Original: C LASS 2S IMI : A N EW P ERSPECTIVE ON L EARNING WITH L ABEL N OISE
Revised: C LASS 2S IMI : A N EW P ERSPECTIVE ON L EARNING WITH L ABEL N OISE

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Label noise is ubiquitous in the era of big data.
Revised: Label noise is ubiquitous in the era of big data.

Original: Deep learning algorithms can easily ﬁt the noise and thus cannot generalize well without properly modeling the noise.
Revised: Deep learning algorithms can easily ﬁt the noise and thus cannot generalize well without properly modeling the noise.

Original: In this paper, we propose a new perspective on dealing with label noise called “ Class2Simi ”.
Revised: In this paper, we propose a new perspective on dealing with label noise called “ Class2Simi ”.

Original: Speciﬁcally, we transform the training examples with noisy class labels into pairs of examples with noisy similarity labels, and propose a deep learning framework to learn robust classiﬁers with the noisy similarity labels.
Revised: Speciﬁcally, we transform the training examples with noisy class labels into pairs of examples with noisy similarity labels, and propose a deep learning framework to learn robust classiﬁers with the noisy similarity labels.

Original: Note that a class label shows the class that an instance belongs to; while a similarity label indicates whether or not two instances belong to the same class.
Revised: Note that a class label shows the class that an instance belongs to; while a similarity label indicates whether or not two instances belong to the same class.

Original: It is worthwhile to perform the transformation: We prove that the noise rate for the noisy similarity labels is lower than that of the noisy class labels, because similarity labels themselves are robust to noise.
Revised: It is worthwhile to perform the transformation: We prove that the noise rate for the noisy similarity labels is lower than that of the noisy class labels, because similarity labels themselves are robust to noise.

Original: For example, given two instances, even if both of their class labels are incorrect, their similarity label could be correct.
Revised: For example, given two instances, even if both of their class labels are incorrect, their similarity label could be correct.

Original: Due to the lower noise rate, Class2Simi achieves remarkably better classiﬁcation accuracy than its baselines that directly deals with the noisy class labels.
Revised: Due to the lower noise rate, Class2Simi achieves remarkably better classiﬁcation accuracy than its baselines that directly deals with the noisy class labels.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: It is expensive to label large-scale data accurately.
Revised: It is expensive to label large-scale data accurately.

Original: Therefore, cheap datasets with label noise are ubiquitous in the era of big data.
Revised: Therefore, cheap datasets with label noise are ubiquitous in the era of big data.

Original: However, label noise will degenerate the performance of trained deep models, because deep networks will easily overﬁt label noise (Zhang et al., 2017; Zhong et al., 2019; Li et al., 2019; Yi & Wu, 2019;
Revised: However, label noise will degenerate the performance of trained deep models, because deep networks will easily overﬁt label noise (Zhang et al., 2017; Zhong et al., 2019; Li et al., 2019; Yi & Wu, 2019;

Original: Zhang et al., 2019; 2018; Xia et al., 2019; 2020).
Revised: Zhang et al., 2019; 2018; Xia et al., 2019; 2020).

Original: In this paper, we propose a new perspective on handling label noise called “ Class2Simi ”, i.e., transformingthe training examples with noisy class labels into pairs of examples with noisy similarity labels.
Revised: In this paper, we propose a new perspective on handling label noise called “ Class2Simi ”, i.e., transforming training examples with noisy class labels into pairs of examples with noisy similarity labels.

Original: A class label shows the class that an instance belongs to, while a similarity label indicates whether or not two instances belong to the same class.
Revised: A class label shows the class that an instance belongs to, while a similarity label indicates whether or not two instances belong to the same class.

Original: This transformation is motivated by the observation that the noise rate becomes lower, e.g., even if two instances have incorrect class labels, their similarity label could be correct.
Revised: This transformation is motivated by the observation that the noise rate becomes lower, e.g., even if two instances have incorrect class labels, their similarity label could be correct.

Original: In the label-noise learning community,  lower noise rate usually results in higher classiﬁcation performance (Bao et al., 2018; Han et al., 2018b; Patrini et al., 2017).
Revised: In the label-noise learning community, a lower noise rate usually results in higher classiﬁcation performance (Han et al., 2018b; Patrini et al., 2017).

Original: Speciﬁcally, we illustrate the transformation and the robustness of similarity labels in Figure 1.
Revised: Speciﬁcally, we illustrate the transformation and the robustness of similarity labels in Figure 1.

Original: Assume we have eight noisy examples { ( x 1 , ¯ y 1 ) , . . .
Revised: Assume we have eight noisy examples { ( x 1 , ¯ y 1 ) , . . .

Original: , ( x 8 , ¯ y 8 ) } as shown in the upper part of the middle column.
Revised: , ( x 8 , ¯ y 8 ) } as shown in the upper part of the middle column.

Original: Their labels are of four classes, i.e., { 1 , 2 , 3 , 4 } .
Revised: Their labels are of four classes, i.e., { 1 , 2 , 3 , 4 } .

Original: The labels marked in red are incorrect labels.
Revised: The labels marked in red are incorrect labels.

Original: We transform the 8 examples into 8 × 8 example-pairs with noisy similarity labels as shown in the bottom part of the middle column, where the similarity label 1 means the two instances have the same class label and 0 means the two instances have different class labels.
Revised: We transform the 8 examples into 8 × 8 example-pairs with noisy similarity labels as shown in the bottom part of the middle column, where the similarity label 1 means the two instances have the same class label and 0 means the two instances have different class labels.

Original: We present the latent clean class labels and similarity labels in the left column.
Revised: We present the latent clean class labels and similarity labels in the left column.

Original: In the middle column, we can see that although the instances x 2 and x 4 both have incorrect class labels, the similarity label of the example-pair ( x 2 , x 4 ) is correct.
Revised: In the middle column, we can see that although the instances x 2 and x 4 both have incorrect class labels, the similarity label of the example-pair ( x 2 , x 4 ) is correct.

Original: Similarity labels are robust because they further consider the information of pairwise relationship.
Revised: Similarity labels are robust because they further consider the information on the pairwise relationship.

Original: We prove that the noise rate in the noisy similarity labels is lower than that of the noisy class labels.
Revised: We prove that the noise rate in the noisy similarity labels is lower than that of the noisy class labels.

Original: For example, if we assume that the noisy class labels in
Revised: For example, if we assume that the noisy class labels in

Original: Figure 1 are generated according to the latent clean labels and the transition matrix shown in the upper part of the right column (the ij -th entry of the matrix denotes the probability that the clean class label i ﬂips into the noisy class label j ), the noise rate for the noisy class labels is 0 .
Revised: Figure 1 are generated according to the latent clean labels and the transition matrix shown in the upper part of the right column (the ij -th entry of the matrix denotes the probability that the clean class label i ﬂips into the noisy class label j ), the noise rate for the noisy class labels is 0 .

Original: 5 while the rate for the corresponding noisy similarity labels is 0 .
Revised: 5 while the rate for the corresponding noisy similarity labels is 0 .

Original: Note that the noise rate is the ratio of the number of noisy examples to the number of total examples, which can be calculated from the noise transition matrix combined with the proportion of each class, i.e., 1 / 6 × 3 / 4 + 1 / 2 × 1 / 4 = 0 .
Revised: Note that the noise rate is the ratio of the number of incorrect labels to the number of total examples, which can be calculated from the noise transition matrix combined with the proportion of each class, i.e., 1 / 6 × 3 / 4 + 1 / 2 × 1 / 4 = 0 .

Original: If we assume the class label noise is generated according to the noise transition matrix presented in the upper part of the right column, it can be calculated that the noise rate for the noisy class labels is 0.5 while the rate for the noisy similarity labels is 0.25.
Revised: If we assume the class label noise is generated according to the noise transition matrix presented in the upper part of the right column, it can be calculated that the noise rate for the noisy class labels is 0.5 while the rate for the noisy similarity labels is 0.25.

Original: Note that the noise transition matrix for similarity labels can be calculated by exploiting the class noise transition matrix as in Theorem 1.
Revised: Note that the noise transition matrix for similarity labels can be calculated by exploiting the class noise transition matrix as in Theorem 1.

Original: It is obvious that Class2Simi suffers information loss because we can not recover the class labels from similarity labels.
Revised: It is obvious that Class2Simi suffers information loss because we can not recover the class labels from similarity labels.

Original: However, since the similarity labels are more robust to noise than the class labels, the advantage of the reduction of noise rate overweighs the disadvantage of the loss of information.
Revised: However, since the similarity labels are more robust to noise than the class labels, the advantage of the reduction of noise rate overweighs the disadvantage of the loss of information.

Original: Intuitively, in the learning process, it is the signal in the information that enhances the performance of the model, while the noise in the information is harmful for the model.
Revised: Intuitively, in the learning process, it is the signal in the information that enhances the performance of the model, while the noise in the information is harmful to the model.

Original: Through Class2Simi, although the total amount of information is reduced, the signal to noise ratio is increased, and so would be the total amount of signals.
Revised: Through Class2Simi, although the total amount of information is reduced, the signal to noise ratio is increased, and so would be the total amount of signals.

Original: Thus, we can beneﬁt from the transformation and achieve better performances.
Revised: Thus, we can beneﬁt from the transformation and achieve better performances.

Original: Theorem 2 and the experimental results will verify the effectiveness of this transformation.
Revised: Theorem 2 and the experimental results will verify the effectiveness of this transformation.

Original: It remains unsolved how to learn a robust classiﬁer from the data with transformed noisy similarity labels.
Revised: It remains unsolved how to learn a robust classiﬁer from the data with transformed noisy similarity labels.

Original: To solve this problem, we ﬁrst estimate the similarity noise transition matrix, a 2 × 2 matrix whose entries denote the ﬂip rates of similarity labels.
Revised: To solve this problem, we ﬁrst estimate the similarity noise transition matrix, a 2 × 2 matrix whose entries denote the ﬂip rates of similarity labels.

Original: Note that the transition matrix bridges the noisy similarity posterior and the clean similarity posterior.
Revised: Note that the transition matrix bridges the noisy similarity posterior and the clean similarity posterior.

Original: The noisy similarity posterior can be learned from the data with noisy similarity labels.
Revised: The noisy similarity posterior can be learned from the data with noisy similarity labels.

Original: Then, given the similarity noise transition matrix, we can infer the clean similarity posterior from the noisy similarity posterior.
Revised: Then, given the similarity noise transition matrix, we can infer the clean similarity posterior from the noisy similarity posterior.

Original: Since the clean similarity posterior is approximated by the inner product of the clean class posterior (Hsu et al., 2019), the clean class posterior (and thus the robust classiﬁer) can thereby be learned.
Revised: Since the clean similarity posterior is approximated by the inner product of the clean class posterior (Hsu et al., 2019), the clean class posterior (and thus the robust classiﬁer) can thereby be learned.

Original: We will empirically show that Class2Simi with the estimated similarity noise transition matrix will remarkably outperform the baselines even given with the ground-truth class noise transition matrix.
Revised: We will empirically show that Class2Simi with the estimated similarity noise transition matrix will remarkably outperform the baselines even given with the ground-truth class noise transition matrix.

Original: The contributions of this paper are summarized as follows:
Revised: The contributions of this paper are summarized as follows:

Original: • We propose a new perspective on learning with label noise, which transforms class labels into similarity labels.
Revised: • We propose a new perspective on learning with label noise, which transforms class labels into similarity labels.

Original: Such  transformationnot only reduces the noise level but also introduces the transition matrix more robust against estimation errors.
Revised: Such a transformation reduces the noise level.

Original: • We provide a way to estimate the similarity noise transition matrix by theoretically establishing its relation to  class noise transition matrix.
Revised: • We provide a way to estimate the similarity noise transition matrix by theoretically establishing its relation to the class noise transition matrix.

Original: We show that even if the class noise transition matrix is inaccurately estimated, the induced similarity noise transition matrix still works well.
Revised: We show that even if the class noise transition matrix is inaccurately estimated, the induced similarity noise transition matrix still works well.

Original: • We design a deep learning method to learn robust classiﬁers from data with noisy similarity labels and theoretically analyze its generalization ability.
Revised: • We design a deep learning method to learn robust classiﬁers from data with noisy similarity labels and theoretically analyze its generalization ability.

Original: • We empirically demonstrate that the proposed method remarkably surpasses the baselines on many datasets with both synthetic noise and real-world noise.
Revised: • We empirically demonstrate that the proposed method remarkably surpasses the baselines on many datasets with both synthetic noise and real-world noise.

Original: The rest of this paper is organized as follows.
Revised: The rest of this paper is organized as follows.

Original: In Section 2, we formalize the noisy multi-class classiﬁcation problem, and in Section 3, we propose the Class2Simi strategy and practical implementation.
Revised: In Section 2, we formalize the noisy multi-class classiﬁcation problem, and in Section 3, we propose the Class2Simi strategy and practical implementation.

Original: Experimental results are discussed in Section 4.
Revised: Experimental results are discussed in Section 4.

Original: We conclude our paper in Section 5.
Revised: We conclude our paper in Section 5.

Original:  P ROBLEM S ETUP AND R ELATED W ORK
Revised:  P ROBLEM S ETUP AND R ELATED W ORK

Original: Let ( X, Y ) ∈
Revised: Let ( X, Y ) ∈

Original: X × { 1 , . .
Revised: X × { 1 , . .

Original: , C } be the random variables for instances and clean labels, where
Revised: , C } be the random variables for instances and clean labels, where

Original: X represents the instance space and C is the number of classes.
Revised: X represents the instance space and C is the number of classes.

Original: However, in many real-world applications (Zhang et al., 2017; Zhong et al., 2019; Li et al., 2019; Yi & Wu, 2019;
Revised: However, in many real-world applications (Zhang et al., 2017; Zhong et al., 2019; Li et al., 2019; Yi & Wu, 2019;

Original: Zhang et al., 2019; Tanno et al., 2019; Zhang et al., 2018), the clean labels cannot be observed.
Revised: Zhang et al., 2019; Tanno et al., 2019; Zhang et al., 2018), the clean labels cannot be observed.

Original: The observed labels are noisy.
Revised: The observed labels are noisy.

Original: Let ¯ Y be the random variable for the noisy labels.
Revised: Let ¯ Y be the random variable for the noisy labels.

Original: What we have is a sample { ( x 1 , ¯ y 1 ) , . . .
Revised: What we have is a sample { ( x 1 , ¯ y 1 ) , . . .

Original: , ( x n , ¯ y n ) } drawn from the noisy distribution D ρ of the random variables ( X, ¯ Y ) .
Revised: , ( x n , ¯ y n ) } drawn from the noisy distribution D ρ of the random variables ( X, ¯ Y ) .

Original: Our aim is to learn a robust classiﬁer that could assign clean labels to test data by exploiting the sample with noisy labels.
Revised: Our aim is to learn a robust classiﬁer that could assign clean labels to test data by exploiting the sample with noisy labels.

Original: Existing methods for learning with noisy labels can be divided into two categories: algorithms that result in statistically inconsistent or consistent classiﬁers.
Revised: Existing methods for learning with noisy labels can be divided into two categories: algorithms that result in statistically inconsistent or consistent classiﬁers.

Original: Methods in the ﬁrst category usually employ heuristics to reduce the side-effect of noisy labels, e.g., selecting reliable examples (Yu et al., 2019;
Revised: Methods in the ﬁrst category usually employ heuristics to reduce the side-effect of noisy labels, e.g., selecting reliable examples (Yu et al., 2019;

Original: Han et al., 2018b; Malach & Shalev-Shwartz, 2017), reweighting examples (Ren et al., 2018; Jiang et al., 2018; Ma et al., 2018; Kremer et al., 2018; Tanaka et al., 2018;
Revised: Han et al., 2018b; Malach & Shalev-Shwartz, 2017), reweighting examples (Ren et al., 2018; Jiang et al., 2018; Ma et al., 2018; Kremer et al., 2018; Tanaka et al., 2018;

Original: Reed et al., 2015), employing side information (Vahdat, 2017; Li et al., 2017; Berthon et al., 2020), and adding regularization (Han et al., 2018a;
Revised: Reed et al., 2015), employing side information (Vahdat, 2017; Li et al., 2017; Berthon et al., 2020), and adding regularization (Han et al., 2018a;

Original: Guo et al., 2018; Veit et al., 2017; Vahdat, 2017; Li et al., 2017).
Revised: Guo et al., 2018; Veit et al., 2017; Vahdat, 2017; Li et al., 2017).

Original: Those methods empirically work well in many settings.
Revised: Those methods empirically work well in many settings.

Original: Methods in the second category aim to learn robust classiﬁers that could converge to the optimal ones deﬁned by using clean data.
Revised: Methods in the second category aim to learn robust classiﬁers that could converge to the optimal ones deﬁned by using clean data.

Original: They utilize the noise transition matrix, which denotes the probabilities that the clean labels ﬂip into noisy labels, to build consistent algorithms (Goldberger & Ben-Reuven, 2017; Patrini et al., 2017; Thekumparampil et al., 2018;
Revised: They utilize the noise transition matrix, which denotes the probabilities that the clean labels ﬂip into noisy labels, to build consistent algorithms (Goldberger & Ben-Reuven, 2017; Patrini et al., 2017; Thekumparampil et al., 2018;

Original: Yu et al., 2018; Liu & Guo, 2020; Zhang & Sabuncu, 2018; Kremer et al., 2018; Liu & Tao, 2016;
Revised: Yu et al., 2018; Liu & Guo, 2020; Zhang & Sabuncu, 2018; Kremer et al., 2018; Liu & Tao, 2016;

Original: Northcutt et al., 2017; Scott, 2015; Natarajan et al., 2013; Yao et al., 2020b).
Revised: Northcutt et al., 2017; Scott, 2015; Natarajan et al., 2013; Yao et al., 2020b).

Original: The idea is that given the noisy class posterior probability and the noise transition matrix, the clean class posterior probability can be inferred.
Revised: The idea is that given the noisy class posterior probability and the noise transition matrix, the clean class posterior probability can be inferred.

Original: Note that the noisy class posterior and the noise transition matrix can be estimated by exploiting the noisy data, where the noise transition matrix additionally needs anchor points (Liu & Tao, 2016;
Revised: Note that the noisy class posterior and the noise transition matrix can be estimated by exploiting the noisy data, where the noise transition matrix additionally needs anchor points (Liu & Tao, 2016;

Original: Patrini et al., 2017).
Revised: Patrini et al., 2017).

Original: Some methods assume anchor points have already been given (Yu et al., 2018).
Revised: Some methods assume anchor points have already been given (Yu et al., 2018).

Original: There are also methods showing how to identify anchor points from the noisy training data (Liu &
Revised: There are also methods showing how to identify anchor points from the noisy training data (Liu &

Original: Tao, 2016; Patrini et al., 2017).
Revised: Tao, 2016; Patrini et al., 2017).

Original:  C LASS 2S IMI MEETS NOISY SUPERVISION
Revised:  C LASS 2S IMI MEETS NOISY SUPERVISION

Original: In this section, we propose a new strategy for learning from noisy data.
Revised: In this section, we propose a new strategy for learning from noisy data.

Original: Our core idea is to transform class labels to similarity labels ﬁrst, and then handle the noise manifested on similarity labels.
Revised: Our core idea is to transform class labels to similarity labels ﬁrst, and then handle the noise manifested on similarity labels.

Original: 3. T RANSFORMATION ON LABELS AND THE TRANSITION MATRIX
Revised: 3. T RANSFORMATION ON LABELS AND THE TRANSITION MATRIX

Original: As in Figure 1, we combine every 2 instances in pairs, and if the two instances have the same class label, we assign this pair a similarity label 1 , otherwise 0 .
Revised: As in Figure 1, we combine every 2 instances in pairs, and if the two instances have the same class label, we assign this pair a similarity label 1 , otherwise 0 .

Original: If the class labels are corrupted, the generated similarity labels also contain noise.
Revised: If the class labels are corrupted, the generated similarity labels also contain noise.

Original: We denote the clean and noisy similarity label of the example-pair ( x i , x j ) by H ij and ¯ H ij respectively.
Revised: We denote the clean and noisy similarity labels of the example-pair ( x i , x j ) by H ij and ¯ H ij respectively.

Original: The deﬁnition of the similarity noise transition matrix is similar to the class one, denoting the probabilities that clean similarity labels ﬂip into noisy similarity labels, i.e., T s,mn = P
Revised: The deﬁnition of the similarity noise transition matrix is similar to the class one, denoting the probabilities that clean similarity labels ﬂip into noisy similarity labels, i.e., T s,mn = P

Original: ( ¯ H ij = n | H ij = m ) .
Revised: ( ¯ H ij = n | H ij = m ) .

Original: The dimension of the similarity noise transition matrix is always 2 × 2 .
Revised: The dimension of the similarity noise transition matrix is always 2 × 2 .

Original: Since the similarity labels are generated from class labels, the similarity noise is also determined and, thus can be calculated, by the class noise transition matrix.
Revised: Since the similarity labels are generated from class labels, the similarity noise is also determined and, thus can be calculated, by the class noise transition matrix.

Original: Theorem 1.
Revised: Theorem 1.

Original: Assume that the dataset is balanced (each class has the same amount of samples) [Equation] and the noise is class-dependent.
Revised: Assume that the dataset is balanced (each class has the same amount of samples, and c classes in total), and the noise is class-dependent.

Original: Given a class noise transition matrix T c , such that T c,ij = P ( ¯ Y =
Revised: Given a class noise transition matrix T c , such that T c,ij = P ( ¯ Y =

Original: j | Y = i ) 
Revised: j | Y = i ) .

Original: [Equation]

Original: The elements of the corresponding similarity noise transition matrix T s can be calculated as
Revised: The elements of the corresponding similarity noise transition matrix T s can be calculated as

Original: [Equation]
Revised: [Equation]

Original: A detailed proof is provided in Appendix A.
Revised: provided in Appendix A

Original: Remark 1.
Revised: Remark 1.

Original: Theorem 1 can easily extend to the setting where the dataset is unbalanced in class by multiplying each T c,ij by a coefﬁcient n i .
Revised: Theorem 1 can easily extend to the setting where the dataset is unbalanced in classes by multiplying each T c,ij by a coefﬁcient n i .

Original: n i is the number of examples from the i -th class.
Revised: n i is the number of examples from the i -th class.

Original: Note thatsince the similarity labels are only dependent on class labels, if the class noise is classdependent, the similarity noise is also “class-dependent” (class means similar and dissimilar).
Revised: Note that the similarity labels are only dependent on class labels. If the class noise is class-dependent, the similarity noise is also “class-dependent” (class means similar and dissimilar).

Original: {+Under classdependent label noise, a binary classiﬁcation is learnable as long as T 
+}
Revised: {+Under classdependent label noise, a binary classiﬁcation is learnable as long as T 
+}

Revised: + T 11 > 1 (Menon et al., 2015), where T is the corresponding binary transition matrix; a multi-class classiﬁcation is learnable if the corresponding transition matrix T c is invertible.

Original: Theorem 2.{+For Class2Simi, in the most general sense, i.e., T c is invertible, T s, 
 + T s, 11 > 1 holds. Namely, the learnability of the pointwise classiﬁcation implies the learnability of the reduced pairwise classiﬁcation. However, the latter cannot imply the former.+}
Revised: {+For Class2Simi, in the most general sense, i.e., T c is invertible, T s, 
 + T s, 11 > 1 holds. Namely, the learnability of the pointwise classiﬁcation implies the learnability of the reduced pairwise classiﬁcation. However, the latter cannot imply the former.+}

Revised: A proof and a counterexample are provided in Appendix F.

Original:  Assume that the dataset is balanced (each class has the same amount of samples), and the noise is class-dependent.
Revised: Theorem 2. Assume that the dataset is balanced (each class has the same amount of samples), and the noise is class-dependent.

Original: When the number of classes c ≥ 8 1 , the noise rate for the noisy similarity labels is lower than that of the noisy class labels.
Revised: When the number of classes c ≥ 8 1 , the noise rate for the noisy similarity labels is lower than that of the noisy class labels.

Original: A detailed proof is provided in Appendix B.
Revised: A detailed proof is provided in Appendix B.

Original: When dealing with label noise,  low noise rate has many beneﬁts.
Revised: When dealing with label noise, a low noise rate has many beneﬁts.

Original: The most important one is that the noise-robust algorithms will consistently achieve higher performance when the noise rate is lower (Bao et al., 2018; Han et al., 2018b; Xia et al., 2019;
Revised: The most important one is that the noise-robust algorithms will consistently achieve higher performance when the noise rate is lower (Bao et al., 2018; Han et al., 2018b; Xia et al., 2019;

Original: Patrini et al., 2017).
Revised: Patrini et al., 2017).

Original: Another beneﬁt is that, when the noise rate is low, the complex instance-dependent label noise can be well approximated by class-dependent label noise (Cheng et al., 2020), which is easier to handle.
Revised: Another beneﬁt is that, when the noise rate is low, the complex instance-dependent label noise can be well approximated by class-dependent label noise (Cheng et al., 2020), which is easier to handle.

Revised: After the Class2Simi transformation, the number of dissimilar pairs is ( c − 1) times as much as that of similar pairs.

Revised: Meanwhile, compared with the original noise rate of class labels, the noise rate of similar pairs (the ratio of the number of mislabeled similar pairs to the number of total real similar pairs) is higher and the noise rate of dissimilar pairs is lower, while the overall noise rate of pairwise examples is lower, which partially reﬂects that the impact of the label noise is less bad.

Revised: Moreover, the ﬂip from dissimilar to similar should be more adversarial and thus more important.

Revised: In practice, it is common that one class has more than one clusters, while it is rare that two or more classes are in the same cluster.

Revised: If there is a ﬂip from similar to dissimilar and based on it we split a (latent) cluster into two (latent) clusters, we still have a high chance to label these two clusters correctly later.

Original:  3. L EARNING WITH NOISY SIMILARITY LABELS
Revised: If there is a ﬂip from dissimilar to similar and based on it we join two clusters belonging to two classes into a single cluster, we nearly have zero chance to label this cluster correctly later. As a consequence, the ﬂip from dissimilar to similar is more adversarial, and thus more important. To sum up, considering the reduction of the overall noise rate is meaningful. 3. L EARNING WITH NOISY SIMILARITY LABELS

Original: In order to learn a multi-class classiﬁer from similarity labeled data, we should establish relationships between class posterior probability and similarity posterior probability.
Revised: In order to learn a multi-class classiﬁer from similarity labeled data, we should establish relationships between class posterior probability and similarity posterior probability.

Original: Here we employ the relationship established in (Hsu et al., 2019), which is derived from a likelihood model.
Revised: Here we employ the relationship established in (Hsu et al., 2019), which is derived from a likelihood model.

Original: As in Figure  2, they denote the predicted clean similarity posterior by the inner product between two categorical distributions: ˆ S ij = f
Revised: As in Figure [Figure] 2, they denote the predicted clean similarity posterior by the inner product between two categorical distributions: ˆ S ij = f

Original: ( X i; θ ) (cid:62) f
Revised: ( X i ) (cid:62) f

Original: ( X j; θ ) .
Revised: ( X j ) .

Original: Intuitively, f ( X; θ ) outputs the predicted categorical distribution of instance X and f ( X i; θ ) (cid:62) f
Revised: Intuitively, f ( X ) outputs the predicted categorical distribution of input data X and f ( X i ) (cid:62) f

Original: ( X j; θ ) can measure how similar the two distributions are.
Revised: ( X j ) can measure how similar the two distributions are.

Original: For clarity, we visualize the predicted similarity posterior in Figure 3.
Revised: For clarity, we visualize the predicted similarity posterior in Figure 3.

Original: If X i and X j are predicted belonging to the same class, i.e., argmax m ∈ C
Revised: If X i and X j are predicted belonging to the same class, i.e., argmax m ∈ C

Original: f m ( X i; θ ) = argmax n ∈ C f n ( X j; θ ) , the predicted similarity posterior should be relatively high ( ˆ S ij = 0 .
Revised: f m ( X i ) = argmax n ∈ C f n ( X j ) , the predicted similarity posterior should be relatively high ( ˆ S ij = 0 .

Original: 30 in Figure 3(a)).
Revised: 30 in Figure 3(a)).

Original: By contrast, if X i and X j are predicted belonging to different classes, the predicted similarity posterior should be relatively low ( ˆ S ij = 0 .
Revised: By contrast, if X i and X j are predicted belonging to different classes, the predicted similarity posterior should be relatively low ( ˆ S ij = 0 .

Original: 0654 in Figure 3(b)).
Revised: 0654 in Figure 3(b)).

Original: Note that the noisy similarity posterior distribution P ( ¯ H ij |
Revised: Note that the noisy similarity posterior distribution P ( ¯ H ij |

Original: X i , X j ) and clean similarity posterior distribution P ( H ij | X i , X j ) satisfy
Revised: X i , X j ) and clean similarity posterior distribution P ( H ij | X i , X j ) satisfy

Original: [Equation]
Revised: [Equation]

Original: Therefore, we can infer noisy similarity posterior ˆ¯ S ij from clean similarity posterior ˆ S ij with  similarity noise transition matrix.
Revised: Therefore, we can infer noisy similarity posterior ˆ¯ S ij from clean similarity posterior ˆ S ij with the similarity noise transition matrix.

Original: To measure the error between the predicted noisy similarity posterior ˆ¯ S ij and noisy similarity label ¯ H ij , we employ a binary cross-entropy loss function (Shannon, 1948).
Revised: To measure the error between the predicted noisy similarity posterior ˆ¯ S ij and noisy similarity label ¯ H ij , we employ a binary cross-entropy loss function (Shannon, 1948).

Original: The ﬁnal optimization function is
Revised: The ﬁnal optimization function is

Original: [Equation]
Revised: [Equation]

Original: [Figure]

Original: The pipeline of the proposed Class2Simi is summarized in Figure 2.
Revised: The pipeline of the proposed Class2Simi is summarized in Figure 2.

Original: The softmax function outputs an estimation for the clean class posterior, i.e., f ( X ) =
Revised: The softmax function outputs an estimation for the clean class posterior, i.e., f ( X ) =

Original: ˆ P ( Y | X ) , where ˆ P ( Y | X ) denotes the estimated class posterior.
Revised: ˆ P ( Y | X ) , where ˆ P ( Y | X ) denotes the estimated class posterior.

Original: Then a pairwise enumeration layer (Hsu et al., 2018) is added to calculate the predicted clean similarity posterior ˆ S ij of every two instances.
Revised: Then a pairwise enumeration layer (Hsu et al., 2018) is added to calculate the predicted clean similarity posterior ˆ S ij of every two instances.

Original: According to Equation 1, by pre-multiplying the transpose of the noise similarity transition matrix, we can obtain the predicted noisy similarity posterior ˆ¯ S ij .
Revised: According to Equation 1, by pre-multiplying the transpose of the noise similarity transition matrix, we can obtain the predicted noisy similarity posterior ˆ¯ S ij .

Original: Therefore, by minimizing L c 2 s , we can learn a classiﬁer for predicting noisy similarity labels.
Revised: Therefore, by minimizing L c 2 s , we can learn a classiﬁer for predicting noisy similarity labels.

Original: Meanwhile, before the transition matrix layer, the pairwise enumeration layer will output a prediction for the clean similarity posterior, which guides f ( X ) to predict clean class labels.
Revised: Meanwhile, before the transition matrix layer, the pairwise enumeration layer will output a prediction for the clean similarity posterior, which guides f ( X ) to predict clean class labels.

Original: 3. I MPLEMENTATION
Revised: 3. I MPLEMENTATION

Original: The proposed algorithm is summarized in Algorithm 1.
Revised: The proposed algorithm is summarized in Algorithm 1.

Original: Since learning only from similarity labels will lose the mapping between the output nodes and the semantic classes, we load the model trained on  data with noisy class labels to learn the class information in Stage 2.
Revised: Since learning only from similarity labels will lose the mapping between the output nodes and the semantic classes, we load the model trained on the data with noisy class labels to learn the class information in Stage 2.

Original: It is worthwhile to mention that Class2Simi increases the computation cost slightly.
Revised: It is worthwhile to mention that Class2Simi increases the computation cost slightly.

Original: Note that the transformation of labels is during the training phase rather than before training.
Revised: Note that the transformation of labels is during the training phase rather than before training.

Original: Speciﬁcally, as in Figure 2, ﬁrst we read a batch of n instances, and generate their corresponding n 2 similarity labels.
Revised: Speciﬁcally, as in Figure 2, ﬁrst, we read a batch of n examples, and generate their corresponding n 2 similarity labels.

Original: Since n is the batchsize, it is usually small.
Revised: Since n is the batch size, it is usually small.

Original: In addition, we only save the labels, not example-pairs, such that it introduces a negligible memory overhead.
Revised: In addition, we only save the labels, not example-pairs, such that it introduces a negligible memory overhead.

Original: Then the neural network outputs the class posterior probabilities of n single instances in the batch of data.
Revised: Then the neural network outputs the class posterior probabilities of n single examples in the batch of data.

Original: After that pairwise enumeration layer calculates the inner products between every two instances, outputting n 2 predicted similarity posterior probabilities.
Revised: After that pairwise enumeration layer calculates the inner products between every two instances, outputting n 2 predicted similarity posterior probabilities.

Original: Then the similarity transition matrix corrects the n 2 predicted similarity posterior probabilities.
Revised: Then the similarity transition matrix corrects the n 2 predicted similarity posterior probabilities.

Original: Finally, the loss is accumulated by n 2 items.
Revised: Finally, the loss is accumulated by n 2 items.

Original: Namely, Class2Simi only does the additional computation on generating similarity labels and calculating the inner products between every two instances in the pairwise enumeration layer, which is time-efﬁcient.
Revised: Namely, Class2Simi only does the additional computation on generating similarity labels and calculating the inner products between every two instances in the pairwise enumeration layer, which is time-efﬁcient.

Original: 3. G ENERALIZATION ERROR
Revised: 3. G ENERALIZATION ERROR

Revised: We formulate the above problem in the traditional risk minimization framework (Mohri et al., 2018).

Revised: The expected and empirical risks of employing estimator f can be deﬁned as

Revised: [Equation]

Revised: and

Original: Theorem 3.
Revised: [Equation] where n is training sample size of the noisy data. Assume that the neural network has d layers with parameter matrices W 1 , . . . , W d , and the activation functions σ 1 , . .

Revised: , σ d − 1 are Lipschitz continuous, satisfying σ j (0)

Revised: = 0 .

Revised: We denote by H : X (cid:55)→ W d σ d − 1 ( W d − 1 σ d − 2 ( . . .

Revised: σ 1 ( W 1 X )))

Revised: ∈ R the standard form of the neural network.

Revised: H = argmax i ∈{ 1 ,...,C } h i .

Revised: Then the output of the softmax function is deﬁned as f i ( X ) =

Revised: exp ( h i ( X )) / (cid:80) Cj =1 exp ( h j ( X )) , i = 1 , . . .

Revised: , C .

Revised: We can obtain the following generalization error bound as follow.

Revised: Theorem 3.

Original: Assume the parameter matrices W 1 , . .
Revised: Assume the parameter matrices W 1 , . .

Original: , W d have Frobenius norm at most M 1 , . . .
Revised: , W d have Frobenius norm at most M 1 , . . .

Original: , M d , and the activation functions are 1-Lipschitz, positive-homogeneous, and applied element-wise (such as the ReLU).
Revised: , M d , and the activation functions are 1-Lipschitz, positive-homogeneous, and applied element-wise (such as the ReLU).

Original: Assume the transition matrix is given, and the instances X are upper bounded by B , i.e., (cid:107) X (cid:107) ≤ B for all X , and the loss function (cid:96) is upper bounded by M 2 .
Revised: Assume the transition matrix is given, and the instances X are upper bounded by B , i.e., (cid:107) X (cid:107) ≤ B for all X , and the loss function (cid:96) is upper bounded by M 2 .

Original: Then, for any δ > 0 , with 
Revised: Then, for any δ > 0 , with probability at least

Original: The assumption holds because deep neural networks will always regulate the objective to be a ﬁnite value and thus the corresponding loss functions are of ﬁnite values.
Revised: [Equation]

Original: probability at least
Revised: δ ,

Original: [Equation]
Revised: [Equation]

Original: Notations and a detailed proof is provided in Appendix C.
Revised: Notation and a detailed proof are provided in Appendix C.

Original: Theorem 3 implies that if the training error is small and the training sample size is large, the expected risk R ( ˆ f ) of the representations for noisy similarity posterior will be small.
Revised: Theorem 3 implies that if the training error is small and the training sample size is large, the expected risk R ( ˆ f ) of the representations for noisy similarity posterior will be small.

Original: If the transition matrix is well estimated, the clean similarity posterior as well as the classiﬁer for the clean class will also have a small risk according to Equation 1 and the Class2Simi relations.
Revised: If the transition matrix is well estimated, the clean similarity posterior as well as the classiﬁer for the clean class will also have a small risk according to Equation 1 and the Class2Simi relations.

Original: This theoretically justiﬁes why the proposed method works well.
Revised: This theoretically justiﬁes why the proposed method works well.

Original: In the experiment section, we will show that the transition matrices are well estimated and that the proposed method signiﬁcantly outperforms the baselines.
Revised: In the experiment section, we will show that the transition matrices are well estimated and that the proposed method signiﬁcantly outperforms the baselines.

Revised: In Class2Simi, a multi-class classiﬁcation is reduced to a pairwise binary classiﬁcation.

Revised: For pairwise examples, if a surrogate loss is classiﬁcation-calibrated, minimizing it leads to minimizing the zeroone loss on the pointwise random variables in the limit case.

Revised: Otherwise, we cannot guarantee the

Revised: [Table]

Revised: worst-case learnability of learning pointwise labels from pairwise examples, but it cannot imply the average-case non-learnability either.

Original: E XPERIMENTS
Revised: Theoretically, Bao et al. (2020) proved that when the pairwise labels are all correct, for the special case c = 2 , a good model for predicting similar/dissimilar pairs must also be a good model for predicting the original classes, under mild assumptions. In practice, it seems ﬁne to use non-classiﬁcation-calibrated losses. According to Tewari & Bartlett (2007), the multi-class margin loss (i.e., one-vs-rest loss) and the pairwise comparison loss (i.e., one-vs-one loss) are proved to be non-calibrated, but they are still the main multi-class losses in Mohri et al.

Revised: Shalev-Shwartz & Ben-David (2014).

Revised:  E XPERIMENTS

Original: Datasets.
Revised: Datasets.

Original: We employ three widely used image datasets, i.e., MNIST (LeCun, 1998), CIFAR-10 , and
Revised: We employ three widely used image datasets, i.e., MNIST (LeCun, 1998), CIFAR-10 , and

Original: CIFAR-100 (Krizhevsky et al., 2009), one text dataset News20 , and one real-world noisy dataset
Revised: CIFAR-100 (Krizhevsky et al., 2009), one text dataset News20 , and one real-world noisy dataset

Original: Clothing1M (Xiao et al., 2015).
Revised: Clothing1M (Xiao et al., 2015).

Original: MNIST has 28 × 28 grayscale images of 10 classes including 60, training images and 10,000 test images.
Revised: MNIST has 28 × 28 grayscale images of 10 classes including 60, training images and 10,000 test images.

Original: CIFAR-10 and CIFAR-100 both have 32 × 32 × 3 color images including 50,000 training images and 10,000 test images.
Revised: CIFAR-10 and CIFAR-100 both have 32 × 32 × 3 color images including 50,000 training images and 10,000 test images.

Original: CIFAR-10 has 10 classes while
Revised: CIFAR-10 has 10 classes while

Original: CIFAR-100 has 100 classes.
Revised: CIFAR-100 has 100 classes.

Original: News20 is a collection of approximately 20,000 newsgroup documents, partitioned nearly evenly across 20 different newsgroups.
Revised: News20 is a collection of approximately 20,000 newsgroup documents, partitioned nearly evenly across 20 different newsgroups.

Original: Clothing1M has 1M images with real-world noisy labels and additional 50k, 14k, 10k images with clean labels for training, validation and test, and we only use noisy training set in the training phase.
Revised: Clothing1M has 1M images with real-world noisy labels and additional 50k, 14k, 10k images with clean labels for training, validation and test, and we only use noisy training set in the training phase.

Original: Note that the similarity learning method of
Revised: Note that the similarity learning method of

Original: Class2Simi is based on Cluster because there is no class information.
Revised: Class2Simi is based on Cluster because there is no class information.

Original: Intuitively, for a noisy class, if most instances in it belong to another speciﬁc class, we can hardly identify it.
Revised: Intuitively, for a noisy class, if most instances in it belong to another speciﬁc class, we can hardly identify it.

Original: For example, assume that a class with noisy labels ¯ i contains n i instances with ground-truth labels i and n j instances with ground-truth labels j .
Revised: For example, assume that a class with noisy labels ¯ i contains n i instances with ground-truth labels i and n j instances with ground-truth labels j .

Original: If n j is bigger than n i , the model will cluster class i into j .
Revised: If n j is bigger than n i , the model will cluster class i into j .

Original: Unfortunately, in
Revised: Unfortunately, in

Original: Clothing1M , most instances with label ‘5’ belong to class ‘3’ actually.
Revised: Clothing1M , most instances with label ‘5’ belong to class ‘3’ actually.

Original: Therefore, we merge the two classes, and denote the ﬁxed dataset by Clothing1M* which contains 13 classes.
Revised: Therefore, we merge the two classes, and denote the ﬁxed dataset by Clothing1M* which contains 13 classes.

Original: For all the datasets, we leave out 10% of the training examples as a validation set, which is for model selection.
Revised: For all the datasets, we leave out 10% of the training examples as a validation set, which is for model selection.

Original: Noisy class labels generation.
Revised: Noisy class labels generation.

Original: For the three clean datasets, we artiﬁcially corrupt the class labels of training and validation sets according to thenoise class  transition matrix.
Revised: For the three clean datasets, we artiﬁcially corrupt the class labels of training and validation sets according to the class noise transition matrix.

Original: Speciﬁcally, for each instance with clean label i , we replace its label by j with a probability of T c,ij .
Revised: Speciﬁcally, for each instance with clean label i , we replace its label by j with a probability of T c,ij .

Original: In this paper, we consider both symmetric and asymmetric noise settings which are deﬁned in Appendix D.
Revised: In this paper, we consider both symmetric and asymmetric noise settings which are deﬁned in Appendix D.

Original: Baselines.
Revised: Baselines.

Original: As mentioned before, Class2Simi is a strategy rather than a speciﬁc algorithm.
Revised: As mentioned before, Class2Simi is a strategy rather than a speciﬁc algorithm.

Original: In this paper, we employ three T -based method, i.e., Forward correction (Patrini et al., 2017), Reweight (Liu & Tao, 2016), and T -revision (Xia et al., 2019), which all utilize a class-dependent transition matrix to model the noise, to implement our approach to show the effectiveness of Class2Simi.
Revised: In this paper, we employ three T -based methods, i.e., Forward correction (Patrini et al., 2017), Reweight (Liu & Tao, 2016), and T -revision (Xia et al., 2019), which all utilize a class-dependent transition matrix to model the noise, to implement our approach to show the effectiveness of Class2Simi.

Original: Besides, we externally conduct experiments on Co-teaching (Han et al., 2018b), which is a representative algorithm of selecting reliable examples for training; APL (Ma et al., 2020), which applies simple normalization on loss functions and makes them robust to noisy labels; S2E (Yao et al., 2020a), which properly controls the sample selection process so that deep networks can beneﬁt from the memorization effect.For fair comparisons, all experiments are conducted on NVIDIA Tesla V100, and all methods are implemented by PyTorch. [Table]
Revised: Besides, we externally conduct experiments on Co-teaching (Han et al., 2018b), which is a representative algorithm of selecting reliable examples for training; APL (Ma et al., 2020), which applies simple normalization on loss functions and makes them robust to noisy labels; S2E (Yao et al., 2020a), which properly controls the sample selection process so that deep networks can beneﬁt from the memorization effect.

Original: Network structure and Optimizer.
Revised: Network structure and Optimizer.

Original: For MNIST , we use LeNet (LeCun et al., 1998).
Revised: For MNIST , we use LeNet (LeCun et al., 1998).

Original: For CIFAR-10 , we use ResNet-32 with pre-activation (He et al., 2016b).
Revised: For CIFAR-10 , we use ResNet-32 with pre-activation (He et al., 2016b).

Original: For CIFAR-100 , we use ResNet-56 with pre-activation (He et al., 2016b).
Revised: For CIFAR-100 , we use ResNet-56 with pre-activation (He et al., 2016b).

Original: For News20 , we use GloVe (Pennington et al., 2014) to obtain vector representations for text, and employ a 3-layer MLP with  Softsign active function.
Revised: For News20 , we use GloVe (Pennington et al., 2014) to obtain vector representations for text, and employ a 3-layer MLP with the Softsign active function.

Original: For Clothing1M* , we use pre-trained ResNet-

Revised: For Clothing1M* , we use pre-trained ResNet-

Original: (He et al., 2016a).
Revised: (He et al., 2016a).

Original: We use the same optimization method as Forward correction to learn the noise transition matrix ˆ T c .
Revised: We use the same optimization method as Forward correction to learn the noise transition matrix ˆ T c .

Original: In Stage 2, we use the Adam optimizer with initial learning rate 0.001.
Revised: In Stage 2, we use the Adam optimizer with initial learning rate 0.001.

Original: On MNIST , the batch size is 128 and the learning rate decays every 20 epochs by a factor of 0.1 with 60 epochs in total.
Revised: On MNIST , the batch size is 128 and the learning rate decays every 20 epochs by a factor of 0.1 with 60 epochs in total.

Original: On CIFAR-10 , the batch size is also 128 and the learning rate decays every 40 epochs by a factor of 0.1 with 120 epochs in total.
Revised: On CIFAR-10 , the batch size is also 128 and the learning rate decays every 40 epochs by a factor of 0.1 with 120 epochs in total.

Original: On CIFAR-100 , the batch size is 1000 and the learning rate drops at epoch 80 and 160 by a factor of 0.1 with 200 epochs in total.
Revised: On CIFAR-100 , the batch size is 1000 and the learning rate drops at epoch 80 and 160 by a factor of 0.1 with 200 epochs in total.

Original: On

Original:  News20 , the batch size is 128 and the learning rate decays every 10 epochs by a factor of 0.1 withepochs in total.
Revised: On News20 , the batch size is 128 and the learning rate decays every 10 epochs by a factor of 0.1 with 30 epochs in total.

Original: On Clothing1M* , the batch size is 32 and the learning rate drops every 5 epochs by a factor of 0.1 with 10 epochs in total.
Revised: On Clothing1M* , the batch size is 32 and the learning rate drops everyepochs by a factor of 0.1 with 10 epochs in total.

Original: Results on noisy image datasets.
Revised: Results on noisy image datasets.

Original: The results in Table 1 and  4 demonstrate that Class2Simi achieves distinguished classiﬁcation accuracy and is robust against the estimation errors on transition matrix.
Revised: The results in Table 1 and Figure 4 demonstrate that Class2Simi achieves distinguished classiﬁcation accuracy and is robust against the estimation errors on transition matrix.

Original: From Table 1, overall, we can see that after the transformation, better performances are achieved due to a lower noise rate and the similarity transition matrix being robust to noise.
Revised: From Table 1, overall, we can see that after the transformation, better performances are achieved due to a lower noise rate and the similarity transition matrix being robust to noise.

Original: Speciﬁcally,
Revised: Speciﬁcally,

Original: On MNIST , as the noise rate increases from Sym-0.1 to Sym-0.5, Forward & Class2Simi maintains remarkable accuracy above 98.20% while the accuracy of Forward decreases steadily.
Revised: On MNIST , as the noise rate increases from Sym-0.1 to Sym-0.5, Forward & Class2Simi maintains remarkable accuracy above 98.20% while the accuracy of Forward decreases steadily.

Original: On CIFAR100 , there are obvious decreases in the accuracy of all methods and our method achieves the best results across all noise rate, i.e., at Sym-0.5, Class2Simi gives accuracy uplifts of about 9.0% compared with those T -based method.
Revised: On CIFAR100 , there are obvious decreases in the accuracy of all methods and our method achieves the best results across all noise rate, i.e., at Sym-0.5, Class2Simi gives accuracy uplifts of about 9.0% compared with those T -based methods.

Original: Results under asymmetric noise are provided in Appendix E.
Revised: Results under asymmetric noise are provided in Appendix E.3.

Original: In Figure [Table] we show that the similarity noise transition matrix is robust against estimation errors.
Revised: In Figure 4, we show that the similarity noise transition matrix is robust against estimation errors.

Original: To verify this, we add some random noise to the ground-truth T c through multiplying every elements in class T c by a random variable α ij .
Revised: To verify this, we add some random noise to the ground-truth T c through multiplying every element in class T c by a random variable α ij .

Original: We control the noise rate on the T c by sampling α ij in different intervals, i [Table] e., 0.1 noise means that α ij is uniformly sampled from ± [1 . 1 , 1 . 2] .
Revised: We control the noise rate on the T c by sampling α ij in different intervals, i.e., 0.1 noise means that α ij is uniformly sampled from ± [1 . 1 , 1 . 2] .

Original: Then we normalize T c to make its row sums equal to 1.
Revised: Then we normalize T c to make its row sums equal to 1.

Original: From Figure 4, we can see that the accuracy of Forward drops dramatically with the increase of the noise on T c on three datasets.
Revised: From Figure 4, we can see that the accuracy of Forward drops dramatically with the increase of the noise on T c on three datasets.

Original: Meanwhile, there is only a slight ﬂuctuation of Forward & Class2Simi on MNIST and CIFAR10 .
Revised: Meanwhile, there is only a slight ﬂuctuation of Forward & Class2Simi on MNIST and CIFAR10 .

Original: On CIFAR100 , the accuracy of Forward & Class2Simi is affected by the noise on T c , but the decline is much lower than Forward.
Revised: On CIFAR100 , the accuracy of Forward & Class2Simi is affected by the noise on T c , but the decline is much lower than Forward.

Original: The reason is that Forward & Class2Simi needs to learn the class information from noisy data which is hard when the number of classes is large.
Revised: The reason is that Forward & Class2Simi needs to learn the class information from noisy data which is hard when the number of classes is large.

Original: [Table]
Revised: [Table]

Original: (CE uses class labels and the cross-entropy loss function.) 
Revised: (CE uses class labels and the cross-entropy loss function.) [Table] [Table] [Table]

Revised: [Table]

Original: {+25 56.17 ± 0.
+}
Revised: {+25 56.17 ± 0.
+}

Original: [Table]
Revised: [Table]

Revised: 71 ±

Original: [Table]
Revised: [Table]

Original: Results on noisy text dataset.
Revised: Results on noisy text dataset.

Original: Results in
Revised: Results in

Original: Table 2 show that the proposed strategy works well on the text dataset under both symmetric and asymmetric noise settings.
Revised: Table 2 show that the proposed strategy works well on the text dataset under both symmetric and asymmetric noise settings.

Original: Results on real-world noisy dataset.
Revised: Results on real-world noisy dataset.

Original: Results in Table 3 show that the proposed strategy signiﬁcantly improves the classiﬁcation accuracy of the T -based methods.
Revised: Results in Table 3 show that the proposed strategy signiﬁcantly improves the classiﬁcation accuracy of the T -based methods.

Original: T -based methods with Class2Simi also outperform those classic methods.
Revised: T -based methods with Class2Simi also outperform those classic methods.

Original: Ablation study.
Revised: Ablation study.

Original: To investigate how the similarity loss function inﬂuences the classiﬁcation accuracy, we conduct experiments with the cross-entropy loss function and the similarity loss function respectively on clean datasets over 3 trails where the
Revised: To investigate how the similarity loss function inﬂuences the classiﬁcation accuracy, we conduct experiments with the cross-entropy loss function and the similarity loss function respectively on clean datasets over 3 trails where the

Original: T c is set to an identity matrix.
Revised: T c is set to an identity matrix.

Original: All other settings are kept the same.
Revised: All other settings are kept the same.

Original: As shown in Table 4, the similarity loss function does not improve the classiﬁcation accuracy, which means the accuracy increase in our paper is beneﬁted from the lower noise rate and the more robust transition matrix.
Revised: As shown in Table 4, the similarity loss function does not improve the classiﬁcation accuracy, which means the accuracy increase in our paper is beneﬁted from the lower noise rate and the more robust transition matrix.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: This paper proposes a new perspective on dealing with class label noise (called Class2Simi) by transforming the training sample with noisy class labels into a training sample with noisy similarity labels.
Revised: This paper proposes a new perspective on dealing with class label noise (called Class2Simi) by transforming the training sample with noisy class labels into a training sample with noisy similarity labels.

Original: We also propose a deep learning framework to learn classiﬁers directly with the noisy similarity labels.
Revised: We also propose a deep learning framework to learn classiﬁers directly with the noisy similarity labels.

Original: The core idea is to transform class information to similarity information, which makes the noise rate lower.
Revised: The core idea is to transform class information into similarity information, which makes the noise rate lower.

Original: We also prove that not only the similarity labels but also the similarity noise transition matrix is robust to noise.
Revised: We also prove that not only the similarity labels but also the similarity noise transition matrix is robust to noise.

Original: Experiments are conducted on benchmark datasets, demonstrating the effectiveness of our method.
Revised: Experiments are conducted on benchmark datasets, demonstrating the effectiveness of our method.

Original: In future work, investigating different types of noise for diverse real-life scenarios might prove important.
Revised: In future work, investigating different types of noise for diverse real-life scenarios might prove important.

Original: A PPENDICES A P ROOF OF T HEOREM
Revised: A PPENDICES A P ROOF OF T HEOREM

Original: Theorem 1.
Revised: Theorem 1.

Original: Assume that the dataset is balanced (each class has the same amount of samples) [Equation] and the noise is class-dependent.
Revised: Assume that the dataset is balanced (each class has the same amount of samples, and c classes in total), and the noise is class-dependent.

Original: Given a class noise transition matrix T c , such that T c,ij = P ( ¯ Y =
Revised: Given a class noise transition matrix T c , such that T c,ij = P ( ¯ Y =

Original: j | Y = i ) .
Revised: j | Y = i ) .

Original: The elements of the corresponding similarity noise transition matrix T s can be calculated as
Revised: The elements of the corresponding similarity noise transition matrix T s can be calculated as

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Assume each class has n samples.
Revised: Assume each class has n samples.

Original: n 2 T c,ij T c,i (cid:48)
Revised: n 2 T c,ij T c,i (cid:48)

Original: j (cid:48) represents the number of sample-pairs generated by ( ¯ Y =
Revised: j (cid:48) represents the number of sample-pairs generated by ( ¯ Y =

Original: j | Y = i ) and ( ¯ Y = j (cid:48)
Revised: j | Y = i ) and ( ¯ Y = j (cid:48)

Original: | Y =
Revised: | Y =

Original: i (cid:48) ) .
Revised: i (cid:48) ) .

Original: For the ﬁrst element T s, 00 , n 2 (cid:80) i (cid:54) =
Revised: For the ﬁrst element T s, 00 , n 2 (cid:80) i (cid:54) =

Original: i (cid:48)
Revised: i (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) is the number of sample-pairs with clean similarity labels H = 0 , while n 2 (cid:80) i (cid:54) =
Revised: j (cid:48) is the number of sample-pairs with clean similarity labels H = 0 , while n 2 (cid:80) i (cid:54) =

Original: i (cid:48) ,j (cid:54) =
Revised: i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) is the number of example-pairs with clean similarity labels S
Revised: j (cid:48) is the number of example-pairs with clean similarity labels S

Original: = 0 and noisy similarity labels ¯ H = 0 .
Revised: = 0 and noisy similarity labels ¯ H = 0 .

Original: Thus the ratio of these two terms is exact the T s, 

Revised: Thus the ratio of these two terms is exact the T s,

Original: = P ( ¯ H = 

Revised: = P ( ¯ H =

Original: | H = 0) .
Revised: | H = 0) .

Original: The remaining three elements are obtained in the same way.
Revised: The remaining three elements can be represented in the same way.

Revised: The primal representations are as follows,

Revised: [Equation]

Revised: Further, note that (cid:88)

Revised: [Equation]

Revised: Substituting above four equations to the primal representations, we have the Theorem 1 proved.

Original: B P ROOF OF T HEOREM
Revised: B P ROOF OF T HEOREM

Original: Theorem 2.
Revised: Theorem 2.

Original: Assume that the dataset is balanced (each class has the same amount of samples), and the noise is class-dependent.
Revised: Assume that the dataset is balanced (each class has the same amount of samples), and the noise is class-dependent.

Original: When the number of classes c ≥ 8 , the noise rate for the noisy similarity labels is lower than that of the noisy class labels.
Revised: When the number of classes c ≥ 8 , the noise rate for the noisy similarity labels is lower than that of the noisy class labels.

Original: Proof.
Revised: Proof.

Original: Assume each class has n samples.
Revised: Assume each class has n samples.

Original: As we state in the proof of Theorem 1, the number of example-pairs with clean similarity labels H = 0 and noisy similarity labels ¯ H = 0 is n 2 (cid:80)
Revised: As we state in the proof of Theorem 1, the number of example-pairs with clean similarity labels H = 0 and noisy similarity labels ¯ H = 0 is n 2 (cid:80)

Original: i (cid:54) =
Revised: i (cid:54) =

Original: i (cid:48) ,j (cid:54) =
Revised: i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) .
Revised: j (cid:48) .

Original: We denote it by N 00 .
Revised: We denote it by N 00 .

Original: Similarly, we have,
Revised: Similarly, we have,

Original: [Equation]{+N 
+}
Revised: {+N 
+}

Original:  i (cid:54) [Equation] (cid:48) [Equation]
Revised: = n(cid:88) i (cid:54) = i (cid:48) ,j (cid:54) =

Original:  (cid:48)
Revised: j (cid:48)

Original: [Equation]
Revised: T c,ij T c,i (cid:48) j (cid:48) , N 01 = n(cid:88) i (cid:54) = i (cid:48) ,j = j (cid:48)

Original:  c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) ,
Revised: j (cid:48) ,

Original: N 10 = n [Equation] (cid:88) i
Revised: N 10 = n(cid:88)

Original:  = i (cid:48) ,j (cid:54) =
Revised: i = i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) 
Revised: j (cid:48) ,

Original: [Equation]

Original: N 11 = n(cid:88)
Revised: N 11 = n(cid:88)

Original: i = i (cid:48) ,j = j (cid:48)
Revised: i = i (cid:48) ,j = j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) .
Revised: j (cid:48) .

Original: The noise rate is the ratio of the number of noisy examples to the number of total examples.
Revised: The noise rate is the ratio of the number of noisy examples to the number of total examples.

Original: Assume that the number of classes is c .
Revised: Assume that the number of classes is c .

Original: We have
Revised: We have

Revised: [Equation]

Revised: cn

Original: [Equation]
Revised: [Equation]

Original: Let S noise minus C noise
Revised: Let S noise minus C noise

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: Let A = (cid:80) i (cid:54)
Revised: Let [Equation]

Original: =
Revised: =

Original: i (cid:48) ,j = j (cid:48) T c,ij T c,i (cid:48) j (cid:48) + (cid:80)
Revised: [Equation]

Original: i = i (cid:48) ,j (cid:54) =
Revised: we have

Original: j (cid:48) T c,ij T c,i (cid:48) j (cid:48) − c (cid:80) i (cid:54) = j T c,ij , we have
Revised: [Equation]

Original: A = (cid:88) i (cid:54) = i (cid:48) ,j = j (cid:48)
Revised: = i (cid:48) ,j = j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48)
Revised: j (cid:48)

Original: + (cid:88)
Revised: + (cid:88)

Original: i = i (cid:48) ,j (cid:54) =
Revised: i = i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) − c (cid:88) i (cid:54) =
Revised: j (cid:48) − c (cid:88) i (cid:54) =

Original: j
Revised: j

Original: T c,ij = (cid:88) i (cid:54)
Revised: T c,ij = (cid:88) i (cid:54)

Original: =
Revised: =

Original: i (cid:48) ,j = j (cid:48)
Revised: i (cid:48) ,j = j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48)
Revised: j (cid:48)

Original: + (cid:88)
Revised: + (cid:88)

Original: i = i (cid:48) ,j (cid:54) =
Revised: i = i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) − c ( (cid:88) i,j
Revised: j (cid:48) − c ( (cid:88) i,j

Original: T c,ij − (cid:88) i =
Revised: T c,ij − (cid:88) i =

Original: j
Revised: j

Original: T c,ij ) = (cid:88)
Revised: T c,ij ) = (cid:88)

Original: i (cid:54) = i (cid:48) ,j = j (cid:48)
Revised: i (cid:54) = i (cid:48) ,j = j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48)
Revised: j (cid:48)

Original: + (cid:88)
Revised: + (cid:88)

Original: i = i (cid:48) ,j (cid:54) =
Revised: i = i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) − c 2 + c (cid:88) i =
Revised: j (cid:48) − c 2 + c (cid:88) i =

Original: j
Revised: j

Original: T c,ij .
Revised: T c,ij .

Original: The second equation holds because the row sum of T c is 1.
Revised: The second equation holds because the row sum of T c is 1.

Original: For the ﬁrst term (cid:80) i (cid:54) = i (cid:48) ,
Revised: For the ﬁrst term (cid:80) i (cid:54) = i (cid:48) ,

Original: j = j (cid:48)
Revised: j = j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) , notice that: (cid:88) i (cid:54) =
Revised: j (cid:48) , notice that: [Equation] i [Equation]

Original: i (cid:48) ,j =j (cid:48)
Revised: =

Original: [Equation]
Revised: [Equation]

Original: (cid:48)
Revised: j

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Revised: T c,ij ( (cid:88) i (cid:48) (cid:54) =

Revised: i

Original: [Equation]
Revised: T c,i (cid:48) j ) = (cid:88) j (cid:88)

Original: i
Revised: i

Revised: T c,ij ( (cid:88) i (cid:48) (cid:54) =

Revised: i

Original: [Equation]
Revised: T c,i (cid:48) j + T c,ij − T c,ij ) = (cid:88) j (cid:88)

Original: i
Revised: i

Original: [Equation]
Revised: T c,ij ( (cid:88) i (cid:48) T c,i (cid:48) j − T c,ij ) = (cid:88) j (cid:88)

Original:  ( S j is the column sum of the j − th column) 
Revised: i T c,ij ( S j − T c,ij ) ( S j is the column sum of the j − th column) =

Revised: (cid:88)

Original: [Equation]
Revised: [Equation]

Revised: (cid:88)

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: Due to the symmetry of i and j , for the second term (cid:80) i =  i(cid:48) ,j (cid:54) =
Revised: = (cid:88) [Equation] (cid:88) i

Revised: [Equation]

Original:  j(cid:48)
Revised: (cid:88) [Equation] = (cid:88) j

Original: T c,ij T c,i (cid:48)
Revised: [Equation]

Original:  j(cid:48) , we have (cid:88) 
Revised: (cid:88) j (cid:88) i

Original: [Equation]
Revised: [Equation]

Revised: Due to the symmetry of i and j , for the second term (cid:80) i = i (cid:48) ,j (cid:54) =

Revised: j (cid:48)

Original: ( R i is the row sum of the
Revised: T c,ij T c,i (cid:48) j (cid:48) , we have (cid:88) i = i (cid:48) ,j (cid:54) = j (cid:48)

Revised: T c,ij T c,i (cid:48)

Revised: j (cid:48) =

Revised: (cid:88)

Revised: j (cid:88)

Original: i  − th row, and R i = 1 ) 
Revised: i T c,ij ( R i − T c,ij ) ( R i is the row sum of the i − th row, and R i = 1 ) = (cid:88) j (cid:88)

Original: [Equation]
Revised: [Equation]

Original: Therefore, substituting Equation (4) and (5) into A , we have
Revised: Therefore, substituting Equation (6) and (7) into A , we have

Original: [Equation]
Revised: [Equation]

Original: To prove S noise − C noise ≤ 0 is equivalent to prove A ≤ 0 .
Revised: To prove S noise − C noise ≤ 0 is equivalent to prove A ≤ 0 .

Original: Let M = c
Revised: Let M = c

Original: 2 − c , N = (cid:80) j S 2 j − 2 (cid:80) j
Revised: 2 − c , N = (cid:80) j S 2 j − 2 (cid:80) j

Original: (cid:80)
Revised: (cid:80)

Original: i T 

Revised: i T

Original: ij + c (cid:80) i =
Revised: ij + c (cid:80) i =

Original: j T ij (we drop the subscript c in T c,ij ), and
Revised: j T ij (we drop the subscript c in T c,ij ), and

Original: A = N − M .
Revised: A = N − M .

Original: Now we utilize the Adjustment method (Su & Xiong, 2015) to scale N .
Revised: Now we utilize the Adjustment method (Su & Xiong, 2015) to scale N .

Original: For every iteration, we denote the original N by N o , and the adjusted N by N a .
Revised: For every iteration, we denote the original N by N o , and the adjusted N by N a .

Original: Since c ≥ 8 , there can not exist three columns with column sum bigger than c/ 2 − 1 .
Revised: Since c ≥ 8 , there can not exist three columns with column sum bigger than c/ 2 − 1 .

Original: Otherwise the sum of the three columns will be bigger than c , which is impossible because the sum of the whole matrix is c .
Revised: Otherwise, the sum of the three columns will be bigger than c , which is impossible because the sum of the whole matrix is c .

Original: Therefore, ﬁrst, we assume that the j, k − th columns have column sum bigger than c/ 2 − 1 .
Revised: Therefore, ﬁrst, we assume that the j, k − th columns have column sum bigger than c/ 2 − 1 .

Original: Then, for the row i , we add the elements l , which are not in j, k − th columns, to the diagonal element.
Revised: Then, for the row i , we add the elements l , which are not in j, k − th columns, to the diagonal element.

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: We do such adjustment to every rows, then N a is getting bigger and the adjusted matrix will only have values on diagonal elements and the j, k − th columns.
Revised: We do such adjustment to every rows, then N a is getting bigger and the adjusted matrix will only have values on diagonal elements and the j, k − th columns.

Original: Since the diagonal elements are dominant in the row, S j + S k < 2 c/ 3 + 2 / 3 (because for i (cid:54)
Revised: Since the diagonal elements are dominant in the row, S j + S k < 2 c/ 3 + 2 / 3 (because for i (cid:54)

Original: =
Revised: =

Original: j, k, T ij + T ik < 2 / 3 ).
Revised: j, k, T ij + T ik < 2 / 3 ).

Original: Assume that the column sum of k − th column is no bigger than that of the j − th column, and thus
Revised: Assume that the column sum of k − th column is no bigger than that of the j − th column, and thus

Original: S k < c/ 3 + 1 / 3 .
Revised: S k < c/ 3 + 1 / 3 .

Original: Then, for a row i , we add the T ik to T ii .
Revised: Then, for a row i , we add the T ik to T ii .

Original: We have
Revised: We have

Original: [Equation]
Revised: [Equation]

Original: We do such adjustment to every rows, then N a is getting bigger and the adjusted matrix will only have values on diagonal elements and the j − th column, which is called ﬁnal matrix.
Revised: We do such adjustment to every rows, then N a is getting bigger and the adjusted matrix will only have values on diagonal elements and the j − th column, which is called ﬁnal matrix.

Original: Note that if there is only one column with  column sum bigger than c/ 2 − 1 , we can adjust the rest c − 1 columns as above and then obtain the ﬁnal matrix as well.
Revised: Note that if there is only one column with a column sum bigger than c/ 2 − 1 , we can adjust the rest c − 1 columns as above and then obtain the ﬁnal matrix as well.

Original: If there is no column with  column sum bigger than c/ 2 − 1 , we can adjust all the elements as above and then obtain a unit matrix .
Revised: If there is no column with a column sum bigger than c/ 2 − 1 , we can adjust all the elements as above and then obtain a unit matrix .

Original: For the unit matrix, A = N − M < N a − M = 0 , the Theorem 2 is proved.
Revised: For the unit matrix, A = N − M < N a − M = 0 , the Theorem 2 is proved.

Original: Now we process the ﬁnal matrix.
Revised: Now we process the ﬁnal matrix.

Original: For simpliﬁcation, we assume j = 0 in the ﬁnal matrix.
Revised: For simpliﬁcation, we assume j = 0 in the ﬁnal matrix.

Original: We denote the T ij by b i and T ii by a i , for i = { 1 , . . .
Revised: We denote the T ij by b i and T ii by a i , for i = { 1 , . . .

Original: , c − 1 } .
Revised: , c − 1 } .

Original: We have
Revised: We have

Original: N [Equation]
Revised: N a = (cid:88)

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: i [Equation] i [Equation]
Revised: i a 2 i − 2 (cid:88) i b 2 i −= 1 + ( (cid:88)

Original: i a{+b i ) 
+}
Revised: i {+b i ) 
+}

Original: [Equation]
Revised: + 2 (cid:88)

Original: ia [Equation] i b [Equation]
Revised: i b i + c (cid:88)

Original: [Equation] i [Equation] i
Revised: i a i + c − (cid:88) i

Original: [Equation] i [Equation]
Revised: a 2 i − 2 (cid:88) i b 2 i −= ( (cid:88)

Original: i a{+b i ) 
+}
Revised: i {+b i ) 
+}

Original: [Equation]
Revised: + 2 (cid:88)

Original: i[Equation] = ( (cid:88) i b i ) [Equation]
Revised: i b i − 2 (cid:88)

Original: + [Equation] (cid:88) i b  i 
Revised: i b 2 i + c (cid:88)

Original: [Equation]
Revised: i a

Original: 2 (cid:88) i b [Equation]
Revised: i − (cid:88) i a 2 i + c −= ( (cid:88)

Original: i a i {+) 
+}
Revised: i b i {+) 
+}

Original: [Equation]
Revised: + 2 (cid:88)

Original: i a 2 i 
Revised: i b i − 2 (cid:88)

Original: [Equation] = ( (cid:88) i  b i ) [Equation]
Revised: i b 2 i + c (cid:88) i (1 − b i ) − (cid:88)

Original: [Equation] i  b {+i ) 
+}
Revised: i (1 − b {+i ) 
+}

Original: [Equation] [Equation]
Revised: + c −= ( (cid:88)

Original: i b {+i ) 
+}
Revised: i b {+i ) 
+}

Original: [Equation] c (cid:88) i ( [Equation]
Revised: + 2 (cid:88) i b i − 2 (cid:88)

Original: [Equation] (cid:88) i(1 − b  i )
Revised: i b 2 i + c 2 − c − c (cid:88)

Original: [Equation]

Original: + c −= ( (cid:88)
Revised: i b i − (cid:88)

Original: [Equation]
Revised: i (1 − 2 b i + b 2 i ) + c −= ( (cid:88) i b i ) 2 + 4 (cid:88) i b i − 3 (cid:88) i b 2 i − c (cid:88) i b i + c

Revised: 2 − c.

Original: Now we prove A = N − M ≤ N a − M ≤ 0 .
Revised: Now we prove A = N − M ≤ N a − M ≤ 0 .

Original: Note that
Revised: Note that

Original: [Equation]
Revised: [Equation]

Original: According to the rearrangement inequality(Hardy et al., 1952), we have
Revised: According to the rearrangement inequality(Hardy et al., 1952), we have

Original: [Equation]
Revised: [Equation]

Original: Note that c ≥ 8 , thus 3 (cid:80) i b i (1 − b i ) − (cid:80) i (1 − b i ) (cid:80) i b i ≤ 0 , and A ≤ 0 .
Revised: Note that c ≥ 8 , thus 3 (cid:80) i b i (1 − b i ) − (cid:80) i (1 − b i ) (cid:80) i b i ≤ 0 , and A ≤ 0 .

Original: Therefore S noise − C noise ≤ 0 , and the equation holds if and only if the noise rate is 0 or every instances have the same noisy class label (i.e., there is one column in the T c , of which every elements are 1, and the rest elements of the
Revised: Therefore S noise − C noise ≤ 0 , and the equation holds if and only if the noise rate is 0 or every instances have the same noisy class label (i.e., there is one column in the T c , of which every elements are 1, and the rest elements of the

Original: T c are 0).
Revised: T c are 0).

Original: Above two extreme situations are not considered in this paper.
Revised: Above two extreme situations are not considered in this paper.

Original: Namely, the noise rate of the noisy similarity labels is lower than that of the noisy class labels.
Revised: Namely, the noise rate of the noisy similarity labels is lower than that of the noisy class labels.

Original: Theorem 2 is proved.
Revised: Theorem 2 is proved.

Original: CN OTATIONS AND P ROOF OF T HEOREM
Revised: C P ROOF OF T HEOREM

Original: We formulate the above problem in the traditional risk minimization framework (Mohri et al., 2018).

Original: The expected and empirical risks of employing estimator f can be deﬁned as

Original: [Equation]

Original: and

Original: [Equation]

Original: where n is training sample size of the noisy data.

Original: Assume that the neural network has d layers with parameter matrices W 1 , . .

Original: . , W d , and the activation functions σ 1 , . .

Original: , σ d − 1 are Lipschitz continuous, satisfying σ j (0)

Original: = 0 .

Original: We denote by h : X (cid:55)→ W d σ d − 1 ( W d − 1 σ d − 2 ( . . .

Original: σ 1 ( W 1 X )))

Original: ∈ R C the standard form of the neural network.

Original: Then the output of the softmax function is deﬁned as f i ( X ) =

Original: exp ( h i ( X )) / (cid:80) Cj =1 exp ( h j ( X )) , i = 1 , . . .

Original: , C .

Original: We can obtain the following generalization error bound as follow.

Original: Theorem 3.
Revised: Theorem 3.

Original: Assume the parameter matrices W 1 , . .
Revised: Assume the parameter matrices W 1 , . .

Original: , W d have Frobenius norm at most M 1 , . . .
Revised: , W d have Frobenius norm at most M 1 , . . .

Original: , M d , and the activation functions are 1-Lipschitz, positive-homogeneous, and applied element-wise (such as the ReLU).
Revised: , M d , and the activation functions are 1-Lipschitz, positive-homogeneous, and applied element-wise (such as the ReLU).

Original: Assume the transition matrix is given, and the instances X are upper bounded by B , i.e., (cid:107) X (cid:107) ≤ B for all X , and the loss function (cid:96) is upper bounded by M 3 .
Revised: Assume the transition matrix is given, and the instances X are upper bounded by B , i.e., (cid:107) X (cid:107) ≤ B for all X , and the loss function (cid:96) is upper bounded by M 3 .

Original: Then, for any δ > 0 , with probability at least 1 − δ ,
Revised: Then, for any δ > 0 , with probability at least 1 − δ ,

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: We have deﬁned
Revised: We have deﬁned

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: where n is training sample size of the noisy data.
Revised: where n is training sample size of the noisy data.

Original: First we bound the generalization error with Rademacher complexity (Bartlett & Mendelson, 2002).
Revised: First, we bound the generalization error with Rademacher complexity (Bartlett & Mendelson, 2002).

Original: Theorem 4 ((Bartlett & Mendelson, 2002)) .
Revised: Theorem 4 (Bartlett & Mendelson (2002)) .

Original: Let the loss function be upper bounded by M .
Revised: Let the loss function be upper bounded by M .

Original: Then, for any δ > 0 , with the probability 1 − δ , we have
Revised: Then, for any δ > 0 , with the probability 1 − δ , we have

Original: [Equation]
Revised: [Equation]

Original: where R n ( (cid:96) ◦ F ) is the Rademacher complexity deﬁned by
Revised: where R n ( (cid:96) ◦ F ) is the Rademacher complexity deﬁned by

Original: [Equation]
Revised: [Equation]

Original: and { σ 1 , · · · , σ n } are Rademacher variables uniformly distributed from {− 1 , 1 } .
Revised: and { σ 1 , · · · , σ n } are Rademacher variables uniformly distributed from {− 1 , 1 } .

Original: Before further upper bound the Rademacher complexity R n ( (cid:96) ◦ F ) , we discuss the special loss function and its Lipschitz continuity w.r.t h k ( X i ) ,
Revised: Before further upper bound the Rademacher complexity R n ( (cid:96) ◦ F ) , we discuss the special loss function and its Lipschitz continuity w.r.t h k ( X i ) ,

Original: k = { 1 , . . .
Revised: k = { 1 , . . .

Original: , C } .
Revised: , C } .

Original: Lemma 1.
Revised: Lemma 1.

Original: Given similarity transition matrix T s , loss function (cid:96) ( f ( X i ) , f ( X j ) , T s , ¯ H ij ) is µLipschitz with respect to h k
Revised: Given similarity transition matrix T s , loss function (cid:96) ( f ( X i ) , f ( X j ) , T s , ¯ H ij ) is µLipschitz with respect to h k

Original: ( X i ) , k
Revised: ( X i ) , k

Original: = { 1 , . . .
Revised: = { 1 , . . .

Original: , C } , and µ = ( T s, 11 − T s, 01 ) /T s,
Revised: , C } , and µ = ( T s, 11 − T s, 01 ) /T s,

Original: [Equation]
Revised: [Equation]

Original: Detailed proof of Lemma 1 can be found in Section C.1.
Revised: Detailed proof of Lemma 1 can be found in Section C.1.

Revised: Lemma 1 shows that the loss function is µ -Lipschitz with respect to h k

Original:  Based on Lemma 1, we can further upper bound the Rademacher complexity R n ( (cid:96) ◦ F ) by the following lemma.
Revised: ( X i ) , k = { 1 , . . . , C } . Based on Lemma 1, we can further upper bound the Rademacher complexity R n ( (cid:96) ◦ F ) by the following lemma.

Original: Lemma 2.
Revised: Lemma 2.

Original: Given similarity transition matrix T s and assume that loss function (cid:96) ( f ( X i ) , f ( X j ) , T s , ¯ H ij ) is µ -Lipschitz with respect to h k
Revised: Given similarity transition matrix T s and assume that loss function (cid:96) ( f ( X i ) , f ( X j ) , T s , ¯ H ij ) is µ -Lipschitz with respect to h k

Original: ( X i ) , k
Revised: ( X i ) , k

Original: = { 1 , . . .
Revised: = { 1 , . . .

Original: , C } , we have
Revised: , C } , we have

Original: [Equation]
Revised: [Equation]

Original: where H is the function class induced by the deep neural network.
Revised: where H is the function class induced by the deep neural network.

Original: Detailed proof of Lemma 2 can be found in Section C.2.
Revised: Detailed proof of Lemma 2 can be found in Section C.2.

Original: The right hand side of the above inequality, indicating the hypothesis complexity of deep neural networks, can be bounded by the following theorem.
Revised: The right-hand side of the above inequality, indicating the hypothesis complexity of deep neural networks and bounding the Rademacher complexity, can be bounded by the following theorem.

Original: Theorem 5.
Revised: Theorem 5.

Original: (Golowich et al., 2018)
Revised: (Golowich et al., 2018)

Original: Assume the Frobenius norm of the weight matrices W 1 , . .
Revised: Assume the Frobenius norm of the weight matrices W 1 , . .

Original: , W d are at most M 1 , . . .
Revised: , W d are at most M 1 , . . .

Original: , M d .
Revised: , M d .

Original: Let the activation functions be 1-Lipschitz, positive-homogeneous, and applied element-wise (such as the ReLU).
Revised: Let the activation functions be 1-Lipschitz, positive-homogeneous, and applied element-wise (such as the ReLU).

Original: Let X is upper bounded by B, i.e., for any X , (cid:107) X (cid:107) ≤ B .
Revised: Let X is upper bounded by B, i.e., for any X , (cid:107) X (cid:107) ≤ B .

Original: Then,
Revised: Then,

Original: E

Original: [Equation]

Original: sup

Original: [Equation]
Revised: [Equation]

Original: Combining Lemma 1,2, and Theorem 4, 5, Theorem 3 is proved.
Revised: Combining Lemma 1,2, and Theorem 4, 5, Theorem 3 is proved.

Original: C. P ROOF OF L EMMA
Revised: C

Revised: P ROOF OF L EMMA

Original: Recall that
Revised: Recall that

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Take the derivative of (cid:96) ( f ( X i ) , f ( X j ) , T s , ¯ H ij = 1) w.r
Revised: Take the derivative of (cid:96) ( f ( X i ) , f ( X j ) , T s , ¯ H ij = 1) w.r.t.

Original: [Equation]

Original: t. h k ( X i )
Revised: h k ( X i )

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Note that the derivative of the softmax function has some properties,
Revised: Note that the derivative of the softmax function has some properties,

Original: i.e., if m (cid:54) =
Revised: i.e., if m (cid:54) =

Original: k , f (cid:48) m
Revised: k , f (cid:48) m

Original: ( X i ) =
Revised: ( X i ) =

Original: − f m ( X i )
Revised: − f m ( X i )

Original: f k ( X i ) and if m = k , f (cid:48) k
Revised: f k ( X i ) and if m = k , f (cid:48) k

Original: ( X i ) =
Revised: ( X i ) =

Original: (1 − f k ( X i ))
Revised: (1 − f k ( X i ))

Original: f k ( X i ) .
Revised: f k ( X i ) .

Original: We denote by V ector
Revised: We denote by V ector

Original: m the m − th element in V ector for those complex vectors.
Revised: m the m − th element in V ector for those complex vectors.

Original: Because 0 < f m ( X i ) < 1 , ∀ m ∈ { 1 , . .
Revised: Because 0 < f m ( X i ) < 1 , ∀ m ∈ { 1 , . .

Original: , c } , we have
Revised: , c } , we have

Original: [Equation]
Revised: [Equation]

Original: Therefore,
Revised: Therefore,

Original: [Equation]
Revised: [Equation]

Original: ∂ [Equation] ij (cid:2) [Equation]
Revised: The second inequality holds because of T s, 11 > T s, 01 (Detailed proof can be found in Section C.1.1) and Equation (20).

Original:  ( X i )[Equation] X i ) (cid:3) (cid:62) [Equation]
Revised: The third inequality holds because of f ( X i ) (cid:62) f

Original: ij ∂f ( X i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

Original: (cid:12)(cid:12)(cid:12) (cid:12) f (cid:48)

Original: ( X i ) (cid:62) f

Original: ( X j ) × T s, 11 − f (cid:48)

Original: ( X i ) (cid:62) f

Original: ( X j ) × T s,f ( X i ) (cid:62) f

Original: ( X j ) × T s, 11 + ( [Equation] f ( X i ) (cid:62) f ( X j ))
Revised: ( X j ) < 1 .

Original: × T s,(cid:12)(cid:12)(cid:12) (cid:12) < (cid:12)(cid:12)(cid:12)(cid:12) f ( X i ) (cid:62) f ( X j ) × T s, 11 − f ( X i ) (cid:62) f
Revised: Similarly, we can prove

Original: ( X j ) × T s,f ( X i ) (cid:62) f

Original: ( X j ) × T s, 11 + (1 − f ( X i ) (cid:62)

Original: [Equation]

Original: ( X

Original: [Equation]

Original: )) ×

Original: [Equation]
Revised: [Equation]

Original: s, 01 (cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12)

Original: [Equation]
Revised: [Equation]

Original: s,

Original: [Equation]

Original: s,

Original: T s,(cid:12)(cid:12)(cid:12)(cid:12) =

Original: [Equation]

Original: [Equation]

Original: The second inequality holds because of T s, 11 > T s, 01 (Detailed proof can be found in Section C.1.1) and Equation (20).

Original: The third inequality holds because of f ( X i ) (cid:62) f

Original: ( X j ) < 1 .

Original: Similarly, we can prove (cid:12)(cid:12)(cid:12)(cid:12)

Original: [Equation]

Original: Combining Equation (21) and Equation (22) [Equation] we obtain
Revised: Combining Equation (21) and Equation (22), we obtain

Original: [Equation]
Revised: [Equation]

Original: C.1. P ROOF OF T s, 11 > T s,
Revised: C.1. P ROOF OF T s, 11 > T s,

Original: As we mentioned in Section B, we have,
Revised: As we mentioned in Section B, we have,

Original: N 

Revised: N

Original: = n(cid:88) i (cid:54) = i (cid:48) ,j (cid:54) =
Revised: = n(cid:88) i (cid:54) = i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) ,
Revised: j (cid:48) ,

Original: N 01 = n(cid:88)
Revised: N 01 = n(cid:88)

Original: i (cid:54) = i (cid:48) ,j = j (cid:48)
Revised: i (cid:54) = i (cid:48) ,j = j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) ,
Revised: j (cid:48) ,

Original: N 10 = n(cid:88)
Revised: N 10 = n(cid:88)

Original: i = i (cid:48) ,j (cid:54) =
Revised: i = i (cid:48) ,j (cid:54) =

Original: j (cid:48)
Revised: j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) ,
Revised: j (cid:48) ,

Original: N 11 = n(cid:88)
Revised: N 11 = n(cid:88)

Original: i = i (cid:48) ,j = j (cid:48)
Revised: i = i (cid:48) ,j = j (cid:48)

Original: T c,ij T c,i (cid:48)
Revised: T c,ij T c,i (cid:48)

Original: j (cid:48) ,
Revised: j (cid:48) ,

Original: T s, 01 =
Revised: T s, 01 =

Original: N
Revised: N

Original: N 00 + N 01 ,
Revised: N 00 + N 01 ,

Original: T s, 11 =
Revised: T s, 11 =

Original: N N
Revised: N N

Original: [Equation]
Revised: [Equation]

Original: N 11 ,
Revised: N 11 ,

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: 01 N(
Revised: 01 N(

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Let us review the deﬁnition of similarity labels: if two instances belong to the same class, they will have similarity label S = 1 , otherwise S = 0 .
Revised: Let us review the deﬁnition of similarity labels: if two instances belong to the same class, they will have similarity label S = 1 , otherwise S = 0 .

Original: That is to say, for a k -class dataset, only 1 k of similarity data has similarity labels S
Revised: That is to say, for a k -class dataset, only 1 k of similarity data has similarity labels S

Original: = 1 , and the rest 1 − 1 k has similarity labels S
Revised: = 1 , and the rest 1 − 1 k has similarity labels S

Original: = 0 .
Revised: = 0 .

Original: We denote the number of data with similarity labels S = 1 by N 1 , otherwise N 0 .
Revised: We denote the number of data with similarity labels S = 1 by N 1 , otherwise N 0 .

Original: Therefore, for the balanced dataset with n samples of each class, N 1 = cn 2 , and N 

Revised: Therefore, for the balanced dataset with n samples of each class, N 1 = cn 2 , and N

Original: =
Revised: =

Original: c ( c − 1) n 2 .
Revised: c ( c − 1) n 2 .

Original: Let A =
Revised: Let A =

Original: T s, 11 − T s, 01 , we have
Revised: T s, 11 − T s, 01 , we have

Original: A = N 11 N 00 − N 01 N= N
Revised: A = N 11 N 00 − N 01 N= N

Original: [Equation]
Revised: [Equation]

Original: > 0 .
Revised: > 0 .

Original: The last equation holds because of ( c − 1) N 11 − N 01 > 0 according to the rearrangement inequality (Hardy et al., 1952).
Revised: The last equation holds because of ( c − 1) N 11 − N 01 > 0 according to the rearrangement inequality (Hardy et al., 1952).

Original: C. P ROOF OF L EMMA
Revised: C. P ROOF OF L EMMA

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: n
Revised: n

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: ) (cid:35)
Revised: ) (cid:35)

Original: =
Revised: =

Original: E (cid:34) sup g n n (cid:88) i
Revised: E (cid:34) sup g n n (cid:88) i

Original: =
Revised: =

Original: σ i (cid:96) (
Revised: σ i (cid:96) (

Original: [Equation]
Revised: [Equation]

Original: ( X i
Revised: ( X i

Original: [Equation]
Revised: [Equation]

Original: ( X j
Revised: ( X j

Original: [Equation]
Revised: [Equation]

Original: ij ) (cid:35) =
Revised: ij ) (cid:35) =

Original: E (cid:34) sup argmax { h 1 ,...,h C } n n (cid:88) i =
Revised: E (cid:34) sup argmax { h 1 ,...,h C } n n (cid:88) i =

Original: σ i (cid:96) (
Revised: σ i (cid:96) (

Original: [Equation]
Revised: [Equation]

Original: ( X i
Revised: ( X i

Original: [Equation]
Revised: [Equation]

Original: ( X
Revised: ( X

Original: [Equation]
Revised: [Equation]

Original: ) (cid:35) =
Revised: ) (cid:35) =

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: sup max { h
Revised: sup max { h

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: h C } n n (cid:88) i =
Revised: h C } n n (cid:88) i =

Original: σ i (cid:96) (
Revised: σ i (cid:96) (

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: ( X j
Revised: ( X j

Original: [Equation]
Revised: [Equation]

Original: ¯
Revised: ¯

Original: [Equation]
Revised: [Equation]

Original: ij ) (cid:35) ≤ E (cid:34)
Revised: ij ) (cid:35) ≤ E (cid:34)

Original: [Equation]
Revised: [Equation]

Original: (cid:88) k
Revised: (cid:88) k

Original: = sup h k ∈ H n n (cid:88) i =
Revised: = sup h k ∈ H n n (cid:88) i =

Original: [Equation]
Revised: [Equation]

Original: i (cid:96) ( f ( X
Revised: i (cid:96) ( f ( X

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ( X j )
Revised: ( X j )

Original: [Equation]
Revised: [Equation]

Original: ¯ H ij ) (cid:35) =
Revised: ¯ H ij ) (cid:35) =

Original: C (cid:88) k
Revised: C (cid:88) k

Original: =
Revised: =

Original: E (cid:34) sup h k ∈
Revised: E (cid:34) sup h k ∈

Original: H n n
Revised: H n n

Original: [Equation]
Revised: [Equation]

Original: where the ﬁrst three equations hold because given T s , f and max { h 1 , . .
Revised: where the ﬁrst three equations hold because given T s , f and max { h 1 , . .

Original: , h C } give the same constraint on h j ( X i ) , j = { 1 , . .
Revised: , h C } give the same constraint on h j ( X i ) , j = { 1 , . .

Original: , C } ; the sixth inequality holds because of the Talagrand Contraction
Revised: , C } ; the sixth inequality holds because of the Talagrand Contraction

Original: Lemma (Ledoux & Talagrand, 2013).
Revised: Lemma (Ledoux & Talagrand, 2013).

Original: D D EFINITION OF NOISE SETTINGS
Revised: D D EFINITION OF NOISE SETTINGS

Original: Symmetric noise setting is deﬁned as follows, where C is the number of classes.
Revised: Symmetric noise setting is deﬁned as follows, where C is the number of classes.

Original: Sym- ρ : T =   1 − ρ ρC −. . .
Revised: Sym- ρ : T =   1 − ρ ρC −. . .

Original: ρC −ρC − 1 ρC −1 − ρ ρC −. . .
Revised: ρC −ρC − 1 ρC −1 − ρ ρC −. . .

Original: ρC − 1 . ρC −. . .
Revised: ρC − 1 . ρC −. . .

Original: ρC −1 − ρ ρC − 1 ρC −ρC −. . .
Revised: ρC −1 − ρ ρC − 1 ρC −ρC −. . .

Original: ρC −1 − ρ   .
Revised: ρC −1 − ρ   .

Original: The 0.3 asymmetric noise setting is set as follow, def asym_transition_matrix_generate(noise_rate=0.3, num_classes=10, seed=1): np.random.seed(seed) t
Revised: The 0.3 asymmetric noise setting is set as follow, def asym_transition_matrix_generate(noise_rate=0.3, num_classes=10, seed=1): np.random.seed(seed) t

Original: =
Revised: =

Original: np.random.rand(num_classes, num_classes)
Revised: np.random.rand(num_classes, num_classes)

Original: i = np.eye(num_classes) t
Revised: i = np.eye(num_classes) t

Original: = t + 1.2 * num_classes
Revised: = t + 1.2 * num_classes

Original: * i for
Revised: * i for

Original: 8return E E XPERIMENTS 
Revised: E E XPERIMENTS E. G IVEN

Original: [Figure]
Revised: [Figure]

Original: From Figure 5, overall, we can see that Class2Simi (Class2Simi TrueT) achieves the best performance whenever class T c is given or estimated.
Revised: From Figure 5, overall, we can see that Class2Simi (Class2Simi TrueT) achieves the best performance whenever class T c is given or estimated.

Original: In most cases, Class2Simi with estimated T c even outperforms baselines with the ground-truth class noise transition matrix, due to lower noise rate and the similarity transition matrix being robust to noise.
Revised: In most cases, Class2Simi with estimated T c even outperforms baselines with the ground-truth class noise transition matrix, due to lower noise rate and the similarity transition matrix being robust to noise.

Original: Speciﬁcally, On MNIST , as the noise rate increases from Sym-0.1 to Sym-0.5, Class2Simi TrueT maintains remarkable accuracy above 99.20% while the accuracy of Class2Simi and Forward TrueT decrease steadily.
Revised: Speciﬁcally, On MNIST , as the noise rate increases from Sym-0.1 to Sym-0.5, Class2Simi TrueT maintains remarkable accuracy above 99.20% while the accuracy of Class2Simi and Forward TrueT decrease steadily.

Original: However, there is a signiﬁcant decrease in the accuracy of Forward.
Revised: However, there is a signiﬁcant decrease in the accuracy of Forward.

Original: On CIFAR10 , the patterns of varying tendencies of four curves are similar to that of MNIST except that the decreases are more dramatic and even Class2Simi TrueT drops slightly at Sym-0.5.
Revised: On CIFAR10 , the patterns of varying tendencies of four curves are similar to that of MNIST except that the decreases are more dramatic and even Class2Simi TrueT drops slightly at Sym-0.5.

Original: On CIFAR100 , there is a obvious decrease in the accuracy of all methods and our method achieves the best results across all noise rate, i.e., at Sym-0.5, Class2Simi gives an accuracy uplift of about 8.0% compared with Forward.
Revised: On CIFAR100 , there is an obvious decrease in the accuracy of all methods and our method achieves the best results across all noise rate, i.e., at Sym-0.5, Class2Simi gives an accuracy uplift of about 8.0% compared with Forward.

Original: E. T RAINING WITH DIFFERENT SAMPLING RATE ON TRAINING DATA
Revised: E. T RAINING WITH DIFFERENT SAMPLING RATE ON TRAINING DATA

Original: In Figure
Revised: In Figure

Original: [Table]
Revised: [Table]

Original: we show that the Class2Simi performs well even with less data.
Revised: we show that the Class2Simi performs well even with fewer data.

Original: The noise in training data is set to Sym-0.5.
Revised: The noise in training data is set to Sym-0.5.

Original: We randomly sample from original training data with  sampling rate from 0. to 1.0 and train the model on the sampled data.
Revised: We randomly sample from original training data with a sampling rate from 0.5 to 1.0 and train the model on the sampled data.

Original: Test datasets remain the same.
Revised: Test datasets remain the same.

Original: At each sampling rate, Class2Simi performs better than the baseline.
Revised: At each sampling rate, Class2Simi performs better than the baseline.

Original: With only 50%, 80% and 80% data on MNIST ,
Revised: With only 50%, 80% and 80% data on MNIST ,

Original: CIFAR10 and CIFAR100 , our method can achieve the same accuracy as Forward.
Revised: CIFAR10 and CIFAR100 , our method can achieve the same accuracy as Forward.

Original: E. R ESULTS ON ASYMMETRIC NOISE SETTING
Revised: E. R ESULTS ON ASYMMETRIC NOISE SETTING

Original: [Table]
Revised: [Table]

Revised: ± 0.

Revised: Revision & Class2Simi 98.24 ± 0.

Original:  In Table 5, we demonstrate the effectiveness of our method under the asymmetric noise setting.
Revised: 85.11 ± 0. 48.69 ± 0. In Table 5, we demonstrate the effectiveness of our method under the asymmetric noise setting.

Revised: F L EARNABILTY CONNECTION BETWEEN PAIRWISE CLASSIFICATION AND POINTWISE CLASSIFICATION

Revised: F.

Revised: Pointwise IMPLIES pairwise

Revised: For an invertible T c , denote by v j the j -th column of T c and 1 the all-one vector.

Revised: Then, (cid:88)

Revised: [Equation]

Revised: where we use the Cauchy–Schwarz inequality (Steele, 2004) in the second step.

Revised: Further, we have

Revised: [Equation]

Revised: Thus the learnability of the pointwise classiﬁcation implies the learnability of the reduced pairwise classiﬁcation.

Revised: F. Pairwise DOES NOT IMPLY pointwise

Revised: Assuming the number of classes is 10 and the dataset is balanced, a singular T c of the pair noise pattern and the transformed T s are shown as follows,

Revised: [Equation]

Revised:  0 .

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: 0 .

Revised: 

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: In this case, the reduced pairwise classiﬁcation is learnable while the original pointwise classiﬁcation is not learnable.

Revised: Thus the learnability of the reduced pairwise classiﬁcation does not imply the learnability of the pointwise classiﬁcation.