International Journal of Computational Intelligence Systems

In Press, Uncorrected Proof, Available Online: 16 September 2020

Deep Learning and Higher Degree F-Transforms: Interpretable Kernels Before and After Learning

Vojtech Molek*, ORCID, Irina PerfilievaORCID
Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Ostrava, 701 03, Czech Republic
*Corresponding author. Email:
Corresponding Author
Vojtech Molek
Received 6 January 2020, Accepted 2 September 2020, Available Online 16 September 2020.
DOI to use a DOI?
F-transform, Convolutional neural network, Deep learning, Interpretability

One of the current trends in the deep neural network technology consists in allowing a man–machine interaction and providing an explanation of network design and learning principles. In this direction, an experience with fuzzy systems is of great support. We propose our insight that is based on the particular theory of fuzzy (F)-transforms. Besides a theoretical explanation, we develop a new architecture of a deep neural network where the F-transform convolution kernels are used in the first two layers. Based on a series of experiments, we demonstrate the suitability of the F-transform-based deep neural network in the domain of image processing with the focus on recognition. Moreover, we support our insight by revealing the similarity between the F-transform and first-layer kernels in the most used deep neural networks.

© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
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Deep neural networks (DNNs) significantly improve classification algorithms in various applications, giving a new impact to computer vision, speech recognition, etc. In the proposed contribution, we consider convolution neural networks (CNNs) and the deep learning (DL) methodology for improving parameters of their basic operations.

We are focused on a smart and conscious initialization of convolutional kernels in the first and second CNN layers where neurons have restricted receptive fields. Our motivation stems from the observation that although CNN is able to accurately generate the classification label, it does not report on features that cause this classification. Without an understanding of how DL comes to a solution, there is no guarantee that the trained networks will move from a laboratory to real systems [1]. The reason is that the inputs can significantly change during exploitation, and there is no guarantee that the machine learning tools will work effectively with these changes.

To fill this gap, we propose an insight into why and how CNN makes a decision, or why a specified object has been given a specific classification label. Our approach can be called “preprocessing of methodology” in the sense that we propose a CNN initialization, which ensures that features with known meaning are extracted. Then, we allow the network to learn the initialization parameters so that they can match the available data better.

Our approach differs from many similar ones, based on fuzzy rules, in which the explanation of the CNN decision is based on a posteriori analysis, i.e., after the features are extracted, see [1] and references therein.

We observe that smart preprocessing becomes more and more important in the DL methodology due to the increasing complexity of datasets and objects therein. It heavily depends on a network assignment and consists of traditional de-noising, regularization, reduction of dimensionality, labeling, etc. In most cases, preprocessing is realized in the first convolution layers, together with feature extraction. In subsequent fully connected layers, the extracted features are used for classification, recognition, etc. Therefore, the initial objects are modeled by the extracted features, so that the former ones can be approximately reconstructed from the latter.

Additionally, we observe that similarly to the above, we can characterize the technique of the higher degree fuzzy (F-) transforms [24]. This observation leads us to the idea that the higher degree F-transform kernels can be used in the first convolution layers of CNNs that perform recognition or classification.

We are based on long-term work on various approximation models in the theory of fuzzy systems, and in particular, on those, based on the theory of higher degree F-transforms [24]. We have collected rich experience regarding creating F-transform-based models in various applications, including image [57]/time series [8,9] processing and DL architecture of NNs [1012].

The principal difference between the DL and the higher degree F-transform is in the criterion of optimality, which is a quality of approximation (F-transform), or a loss function (DL methodology). To reach optimality, it is recommended to refine fuzzy partition and increase the degree of the F-transform [4], or to increase the number of kernels in convolutional layers and increase the number of layers. Both recommendations have the same nature.

The mentioned similarity between the higher degree F-transform and the DL, motivated us to confirm these theoretical considerations by experiments. The latter were conducted in two opposite directions: at first, we trained a neural network with F-transform kernels and estimated its success, and at second, we analyzed kernels of already trained known neural networks and compared them with the F-transform ones. We discussed the obtained results in several conference papers [1012], where we fully confirmed the hypothesis we made. In the proposed manuscript, we summarize all the results and give extended explanations to the theoretical backgrounds and experimental tests.

The paper is organized as follows: in Section 2, we briefly explain the information-theoretical principles of DNN and the impact of our contribution; in Section 3, we recall essential facts about the higher degree F-transform with the focus on the F 2-transform; in Section 4, we give details of our new neural network (FTNet) design. In Section 4.2, the performance of the proposed FTNet is discussed from various angles and compared with the baseline network. In Section 5.1, we discussed the problem of the kernel interpretability, where we examined the first convolutional layers of several well-known networks.


In this section, we explain the essence of our proposal from a theoretical point of view. We will start with a brief and focused description of DNN, and then explain how we contribute to the current state.

We refer to [13], where the very general characterization of a DNN as a particular computing machine is given: a DNN is a parametric model that performs sequential operations on inputs. Each such operation consists of a linear transformation (e.g., convolution in CNN types), followed by a nonlinear “activation.” An essential factor for DNN success is the availability of large data sets, such as ImageNet and hardware, with a graphics processor, solve to the problem of multidimensional optimization.

In our understanding and approach, we distinguish three key elements in the design of DNN and its corresponding functioning strategy: architecture, the ability to create a good representation of input data, and optimization algorithms. Leaving architecture and optimization aside, we will give a brief description of the second key.

Roughly speaking [13], representation of the input data is any function of it that is useful for the task. If we focus on the most useful (the “best”) representation, then we think of some quantitation, e.g., in terms of complexity or invariance. The relevant line of research is known as representative learning. Despite great interest in this, a comprehensive theory that explains how deep networks with DL methodology contribute to this still does not exist.

However, one thing is clear—the crucial role of the dataset, which is used for training. There is a close connection between the DNN architecture (the number of levels, frames, activations, etc.) and the dataset, which is used to train network parameters. An interesting phenomenon has been reported in [14], where the almost linear relationship was revealed between the sizes of the DNN computational model and the required amount of training data. Obviously, large and multi-object databases require more levels and more complex learning and optimization process.

In a CNN, representation of the input data is realized in the form of a collection of features; the latter are results of convolutions. The collection of features should be complete in the sense of a possible reconstruction (backward representation) of any input object.

Mathematically, the backward “lossy” representation of an object is its approximation. A neural network’s ability to produce approximate representations of initial data objects was reported in many papers. However, as shown by earlier work, even neural networks with one hidden layer and sigmoidal activations are universal approximators of functions, see, e.g., [15]. Therefore, the question of why DNNs are advantageous in this regard is still open [13].

One possible explanation is that deeper architectures are better than their shallower counterparts because they are capable of covering not only the requirement of a suitable approximation but also invariance with respect to some rigid transformations. As an example, scattering networks [16] are a class of deep networks whose convolution filter banks are defined by multiple-resolution wavelet families and whose stability and local invariance are confirmed.

This fact supports our initiative in a creation of a “convolutional filter bank” whose kernels are taken from the theory of higher degree F-transforms. Comparing with wavelet kernels, the F-transform ones have clear interpretability in a single and sequential layers in a DNN computation.

It has been proven in many papers [24,7] that the higher degree F-transforms are universal approximators of smooth and discrete functions. The approximation on a whole domain is a combination of locally best approximations called F-transform components. They are represented by higher degree polynomials and parametrized by coefficients that correspond to average values of local and nonlocal derivatives of various degrees. If the F-transform is applied to images, then its parameters are used in regularization, edge detection, characterization of patches [7,17], etc. Their computation can be performed by discrete convolutions with kernels that, up to the second degree, are similar to those widely used in image processing, namely Gaussian, Sobel, Laplacian. Thus, we can draw an analogy with the DNN method of computation and call the parameters of the higher degree F-transform features. Moreover, based on a clear understanding of these features’ semantic meaning, we say that a DNN with the F-transform kernels extracts features with a clear interpretation. In addition, the sequential application of F-transform kernels with an up to the second degree gives average (nonlocal) derivatives of higher and higher degrees.

Last but not least, we note that after training DNN, initialized by the F-transform kernels, the shapes of the kernels were not significantly distorted. This fact has been empirically verified on the two datasets: MNIST and CIFAR-10, see Section 4.2 where we compare kernels of various known DNNs after being trained on the same datasets. We observe a similarity of kernel shapes of all considered DNNs. This confirms the stability of the proposed DNN and its sufficiency with respect to the selected datasets.


In this section, we recall the main facts (see [4,18] for more details) about the higher degree F-transform and specifically F2-transform—the technique, which will be used in the proposed below CNN with the F-transform kernels (FTNet).

3.1. Fuzzy Partition

The F-transform components are the result of a convolution of an object function (image, signal, etc.) and a generating function of what is regarded as a fuzzy partition of a universe.

Definition 1.

Let n>2, a=x0=x1<<xn=xn+1=b be fixed nodes within [a,b]. Fuzzy sets A1,,An:[a,b][0,1], identified with their membership functions defined on [a,b], establish a fuzzy partition of [a,b], if they fulfill the following conditions for k=1,,n:

  1. Ak(xk)=1;

  2. Ak(x)=0 if x[a,b]\(xk1,xk+1);

  3. Ak(x) is continuous on [xk1,xk+1];

  4. Ak(x) for k=2,,n strictly increases on [xk1,xk] and for k=1,,n1 strictly decreases on [xk,xk+1];

  5. for all x[a,b] holds the Ruspini condition


The elements of fuzzy partition {A1,,An} are called basic functions.

In particular, an h-uniform fuzzy partition of [a,b] can be obtained using the so called generating function

which is defined as an even, continuous and positive function everywhere on [1,1] except for on boundaries, where it vanishes. Basic functions A2,,An1 of an h-uniform fuzzy partition are rescaled and shifted copies of A in the sense that for all k=2,,n1;

Below, we will be working with one particular case of an h-uniform fuzzy partition that is generated by the triangular-shaped function Atr and its h-rescaled version Ahtr, where


A fuzzy partition generated by the triangular-shaped function Atr will be referred to as triangular shaped.

3.2. Space L2(Ak)

Let us fix [a,b] and its h-uniform fuzzy partition A1,,An, where n2 and h=ban11. Let k be a fixed integer from {1,,n}, and let L2(Ak) be a set of square-integrable functions f:[xk1,xk+1]. Denote L2(A1,,An) a set of functions f:[a,b] such that for all k=1,,n, f|[xk1,xk+1]L2(Ak). In L2(Ak), we define an inner product of f and g


The space (L2(Ak,f,gk)) is a Hilbert space. We apply the Gram–Schmidt process to the linearly independent system of polynomials {1,x,x2,xm} restricted to the interval [xk1,xk+1] and convert it to an orthogonal system in L2(Ak). The resulting orthogonal polynomials are denoted by Pk0,Pk1,Pk2,,Pkm.

Example 1.

Below, we write the first three orthogonal polynomials P0,P1,P2 in L2(A), where A is the generating function of a uniform fuzzy partition, and ,0 is the inner product:

P0(x)=1,P1(x)=x,P2(x)=x2I2, where I2=h211x2A(x)dx,

If generating function Atr is triangular shaped and h-rescaled, then the polynomial P2 can be simplified to the form


We denote L2m(Ak) a linear subspace of L2(Ak) with the basis Pk0,Pk1,Pk2,Pkm.

3.3. Fm-Transform

In this section, we define the Fm-transform, m0, of a function f with polynomial components of degree m. Let us fix [a,b] and its fuzzy partition A1,,An, n2.

Definition 2. [4]

Let f:[a,b] be a function from L2(A1,,An), and let m0 be a fixed integer. Let Fkm be the k-th orthogonal projection of f|[xk1,xk+1] on L2m(Ak), k=1,,n. We say that the n-tuple (F1m,,Fnm) is an Fm-transform of f with respect to A1,,An, or formally,


Fkm is called the kth Fm-transform component of f.

Explicitly, each kth component is represented by the mth degree polynomial


Remark 1.

By the orthogonality of basis polynomials Pk0,Pk1,Pk2,Pkm, the kth Fm-transform component Fkm can be decomposed as follows:


This fact shows that all subsequent Fm-transform components, starting with m1, include the preceding ones. In addition, the higher m, the better the quality of local approximation of f on [xk1,xk+1] by kth Fm-transform component Fkm[4], i.e.,

where k is the norm in L2(Ak).

Definition 3.

Let Fm[f]=(F1m,,Fnm) be the direct Fm-transform of f with respect to A1,,An. Then the function

is called the inverse Fm-transform of f.

The following theorem proved in [4] estimates the quality of approximation by the inverse Fm-transform in a normed space L1.

Theorem 1.

Let A1,,An be an h-uniform fuzzy partition of [a,b]. Moreover, let functions f and Ak, k=1,,n be four times continuously differentiable on [a,b], and let f̂nm be the inverse Fm-transform of f, where m1. Then

where L1 is the Lebesgue space on [a+h,bh].

3.4. F2-Transform in the Convolutional Form

Let us fix [a,b] and its h-uniform fuzzy partition A1,,An, n2, generated from A:[1,1][0,1] and its h-rescaled version Ah, so that Ak(x)=A(xxkh)=Ah(xxk),x[xkh,xk+h], and xk=a+kh. The F2-transform of a function f from L2(A1,,An) has the following representation:

where for all k=1,,n,

I2=h211x2A(x)dx, and coefficients are as follows:


In [4,18], it has been proved that

where is meant up to O(h2).

Without going into technical details, we rewrite (911) into the following discrete representations

where k=1,,n, n=ls, s is the so called “stride” and g0, g1, g2 are normalized functions that correspond to generating functions Ah, (xAh) and ((x2I2)Ah). It is easy to see that if s=1, then coefficients ck,0, ck,1, ck,2 are the results (k-th coordinates) of the corresponding discrete convolutions fg0, fg1, fg2, written in vector form. Thus, we can rewrite the representation of F2 in (8), using the following vector form:
where P0, P1, P2 are vectors of the corresponding polynomials, and s denotes the convolution with the stride s,s1.


In this section, we discuss the details of our neural network design. We chose the LeNet-5 [19] as an architecture prototype and composed a new CNN—FTNet with the kernels initialization taken from the higher degree F-transforms theory [11]. We applied FTNet to several datasets and evaluated the results. For simplicity, we restricted the FTNet architecture to the fixed number of convolutional kernels in the first and second convolutional layers, making the one-to-one correspondence between set of convolutional kernels and set of F-transform kernels K. We use up to second-degree F-transform kernels and some of their modifications based on principal geometrical transformations (rotation and flipping).

In detail, we replace convolutional kernels in the first and second convolutional layers C1 and C3 with the F-transform kernels according to Eq. (13) adapted to functions of two variables. Together with negative versions of kernels, The FTNet C1-layer has 8 distinct kernels. Each of C1 feature maps is further processed in C3 with the same 8 kernels, and as a result, 64 feature maps are obtained.

The details of the FTNet architecture are given below in Table 1. Note that layer FC6 has variable number of neurons, according to number of classes in dataset.

Hyper-parameter Layers

C1 S2 C3 S4 FC5 FC6
Kernel size 5×5 - 5×5 - - -
# Kernels 8 - 64 - - -
Stride 1×1 2×2 1×1 2×2 - -
Pooling size - 2×2 - 2×2 -
# FC units - - - - 500 var
Table 1

FTNet architecture.

4.1. Datasets

All the discussed experiments were realized on the following databases: MNIST [20], CIFAR-10 [21], Caltech 101 [16], and Intel Image classification.2 Datasets details are shown in Table 2

MNIST CIFAR-10 Caltech 101 Intel
Res. 28×28 32×32 var 150×150
Color Gray RGB RGB RGB
Train 60k 50k 7281 14034
Test 10 k 10k 1863 3000
Classes 10 10 101 6
Table 2

Datasets used for experiments.

We convert all datasets to the grayscale because the F-transform kernels extract features with functional meaning and are insensitive to colors. Moreover, we downscale Caltech 1O1 and Intel to 64×64 resolution as FTNet is not suited for higher resolution. Additionally we normalize [22] both datasets. Since Caltech 101 does not have an official train/test split, we split it in 4:1 ratio.

Below, we give a short overview of the known neuro-fuzzy networks, trained on MNIST, and designed for the pattern recognition. We will use MNIST to compare some of the following approaches with FTNet.

Authors of [23] improved the MNIST recognition by optimizing features and architecture, and reached an accuracy of 99.52% on 10 000 testing images. This accuracy is similar to that claimed in [24,25].

In [26], the feature selection is based on the wavelet transform that uses 2D scaling moments and various classifiers (support vector machines/classifiers, artificial neural networks, neuro-fuzzy classifiers, and others). The SVM classifier demonstrates the best accuracy of 99.39%, while the neuro-fuzzy one achieves accuracy 98.72%.

Similar to MNIST, the database of handwritten characters Chars74k [27] has been studied in [28]. The authors used a three-fold cross-validation and achieved 97.22% accuracy.

In the recent publication [29], the Fuzzy Deep Belief Net (FDBN) was proposed to classify MNIST with different types and levels of noise. The FDBN architecture is described in [30,31], where the authors declared better results than using the standard Deep Belief Networks.

4.2. Performance of FTNet and Comparison with the He Initialization

In this section, we compare FTNet initialized with F-transform kernels with Baseline network using the same architecture and He initialization [32]—one of the most common initializations.

We follow He initialization and scale F-transform kernels to 2fanin,+2fanin, where fanin is number of incoming neurons to the layer. C1 initialization is straight forward as the input is a single channel image. To initialize C3 we set majority of the kernel values to zeros, effectively turning them into 64 5×5×1. This is the same for both FTNet and Baseline network.

We remark that in the FTNet, the C3-layer uses the same kernels as in the C1, i.e., C1 computes feature maps fm1C1,,fm8C1 using F-transform kernels K and C3 creates fm1C3,,fm64C3 feature maps applying all 8 F-transform kernels fmiC3=fmi|K|C1Kimod|K|. This way C1 and C3 performs two successive convolutions with all possible kernel combinations.

Figure 1 shows normalized results of training Baseline network and two variants of FTNet. FTNet (red graph curve) corresponds to F-transform kernel initialization, where kernels are allowed to learn. During the learning the kernels are modified to a certain degree. From the graphs, we can see that F-transform kernels initialization is advantageous over He initialization. The Second variation does not allow F-transform kernels to learn and kept unchanged. We can observe that while it has lower loss values at the beginning of training, it falls behind later on. The advantage of static FTNet is higher training speed and lower number of trainable parameters; this can be particularly beneficial to overfitting problem and problem of small datasets. Lastly, static FTNet kernels and features they extract are clearly interpretable.

Figure 1

Results of 2 epoch training on datasets. In the left column are results of training with FTNet C1 initialized with F-transform kernels. Second column contains results of training with both C1 and C3 initialized with F-transform kernels. Note that both accuracy and loss are scaled to [0, 1].

We trained each network 10 times in 2 epochs. For the training, we used Adam (α=1e3, γ=1e3)3 optimizer, cross entropy loss, FC5 and FC6 with L2(λ=1e3)3 regularization and batch size = 50. For initialization of any layer not initialized with F-transform kernels, we used He initialization.

Since disabling trainability of C1 and C3 decreases the number of free parameters in the network, the training time decreases accordingly. The average training time for each network setting and each database is shown in Table 3.

Dataset FTNet
Trainable Nontrainable
MNIST 320s 299s 315s
CIFAR-10 290s 252s 289s
Caltech 101 20s 15s 20s
Intel 65s 53s 66s
Table 3

Average training times for FTNet and its variants and baseline network.

Due to MNIST being the most common dataset, we use it to compare FTNet with other approaches. In Table 4 you can see comparison of FTNet and results reported in selected publications (the latter are indicated by their reference numbers in the first row).

[26] [29] [23] [33] [34] FTNet
99.39% 99.01% 99.52% 99.23% 99.58% 99.39%
Table 4

Comparison of FTNet with other approaches on MNIST.

3×3 None Trainable F-transform kernels
5×5 Max-pool Nontrainable Kernels N(0,1)
7×7 Stride - -
VACL - - -
Table 5

The four hyperparameters values: D, T, and I are associated with C1 and C3 layer; S is associated with C1,S2,C3 and S4 layer.

4.3. F-Transform Kernels as Preprocessing

One can use the F-transform to process data before feeding them into a network. We perform a comparative experiment between the F-transform and recently published fuzzy preprocessing technique—IRFF [35]. IRFF processes images in a convolutional manner, saving minimum, maximum, and central pixel values from a selected neighborhood. In general, the IRFF preprocessing increases the accuracy of a network.

We conducted experiment, comparing the IRFF and F-transform performances on CIFAR-10 using ResNet34 [36] with Shake-Shake regularization [37]4 that achieved accuracy up to 97.14%. From Figure 2, we see that the preprocessing of the F-transform is on the same level as IRFF.

Figure 2

Average accuracy (over 10 runs) of ResNet34 with Shake-Shake regularization on CIFAR-10 over 100 epochs. Network uses Adam(α = 1e − 3), cross entropy loss, early stopping, batch size 128, and light augmentation (width/height shift up to 0.1 and vertical flipping).


To assure that we use a proper combination of hyperparameters, we searched through the hyperparameters space, determined by four hyperparameters: Initialization I, kernels size D, Trainability T, and Subsampling S. These hyperparameters applies to C1,S2,C3 and S4 layers.

Let us describe the functionality of the hyperparameters mentioned above:

  • I determines whether layer C1 and C3 are initialized with F-transform kernels or random ones with N(0,1).

  • D determines the size of C1 and C3 convolution kernels.

  • T determines C1 and C3 trainability.

  • S determines whether C1 and C3 are followed by maxpooling layer5 or are strided convolutions[40] or neither.

Scale-space [41] inspired VAriable Convolutional Layer (VACL) value of D specifies C1 convolution sizes such that it has all three sizes (3×3, 5×5 and 7×7). With VACL, we force the network to process the input image through multiple scales (resolutions). VACL schema is in Figure 3.

Figure 3

Scheme of the convolutional layer with variable kernels sizes, realized as multiple convolution layers with their outputs concatenated.

Using VACL in C3 leads to huge increase in trainable parameters and results in overfitting. For this reason we remove D=VACL option for C3. The results of our searching are sorted with respect to the accuracy on the testing portion of datasets. Figures 4 and 5 contain relative frequencies of hyperparameters of the 500 best combinations. In the case of MNIST (Figure 4), the most prevalent kernel size for C1 is VACL; this confirms our hypothesis regarding the scale-space methodology beneficence to the learning process. Among the optimal initialization of C1 and C3 we see significantly more combinations with trainable F-transform kernels. We conclude that the F-transform kernels initialization has serious, positive impact on the accuracy of FTNet. The results on CIFAR-10 dataset have the same statistics and additionally prove the usefulness of VACL and F-transform initialization.

Figure 4

Relative frequencies of the hyperparameters values for C1 and C3 within the first 500 best combinations in terms of accuracy after 3 epochs of learning on MNIST.

An unexpected result is a high relative frequency of the ”no subsampling” in both cases while stride being worst out of the three.

Figure 5

Relative frequencies of the hyperparameters values for C1 and C3 within the first 500 best combinations in terms of accuracy after 3 epochs of learning on CIFAR-10.

As an additional argument in favor of our technique, we visualize the F-transform kernels in C1 before and after 100 epochs of training with the following setting: I = F-transform kernels, D = { C1=VACL, C3=3×3}, T = trainable, S = max pool and dropout [42]6 between FC5 and FC6 combination. In Figure 6, we see that

Figure 6

Visualization of the eight F-transform kernels before and after training (100 epochs) in C1. The first two rows contain 3 × 3 kernels (training does not change the kernels significantly). The second two rows contain 5 × 5 kernels (the training added some variational details) and the third two rows contain 7 × 7 kernels (the training changed the F0-transform kernels transforming them to the rotated F1-fransform).

  1. Up to the small contrast changes, 3×3 kernels remain unchanged.

  2. The shapes of 5×5 kernels are generally preserved; however, some variational details were added after training.

  3. The shapes of 7×7 are preserved for F1 and F2 kernels; however the F0 kernels became similar to the rotated F1.

Thus, we can summarize achieved results:

  1. The scale-space inspired VACL is the most frequent size of the convolution kernels in the first convolutional layer of the considered CNN, trained on both datasets.

  2. The initialization of C1 and C3 convolutional layers with the F-transform kernels leads to the higher network accuracy.

  3. Excluding subsampling from CNN’s architecture increases network accuracy; however, it contributes to an undesirable effect of overfitting.

  4. Including subsampling into CNN’s architecture leads to a quicker decrease of a loss function.

  5. The F-transform kernels in the first layer C1, do not significantly change their shapes during training. Therefore they are an ideal choice for feature extraction.

5.1. Semantic Meaning of Principal Kernels in Convolutional Layers

In this subsection, we tackle the problem of interpretability from the opposite angle. Instead of initializing a network with predefined kernels, we examine the first convolutional layers and the corresponding to them (already trained) kernels taken from several well-known networks. The purpose is to find general semantic meanings of kernels and through them compare with the F-transform kernels.

We tried to assign interpretation to kernels of already trained CNNs: We based on the known interpretation of the higher degree F-transform kernels and wished to reveal the similar meaning of kernels extracted from the first convolutional layer of several frequently used CNNs, trained on the ImageNet [43]. Let us review some of the existing contributions that connect both DL and fuzzy disciplines.

We selected 6 networks: VGG16 [44], VGG19 [44], InceptionV3 [45], MobileNet [46], ResNet [36], and AlexNet [47] as the representative examples of CNNs. All of the networks were trained on ImageNet [43], using the same training database, consisting of 1.2M RGB images7 with various resolution (usually downsampled to 256×256).

We extracted kernels from the first convolutional layer of all considered networks and analyzed whether there are similarities among kernels across the networks. To reduce the space of kernels, we first apply the hierarchical clustering on every network kernel set separately, and then, look for the similarities among clusters. The medoids of the found clusters are shown in Figure 7.

Figure 7

From top to down: the medoids of the clusterized kernels from the first convolutional layer of AlexNet, InceptionV3, MobileNet, ResNet, VGG 16, and VGG 19.

We observed that the extracted clusters contain similar elements (kernels) across the different networks that share one of the following characteristics/functionality: gaussian-like; edge detection (with various angle specifications); texture detection; color blobs.

If we compare the semantic meaning of the extracted clusters (in terms of the above-given characteristics/functionality) with that of the F-transform kernels in the FTNet, then we see the coincidence in the first two items from the above-given list. To be more precise, the F 0-transform kernels are the Gaussian-like, and the F 1-transform kernels are (horizontal or vertical) edge detectors. This again supports our conclusion regarding the suitability of the F-transform kernels in the first layers of CNNs.

The above-formulated general conclusion relates to the disclosed semantic meaning of convolutional kernels. This knowledge is helpful for the optimal and nonexhaustive design of CNNs.


We have proposed a new CNN learning methodology that is focused on a smart preprocessing with the meaningful initialization of CNN kernels in the first two layers. The methodology is based on the fuzzy modeling technique—F-transform. As a result, we have designed a new CNN-type network called FTNet.

The performance of FTNet was examined on several datasets and on them it converges faster in terms of accuracy/loss than the baseline network, subject to the same number of steps.

We compared the F-transform kernels in the first layer before and after training. We observed that the kernels remain unchanged. Moreover, their shapes are similar to the shapes of extracted kernel groups from the most known CNNs.

All these facts confirm our hypothesis that the smart initialization of the first layers kernels can be proposed based on their semantic meaning and the general network designation.

Our future work will be focused on neural nets with larger number of layers and other than recognition objectives.


The work is supported by ERDF/ESF “Center for the development of Artificial Intelligence Methods for the Automotive Industry of the region” (No. CZ.02.1.01/0.0/0.0/17_049/0008414).



The text of this and the following subsection is a free version of a certain part of [4] where the theory of a higher degree F-transform was introduced.


α - learning rate, γ - decay, λ - strength of regularization.


Subsampling operation originates from Hubel and Wiesel [38]; comparison of pooling can be found in Ref. [39].


We have employed dropout to reduce network overfitting.


ImageNet database content depends on the year of ILSVRC competition.


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Cite this article

AU  - Vojtech Molek
AU  - Irina Perfilieva
PY  - 2020
DA  - 2020/09
TI  - Deep Learning and Higher Degree F-Transforms: Interpretable Kernels Before and After Learning
JO  - International Journal of Computational Intelligence Systems
SN  - 1875-6883
UR  -
DO  -
ID  - Molek2020
ER  -