Original: L ABEL E NCODING FOR R EGRESSION N ETWORKS
Revised: L ABEL E NCODING FOR R EGRESSION N ETWORKS

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: Deep neural networks are used for a wide range of regression problems and are typically trained by minimizing the squared or absolute error of output labels.
Revised: Deep neural networks are used for a wide range of regression problems and are typically trained by minimizing the squared or absolute error of output labels.

Original: Solving a regression problem with a set of binary classiﬁers can improve the accuracy and also utilizes well-studied binary classiﬁcation algorithms.
Revised: Solving a regression problem with a set of binary classiﬁers can improve the accuracy and also utilizes well-studied binary classiﬁcation algorithms.

Original: However, a limited design space of regression by binary classiﬁcation has been studied by prior works.
Revised: However, a limited design space of regression by binary classiﬁcation has been studied by prior works.

Original: This work introduces a novel approach using binary-encoded labels (BEL) to generalize the application of binary classiﬁcation to regression, and a taxonomy identifying the key design aspects of such formulation.
Revised: This work introduces a novel approach using binary-encoded labels (BEL) to generalize the application of binary classiﬁcation to regression, and a taxonomy identifying the key design aspects of such formulation.

Original: Of these design aspects, we primarily explore and identify the desirable properties of suitable encoding and decoding functions used for the conversion between real-valued and binary-encoded labels based on theoretical and empirical analysis.
Revised: Of these design aspects, we primarily explore and identify the desirable properties of suitable encoding and decoding functions used for the conversion between real-valued and binary-encoded labels based on theoretical and empirical study.

Original: BEL can be combined with off-the-shelf task-speciﬁc feature extractors and trained end-toend.
Revised: BEL can be combined with off-the-shelf task-speciﬁc feature extractors and trained end-toend.

Original: We further propose a series of sample encoding, decoding, and training loss functions applicable in this domain and demonstrate that our generic regression approach results in lower error than direct regression and specialized approaches while being suitable for various regression problems, network architectures, and evaluation metrics.
Revised: We further propose a series of sample encoding, decoding, and training loss functions applicable in this domain and demonstrate that our generic regression approach results in lower error than direct regression and specialized approaches while being suitable for a diverse set of regression problems, network architectures, and evaluation metrics.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Deep regression networks, in which a continuous output is predicted for a given input, are traditionally trained by minimizing squared/absolute error of a logit, which we refer to as direct regression .
Revised: Deep regression networks, in which a continuous output is predicted for a given input, are traditionally trained by minimizing squared/absolute error of a logit, which we refer to as direct regression .

Original: More recently, several works have proposed specialized regression approaches for regression problems such as head pose estimation, age estimation, and facial landmark estimation, and shown signiﬁcant accuracy improvements over direct regression.
Revised: More recently, several works have proposed specialized regression approaches for regression problems such as head pose estimation, age estimation, and facial landmark estimation, and shown signiﬁcant accuracy improvements over direct regression.

Original: However, developing task-agnostic generic approaches to improve accuracy for regression is desirable.
Revised: However, developing task-agnostic generic approaches to improve accuracy for regression is desirable.

Original: A regression problem can be posed as a set of binary classiﬁcation sub-problems.
Revised: A regression problem can be posed as a set of binary classiﬁcation sub-problems.

Original: A similar approach has been applied to other domains such as ordinal regression (Li & Lin, 2006) and multiclass classiﬁcation (Dietterich & Bakiri, 1995).
Revised: A similar approach has been applied to other domains such as ordinal regression (Li & Lin, 2006) and multiclass classiﬁcation (Dietterich & Bakiri, 1995).

Original: Such formulation allows the use of well-studied binary classiﬁcation approaches.
Revised: Such formulation allows the use of well-studied binary classiﬁcation approaches.

Original: Further, new generalization bounds for ordinal regression or multiclass classiﬁcation can be derived from the known generalization bounds of binary classiﬁcation.
Revised: Further, new generalization bounds for ordinal regression or multiclass classiﬁcation can be derived from the known generalization bounds of binary classiﬁcation.

Original: This reduces the efforts for design, implementation, and theoretical analysis signiﬁcantly (Li & Lin, 2006).
Revised: This reduces the efforts for design, implementation, and theoretical analysis signiﬁcantly (Li & Lin, 2006).

Original: Dietterich & Bakiri (1995) demonstrated that posing multiclass classiﬁcation as a set of binary classiﬁcation problems can increase error tolerance and improve accuracy.
Revised: Dietterich & Bakiri (1995) demonstrated that posing multiclass classiﬁcation as a set of binary classiﬁcation problems can increase error tolerance and improve accuracy.

Original: However, the proposed approaches for multiclass classiﬁcation do not apply to regression due to the differences in task objective and properties of the error probability distribution of classiﬁers (Section 2).
Revised: However, the proposed approaches for multiclass classiﬁcation do not apply to regression due to the differences in task objective and properties of the error probability distribution of classiﬁers (Section 2).

Original: On the other hand, prior works on ordinal regression have explored the application of binary classiﬁers in a more restricted way which limits its application to a wide range of complex regression problems (Section 2).
Revised: On the other hand, prior works on ordinal regression have explored the application of binary classiﬁers in a more restricted way which limits its application to a wide range of complex regression problems (Section 2).

Original: There exists a lack of a generic framework that uniﬁes possible formulations for using binary classiﬁcation to solve regression.
Revised: There exists a lack of a generic framework that uniﬁes possible formulations for using binary classiﬁcation to solve regression.

Original: In this work, we propose binary-encoded labels (BEL) which improves regression by generalizing the application of binary classiﬁcation.
Revised: In this work, we propose binary-encoded labels (BEL) which improves regression by generalizing the application of binary classiﬁcation.

Original: In BEL, a target label is quantized and converted to a binary code of length M , and M binary classiﬁers are then used to learn binary-encoded labels.
Revised: In BEL, a target label is quantized and converted to a binary code of length M , and M binary classiﬁers are then used to learn binary-encoded labels.

Original: An encoding function is introduced to convert the target label to a binary code, and a decoding function is introduced to decode the output of binary classiﬁers to the predicted label.
Revised: An encoding function is introduced to convert the target label to a binary code, and a decoding function is introduced to decode the output of binary classiﬁers to the predicted label.

Original: BEL allows using an adjustable number of binary classiﬁers depending upon the quantization, encoding, and decoding functions.
Revised: BEL allows using an adjustable number of binary classiﬁers depending upon the quantization, encoding, and decoding functions.

Original: BEL opens possible avenues to improve the accuracy of regression problems with a large design space of the quantization function, encoding/decoding functions, and optimization algorithm.
Revised: BEL opens possible avenues to improve the accuracy of regression problems with a large design space of the quantization function, encoding/decoding functions, and optimization algorithm.

Original: We focus on the encoding and decoding functions and theoretically study the relations between the absolute error of label and binary classiﬁers’ errors for sample encoding and decoding functions.
Revised: We focus on the encoding and decoding functions and theoretically study the relations between the absolute error of label and binary classiﬁers’ errors for sample encoding and decoding functions.

Original: This analysis demonstrates the impact of binary classiﬁers’ error distribution over the numeric range of target labels on the suitability of different encoding and decoding functions.
Revised: This analysis demonstrates the impact of binary classiﬁers’ error distribution over the numeric range of target labels on the suitability of different encoding and decoding functions.

Original: Based on our analysis and empirically observed binary classiﬁers’ error distribution, we propose properties of suitable encoding functions for regression and explore various encoding functions on a wide range of tasks.
Revised: Based on our analysis and empirically observed binary classiﬁers’ error distribution, we propose properties of suitable encoding functions for regression and explore various encoding functions on a wide range of tasks.

Original: We also propose an expected correlation-based decoding function for regression that can effectively reduce the quantization error introduced by the encoding.
Revised: We also propose an expected correlation-based decoding function for regression that can effectively reduce the quantization error introduced by the encoding.

Original: A deep regression network consists of a feature extractor and a regressor and is trained end-to-end.
Revised: A deep regression network consists of a feature extractor and a regressor and is trained end-toend.

Original: Our proposed regression approach (BEL) can be combined with off-the-shelf task-speciﬁc feature extractors by increasing the regressor’s output logits.
Revised: Our proposed regression approach (BEL) can be combined with off-the-shelf task-speciﬁc feature extractors by increasing the regressor’s output logits.

Original: We ﬁnd that the correlation between multiple binary classiﬁers’ outputs can be exploited to reduce the size of the feature vector and consequently reduce the number of parameters in the regressor.
Revised: We ﬁnd that the correlation between multiple binary classiﬁers’ outputs can be exploited to reduce the size of the feature vector and consequently reduce the number of parameters in the regressor.

Original: The training loss function can be formulated according to the decoding function used; we study the binary cross-entropy, cross-entropy,MAE, and MSE loss functions for different decoding functions.
Revised: We explore the use of decoding functions for training loss formulation and evaluate the use of the binary cross-entropy, cross-entropy, and squared/absolute error loss functions for BEL.

Original: We evaluate our approach on four complex regression problems: head pose estimation, facial landmark detection, age estimation, and end-to-end autonomous driving.
Revised: We evaluate our approach on four complex regression problems: head pose estimation, facial landmark detection, age estimation, and end-to-end autonomous driving.

Original: We make the following contributions in this work
Revised: We make the following contributions in this work

Original: : • We propose binary-encoded labels for regression and introduce a general framework and a taxonomy for the design aspects of regression by binary classiﬁcation.
Revised: : • We propose binary-encoded labels for regression and introduce a general framework and a taxonomy for the design aspects of regression by binary classiﬁcation.

Original: We propose desirable properties of encoding and decoding functions suitable for regression problems.
Revised: We propose desirable properties of encoding and decoding functions suitable for regression problems.

Original: • We present a series of suitable encoding, decoding, and loss functions for regression with BEL.
Revised: • We present a series of suitable encoding, decoding, and loss functions for regression with BEL.

Original: We present an end-to-end learning approach and regression layer architecture for BEL.
Revised: We present an end-to-end learning approach and regression layer architecture for BEL.

Original: We combine BEL with task-speciﬁc feature extractors for four tasks and evaluate multiple encoding, decoding, and loss functions.
Revised: We combine BEL with task-speciﬁc feature extractors for four tasks and evaluate multiple encoding, decoding, and loss functions.

Original: BEL outperforms direct regression for all the problems and specialized approaches for several tasks.
Revised: BEL outperforms direct regression for all the problems and specialized approaches for several tasks.

Original: • We theoretically and empirically demonstrate the effect of different design parameters on the accuracy, how it varies across different tasks, datasets, and network architectures, and provide preliminary insights and motivation for further study.
Revised: • We theoretically and empirically demonstrate the effect of different design parameters on the accuracy, how it varies across different tasks, datasets, and network architectures, and provide preliminary insights and motivation for further study

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Binary classiﬁcation for regression: Prior works have proposed binary classiﬁcation-based approaches for ordinal regression (Crammer & Singer, 2001; Chu & Keerthi, 2005; Li & Lin, 2006).
Revised: Binary classiﬁcation for regression: Prior works have proposed binary classiﬁcation-based approaches for ordinal regression (Crammer & Singer, 2001; Chu & Keerthi, 2005; Li & Lin, 2006).

Original: Ordinal regression is a class of supervised learning problems, where the samples are labeled by a rank that belongs to an ordinal scale.
Revised: Ordinal regression is a class of supervised learning problems, where the samples are labeled by a rank that belongs to an ordinal scale.

Original: Ordinal regression approaches can be applied to regression by discretizing the numeric range of the real-valued labels (Fu et al., 2018; Berg et al.
Revised: Ordinal regression approaches can be applied to regression by discretizing the numeric range of the real-valued labels (Fu et al., 2018; Berg et al.

Original: ; Niu et al., 2016).
Revised: ; Niu et al., 2016).

Original: In the existing works on ordinal regression by binary classiﬁcation, N − 1 binary classiﬁers are used for target labels ∈ { 1 , 2 , ..., N } , where classiﬁer- k predicts if the label is greater than k or not for a given input.
Revised: In the existing works on ordinal regression by binary classiﬁcation, N − 1 binary classiﬁers are used for target labels ∈ { 1 , 2 , ..., N } , where classiﬁer- k predicts if the label is greater than k or not for a given input.

Original: Li & Lin (2006) provided a reduction framework and generalization bound for the same.
Revised: Li & Lin (2006) provided a reduction framework and generalization bound for the same.

Original: However, the proposed binary classiﬁcation formulation is restricted.
Revised: However, the proposed binary classiﬁcation formulation is restricted.

Original: It requires several binary classiﬁers if the numeric range of output is extensive, whereas reducing the number of classiﬁers by using fewer quantization levels increases quantization error.
Revised: It requires several binary classiﬁers if the numeric range of output is extensive, whereas reducing the number of classiﬁers by using fewer quantization levels increases quantization error.

Original: Thus, a more generalized approach for using binary classiﬁcation for regression is desirable to allow  theﬂexible design ofoutput classiﬁers.
Revised: Thus, a more generalized approach for using binary classiﬁcation for regression is desirable to allow ﬂexibility in the design of classiﬁers.

Original: Binary classiﬁcation for multiclass classiﬁcation: Dietterich & Bakiri (1995) proposed the useand properties of error-correcting output codes  to convert a multiclass classiﬁcation to a set of binary classiﬁcation sub-problems.
Revised: Binary classiﬁcation for multiclass classiﬁcation: Dietterich & Bakiri (1995) proposed the use of error-correcting output codes (ECOC) to convert a multiclass classiﬁcation to a set of binary classiﬁcation sub-problems.

Original: This improves the accuracy of the network as it introduces tolerance to  errorsin output logits depending upon the hamming distance (i.e., number of bits changed between two binary strings) between two codes.
Revised: This improves the accuracy of the network as it introduces tolerance to binary classiﬁers’ errors depending upon the hamming distance (i.e., number of bits changed between two binary strings) between two codes.

Original: Allwein et al.
Revised: Allwein et al.

Original: (2001) provided a unifying framework and multiclass loss bounds in terms of binary classiﬁcation loss for this formulation.
Revised: (2001) provided a unifying framework and multiclass loss bounds in terms of binary classiﬁcation loss.

Original: More recent works have also used Hadamard code, a widely used error-correcting code (Song et al., 2021; Verma & Swami, 2019).
Revised: More recent works have also used Hadamard code, a widely used error-correcting code (Song et al., 2021; Verma & Swami, 2019).

Revised: Other works have focused on the problem of extreme classiﬁcation and the use of ECOC and compact codes (sublinear increase in the length of codes with the number of classes) for a classiﬁcation problem with a large number of classes (Cissé et al., 2012; Evron et al., 2018).

Original: However, the proposed encoding and decoding approaches do not consider the ordinality of labels for regression.
Revised: However, the proposed encoding and decoding approaches do not consider the ordinality of labels for regression.

Original: Further,the binary classiﬁers possess distinct error probability distribution properties for regression problems as observed empirically (Section 3.1), which can be exploited to design codes suitable for regression.
Revised: Further,

Original: [Figure]
Revised: [Figure]

Revised: the binary classiﬁers possess distinct error probability distribution properties for regression problems as observed empirically (Section 3.1), which can be exploited to design codes suitable for regression.

Original: Multiclass classiﬁcation and ordinal regression by binary classiﬁcation can be viewed as special cases falling under the BEL framework.
Revised: Multiclass classiﬁcation and ordinal regression by binary classiﬁcation can be viewed as special cases falling under the BEL framework.

Original: As shown in Section 4, other BEL designs yield improvements in accuracy over these approaches.
Revised: As shown in Section 4, other BEL designs yield improvements in accuracy over these approaches.

Original: Task-speciﬁc regression techniques are well explored as summarized below (see also Appendix D-G).
Revised: Task-speciﬁc regression techniques are well explored as summarized below (see also Appendix D.

Original: While effective, task-speciﬁc approaches lack generality by design.
Revised: While effective, task-speciﬁc approaches lack generality by design.

Original: Head pose estimation:
Revised: Head pose estimation:

Original: SSR-Net (Rothe et al., 2015) and FSA-Net (Yang et al., 2019) used softmax to reﬁne labels.
Revised: SSR-Net (Rothe et al., 2015) and FSA-Net (Yang et al., 2019) used softmax to reﬁne labels.

Original: HopeNet (Ruiz et al., 2018) used a combination of classiﬁcation and regression loss.
Revised: HopeNet (Ruiz et al., 2018) used a combination of classiﬁcation and regression loss.

Original: QuatNet (Hsu et al., 2019) used a combination of regression and ordinal regression loss.
Revised: QuatNet (Hsu et al., 2019) used a combination of regression and ordinal regression loss.

Original: Facial landmark detection: Wang et al.
Revised: Facial landmark detection: Wang et al.

Original: (2020) minimize L2 loss between predicted and target 2D heatmaps with the latter formed using small variance Gaussians centered on ground truth landmarks.
Revised: (2020) minimize L2 loss between predicted and target 2D heatmaps with the latter formed using small variance Gaussians centered on ground truth landmarks.

Original: AWing (Wang et al., 2019) modiﬁed loss for different pixels in the heatmap.
Revised: AWing (Wang et al., 2019) modiﬁed loss for different pixels in the heatmap.

Original: LUVLi (Kumar et al., 2020) proposed a landmark’s location, uncertainty, and visibility likelihood-based loss.
Revised: LUVLi (Kumar et al., 2020) proposed a landmark’s location, uncertainty, and visibility likelihood-based loss.

Original: Bulat &
Revised: Bulat &

Original: Tzimiropoulos (2016) used binary heatmaps with pixel-wise binary cross-entropy loss.
Revised: Tzimiropoulos (2016) used binary heatmaps with pixel-wise binary cross-entropy loss.

Original: Age estimation: OR-CNN (Niu et al., 2016)  used ordinal regression via binary classiﬁcation.CORAL-CNN (Cao et al., 2020) reﬁned this approach by enforcing the ordinality of the model output.
Revised: Age estimation: OR-CNN (Niu et al., 2016) and CORAL-CNN (Cao et al., 2020) used ordinal regression via binary classiﬁcation.

Original: MV-Loss (Pan et al., 2018) extended the soft regression approach by penalizing the output of the model  based on thevariance of the age distribution, while DLDL (Gao et al., 2018) proposed to use the KL-divergence between the softmax output and a generated label distribution to train a model.
Revised: MV-Loss (Pan et al., 2018) proposed to penalize the model output based on the age distribution ’s variance, while Gao et al. (2018) proposed to use the KL-divergence between the softmax output and a generated label distribution for training.

Original:  B INARY - ENCODED L ABELS FOR R EGRESSION (BEL)
Revised:  B INARY - ENCODED L ABELS FOR R EGRESSION (BEL)

Original: We consider regression problems where the goal is to minimize the error between real-valued target labels y i and predicted labels ˆ y i , over a set of training samples i .
Revised: We consider regression problems where the goal is to minimize the error between real-valued target labels y i and predicted labels ˆ y i , over a set of training samples i .

Original: We transform this problem to a set of binary classiﬁcation sub-problems by converting a real-valued label to a binary code.
Revised: We transform this problem to a set of binary classiﬁcation sub-problems by converting a real-valued label to a binary code.

Original: Figure 1a shows the training and inference ﬂow for BEL.
Revised: Figure 1 shows the training and inference ﬂow for BEL.

Original: The red-colored blocks highlight functions that vary under BEL.
Revised: The red-colored blocks highlight functions that vary under BEL.

Original: A real-valued label y i ∈ R is quantized to level Q i ∈ { 1 , 2 , ..., N } .
Revised: A real-valued label y i ∈ R is quantized to level Q i ∈ { 1 , 2 , ..., N } .

Original: The quantized label is then converted to a binary sequence B i ∈ { 0 , 1 } M using an encoding function E .
Revised: The quantized label is then converted to a binary sequence B i ∈ { 0 , 1 } M using an encoding function E .

Original: The binary-encoded labels B i are used to train M classiﬁers.
Revised: The binary-encoded labels B i are used to train M classiﬁers.

Original: In the prediction phase, the classiﬁers predict a binary code ˆ B i ∈ { 0 , 1 } M for an input x i .
Revised: In the prediction phase, the classiﬁers predict a binary code ˆ B i ∈ { 0 , 1 } M for an input x i .

Original: The predicted binary code ( ˆ B i ) or the real-valued conﬁdence vector of classiﬁers’ prediction ( ˆ Z i ) is then decoded to a predicted label ˆ y i ∈
Revised: The predicted binary code ( ˆ B i ) or the real-valued conﬁdence vector of classiﬁers’ prediction ( ˆ Z i ) is then decoded to a predicted label ˆ y i ∈

Original: R using a decoding function D .
Revised: R using a decoding function D .

Original: We explore decoding functions that yield either quantized or continuous predicted outputs.
Revised: We explore decoding functions that yield either quantized or continuous predicted outputs.

Original: For the latter, we employ an expected correlation-based decoding function that decodes classiﬁers’ output to a continuous value, reducing the quantization error (Section 3.3).
Revised: For the latter, we propose an expected correlation-based decoding function that decodes classiﬁers’ output to a continuous value, reducing the quantization error (Section 3.3).

Original: There are ﬁve key design parameters when applying BEL: quantization, encoding, decoding, regressor network architecture, and training loss formulation yielding a large design space.
Revised: There are ﬁve key design parameters when applying BEL: quantization, encoding, decoding, regressor network architecture, and training loss formulation yielding a large design space.

Original: Figure 1b illustrates how encoding and decoding functions {E , D} can impact accuracy.

Original: The ﬁgure shows two distinct lookup table based encoding/decoding functions {E 1 , D 1 } and {E 2 , D 2 } .

Original: Case 1 and 2 illustrate examples of encoding a given target value and decoding an erroneous predicted code (red bits indicate misclassiﬁcation).

Original: For a given classiﬁcation error, {E 2 , D 2 } exhibits lower regression error versus {E 1 , D 1 } .

Original: In this work we consider only uniform quantization while leaving nonuniform quantization (Fu et al., 2018) to future work.
Revised: In this work we consider only uniform quantization while leaving nonuniform quantization (Fu et al., 2018) to future work.

Original: Section 3.2 and 3.3 explore the characteristics of suitable [Figure] [Figure] [Figure]
Revised: Section 3.2 and 3.3 explore the characteristics of suitable encoding, decoding, and training loss functions.

Original: [Figure] [Figure] [Figure] [Figure]
Revised: Section 3.4 explore the impact of regression network architecture.

Original: [Figure] [Figure] C 1 C 2 C 3 C 4 C [Figure]
Revised: We ﬁnd varying any of these aspects can improve accuracy.

Original: C

Original: [-[Figure] C [Figure] Unary 0 0 0 1 0 0 10 1 1 1 1 1 11 
-]
Revised: [-[Figure] C [Figure] Unary 0 0 0 1 0 0 10 1 1 1 1 1 11 
-]While BEL provides a framework and some codes appear generally better than others, the most suitable BEL code to employ appears to vary across task, dataset, and network architecture, as we show both theoretically (Section 3.1) and empirically (Section 4).

Original: [Figure] C 1 C 2 C [Figure] C(
Revised: 3. A NALYSIS OF E NCODING /D ECODING F UNCTIONS

Original: [Figure] [Figure] [Figure] [Figure]
Revised: This section analytically analyzes the potential impact of encoding/decoding functions on regression error based upon empirically observed error distributions for the underlying classiﬁers.

Original: C 1 C 2 C( [Figure] C 1 C 2 C(e [Figure]
Revised: With this

Original: C 1 C 2 C(

Original: [Figure]

Original: [Figure] encoding and decoding functions. Section 3.4 explore the impact of regression network architecture and training loss for given encoding/decoding functions. We ﬁnd varying any of these aspects can improve regression accuracy.
Revised: [Table]

Original: While BEL provides a framework and some codes appear generally better than others, the most suitable BEL code to employ appears to vary across task, dataset, and network architecture, as we show both theoretically (Section 3.1) and empirically (Section 4).
Revised: [Figure]

Original: This section analytically analyzes the potential impact of  {E , D}  on regression accuracy based upon empirically observed error distributions for the underlying classiﬁers.
Revised: analytical study, we aim to understand the properties of classiﬁers for ordinal labels and their effect on regression error using simple {E , D} functions and draw insights on desirable properties of these functions for regression and empirical results.

Original: We  divide our analysis into three parts.
Revised: We use unary (Figure 2b) and Johnson code (Figure 2c) for the analysis. The design of these codes and intuition for trying these codes are discussed in Section 3.2. We divide our analysis into three parts.

Original: First,we derive the relation between the expected error of predicted labels and classiﬁers’ errors for two {E , D} functions.
Revised: First, the expected error of predicted labels is derived in terms of classiﬁers’ errors for two {E , D} functions.

Original: Next, we propose an approximate classiﬁer’s error probability distribution over the numeric range of target labels for regression based on empirical study.
Revised: Next, we propose an approximate classiﬁer’s error probability distribution over the numeric range of target labels for regression based on empirical study.

Original: Last, we compare the expected error of sample {E , D} functions based on our analysis.
Revised: Lastly, we compare the expected error of sample {E , D} functions based on our analysis.

Original: We use labels y i ∈ [1 , N − 1] , with quantization levels Q i ∈ { 1 , 2 , ..., N − 1 } .
Revised: We use labels y i ∈ [1 , N − 1] , with quantization levels Q i ∈ { 1 , 2 , ..., N − 1 } .

Original: Quantization error is not included as it is not affected by {E , D} functions.
Revised: Quantization error is not included as it is not affected by {E , D} functions.

Original: We compare the impact of a unary (Figure 2b) and a Johnson code (Figure 2c).

Original: The intuition for trying these codes is discussed in Section 3.2.

Original: Expected absolute error bounds in terms of classiﬁcation error:
Revised: Expected absolute error bounds in terms of classiﬁcation error:

Original: First, we analyze the expected absolute error given the classiﬁcation error for a unary code,
Revised: First, we analyze the expected absolute error given the classiﬁcation error for a unary code,

Original: BEL-U.
Revised: BEL-U.

Original: The encoding function E BEL-U converts Q i to B i =
Revised: The encoding function E BEL-U converts Q i to B i =

Original: b 1 i , b 2 i , ..., b N − 2 i , where b ki = 1 for k < Q i , else 0 .
Revised: b 1 i , b 2 i , ..., b N − 2 i , where b ki = 1 for k < Q i , else 0 .

Revised: A single-bit error changes the predicted value by one quantization level for a decoding function that counts the number in the binary code.

Original: The decoding function is:
Revised: Hence the decoding function is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: Let e k ( n ) be the error probability of classiﬁerC k for target quantized label Q i = n .
Revised: Let e k ( n ) be the error probability of classiﬁer k for target quantized label Q i = n .

Original: For a uniform distribution of y i in the range [1 , N − 1] , the expected error for BEL-U is:
Revised: For a uniform distribution of y i in the range [1 , N − 1] , the expected error for BEL-U is:

Original: [Equation]
Revised: [Equation]

Original: A similar analysis can be applied for the Johnson code  in whichthe encoding function E BEL-J converts Q i  to a binary code B i=
Revised: A similar analysis is applied for the Johnson code (referred to as BEL-J) in which Q i is encoded to a binary code B i

Original:  b 1 i , b 2 i , ..., b N/ 2 i , where, b ki = 1 for N 2 − Q i < k (cid:54) N − Q
Revised: = b 1 i , b 2 i , ..., b N/ 2 i , where, b ki = 1 for N 2 − Q i < k (cid:54) N − Q

Original: i , else(Appendix B).
Revised: i , else 0 (Appendix B).

Original: Error probability ofbinary classiﬁers:
Revised: Error probability of classiﬁers:

Original: A classiﬁer’s target output is 0 or 1 fordifferent quantized target labels on an ordinal scale, and consists of one or more decision boundaries.
Revised: A classiﬁer’s target output is 0 or 1 for quantized target labels on an ordinal scale, and consists of one or more bit transitions .

Revised: The bit transition is a 0 → 1 or 1 →transition in the classiﬁer’s target output over the numeric range of target labels.

Original: For example, for the unary code shown in Figure 2b, the target output of classiﬁerC 2 is 1 for Q i > 2 and 0 otherwise.
Revised: For example, for the unary code (Figure 2b), the target output of classiﬁer 2 ( b 2 ) changes from 1 to 0 at a bit transition between Q i = 2 and Q i = 3 .

Original: This classiﬁer should learn a decision boundary between target labels in [1 , 2] and [3 , 8] .
Revised: This classiﬁer should learn a decision boundary between targets in [1 , 2 . 5] and [2 . 5 , 8] .

Original: Thus, ideally, a classiﬁer learns a decision boundary for [-each 
 → 1 or 1 → 0 transition in-] the classiﬁer’s target output over the range of target labels.
Revised: Thus, ideally, a classiﬁer learns a decision boundary for [-each 
 → 1 or 1 → 0 transition in-]all bit transitions, and the complexity of the decision boundary increases with the number of bit transitions.

Original: It is known that the error/misclassiﬁcation probabilitye k ( n ) of a classiﬁer tends to increase as the target labelvalue n is closer to the classiﬁer’s decision boundaries (Cardoso & Pinto da Costa, 2007).
Revised: It is known that the error/misclassiﬁcation probability of a classiﬁer tends to increase as the target label is closer to the classiﬁer’s decision boundaries (Cardoso & Pinto da Costa, 2007).

Original: Thus, we approximate e k ( y ) for a classiﬁerC k with t [Figure] decision boundaries as a linear combination of t Gaussian distributions.
Revised: Thus, we approximate e k ( y ) for a classiﬁer k with t bit transitions as a linear combination of t Gaussian distributions.

Original: Here, each Gaussian term is centered around a decision boundary.
Revised: Here, each Gaussian term is centered around a bit transition.

Original: Let f N ( µ,σ 2 ) ( y ) denote the probability density of a normal distribution with mean µ and variance σ 2 .
Revised: Let f N ( µ,σ 2 ) ( y ) denote the probability density of a normal distribution with mean µ and variance σ 2 .

Original: Each classiﬁer for BEL-U encoding has one decision boundary, whereas, each classiﬁer for BEL-J encoding has two decision boundaries (except the ﬁrst and last classiﬁers).
Revised: Each classiﬁer for BEL-U encoding has one bit transition, whereas, each classiﬁer for BEL-J encoding has two bit transitions (except the ﬁrst and last classiﬁers).

Original: e k ( y ) of a classiﬁerC k for BEL-U and BEL-J encoding is approximated as:
Revised: e k ( y ) of a classiﬁer k for BEL-U and BEL-J encoding is approximated as:

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: y

Original: [Equation]
Revised: [Equation]

Original: , where, µ [Equation] = N
Revised: where, µ k = k + 0 .(3)

Original: [Equation]
Revised: [Figure]

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

Original: = N
Revised: [Figure]

Original: [Equation]
Revised: [Equation]

Original: Here, r is a variable that scales up/down the error probabilities.
Revised: Here, r is a scaling factor for the error probabilities.

Original: Figure 3a and Figure 3b show the empirically observed and proposed approximate error probability distributions for two classiﬁers trained using HRNetV2-W18 (Wang et al., 2020) and COFW facial landmark detection dataset (Burgos-Artizzu et al., 2013) for labels y ∈ [1 , 256] .
Revised: Figure 3a and Figure 3b show the empirically observed and proposed approximate error probability distributions for two classiﬁers trained using HRNetV2-W18 (Wang et al., 2020) and COFW facial landmark detection dataset (Burgos-Artizzu et al., 2013) for labels y ∈ [1 , 256] .

Original: Comparison of expected absolute error for BEL-U and BEL-J:
Revised: Comparison of expected absolute error for BEL-U and BEL-J:

Original: Based on equation 2equation 4, we compare expected absolute errors of BEL-U and BEL-J encoding/decoding functions.
Revised: Based on equation 2equation 4, we compare expected absolute errors of BEL-U and BEL-J encoding/decoding functions.

Original: Figure 3c represents the percentage increase in absolute error for BEL-U compared to BEL-J for valid values of variance σ 2 ( y − axis) and scaling factor r ( x − axis) from equation 3 and equation 4.
Revised: Figure 3c represents the percentage increase in absolute error for BEL-U compared to BEL-J for valid values of variance σ 2 ( y − axis) and scaling factor r ( x − axis) from equation 3 and equation 4.

Original: Here,
Revised: Here,

Original: BEL-J has a lower error in the red-colored region (%increase > 0 ), whereas BEL-U has a lower error in the blue-colored region (%increase > 0 ).
Revised: BEL-J has a lower error in the red-colored region (%increase > 0 ), whereas BEL-U has a lower error in the blue-colored region (%increase < 0 ).

Original: The ﬁgure shows that BEL-J or BEL-U has lower error depending upon the values σ and r used to approximate the classiﬁers’ error probability distribution.
Revised: The ﬁgure shows that BEL-J or BEL-U has lower error depending upon the values σ and r used to approximate the classiﬁers’ error probability distribution.

Original: We leave the derivation of these parameters from the ﬁrst principles to future work.
Revised: We leave the derivation of these parameters from the ﬁrst principles to future work.

Original: Further theoretical study of the properties of classiﬁers for ordinal labels can provide bounds on these values, which will also help with encoding/decoding design.
Revised: Further theoretical study of the properties of classiﬁers for ordinal labels can provide bounds on these values, which will also help with encoding/decoding design.

Original: Thus, the choice of {E , D} functions heavily depends upon the classiﬁers’ error probability distribution, which depends upon many factors, including the task objective, dataset, label distribution,  and optimization approach.
Revised: Thus, the choice of {E , D} functions heavily depends upon the classiﬁers’ error probability distribution, which depends upon many factors, including the task, dataset, label distribution, network architecture, and optimization approach.

Original: We provide the derivation of expected error for BEL-U and BEL-J and classiﬁers’ empirical error probability distributions for different architectures, datasets, and encodings in Appendix C-D.
Revised: We provide the derivation of expected error for BEL-U and BEL-J and classiﬁers’ empirical error probability distributions for different architectures, datasets, and encodings in Appendix B-C.

Original: 3. D ESIGN OF E NCODING F UNCTIONS
Revised: 3. D ESIGN OF E NCODING F UNCTIONS

Revised: The distance between two codes (e.g., Hamming distance) gives a measure of error correction capability.

Original: The objective of regression is to minimize the absolute/squared error between predicted and target values, and the encoding function  can be designed to take advantage of this fact.
Revised: The objective of regression is to minimize the absolute/squared error between predicted and target values. The encoding function for BEL can be designed to take advantage of this fact.

Original: Thus, a desirable property for a BEL encoding function is that: the distance between two quantized values should be proportional to the distance between their codes.
Revised: Thus, a desirable property for a BEL encoding function is that: the distance between two quantized values should be proportional to the distance between their codes.

Original: Further, the error probability of classiﬁers tends to increase as the target gets closer to a decision boundary (Figure 3a-3b).
Revised: Further, the error/misclassiﬁcation probability of classiﬁers tends to increase as the target gets closer to a bit transition (Figure 3a-3b).

Original: If the target value is far from the decision boundaries of a given classiﬁer, its output is less likely to be erroneous.
Revised: Since hamming distance (i.e., number of bits changed between two codes) weighs all the bits equally, hamming distance-based code design provides equal error protection capability to all the bits (Wu, 2018;

Original: We note that a hamming distance does not account for which classiﬁers are more likely to mispredict.
Revised: Xie et al., 2002) and does not account for which classiﬁers are more likely to mispredict for given target value.

Original: This gives the second property of a suitable encoding: the distance metric must consider how likely each classiﬁer is to mispredict for some target value to account for classiﬁers’ nonuniform error probability distribution for a regression problem.
Revised: This gives the second property of a suitable encoding: the distance metric must consider how likely each classiﬁer is to mispredict for some target value to account for classiﬁers’ nonuniform error probability distribution for a regression problem.

Revised: This design property covers the impact of bit transitions in codes on continuous predictions.

Original: We evaluate several encodings that to greater or lesser extent exhibit these desirable properties.
Revised: We evaluate several encodings that to greater or lesser extent exhibit these desirable properties.

Original: We focus on encodings that reduce the number of classiﬁers (bits) while balancing the separation between the decision boundaries per classiﬁer.
Revised: We focus on encodings that reduce the number of classiﬁers (bits) while balancing the separation between the bit transitions per classiﬁer.

Original: Below we empirically evaluate the impact of these different encodings on previously proposed deep regression networks.
Revised: Below we empirically evaluate the impact of these different encodings on previously proposed deep regression networks.

Original: We leave the development of procedures for automatically discovering optimized encodings to future work.
Revised: We leave the development of procedures for automatically discovering optimized encodings to future work.

Original: Figure 2 shows example codes for encodings explored.
Revised: Figure 2 shows example codes for encodings explored.

Original: We explain all encodings in detail below.
Revised: We explain all encodings in detail below.

Original: Unary code (U): The unary code(Wikipedia, b) (Figure 2b) is a simple code with one decision boundary per classiﬁer and requires N − 1 bits to encode N values.
Revised: Unary code (U): The unary code (Figure 2b) is a simple code with one bit transition per classiﬁer and requires N − 1 bits to encode N values.

Original: Prior works on ordinal regression by binary classiﬁcation (Li & Lin, 2006;Fu et al., 2018; Niu et al., 2016) uses a similar formulation of classiﬁers as the unary code.
Revised: Prior works on ordinal regression by binary classiﬁcation (Li & Lin, 2006; Niu et al., 2016) uses a similar formulation of classiﬁers as the unary code.

Original: Johnson code (J): The Johnson code sequence (Figure 2c) isused in the design of counters in digital circuits and is based on Libaw-Craig code (Libaw & Craig, 1953).
Revised: Johnson code (J): The Johnson code sequence (Figure 2c) is based on Libaw-Craig code (Libaw & Craig, 1953).

Original: We select this code as it has well-separated decision boundaries and requires N 2 bits compared to N required for unary code.
Revised: We select this code as it has well-separated bit transitions and requires N 2 bits compared to N required for unary code.

Original: This code exempliﬁes the difference between hamming distance and a distance that considers classiﬁers’ error probability distribution.
Revised: This code exempliﬁes the difference between hamming distance and a distance that considers classiﬁers’ error probability distribution.

Original: Although the hamming distance between 1 and 7 is small, since the differing bits ( C 1 and C 4 ) are far from decision boundaries and less likely to be mispredicted, a 1 is less likely to be mispredicted as a 7 .
Revised: Although the hamming distance between 1 and 7 is small, since the differing bits ( C 1 and C 4 ) are far from bit transitions and less likely to be mispredicted, a 1 is less likely to be mispredicted as a 7 , reducing the regression error.

Original: Base+displacement based code (B1JDJ/B2JDJ): We further reduce the number of bits using a base+displacement-based representation.
Revised: Base+displacement based code (B1JDJ/B2JDJ): We further reduce the number of bits using a base+displacement-based representation.

Original: In this representation, a value is represented in base-k using a radix-term b and displacement d via b * k + d. We represent both the base and displacement using Johnson codes.
Revised: In this representation, a value is represented in base-k using a radix-term r and displacement d via r * k + d . r and d are represented using Johnson codes.

Original: Further, to minimize the misclassiﬁcation probability around the design boundary for the radix-term, we adapt reﬂected binary codes  (Gray, 1953).
Revised: Further, to improve the distance between two remote codes, we adapt reﬂected binary codes for term d (Gray, 1953).

Original: We evaluate base-2 (B1JDJ - Figure 2e) and base-4 representations (B2JDJ - Figure 2f).
Revised: We evaluate base-2 (B1JDJ - Figure 2e) and base-4 codes (B2JDJ - Figure 2f).

Original: Binary coded hex - Johnson code (HEXJ): A numberk is converted toits hex value in this code, and each hex digit ( 0 − F) is encoded separately using an 8 − bit Johnson code.
Revised: Binary coded hex - Johnson code (HEXJ): In HEXJ, each digit (0-F) of the hexadecimal representation of a number is converted to an 8-bit binary code using Johnson code.

Original: Figure 2g represents examples of quantized labels, their hex values, and encoded values.
Revised: For example, for the decimal number 47 (i.e., 2F in hex), HEXJ(47) = Concetanate(Johnson(2), Johnson(F)).

Original: A 16 -bit HEXJ code can represent 256 values, reducing the  number of classiﬁers signiﬁcantly for a large number of quantization  levels.
Revised: A 16-bit HEXJ code can represent numbers in the range of 00 to FF (a total of 256). The number of bits increases sublinearly with the number of quantization levels for HEXJ, making it suitable for regression problems with many quantization levels.

Original: Hadamard code (HAD): Hadamard codes (Wikipedia, a) are widely used as error-correcting codes due to their fault-tolerant properties and also have been used to encode classes in multiclass classiﬁcationto improve the accuracy and robustness to adversarial attacks (Dietterich & Bakiri, 1995; Verma & Swami, 2019).
Revised: Hadamard code (HAD): Hadamard codes (Bose & Shrikhande, 1959) are widely used as errorcorrecting codes due to their fault-tolerant properties and also have been used to encode classes in multiclass classiﬁcation (Dietterich & Bakiri, 1995; Verma & Swami, 2019).

Original: It requires N bits to encode N values.
Revised: It requires N bits to encode N values.

Revised: However, in the Hadamard code, each column has many bit transitions, increasing the complexity of the decision boundary to be learned by each binary classiﬁer.

Revised: Further, the Hadamard code does not follow the desirable properties proposed in this work for regression tasks.

Original: We evaluate the Hadamard code to show that it is not suitable for regression.
Revised: We evaluate the Hadamard code to show that it is not suitable for regression (Section A).

Original: 3. D ESIGN OF D ECODING F UNCTIONS
Revised: 3. D ESIGN OF D ECODING F UNCTIONS

Original: We explore three decoding functions: custom decoding, correlation-based decoding, and expected value-based decoding.
Revised: We explore three decoding functions: custom decoding, correlation-based decoding, and expected correlation-based decoding.

Original: Custom decoding functions are speciﬁc to the encoding function, and are only evaluated for unary and Johnson codes.
Revised: Custom decoding functions are speciﬁc to the encoding function, and are only evaluated for unary and Johnson codes.

Original: In contrast, correlation-based decoding, ﬁrst explored in prior work studying ECOC for multiclass classiﬁcation (Allwein et al., 2001), can be applied to all codes.
Revised: In contrast, correlation-based decoding, ﬁrst explored in prior work studying ECOC for multiclass classiﬁcation (Allwein et al., 2001), can be applied to all codes.

Original: For labels in { 1 , 2 , ..., N } , we deﬁne a code matrix C of size N × M , where M is the number of bits/classiﬁers used for the binary-encoded label.
Revised: For labels in { 1 , 2 , ..., N } , we deﬁne a code matrix C of size N × M , where M is the number of bits/classiﬁers used for the binary-encoded label.

Original: Each row C k, : in this matrix represents the binary code for label Q i = k .
Revised: Each row C k, : in this matrix represents the binary code for label Q i = k .

Original: For example, Figure 2b can be considered a code matrix, where the ﬁrst row represents code for label Q i = 1 .
Revised: For example, Figure 2b can be considered a code matrix, where the ﬁrst row represents code for label Q i = 1 .

Original: Let ˆ Z
Revised: Let ˆ Z

Original: i ∈
Revised: i ∈

Original: R M denote the output logit values of the classiﬁers.
Revised: R M denote the output logit values of the classiﬁers.

Original: For decoding, the row with the highest correlation with the output ˆ
Revised: For decoding, the row with the highest correlation with the output ˆ

Original: Z i is selected as the decoded label.
Revised: Z i is selected as the decoded label.

Original: Output logit value vector ˆ Z i is used instead of output binary code ˆ B i to ﬁnd the correlation as it uses the conﬁdence of a classiﬁer to make a more accurate prediction.
Revised: Output logit value vector ˆ Z i is used instead of output binary code ˆ B i to ﬁnd the correlation as it uses the conﬁdence of a classiﬁer to make a more accurate prediction.

Original: For target quantized labels Q i ∈  1 , 2 , ..., N  , the decoding function is deﬁned as:
Revised: For target quantized labels Q i ∈ { 1 , 2 , ..., N } , the decoding function is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: However, D GEN outputs a quantized prediction, introducing quantization error.
Revised: However, D GEN outputs a quantized prediction, introducing quantization error.

Original: To remedy this concern and demonstrate the potential of more sophisticated decoding rules, we propose and evaluate an expected value-based decoding function, which allows prediction of real-valued label ˆ y i .For [Figure] [Table]
Revised: To remedy this concern and demonstrate the potential of more sophisticated decoding rules, we propose and evaluate an expected correlation-based decoding function, which allows prediction of real-valued label ˆ y i .

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table] [Figure] [Table] target labels y i ∈ [1 , N ] , the decoding function is deﬁned as:
Revised: For target labels y i ∈ [1 , N ] , the decoding function is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: GEN- [Equation] kσ k , whereσ k = e ˆ Z i · C k, : (cid:80)
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Training loss functions:
Revised: Training loss functions:

Original: A deep neural network with multiple output binary classiﬁers can be trained using the binary cross-entropy (BCE) loss L BCE (cid:0)
Revised: A deep neural network with multiple output binary classiﬁers can be trained using the binary cross-entropy (BCE) loss L BCE (cid:0)

Original: ˆ Z i , E ( Q i ) (cid:1) .
Revised: ˆ Z i , E ( Q i ) (cid:1) .

Original: However, this loss minimizes the mismatch between predicted and target code but does not directly minimize the error between the target and predicted values.
Revised: However, this loss minimizes [Figure] [Table] [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Revised: [Figure]

Original: For decoding function D GEN , cross-entropy loss L CE (cid:0) C ˆ Z i , Q i (cid:1) can be used for training.
Revised: [Table] [Figure] [Table] [Figure]

Revised: [Table]

Original: C represents the  code matrix for given encoding.
Revised: [Figure] [Table] [Figure] the mismatch between predicted and target code but does not directly minimize the error between the target and predicted values.

Original: Similarly, for decoding function 
Revised: Decoding functions D GEN and D GEN-EX can be used to calculate the loss and minimize the mismatch between decoded predictions and target values directly. Decoding function D GEN ﬁnds the correlation between each row of the code matrix ( C k, : ) and the output ˆ Z i . C ˆ Z i gives the correlation vector, and the index with the highest correlation is used as the predicted label. In this case, cross-entropy loss L CE (cid:0) C

Original:  D GEN-EX , which predicts a continuous value, L1 or L2 loss L L1/L2 (cid:0) D GEN-EX ( ˆ Z i ) , y i (cid:1) can also be used for training.
Revised: ˆ Z i , Q i (cid:1) can be used to train the network. Similarly, for decoding function D GEN-EX , which predicts a continuous value, L1 or L2 loss L L1/L2 (cid:0) D GEN-EX ( ˆ Z i ) , y i (cid:1) can also be used for training.

Original: Wefurther evaluate multiple combinations of decoding and loss functions in Section 4.
Revised: We evaluate multiple combinations of decoding and loss functions in Section 4.

Original: 3. R EGRESSION N ETWORK A RCHITECTURE FOR BEL
Revised: 3. R EGRESSION N ETWORK A RCHITECTURE FOR BEL

Original: A deep regression network typically consists of a feature extractor and regressor.
Revised: A deep regression network typically consists of a feature extractor and regressor.

Original: Figure 4a represents a network architecture for direct regression.

Original: The regressor consists of a fully connected layer between theoutput of the feature extractor (i.e., feature vector) and output logits.
Revised: The regressor consists of a fully connected layer between the feature extractor’ output (i.e., feature vector) and output logits.

Original: Figure 4b shows the modiﬁed neural network architecture for BEL.
Revised: Figure 4a represents a network architecture for direct regression.

Original: The number of  logits is increased to the number of classiﬁers (bits) used.
Revised: In BEL, the number of output logits is increased to the number of classiﬁers (bits) used.

Original: The size of the fully connected regression layer increases with the number of output logits.
Revised: The size of the fully connected regression layer increases with the number of output logits.

Original: However, empirically, we ﬁnd small feature vectors sufﬁce as the output logits are highly correlated for the encoding functions we explore.
Revised: However, empirically, we ﬁnd small feature vectors sufﬁce as the output logits are highly correlated for the encoding functions we explore.

Original: Adding a fully connected layer to reduce feature vector size to θ can reduce the number of parameters and provide a trade-off between the model size and accuracy.
Revised: Adding a fully connected layer to reduce feature vector size to θ reduces the number of parameters and provide a trade-off between the model size and accuracy.

Revised: Figure 4b shows the modiﬁed neural network architecture for BEL.

Original:  E VALUATION
Revised:  E VALUATION

Original: Table 1 summarizes the tasks, datasets, and network architectures used for the evaluation of BEL.
Revised: Table 1 summarizes the tasks, datasets, and network architectures used for the evaluation of BEL.

Original: These tasks are commonly used for evaluation of regression approaches by prior works due to the complexity of problem and network architectures (Díaz & Marathe, 2019).
Revised: These tasks are commonly used for evaluation of regression approaches by prior works due to the complexity of problem and network architectures (Díaz & Marathe, 2019).

Original: Landmark-free 2D head pose estimation (HPE) aims to ﬁnd a human head’s pose in terms of three angles: yaw, pitch, and roll from a 2D image without landmarks.
Revised: Landmark-free 2D head pose estimation (HPE) aims to ﬁnd a human head’s pose in terms of three angles: yaw, pitch, and roll from a 2D image without landmarks.

Original: Facial landmark detection (FDL) is a problem of detecting the ( x, y ) coordinates of keypoints in a given face image.
Revised: Facial landmark detection (FLD) is a problem of detecting the ( x, y ) coordinates of keypoints in a given face image.

Original: Age estimation aims to predict the age of a 
Revised: Age estimation aims to predict the age of a person from an image.

Revised: In end-to-end autonomous driving steering

Original: [Figure]
Revised: [Figure]

Original: person from an image.
Revised: [Figure] rr [Figure]

Original: In end-to-end autonomous driving, the steering wheel’s next angle is predicted from an image of the road.
Revised: ( M

Original: Normalized Mean Error (NME) and Mean Absolute Error (MAE) with resepct to raw real-valued labels are used as the evaluation metric for FDL and the rest, respectively.
Revised: [Figure] / N M E ) HPE D 1 D 2 D

Revised: [Figure]

Revised: D

Revised: [Figure]

Revised: D

Revised: [Figure]

Revised: [Figure]

Revised: HPE

Revised: D 1 D 2 D 3 D 4 D

Revised: [Figure]

Revised: HPE

Revised: D 1 D

Revised: [Figure]

Revised: HPE

Revised: D

Revised: [Figure]

Revised: D

Revised: [Figure]

Revised: D

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: FLD

Revised: D

Revised: [Figure]

Revised: FLD

Revised: [Figure]

Revised: ( M A E / N M E )

Revised: FLD

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: [Figure]

Revised: FLD

Revised: [Figure]

Revised: AE

Revised: [Figure]

Revised: AE

Revised: [Figure]

Revised: PN

Revised: U

Revised: J

Revised: B1JDJ

Original: Our main goal is to compare the regression approach; hence we keep the feature extractor, data augmentation, evaluation protocols, and the number of training iterations  uniform across different methods for each benchmark. 
Revised: B2JDJ HEXJ The feature extractor, data augmentation, evaluation protocols, and the number of training iterations are kept uniform across different methods for each benchmark. We also evaluate direct regression and multiclass classiﬁcation as baseline regression approaches.

Revised: For direct regression, L1 or L2 loss functions are used.

Revised: Label values are scaled to reduce the range of labels.

Revised: The loss function and the scaling factors are set using hyperparameter tuning.

Original:  We report average of ﬁve training runs and error margin of 95% conﬁdence interval.
Revised: In the multiclass classiﬁcation-based regression, the target values are quantized and converted to a class. The network is trained using cross-entropy loss in this case. In our evaluation, the entire network (i.e., feature extractor and regressor) are trained end-to-end for direct regression, multiclass classiﬁcation, and BEL. We report average of ﬁve training runs and error margin of 95% conﬁdence interval.

Original: Details on datasets, training parameters, related work for speciﬁc tasks, and other evaluation metrics are provided in Appendix D-G. BEL introduces several design parameters for regression by binary classiﬁcation.
Revised: Details on datasets, training parameters, related work for speciﬁc tasks, and other evaluation metrics are provided in Appendix C. BEL introduces several design parameters for regression by binary classiﬁcation.

Original: We evaluate different encoding ( E ), decoding ( D ), and training loss ( L ) functions for BEL across all the benchmarks as shown in Figure 5. We study if design parameters impact accuracy and analyze the relationship between design parameters and error.
Revised: We evaluate different encoding ( E ), decoding ( D ), and training loss ( L ) functions for BEL across all the benchmarks and study the extent and nature of the impact of these design parameters on accuracy.

Original: Encoding function ( E ): Figure 5 plots MAE/NME using different encodings.
Revised: Encoding function ( E ): Figure 5 plots error (MAE or NME) using different encodings.

Original: We do not show results for Hadamard codes in the ﬁgure as it results in signiﬁcantly higher error than other encodings.
Revised: We do not show results for Hadamard codes in the ﬁgure as it results in signiﬁcantly higher error than other encodings (Appendix A).

Original: On average, Hadamard codes result in 60% higher error than J encoding for GEN decoding and CE loss function, which shows that these codes are unsuitable for regression.
Revised: On average, Hadamard codes result in 60% higher error than J encoding, which shows that these codes are unsuitable for regression.

Original: The results show the encoding function has a signiﬁcant impact on the accuracy of several benchmarks and the bestperforming encoding function varies across datasets and network architecturesfor a given task (e.g., HPE1 and HPE3).
Revised: The results show the encoding function has a signiﬁcant impact on the accuracy of several benchmarks and the best-performing encoding function varies across tasks, datasets, and network architectures (e.g., HPE1 and HPE3 are trained on the same dataset and different architecture).

Original: In Section 3.1 we observed that which encoding/decoding functions result in lower error depends upon the  error probability distribution of the classiﬁers.
Revised: In Section 3.1 we observed that which encoding/decoding functions result in lower error depends upon the classiﬁers’ error distribution.

Original: For HPE4, FDL1, andAE1,
Revised: For HPE3, FLD1, and

Original: Johnson does better than unary; we attribute this to misclassiﬁcation errors occurring more frequently near decision boundaries in these tasks.
Revised: AE1, J does better than U for D1 (decoding functions used for analytical comparison in Section 3.1); we attribute this to misclassiﬁcation errors occurring more frequently near bit transitions in these tasks based on the analytical study.

Original: The encoding function impacts the number ofoutput classiﬁers and the complexity of  function to be learned by a classiﬁer (i.e., number of decision boundaries).
Revised: The encoding function impacts the number of classiﬁers and the complexity of the function to be learned by a classiﬁer.

Original: We observe a trade-off between these two parameters.
Revised: We observe a trade-off between these two parameters.

Original: For some benchmarks, the availability of sufﬁcient training data and network capacity facilitates the learning of complex classiﬁers such as B2JDJ.
Revised: For some benchmarks, the availability of sufﬁcient training data and network capacity facilitates the learning of complex classiﬁers such as B2JDJ.

Original: In such a case, a reduced number ofbinary classiﬁers compared to U, J, or B1JDJ codes results in a lower error for B2JDJ.
Revised: In such a case, a reduced number of classiﬁers compared to U, J, or B1JDJ codes results in a lower error.

Original: We provide empirical results for the same in Appendix B.
Revised: We provide empirical results for the same in Appendix A.

Original: Decoding ( D ) and training loss ( L ) functions: We explore three decoding functions (Section 3.3).
Revised: Decoding ( D ) and training loss ( L ) functions: We explore three decoding functions (Section 3.3).

Original: The training loss functions can be selected from BCE, CE, L1, or L2 loss functions.
Revised: The training loss functions can be selected from BCE, CE, and L1/L2 loss.

Original: However, not all the combinations of decoding and loss functions perform well.
Revised: However, not all the combinations of decoding and loss functions perform well.

Original: For example, CE, L1, or L losses do not use BEL-J or BEL-U decodings.
Revised: For example, CE, L1, or L2 losses do not use BEL-J or BEL-U decodings.

Original: Therefore, optimizing the network for these losses does not  minimize the absolute error between the target and decoded predictions.
Revised: Therefore, optimizing the network for these losses does not directly minimize the absolute error between the target and decoded predictions.

Original: In contrast, GENEX decoding is used for L1/L2 loss function calculation; in such a case, the training loss directly
Revised: In contrast, GENEX decoding is used for L1/L2 loss function calculation; in such a case, the training loss directly

Original: [Table]
Revised: [Table]

Original: minimized the error between target values and decoding values.
Revised: minimizes the error between target values and decoded values.

Original: Out of such twelve combinations, we present results for ﬁve combinations.
Revised: Out of such nine combinations, we present results for ﬁve combinations.

Original: Figure 5 compares MAE/NME achieved by different decoding/loss functions combinationsacross several benchmarks and highlights the range of error variations across decoding/loss functions.
Revised: Figure 5 compares error (MAE or NME) achieved by different D / L combinations and highlights the range of error variations across D / L .

Original: D GEN-EX results in the lowest error for the majority of the benchmarks as it reduces quantization error and also takes advantage of the output logit conﬁdence values to decode the output code.
Revised: D GEN-EX results in the lowest error for the majority of the benchmarks as it reduces quantization error and also takes advantage of the output logit conﬁdence values.

Original: D GEN-EX consistently perform better than D GEN function that has been used for multiclass classiﬁcaiton by prior works (Allwein et al., 2001).
Revised: D GEN-EX consistently perform better than D GEN function that has been used for multiclass classiﬁcation by prior works (Allwein et al., 2001).

Original: For loss function, we observe that the use of CE or L1/L2 loss results in a lower error with D GEN-EX  as the training loss function objective minimized the error between decoded labels and target labels.
Revised: The use of CE or L1/L2 loss results in a lower error with D GEN-EX for most benchmarks as the training loss function minimizes the error between decoded labels and target labels.

Original: In contrast, BCE loss does not directly minimize the error between decoded and target labels.

Original: Comparison of BEL with regression approaches:
Revised: Comparison of BEL with regression approaches:

Original: Table 2 compares BEL with other approaches for different benchmarks (Table 1).
Revised: Table 2 compares BEL with other approaches for different benchmarks (Table 1).

Original: We explore and evaluate multiple combinations of encoding ( E ), decoding ( D ), and loss ( L ) functions for BEL in this work.
Revised: We explore and evaluate multiple combinations of encoding ( E ), decoding ( D ), and loss ( L ) functions for BEL in this work.

Original: The best combination of these functions and its error is reported in Table 2.
Revised: The 20% of the training set is used as validation set and the validation error is used to choose the best BEL approach. An ablation study for using more fully connected layers for direct regression and multiclass classiﬁcation in Appendix A.

Original: BEL results in lower error than direct regression and multiclass classiﬁcation and even outperforms task-speciﬁc regression approaches for several benchmarks.
Revised: BEL results in lower error than direct regression and multiclass classiﬁcation and even outperforms task-speciﬁc regression approaches for several benchmarks.

Original: The results show no single combination of encoding/decoding/loss functions evaluated was best for all benchmarks but also demonstrate BEL enables accuracyimprovements across a range of regression problems.
Revised: The results show no single combination of encoding/decoding/loss functions evaluated was best for all benchmarks but also demonstrate BEL improves accuracy across a range of regression problems.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: This work proposes binary-encoded labels  to pose regression as binary classiﬁcation.
Revised: This work proposes binary-encoded labels (BEL) to pose regression as binary classiﬁcation.

Original: We propose a taxonomy identifying the key design aspects for regression by binary classiﬁcation and demonstrate the impact of classiﬁcation error and encoding/decoding functions on the expected label error.
Revised: We propose a taxonomy identifying the key design aspects for regression by binary classiﬁcation and demonstrate the impact of classiﬁcation error and encoding/decoding functions on the expected label error.

Original: Different encoding, decoding, and loss functions are explored to evaluate our approach using four complex regression tasks.
Revised: Different encoding, decoding, and loss functions are explored to evaluate our approach using four complex regression tasks.

Original: Binary-encoded labels result in an average 9 . 1% , 8 . 4% , and 12 .
Revised: BEL results in an average 9 . 9% , 15 .

Original: 9% lower error than direct regression, multiclass classiﬁcation, and task-speciﬁc regression approaches, respectively.
Revised: 5% , and 7 . 2% lower error than direct regression, multiclass classiﬁcation, and task-speciﬁc regression approaches, respectively.

Original: BEL improves accuracy over state-of-the-art approaches for head pose estimation (BIWI, AFLW2000), facial landmark detection (COFW), age estimation (AFAD), and end-to-end autonomous driving (PilotNet).
Revised: BEL improves accuracy over state-of-the-art approaches for head pose estimation (BIWI, AFLW2000), facial landmark detection (COFW), age estimation (AFAD), and end-to-end autonomous driving (PilotNet).

Original: Our analysis and empirical evaluation in this work demonstrate the potential of the vast design space of BEL for regression problems and the importance of ﬁnding suitable design parameters for a given task.
Revised: Our analysis and empirical evaluation in this work demonstrate the potential of the vast design space of BEL for regression problems and the importance of ﬁnding suitable design parameters for a given task.

Original: The best performing encoding/decoding function pair may be task, dataset, and network speciﬁc.
Revised: The best performing encoding/decoding function pair may be task, dataset, and network speciﬁc.

Original: A possibility this suggests, which we leave to future work, is that it may be beneﬁcial to develop automated approaches to optimizing these functions,
Revised: A possibility this suggests, which we leave to future work, is that it may be beneﬁcial to develop automated approaches to optimizing these functions,

Original: Reproducibility:
Revised: Reproducibility:

Original: We have provided a detailed discussion about training hyperparameters, experimental setup, and modiﬁcations made in publicly available network architectures in Appendix D-E for all benchmarks.
Revised: We have provided a detailed discussion about training hyperparameters, experimental setup, and modiﬁcations made in publicly available network architectures in Appendix D.1-D. for all benchmarks.

Original: We will add a comment directed to the reviewers and area chairs to provide a link to our code.
Revised: We have added a comment directed to the reviewers and area chairs to provide a link to our training and inference code with trained models.

Revised: We will publicly release our code if accepted.

Original: Code
Revised: Code

Original: of Ethics
Revised: of Ethics

Original: Some of the major applications of regression problems are artiﬁcial intelligence and autonomous machines, and regression improvement can accelerate the development of autonomous systems.
Revised: Some of the major applications of regression problems are artiﬁcial intelligence and autonomous machines, and regression improvement can accelerate the development of autonomous systems.

Original: However, depending upon the use, autonomous systems can have some negative societal impacts, such as job loss in some sectors and ethical concerns
Revised: However, depending upon the use, autonomous systems can have some negative societal impacts, such as job loss in some sectors and ethical concerns

Original: A A BLATION STUDY
Revised: A A BLATION STUDY

Original: Impact of quantization and decoding functions:
Revised: Impact of BEL encoding, decoding, and loss functions:

Revised: We propose multiple combinations of encoding, decoding

Revised: [Table]

Revised: and loss functions that can be used with BEL.

Revised: In Tables 3- 13, we show the effect of each combination of encoding, decoding, and loss function on the error of the model.

Revised: Although general trends exist and some combinations perform consistently well across dataset, the optimal combination varies based on the dataset.

Revised: [Table]

Revised: [Table]

Revised: [Table]

Revised: [Table]

Original: Table 3 compares decoding functions equation 5 and equation 6 that predict quantized and smooth value respectively for different number of quantization intervals.
Revised: [Table] -W18 feature extractor (FLD2) [Table] [Table]

Original: This analysis aims to study the impact of quantization levels on accuracy and the use of expected value-based decoding ( D GEN − EX ) to reduce quantization error.
Revised: [Table] [Table] [Table] [Table]

Original: As shown in Table 3,  the use of  ( D GEN − EX ) consistently results in lower errorcompared to ( D GEN ) for all quantization levels, where D GEN − EX and  D GEN predict continuous  and quantized value, respectively.
Revised: As discussed in Section 3, a real-valued label is quantized to a discrete value in { 1 , 2 , ..., N } . In Table 14, we show the effect of increasing the number of quantization levels ( N ) on the error for correlation-based decoding ( D GEN , which returns a quantized prediction) and expected correlation-based decoding ( D GEN-EX , which returns a continuous prediction). We vary the number of quantization levels from 64 to 256 and observe that there exists a tradeoff between reducing quantization error and using fewer classiﬁers.

Original: Moreover, the error is lower for 128 quantization levels than  256 as the improvement resulting from fewer binary classiﬁers is higher than the increase inthe quantization error.
Revised: The error is lower forquantization levels than it is for 256 as the improvement resulting from fewer binary classiﬁers is higher than the increase in quantization error.

Original: However, as quantization levels are decreased further, the advantage of using fewer classiﬁers is diminished by high quantization error. Thus there exists a trade-off between the number of binary classiﬁers and quantization error.
Revised: Moreover, the use of D GEN-EX consistently results in lower error compared to D GEN .

Original: [Table]
Revised: [Table]

Original: Effect of the number of training samples on BEL: We further analyze the effect of  the number of  training samples on the suitable encoding. The encoding functions explored in this work have varying complexity.
Revised: Effect of the number of training samples on BEL: As discussed in Section 4, the performance of different encoding functions varies depending on the availability of sufﬁcient training data.

Original: Using a more complex encoding such as B2JDJ or HEXJ reduces the number of classiﬁers; however, it increases each classiﬁer’s complexity (i.e., the number of decision boundaries to be learned).
Revised: In

Original: Table 4 compares the error for different training dataset sizes and encodings for facial landmark detection benchmarks.
Revised: Table 15, we analyze the effect of the number of available training samples for both simple and complex encodings.

Original: Empirical results suggest that simpler encodings (e.g., J) are more suitable than  complex encodings as the training dataset size decreases.
Revised: As the number of training samples decreases, simpler encodings (U and J) perform better than more complex encodings (B1JDJ, B2JDJ, and Hex). Using a more complex encoding reduces the number of classiﬁers; however, it increases each classiﬁer’s complexity (i

Original: [Table]
Revised: [Table]

Original: [Table]
Revised: e. the number of bit transitions) and thus performs poorly with less training data.

Original: [Table]
Revised: [Table]

Original: Effect of reﬂected binary conversion:
Revised: Effect of reﬂected binary conversion:

Original: We use reﬂected binary to increase the distance between distant labels based on the design properties of suitable regression encodings we proposed.
Revised: As mentioned in Section 3.2, we use reﬂected binary to increase the distance between distant labels based on the design properties of suitable regression encodings we proposed.

Original: Table 5 shows the impact of using reﬂected binary conversion on error for facial landmark detection benchmarks.
Revised: Table 16 shows the impact of using reﬂected binary conversion on error for facial landmark detection benchmarks.

Original: As shown in the table, the use of reﬂected binary signiﬁcantly reduces the error. 
Revised: As shown in the table, the use of reﬂected binary signiﬁcantly reduces the error. [Table]

Original: Use of binary heamaps:Facial landmark detection approaches typically use heatmap regression.
Revised: Use of binary heamaps:

Revised: Facial landmark detection approaches typically use heatmap regression.

Original: We also evaluate BEL-H-x, in which the real-valued heatmaps are converted to binary heatmaps with 8 quantization levels.
Revised: We also evaluate BEL-H-x, in which the real-valued heatmaps are converted to binary heatmaps with 8 quantization levels.

Original: Table 6 shows the impact of usingreﬂected binary conversion on error for facial landmark detection benchmarks.
Revised: Table 17 shows the impact of using binary heatmaps on error for facial landmark detection benchmarks.

Original: In BEL-H-x, each real-valued pixel of the target heatmap is converted to a binary code using eight quantization levels.
Revised: In BEL-H-x, each real-valued pixel of the target heatmap is converted to a binary code using eight quantization levels.

Original: For unary code, a 64 × 64 real-valued heatmap of one facial landmark is converted to eight 64x64 binary heatmaps, resulting in 32 ,( 8 × 64 × 64 ) binary classiﬁers compared to 512 for BEL-U.
Revised: For unary code, a 64 × 64 real-valued heatmap of one facial landmark is converted to eight 64x64 binary heatmaps, resulting in 32 , 768 ( 8 × 64 × 64 ) binary classiﬁers compared to 512 for BEL-U.

Original: We believe that training a high number of binary classiﬁers results in poor accuracy for BEL-H-x.
Revised: We believe that training a high number of binary classiﬁers results in high error for BEL-H-x.

Original:  Hyperparameter θ : We introduce ahyperparameter θ in the proposed learning approach, where θ denotes the size of the feature vector  before the output layer.
Revised: [Table] Hyperparameter θ : As shown in Figure 4b, we introduce a feature vector of size θ before the output layer.

Original: A lower value of θ can be used for encodings with a high correlation between the classiﬁers’ output.

Original: Increasing the value of θ decreases the error but increases the number of network parameters.

Original: Thus there exists a trade-off between the number of model parameters and error.

Original: Figure 6 compares the decrease in the error for different encodings and θ values.
Revised: Figure 6 compares the decrease in the error for different encodings and θ values.

Original: We observe that more complex encodings that show lower correlation between the classiﬁers’ outputs beneﬁt more from an increase in the  θ value.
Revised: We observe that more complex encodings that show lower correlation between the classiﬁers’ outputs beneﬁt more from an increase in the value of θ , while a lower value of θ can be used for encodings with a high correlation between the classiﬁers’ outputs.

Original: [Figure]
Revised: [Figure]

Revised: Effect of increasing the number of fully connected layers: For BEL, we propose to add a fully connected layer in the regressor to reduce the feature vector size to θ and thus decrease the number of parameters in the regressor.

Revised: We perform an ablation study to study the impact of these added fully connected layers on direct regression, multiclass classiﬁcation, and binary encoded labels.

Revised: Tableprovides the error (MAE or NME) for direct regression and multiclass classiﬁcation with one or two fully connected layers after the feature extractor.

Revised: Table 19 shows the impact of increasing the number of fully connected layers up to three on the error (MAE or NME) for different regression approaches.

Revised: We observe that increasing the number of fully connected layers in direct regression and multiclass classiﬁcation does not improve the accuracy for most benchmarks (possibly due to overparameterization).

Revised: BEL with two fully connected layers outperforms direct regression and multiclass classiﬁcation in both cases.

Revised: Furthermore, even for a higher number of fully connected layers in BEL, different encoding functions result in the lowest error.

Revised: [Table]

Revised: [Table]

Revised: Training-validation set based evaluation Ideally, a validation set should be used for model selection.

Revised: Hence we have reevaluated the benchmarks with 20% of the training data being used as a validation set to select the best model (i.e., which model is the best over multiple epochs).

Revised: Datasets used in benchmarks do not provide separate validation datasets.

Revised: We use 80% of the training data for training and use validation error to select the best design parameters for BEL and the best model for all approaches.

Revised: Since earlier works use 100% training data for the reported results and use test error for model selection, we have re-run specialized approaches (if possible), direct regression, and multiclass classiﬁcation.

Revised: It was not possible for us to re-run experiments for all specialized approaches due to resource constraints, and the comparison is conservative for many benchmarks since we use less data while selecting the best BEL approach than prior works used for model selection.

Revised: Table 20 compares different regression approaches for this evaluation setup.

Revised: Note that the additional results do not diminish the effectiveness of BEL and BEL outperforms direct regression and multiclass classiﬁcation for all benchmarks and specialized approaches for several benchmarks.

Revised: Here, r is a scaling factor for the error probabilities.

Original: B E XPECTED E RROR D ERIVATION
Revised: B E XPECTED E RROR D ERIVATION

Original: This section explains the expected error equations used to compare BEL-U and BEL-J in Section 3.1.
Revised: This section explains the expected error equations used to compare BEL-U and BEL-J in Section 3.1.

Original: First we explain the decoding function used for BEL-U and derive the relation between the expected error of labels and binary classiﬁers.
Revised: We ﬁrst explain the decoding function used for BEL-U and derive the relation between the expected error of labels and binary classiﬁers for BEL-U.

Original: Lastly, we explain the encoding/decoding functions and expected error relation for BEL-J.
Revised: Then, we explain the encoding/decoding functions and expected error relation for BEL-J.

Original: B. P RELIMINARIES
Revised: B. P RELIMINARIES

Original: Figure 7 represent the training and inference ﬂow for binary-encoded labels.
Revised: We compose binary-encoded labels as a quantization function q , an encoding function E , and a decoding function D .

Original: Here, y i ∈ R represent the target label for sample i .
Revised: Given a sample i drawn from a dataset with minimum label a and maximum label b , let y i ∈ [ a, b ] represent the target label for that sample. Assuming uniform quantization, the range of target labels can be quantized using q : [ a, b ] → { 1 , 2 , ..., N − 1 } through equation 7. [Equation]

Original: Q i represent the  target quantized level, which is converted to the target binary code B i ∈ { 0 , 1 } Musing an encoding function E .C 1 , ..., C M binary classiﬁers are used in the proposed approach.
Revised: [Table] We deﬁne the encoding function E : { 1 , 2 , ..., N − 1 } → { 0 , 1 } M to convert a target quantized level Q i ∈ { 1 , 2 , ..., N − 1 } to a binary code B i ∈ { 0 , 1 } M .

Original: ˆ B i ∈ { 0 , 1 } M is the predicted binary code, which is then decoded to  ˆ y i∈ R using a decoding function D .
Revised: We further deﬁne the decoding function D : { 0 , 1 } M → [ a, b ] to convert the predicted binary code ˆ B i to the predicted label ˆ y i .

Original: For theanalysis in this section, we consider target label y i ∈ [1 , N − 1] and the quantized target label Q i ∈ { 1 , 2 , ..., N − 1 } . The decoding functions used in this analysis predict the quantized label, which introduces quantization error. We do not include quantization error in the expected label error for our analysis as it is constant for  BEL-U and BEL-J. The expected value of absolute error between the target y and predicted labels ˆ y is used for the analysis as typically mean absolute error is used as the evaluation metric in regression problems. 
Revised: Although the decoding functions used in this analysis predict the quantized label and introduce quantization error, we do not include quantization error in the expected label error for our analysis and it is constant for both BEL-U and BEL-J. The expected value of absolute error between the target y and predicted labels ˆ y is used for the analysis as typically mean absolute error is used as the evaluation metric in regression problems. Let us denote the error probability of a binary classiﬁer C k used to predict bit k in a binary code

Original: ˆ b ki = T indicates correct binary classiﬁcation by classiﬁer C k ( ˆ b ki =
Revised: B i = E ( n ) as e k ( n ) , where n is the target quantized label Q i . Then, e [Equation]

Original: =  b ki  ) for sample i. ˆ b ki = F indicates incorrect binary classiﬁcation by classiﬁer C k ( ˆ b ki (cid:54) =
Revised: = Pr (ˆ b ki = F ) for target label Q i = n and target binary code B i

Original: b ki ) for sample i. Let e k ( n ) denote the error probability of a binary classiﬁer C k , when target quantized label Q i = n and B i = b 1 i , b 2 i , ..., b
Revised: [Equation]

Original: Mi = E( n ) .
Revised: E

Original: [Equation]
Revised: [Equation]

Original: target label Q i = n and target binary code [Equation]
Revised: (8) where ˆ b ki = T indicate a correct binary classiﬁcation by classiﬁer C k ( ˆ b ki ==

Original: i [Equation] Q
Revised: b ki ) for sample

Original: [Equation]
Revised: i and ˆ b ki = F indicat an incorrect binary classiﬁcation by classiﬁer C k ( ˆ b ki (cid:54) =

Original: [Equation]
Revised: b ki ) for sample i.

Original:  E XPECTED E RROR FOR BEL-U
Revised: B. E XPECTED E RROR FOR BEL-U

Original: Encoding and decoding functions The encoding and decoding functions for BEL-U are deﬁned as:E BEL-U ( Q i ) = b
Revised: Encoding and decoding functions The encoding and decoding functions for BEL-U are deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: i , b 2 i , .., b N − 2 i , where b ki = (cid:26) 1 k < Q i 

Revised: , where b ki = (cid:26) 1 k < Q i

Original: Otherwise
Revised: Otherwise

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

Original: [Figure]

Original: [Equation]

Original: [Figure]

Original: [Equation]

Original: [Figure]

Original: [Equation]

Original: [Figure] [Equation] [Figure] Expected error For target quantized label Q i = n ( n ∈ { 1 , 2 , ..., N − 1 } ), ignoring the quantization error, the expected error between target y i and predicted label ˆ y i can be derived as:
Revised: Expected error For target quantized label Q i = n ( n ∈ { 1 , 2 , ..., N − 1 } ), ignoring the quantization error, the expected error between target y i and predicted label ˆ y i can be derived as:

Original: [Equation]
Revised: [Equation]

Original: equation 7)
Revised: equation 8)

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

Original: For a uniform distribution of target labels in the range [1 , N − 1] , the expected error can be derived as:
Revised: For a uniform distribution of target labels in the range [1 , N − 1] , the expected error can be derived as:

Original: [Equation]
Revised: [Equation]

Original: . E XPECTED E RROR FOR BEL-J
Revised: B. E XPECTED E RROR FOR BEL-J

Original: Encoding and decoding functions For target quantized label Q i ∈ { 1 , 2 , ..., N − 1 } , BEL-J encoding requires N 2 bits/binary classiﬁers.
Revised: Encoding and decoding functions For target quantized label Q i ∈ { 1 , 2 , ..., N − 1 } , BEL-J encoding requires N 2 bits/binary classiﬁers.

Original: The encoding for BEL-J can be deﬁned as:
Revised: The encoding for BEL-J can be deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: 2 i , .., b N 2 i ,where b ki = (cid:26)
Revised: 2 i , .., b N 2 i ,where b ki = (cid:26)

Original: N 2 − Q
Revised: N 2 − Q

Original: i < k (cid:54) N − Q i 

Revised: i < k (cid:54) N − Q i

Original: Otherwise (12)
Revised: Otherwise (13)

Original: B i
Revised: B i

Original: C 1 C 2 C 3 C
Revised: C 1 C 2 C 3 C

Original: Q i - - - - - - - + + + + + + + + + + + + + +
Revised: Q i - - - - - - - + + + + + + + + + + + + + +

Original: Tl Tf Tc
Revised: Tl Tf Tc

Original: Decodingfunction D
Revised: Decodingfunction D

Original: Decodedoutput
Revised: Decodedoutput

Original: Tl + Tf + Tc
Revised: Tl + Tf + Tc

Original: [Figure]

Original: [Figure]

Original: N2-n+  

Revised: N2-n+

Original: Bit position 1 B
Revised: Bit position 1 B

Original: i
Revised: i

Original: N2-n+  .....
Revised: N2-n+  .....

Original: ..... N2-n
Revised: ..... N2-n

Original: N2-1 N2 .....
Revised: N2-1 N2 .....

Original: N2-nk .....
Revised: N2-nk .....

Original: .....k-
Revised: .....k-

Original:  - - - - Case 1: B i - ..... ..... - 1 .....|
Revised:  - - - - Case 1: B i - ..... ..... - 1 .....|

Original: Tf i -
Revised: Tf i -

Original: Tf i | =
Revised: Tf i | =

Original: ( N / 2)- n +1- k - -0 Case 2: B i ..... .....
Revised: ( N / 2)- n +1- k - -0 Case 2: B i ..... .....

Original: .....|
Revised: .....|

Original: Tf i - Tf i | =
Revised: Tf i - Tf i | =

Original: N2-n+  - 1 ^ ^ ^ ^
Revised: N2-n+  - 1 ^ ^ ^ ^

Original: Similarly, the decoding functions for BEL-J can be deﬁned as:
Revised: Similarly, the decoding functions for BEL-J can be deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where, Tl (
Revised: where, Tl (

Original: [Equation]
Revised: [Equation]

Original: In equation 13, Tl () ﬁnds the location of the last occurrence of “ 1 ” in the predicted binary code ˆ B i .
Revised: In equation 14, Tl () ﬁnds the location of the last occurrence of “ 1 ” in the predicted binary code ˆ B i .

Original: Similarly, Tf () ﬁnds the location of the ﬁrst occurrence of “ 1 ” in the binary code ˆ B i .
Revised: Similarly, Tf () ﬁnds the location of the ﬁrst occurrence of “ 1 ” in the binary code ˆ B i .

Original: Figure 8 gives examples of binary codes for label Q i ∈ { 1 , 2 , ..., 7 } and the corresponding values of the different terms in equation 13.
Revised: Figure 7 gives examples of binary codes for label Q i ∈ { 1 , 2 , ..., 7 } and the corresponding values of the different terms in equation 14.

Original: For example, for label Q i = 3 , the binary code is “ 0111 ”.
Revised: For example, for label Q i = 3 , the binary code is “ 0111 ”.

Original: Here, the last occurrence of “ 1 ” is at position 4 , and Tl = − 4 .
Revised: Here, the last occurrence of “ 1 ” is at position 4 , and Tl = − 4 .

Original: Similarly, the ﬁrst occurrence of “ 1 ” is at position 2 , and Tf = (4 + 1) − 2 = 3 .
Revised: Similarly, the ﬁrst occurrence of “ 1 ” is at position 2 , and Tf = (4 + 1) − 2 = 3 .

Original: Expected error For BEL-J code, binary classiﬁers ( C 1 , C 2 , ..., C N 2 ) are used.
Revised: Expected error For BEL-J code, binary classiﬁers ( C 1 , C 2 , ..., C N 2 ) are used.

Original: For a given input sample i, an error in any of the binary classiﬁers’ outputs (ˆ b 1 i , ˆ b 2 i , ..., ˆ b N 2 ) will result in an error between Tf ( ˆ B i ) /Tl
Revised: For a given input sample i, an error in any of the binary classiﬁers’ outputs (ˆ b 1 i , ˆ b 2 i , ..., ˆ b N 2 ) will result in an error between Tf ( ˆ B i ) /Tl

Original: ( ˆ B i ) and Tf ( B i ) /Tl
Revised: ( ˆ B i ) and Tf ( B i ) /Tl

Original: ( B i ) in equation 13.
Revised: ( B i ) in equation 14.

Original: We refer to Tf ( ˆ B i ) and Tl ( ˆ B i ) as ˆ Tf i and ˆ Tl
Revised: We refer to Tf ( ˆ B i ) and Tl ( ˆ B i ) as ˆ Tf i and ˆ Tl

Original: i (predicted binary code), and Tf ( B i ) and Tl ( B i ) as Tf i and Tl i (target binary code) for brevity.
Revised: i (predicted binary code), and Tf ( B i ) and Tl ( B i ) as Tf i and Tl i (target binary code) for brevity.

Original: Expected value of the absolute error can be further expanded as:
Revised: Expected value of the absolute error can be further expanded as:

Original: [Equation]
Revised: [Equation]

Original: Thus, the sum of the expected error of Tf () and Tl () is the upper bound of the label’s expected error.
Revised: Thus, the sum of the expected error of Tf () and Tl () is the upper bound of the label’s expected error.

Original: Further, we derive the relation between binary classiﬁers’ error probabilities and E ( | ˆ Tf i − Tf i | ) and
Revised: Further, we derive the relation between binary classiﬁers’ error probabilities and E ( | ˆ Tf i − Tf i | ) and

Original: E ( | ˆ Tl i − Tl i | ) .
Revised: E ( | ˆ Tl i − Tl i | ) .

Original: We consider Q i = n, where 1 (cid:54) n (cid:54) N 2 for our derivation.
Revised: We consider Q i = n, where 1 (cid:54) n (cid:54) N 2 for our derivation.

Original: In such a case, Tf i = n and Tl i = − N 2 .
Revised: In such a case, Tf i = n and Tl i = − N 2 .

Original: However, as the code is symmetric around Q i = N 2 , it can be shown that the derived equation for E | ˆ y i − y i | can be used for 1 (cid:54) Q i (cid:54) N − 1 .
Revised: However, as the code is symmetric around Q i = N 2 , it can be shown that the derived equation for E | ˆ y i − y i | can be used for 1 (cid:54) Q i (cid:54) N − 1 .

Original: Derivation of E | ˆ
Revised: Derivation of E | ˆ

Original: Tf i − Tf i | : As shown in equation 13, Tf () ﬁnds the location k of the ﬁrst occurrence of “ 1 ” in the binary sequence.
Revised: Tf i − Tf i | : As shown in equation 14, Tf () ﬁnds the location k of the ﬁrst occurrence of “ 1 ” in the binary sequence.

Original: In the case of an erroneous binary sequence, the position of the ﬁrst occurrence of “ 1 ” might shift, which results in an error between ˆ Tf i and Tf i .
Revised: In the case of an erroneous binary sequence, the position of the ﬁrst occurrence of “ 1 ” might shift, which results in an error between ˆ Tf i and Tf i .

Original: Figure 9 shows examples of the correct and erroneous outputs of classiﬁers for label Q i = n .
Revised: Figure 8 shows examples of the correct and erroneous outputs of classiﬁers for label Q i = n .

Original: For label Q i = n , b ki = 0 for k ∈ { 1 , 2 , ..., N 2 − n } and b ki = 1 for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } .
Revised: For label Q i = n , b k i = 0 for k ∈ { 1 , 2 , ..., N 2 − n } and b k i = 1 for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } .

Original: For case 1, error in a classiﬁer C k , k ∈ { 1 , 2 , ..., N 2 − n } is considered, where b ki = 0 and ˆ b ki = 1 .
Revised: For case 1, error in a classiﬁer C k , k ∈ { 1 , 2 , ..., N 2 − n } is considered, where b ki = 0 and ˆ b ki = 1 .

Original: For k ∈ { 1 , 2 , ..., N 2 − n } , an error at classiﬁer C k will result in erroneous ˆ Tf i only if all proceeding classiﬁers are correct, since if any of the proceeding classiﬁer z is incorrect, i.e. ˆ b z i = 1 , then the location of the ﬁrst occurrence of “ 1 ” will be shifted to z , and any error in the following classiﬁers will not affect the value of ˆ Tf .
Revised: For k ∈ { 1 , 2 , ..., N 2 − n } , an error at classiﬁer C k will result in erroneous ˆ Tf i only if all proceeding classiﬁers are correct, since if any of the proceeding classiﬁer z is incorrect, i.e. ˆ b zi = 1 , then the location of the ﬁrst occurrence of “ 1 ” will be shifted to z , and any error in the following classiﬁers will not affect the value of ˆ Tf .

Original: Such a case ( ˆ b 1 i =T,
Revised: Such a case ( ˆ b 1 i =

Original: [Equation] b 2 i = T, ..., ˆ b k 
Revised: T, ˆ b 2 i = T, ..., ˆ b k − 1 i = T, ˆ b ki

Original: [Equation]
Revised: [Equation]

Original: i
Revised: F,

Original: [Equation]
Revised: [Equation]

Revised: =

Original: T/F, ..., ˆ b
Revised: T/F, ..., ˆ b

Original: N 

Revised: N

Original: i
Revised: i

Original: =
Revised: =

Original: T/F ) considers a total of 2 N 2 − k combinations out of 2 N 2 for k ∈ { 1 , 2 , ..., N 2 − n } .
Revised: T/F ) considers a total of 2 N 2 − k combinations out of 2 N 2 for k ∈ { 1 , 2 , ..., N 2 − n } .

Original: Assuming that the binary classiﬁers are mutually independent, the error value and the probability of this combination can be shown to be:
Revised: Assuming that the binary classiﬁers are mutually independent, the error value and the probability of this combination can be shown to be:

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: The above term considers combinations ( b (cid:48) 1 = T, b
Revised: The above term considers combinations ( b (cid:48) 1 = T, b

Original: [Equation]
Revised: [Equation]

Revised: 2 = T, ..., b

Original: [Equation]
Revised: [Equation]

Original: ..., b (cid:48)

Original: k − 1 =
Revised: k [Equation]

Original:  T, b (cid:48) k
Revised: = T, b [Equation]

Original: = F, b (cid:48)
Revised: = F, b [Equation]

Original: k +1 =
Revised: k + [Equation] =

Original: T/F, ..., b (cid:48)
Revised: T/F, ..., b (cid:48)

Original: N 2 = T/F ) for k ∈ { 1 , 2 , ..., N 2 − n } , which constitutes to a total of (cid:80) N 2 − n k
Revised: N 2 = T/F ) for k ∈ { 1 , 2 , ..., N 2 − n } , which constitutes to a total of (cid:80) N 2 − n k

Original: =1 2 N 2 − k combinations out of 2 N 2 .
Revised: =1 2 N 2 − k combinations out of 2 N 2 .

Original: For case 2, error in a classiﬁer C k , k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } is considered, where b ki =and ˆ b ki = 0 .
Revised: For case 2, error in a classiﬁer C k , k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } is considered, where b ki =and ˆ b ki = 0 .

Original: We consider a combination ( ˆ b 1 i
Revised: We consider a combination ( ˆ b 1 i

Original: = T, ˆ b 2 i
Revised: = T, ˆ b 2 i

Original: = T, ..., ˆ b N 2 − n i =
Revised: = T, ..., ˆ [Equation] =

Original: T, ˆ b N 2 − n +1 i = F, ..., ˆ b k − 1 i =
Revised: [Equation]

Original: F, ˆ b ki = T, ˆ b k
Revised: F, ˆ b ki = T, ˆ b k

Original: +1 i
Revised: +1 i

Original: = T/F, ..., ˆ b N 2 = T/F i ).
Revised: = T/F, ..., ˆ [Equation] N [Equation]

Original:  For this case, the position of the ﬁrst occurrence of “ 1 ” will be moved to k , which will result in erroneous
Revised: T/ [Equation] i ). For this case, the position of the ﬁrst occurrence of “ 1 ” will be moved to k , which will result in erroneous

Original: ˆ Tf
Revised: ˆ Tf

Original: i .
Revised: i .

Original: Such a case would cover 2 N 2 − k combinations out of 2 N 2 for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } .
Revised: Such a case would cover 2 N 2 − k combinations out of 2 N 2 for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } .

Original: The error value and the probability of this combination can be shown to be:
Revised: The error value and the probability of this combination can be shown to be:

Original: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: The above term considers combinations (ˆ b 1 i
Revised: The above term considers combinations (ˆ b 1 i

Original: =
Revised: =

Original: T, ˆ b 2 i
Revised: T, ˆ b 2 i

Original: = T, ..., ˆ b N 2 − n i[Equation] N
Revised: = T, ..., ˆ b N 2 − n i

Original: [Equation]
Revised: [Equation]

Original: − n

Original: [Equation]

Original: = F
Revised: = F

Original: [Equation]
Revised: [Equation]

Original: ..., ˆ b k − 1 i =
Revised: ..., ˆ b k − 1 i =

Original: F, ˆ b ki = T, ˆ b k
Revised: F, ˆ b ki = T, ˆ b k

Original: +1 i
Revised: +1 i

Original: = T/F, ..., ˆ b N 2 = T/F i ) , which constitutes to a total of (cid:80) N 
Revised: = T/F, ..., ˆ b N 2 = T/F i ) , which constitutes to a total of (cid:80) N 2 k

Original: [Equation] +1 2 N 2 − k combinations out of 2 N 2 for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } .
Revised: = N 2 − n +1 2 N 2 − k combinations out of 2 N 2 for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } .

Original: Combining equation 15-equation 18, the expected value of | ˆ Tf i − Tf i | can be derived as: 
Revised: Combining equation 16-equation 19, the expected value of | ˆ Tf i − Tf i | can be derived as: E

Original: [Equation]
Revised: [Equation]

Original: The ﬁrst term in equation 19 covers (cid:80) N 2 − n k
Revised: The ﬁrst term in equation 20 covers (cid:80) N 2 − n k

Original: =1 2 N 2 − k combinations and the second term considers (cid:80) N 2 k
Revised: =1 2 N 2 − k combinations and the second term considers (cid:80) N 2 k

Original: = N 2 − n +1 2 N 2 − k combinations.
Revised: = N 2 − n +1 2 N 2 − k combinations.

Original: Adding one combination where all the classiﬁers are correct, equation 19 considers all of the possible combinations 2 N 2 to ﬁnd expected value of | ˆ Tf i − Tf i | .
Revised: Adding one combination where all the classiﬁers are correct, equation 20 considers all of the possible combinations 2 N 2 to ﬁnd expected value of | ˆ Tf i − Tf i | .

Original: N2-n+  1 

Revised: N2-n+  1

Original: N2-n+ 1 .....
Revised: N2-n+ 1 .....

Original: ..... N2-n
Revised: ..... N2-n

Original: N2-1 N2 .....
Revised: N2-1 N2 .....

Original: N2-nk .....
Revised: N2-nk .....

Original: .....k- - 0 0 - - - ..... ..... - - ..... - |
Revised: .....k- - 0 0 - - - ..... ..... - - ..... - |

Original: Tl i - Tl
Revised: Tl i - Tl

Original: i | =
Revised: i | =

Original: ( N / 2)- k +  0 0 

Revised: ( N / 2)- k +  0 0

Original: | Tl i - Tl i | =
Revised: | Tl i - Tl i | =

Original: N2Bit position
Revised: N2Bit position

Original: B i
Revised: B i

Original: Case 1: B i
Revised: Case 1: B i

Original: Case 2: B i ^ ^
Revised: Case 2: B i ^ ^

Original: ^ ^
Revised: ^ ^

Original: Derivation of E | ˆ Tl i − Tl i | : As shown in equation 13, Tl () ﬁnds the location k of the last occurrence of “ 1 ” in the binary sequence.
Revised: Derivation of E | ˆ Tl i − Tl i | : As shown in equation 14, Tl () ﬁnds the location k of the last occurrence of “ 1 ” in the binary sequence.

Original: In the case of an erroneous binary sequence, the position of the last occurrence of “ 1 ” might shift, which results in an erroneous value of ˆ Tl i .
Revised: In the case of an erroneous binary sequence, the position of the last occurrence of “ 1 ” might shift, which results in an erroneous value of ˆ Tl i .

Original: Figure 10 shows examples of correct and erroneous outputs of classiﬁers for label Q i = n .
Revised: Figure 9 shows examples of correct and erroneous outputs of classiﬁers for label Q i = n .

Original: For case 1, an error in a classiﬁer C k , k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } is considered, where b ki = 1 and ˆ b ki = 0 .
Revised: For case 1, an error in a classiﬁer C k , k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } is considered, where b ki = 1 and ˆ b ki = 0 .

Original: We consider a combination (ˆ b N 2 i =
Revised: We consider a combination (ˆ b N 2 i =

Original: F, ˆ b N 2 − 1 i = F, ..., ˆ b k
Revised: F, ˆ b N 2 − 1 i = F, ..., ˆ b k

Original: +1 i
Revised: +1 i

Original: = F, ˆ b ki = T, ˆ b k − 1 i =
Revised: = F, ˆ b ki = T, ˆ b k − 1 i =

Original: T/F, ..., ˆ b 

Revised: T/F, ..., ˆ b

Original: i = T/F ) .
Revised: i = T/F ) .

Original: For this case, position of the last occurrence of “ 1 ” will be moved to k , which will result in erroneous ˆ Tl i .
Revised: For this case, position of the last occurrence of “ 1 ” will be moved to k , which will result in erroneous ˆ Tl i .

Original: Such a case would cover 2 k − 1 combinations out of 2 N 2 .
Revised: Such a case would cover 2 k − 1 combinations out of 2 N 2 .

Original: The error value and the probability of this combination can be shown to be:
Revised: The error value and the probability of this combination can be shown to be:

Original: [Equation]
Revised: [Equation]

Original: The above term considers combinations (ˆ b N 

Revised: The above term considers combinations (ˆ b N

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: T/F, ..., ˆ b 

Revised: T/F, ..., ˆ b

Original: i = T/F ) for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } , which constitutes to a total of (cid:80) N 2 k = N 2 − n +1 2 k − 1 combinations out of 2 N 2 .
Revised: i = T/F ) for k ∈ { N 2 − n + 1 , N 2 − n + 2 , ..., N 2 } , which constitutes to a total of (cid:80) N 2 k = N 2 − n +1 2 k − 1 combinations out of 2 N 2 .

Original: For case 2, an error in a classiﬁer C k , k ∈ { 1 , 2 , ..., N 2 − n } is considered, where b ki = 0 and ˆ b ki = 1 .
Revised: For case 2, an error in a classiﬁer C k , k ∈ { 1 , 2 , ..., N 2 − n } is considered, where b ki = 0 and ˆ b ki = 1 .

Original: We consider a combination ( ˆ b N 2 i = F, ..., ˆ b N 2 − n +1 i
Revised: We consider a combination ( ˆ b N 2 i = F, ..., ˆ b N 2 − n +1 i

Original: =
Revised: =

Original: F, ˆ b N 2 − n i
Revised: F, ˆ b N 2 − n i

Original: [Equation]

Original:  T, ..., ˆ b k +{++
 i+}
Revised: = T, ..., ˆ b k {++
 i+}

Original: [Equation]
Revised: [Equation]

Original: T, ˆ b ki = F, ˆ b k − 1 i =
Revised: T, ˆ b ki = F, ˆ b k − 1 i =

Original: T/F, ..., ˆ b 

Revised: T/F, ..., ˆ b

Original: i = T/F ).
Revised: i = T/F ).

Original: For this case, position of the last occurrence of “ 1 ” will be moved to k , which will result in erroneous ˆ Tl i .
Revised: For this case, position of the last occurrence of “ 1 ” will be moved to k , which will result in erroneous ˆ Tl i .

Original: Such a case would cover 2 k − 1 combinations out of 2 N 2 .
Revised: Such a case would cover 2 k − 1 combinations out of 2 N 2 .

Original: The error value and the probability of this combination can be shown to be:
Revised: The error value and the probability of this combination can be shown to be:

Original: [Equation]
Revised: [Equation]

Original: The above term considers combinations (ˆ b N 

Revised: The above term considers combinations (ˆ b N

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: T, ˆ b ki = F, ˆ b k − 1 i = T/F, ..., ˆ b 1 i = T/F ) for k ∈ { 1 , 2 , ..., N 2 − n } , which constitutes to a total of (cid:80) N 2 − n k
Revised: T, ˆ b ki = F, ˆ b k − 1 i = T/F, ..., ˆ b 1 i = T/F ) for k ∈ { 1 , 2 , ..., N 2 − n } , which constitutes to a total of (cid:80) N 2 − n k

Original: =
Revised: =

Original: k − 1 combinations out of 2 N 2 .
Revised: k − 1 combinations out of 2 N 2 .

Original: Combining equation 20-equation 23, the expected value of | ˆ Tl i − Tl i | can be derived as:
Revised: Combining equation 21-equation 24, the expected value of | ˆ Tl i − Tl i | can be derived as:

Original: [Equation]
Revised: [Equation]

Original: The ﬁrst term in equation 24 covers (cid:80) N 2 k = N 2 − n +1 2 k − 1 combinations and the second term considers (cid:80) N 2 − n k
Revised: The ﬁrst term in equation 25 covers (cid:80) N 2 k = N 2 − n +1 2 k − 1 combinations and the second term considers (cid:80) N 2 − n k

Original: =1 2 k − 1 combinations.
Revised: =1 2 k − 1 combinations.

Original: Adding one combination where all the classiﬁers are correct, equation 24 considers all of the possible combinations 2 N 2 to ﬁnd expected value of | ˆ Tl i − Tl i | .
Revised: Adding one combination where all the classiﬁers are correct, equation 25 considers all of the possible combinations 2 N 2 to ﬁnd expected value of | ˆ Tl i − Tl i | .

Original: Combining equation 14, equation 19, and equation 24, the expected value of average error in terms of classiﬁers’ error probabilities can de derived as:
Revised: Combining equation 15, equation 20, and equation 25, the expected value of average error in terms of classiﬁers’ error probabilities can de derived as:

Original: [Equation]
Revised: [Equation]

Original: As the binary code is symmetric around N 2 as shown in Figure 8, the expected errors for label y i ∈ [1 , N 2 ] can be mirrored to ﬁnd expected errors for label y i ∈
Revised: As the binary code is symmetric around N 2 as shown in Figure 7, the expected errors for label y i ∈ [1 , N 2 ] can be mirrored to ﬁnd expected errors for label y i ∈

Original: [ N 2 , N − 1] .
Revised: [ N 2 , N − 1] .

Original: For a uniform distribution of target labels in the range [1 , N − 1] , the expected error can be derived as:
Revised: For a uniform distribution of target labels in the range [1 , N − 1] , the expected error can be derived as:

Original: [Equation]
Revised: [Equation]

Original: We also verify the equation by comparing the expected value of error based on equation 25 for Q i ∈ 1 , 2 , ..., N − 1 with the expected error calculated by 100 , 000 random samples of binary sequences for the same error probabilities e k ( n ) .
Revised: We also verify the equation by comparing the expected value of error based on equation 26 for Q i ∈ 1 , 2 , ..., N − 1 with the expected error calculated by 100 , 000 random samples of binary sequences for the same error probabilities e k ( n ) .

Original: Figure 11 compares the expected error from equation 25 and measured from statistical samples, and validates error upper bounds calculated using equationand equation 26.
Revised: Figure 10 compares the expected error from equation 26 and measured from statistical samples, and validates error upper bounds calculated using equationand equation 27.

Original: [Figure]
Revised: [Figure]

Original: C E RROR P ROBABILITY OF C LASSIFIERS
Revised: C E RROR P ROBABILITY OF C LASSIFIERS

Original: It is known that the error/misclassiﬁcation probability e k ( n ) of a classiﬁer tends to increase as the target label value n is closer to the classiﬁer’s decision boundaries Cardoso & Pinto da Costa (2007).
Revised: It is known that the error/misclassiﬁcation probability e k ( n ) of a classiﬁer tends to increase as the target label value n is closer to the classiﬁer’s decision boundaries Cardoso & Pinto da Costa (2007).

Original: We approximate e k ( y ) for a classiﬁer C k with t decision boundaries as a linear combination of t
Revised: We approximate e k ( y ) for a classiﬁer C k with t bit transitions as a linear combination of t

Original: Gaussian distributions.
Revised: Gaussian distributions.

Original: Here, each Gaussian term is centered around a decision boundary.
Revised: Here, each Gaussian term is centered around a bit transition.

Original: Figureshows the empirically observed error probability distributions for different classiﬁers trained for different combinations of network and dataset.
Revised: Figure 11 shows the empirically observed error probability distributions for different classiﬁers trained for different combinations of network and dataset.

Original: We also show the approximate error probability distribution using a linear combination of Gaussian distributions.
Revised: We also show the approximate error probability distribution using a linear combination of Gaussian distributions.

Original: Here r is a scalar multiplied with probability density of gaussian distribution and σ 2 is the variance (equation 3 and equation 4).
Revised: Here r is a scalar multiplied with probability density of gaussian distribution and σ 2 is the variance (equation 3 and equation 4).

Original: [Figure]
Revised: [Figure]

Original: DH EAD P OSE E STIMATION
Revised: D E XPERIMENTAL M ETHODOLOGY

Original: Head pose estimation aims to ﬁnd a human head’s pose in terms of three angles: yaw, pitch, and roll. In this work, we consider landmark-free 2D head pose estimation.
Revised: All experiments are conducted on a Linux machine with an Intel i9-9900X processor and a Nvidia RTX 2080 Ti GPU with 11GB of memory.

Original: We use two evaluation protocols, Protocol-1 (Train dataset: 300W-LP (Zhu et al., 2016), Test dataset: AFLW2000 (Köstinger et al., 2011)) and Protocol-2 (70-30% train-test split of BIWI dataset (Fanelli et al., 2013)) with mean absolute error (MAE) as the evaluation metric.
Revised: Our code is implemented using Python 3.8.3 with Pytorch 1.5.1 using CUDA 10.2. Our evaluation is averaged over 5 training runs with separate seeds. D. H EAD P OSE E STIMATION

Original: Head pose estimation aims to ﬁnd
Revised: Head pose estimation aims to ﬁnd

Original: a human head’s pose in terms of three angles: yaw, pitch, and roll.
Revised: a human head’s pose in terms of three angles: yaw, pitch, and roll.

Original: This work considers landmark-free 2D head pose estimation, which performs the task based on RGB image information without using facial landmark information. D. E XPERIMENTAL SETUP
Revised: In this work, we consider landmark-free 2D head pose estimation.

Original:  Weuse two evaluation protocols with three widely used datasets: 300W-LP (Zhu et al., 2016), BIWI (Fanelli et al., 2013), AFLW2000 (Zhu et al., 2016) and follow the evaluation setting of Hopenet (Ruiz et al., 2018) and FSA-Net (Yang et al., 2019).
Revised: Datasets: We follow the evaluation setting of Hopenet (Ruiz et al., 2018) and FSA-Net (Yang et al., 2019) and use two evaluation protocols with three widely used datasets: 300W-LP (Zhu et al., 2016),

Original: Mean absolute error (MAE) is used as the evaluation metric and the individual yaw, pitch, and roll and overall MAE are evaluated. NVIDIA 2080Ti GPU is used for all the experiments. Bold numbers represent the lowest error for the same feature extractor and data augmentation, and the underlined numbers represent the lowest error overall.
Revised: BIWI (Fanelli et al., 2013), and AFLW2000 (Zhu et al., 2016).

Original: Evaluation metrics for BEL and direct regression are averaged over ﬁve training runs with random seeds. We further explain two protocols in detail. Bold numbers represent the lowest error for the same feature extractor and data augmentation, and the underlined numbers represent the lowest error overall. Evaluation metrics for BEL and direct regression are averaged over ﬁve training runs with random seeds.
Revised: We use two evaluation protocols.

Original: Protocol 1:
Revised: Protocol 1:

Original: In this setting, the synthetic 300W-LP dataset is used for training, consisting of 122 ,samples.
Revised: In this setting, the synthetic 300W-LP dataset is used for training, consisting of 122 ,samples.

Original: The trained network is tested on a real-world AFLW2000 dataset.
Revised: The trained network is tested on a real-world AFLW2000 dataset.

Original: Yaw, pitch, and roll angles are in the range [ − 99 , 99] for both datasets.
Revised: Yaw, pitch, and roll angles are in the range [ − 99 , 99] for both datasets.

Original: Table 7 provides a summary of the training parameters.

Original: For head pose estimation with ResNet50 architecture (Direct regression and BEL), two runs with

Original: [Table]

Original: different random seeds for each combination of learning rate {

Original: [Table]

Original: [Table]

Original: and batch size { 8 , 16 } are used for hyperparameter tuning.

Original: For data augmentation, images are loosely cropped around the center in the training and testing datasets with random ﬂipping and blurring.

Original: Samples with angles in the range [ − 99 ◦ , 99 ◦ ] are considered for training and testing.

Original: 200 quantization levels for binary-encoded labels.

Original: For head pose estimation with RAFA-Net, we use the training parameters used in (Behera et al., 2021).

Original: For data augmentation, we use the approach used in the previous work on RAFA-Net for head pose estimation using direct regression.

Original: Angles inthe range [ − 180 ◦ , 180 ◦ ] with 360 quantization levels are considered for BEL.
Revised: Angles in range [ − 180 ◦ , 180 ◦ ] with 360 quantization levels are considered for BEL.

Original: Protocol 2: BIWI dataset is used for training and evaluation in this protocol.
Revised: Protocol 2: BIWI dataset is used for training and evaluation in this protocol.

Original: BIWI dataset consists of 24 videos of 20 subjects with total 15 , 128 frames.
Revised: BIWI dataset consists of 24 videos of 20 subjects with total 15 , 128 frames.

Original: Three random splits of 70%-30% images are used for training and evaluation.
Revised: Three random splits of 70%-30% images are used for training and evaluation.

Original: Table 8 provides a summary of the  training parameters.
Revised: For the BIWI dataset, the yaw angle is in the range [ − 75 , 75] , the pitch is in the range [ − 65 , 85] , and the roll angle is in the range [ − 55 , 45] . 150 quantization levels are used for yaw and pitch, and 100 quantization levels are used for roll. Network Architecture and training parameters: We evaluate our approach on two models:

Original: For head pose estimation with ResNet50 architecture (Direct regression and BEL), two runs with different random seeds for each combination of learning rate { 0 . 001 , 0 . 0001 , 0 . 00001 } and batch size { 8 , 16 } are used for hyperparameter tuning.
Revised: ResNet-50 and RAFA-Net. With ResNet-50 (Direct regression and BEL), two runs with different random seeds for each combination of learning rate { 0 . 001 , 0 . 0001 , 0 . 00001 } and batch size { 8 , 16 } are used for hyperparameter tuning.

Original: For data augmentation, images are loosely cropped around the center in the training and testing datasets with random ﬂipping.For the BIWI dataset, the yaw angle is in the range [ − 75 , 75] , the pitch is in the range [ − 65 , 85] , and the roll angle is in the range [ − 55 , 45] . Total 150 quantization levels are used for yaw and pitch, and 100 quantization levels are used for roll.
Revised: For data augmentation, images are loosely cropped around the center in the training and testing datasets with random ﬂipping.

Original: For head pose estimation with RAFA-Net, we use the training parameters used in (Behera et al., 2021).
Revised: With RAFA-Net, we use the training parameters used in Behera et al.

Original: For data augmentation, we use the approach used inthe previous work on RAFA-Net for head pose estimation using direct regression.
Revised: For data augmentation, we use the approach described in RAFA-Net Behera et al.

Original: Angles in range [ − 180 ◦ , 180 ◦ ] with 360 quantization levels are considered for BEL.
Revised: We refer to Protocol 1 evaluated with ResNet-50 as HPE1 , Protocol 1 evaluated with RAFA-Net as HPE3 , Protocol 2 evaluated with ResNet-50 as HPE2 , and Protocol 2 evaluated with RAFA-Net as HPE4 . Table 21 provides a summary of the training parameters used with protocol 1. Tableprovides a summary of the training parameters used with protocol 2.

Original: [Table] [Table]
Revised: [Table] [Table]

Original:  Existing approaches for head pose estimation includes stage-wise soft regression (Rothe et al., 2015; Yang et al., 2019),  combination of classiﬁcation and regression (Mukherjee & Robertson, 2015; Ruiz et al., 2018)
Revised: Related Work Existing approaches for head pose estimation include stage-wise soft regression (Rothe et al., 2015; Yang et al., 2019), a combination of classiﬁcation and regression (Mukherjee & Robertson, 2015; Ruiz et al., 2018)

Original: [Table]
Revised: [Table]

Original: and ordinal regression (Hsu et al., 2019).
Revised: and ordinal regression (Hsu et al., 2019).

Original: SSR-Net (Rothe et al., 2015) proposed the use of stage-wise soft regression that uses softmax values of classiﬁcation output to reﬁne the label.
Revised: SSR-Net (Rothe et al., 2015) proposes the use of stage-wise soft regression to use the softmax values of classiﬁcation output to reﬁne the label.

Original: FSA-Net (Yang et al., 2019) proposes extending stage-wise estimation to head pose estimation using feature aggregation.
Revised: FSA-Net (Yang et al., 2019) proposes extending stage-wise estimation to head pose estimation using feature aggregation.

Original: HopeNet (Ruiz et al., 2018) uses a combination of classiﬁcation and regression loss to train a model for head pose estimation. Whereas QuatNet (Hsu et al., 2019) solves head pose estimation using a combination ofregression L2 loss and  ordinal regression loss.
Revised: HopeNet (Ruiz et al., 2018) uses a combination of classiﬁcation and regression loss to train a model for head pose estimation whereas QuatNet (Hsu et al., 2019) proposes a combination of L2 loss and a custom ordinal regression loss.

Original: RAFA-Net (Behera et al., 2021) uses an attention based approach for feature extraction with direct regression.
Revised: RAFA-Net (Behera et al., 2021) uses an attention based approach for feature extraction with direct regression.

Original: D. E VALUATION

Original: Protocol 1 (HPE2 and HPE4) Table 9 compares different combinations of encoding, deocding, and loss functions for BEL approach on protocol 1 and ResNet50 feature extractor. Table 10 compares different combinations of encoding, deocding [Table] and loss functions for BEL approach on protocoland RAFA-Net feature extractor.
Revised: We compare BEL with the performance of related work in Table 23 and Table 24.

Original: Table 11 shows results for protocol 1 with error margin for 95% 
Revised: 95% conﬁdence intervals are given.

Original: [Table]
Revised: [Table]

Original: conﬁdence interval.

Original: For ResNet50 feature extractor, BEL-U/BCE/GEN-EX provides the lowest MAE compared to direct regression and HopeNet (Ruiz et al., 2018).

Original: Whereas, for RAFA-Net feature extractor, BEL-U/BCE/GEN-EX provides the lowest MAE compared to direction regression (L loss).

Original: [Table]
Revised: [Table]

Original: Protocol 2 (HPE1 and HPE3) Table 12 compares different combinations of encoding, deocding, and loss functions for BEL approach on protocol 2 and ResNet50 feature extractor.

Original: Table 13 compares different combinations of encoding, deocding, and loss functions for BEL approach on protocoland RAFA-Net feature extractor.

Original: Table 14 shows results for protocol 2 with error margin for 95% conﬁdence interval.

Original: For ResNet50 feature extractor, BEL-U/BCE/GEN-EX provides the lowest

Original: MAE compared to direct regression (L2 loss).

Original: Whereas, for RAFA-Net feature extractor, BELJ/BCE/GEN-EX provides the lowest MAE compared to direction regression (L2 loss).

Original: For both the feature extractors, BEL results in signiﬁcant decrease in the MAE for pitch compared to direct regression.

Original: [Table]

Original: [Table]

Original: [Table]

Original: E F ACIAL L ANDMARK D ETECTION
Revised: D. F ACIAL L ANDMARK D ETECTION

Original: Facial landmark detection is a problem of detecting the ( x, y ) coordinates of keypoints in a given face image.
Revised: Facial landmark detection is a problem of detecting the ( x, y ) coordinates of keypoints in a given face image.

Original: Normalized mean error (NME) and failure rate (f@8% and f@10%) are used as evaluation metrics. The state-of-the-art approach with HRNetV2-W18 (Wang et al., 2020) uses heatmap regression, which minimizes the pixel-level loss between the predicted and target heatmaps.
Revised: Datasets We use the COFW (Burgos-Artizzu et al., 2013), 300W (Sagonas et al., 2013), WFLW (Wu et al., 2018)

Original: The target heatmap is generated by Gaussian distribution around the landmark location. BEL-x predicts of ( x, y ) coordinates directly with 256 quantization levels. E. E XPERIMENTAL SETUP
Revised: [Table]

Original: We use the COFW (Burgos-Artizzu et al., 2013), 300W (Sagonas et al., 2013), WFLW (Wu et al., 2018), and AFLW (Köstinger et al., 2011) datasetsfor the evaluation with data augmentation and evaluation protocol used in (Wang et al., 2020).
Revised: and AFLW (Köstinger et al., 2011) datasets with data augmentation and evaluation protocols described in (Wang et al., 2020).

Original: Normalized mean error (NME) and failure rate (f@10%) are used as evaluation metrics. 
Revised: Normalized mean error (NME) and failure rate (f@10%) are used as evaluation metrics. .

Original: Inter-ocular distancefor normalization is used for the
Revised: Inter-ocular distance normalization is used for the

Original: COFW, 300W, and WFLW datasets andthe bounding box-based normalization is used for the AFLW dataset.
Revised: COFW, 300W, and WFLW datasets and bounding box-based normalization is used for the AFLW dataset.

Original: Further, 256 quantization levels are used for binary-encoded labels.

Original: Data augmentation is performed by random ﬂipping, 0 .
Revised: Data augmentation is performed by random ﬂipping, 0 .

Original: 75 − 1 .
Revised: 75 − 1 .

Original: 25 scaling, and ± 30 degrees in-plane rotation for all the datasets.
Revised: 25 scaling, and ± 30 degrees in-plane rotation for all the datasets.

Revised: We use 256 quantization levels for binary-encoded labels.

Original: COFW
Revised: COFW

Original: The COFW dataset (Burgos-Artizzu et al., 2013) consists of 1 , 345 training and 507 testing images.
Revised: The COFW dataset (Burgos-Artizzu et al., 2013) consists of 1 , 345 training and 507 testing images.

Original: Each image is annotated with 29 facial landmarks.
Revised: Each image is annotated with 29 facial landmarks.

Original: 300W This dataset is a combination of HELEN, LFPW, AFW, XM2VTS, and IBUG datasets.
Revised: 300W This dataset is a combination of HELEN, LFPW, AFW, XM2VTS, and IBUG datasets.

Original: Each image is annotated with 68 facial landmarks.
Revised: Each image is annotated with 68 facial landmarks.

Original: The training dataset consists of 3 , 148 images.
Revised: The training dataset consists of 3 , 148 images.

Original: We evaluate the trained model on four test sets: full test set with 689 images, common subset withimages from HELEN and LFPW, challenging subset with 135 images from IBUG,  the ofﬁcial test set with 300 indoor and 300 outdoor images.
Revised: We evaluate the trained model on four test sets: full test set with 689 images, common subset withimages from HELEN and LFPW, challenging subset with 135 images from IBUG, and the ofﬁcial test set with 300 indoor and 300 outdoor images.

Original: WFLW WFLW dataset consists of 7 , 500 training images where each image is annotated withfacial landmarks.
Revised: WFLW WFLW dataset consists of 7 , 500 training images where each image is annotated withfacial landmarks.

Original: Full test dataset consists of 2 , 500 images.
Revised: Full test dataset consists of 2 , 500 images.

Original: We use test subsets: large pose (images), expression ( 314 images), illumination ( 698 images), make-up ( 206 images), occlusion (images) and blur ( 773 images).
Revised: We use test subsets: large pose (images), expression ( 314 images), illumination ( 698 images), make-up ( 206 images), occlusion (images), and blur ( 773 images).

Original: AFLW Each image has 19 annotated facial key points in this dataset.
Revised: AFLW Each image has 19 annotated facial key points in this dataset.

Original: AFLW dataset consists of 20 , 000 training images where each image is annotated with 19 facial landmarks.
Revised: AFLW dataset consists of 20 , 000 training images where each image is annotated with 19 facial landmarks.

Original: The full test dataset consists of 4 , 836 images, and the frontal test set consists of 1 , 314 images.
Revised: The full test dataset consists of 4 , 836 images, and the frontal test set consists of 1 , 314 images.

Original: Training parameters Two runs with different random seeds for a learning rate to decide the learning rate.
Revised: Network Architecture and training parameters:

Original: We considered learning rate { 0 . 0003 , 0 . 0005 , 0 . 0007 } . θ ∈ 10 , 30 is considered for all the codes, and the value is selected for a trade-off between the NME and the model size as a higher value of θ increases the model size. Table 15 provides a summary of all the training parameters. [Table]
Revised: We evaluate BEL by applying it on HRNetV2W18.

Original: Network Architecture HRNetV2-W18 feature extractor’s output is 240 channels of size 64 × 64 .
Revised: HRNetV2-W18 feature extractor’s output is 240 channels of size 64 × 64 .

Original: For heatmap regression, a 1 × 1 convolution is used to get P heatmaps of size 64 × 64 , where P is the number of landmarks.
Revised: For heatmap regression, a 1 × 1 convolution is used to get P heatmaps of size 64 × 64 , where P is the number of landmarks.

Original:  BEL-x predictsof ( x, y ) coordinates directly with 256 quantization levels.
Revised: Since BEL-x predicts ( x, y ) coordinates directly we modify the architecture of

Revised: HRNetV2-W18 to support direct prediction of landmarks.

Original: Figure 13 shows the modiﬁed architecture of HRNetV2-W18 for BEL-x.
Revised: Figure 12 shows the modiﬁed architecture of HRNetV2-W18 for BEL-x.

Revised: The state-of-the-art approach with HRNetV2-W18 (Wang et al., 2020) uses heatmap regression, which minimizes the pixel-level loss between the predicted and target heatmaps.

Original: E. R ELATED WORK
Revised: We evaluate the applicability of BELon heatmap regression in Appendix A. In contrast, BEL-x predicts ( x, y ) coordinates directly with 256 quantization levels. [Figure] We use two runs with different random seeds to decide the learning rate.

Revised: We consider learning rates { 0 . 0003 , 0 . 0005 , 0 . 0007 } and θ ∈ 10 , 30 .

Revised: The value of θ is selected for NME, but

Revised: Table 25 provides a summary of all the training parameters.

Revised: We refer to HRNetV2-W18 evaluated on

Revised: COFW as FLD1 , on 300W as FLD2 , on WFLW as FLD3 and on AFLW as FLD4 .

Original:  Facial landmark detection is an extensively studied problem used for facial analysis and modeling.
Revised: Related Work Facial landmark detection is an extensively studied problem used for facial analysis and modeling.

Original: Common regression approaches for this tasks includes regression using MSE loss (Xiong & De la Torre, 2013; Lv et al., 2017), cascaded regression (Miao et al., 2018; Tzimiropoulos, 2015; Zhu et al., 2016; Sun et al., 2013b), and coarse to ﬁne regression (Sun et al., 2013a; Shizhan Zhu et al., 2015; Zhang et al., 2014).
Revised: Common regression approaches for this tasks includes regression using MSE loss (Xiong & De la Torre, 2013; Lv et al., 2017), cascaded regression (Miao et al., 2018; Tzimiropoulos, 2015; Zhu et al., 2016; Sun et al., 2013b), and coarse-to-ﬁne regression (Sun et al., 2013a; Shizhan Zhu et al., 2015; Zhang et al., 2014).

Original: State-of-the-art methods for this task learns heatmaps by regression to ﬁnd facial landmarks.
Revised: State-of-the-art methods for this task learn heatmaps by regression to ﬁnd facial landmarks.

Original: SAN (Dong et al., 2018) augments training data using temporal information and GAN-generated faces.
Revised: SAN (Dong et al., 2018) augments training data using temporal information and GAN-generated faces.

Original: DVLN (Wu & Yang, 2017), CFSS (Shizhan Zhu et al., 2015), LAB (Wu et al., 2018), DSRN (Miao et al., 2018) take advantage of correlations between facial landmarks.
Revised: DVLN (Wu & Yang, 2017), CFSS (Shizhan Zhu et al., 2015), LAB (Wu et al., 2018), DSRN (Miao et al., 2018) take advantage of correlations between facial landmarks.

Original: DAN (Kowalski et al., 2017) introduces a progressive reﬁnement approach using predicted landmark heatmaps.
Revised: DAN (Kowalski et al., 2017) introduces a progressive reﬁnement approach using predicted landmark heatmaps.

Original: LAB (Wu et al., 2018) also exploits extra boundary information to improve the accuracy.
Revised: LAB (Wu et al., 2018) also exploits extra boundary information to improve the accuracy.

Original: LUVLi (Kumar et al., 2020) proposes a landmark’s location, uncertainty, andvisibility likelihoodbased loss. Bulat & Tzimiropoulos (2016) proposes the use of binary heatmaps with pixel-wise [Figure]
Revised: LUVLi (Kumar et al., 2020) proposes a landmark’s location, uncertainty, and

Original: [Table]
Revised: [Table]

Original: binary cross-entropy loss.
Revised: visibility likelihood-based loss.

Revised: Bulat & Tzimiropoulos (2016) proposes the use of binary heatmaps with pixel-wise binary cross-entropy loss.

Original: AWing (Wang et al., 2019) proposes adapted wing loss to improve the accuracy of heatmap regression.
Revised: AWing (Wang et al., 2019) proposes adapted wing loss to improve the accuracy of heatmap regression.

Original: AnchorFace (Xu et al., 2020) demonstrates that anchoring facial landmarks on templates improves regression performance for large poses.
Revised: AnchorFace (Xu et al., 2020) demonstrates that anchoring facial landmarks on templates improves regression performance for large poses.

Original: HRNet (Wang et al., 2020) proposes a CNN architecture to maintain high-resolution representations across the network
Revised: HRNet (Wang et al., 2020) proposes a CNN architecture to maintain high-resolution representations across the network

Original: [Table]
Revised: [Table]

Original: and uses heamap regression.
Revised: and uses heatmap regression.

Original: E. E VALUATION

Original: COFW (FDL1) Table 16 compares different combinations of encoding, deocding, and loss functions for BEL approach on COFW dataset. Table 17 compares different approaches for COFW dataset. BEL-HEXJ/CE/GEN-EX results in the lowest error.
Revised: The target heatmap is generated by assuming a Gaussian distribution around the landmark location. We compare BEL with related work in Table 26- 28.

Original: [Table] [Table] 300W (FDL2) Table 18 compares different combinations of encoding, deocding, and loss functions for BEL approach on COFW dataset. Table 19 compares different approaches for 300W dataset.approaches.
Revised: 95% conﬁdence intervals are provided.

Original: [Table]
Revised: [Table]

Original: WFLW (FDL3) Table 20 compares different combinations of encoding, deocding, and loss functions for BEL approach on COFW dataset.

Original: Table 21 compares different approaches for WFLW dataset

Original: [Table][Table] [Table] [Table]
Revised: [Table]

Original: AFLW (FDL4) Table 22 compares different combinations of encoding, deocding

Original: [Table]
Revised: [Table]

Original: and loss functions for BEL approach on COFW dataset.

Original: Table 23 compares different approaches for AFLW dataset

Original: [Table]

Original: [Table]
Revised: [Table]

Original: F A GE E STIMATION
Revised: D. A GE E STIMATION

Original: This task aims to predict the age  of a person from an image.
Revised: Age estimation aims to predict the age given an image of a human head.

Original: We use MORPH-II (Ricanek & Tesafaye, 2006) an AFAD (Niu et al., 2016) datasets with data augmentation and evaluation protocols used in (Pan et al., 2018).

Original: F. E XPERIMENTAL SETUP

Original:  We use the MORPH-2 (Ricanek & Tesafaye, 2006) and AFAD (Niu et al., 2016) datasets for our evaluation.
Revised: Datasets We use the MORPH-II (Ricanek & Tesafaye, 2006) and AFAD (Niu et al., 2016) datasets for our evaluation.

Original: Cumulative Score (CS) and MAE are used as evaluation metrics.
Revised: Cumulative Score (CS) and MAE are used as evaluation metrics.

Original: We preprocess the MORPH-2 dataset by aligning images ﬁrst along the average eye position (Raschka, 2018), then by re-aligning so that the tip of the nose is in the center of each image.
Revised: We preprocess the MORPH-II dataset by aligning images ﬁrst along the average eye position (Raschka, 2018), then by re-aligning so that the tip of the nose is in the center of each image.

Original: We do not preprocess the AFAD dataset as faces are already centered.
Revised: We do not preprocess the AFAD dataset as faces are already centered.

Original: Afterwards, face images are resized into 256 × 256 × 3 and randomly cropped to 224 × 224 × 3 for training.
Revised: Afterwards, face images are resized to 256 × 256 ×and randomly cropped to 224 × 224 × 3 for training.

Original: For testing, a center crop of 224 × 224 × 3 is taken.
Revised: For testing, a center crop of 224 × 224 × 3 is taken.

Original: [-MORPH-
-] This dataset consists of 55,608 face images with age labels between 16 and 70.
Revised: [-MORPH-
-]MORPH-II: This dataset consists of 55,608 face images with age labels between 16 and 70.

Original: The dataset is randomly divided into 39,617 training, 4,398 validation, and 11,001 testing images.
Revised: The dataset is randomly divided into 39,617 training, 4,398 validation, and 11,001 testing images.

Original: AFAD
Revised: AFAD:

Original: This dataset consists of 164,432 Asian facial images and age labels between 15 and 40.
Revised: This dataset consists of 164,432 Asian facial images and age labels between 15 and 40.

Original: The dataset is randomly divided into 118,492 training, 13,166 validation, and 32763 testing images.
Revised: The dataset is randomly divided into 118,492 training, 13,166 validation and 32,763 testing images.

Revised: Network architecture and training parameters:

Revised: We evaluate our approach on ResNet50.

Original: Training parameters We performed two runs with different random seeds to determine the learning rate between [0 . 00001 , 0 . 0001 , 0 . 001] and use a batch size of 64 for all experiments.
Revised: We perform two runs with different random seeds to determine the learning rate between [0 . 00001 , 0 . 0001 , 0 . 001] and use a batch size of 64 for all experiments.

Original: We use ImageNet pretrained weights to initialize the network.
Revised: We use ImageNet pretrained weights to initialize the network.

Original: [Table]
Revised: Full training parameters are described in Table 30. We refer to our evaluation on MORPH-II as AE1 and AFAD as AE2 .

Original: [Table]
Revised: [Table]

Original:  Existing approaches for age estimation include ordinal regression (Niu et al., 2016; Cao et al., 2020), soft regression (Rothe et al., 2015), and expected value ordinal regression (Pan et al., 2018; Gao et al., 2018).
Revised: Related Work Existing approaches for age estimation include ordinal regression (Niu et al., 2016; Cao et al., 2020), soft regression (Rothe et al., 2015), and expected value ordinal regression (Pan et al., 2018; Gao et al., 2018).

Original: OR-CNN (Niu et al., 2016) proposed the use of ordinal regression via binary classiﬁcation to predict the label.
Revised: OR-CNN (Niu et al., 2016) proposed the use of ordinal regression via binary classiﬁcation to predict the label.

Original: CORAL-CNN (Cao et al., 2020) reﬁned this approach by enforcing the ordinality of the model output.
Revised: CORAL-CNN (Cao et al., 2020) reﬁned this approach by enforcing the ordinality of the model output.

Original: SSR-Net (Rothe et al., 2015) proposed the use of stage-wise soft regression using the softmax of the classiﬁcation output to reﬁne the predicted label.
Revised: SSR-Net (Rothe et al., 2015) proposed the use of stage-wise soft regression using the softmax of the classiﬁcation output to reﬁne the predicted label.

Original: MV-Loss (Pan et al., 2018) extended the soft regression approach by penalizing the output of the model based on the variance of the age distribution, while DLDL (Gao et al., 2018) proposed to use the KL-divergence between the softmax output and a generated label distribution to train a model.F. E VALUATION
Revised: MV-Loss (Pan et al., 2018) extended the soft regression approach by penalizing the output of the model based on the variance of the age distribution, while DLDL (Gao et al., 2018) proposed to use the KL-divergence between the softmax output and a generated label distribution to train a model.

Original: Table 25 compares different combinations of encoding, deocding, and loss functions for BEL approach on MORPH-II dataset and ResNet50 feature extractor. Table 27 compares different combinations of encoding, deocding, and loss functions for BEL approach on AFAD dataset and ResNet50 feature extractor. Table 26 and Table 28 compare BEL with other regression approaches for MORPHII and AFAD datasets. BEL-x provides lower MAE than the direct regression for both datasets, and BEL-B1JDJ/MAE/GEN-EX results in the lowest MAE for the AFAD dataset compared to the state-of-the-art approaches.
Revised: We compare BEL with related work in Table 31 and Table 32.

Original: OR-CNN (Niu et al., 2016) and CORAL-CNN (Cao et al., 2020) use ordinal regression by binary classiﬁcation approach.

Original: We achieve signiﬁcant improvement over these approaches due to the differences in encoding and decoding function, regressor architecture, [Table] [Table] and training framework.
Revised: 95% conﬁdence intervals are provided.

Original: This improvement shows the potential of these design aspects to improve regression accuracy

Original: [Table]
Revised: [Table]

Original: [Table]

Original: [Table]
Revised: [Table]

Original: G E ND - TO - END A UTONOMOUS D RIVING
Revised: D. E ND - TO - END S ELF D RIVING

Original: We evaluate our approach on the NVIDIA PilotNet dataset  for end-to-end autonomous driving (Bojarski et al., 2016).
Revised: We evaluate our approach on the NVIDIA PilotNet dataset and PilotNet model for end-to-end autonomous driving (Bojarski et al., 2016).

Original: In this task, the steering wheel’s next angle is predicted from an image of the road
Revised: In this task, the steering wheel’s next angle is predicted from an image of the road.

Original: [Table] G. E XPERIMENTAL SETUP End-to-end autonomous driving is the problem of predicting the target wheel state directly from an image representing the view in front of a car.
Revised: We refer to these experiments as PN .

Original:  We use a driving dataset consisting of 45, images taken around Rancho Palos Verdes and San Pedro, California (Chen).
Revised: Dataset We use a driving dataset consisting of 45,500 images taken around Rancho Palos Verdes and San Pedro, California (Chen).

Original: We crop images to 256 × 70 × 3 then resized them to 200 × 66 × 3 .
Revised: We crop images to 256 × 70 × 3 then resize them to 200 × 66 × 3 .

Original: We randomly vary the brightness of the image between [0 . 2 × , 1 . 5 × ] and randomly perturb the steering direction.
Revised: We randomly vary the brightness of the image between [0 . 2 × , 1 . 5 × ] , randomly ﬂip images [Table] and make random minor perturbations on the steering direction.

Original: We usePilotNet architecture (Bojarski et al., 2017) and use θ = 10 with 670 quantization levels for BEL.
Revised: We use θ = 10 with 670 quantization levels for BEL.

Original: Training parameters We performed two runs with different random seeds to determine the learning rate between [0 . 00001 , 0 . 0001 , 0 . 001] and use a batch size of 64 for all experiments
Revised: Training parameters We perform two runs with different random seeds to determine the learning rate between [0 . 00001 , 0 . 0001 , 0 . 001] and use a batch size of 64 for all experiments.

Original: [Table] [Table] decay=1e-5,
Revised: Full training parameters are described in Table 33.

Original: [Table]
Revised: [Table]

Original:  End-to-end autonomous driving is a novel task that has become increasingly relevant due to the rise of self-driving vehicles.
Revised: Related Work End-to-end autonomous driving is a novel task that has become increasingly relevant due to the rise of self-driving vehicles.

Original: The autonomous driving model’s task is to predict the future driving angle based on a forward-facing image from the perspective of the vehicle.
Revised: The autonomous driving model’s task is to predict the future driving angle based on a forward-facing image from the perspective of the vehicle.

Original: PilotNet (Bojarski et al., 2017) used a small, application-speciﬁc network to succeed within the time constraints of autonomous driving.
Revised: PilotNet (Bojarski et al., 2017) used a small, application-speciﬁc network to succeed within the time constraints of autonomous driving

Original: G. E VALUATION
Revised: [Table]

Original: Table 30 compares different combinations of encoding, deocding, and loss functions for BEL approach on PilotNet dataset and feature extractor.
Revised: We compare BEL with the baseline PilotNet architecture in Table 34.

Original: Table 31 compares binary encoded labels with the PilotNet which uses direct regression (Bojarski et al., 2017). BEL-x results in a signiﬁcant reduction in the MAE compared to the direct regression approach.
Revised: 95% conﬁdence intervals are provided

Original: [Table]
Revised: [Table]

Original: [Table]