Neural Random Projection: From the Initial Task To the Input Similarity Problem
Authors: Alan Savushkin
Nikita Benkovich
Dmitry Golubev
Venue:  (2021)

Abstract: The data representation plays an important role in evaluating similarity between objects. In this paper, we propose a novel approach for implicit data representation to evaluate similarity of input data using a trained neural network. In contrast to the previous approach, which uses gradients for representation, we utilize only the outputs from the last hidden layer of a neural network and do not use a backward step. The proposed technique explicitly takes into account the initial task and significantly reduces the size of the vector representation, as well as the computation time. Generally, a neural network obtains representations related only to the problem being solved, which makes the last hidden layer representation useless for input similarity task.
In this paper, we consider two reasons for the decline in the quality of representations: correlation between neurons and insufficient size of the last hidden layer. To reduce the correlation between neurons we use orthogonal weight initialization for each layer and modify the loss function to ensure orthogonality of the weights during training. Moreover, we show that activation functions can potentially increase correlation. To solve this problem, we apply modified Batch-Normalization with Dropout. Using orthogonal weight matrices allow us to consider such neural networks as an application of the Random Projection method and get a lower bound estimate for the size of the last hidden layer. We perform experiments on MNIST and physical examination datasets. In both experiments, initially, we split a set of labels into two disjoint subsets to train a neural network for binary classification problem, and then use this model to measure similarity between input data and define hidden classes. We also cluster the inputs to evaluate how well objects from the same hidden class are grouped together. Our experimental results show that the proposed approach achieves competitive results on the input similarity task while reducing both computation time and the size of the input representation.

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NEURAL RANDOM PROJECTION: FROM THE INITIAL TASK To THE INPUT SIMILARITY PROBLEM
Anonymous authors Paper under double-bling review
ABSTRACT
The data representation plays an important role in evaluating similarity between objects. In this paper, we propose a novel approach for implicit data representa- tion to evaluate similarity of input data using a trained neural network. contrast to the previous approach, which uses gradients for representation, we utilize only the outputs from the last hidden layer of a neural network and do not use a back- ward step. The proposed technique explicitly takes into account the initial task and significantly reduces the size of the vector representation, as well as the com- putation time. Generally, neural network obtains representations related only to the problem being solved, which makes the last hidden layer representation useless for input similarity task. In this paper, consider two reasons for the decline in the quality of representations: correlation between neurons and insuffi- cient size of the last hidden layer. To reduce the correlation between neurons we use orthogonal weight initialization for each layer and modify the loss function to ensure orthogonality of the weights during training. Moreover, we show that ac- tivation functions can potentially increase correlation. To solve this problem, we apply modified Noroallommliizttion with Dropout. Using orthogonal weight ma- trices allow us to consider such neural networks as an application of the Random Projection method and get a lower bound estimate for the size of the last hidden layer. We perform experiments both experiments, initially, we split set of labels into two disjoint subsets to train neural network for binary classification problem, and then use this model measure similarity between input data and define hidden classes. We also cluster the inputs to evaluate how well objects from the same hidden class grouped together. Our experimental results show that the proposed approach achieves com- petitive results the input similarity task while reducing both computation time and the size of the input representation.
MNIST and physical examination datasets. In
INTRODUCTION
Evaluating object similarity is an important area machine learning literature. It is used in various applications such search query matching, image similarity search, recommender systems, clus- tering, classification. In practice, the quality of similarity evaluation methods depends on the data representation.
For a long time neural networks show successful results in many tasks and one such task is obtaining good representations. Many of these methods can be considered in terms of domain and task. The first case is when we have only unlabeled dataset. Then we can use autoencoders (Bank et al. 2020) or self-supervised approaches (Chen et al. 2020 Devlin al. 2018 Doersch et 2015 Dosovitskiy et al.] 2014 Gidaris et 2018 Noroozi [ & Favaro 2016, Noroozi et al.) 2017 Oord al. 2018 Peters et al. (2018), which require formulation of pretext task, which in most cases depends on the data type. These methods can be called explicit because they directly solve the problem of representation learning. Moreover, these models can be used for transfer knowledge when we have labeled data only in the target domain. The second case is where we have labeled data in the source and target domains. Then we can apply a multi-task learning approach (Ruder) 2017) fine-tune the models (Simonyan & Zisserman, 2014 He et al. 2016) trained dataset like ImageNet. Finally, there is the domain adaptation approach (Wang & Deng. 2018) where we have single task but different source and target domains with labeled data in the target
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domain (Hu al., 2013) with unlabeled data (Li et al. 2016 Yan et al. 2017 Zellinger al. 2017).
In our study the target task is to measure similarity between objects and to define hidden classes based on it. We are interested in studying the issue of implicit learning of representations. Can the neural networks store information about subcategories if we don't I explicitly train them to do this? More formally, we have the same source and target domains but different tasks and we don't t have labeled data in the target domain. That makes our case different from the cases of transfer learning.
A solution to this problem could be useful in many practical cases. For example, we could train model to classify whether messages are spam not and then group spam campaigns kind of attacks (phishing. spoofing, etc.) based on similarity measuring by trained neural network. Similar cases could be the medicine (classifying patients into healthy/sick and grouping them by the disease) or in financial (credit scoring) area. The benefits are that we do not depend on the data type and, more importantly, we use only one model for different tasks without fine-tuning, which significantly reduces time for developing and supporting of several models.
Similar study was done in (Hanawa et al., 2020), where authors proposed evaluation criteria for instance-based explanation of decisions made by neural network and tested several metrics for mea- suring input similarity. In particular, they proposed the Identical subclass test which checks whether two objects considered similar are from the same subclass. According to the results of their exper- iments, the most qualitative approach is the approach presented in (Charpiat et al, 2019). which proposed measure similarity between objects using the gradients of neural network. In experi- ments, the authors applied their approach to the analysis of the self-denoising phenomenon. Despite the fact that this method has theoretical guaranties and does not require modify the model to use it, in practice, especially in real-time tasks, using gradients tends to increase the computation time and size of vector representation. This approach will be described in more detail in Section
2 To avoid these problems, we propose method that only uses outputs from the last hidden layer of a neural network and does not use backward step vectorize the input. In our research, we found that correlation of neurons and insufficient width of the last hidden layer influence on the quality of representations obtained in implicit way. To solve these issues, we propose several modifications. First, we show that the weight matrix should be orthogonal. Second, we modify Batch-Normalization (Toffe & Szegedy 2015) to obtain the necessary mathematical properties, and use it with (ropouu(Srivastave et al. 2014) to reduce the correlation caused by nonlinear activation functions. Using orthogonal weight matrices allows us to consider the neural network in terms of Random Projection method and evaluate the lower bound of the width of the last hidden layer. Our approach will be discussed in detail in Section 3 Finally, in Section 4 we perform experiments on MNIST dataset and physical examination dataset (Maxwell et al.) (017). We used these datasets to show that our approach can be applied for any type of data and combined with different architectures of neural networks. In both experiments, we split set of labels into two disjoint subsets to train neural network for binary classification problem, and then use this model to measure the similarity between input data and define hidden classes. We also cluster the inputs to evaluate how well objects from the same class are grouped together. Our experimental results show that the proposed approach achieves competitive results on the input similarity task while reducing both computation time and the size of the input representation.
RELATED WORKS
Using a trained neural network to measure similarity of inputs is a new research topic. In (Charpiat) al. 2019) the authors introduce the notion of object similarity from the neural network perspective. The main idea is as follows: how much would parameter variation that changed the output for æ impact the output for x'? In principle, if the objects æ and x' are similar, then changing parameters should affect the outputs in a similar way. The following is a formal description for one- and multi- dimensional cases of the output value of neural network.
One-dimensional case Let fa(æ:) E R be parametric function, in particular neural network, x, x' be input objects, 0 E R" model parameters, ng number of parameters. The authors proposed the following metric:
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Vofo(='\Vofo(z) (V)) ([ooe
po( x²)
In this way, the object similarity is defined the cosine similarity between the gradients computed at these points.
Let fo(x) E Rd, > 1. In this case, the authors obtained the following Multi-dimennional case
metric:
po,a(x, x') = Tr(Ko(z,
where Ko(x,, x²)
K (/21/K,, Kn ne Kalld 1/2
and Ka',
Ofe 80
0fe
calculated using the Jaco-
x 80
bian matrix afe 30
Summary Unlike using third-party models for vectorization, such as VGG, this approach allows us use any pre-trained neural network to calculate the similarity of input data. To achieve this, authors use gradients of neural network as illustrated in equation | and equation 2 This gradient- based solution takes into account all activations, which does not require the selection of a hidden layer for vectorization. However, it has number of drawbacks. First of all, fast computation of gradients require additional computational resources. Second, the size of objects representation is ni for micddimesioonll output and ne d for multi-dimensional case. This means that increasing model complexity increases representation size and, as a result, can lead to large memory consumption. Motivation of this research is to develop approach that reduces the size of the representation and uses only the forward pass of a trained neural network to work. In the next section, we will discuss the proposed approach in detail.
PROPOSED METHOD
In this paper, we suggest using outputs from the last hidden layer of a trained neural network to encode inputs. This representation has several advantages. First, the output of the last layer is usually low-dimensional, which allows us to get a smaller vector dimension. Second, and more importantly, the last hidden layer the semantics of the original problem is taken into account to the greatest extent. For example, for classification problem, data is theoretically linearly separable on the last hidden layer. Informally, this allows us measure similarities within each class, which reduces false cases in terms of the original problem. This is extremely important for clustering spam campaigns, because this representation reduces the probability group spam and legitimate messages together. However, as already mentioned in the section ! a neural network that is trained solve a simple problem does not retain the quality of representaions needed solve a more complex one. Due to this fact, it's impossible to apply this representation as is. We have identified the main causes of this: strong correlation of neurons and an insufficient size of the last hidden layer. Our goal is to avoid these in order to make the proposed vectorization reasonable. In practice, last hidden layers are usually fully connected. For this reason, we consider only this type of the last layer. In 3.1, we show how to reduce correlation between neurons, and in 3.2 we offer an estimate of the lower bound of the size of the last hidden layer and prove that the proposed representation can be used for the input similarity problem. We introduce the following notation:
1. E (0, I,...,II) [.}- layer number, where 1 = 0 - input layer (source space), / = L output layer, other - hidden layers.
2. Nr number of units in layer
3. hy E R -rre--ctiiviaiinn vector
4. (hi) = (d(tu,1),.... ((h),))) activation vector
5. Wi RN-IIN,, bi TR:: weight matrix and bias vector
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3.1 NEURON CORRELATION
The benefits of decorrelated representations have been studied in (LeCun et al. 2012) from op- timization viewpoint and (Cogswell et al. 2013) for reducing overfitting. We consider the decor- related representations from information perspective. Basically, correlation of neurons means that neurons provide similar information. Therefore, we gain less information from observing two neu- rons at once. This phenomenon may occur due to the fact that the neural network does not retain more information than is necessary to solve particular task. Thus, only important features are highlighted. The example in Fig. IA illustrates that the output values of two neurons linearly dependent, which entails that many objects this space are indistinguishable On the contrary (Fig. [B), decorrelation of neurons provides more information and, result, the ability to distinguish most objects.
class
class
AC-11
Figure 1: A) Correlated neurons. Only end task semantics are taken into account. B) Uncorrelated neurons. More objects are distinguishable
In the following paragraphs, we identify the main causes of the correlation of neurons and suggest ways to prevent it. We decided to consider the correlations before and after activation of neurons separately.
Reducing correlation of neurons before activation correlation that occurs before activation.
Statement explains the main reason for the
Statement 1. Suppose that for some layer the following conditions are satisfied:
1. E (ba)) =0
2. (lo(h:) (bx)) = of I covariance matrix as we required B[((h::)
Then, in order for correlation not to occur on the layer 1 + 1, it is necessary that the weight matrix W141 be orthogonal, that is, satisfy the condition WT W(+
The statement shows that the first correlation factor on a fully connected layer a non-orthogonal weight matrix, given that, if the input neurons do not correlate. See Appendix A for proof of the statement T During training, the network does not try to maintain this property, solving the problem at hand. Later in this paper, we will add regularization to penalize the loss function if the weights are not orthogonal
The corollary follows from the statement 1 which gives the second condition for preventing corre- lation. This corollary states that the dimension of the layer / + should be no greater than on the layer Otherwise, this leads to correlation and does not increase information. See Appendix A.] for proof this corollary.
Corollary 1. Suppose that the conditions of statement 7 are satisfied, then if the dimension N1+1 Nú. then there is certainly pair of neurons h441,5, h1415 i + Cov (.04hh,,,,, #
It should be noted that there are also the studies addressing orthogonal weight matrices (Huang al. 2017 Jia et al. 2019; Xie et al., (017). However all these works consider this topic from opti- mization perspective. In particular, in (Cho & Lee 2017) was proposed an approach for optimization of loss on the Stiefel manifold (I("))) W xxp
W I} to ensure orthonormality of weight matrices throughout training. To achieve this, they applied the orthogonality regularization
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require the Gram matrix of the weight matrix to be close identity matrix. In our study also use regularization (3) to ensure orthonormality of weight matrices.
A -
2
1=1
Providing the necessary moments of neurons after activation In the statement | we relied on the zero expected value and the same variance of units the layer. But the nonlinear activation function does not guarantee the preservation of these properties. Due to this fact, we cannot reason in the same way for the following layers. Therefore, we propose using an activation normalization approach similar to Batch-Normalizatio
((hrs) ((u,,)) - Ma(hya))
(4)
Vo +E
where " is trainable scale parameter, Ma(hr,s): ())),,)) parameters that evaluated, Infoffe & Szegedy (2013). The difference compared the standard Namma-oormalization is that the " is the same for all neurons and we removed the St parameters. This leads to an expected value of zero and the same variance of of each unit in the layer.
Reducing correlation of neurons after activation should be noted that an activation function also impact the formation of redundant features (Ayinde et al. 2019). In particular, in this work we use lanh(x:) E (-1,1) 1) as the activation function. There are several methods that prevent for- mation of redundant features. In (Cogswell al., 2013) was proposed DeCov loss which penalizes non-diagonal elements of estimated covariance matrix of hidden representation. In (Desjardins et al., 2015 Blanchette & Laganière 2018 Huang al. 2018) were proposed approaches for learning
decorrelation layers that perform the following transformation: (hh)) (Thhu) (k))))) ( All of these methods have common drawback: they require estimating covariance matrices. Often in practice the size of mini- mini-batch is much smaller than is needed for estimating covariance matrices. Therefore, the covariance matrix often singular. Moreover, methods that use the decorrelation layer are computationally expensive when it comes to high-din nigh-dimensional embeddings, since it is nec- essary to calculate the square root of the inverse covariance matrix. This is especially evident in wide neural networks. Besides, these techniques add significant amount of parameters Σ/ N?-
As alternative, suggest using Dropout (Srivastava et al. 2014). which prevents units from co-adapting too much and reduces the correlation between neurons during training stage in the layer in proportion to p the probability of retaining unit in the network. See Appendix A.2 for the proof of this statement.
important note that we apply normalization the input data (input layer = 0) as well as Dropout, since it is not always possible to decorrelate data so that this does not affect the quality of training. Moreover, fully-connected layers are often used in more complex architectures, such as convolutional neural networks. Obviously, after convolution operations. transformed data will be correlated. In this case, we must apply dropout after the convolutional block in order to reduce the correlation of neurons and use the proposed approach for vectorization.
3.2 NEURAL NETWORK FROM THE RANDOM PROJECTION PERSPECTIVE
In the previous section, proposed techniques of minimizing correlation between neurons. How- ever, it is not guaranteed that the obtained representations are useful or sufficient for the similarity measuring of objects. In order to prove this, consider neural network application of the Random Projection method that allow us to estimate a lower bound of the representation size and define metric measure similarity.
The Random Projection method (Mtouuskk 2013) is one of the methods of dimensionality reduction, which is based on the Johnson-I Lindenstaasssssssss lemma (Matoušek 2013):
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Lemma 1. Let (0,1) and X f.i,.... "ei....,,,,.) of points in space IR:; k 2 Cloog
=²
where C > is large enough constant. Then there exists a linear map f Rdd - RR such that x' E X:
--=a]|la- æ'1l - (1 +)]l =x'll
æ'1l
(5)
As can be seen, Lemma| 1 states only the existence of a linear map. However, there is also proba- bilistic formulation the Lemmal) [
which states that if we take a sufficiently large ks and consider random orthogonal projection onto the space RR then inequality in equation 5 holds with high probability. This is the cornerstone this work, allowing us to obtain Neural Random Projection, which is explained below.
In this work, we use the tanh activation function and assume that we are working in the linear region of the activation function. Due to this, can make the following approximation:
h hz-1(x) = W. W2 ... WL- + W
(6)
where Wi, b means that consistent use of scales and shifts taken into account, respectively. Just
WL-1 means that we consider the output before activation and hence before applying equation 4
And y is common multiplier after all modified batch- norrddniaaaiiziinns.. It is obvious that the final matrix W is still orthonormal.
According to approximation in equation 6 pre-activation outputs of the last layer are an orthogonal projection of the input data. Moreover, the process of random orthogonal initialization and opti- mization on the Stiefel manifold can be seen as a random sampling of an orthogonal matrix. For this reasons, we consider the neural network from the point of view of the Random Projection method. Due to this fact, we use the similarity metric L2 and get a lower bound estimate for the size of the last hidden layer (k > Clorn), ), although in practice this estimate is often too conservative, since Lemma 1 does not take data structure into account. Therefore a small C and æ closer one can be considered, as will be shown in the experiments section.
EXPERIMENTS
In this section present the experiments two datasets with different data types. Both experi- ments have the same structure. Initially, we group a set of labels into two disjoint subsets to train neural network for binary classification problem, and then use this model to measure similarity between input data and define hidden classes. To evaluate the quality of defining hidden classes and hence the quality of similarity measure we use kNN classifier. To evaluate how well objects from the same hidden class are grouped together we use KMeans approach and v-measure (Rosenberg & Hirschberg 2007). We use source data representation as baseline. To compare our approach, we consider 3 models (A, B, C) that have the same architectures but different regularizations in fully- connected layers. We also compare the representations from the last hidden layer with the previous gradient-based approach (Charpiat et al. 2019) using Model A, since the authors did not impose additional requirements on the neural network.
4.1 MNIST
We performed experiments on MNIST dataset, which consists of 70,000 images of size 28x28 with 6-1 thining-testing split. This dataset chosen to show that our approach works with images and can be plugged-in in after convolution blocks. See Appendix B. for more detailed description of the experiments.
Results As illustrated in Table we achieved results comparable (kNN digit accuracy) with the previous approach and much better results from y-measure perspective. We also obtained much smaller dimension of the representation (30 vs 14.9k), and in addition it is smaller than the original dimension (30 vs 784). This allowed us to drastically reduce the time for vectorization, using only the forward pass, and the time to measure the similarity of objects (kNN prediction time) and search

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Meder A
WodeiB
Motel C (Our morelt
Hocel & AGracienn
Figure t-SNE last hidden layer (test set)
Table 1: Experimental results (MNIST dataset)
Model
Model accuracy (even/odd)
kNN digit accuracy
KNN prediction
V
Vec. time (s)
Space dimension
(object
rep.)
measure
time (s)
A aVafa)
96.8
0.542
677.04
188.79
14911
A (h)
97.
52.9433.7
0.002
1.93:.0.07
25.23-10.97
275
B (hI
98.6+
93.1++0.4
0.588 1.006
0.95+-0.10
20.96
30
C (h1.-1)
9..910.1
97.7.10.3
0.812
188310.08
21.04
30
ho = x
94.29
0.4
21.48
784
Model A
Model B
Model C
denaty
Figure 3; Distribution of correlations between pairs of neurons of the last hidden layer
Table 2: Spearman's 's correlatioon between median of correlation distribution of neurons and target
metries
(dim h1,-1)
KNN digit accuracy
V-measure
0.2 (275)
-0.28
-0).47
0.4 (68)
-0.86
0.13
0.6 (30)
-0.72
-0.82
0.8 (17)
-0.57
-0.81
1.0 (11)
-0.46
-0.83
for the closest object. should be noted that only our model (Model C) achieves comparable quality with the previous approach. Moreover we could improve evaluation of similarity in comparison to source representation.
Although the classic Batch-Normalization (Model B) gives good increase in the quality of digit recognition compared to Model A, still fails to achieve sufficient results on similarity task. This is well illustrated by the visualization of the representation of the last hidden layer (Fig. 2 As can be seen, Model A can only take into account the semantics of the original problem. Model B, for the most part, the numbers are locally distinguishable, but some classes strongly mixed, which confirmed by the presence of high correlation (Fig. 3). Our model (Model C) can significantly reduce the correlation and gives explicit separation with large margin between the classes. In addition, the proposed modifications do not impair the quality of the model, as seen in Table (Model accuracy).
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As can be from Table 2, generally, there is negative statistical relationship between target metrics and neuronal correlation.
4.2 HENAN RENMIN HOSPITAL DATA
Next, used physical examination dataset (Maxwell et al.] 2017), which contains 110,300 anony- mous medical examination records. We retained only four most frequent classes (Normal, Fatty Liver, Hypertension and Hypertension Fatty Liver), because this dataset highly imbalanced. As a result, we got 100140 records. After that we split four classes on two corresponding groups "Sick" (Fatty Liver, Hypertension and Hypertension & Fatty Liver) and "Healthy" (Normal). After that divided data set into train and test subsets in proportion 4-1.
should be noted that this task is more difficult in comparison to the previous experiment because hidden classes are semantically related which makes it easier for the neural network to mix these classes into one.
This dataset was chosen to show that our approach works well with categorical features. Also we show that we can make the first layer wider than the source dimension and after that plug-in our approach. As mentioned above, the original classes are imbalanced so instead of accuracy metric we used F imiiro score to evaluate the quality of the similarity measurement (kNN Pimicro). See Appendix B.2 for more detailed description of the experiments.
Figure 4: t-SNE last hidden layer (test set)
Table 3: Experimental results (Henan Renmin Hospital Data)
Model
Model (object rep.)
kNN disease F Fimicro
kNN
acc.
V measure
Vec. time (s)
pred. time (s)
Space dim
(sick/
healthy)
(Vaajo)
79.1
0.316
169.2
980.8
29111
A (h1-1)
88.25-10.06
75.0+22.7
0.371=0.000
0.1610.01
62.22.+1.27
31
B (hL-1)
86.7-0.1
79.6:10.2
0.011
0.28+00.2
93.26+22.59
31
C (h1-11
87.4-10.2
0.2
0.26+00.0
75.87+1.44
31
hg
61.1
0.066
45.91
62
Model A
Model B
Model C
density
Figure 5: Distribution of correlations between pairs of neurons of the last hidden layer
Results As illustrated Table 3 we achieved results comparable (kNN disease Fimiero score)
with the previous approach. As in the previous experiments obtained much smaller dimension
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of the representation (31 29k) and drastically reduced the time for vectorization and the time to measure the similarity of objects (kNN prediction time). should be noted that we noticeably improved the quality of similarity measurement in comparison to the source space (61.1 vs 79.6) as can be seen also in Fig. 4 obtain more quality representaions. This result be explained by the fact that the hidden classes are strongly semantically related, in particular in "Sick" class. Therefore, it is easier for the neural network mix these hidden classes into one. Probably, the data have more complicated structure in source space (see Table for ho) and it's not enough to only reduce the correlation. See Table for detailed results.
however it did not help We also reduced correlation comparison of the Model A and Model B,
Table 4: Spearman's 'ss correlattion between median of correlation distribution of neurons and target metrics
& (dim h1-1)
KNN F Iiiero disease
V-measure
0.2 (282)
-0.83
-0.6
0.4 (70)
-0.81
-0.13
0.6 (31)
-0.11
-0.34
0.8 (17)
0.1
-0.1
1.0 (11)
0.02
-0,15
As can seen from Tabel 4 in this case, the statistical relationship between neuron correlation and target metrics is not strong. Especially for the small dimension of the last hidden layer. This may be due to the fact that if the layer size is insufficient, there are not enough dimensions to describe the variability of the data, even if the data is not highly correlated.
5 CONCLUSION
In this work, we studied an approach for obtaining implicit data representation. In order to obtain implicit data representation, we introduced the Neural Random Projection method, which includes regularization of the neural network in the form of optimization among orthogonal matrices, a mod- ification of Naach--ormalization and its combination with Dropout. This allowed obtain repre- sentations applicable for input similarity measure using trained neural network. We experimentally compared our approach to the previous one and showed that it significantly reduced both computa- approach allowed us introduce the
tion time and the size of the input representation. Finally, L2 metric on the representations and improve the quality of the similarity measurement compar- ison with the source space. And how be seen from the experiments the correlation and the size of the last hidden layer not yet all factors affecting on the final implicit representation. This remains an open question, and in the future we are planning research it more. We expect that our study will be useful for many applied problems, where it is initially very difficult to determine a method for measuring data similarity in its original form.
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Under review as a conference paper at ICLR 2021
A THEORETICAL PART
A.1 REDUCING CORRELATION OF NEURONS BEFORE ACTIVATION
Statement 1. Suppose that for some layer the following conditions satisfied:
1. E (h:)) =0
2. (h.)) = of covariance matrix as we required E E[((hi)] =0
Then, in order for correlation not to occur the layer I + 1, it is necessary that the weight matrix W1+1 be orthogonal, that is, satisfy the condition W W4+1 = I.
Proof. To prove this, consider the expected value and the covariance matrix of the vector h4+1, using the relation h441 = T W(+1 + b and let the condition of orthogonality be satisfied:
1. Expected value:
E[h (E(((h)))))))))))))) W1+1 bI+1] ([ W141 + E[b++1] = b
2. Covariance matrix:
E[(4++ - E(h (h141 1[) = (h.)
W441
ofW W4+1 of
Corollary 1. Suppose that the conditions of statement I are satisfied, then if the dimension Ni+1 Nú, then there is certainly pair of neurons h441,5, h141,5; i # Cov((1 hit41) +
Proof.
Suppose, towards a contradiction, that N441 > N₁ and Vi.j E By the statement [ this is possible only W an orthogonal matrix, which means that there is linearly independent system of N4t1 vectors of dimension N It follows that rank W++1 min(N,, N++1) N₁, which is impossible, therefore the matrix W 1 not orthogonal, thus we have a contradiction.
j
Cov(h h(41,)
A.2 REDUCING CORRELATION OF NEURONS AFTER ACTIVATION
For further discussion, we describe the Dropout method in our notation: 2; = Tt- ((rr). where (fi,1...., PiNN;); Tı, Bernoulli(p) and the sign means element-wing multiplication, the probability of retaining a unit in the network and & denotes activation after modified Batch- Normalization. The following statement explains how Dropout reduces correlation.
Statement Dropout reduces the correlation proportion p
Proof. Let Cijy i + j correlation value between neurons i,j. Given the properties of ((hu),
B B)a(ha) Now consider the x2
we obtain the following expression for the correlation: Ci5 = correlation after applying Dropout:
E E mig
B Blrud(taa) Tú, [(11;)]
Oe
E Blr 32 (hr)) B (his))
Bma B[rg
Blo( ). d (hi;))
B[r Biri)) BiO² (fo:)) ) 1662 (h1,))
p ((o))))) (( pCAs
mr2
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Under review as a conference paper at ICLR 2021
Here used the fact that Pass Mty independent Bernoulli random variables, and they are also independent of ((ha); (h11)).
B DETAILS OF EXPERIMENTS
Computing infrastructure All experiments performed Intel(R) Xeon(R) CPU E5-2650 O 2.00GHz, 189GB RAM and CentOS Linux (Core) x86-64 operating system. The models implemented using Python 3.6v and TensorFlow 2.2.0v library.
B. MNIST
Experiments description
1. Models (A, B, C) trained to solve binary classification problem (even/odd digit). Number of epochs 100, optimizer Adam with learning rate 0.001; batch size 256; validation split 0.1,, early stopping: min delta le-3 and patience 5 epochs.
2. KNN classifier trained (A: 9) to solve digit classification problem, using pre-trained model A and metric from equation [ from previous approach. We measured vectorization time on the complete dataset and evaluated digit accuracy, even/odd accuracy and prediction time this KNN model on test set.
3. Everything is similar to the previous item, only the representations of the last hidden layer of each model and the L2 metric were used.
4. Like previous item, only the source space representation and the L2 metric were used.
We used models with the following base architecture:
D(16 x 3) ReLU - Mon 2) C x - ReLU Max Pollingg2) 2) + Dense(64) > tanh + Dense(dim h1-1) - tanh. - Dense(
The size of the last hidden layer was chosen corresponding the lower bound from section 3.2]
dim h log 6- 10
E2
In all models, only la with À 0.01 regularization was used in the convolution block. Further, the difference between models in fully-connected layers will be given.
Model A: simple network, only l2 with À 0.01 regularization is used. And the variable ranges from 0.2 to 1.0 with step 0.2.
Model B: orthogonal matrix initialization, regularization from equation with A E (0.0, 0.01, 0.1, 1) and l2 regularization on bias vector; standard Normalizionii used before activation and Dropout with p E {0.0, 0.1, 0.2, 0.3) after activation.
0.2,
Model (proposed model): modified Normaaaiimaii is used after activation and Dropout after Batch-Normalization..
Each model with each combination of hyperparameters is trained 20 times. needed to obtain 95% confident intervals for target metrics (KNN accuracy digit and v-measure).
B.2 HENAN RENMIN HOSPITAL DATA
Experiments description
Models (A, B, C) trained to solve binary classification problem (sick/healthy). Number of epochs 200, optimizer Adam with initial learning rate 0.001 and exponential decay factor 0.99 every 500 iterations: batch size 128; validation split 0.1, early stopping: min delta 1e-3 and patience epochs.
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Under review as a conference paper at ICLR 2021
2. KNN classifier (k 9) trained to classify type of disease, using pre-trained model A and metric from equation [ We measured vectorization time on the complete dataset. And we evaluated Fimiero score on hidden classes because they are imbalanced, sick/healthy accuracy and prediction time this kNN model on test set.
Everything is similar to the previous item, only the representations of the last hidden layer of each model and the L2 metric were used.
4. Like previous item, only the source space representation and the L2 metric were used.
We used models with the following base architecture: Dense(128) + tanh + Dense(96) tanh Dense(64) tanh (ense(32) tanh Dense (dim h1--1) (1)se(1)
The size of the last hidden layer was chosen corresponding to the lower bound from previous section:
log 8- 10*
dim hy
g²
Each model with each combination of hyperparameters is trained 20 times. It is needed to obtain 95% confident intervals for target metrics (KNN Fimiero disease and v-measure).
Model A: simple network, only l2 with À 0.001 regularization is used.
Model B: orthogonal matrix initialization, regularization from equation with À E (0.0, 0.01, 0.01, except the first layer and regularization bias vector; standard Batch- Normalization is used before activation and Dropout with p E {0.0, 0.2, 0.3] after activation.
Model C (proposed model): modified Batch-Normalization is used after activation and Dropout with after Batch-Normalization.
14


Reviews:
[{"review_id": "qP-Hjb9iFjq", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper studies the usage of the representations developed in the last layer of a neural network as a way to measure the similarity between input patterns. The fundamental idea revolves around the concept of orthogonal weight matrices, to decorrelate the activations of the neurons, and which would definitely enrich the internal representations developed by the neurons in the last hidden layer. In the paper, it is suggested to regularize during training to maintain the orthogonality of the weight matrices. Moreover, a variant of Batch Normalization is proposed. The method proposed in the paper is then experimentally applied and evaluated on two benchmark datasets (MNIST and  Henan Renmin).\n\nPROs:\n- The study of intrinsic richness in neural network architectures related to the shape of the weight matrices, in this case orthogonality, is of great appeal.\n\nCONs:\n- In the experimental part I could not see (even in the appendix and in the code) any consideration made in relation to the hyper-parameters of the neural networks, and of the used learning algorithms. The choices made in the paper (and reported in the appendix), e.g. on the learning rate, on the number of epochs, on the regularizer coefficients, seems rather arbitrary and would be better selected on a validation set.\n- Being (at least in my view) the paper based on suggesting and using orthogonality as a design choice for the weight matrices in a neural network, the novelty content seems rather limited\n- Personally, I found the manuscript a bit hard to follow in some parts, limiting the overall readability. For example, the notation is first introduced in the appendix (and I suggest to introduce it at least before section 3.1). Moreover, I suggest to re-phrase some overstating sentences, e.g. on page 3 where it is reported that \" [...] there is usually a significant information loss on the last hidden layer, which makes it impossible to apply this representation as is. We have identified the main causes of this: a strong correlation of neurons and an insufficient size of the last hidden layer. [...]\". While the intuition is clear here, I could not see in the paper a clear evidence of the fact that the information loss in the last hidden layer is caused by the correlation of small-sized layers.\n\n-- EDIT: I am thankful to the authors for their insightful answer to my concerns, and for the though work in reviewing the manuscript. At this stage, I would keep my score (also in light of the other comments), but I hope that the authors find the suggestions from all the reviewers useful to re-present a more consolidated work soon.", "extra_scores": {}, "extra_text": {}}, {"review_id": "PBBk9cCtxFR", "reviewer": "AnonReviewer5", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper proposed two methods for reducing the correlation of neurons within a layer so that the effective dimension of the last layer can be reduced as well, and the two methods include enforcing orthogonality on weight matrices, and batchnorm after activation functions, There are many issues regarding the claims and the experiments presented in the paper. \n\n1. Is the paper trying to propose a novel way of measuring the similarity between inputs or to improve the vector representations generated from the last layer to be more reflective and indicative in terms of the fine-grained structure of the data? Those two topics seemed intertwined with each other, and also it seemed that the authors are using one as the supporting argument for the other. \n\nFor the first one, the authors claimed that the reasons for the information loss from bottom to the top layer in a neural networks include (1) correlation of neurons, and (2) insufficient width of the last hidden layer. \n\nLet's assume that both reasons are valid, then why bother using neural networks for comparing similarity between datasets? One can certainly consider kernel methods for comparison, and specifically kernel two-sample tests with characteristic or universal kernels. \n\nIMO, I don't agree with the reasons listed by the authors. In fact, let's take VGGNet and ResNet34 pretrained on ImageNet for comparison. The dimension of the top layer of VGGNet is 4096, while it is 512 for ResNet34, and as known, ImageNet classification task has 1000 different labels. It means that those 1000 512-D vectors learnt in the top layer in ResNet34 are correlated with each other to some degree, while those 1000 4096-D vectors from VGGNet could be orthogonal to each other. However, as shown, ResNet34 generalises better than VGGNet not only on ImageNet tasks, but also on other vision-related tasks. In this comparison, clearly the architecture matters more than the dimension of the last layer and the correlation of neurons.\n\nFor the second one, let's consider a linear system, ridge regression in particular. By adding a certain level of l2 regularisation on the parameter vector or matrix, it effectively moves the covariance matrix of the input data close to an identity matrix, which improves the orthogonality of the covariance matrix. It seems that adding l2 regularisation is a simpler way than ones proposed in this paper to reduce the correlation of neurons. \n\n2. The example presented in Fig 1. is not a valid piece of evidence for the claims made in this paper.\n\nFrom the perspective of information theory, the definition of information is the thing/observation that reduces the uncertainty. It heavily depends on the subject of the stufy. Therefore, from classification perspective, one can say that, in Fig. 1a, the model has observed enough amount of information so that the uncertainty of the two neurons' behaviours is drastically reduced, while Fig. 2b still presents a large amount of uncertainty. The paper referred to the term 'information' many times without consolidating specific occasions. \n\n3. The activation function applied in the experiments is tanh, which squashes values to be in between -1 and 1, and the authors proposed to apply BatchNorm after tanh activation function. I don't see how the batchnorm after tanh is helpful in anyways as both mean and variance of the neurons are bounded already.\n\n4. The comparison conducted on the MNIST dataset seemed missing some fundimental vectorisation and quantisation methods, including both learning-based or non-learning-based. If the goal was to find a low-D vector representation of data, one can consider many approaches without using neural networks. This comment goes back to my first point in a way that I am not super clear about what the paper is trying to solve. ", "extra_scores": {}, "extra_text": {}}, {"review_id": "AXISW-U9RxY", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "I love the problem of neural information decorrelation which is so widespread in neural systems precisely for the reasons that the authors point out to detect novel information.\nDespite the paper is nicely written, it’s perhaps not best organized, because I had to go back and forth to find out the main author’s contributions. I think the main contribution of the authors is combining random projection initialization, training with orthonormality regularization (as proposed by others) and drop out. The authors also claim that they can reduce the dimensionality by applying Lemma 1.\n\nThe results in Fig. 3 illustrate how the model achieves decent to batch normalization but with a flatter distribution of correlations, which I agree with the authors is the goal.\n\nThe paper could benefit from a small summary at the introduction that lists with clarity the components of the model in a succinct manner. \n\nThe three models compared should be explained better in the experimental section. So, people can read the paper faster.\n\nI’d be interesting to know what component on the model is more important and for what. Is it the regularization, is it the random projection initialization used for faster convergence and information compression? Is it the dropout necessary to ensure a flatter distribution of correlations? An experimental section illustrating the impact on the model performance with each of these components would help us learn more.\n", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Techniques are introduced for improving representation learning
capabilities of neural networks, and the result is interpreted in
terms of random projections.

In further discussion, even the reviewer with the highest grade said
that the paper does not yet have enough clarity to address the
reviewers' comments. Particularly important would be to isolate the
causal impact of the proposed components in the final result. But also
several technical details would need to be clarified including
comparing to simple l2 regularization and precise implications of Fig
1.

Positive aspects: The problem of learning representations and
decorrelation is of course important. The authors have imagination,
and the authors are encouraged to improve the ideas by taking the
reviewer feedback into account.

Decision: Reject