Convolutional neural networks (CNNs) have made great progress in various fields, such as object classification [Krizhevsky, Sutskever, and Hinton2012], detection [Girshick et al.2013] and character recognition [Cireşan et al.2011]. One of the key factors contributing to the success of the modern deep learning models is using the non-saturated activation function (e.g.ReLU) to replace its saturated counterpart (e.g. sigmoid and tanh), which not only solves the problem of “exploding/vanishing gradient” but also makes the deep networks converge fast. Among all the proposed non-saturated activation functions, the Rectified Linear Unit (ReLU) [Nair and Hinton2010] is widely viewed as one of the several reasons for the remarkable performance of deep networks [Krizhevsky, Sutskever, and Hinton2012].
Recently, there are some other activation functions proposed to boost the performance of CNNs. Leaky ReLU (LReLU) [Maas, Hannun, and Ng2013] assigns the negative part with a non-zero slope. [He et al.2015] proposed the parametric rectified linear unit (PReLU), which requires learning the negative part instead of using predefined values. Adaptive piecewise linear activation (APL) proposed in [Agostinelli et al.2014] sums up several hinge-shared linear functions. [Goodfellow et al.2013] proposed the “maxout” activation function, which approximates arbitrary convex functions by computing the maximum of
linear functions for each neuron as the output.
Although the activation functions mentioned above have been reported to achieve good performance in CNNs, they all suffer from a weaknesses, i.e.,
their limited ability to learn non-linear transformation. For example, none of ReLU, LReLU, PReLU and maxout can learn the non-convex functions since they are essentially all convex functions. Although APL can approximate non-convex function, it requires the rightmost linear function in all the component functions to have a unit slope and bias 0, which is an inappropriate constraint and undermines its representation ability.
Inspired by the fundamental Webner-Fechner law [Weber1851] and Stevens law [Stevens1957] in psychophysics and neural sciences, we propose a novel kind of activation unit, namely the S-shaped rectified linear unit (SReLU). Two examples of SReLU’s function forms are shown in Figure 1(c)(d). Briefly speaking, both the Webner-Fechner law and the Stevens law describe the relationship between the magnitude of a physical stimulus and its perceived intensity or strength [Johnson, Hsiao, and Yoshioka2002]. The Webner-Fechner law holds that the perceived magnitude is a logarithmic function of the stimulus intensity multiplied by a modality and a dimension specific constant . That is,
And the Stevens law explains the relationship through a power function, i.e.,
where all the parameters have the same definitions as in the Webner-Fechner law, except for an additional parameter which is an exponent depending on the type of the stimulus. The function forms proposed by the two laws are shown in Figure 1(a)(b). These laws are usually valid for general sensory phenomena and can account for many properties of sensory neurons [Randall et al.2002]. More detailed discussions will be presented in Related Work.
Roughly, SReLU consists of three piecewise linear functions constrained by four learnable parameters as shown in Eqn. (7
). The usage of SReLU brings two advantages to the deep network. Firstly, SReLU can learn both convex and non-convex functions, without imposing any constraints on its learnable parameters, thus the deep network with SReLU has a stronger feature learning capability. Secondly, since SReLU utilizes piecewise linear functions rather than saturated functions, thus it shares the same advantages of the non-saturated activation functions: it does not suffer from the “exploding/vanishing gradient” problem and has a high computational speed during the forward and back-propagation of deep networks. To verify the effectiveness of SReLU, we test it with two popular deep architectures, Network in Network and GoogLeNet, on four datasets with different scales, including CIFAR10, CIFAR100 and MNIST and ImageNet. The experimental results have shown remarkable improvement over other activation functions.
In this section, we first review some activation units including ReLU, LReLU, PReLU, APL and maxout. Then we introduce two basic laws in psychophysics and neural sciences: Webner-Fechner law [Fechner1965] and Stevens law [Stevens1957], as well as our motivation.
Rectified Linear Unit (ReLU) and Its Generalizations
ReLU [Nair and Hinton2010] is defined as
where is the input and is the output. LReLU [Maas, Hannun, and Ng2013] assigns a slope to its negative input. It is defined as
where is a predefined slope. PReLU is only different from LReLU in that the former needs to learn the slope parameter via back propagation during the training phase.
Adaptive Piecewise Linear Units (APL)
APL is defined as a sum of hinge-shared functions:
where is the number of hinges, and the variables are parameters of linear functions.
One disadvantage of APL is it explicitly forces the rightmost line to have unit slope 1 and bias 0. Although it is stated that if the output of APL serves as the input to a linear function , the linear function will restore the freedom of the rightmost line which is lost due to the constraint, we argue that this does not always hold because in many cases for deep networks, the function taking the output of APL as the input is non-linear or unrestorable, such as local response normalizatioin [Krizhevsky, Sutskever, and Hinton2012] and dropout [Krizhevsky, Sutskever, and Hinton2012].
Maxout unit takes as the input the output of multiple linear functions and returns the largest:
In theory, maxout can approximate any convex
function [Goodfellow et al.2013], but unfortunately, it lacks the ability to learn non-convex
functions. Moreover, a large number of extra parameters introduced by the linear functions of each
hidden maxout unit result in large storage memory cost
and considerable training time, which affect the training efficiency of very deep
CNNs, e.g. GoogLeNet [Szegedy et al.2014].
Basic Laws in Psychophysics and Neural Sciences
The Webner-Fechner law [Fechner1965] and the Stevens law [Stevens1957] are two basic laws in psychophysics [Randall et al.2002] [Johnson, Hsiao, and Yoshioka2002] and neural sciences [Dayan and Abbott2001]. Webner first observed through experiments that the amount of change needed for sensory detection to occur increases with the initial intensity of a stimulus, and is proportional to it [Weber1851]. Based on Webner’s work, Fechner proposed the Webner-Fechner law which developed the theory by stating that the subjective sense of intensity is related to the physical intensity of a stimulus by a logarithmic function, which is formulated as Eqn. (1) and shown in Figure 1(a). Stevens refuted the Webner-Fechner law by arguing that the subjective intensity is related to the physical intensity of a stimulus by a power function [Johnson, Hsiao, and Yoshioka2002], which is formulated as Eqn. (2) and shown in Figure 1(b). The two laws have been verified through lots of experiments [Nieder2005]. For example, in vision, the amount of change in brightness with respect to the present brightness accords with the Webner-Fechner law, i.e., Eqn. (2). [Stevens1957] shows various examples, one of which is that the perception of pain and taste follows the Stevens law but with different exponent values. In neural sciences, these two laws also explain many properties of sensory neurons and the response characteristics of receptor cells [Nieder2005] [Dayan and Abbott2001]. The more detailed discussion is beyond the range of this paper.
Motivated by the previous research on these two laws, we propose SReLU which imitates the logarithm function and the power function given by the Webner-Fechner law and the Stevens law, repectively, and uses piecewise linear functions to approximate non-linear convex and non-convex functions. Through experiments, we find that our method can be universally used for current deep networks, and significantly boosts the performance.
S-shaped Rectified Linear Units (SReLU)
In this section, we introduce
in detail our proposed SReLU. Firstly, we present the definition and the training process of SReLU. Secondly, we propose a method to initialize the parameters of SReLU as a good starting point for training. Finally, we discuss the relationship of SReLU with other activation
Definition of SReLU
SReLU is essentially defined as a combination of three
linear functions, which perform mapping with the
where are four learnable parameters used to model an individual SReLU activation unit. The subscript indicates that we allow SReLU to vary in different channels. As shown in Figure 1(c)(d), in the positive direction, is the slope of the right line when the inputs exceed the threshold . Symmetrically, is used to represent another threshold in the negative direction. When the inputs are smaller than , the outputs are calculated by the left line. When the inputs of SReLU fall into the range of , the outputs are linear functions with unit slope 1 and bias 0.
By designing SReLU in this way, we hope that it can imitate the formulations of multiple non-linear functions, including the logarithm function (Eqn. (1)) and the power function (Eqn. (2)) given by the Webner-Fechner law and the Stevens law, respectively. As shown in Figure 1(c), when , the positive part of SReLU imitates the power function with the exponent larger than 1; when , the positive part of SReLU imitates the logarithm function; when , SReLU follows the power function with the exponent 1. For the negative part of SReLU, we have a similar observation except for the inverse representation of the logarithm function and the power function as analyzed for its positive counterpart. The reason for setting the middle line to be a linear function with slope 1 and bias 0 when the input is within the range is that it can better approximate both Eqn. (1) and Eqn. (2) using such a function, because the change of the outputs with respect to the inputs is slow when the inputs are in small magnitudes.
Unlike APL which restricts the form of rightmost line, we do not apply any constraints or regularization to the parameters, thus both the thresh parameters and slope parameters can be learned freely as the training goes on. It is noteworthy that no divergence of deep networks occurs although SReLU is allowed to be trained without any constraints in all of our experiments. As shown in Table 3, the learned parameters are all in reasonable condition.
In our method, we learn an independent SReLU following each channel of kernels. Thus the number of the parameters for SReLU in the deep networks is only , where is the overall number of kernel channels in the whole network. Compared with the large number of parameters in CNNs, e.g. 5 million parameters in GooLeNet [Szegedy et al.2014], such an increase in the number of parameters (21.7K in GoogLeNet with SReLU, as shown in Table 5) is negligible. This is a good property of SReLU, because on one hand, we avoid the overfitting effectively by increasing only a negligible number of parameters, and on the other hand we keep the memory size and the computing time almost unchanged. Similar to PReLU [He et al.2015], we also try the channel-shared variant of SReLU. In this case the number of SReLU is equal to the overall number of layers in the deep network. In Tabel 1, we compare the performance of these two variants of SReLU on CIFAR-10 without data augmentation and find that the channel-wise version performs slightly better than the channel-shared version.
|NIN + ReLU [Lin et. al.]||10.43%|
|NIN + SReLU (channel-shared)||9.01%|
|NIN + SReLU (channel-wise)||8.41%|
With respect to the training of SReLU, we use the gradient descent algorithm and jointly train the parameters of SReLU with the deep networks. The update rule of
is derived by the chain rule:
where and represents the objective function of the deep network. The term is the gradient back-propagated from the higher layer of SReLU. The summation is applied in all positions of the feature map. For the channel-shared variant, the gradient of is , where is the sum over all channels in each layer. Specifically, the gradient for each parameter of SReLU is given by
where is an indicator function and when the expression inside holds true, otherwise . By this way, the gradient of the input is
The rule for updating by momentum method is:
Here is the momentum and is the learning
rate. Because the weight decay term tends to pull the parameters to zero, we do
not use weight decay ( regularization) for .
Adaptive Initialization of SReLU
One problem we are faced in training SReLU is how to initialize the parameters in SReLU. An intuitive way is to set the parameters manually. However, such an initialization method is cumbersome. Furthermore, if the manually set initialization values are not appropriate, e.g. too large or too small compared with the real value of its input, SReLU may not work well. For example, if is set to be very large, based on Eqn. (9), nearly all the inputs for the SReLU will lie in the left part of , which will cause and to be insufficiently learned. In current deep networks, the magnitude of the inputs in each layer varies a lot (see Figure 3), making it more difficult to manually set parameters. To deal with this problem, we propose to firstly initialize each to be in all layers, where is any positive real number and , and we “freeze” the update of the parameters of SReLU during the initial several training epochs. By this method, SReLU is degenerated into a conventional LReLU at the beginning of the training. Then upon the end of the “freezing” phase, we set to be the largest value of each SReLU’s input from all training data, i.e.,
where calculates the largest value from the set X, and represents all the input values of an individual SReLU. Our initialization method offers following two advantages. Firstly, it learns adaptively the initial values of to fit better to the real distributions of the training data, thus providing a good starting point for the training of SReLU. Secondly, it enables SReLU to re-use the per-trained model with LReLU, thus it can reduce the training time compared with training the whole network from the scratch.
Comparison with Other Activation Functions
In this part, we compare our method with five published nonlinear activation functions: ReLU, LReLU, PReLU, APL and maxout.
By checking Eqn. (3), Eqn. (4) and Eqn. (7), it can be easily concluded that ReLU, LReLU and PReLU can be seen as special cases of SReLU. Specifically, when , SReLU is degenerated into ReLU; when , SReLU is transformed to LReLU and PReLU. However, ReLU, LReLU and PReLU can only approximate convex functions, while SReLU is able to approximate both convex and non-convex functions. Compared with APL, when the inputs have large magnitudes and lie in the rightmost region of the activate function, SReLU allows its parameters to take more flexible values and gives output features with adaptive scaling over the inputs. This is similar to the Webner-Fechner law that has logarithm function form to suppress the outputs for the input with too large magnitude. SReLU models such suppression effect by learning the slope of its rightmost line adaptively. In contrast, APL constrains the output to be same as input even when the inputs have very large magnitudes. This is the key difference between SReLU and APL and also the main reason why SReLU consistently outperforms APL. The experimental results shown in Table 2 clearly demonstrate this point. Without data augmentation and the proposed initialization strategy, NIN + SReLU outperforms NIN + APL by 0.98% and 3.04% on CIFAR-10 and CIFAR-100, respectively. Compared to maxout, which can only approximate convex functions and introduces a large number of extra parameters, SReLU needs much less parameters, therefore SReLU is more suitable for training very deep networks, e.g. GoogLeNet.
Experiments and Analysis
To evaluate our method thoroughly, we conduct experiments on four datasets with different scales, including CIFAR-10, CIFAR-100 [Krizhevsky and Hinton2009], MNIST [LeCun et al.1998] and a much larger dataset, ImageNet [Deng et al.2009] with two popular deep networks, i.e., NIN [Lin, Chen, and Yan2013] and GoogLeNet [Szegedy et al.2014]
. NIN is used on CIFAR-10, CIFAR-100 and MNIST and GoogLeNet is used on ImageNet. NIN replaces the single linear convolution layers in the conventional CNNs by multilayer perceptrons, and uses the global average pooling layer to generate feature maps for each category. Compared to NIN, GoogLeNet is much larger with 22 layers built onInception model, which can be seen as a deeper and wider extension of NIN. Both these two networks have achieved state-of-the-art performance on the datasets we use.
Since we mainly focus on testing the effects of SReLU on the performance of deep networks, in all our experiments, we only replace the ReLU in the original networks with SReLU and keep the other parts of networks unchanged. For the setting of hyperparameters (such as learning rate, weight decay and dropout ratio, etc.), we follow the published configurations of original networks. To compare LReLU with our method, we try different slope values in Eqn. (4) and picks the one that gets the best performance on validation set. For PReLU in our experiments, we follow the initialization methods presented in [He et al.2015]. For every dataset, we randomly sample 20% of the total training data as the validation set to configure the needed hyperparameters in different methods. After fixing hyperparameters, we train the model from the scratch with the whole training data. For SReLU, we use and for all datasets. In all experiments, we ONLY use single model and single view test.
We choose Caffe[Jia et al.2014] as the platform to conduct our experiments. To reduce the training time, four NVIDIA TITAN GPUs are employed in parallel for training. Other hardware information of the PCs we use includes Intel Core i7 3.3GHz CPU, 64G RAM and 2T hard disk. The codes of SReLU are available at https://github.com/AIROBOTAI/caffe/tree/SReLU.
|Model||No. of Param.(MB)||CIFAR-10||CIFAR-100|
|Without Data Augmentation|
|Tree based priors||-||-||36.85%|
|NIN + ReLU||0.97M||9.67%||35.96%|
|NIN + LReLU||0.97M||9.75%||36.00%|
|NIN + PReLU(ours)||0.97M + 1.42K||9.74%||35.95%|
|NIN + APL||0.97M + 5.68K/2.84K||9.59%||34.40%|
|NIN + SReLU111Manually set initialization parameters in SReLU(ours)||0.97M + 5.68K||8.61%||31.36%|
|NIN + SReLU (ours)||0.97M + 5.68K||8.41%||31.10%|
|With Data Augmentation|
|NIN + ReLU||0.97M||7.73%||32.75%|
|NIN + LReLU||0.97M||7.69%||32.70%|
|NIN + PReLU (ours)||0.97M + 1.42K||7.68%||32.67%|
|NIN + APL||0.97M + 5.68K/2.84K||7.51%||30.83%|
|NIN + SReLU (ours)||0.97M + 5.68K||6.98%||29.91%|
The CIFAR-10 and CIFAR-100 datasets contain color images with size of 32x32 from 10 and 100 classes, respectively. Both of them have 50,000 training images and 10,000 testing images. The preprocessing methods follow the way used in [Goodfellow et al.2013]. The comparison results of SReLU with other methods (including maxout [Goodfellow et al.2013], prob maxout [Springenberg and Riedmiller2013], APL [Agostinelli et al.2014], DSN [Lee et al.2014], tree based priors [Srivastava and Salakhutdinov2013], NIN [Lin, Chen, and Yan2013], etc.) on these two datasets either when the data augmentation is applied or not are shown in Tabel 2, from which we can see that our proposed SReLU achieves the best performance against all the compared methods.
When no data augmentation is used, compared with ReLU, LReLU and PReLU, our method reduces the error significantly by 1.26%, 1.34%, 1.33% on CIFAR-10, respectively. On CIFAR-100, the error reduction is 4.86%, 4.90%, 4.85%, respectively. SReLU also demonstrates superiority by surpassing other activation functions including APL and maxout. When compared with other deep network methods, such as tree based priors and DSN, our method also beats them by a remarkable gap, demonstrating a promising ability to help boost the performance of deep models. We also compare the number of parameters used in each method, from which we notice that SReLU only incurs a very slight increase (5.68K) to the total number of parameters (0.97M in original NIN). APL uses the same number of additional parameters as SReLU on CIFAR-10, but its performance in either case of applying data augmentation or not is inferior to our method. The convergence curve of SReLU with other methods on CIFAR-10 and CIFAR-100 are shown in Figure 4(a) and Figure 4(b), respectively.
To observe the learned parameters of SReLU, we list in Table 3 the parameters’ values after the training phase. Since the SReLUs we use are channel-wise, we simply calculate the average of the input for all SReLUs in the same layer. It is interesting to observe that SReLUs in different layers learn meaningful parameters in coincide with our motivations. For example, the SReLUs following conv1 and cccp1 learns less than 1 (0.81 and 0.77, respectively) on CIFAR-10, while SReLUs following conv3 and cccp5 on CIFAR-100 learns larger than 1 (1.42 and 1.36, respectively). SReLU following conv2 on CIFAR-10 learns nearly equal to 1 (1.01). These experimental results verify that SReLU has a strong ability to learn various forms of nonlinear functions, which can either be convex or non-convex. Moreover, in Table 3, we can see that is of very large value in higher layers. It’s because that the inputs of SReLU have higher average values than the ones in lower layers. Therefore, SReLU in higher layers learns larger for adapting to inputs. This demonstrates the strong adaptive ability of SReLU to distribution of its inputs.
In the experiments on the augmented version of CIFAR-10 and CIFAR-100, we simply use random horizontal reflection during training for both datasets. In this case, SReLU still consistently outperforms other methods.
|layers||CIFAR-10 / CIFAR-100|
|conv1||0.91 / 0.73||-0.48 / -0.68||0.81 / 0.62||-0.25 / -0.22|
|cccp1||1.06 / 0.52||-0.36 / -0.34||0.77 / 0.38||-0.04 / 0.04|
|cccp2||1.27 / 0.37||-0.20 / -0.26||0.47 / 0.51||0.39 / 0.44|
|conv2||5.32 / 4.02||-0.31 / -0.51||1.01 / 0.88||0.07 / 0.06|
|cccp3||6.95 / 4.73||-0.21 / -0.79||0.92 / 0.64||-0.01 / 0.05|
|cccp4||8.18 / 5.79||-0.08 / -0.13||0.77 / 0.56||0.61 / 0.45|
|conv3||25.17 / 23.72||-0.15 / -0.61||1.21 / 1.42||0.05 / 0.07|
|cccp5||31.09 / 36.44||-0.47 / -0.46||0.97 / 1.36||-0.16 / -0.02|
|cccp6||72.03 / 66.13||-0.13 / -0.21||1.53 / 1.23||-0.44 / -0.35|
|Model||No. of Param.(MB)||Error Rates|
|NIN + ReLU||0.35M||0.47%|
|NIN + LReLU (ours)||0.35M||0.42%|
|NIN + PReLU (ours)||0.35M + 1.42K||0.41%|
|NIN + SReLU (ours)||0.35M + 5.68K||0.35%|
MNIST [LeCun et al.1998] contains 70,000 28x28 gray scale images of numerical digits from 0 to 9, divided as 60,000 images for training and 10,000 images for testing. In this dataset, we do not apply any preprocessing to the data and only compare models without data augmentation. The experiment results on this dataset are shown in Tabel 4, from which we see SReLU performs better than other methods.
|Model||No. of Param.(MB)||Error Rates|
|GoogLeNet + SReLU (ours)||5M + 21.6K||9.86%|
To further evaluate our method on large-scale datasets, we perform a much more challenging image classification task on 1000-class ImageNet dataset, which contains about 1.2 million training images, 50,000 validation images and 100,000 test images. Our baseline model is GoogLeNet model, which achieved the best performance on image classification in ILSVRC 2014 [Russakovsky et al.2015]. We run experiments using the publicly available configurations in Caffe [Jia et al.2014]. For this dataset, no additional preprocessing method is used except subtracting the image mean from each input raw image.
Table 5 compares the performance of GoogLeNet using SReLU and the original GoogLeNet released by Caffe. The GoogLeNet with SReLU achieves significant improvement (1.24%) on this challenging dataset compared with the original GoogLeNet using ReLU, at the cost of only 21.6K additional parameters (versus the total number of 5M parameters in the original GoogLeNet).
In this paper, inspired by the fundamental laws in psychophysics and neural sciences, we proposed a novel S-shaped rectified linear unit (SReLU) to be used in deep networks. Compared to other activation functions, SReLU is able to learn both convex and non-convex functions, and can be universally used in existing deep networks. Experiments on four datasets including CIFAR-10, CIFAR-100, MNIST and ImageNet with NIN and GoogLeNet demonstrate that SReLU effectively boosts the performance of deep networks. In our future work, we will exploit the applications of SReLU in other domains beyond vision, such as NLP.
- [Agostinelli et al.2014] Agostinelli, F.; Hoffman, M.; Sadowski, P. J.; and Baldi, P. 2014. Learning activation functions to improve deep neural networks. CoRR abs/1412.6830.
- [Cireşan et al.2011] Cireşan, D. C.; Meier, U.; Gambardella, L. M.; and Schmidhuber, J. 2011. Convolutional neural network committees for handwritten character classification. In Document Analysis and Recognition (ICDAR), 2011 International Conference on, 1135–1139. IEEE.
- [Dayan and Abbott2001] Dayan, P., and Abbott, L. F. 2001. Theoretical neuroscience, volume 806. Cambridge, MA: MIT Press.
- [Deng et al.2009] Deng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei-Fei, L. 2009. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, 248–255. IEEE.
- [Fechner1965] Fechner, G. 1965. Elements of psychophysics.
- [Girshick et al.2013] Girshick, R. B.; Donahue, J.; Darrell, T.; and Malik, J. 2013. Rich feature hierarchies for accurate object detection and semantic segmentation. CoRR abs/1311.2524.
- [Goodfellow et al.2013] Goodfellow, I. J.; Warde-Farley, D.; Mirza, M.; Courville, A.; and Bengio, Y. 2013. Maxout networks. arXiv preprint arXiv:1302.4389.
- [He et al.2015] He, K.; Zhang, X.; Ren, S.; and Sun, J. 2015. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. CoRR abs/1502.01852.
- [Jia et al.2014] Jia, Y.; Shelhamer, E.; Donahue, J.; Karayev, S.; Long, J.; Girshick, R. B.; Guadarrama, S.; and Darrell, T. 2014. Caffe: Convolutional architecture for fast feature embedding. CoRR abs/1408.5093.
- [Johnson, Hsiao, and Yoshioka2002] Johnson, K. O.; Hsiao, S. S.; and Yoshioka, T. 2002. Book review: neural coding and the basic law of psychophysics. The Neuroscientist 8(2):111–121.
- [Krizhevsky and Hinton2009] Krizhevsky, A., and Hinton, G. 2009. Learning multiple layers of features from tiny images.
- [Krizhevsky, Sutskever, and Hinton2012] Krizhevsky, A.; Sutskever, I.; and Hinton, G. E. 2012. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, 1097–1105.
- [LeCun et al.1998] LeCun, Y.; Bottou, L.; Bengio, Y.; and Haffner, P. 1998. Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11):2278–2324.
- [Lee et al.2014] Lee, C.-Y.; Xie, S.; Gallagher, P.; Zhang, Z.; and Tu, Z. 2014. Deeply-supervised nets. arXiv preprint arXiv:1409.5185.
- [Lin, Chen, and Yan2013] Lin, M.; Chen, Q.; and Yan, S. 2013. Network in network. CoRR abs/1312.4400.
- [Maas, Hannun, and Ng2013] Maas, A. L.; Hannun, A. Y.; and Ng, A. Y. 2013. Rectifier nonlinearities improve neural network acoustic models. In Proc. ICML, volume 30.
[Nair and Hinton2010]
Nair, V., and Hinton, G. E.
Rectified linear units improve restricted boltzmann machines.In
Proceedings of the 27th International Conference on Machine Learning (ICML-10), 807–814.
- [Nieder2005] Nieder, A. 2005. Counting on neurons: the neurobiology of numerical competence. Nature Reviews Neuroscience 6(3):177–190.
- [Randall et al.2002] Randall, D.; Burggren, W. W.; French, K.; and Eckert, R. 2002. Eckert animal physiology. Macmillan.
- [Russakovsky et al.2015] Russakovsky, O.; Deng, J.; Su, H.; Krause, J.; Satheesh, S.; Ma, S.; Huang, Z.; Karpathy, A.; Khosla, A.; Bernstein, M.; Berg, A. C.; and Fei-Fei, L. 2015. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV) 1–42.
- [Springenberg and Riedmiller2013] Springenberg, J. T., and Riedmiller, M. 2013. Improving deep neural networks with probabilistic maxout units. arXiv preprint arXiv:1312.6116.
Srivastava, N., and Salakhutdinov, R. R.
Discriminative transfer learning with tree-based priors.In Burges, C.; Bottou, L.; Welling, M.; Ghahramani, Z.; and Weinberger, K., eds., Advances in Neural Information Processing Systems 26. Curran Associates, Inc. 2094–2102.
- [Stevens1957] Stevens, S. S. 1957. On the psychophysical law. Psychological review 64(3):153.
- [Szegedy et al.2014] Szegedy, C.; Liu, W.; Jia, Y.; Sermanet, P.; Reed, S.; Anguelov, D.; Erhan, D.; Vanhoucke, V.; and Rabinovich, A. 2014. Going deeper with convolutions. arXiv preprint arXiv:1409.4842.
- [Weber1851] Weber, E. 1851. Annotationes anatomicae et physiologicae [anatomical and physiological annotations]. Leipzig: CF Koehler.