Classification of 3D Shapes with Convolutional Neural Networks
Leonid Keselman
Stanford University
leonidk@stanford.edu

various designs and tests of convolutional neural networks
to solve this object classification challenge. In this case, wewill present both 2D and 3D convolutional networks, bothtrained from scratch and refined on existing datasets.

1.1. Dataset format
In this paper, we will train and test on the ModelNet40
dataset released from the ShapeNet dataset [1, 22]. The
dataset is already split into a test set and a training set. In
my own use, the training set is further split into a training
set and a validation set, and the validation set is used for
hyper-parameter tuning, although the reported results in thepaper are test set accuracies. The authors of the original paper have kept track of published results on this dataset andtheir table is replicated in full, with citations as Table 1 on
page 7.
Unlike images, in which there is a single standard representation, namely pixel grids of luminosity, shapes have awide range of representations, many of them incompatible
in nature [12]. In fact, shapes represent a very different typeof object than an image. While images represent the projection of a three-dimensional scene (four if you count motionblur), shapes represent an R2 surface embedded in R3 . Traditionally, shapes are stored as meshes, which are a list of
triangular or quadrilateral components, unordered and withno connectivity information. This is similar to using vector shapes to represent 2D content, and has the downsideof providing minimal locality and connectivity information.
A straightforward extension of the pixel model of images(that is, a quantized representation as a rank-2 tensor) to
shapes is to represent them as a rank-3 tensor, or volume.
While many volume models exist, we chose to use occupancy grids, which are represent the inside of a shape withones and all other elements as zero. An alternative volume
representation that we wish to explore in future work is thatof level sets or signed distance fields [12], which are more
information dense, but are less common in 2D. Our volume
size is 30 x 30 x 30, which was used in the original paper
and will be replicated for consistent results [1]. However,as seen in figure 1, such volumes are fairly low resolution,
but still require the same memory footprint as an 165x165

Figure 1. A visual example of the data used in this report. On theleft is a mesh from the ShapeNet dataset (from the ’chair’ class),
while the examples on the right are 30 x 30 x 30 voxel grids. Examples of the ’desk’, ’bench’, ’airplane’ and ’chair’ classes areprovided. Only the surface voxels are visible and the blue color isfor visualization, as the data itself is simply binary.

Abstract
This project explores how to apply convolutional neural networks to efficiently classify 3D shapes. Using 3D
shapes from the ShapeNet project [1], they are first centered, rescaled and resampled to occupancy voxel grids ofdimension 30 x 30 x 30. We implement 3D convolutional
neural networks operating directly on these three dimensional volumes, as well as 2D convolutional networks operating on a learned embedded from the 3D model. Resultsfrom both methods are state-of-the-art compared to existing, published methods.

1. Introduction
Recently, with the the ImageNet challenge in image classification [15], many other large scale datasets have beencollected for the purpose of creating competitive, quantitative progress in machine learning research. Specifically,
this project focuses on ShapeNet [1], a recently released
dataset that includes 3D CAD models of forty different object categories, dubbed ModelNet40. This paper presents1across the R2 surface of a shape, this theory is neither tested
nor elaborated upon in this project.
However, it seems that a natural question to ask is, canwe come up with a way to learn a robust 3D to 2D projection? That is the motivating example behind most of themethods implemented below, which seem to implement anembedding network to transform 30 x 30 x 30 volumes directly into an image sized dimension for shape classification. The goal is that this non-linear transformation willarise from gradient learning methods.
A table summarized the performance of all previously
published methods and all methods implemented herein isavailable as Table 1 on page 7.

image. As an implementation detail, the datasets were converted and stored as shuffled HDF5 files. The dataset contains 12,311 files, with a roughly 70%/30% training and test
split already provided. A smaller subset of the training set
was used for validation, but all the results reported here areon the standard test set provided.

2. Related Work
The original ShapeNets paper, which introduced theModelNet40 dataset [22], implemented a 3D volumetricconvolution approach to classify the dataset. They report
77% top one accuracy, which suggests that the dataset isfairly easy for convolutional neural networks to fit, as top
one accuracy performance on ImageNet is still below 30%[5]. Of interest, they report pre-training the data with a
deep-belief-net architecture and contrastive divergence [6].
They also report that using trained neural networks significantly reports the hand-designed features used in the previous literature, namely Light Field descriptors and Spherical
Harmonic descriptors [2].
The other published work using 3D convolutions is
VoxNet [13], which uses a more traditional CNN architecture, with only gradient-based training methods. This mirrors those used in seminal works in digit [10] and image
classification [9]. Our implemented 3D CNN work draws
on a similar architecture, design, and training procedure.
Their work was sufficient to bring state-of-the-art to 83.0%
top one accuracy.
However, there has been another line of work on thisdataset, that focusing on using two dimensional convolutional networks. This circumvents the intense memory requirements of 3D CNNs, where data is typically in R5(width, length, height, filter number, minibatch size). The
first work, DeepPano [17], used a cylindrical projection of
the mesh, and only fed the network a single view. This circumvented the need to select a viewpoint.The results, obtaining a 77.6% accuracy, were an improvement on the originally published ShapeNet paper.
Most recently, at ICCV 2015, Multiview CNNs were
published, which used a set of sampled projections of themesh. Their results only used 2D convolutions, were pretrained using ImageNet based classifiers, and were capableof accepting multiple viewpoints. Their best performancewith a single viewpoint, of 85.1%, was a significant improvement on both the 3D CNNs and single view 2D CNNs.
As an even more impressive result, their performance with
80 views was 90.1%.
Based on this impressive performance, there may be evidence to believe that 2D projections of 3D shapes, classified
with 2D CNNs, may perform better than using straightforward 3D CNNs. While the authors have some thoughts on
this matter, perhaps connecting the locality of CNN models with how projections create a small-world network [21]3. Methods
In general, this work uses convolutional neural networks,
with pooling, convolution, and fully connected layers [10].
For this work there are two general classes of model implemented, 3D CNNs such as those implemented in [13],and 2D CNNs such as those implemented in [18], with the
caveat that a 3D to 2D embedded is learned rather than developed using a heuristic. An appropriate section is available to describe each class of methods, but they both haveseveral things in common. Namely, all models are learned
using a ReLU [14] nonlinearlity, convolutional layers include L2 regularization, and the model is a loss function isa softmax training loss, which implements a cross-entropy
loss. Formally,
1 X
Li + λR(W )Loss =
N ie fy iLi = − log( P fi )
je
XX
2
Wk,l
R(W ) =
kl

This should provide a more unique solution across thespace of possible weights, while also preferring a high probability value for the correct answer and a low probability
for all incorrect answers. Additionally, all models were initialized with mostly Xavier [4] initialization, and slightly
positive biases, unless otherwise noted. Models were all
trained directly with gradient-based learning, using backpropagation and mini-batch gradient descent. Batches weretypically sized to fit in memory and gradient were accumulated until batches were at least size 32. Learning rates (α)
were set to 0.001, and dropped by a power of ten every 2
epochs. Typically convergence is quick with 5-8 epochs required. The update rule was typically SGD with momentum(γ set to 0.9), as follows, although some alternatives werementioned below.
v = γv + α∇Loss2

40 classes
Fully Connected. 1200 Dimensions
Fully Connected. 2048 Dimensions

512 filters, 4x4,
Stride 1

160 filters, 5x5Stride 2
3x3, Stride 2 Pooling

Figure 3. An example of our Upconv2 network being trained
directly from VGG-16 weights. The plot shows test accuracy andloss, but this was simply because the graph was generated frommodel snapshots. Actual results during design were only on thevalidation set

48 filters, 3x3
Stride 130
3030

consultation provided by Charles Qi and Hao Su from Professor Guibas’ Geometric Computation group. Some interesting results were that Adam converged very quickly to agood result in a single epoch, but saturated in performancecompared to vanilla SGD, which took longer to converge
but did so to a better result. Additionally, dropout provided
an improvement of a few percentage points, and was applied at the fully connected layers only. Additionally, asmall amount of max pooling was beneficial after the bottom convolution layer. The best model generated obtained
86.7% top one accuracy on the ModelNet40 dataset.

Figure 2. A view of the architecture used for the 3D CNN. Thisshows the most complete and best performing architecture, although others were tried, with larger convolutional filters, smaller
fully connected layers, and without pooling. Some selected results
are reported in Table 1. Additionally, dropout of 50% probability
was used before both fully connected layers.

W =W −v
In general, most of this work will be done using existing
CNN (convolutional neural network) packages. I’ve used
Torch [3] for my 3D CNN networks, and Caffe [7] for myfine-tuned embedded networks. This comes from Caffe’s
convenience for tweaking layerwise learning rates and finetuning from the Model Zoo. On the other hand, Torch has
better support for volumeric operations (namely, Torch has
VolumetricPooling while Caffe did not).

3.2. Learning a 3D to 2D embedding
Since the current best performer on ModelNet40 uses
pre-trained CNNs on multi-view renders of ModelNet, it
seems natural to propose that we use a gradient-trainabletransformation to directly feed ModelNet data to a pretrained image network, with no intermediate steps in between. There are multiple advantages of those approach.
First, the memory hit of using volumetric convolution(which is typically N 5 with minibatches, width, height,
length, and filter bank size) is avoided. Second, this allows us to learn a transformation that may be better than theheuristic transformations used in existing 2D-based shapeclassifiers. Third, we can exploit ImageNet’s pre-training
as a regularizer, or prior, so that the trained embedding
layer will create images that follow natural image statistics
and don’t overfit our small dataset as quickly. Additionally,systems which deploy both ShapeNet and ImageNet classifiers can reuse weights and a computational routine between
them, allowing for an increased value and use of accelerating existing ImageNet classifiers.
The basic idea is straightforward, learn an approach totake 30 x 30 x 30 volumetric data and find a way to output
224 x 224 x 3 image data. This embedding network is then

3.1. 3D Convolutional Neural Network
As seen in figure 2, a fairly straightforward CNN architecture, with three convolutional layers and two fullyconnected layers was used to train a network. While
two convolutional layer models have been published before[13], we decided to follow the recent trend of going deeper[19] and build a three convolutional layer model; we also
used a traditional two layer fully connected network on the
top [10] After some experimentation with different solvers(SGD, Adam [8], RMSProp[20]), we were able to get very
good performance, exceeding the previously published results for 3D CNNs. Using computing resources from theGraphics Lab at Stanford, it takes about hour to train amodel to convergence on a single Tesla K40. These models were not heavily tuned or optimized, as their goal was
primarily to serve a baseline for the 2D embedding models below. The tuning of hyperparameters received help and33.2.1

Learning a full-connected embedding

Image-based classifier

224
224
3

The primary challenge of connecting an ImageNet network
to ModelNet is that the dimensions of the data are very different, and the naive solution is the use of a fully-connected
layer. As seen in figure 4, it’s fairly staightforward to use a
traditional, single layer fully connected layer (with a nonlinearility) to perform embedding. However, this creates about
80 million parameters, even by doing this over greyscale
images and duplicating the image across all channels. Our
initial experiments fine-tuned a pre-trained CaffeNet [7],with a fully-connected layer hooking up the ModelNet input to an ImageNet sized image. The challenge here is thatthe representation is very low dimensional (1024), the number of parameters is very large, and the images look likenoise. We were able to tune these networks to begin training and converge, but their performance was significantly
worse than that of 3D CNNs.
To further reduce the number of parameters, we moved
to fine-tuning a Network-in-Network [11] architecture,
which has no fully-connected layers at the top, and instead
uses global average pooling and 1x1 convolutions to reduce
down to a classifier. We were also able to modify and train
this network to converge, although it took longer and performed worse than the CaffeNet model. These fine-tuned
networks were trained on a CPU, specifically, a four coredesktop computer in Caffe (on Windows!) and took roughly
3-4 hours to converge. GoogleNet was also fine-tuned, andalso performed poorly.
To enhance the performance of the FC layer, we locked
the CaffeNet weights, and only fine-tuned the FC7 features
and the embedding network, along with trying 3 1x1 convolutions on the volume (one down each cardinal axis of
the volume, equivalent to 1x1x30, 1x30x1,30x1x1 convolutions), to generate a 30x30x3 volume, which was thenreduced to size 30x30x1 and learned fully connected withthe single channel image size. While this slightly reduced
the dimensionality of the vector (900 dimensions), the morestructured learning allowed for a large increase in performance (up to 75%). An example of these embeddings canbe seen in 5.

1x1
Convolutions
Stride 1, Pad 7

240
240
3
7x7
Up-Convolutions
Stride 7

30
30
3

3x3
Convolutions
Stride 1

Image-based classifier

30
30
48

227
227
3

Concatenate

Duplicate 3 times
227
227
1

30
30
16

30
30
16

30
30
16

30
30
16

30
30
16

30
30
16

Fully Connected. 1200 Dimensions

30
30
30

3x3
Convolutions
Stride 1

16 30 x 1 x 1
16 1 x 30 x 1
16 1 x 1 x 30
Convolutions

30
30
30

Figure 4. A view of the various architectures for 3D to 2D Embedding networks. On the left is a basic fully connected embedding
with just a single matrix multiply. To reduce the parameter space,this network is only done on a single channel and then duplicatedthrice for a three channel input image. The number of parameters
for such a network is easily 80 million, even for the single channel
version. To the right is an upsampling-based architecture, which
has far fewer parameters and generates more natural looking images; this is our Upconv1 architecture

Figure 5. Two example images generated by using a fully connected network. The images look like complete noise, but stillachieve 75% top one accuracy in classification performance byusing CaffeNet with fine-tuned fc7 features [7]. The images arenormalized for 0 to 1 for display.

3.2.2

Learning a convolutional embedding

In order to reduce the burden of a fully-connected layer, webegan to explore a more convolutional architecture to creating an embedding between the 3D ModelNet and 2D ImageNet data. The basic building block was a deconvolution,or an upconvolution layer as we prefer to call it, where a single input pixel is mapped to multiple output pixels. These
were recently introduced in the scope of full-frame segmentation [23], and are being deployed here to avoid the highdimensional fully connected layer.
One challenge to moving to these upconvolution layers

used to generate images which can be fed into a normal
ImageNet network. The network can then either be trained
from scratch, fine-tuned entirely, or fine-tuned for just the
top layer features. There are multiple possible architectures
for such an embedding and three at explored in detail here.
The two basic differences are summarized in figure 4. All
these models converge fairly smoothly and quickly and an
example of their performance is shown in figure 3. The
models were all trained on Amazon’s AWS EC2 instanceswith Nvidia K520 GPUs.
4

Image-based classifier

224
224
3

1x1
Convolutions
Stride 1, Pad 7

240
240
16
13x13
Up-Convolutions
Stride 7, Pad 3
30
30
64

Figure 6. Upconv1 results. These examples are all predicted correctly and are of classes (left-to-right) of ’cup’, ’bowl’, ’desk’.
There are visible holes from using non-overlapping convolutions
for our upsampling, and these are fixed by our upconv2 architecture. The images are normalized for 0 to 1 for display.

3x3
Convolutions
Stride 130
30
48Concatenate

30
30
16

30
30
16

30
30
16

30
30
16

30
30
16

30
30
16

3x3
Convolutions
Stride 1

16 30 x 1 x 1
16 1 x 30 x 1
16 1 x 1 x 30
Convolutions

30
30
30

Figure 8. An overview of our improved upconvolution architecture: Upconv2.

Figure 7. Upconv2 examples. These examples are all predicted
correctly and are of classes (left-to-right) of ’cup’, ’bowl’, ’desk’.
The images are normalized for 0 to 1 for display.

is that the dimensions of our data is even, which requires aneven sized output image layer if we want symmetric edge
performance in between. To achieve this, we only implement these upconvolution models on 224 sized ImageNet
networks, such as ResNet[5] and VGG [16], but not CaffeNet, which is sized 227x227.
Our basic design idea (as seen in figure 4) was to continue with the 1x1 convolution design we used in the fully
connected case, where 1x1x30, 1x30x1 and 30x1x1 convolutions are used down each cardinal axis. They’re thentransformed with another convolution layer, concatenated,and passed through upsampling layers. These then have tobe cropped with a 1x1 convolution to ImageNet size. The
upconvolution layer has to be carefully initialized with bilinear weights, as using Xavier initialization leads to a failure of the optimization to converge in our testing. An example of these embeddings can be seen in 6.
Our initial results with the upconvolution layer in werevery successful, but we noticed some issues with holes inthe data due to our choice of non-overlapping convolution
kernels. We thus implemented 8 to alleviate these issues.
It is denoted as Upconv2 in our results, and some example
embeddings can be seen in 7.
3.2.3

Figure 9. Confusion matrix for a 1x1 + Locked CaffeNet stylemethod (2D convolutions)ter ImageNet performance but were difficult to train in our
testing.

4. Results
The summary table of results can be seen in table 1. The
previously published methods are at the top with citations
numbers. The 3D CNN networks, as described in figure 2,
built in Torch and trained from scratch (on a GPU) are listed
in the middle section. While results from three different
embedding architectures, as shown in figures 4 and 8 areseen at the end.

Picking an ImageNet network

We tried our embedding results across many various ImageNet architectures. In general, we found VGG-16 to perform best in our testing. ResNet and GoogleNet report bet5

ding may be very similar (a flat surface). However, eventhis is an understandably difficult case of semantic labels.
There are many other interesting lessons learned. First,
certain models are difficult to learn an embedding for,such as ResNets and GoogleNet. Initializing the upconvolution layers in the embedding with anything other thanbilinear interpolation is disastrous. Fine-tuning from an
ImageNet model is significantly superior to training fromscratch. Another interesting results is these embeddings
generate natural enough images that even models without
fully-connected classifier layers, such as the network-innetwork or GoogleNet models, are able to train very highperforming classifiers on the dataset.
All of these results are clearly demonstrated in the tableof results on page 7.

5. Conclusions and Future WorkThe results in this project show that CNNs have the ability to learn an embedding from 3D to 2D which is superior
to the single view render and classification results publishedin the current state-of-the-art papers [17] [18]. This showsthat gradient-trainable transformations across various data
formats are very powerful if the goal is only classification
loss.
There are many interesting questions left. Such as, dosigned distance fields perform better than occupancy voxels? What is the nature of 2D convolutions being more
powerful than 3D convolutions? Is it in the model ensemble
or is there something inherent in the data? We would liketo also explore improved typologies for building an embedding network.
The last line of further work would be to improve theapplicability of these models. For example, if we had additional time, we’d like to demonstrate that they can be usedon live depth data (from a single viewpoint), using a data
source such as a Microsoft Kinect. Additionally, we’d liketo perhaps learning a 3D embedding from 2D data. This is
the direct inverse of the data transformations explored here
and would allow us to perform shape classification from images, using only annotated sets of CAD models.

Figure 10. Confusion matrix for a volumetric convolution method(3D convolutions)

The 3D CNNs are stronger (86.7%) than the previously
published 3D CNN results on this dataset [13, 22] at 83.1%.
However, they don’t perform as well as the published workon multiview 2D CNNs. The confusion matrix over the test
set is shown in figure 10. Most of the errors are sensible,
such as confusing wardrobes for dressers, or flowers pots
for plants.
The 2D CNNs, attempting to create a learned embedding layer to meet and surpass the heuristic techniques presented in previous work, perform very well. Their generated images, shown in figures 6 and 7, are seemingly natural. Additionally, their performance (85.5%) is only a hairbehind the trained 3D CNN on the same data, and is better than all previously published results with just a single
view. That is, DeepPano’s 77.6% or even MVCNN’s 85.1%
on a single image. This suggests that a machine-learnedtransformation may be superior to an old standby of using
a rendering technique, even though the input resolution to
the transformation (30x30x30) is very low, compared to arender of the original mesh, as seen in figure 1. This isvery impressive considering the ambiguity in the classes ofthe dataset (wardrobe/dresser, table/tv-stand/xbox) and thelow resolution images generated by the model (see figure 6
7. There are many possible ways to mitigate this, including
the use of higher resolution voxel grids, or possibly learning
an embedding from another, more compact data representation. There is a confusion matrix for a 2D embedding model
shown in figure 9. It’s errors are very similar to those in the
3D CNN, confusing flower pots for plants and so forth. The
biggest difference is that the 2D CNN struggles separatingtables from desks, a suggestion that their learned embed6

Architecture

Conv

EmbeddingPretrain

Regularization

Top 1 Accuracy

MVCNN 80 Images [18]MVCNN 1 Image [18]VoxNet [13]DeepPano [17]3DShapeNets [22]

2D
2D
3D
2D
3D

ImageNet
ImageNetNone

3D CNN
3D CNN
3D CNN + Pooling

3D
3D
3D

None
None
None

None
Dropout
Dropout

83.8%
84.6%
86.7%

10000
10000
10000

Embedding + CaffeNet [7]Embedding + NetworkInNetwork [11]Embedding + GoogleNet [19]Embedding + 1x1 Conv + CaffeNet
Embedding + 1x1 Conv + CaffeNet (Locked)
Embedding + 1x1 Conv + CaffeNet (from scratch)

2D
2D
2D
2D
2D
2D

FC
FC
FC
FC
FC
FC

ImageNet
ImageNet
ImageNet
ImageNet
ImageNet
ImageNet

Dropout
Dropout
Dropout
Dropout
Dropout

66.5%
63.0%
57.0%
67.3%
75.6%
63.1%

3000
5000
5000
3750
4750
10000

Embedding + ResNet-50 [5]Embedding + GoogleNet [19]Embedding + VGG[16] + Xavier[4] for Upconv
Embedding + VGG-16 [16]Embedding + VGG-16 [16] + ft

2D
2D
2D
2D
2D

Upconv1
Upconv1
Upconv1
Upconv1
Upconv1

ImageNet
ImageNet
ImageNet
ImageNet
ImageNet

Dropout
Dropout
Dropout
Dropout
Dropout

50.3%
31.0%
0.03%
78.0%
83.5%

10000
10000
10000
10000
10000

Embedding + VGG-16 [16]2DUpconv2

ImageNet

Dropout

85.5%

10000

90.1%
85.1%
83.0%
77.6%
77.0%

None

Table 1. Table of results. The results in bold are the best performers when given a single input model. The presented methods in this project
are the best in both 2D and 3D CNNs when evaluating just a single image.

References[9] A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet
classification with deep convolutional neural networks. In
Advances in neural information processing systems, pages
1097–1105, 2012.
[10] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradientbased learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
[11] M. Lin, Q. Chen, and S. Yan. Network in network. arXiv
preprint arXiv:1312.4400, 2013.
[12] B. C. Lucas, M. Kazhdan, and R. H. Taylor. Springls: adeformable model representation to provide interoperability
between meshes and level sets. In Medical Image Computing
and Computer-Assisted Intervention–MICCAI 2011, pages
442–450. Springer, 2011.
[13] D. Maturana and S. Scherer. Voxnet: A 3d convolutional
neural network for real-time object recognition. In IEEE/RSJ
International Conference on Intelligent Robots and Systems,
September 2015.
[14] V. Nair and G. E. Hinton. Rectified linear units improve
restricted boltzmann machines. In Proceedings of the 27th
International Conference on Machine Learning (ICML-10),
pages 807–814, 2010.
[15] O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh,
S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. S. Bernstein,
A. C. Berg, and F. Li. Imagenet large scale visual recognition
challenge. CoRR, abs/1409.0575, 2014.
[16] K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. CoRR,
abs/1409.1556, 2014.

[1] A. X. Chang, T. A. Funkhouser, L. J. Guibas, P. Hanrahan,
Q. Huang, Z. Li, S. Savarese, M. Savva, S. Song, H. Su,
J. Xiao, L. Yi, and F. Yu. Shapenet: An information-rich 3dmodel repository. CoRR, abs/1512.03012, 2015.
[2] D.-Y. Chen, X.-P. Tian, Y.-T. Shen, and M. Ouhyoung. On
visual similarity based 3d model retrieval. In Computer
graphics forum, volume 22, pages 223–232. Wiley Online
Library, 2003.
[3] R. Collobert, K. Kavukcuoglu, and C. Farabet. Torch7: A
matlab-like environment for machine learning. In BigLearn,
NIPS Workshop, number EPFL-CONF-192376, 2011.
[4] X. Glorot and Y. Bengio. Understanding the difficulty of
training deep feedforward neural networks. In International
conference on artificial intelligence and statistics, pages
249–256, 2010.
[5] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learningfor image recognition. CoRR, abs/1512.03385, 2015.
[6] G. E. Hinton, S. Osindero, and Y.-W. Teh. A fast learning algorithm for deep belief nets. Neural computation,
18(7):1527–1554, 2006.
[7] Y. Jia, E. Shelhamer, J. Donahue, S. Karayev, J. Long, R. Girshick, S. Guadarrama, and T. Darrell. Caffe: Convolutionalarchitecture for fast feature embedding. In Proceedings of
the ACM International Conference on Multimedia, pages
675–678. ACM, 2014.
[8] D. P. Kingma and J. Ba. Adam: A method for stochastic
optimization. CoRR, abs/1412.6980, 2014.

7

Iteration

[17] S. Song and J. Xiao. Deep sliding shapes for amodal
3d object detection in rgb-d images.
arXiv preprint
arXiv:1511.02300, 2015.
[18] H. Su, S. Maji, E. Kalogerakis, and E. Learned-Miller. Multiview convolutional neural networks for 3d shape recognition. In Proceedings of the IEEE International Conference
on Computer Vision, pages 945–953, 2015.
[19] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed,
D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich.
Going deeper with convolutions. CoRR, abs/1409.4842,
2014.
[20] T. Tieleman and G. Hinton. Lecture 6.5—RmsProp: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning,
2012.
[21] D. J. Watts and S. H. Strogatz. Collective dynamics of smallworldnetworks. nature, 393(6684):440–442, 1998.
[22] Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang, andJ. Xiao. 3d shapenets: A deep representation for volumetricshapes. In Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition, pages 1912–1920, 2015.
[23] L. Xu, J. S. Ren, C. Liu, and J. Jia. Deep convolutional
neural network for image deconvolution. In Z. Ghahramani,
M. Welling, C. Cortes, N. D. Lawrence, and K. Q. Weinberger, editors, Advances in Neural Information Processing Systems 27, pages 1790–1798. Curran Associates, Inc.,2014.

8