This paper has been published in Proceeding of International
Conference on Parallel Problem Solving from Nature, 2022.
HVC-Net: Deep Learning based Hypervolume
Contribution Approximation
Ke Shang, Weiduo Liao, and Hisao Ishibuchi?
Guangdong Provincial Key Laboratory of Brain-inspired Intelligent Computation,
Department of Computer Science and Engineering,Southern University of Science and Technology, Shenzhen, Chinakshang@foxmail.com, {liaowd@mail.,hisao@}sustech.edu.cn

Abstract. In this paper, we propose HVC-Net, a deep learning based
hypervolume contribution approximation method for evolutionary multiobjective optimization. The basic idea of HVC-Net is to use a deep neuralnetwork to approximate the hypervolume contribution of each solution ina non-dominated solution set. HVC-Net has two characteristics: (1) It ispermutation equivalent to the order of solutions in the input solution set,and (2) a single HVC-Net can handle solution sets of various size (e.g., solution sets with 20, 50 and 100 solutions). The performance of HVC-Net
is evaluated through computational experiments by comparing it withtwo commonly-used hypervolume contribution approximation methods(i.e., point-based method and line-based method). Our experimental results show that HVC-Net outperforms the other two methods in termsof both the runtime and the ability to identify the smallest (largest) hypervolume contributor in a solution set, which shows the superiority of
HVC-Net for hypervolume contribution approximation.
Keywords: Hypervolume contribution · Approximation · Evolutionary
multi-objective optimization · Deep learning.

1

Introduction

Hypervolume [15, 11] is a popular performance indicator in the field of evolutionary multi-objective optimization (EMO). It possesses rich theoretical properties(e.g., Pareto compliance [16], submodularity [13]), which make it attractive to
use in practice. For example, it has been used to design EMO algorithms (e.g.,
SMS-EMOA [6, 2]) and subset selection algorithms (e.g., greedy hypervolume
subset selection [7, 4]).
In SMS-EMOA, the population evolves in a steady-state manner. In eachgeneration, one offspring is generated and added to the population, then one solution is removed from the population so that the hypervolume of the remaining
population is maximized. In greedy hypervolume subset selection, in each step,
one solution is selected from a candidate set and added to the subset so that thehypervolume of the subset is maximized. In these two cases, in order to remove?

Corresponding author.

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K. Shang et al.

(select) the correct solution, we need to calculate the hypervolume contributionof each solution. Hypervolume contribution is an important concept which describes the amount of hypervolume contributed by one solution to a solution set.
In SMS-EMOA, we need to identify the solution with the smallest hypervolume
contribution to the population. In greedy hypervolume subset selection, we need
to identify the solution with the largest hypervolume contribution to the subset.
However, the calculation of the hypervolume contribution is #P-hard [3],which limits its applicability in many-objective optimization. In order to overcome this drawback, some hypervolume contribution approximation methods
have been proposed [1, 12, 5]. Two representative methods are the point-based
method [1] and the line-based method [12]. The point-based method is also
known as the Monte Carlo sampling method. In this method, in order to approximate the hypervolume contribution of a solution, a sampling space is firstly
determined. After that, a large number of samples are uniformly drawn in thesampling space to do the approximation. The line-based method is also known
as the R2 indicator variant. In this method, a large number of line segments are
uniformly drawn in the hypervolume contribution region of a solution to do theapproximation.
In this paper, we propose a new hypervolume contribution approximation
method. The proposed method, named HVC-Net, uses a deep neural network to
do the approximation. The input of HVC-Net is a non-dominated solution set,
and the output of HVC-Net is the approximated hypervolume contribution ofeach solution in the input solution set. HVC-Net has two characteristics. One isthat it is permutation equivalent [14]. That is, a change of the order of solutionsin the input solution set will cause the same change of the order of the outputs(i.e., the same results are obtained for any permutation of solutions in the input
solution set). The other is that it can handle solution sets of different size (e.g.,20, 50, 100 solutions). That is, a single HVC-Net is trained using solution sets ofvarious size. These two characteristics guarantee high applicability of HVC-Net
for hypervolume contribution approximation.
The rest of this paper is organized as follows. Section 2 presents the preliminaries of the study. Section 3 introduces a new hypervolume contribution approximation method, HVC-Net. Section 4 conducts experimental studies. Section 5
concludes the paper.

2

Preliminaries

In this section, we present the preliminaries of this paper, including the definitions of hypervolume and hypervolume contribution, and two representative
hypervolume contribution approximation methods.
2.1

Hypervolume and Hypervolume Contribution

Hypervolume The hypervolume is a widely used performance indicator in thefield of evolutionary multi-objective optimization. Formally, for a solution set S

HVC-Net: Deep Learning based HVC Approximationin the objective space, the hypervolume of S is defined as!
[
HV (S, r) = L{s0 |s ≺ s0 ≺ r} ,

3

(1)

s∈S

where L(·) is the Lebesgue measure of a set, r is a reference point which isdominated by all solutions in S, and s ≺ s0 denotes that s dominates s0 (i.e.,si ≤ s0i for all i = 1, ..., m and sj < s0j for at least one j = 1, ..., m in theminimization case, where m is the number of objectives).
Fig. 1 (a) gives an illustration of the hypervolume of a solution set {a1 , a2 , a3 }in a two-dimensional objective space, where each objective is to be minimized.
Hypervolume Contribution The hypervolume contribution is an importantconcept based on the hypervolume indicator. It describes the amount of thehypevolume value contributed by a solution to the solution set. Formally, for asolution s ∈ S, the hypervolume contribution of s to S is defined as
HV C(s, S, r) = HV (S, r) − HV (S \ {s}, r).

(2)

Fig. 1 (b) gives an illustration of the hypervolume contribution of each solution to the solution set {a1 , a2 , a3 }.

𝐚!

𝐫
min 𝑓2min 𝑓2

𝐫

𝐚#
𝐚"

0

min 𝑓1(a) Hypervolume

𝐚!
𝐚#
𝐚"

0

min 𝑓1(b) Hypervolume Contribution

Fig. 1. Illustrations of the hypervolume and the hypervolume contribution. The shaded
area in (a) is the hypervolume of the solution set {a1 , a2 , a3 }, and each shaded area in(b) is the hypervolume contribution of the corresponding solution to the solution set{a1 , a2 , a3 }.

2.2

Hypervolume Contribution Approximation

Two representative hypervolume contribution approximation methods are thepoint-based method and the line-based method. These two methods are briefly
explained as follows.

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K. Shang et al.

Point-based Method The point-based method is also known as the Monte
Carlo sampling method [1]. Fig. 2 (a) illustrates this method. The basic idea is
as follows. To approximate the hypervolume contribution of a solution s ∈ S, asampling space (i.e., a hyperrectangle) which contains s’s hypervolume contribution region is firstly determined (e.g., the rectangle bounded by a2 and p in
Fig. 2 (a)). Then a large number of samples are uniformly drawn in the sampling
space (e.g., k samples). Suppose there are k 0 samples uniquely dominated by s(e.g., the red samples in Fig. 2 (a)), then the hypervolume contribution of s isapproximated as
k0(3)
HV C(s, S, r) ≈ V,kwhere V is the volume of the sampling space (i.e., the hyperrectangle).
In practice, the sampling space should be as tight as possible. In [1], thetightest sampling space is theoretically derived. The lower bound of the sampling
space is the solution itself (e.g., a2 in Fig. 2 (a)). The upper bound (e.g., p in
Fig. 2 (a)) is determined as follows:
pi = min {s0i |s0 ∈ S \ {s} and s0 ≺i s} , i = 1, ..., m,(4)

where s0 ≺i s denotes that s0 dominates s in all but the ith objective.
Therefore, the tightest sampling space in Fig. 2 (a) is exactly the hypervolume
contribution region of a2 (i.e., p = (a31 , a12 )). We did not put p exactly at position(a31 , a12 ) (i.e., the red point) in Fig. 2 (a) for easy illustration.
*

𝐚!
𝐚#

- - - + + + ++ + ++ ++ + + -𝐩
min %2min 𝑓2𝐫

min 𝑓1(a) Point-based method

(,
(+

𝐚"
0

()

0

min %1(b) Line-based method

Fig. 2. Illustrations of the point-based and line-based methods for hypervolume contribution approximation.

Line-based Method The line-based method is also known as the R2 indicator variant [12]. Fig. 2 (b) illustrates this method. The basic idea is as follows.
To approximate the hypervolume contribution of a solution s ∈ S, a set of line
segments starting from s and with different directions are drawn in its hypervolume contribution region. Suppose there are n line segments and the lengthof each line segment is li , i = 1, ..., n, then the hypervolume contribution of s is

HVC-Net: Deep Learning based HVC Approximation

5

approximated asn

HV C(s, S, r) ≈1X m(li ) ,
n i=1

(5)

where m is the number of objectives.
The directions of the line segments can be defined using a direction vector
set Λ = {λ1 , ..., λn } where each direction vector satisfies λi 2 = 1, λij ≥ 0,
i = 1, ..., n, j = 1, ..., m. The length of each line segment can be calculated asnoli = min 0 min
g *2tch (s0 |λi , s) , g mtch (r|λi , s) , i = 1, ..., n,(6)s ∈S\{s}where g *2tch (·)1 and g mtch (·) are defined as follows:()
s0j − sj
*2tch 0 ig(s |λ , s) = max,
j∈{1,...,m}λij(
g mtch (r|λi , s) =minj∈{1,...,m}3|sj − rj |
λij

(7)

)
.

(8)

HVC-NetMotivated by DeepSets [14], which is a deep neural network for dealing with a set
as its input, we design HVC-Net to approximate the hypervolume contributionof each solution in a solution set. The architecture of HVC-Net is shown in Fig.
3. The input of HVC-Net is a non-dominated solution set S = {s1 , s2 , ..., sN }.
The output of HVC-Net is the hypervolume contribution approximation of eachsolution in S. The working mechanism of HVC-Net can be described in thefollowing three steps.
– Step 1: Each of N solutions si (i = 1, ..., N ) is presented to the network φ
and transformed to ai = φ(si ).
PN– Step 2: N vectors ai are averaged as one vector b = N1 i=1 ai . Vector b
is further presented to network η and transformed to c = η(b). Vector c is
added to each of N vectors ai and N vectors di = c + ai are obtained.
– Step 3: Each of N vectors di is presented to network ρ and N outputs^
HVC θ (si , S, r) = ρ(di ) are obtained, where θ is the parameter vector of
HVC-Net.
It should be noted that Step 2 can be stacked multiple times (i.e., K) as
shown in Fig. 3. Step 2 is used to learn the relation between a solution and thewhole solution set. The two main characteristics of HVC-Net are as follows:1

The g *2tch (·) function defined in Eq. (7) is used in minimization case. For maximization case, s0j and sj should be swapped in Eq. (7). Please refer to [12] for moredetailed explanations.

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K. Shang et al.
𝑆
𝜙

𝐬!
𝐬"

⋱

𝔼

𝐛

𝐚!
𝐚"

𝜂

𝐜

𝜌

+

&𝜽 (𝐬 ", 𝑆, 𝐫)
𝐻𝑉𝐶&𝜽 (𝐬 # , 𝑆, 𝐫)
𝐻𝑉𝐶

⋱

𝐝!
𝐝"
⋱

⋱

×𝐾

Fig. 3. The architecture of HVC-Net.

1. It is permutation equivalent. That is, for any permutation π of the input solutions (i.e., S = {sπ(1) , ..., sπ(N ) }), the outputs of HVC-Net are^^
HVC θ (sπ(1) , S, r), ..., HV
C θ (sπ(N ) , S, r). This means that the approximatedhypervolume contribution value for each solution is not affected by the orderof the input solutions. This is because the change of the order of the input
solutions will lead to the change of the order of the approximated values
consistently. This characteristic guarantees the robustness of HVC-Net.
2. A single HVC-Net can handle solution sets of various size. For example, we
can use a trained HVC-Net to handle a solution set with 10 solutions. We
can also use the same HVC-Net to handle a solution set with 100 solutions.
This characteristic guarantees high applicability of HVC-Net.
3.1

How to Train HVC-Net

In HVC-Net, we implicitly assume the minimization case where the reference
point for hypervolume contribution calculation is set to r = (1, ..., 1) and all
solutions in S are located in [0, 1]m . For the training of HVC-Net, we prepare
the training data as follows. First, we prepare L non-dominated solution sets
{S1 , S2 , ..., SL } where each solution set is located in [0, 1]m . Then, we calculate
the hypervolume contribution of each solution in each solution set based on thereference point r = (1, ..., 1). That is, we obtain the target output HV C(si , Sj , r)for each solution si ∈ Sj (i = 1, ..., |Sj |, j = 1, ..., L).
Based on the training data (i.e., the pairs of the solution sets and the corresponding hypervolume contributions), we define the loss function of HVC-Net
as follows:
|Sj |
L
2
1 X 1 X^
L(θ) =log HVC θ (si , Sj , r) − log HV C(si , Sj , r) .
L j=1 |Sj | i=1(9)

The loss function defined in Eq. (9) is similar to the mean squared error(MSE) loss function. The only difference is that we add log function to thehypervolume contribution (approximation) values. This is because the hypervolume contribution values are usually very small (e.g., in the magnitude of 10−4 ).
Using log values can make the training easier and better. More details about thetraining of HVC-Net are described in Section 4.1.

HVC-Net: Deep Learning based HVC Approximation

3.2

7

How to Use HVC-Net

After we train a HVC-Net, we can use it to approximate the hypervolume contribution if the solution set is in [0, 1]m and the reference point is r = (1, ..., 1). The
question is how to use it for hypervolume contribution approximation when thesolution set and the reference point are both arbitrarily given. Before answering
this question, we need the following properties.
Qm1
HV C(αProperty 1. For any positive vector α ∈ Rm
>0 , HV C(s, S, r) =i=1 αi2s, α S, α r), where denotes the element-wise multiplication .

Property 2. For any real vector β ∈ Rm , HV C(s, S, r) = HV C(s + β, S + β, r +β).
Property 3. HV C(s, S, r) = HV C(−s, −S, −r) where HV C(−s, −S, −r) is calculated for maximization problems whereas HV C(s, S, r) is calculated for minimization problems.
The above three properties can be easily obtained from the properties of thehypervolume indicator obtained in [10]. Based on these properties, we can firsttransform the solution set and the reference point so that the reference pointis r = (1, ..., 1) and the solution set is located in [0, 1]m . Then we use HVCNet to approximate the hypervolume contribution for the transformed solution
set. Lastly, we calculate the hypervolume contribution approximation for theoriginal solution set based on the output of HVC-Net. The last step is not needed
when our task is to find the solution with the smallest or largest hypervolume
contribution in a solution set.
Thus, although HVC-Net is trained based on solution sets in [0, 1]m andreference point r = (1, ..., 1), it can be used for any solution set with any reference
point.

4

Experiments

In this section, we conduct computational experiments to examine the performance of HVC-Net by comparing it with the point-based and line-based methods
for hypervolume contribution approximation.
4.1

Experimental Settings

HVC-Net Specifications In HVC-Net in Fig. 3, three networks φ, η, andρ need to be specified. In our experiments, all three networks are specified as
feedforward neural networks. Fig. 4 shows the structures of φ, η, and ρ used inour experiments. All the networks have three hidden layers. In the hidden layersof the three networks, we use ReLU activation function for efficient training.
2

For two vectors a = (a1 , ..., am ) and b = (b1 , ..., bm ), a
set B, a B means a b for all b ∈ B.

b = (a1 b1 , ..., am bm ). For a

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In the output layer of network ρ, we use Sigmoid activation function since thehypervolume contribution values are in [0, 1] for the training solution sets. In
HVC-Net, Step 2 can be stacked K times as shown in Fig. 3. In our experiments,
we set K = 10. That is, we have 10 different η networks in HVC-Net.
Input layer (𝑚)Hidden layers (128)Output layer (128)Output layer (128)

…

ReLU

ReLU

ReLU

…

…

(a) Network φ

Hidden layers (128)

…

…
ReLU

…

…
ReLU

…

…

…

ReLU

Input layer (128)(b) Network η

Input layer (128)

Hidden layers (128)Output layer (1)

…

…

…

…

ReLU

ReLU

ReLU

Sigmoid(c) Network ρ
Fig. 4. Networks φ, η, and ρ in HVC-Net. The number in the parentheses indicates thenumber of neurons in each layer. The activation function used in each layer is shown
under each layer.

Training and Testing Data Generation We examine 5, 8, and 10-objective
cases (i.e., m = 5, 8, 10). We generate training data with L solution sets {S1 , ..., SL }for each case where L = 1, 000, 000. Each solution set is generated using the following procedure:– Step 1: Randomly sample an integer num ∈ [1, 100] where num denotes thenumber of solutions in the solution set.
– Step 2: Randomly sample 1000 solutions in [0, 1]m as candidate solutions.
– Step 3: Apply non-dominated sorting to these 1000 solutions and obtain
different fronts {F1 , ..., Fl } where F1 is the first front (i.e., the set of nondominated solutions in the 1000 solutions) and Fl is the last front.
– Step 4: Identify all the fronts Fi with |Fi | ≥ num. If no front satisfies thiscondition, go back to Step 2.
– Step 5: Randomly pick one front Fi with |Fi | ≥ num and randomly selectnum solutions from this front to construct one solution set.
This procedure is used in order to select a wide variety of non-dominatedsolution sets for training. On average, about 10,000 solution sets with the same
size are generated (1,000,000 solution sets in total for 100 different sizes).

HVC-Net: Deep Learning based HVC Approximation

9

We generate two types of testing solution sets. Type-I testing solution sets
are generated using exactly the same procedure as described above. We generate
10,000 Type-I testing solution sets for each case (i.e., m = 5, 8, 10). These 10,000
solution sets form one group. We generate 10 different groups of Type-I testing
solution sets for each case of m. Type-II testing solution sets are generated using
a similar procedure. The only difference is that an integer num ∈ [101, 200] is
randomly sampled in Step 1 and 10,000 solutions are randomly sampled in Step 2.
We generate 10,000 Type-II testing solution sets for each case of m. These 10,000
solution sets form one group. We generate 10 different groups of Type-II for eachcase of m. Type-II testing solution sets are used to test the generalization abilityof HVC-Net since we train HVC-Net using solution sets with 1-100 solutions andtest HVC-Net using solution sets with 101-200 solutions.
Parameter Settings For the training of HVC-Net, we use Adam [8], an effective gradient-based optimization method with an adaptive learning rate. The
initial learning rate is set to 10−4 . For all the other parameters in Adam, we use
their default settings in PyTorch [9]. The batch size during training is set to 100.
The number of epochs for training is set to 100.
For the number of sampling points in the point-based method, we examine 20
different settings: 5, 10, ..., 100. For the number of lines in the line-based method,
we examine 20 different settings: 5, 10, ..., 100. We use the unit normal vector
method [5] to generate the direction vector set Λ in the line-based method.
Performance Metrics To compare the performance of different hypervolume contribution approximation methods, we use the correct identification rate(CIR). CIR is a metric which can evaluate the ability of a method to identify
the smallest (largest) hypervolume contributor in a solution set. For example,
suppose we have P solution sets. If a method can correctly identify the smallest (largest) hypervolume contributor for Q out of P solution sets, then CIR is
calculated as Q/P . In our experiments, we use CIRmin (i.e., CIR for identifying
the smallest hypervolume contributor) and CIRmax (i.e., CIR for identifying thelargest hypervolume contributor). A larger CIR value means better approximation quality of a method.
We also record the runtime of the three methods to compare their efficiency.
Here the runtime of HVC-Net means its evaluation time on the testing solution
sets, not the training time.
Platforms All the methods are coded in Python and tested on a server withIntel(R) Xeon(R) Gold 6130 CPU @ 2.10GHz, GeForce RTX 2080 GPU, andUbuntu 18.04.6 LTS. HVC-Net is implemented based on PyTorch version 1.9.0.
4.2

The Training of HVC-Net

Fig. 5 shows the training curve of HVC-Net in each case of m. We can see thatthe loss decreases sharply in the first 20 epochs. This is mainly because we use

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K. Shang et al.

a batch size of 100 for training 1M solutions, which means that the parameters
of HVC-Net can be updated 10K times in each epoch. We can also observe thatthe loss becomes very small at the end of the training process in each figure,which shows the success of the training of HVC-Net.

(a) 5-objective(b) 8-objective(c) 10-objective

Fig. 5. The training curve of HVC-Net in each case.

Table 1 shows the time used for training HVC-Net. Although the training of
HVC-Net needs quite a substantial time, the trained HVC-Net models can besaved and are ready to use at any time. That is, once we obtain a well trained
HVC-Net model, we can save it and use it directly in the future without spending
a lot of time to retrain it.
Table 1. The time (GPU hours) used for training HVC-Net in each case.
5-objective

8-objective

10-objective

369.13370.47383.79Next, we will examine the performance of our trained HVC-Net models on
the testing solution sets.
4.3

Testing on Type-I Solution Sets

We apply the three hypervolume contribution approximation methods to Type-I
testing solution sets, i.e., solution sets with 1-100 solutions. For fair comparison,
all the methods are tested on CPU. That is, we disable GPU when using HVCNet for evaluation.
Fig. 6 shows the results of CIRmin and CIRmax for each case. We can see that
HVC-Net clearly dominates the other two methods in most cases in terms of boththe correct identification rate and the runtime, which shows the advantage ofusing HVC-Net for identifying the smallest (largest) hypervolume contributor ina solution set. It is worth noting that the point-based and line-based methods arevery time-consuming compared with HVC-Net. HVC-Net is able to process thetesting solution sets in less than 10 seconds and achieves a good CIR value. The

HVC-Net: Deep Learning based HVC Approximation(a) 5-objective(b) 8-objective(c) 10-objective(d) 5-objective(e) 8-objective(f) 10-objective

11

Fig. 6. Experimental results of the three hypervolume contribution approximation
methods on Type-I testing solution sets in each case. The runtime (in seconds) means
the total time for evaluating 10,000 testing solution sets. The CIR (CIRmin in (a)-(c)and CIRmax in (d)-(f)) means the correct identification rate over 10,000 testing solution
sets. All the results are the average over 10 groups of Type-I testing solution sets.

other two methods even perform worse (i.e., a smaller CIR value) than HVCNet by consuming 1000 seconds. Of course, the other two methods can achieve
better CIR values than HVC-Net using more points or lines in some cases (e.g.,
in Fig. 6 (a) the other two methods achieve better CIR values than HVC-Net
by consuming more than 1000 seconds). However, the runtime cost is too highto realize this goal for the point-based and line-based methods.

4.4

Testing on Type-II Solution Sets

We also apply the three hypervolume contribution approximation methods toType-II testing solution sets, i.e., solution sets with 101-200 solutions. We use
Type-II testing solution sets to examine the generalization ability of HVC-Net.
Fig. 7 shows the results of CIRmin and CIRmax for each case. We can observe that HVC-Net performs well in general. It can still dominate the othertwo methods in terms of the correct identification rate and the runtime in mostcases. There is almost no runtime increase for HVC-Net compared with the results on Type-I solution sets. However, there is a significant runtime increase forthe point-based and line-based methods. These results show the strong generalization ability and the efficiency of HVC-Net. Although HVC-Net is trainedusing solution sets with 1-100 solutions, it is able to handle solution sets with
101-200 solutions effectively and efficiently.

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K. Shang et al.

(a) 5-objective(b) 8-objective(c) 10-objective(d) 5-objective(e) 8-objective(f) 10-objective

Fig. 7. Experimental results of the three hypervolume contribution approximation
methods on Type-II testing solution sets in each case. The runtime (in seconds) means
the total time for evaluating 10,000 testing solution sets. The CIR (CIRmin in (a)-(c)and CIRmax in (d)-(f)) means the correct identification rate over 10,000 testing solution
sets. All the results are the average over 10 groups of Type-II testing solution sets.

5

Conclusions

In this paper, we proposed HVC-Net, a deep learning based method for hypervolume contribution approximation. We compared HVC-Net with the point-based
method and the line-based method. The experimental results showed that HVCNet outperformed the other two methods in terms of both the correct identification rate and the runtime, which showed the potential of using deep learningtechnique for hypervolume contribution approximation.
Our future work is the development of a hypervolume-based EMO algorithm
and a hypervolume subset selection algorithm based on HVC-Net for manyobjective optimization. Of course, we will also try to further improve the performance of HVC-Net by improving its structure, parameter settings, training
method, and so on.
All the source codes and the trained HVC-Net models are available athttps://github.com/HisaoLabSUSTC/HVC-Net.
Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 62002152, 61876075), Guangdong Provincial Key Laboratory(Grant No. 2020B121201001), the Program for Guangdong Introducing Innovative andEnterpreneurial Teams (Grant No. 2017ZT07X386), The Stable Support Plan Programof Shenzhen Natural Science Fund (Grant No. 20200925174447003), Shenzhen Scienceand Technology Program (Grant No. KQTD2016112514355531).

HVC-Net: Deep Learning based HVC Approximation

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