is useful when is large and exact occurences are rare.

2.1 Substring kernels

As [19], we consider the substring kernel introduced in [23], which allows to model the presence of gaps when trying to match a substring to a sequence . Modeling gaps requires introducing the following notation: denotes the set of indices of sequence with elements satisfying , where is the length of . For an index set in , we may now consider the subsequence of indexed by . Then, the substring kernel takes the same form as the mismatch and spectrum kernels, but counts all—consecutive or not—subsequences of equal to , and weights them by the number of gaps. Formally, we consider a parameter in , and , where if and only if , and otherwise, and is the number of gaps in the index set . When is small, gaps are heavily penalized, whereas a value close to  gives similar weights to all occurrences. Ultimately, the resulting kernel between two sequences and is

(1)

As we will see in Section 3, our RKN model relies on (1), but unlike [19], we replace the quantity that matches exact occurrences by a relaxation, allowing more subtle comparisons. Then, we will show that the model can be interpreted as a gap-allowed extension of CKNs for sequences. We also note that even though  seems computationally expensive at first sight, it was shown in [23] that (1) admits a dynamic programming structure leading to efficient computations.

2.2 The Nyström method

When computing the Gram matrix is infeasible, it is typical to use kernel approximations  [29, 37], consisting in finding a -dimensional mapping such that the kernel can be approximated by a Euclidean inner-product . Then, kernel methods can be simulated by a linear model operating on , which does not raise scalability issues if is reasonably small. Among kernel approximations, the Nyström method consists in projecting points of the RKHS onto a -dimensional subspace, allowing to represent points into a -dimensional coordinate system.

Specifically, consider a collection of points in and consider the subspace

where is the Gram matrix of restricted to the samples and in  carries the kernel values . This approximation only requires kernel evaluations and often retains good performance for learning. Interestingly as noted in [24], is exactly the inner-product in between the projections of and onto , which remain in .

When

is a Euclidean space—this can be the case for sequences when using a one-hot encoding representation, as discussed later— a good set of anchor points

can be obtained by simply clustering the data and choosing the centroids as anchor points [38]. The goal is then to obtain a subspace that spans data as best as possible. Otherwise, previous work on kernel networks [4, 24] have also developed procedures to learn the set of anchor points end-to-end by optimizing over the learning objective. This approach can then be seen as performing feature selection in the RKHS.

3 Recurrent Kernel Networks

With the previous tools in hand, we now introduce RKNs. We show that it admits variants of CKNs, substring and local alignment kernels as special cases, and we discuss its relation with RNNs.

3.1 A continuous relaxation of the substring kernel allowing mismatches

From now on, and with an abuse of notation, we represent characters in as vectors in . For instance, when using one-hot encoding, a DNA sequence of length can be seen as a 4-dimensional sequence where each in has a unique non-zero entry indicating which of is present at the -th position, and we denote by the set of such sequences. We now define the single-layer RKN as a generalized substring kernel (1) in which the indicator function is replaced by a kernel for -mers (which is a first difference with [19]):

(2)

where we assume that the vectors representing characters have unit -norm, such that is a dot-product kernel, and if we follow (1).

For and using the convention , all the terms in these sums are zero except those with no gap, and we recover the kernel of the CKN model of [4] with a convolutional structure—up to the normalization, which is done -mer-wise in CKN instead of position-wise.

Compared to (1), the relaxed version (2) accommodates inexact -mer matching. This is important for protein sequences, where it is common to consider different similarities between amino acids in terms of substitution frequency along evolution [12]. This is also reflected in the underlying sequence representation in the RKHS illustrated in Figure 1: by considering the kernel mapping and RKHS  such that , we have

(3)

A natural feature map for a sequence is therefore : using the RKN amounts to representing  by a mixture of continuous neighborhoods centered on all its -subsequences , each weighted by the corresponding (e.g., ).

Figure 1: Representation of sequences in a RKHS based on  with and .

3.2 Extension to all -mers and relation to the local alignment kernel

Dependency in the hyperparameter

can be removed by summing over all possible values:

Interestingly, we note that admits the local alignment kernel of [35]

as a special case. More precisely, local alignments are defined via the tensor product set

, which contains all possible alignments of positions between a pair of sequences . The local alignment score of each such alignment in is defined, by [35], as , where is a symmetric substitution function and is a gap penalty function. The local alignment kernel in [35] can then be expressed in terms of the above local alignment scores (Thrm. 1.7 in [35]):

(4)

When the gap penalty function is linear—that is, with , becomes . When can be written as an inner-product between normalized vectors, we see that becomes a special case of (2)—up to a constant factor—with , .

This observation sheds new lights on the relation between the substring and local alignment kernels, which will inspire new algorithms in the sequel. To the best of our knowledge, the link we will provide between RNNs and local alignment kernels is also new.

3.3 Nyström approximation and recurrent neural networks

As in CKNs, we now use the Nyström approximation method as a building block to make the above kernels tractable. According to (3), we may first use the Nyström method described in Section 2.2 to find an approximate embedding for the quantities , where is one of the -mers represented as a matrix in . This is achieved by choosing a set of anchor points in , and by encoding as —where is the kernel of . Such an approximation for -mers yields the -dimensional embedding for the sequence :

(5)

Then, an approximate feature map  for the kernel can be obtained by concatenating the embeddings for large enough.

The anchor points as motifs.

The continuous relaxation of the substring kernel presented in (2) allows us to learn anchor points that can be interpreted as sequence motifs, where each position can encode a mixture of letters. This can lead to more relevant representations than -mers for learning on biological sequences. For example, the fact that a DNA sequence is bound by a particular transcription factor can be associated with the presence of a T followed by either a G or an A, followed by another T, would require two -mers but a single motif [4]. Our kernel is able to perform such a comparison.

Efficient computations of and approximation via RNNs.

A naive computation of would require enumerating all substrings present in the sequence, which may be exponentially large when allowing gaps. For this reason, we use the classical dynamic programming approach of substring kernels [23]. Consider then the computation of defined in (5) for as well as a set of anchor points with the ’s in . We also denote by  the set obtained when keeping only -th first positions (columns) of the ’s, leading to , which will serve as anchor points for the kernel to compute . Finally, we denote by in  the -th column of such that . Then, the embeddings  can be computed recursively by using the following theorem:

Theorem 1.

For any and ,

(6)

where and form a sequence of vectors in indexed by such that , and is a vector that contains only ones, while the sequence obeys the recursion

(7)

where is the elementwise multiplication operator and is a vector in whose entry  in is and is the -th character of .

A proof is provided in Appendix A and is based on classical recursions for computing the substring kernel, which were interpreted as RNNs by [19]. The main difference in the RNN structure we obtain is that their non-linearity is applied over the outcome of the network, leading to a feature map formed by composing the feature map of the substring kernel of [23] and another one from a RKHS that contains their non-linearity. By contrast, our non-linearities are built explicitly in the substring kernel, by relaxing the indicator function used to compare characters. The resulting feature map is a continuous neighborhood around all substrings of the described sequence. In addition, the Nyström method yields an orthogonalization factor to the output of the network to compute our approximation, which is perhaps the only non-standard component of our RNN. This factor provides an interpretation of as a kernel approximation. As discussed next, it makes it possible to learn the anchor points by -means, see [4], which also makes the initialization of the supervised learning procedure simple without having to deal with the scaling of the initial motifs/filters .

Learning the anchor points .

We now turn to the application of RKNs to supervised learning. Given sequences in and their associated labels in , e.g., for binary classification or for regression, our objective is to learn a function in the RKHS of by minimizing

where

is a convex loss function that measures the fitness of a prediction

to the true label and controls the smoothness of the predictive function. After injecting our kernel approximation , the problem becomes

(8)

Following [4, 24], we can learn the anchor points without exploiting training labels, by applying a -means algorithm to all (or a subset of) the -mers extracted from the database and using the obtained centroids as anchor points. Importantly, once has been obtained, the linear function parametrized by is still optimized with respect to the supervised objective (8). This procedure can be thought of as learning a general representation of the sequences disregarding the supervised task, which can lead to a relevant description while limiting overfitting.

Another strategy consists in optimizing (8) jointly over , after observing that is a smooth function of . Learning can be achieved by using backpropagation over , or by using an alternating minimization strategy between and . It leads to an end-to-end scheme where both the representation and the function defined over this representation are learned with respect to the supervised objective (8). Backpropagation rules for most operations are classical, except for the matrix inverse square root function, which is detailed in Appendix B

. Initialization is also parameter-free since the unsupervised learning approach may be used for that.

3.4 Extensions

Multilayer construction.

In order to account for long-range dependencies, it is possible to construct a multilayer model based on kernel compositions similar to [19]. Assume that is the -th layer kernel and its mapping function. The corresponding -th layer kernel is defined as

(9)

where will be defined in the sequel and the choice of weights slightly differs from the single-layer model. We choose indeed only for the last layer of the kernel, which depends on the number of gaps in the index set but not on the index positions. Since (9) involves a kernel operating on the representation of prefix sequences from layer , the representation makes sense only if carries mostly local information close to position . Otherwise, information from the beginning of the sequence would be overrepresented. Ideally, we would like the range-dependency of (the size of the window of indices before that influences the representation, akin to receptive fields in CNNs) to grow with the number of layers in a controllable manner. This can be achieved by choosing for , which assigns exponentially more weights to the -mers close to the end of the sequence.

For the first layer, we recover the single-layer network defined in (2) by defining and . For , it remains to define to be a homogeneous dot-product kernel, as used for instance in CKNs [24]:

(10)

Note that the Gaussian kernel used for 1st layer may also be written as (10) since characters are normalized. As for CKNS, the goal of homogenization is to prevent norms to grow/vanish exponentially fast with , while dot-product kernels lend themselves well to neural network interpretations.

As detailed in Appendix C, extending the Nyström approximation scheme for the multilayer construction may be achieved in the same manner as with CKNs—that is, we learn one approximate embedding at each layer, allowing to replace the inner-products by their approximations , and it is easy to show that the interpretation in terms of RNNs is still valid since has the same sum structure as (2).

Max pooling in RKHS.

Alignment scores (e.g. Smith-Waterman) in molecular biology rely on a max operation—over the scores of all possible alignments—to compute similarities between sequences. However, using max in a string kernel usually breaks positive definiteness, even though it seems to perform well in practice. To solve such an issue, sum-exponential is used as a proxy in [31], but it leads to unexpected diagonal dominance issue and makes SVM solvers unable to learn. For RKN, the sum in (3) can also be replaced by a max

(11)

which empirically seems to perform well, but breaks the kernel interpretation, as in [31]. The corresponding recursion amounts to replacing all the sum in (7) by a max.

An alternative way to aggregate local features is the generalized max pooling (GMP) introduced in [27], which can be adapted to the context of RKHSs. Assuming that before pooling is embedded to a set of local features , GMP builds a representation whose inner-product with all the local features is one: . coincides with the regular max when each is an element of the canonical basis of a finite representation—i.e., assuming that at each position, a single feature has value 1 and all others are .

Since GMP is defined by a set of inner-products constraints, it can be applied to our approximate kernel embeddings by solving a linear system. This is compatible with CKN but becomes intractable for RKN which pools across

positions. Instead, we heuristically apply GMP over the set

for all  with , which can be obtained from the RNN described in Theorem 1. This amounts to composing GMP with mean poolings obtained over each prefix of . We observe that it performs well in our experiments. More details are provided in Appendix D.

4 Experiments

We evaluate RKN and compare it to typical string kernels and RNN for protein fold recognition. Pytorch code is provided with the submission and additional details given in Appendix 

E.

4.1 Datasets and implementation details

Sequencing technologies provide access to gene and, indirectly, protein sequences for yet poorly studied species. In order to predict the 3D structure and function from the linear sequence of these proteins, it is common to search for evolutionary related ones, a problem known as homology detection. When no evolutionary related protein with known structure is available, a—more difficult—alternative is to resort to protein fold recognition. We evaluate our RKN on such a task, where the objective is to predict which proteins share a 3D structure with the query [30].

Here we consider the Structural Classification Of Proteins (SCOP) version 1.67 [28]. We follow the preprocessing procedures of [10] and remove the sequences that are more than 95% similar, yielding 85 fold recognition tasks. Each positive training set is then extended with Uniref50 to make the dataset more balanced, as proposed in [14]. The resulting dataset can be downloaded from http://www.bioinf.jku.at/software/LSTM_protein. The number of training samples for each task is typically around 9,000 proteins, whose length varies from tens to thousands of amino-acids. In all our experiments we use logistic loss. We measure classification performances using auROC and auROC50 scores (area under the ROC curve and up to 50% false positives).

For CKN and RKN, we evaluate both one-hot encoding of amino-acids by 20-dimensional binary vectors and an alternative representation relying on the BLOSUM62 substitution matrix [12]

. Specifically in the latter case, we represent each amino-acid by the centered and normalized vector of its corresponding substitution probabilities with other amino-acids. The local alignment kernel (

4), which we include in our comparison, natively uses BLOSUM62.

Hyperparameters.

We follow the training procedure of CKN presented in [4]. Specifically, for each of the tasks, we hold out one quarter of the training samples as a validation set, use it to tune , gap penalty and the regularization parameter

in the prediction layer. These parameters are then fixed across datasets. RKN training also relies on the alternating strategy used for CKN: we use an Adam algorithm to update anchor points, and the L-BFGS algorithm to optimize the prediction layer. We train 100 epochs for each dataset: the initial learning rate for Adam is fixed to 0.05 and is halved as long as there is no decrease of the validation loss for 5 successive epochs. We fix

to 10, the number of anchor points to 128 and use single layer CKN and RKN throughout the experiments.

Implementation details for unsupervised models.

The anchor points for CKN and RKN are learned by k-means on 30,000 extracted

-mers from each dataset. The resulting sequence representations are standardized by removing mean and dividing by standard deviation and are used within a logistic regression classifier.

in Gaussian kernel and the parameter are chosen based on validation loss and are fixed across the datasets. for regularization is chosen by a 5-fold cross validation on each dataset. As before, we fix to 10 and the number of anchor points to 1024. Note that the performance could be improved with larger as observed in [4], at a higher computational cost.

Comparisons and results.

The results are shown in Table 1. The blosum62 version of CKN and RKN outperform all other methods. Improvement against the mismatch and LA kernels is likely caused by end-to-end trained kernel networks learning a task-specific representation in the form of a sparse set of motifs, whereas data-independent kernels lead to learning a dense function over the set of descriptors. This difference can have a regularizing effect akin to the -norm in the parametric world, by reducing the dimension of the learned linear function while retaining relevant features for the prediction task. GPkernel also learns motifs, but relies on the exact presence of discrete motifs. Finally, both LSTM and [19] are based on RNNs but are outperformed by kernel networks. The latter was designed and optimized for NLP tasks and yields a auROC50 on this task.

RKNs outperform CKNs, albeit not by a large margin. Interestingly, as the two kernels only differ by their allowing gaps when comparing sequences, this results suggests that this aspect is not the most important for identifying common foldings. In particular, the advantage of the LA-kernel against its mismatch counterpart is more likely caused by other differences than gap modelling, namely using a max rather than a mean pooling of -mer similarities across the sequence, and a general substitution matrix rather than a Dirac function to quantify mismatches. Consistently, within kernel networks GMP systematically outperforms mean pooling, while being slightly behind max pooling.

Additional details and results, scatter plots, and pairwise tests between methods to assess the statistical significance of our conclusions are provided in Appendix E.

Method pooling one-hot BLOSUM62
auROC auROC50 auROC auROC50
GPkernel [10] 0.844 0.514
SVM-pairwise [22] 0.724 0.359
Mismatch [21] 0.814 0.467
LA-kernel [31] 0.834 0.504
LSTM [14] 0.830 0.566
CKN-seq [4] mean 0.827 0.536 0.843 0.563
CKN-seq [4] max 0.837 0.572 0.866 0.621
CKN-seq GMP 0.838 0.561 0.856 0.608
CKN-seq (unsup)[4] mean 0.804 0.493 0.827 0.548
RKN () mean 0.829 0.542 0.838 0.563
RKN mean 0.829 0.541 0.840 0.571
RKN () max 0.840 0.575 0.862 0.618
RKN max 0.844 0.587 0.871 0.629
RKN () GMP 0.840 0.563 0.855 0.598
RKN (unsup) mean 0.805 0.504 0.833 0.570
Table 1: Average auROC and auROC50 for SCOP fold recognition benchmark. LA-kernel uses BLOSUM62 to compare amino acids which is a little different from our encoding approach.

Acknowledgements

This work has been supported by the grants from ANR (FAST-BIG project ANR-17-CE23-0011-01) and by the ERC grant number 714381 (SOLARIS).

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Appendix A Nyström Approximation for Single-Layer RKN

We detail here the Nytröm approximation presented in Section 3.3, which we recall here for a sequence :

(12)

Consider then the computation of defined in (12) for given a set of anchor points with the ’s in . Given the notations introduced in Section 3.3, we are now in shape to prove Theorem 1.

Proof.

The proof is based on Theorem 1 of [19] and definition 2 of [23]. For , let us denote by the first entries of . We first notice that for the Gaussian kernel , we have the following factorization relation for

Thus

with defined as in the theorem.

Let us denote by if and by if . We want to prove that and . First, it is clear that for any . We show by induction on that . When , we have

and have the same recursion and initial state thus are identical. When and suppose that , then we have

and have the same recursion and initial state. We have thus proved that . Let us move on for proving by showing that they have the same initial state and recursion. It is straightforward that , then for we have

Therefore . ∎

Appendix B Back-propagation for Matrix Inverse Square Root

In Section 3.3, we have described an end-to-end scheme to jointly optimize and . The back-propagation of requires computing that of the matrix inverse square root operation as it is involved in the approximate feature map of as shown in (12). The back-propagation formula is given by the following proposition, which is based on an errata of [24] and we include it here for completeness.

Proposition 1.

Given a symmetric positive definite matrix in and the eigencomposition of is written as where is orthogonal and

is diagonal with eigenvalues

. Then

(13)
Proof.

First, let us differentiate with respect to the inverse matrix :

Then, by applying the same (classical) trick,

By multiplying the last relation by on the left and by on the right.

Note that is diagonal. By introducing the matrix such that , it is then easy to show that

where is the Hadamard product between matrices. Then, we are left with

When doing back-propagation, one is usually interested in computing a quantity such that given (with appropriate dimensions), we have

see [9], for instance. Here, denotes the Frobenius inner product. Then, it is easy to show that

Appendix C Multilayer Construction of RKN

For multilayer RKN, assume that we have defined the -th layer kernel. To simplify the notation below, we consider that an input sequence is encoded at layer  as where the feature map at position is . The -layer kernel is defined by induction by

(14)

where is defined in (10. With the choice of weights described in Section 3.4, the construction scheme for an -layer RKN is illustrated in Figure 2

Figure 2: Multilayer construction of RKN: an example with .

The Nyström approximation scheme for multilayer RKN is straightforward by inductively applying the Nytröm method to the kernels from bottom to top layers. Specifically, assume that is approximated by such that the approximate feature map of at position is . Now Consider a set of anchor points with the ’s in which have unit norm at each column. We use the same notations as in single-layer construction. Then very similar to the single-layer RKN, the embeddings are given by the following recursion

Theorem 2.

For any and ,

where