This letter deals with neural networks as dynamical systems governed by finite difference equations. It shows that the introduction of -many skip connections into network architectures, such as residual networks and additive dense networks, defines th order dynamical equations on the layer-wise transformations. Closed-form solutions for the state-space representations of general th order additive dense networks, where the concatenation operation is replaced by addition, as well as th order smooth networks, are found. The developed provision endows deep neural networks with an algebraic structure. Furthermore, it is shown that imposing th order smoothness on network architectures with -many nodes per layer increases the state-space dimension by a multiple of , and so the effective embedding dimension of the data manifold by the neural network is -many dimensions. It follows that network architectures of these types reduce the number of parameters needed to maintain the same embedding dimension by a factor of when compared to an equivalent first-order, residual network. Numerical simulations and experiments on CIFAR10, SVHN, and MNIST have been conducted to help understand the developed theory and efficacy of the proposed concepts.
The way in which deep learning was initially used to transform data representations was by nested compositions of affine transformations followed by nonlinear activations. The affine transformation can be, for example, a fully connected weight matrix or convolution operation. Residual networks (He, Zhang, Ren, & Sun, 2016) introduce an identity skip connection that bypasses these transformations, thus allowing the nonlinear activation to act as a perturbation term from the identity. Veit, Wilber, and Belongie (2016) introduced an algebraic structure showing that residual networks can be understood as the entire collection of all possible forward pass paths of subnetworks, although this algebraic structure ignores the intuition that the nonlinear activation is acting as a perturbation from identity. Lin and Jegelka (2018) showed that a residual network with a single node per layer and ReLU activation can act as a universal approximator, where it is learning something similar to a piecewise linear finite-mesh approximation of the data manifold.
Recent work consistent with the original intuition of learning perturbations from the identity has shown that residual networks, with their first-order perturbation terms, can be formulated as finite difference approximations of first-order differential equations (Hauser & Ray, 2017). This has the interesting consequence that residual networks are smooth dynamic equations through the layers of the network. In addition, one may then define entire classes of differentiable transformations over the layers and then induce network architectures from their finite difference approximations.
Chang, Meng, Haber, Tung, and Begert (2017) considered residual neural networks as forward-difference approximations to transformations as well. This work has been extended to develop new network architectures by using central differencing, as opposed to forward differencing, to approximate the set of coupled first-order differential equations, called the midpoint network (Chang, Meng, Haber, Ruthotto et al., 2017). Similarly, other researchers have used different numerical schemes to approximate the first-order ordinary differential equations, such as the linear multistep method to develop the linear multistep-architecture (Lu, Zhong, Li, & Dong, 2017). This is different from previous work (Hauser & Ray, 2017), where entire classes of finite-differencing approximations to th order differential equations are defined. Haber and Ruthutto (2017) considered how stability techniques from finite difference methods can be applied to improve first- and second-order smooth neural networks. For example, they suggest requiring that the real part of the eigenvalues from the Jacobian transformations be approximately equal to zero. This ensures that little information about the signal is lost and that the input data not diverge in progressing through the network.
In current work set out in section 2, closed-form solutions are found for the state-space representations for both general network architectures as well as general additive densely connected network architectures (Huang, Liu, Weinberger, & van der Maaten, 2017), where a summation operation replaces the concatenation operation. The reason for this is that the concatenation operation explicitly increases the embedding dimension, while the summation operation implicitly increases the embedding dimension. We then show in section 3 that the embedding dimension for a network is increased by a factor of when compared to an equivalent (standard) network and (residual) network, and thus the number of parameters needed to learn is reduced by a factor of to maintain transformations on the same embedding dimension. Section 4 presents the results of experiments for validation of the proposed theory, and the details are provided in the appendix. The paper concludes in section 5, along with recommendations for future research.
2 Smooth Network Architectures
This section develops a relation between skip connections in network architectures and algebraic structures of dynamical systems of equations. The network architecture can be thought of as a map , where is the data manifold, is the set of input data/initial conditions, and is the set for an -layer deep neural network. We write to denote the coordinate representation for the data manifold at layer . In fact, the manifold is a Riemannian manifold as it has the additional structure of possessing a smoothly varying metric on its cotangent bundle (Hauser & Ray, 2017); however for the current purpose, we will only consider the manifold's structure to be .
Section 2.1 defines and reviews smooth residual (He et al., 2016) architectures. Section 2.2 expands on the section 2.1 to define and study the entire class of architectures (Hauser and Ray, 2017) and develop the state-space formulation for these architectures to show that the effective embedding dimension increases by a multiple of for architectures of these types. Similarly, section 2.3 develops the state-space formulation for densely connected networks (Huang et al., 2017) and shows that for these dense networks with -many layer-wise skip connections, the effective embedding dimension again increases by a multiple of .
2.1 Residual Networks as Dynamical Equations
2.2 Architectures Induced from Smooth Transformations
Following the work of Hauser and Ray (2017), we call network architectures as being architectures depending on how many times the finite difference operators have been applied to the map .
The notation is defined as -many applications of the operator , and is the binomial coefficient, read as -choose-. We take one forward difference and the remaining as backward differences so that the next layer (forward) is a function of the previous layers (backward).
From this formulation, depending on the order of smoothness, the network is implicitly creating interior or ghost elements, borrowing language from finite difference methods, to properly define the initial conditions. One can view a ghost element as a pseudo-element that lies outside the domain used to control the gradient. For example with a architecture from equation 2.4b, one needs the initial position and velocity in order to be able to define as a function of and . In the section 2.3, we show that the dense network (Huang et al., 2017) can be interpreted as the interior or ghost elements needed to initialize the dynamical equation.
We use the notation where is the identity matrix and is the matrix of all zeros. From equation 2.7 and, equivalently, equation 2.8, it is understood that if there are -many nodes at layer (i.e., maps to ), then a th-order smooth neural network can be represented in the state-space form as , which maps to . Furthermore, it is seen that the -many state variables are transformed by the shared activation function , which has a -parameter matrix, as opposed to a full -parameter matrix, thus reducing the number of parameters by a factor of .
The schematic of the architecture, with its equivalent first-order state-space representation, is given in Figure 1. The architecture is given by equation 2.4b, which can be conveniently rewritten as . Setting and , the state-space model is updated as and . Thus, given maps to , will map to .
2.3 Additive Dense Network for General
The left-hand side of equation 2.13 is the definition of states from equation 2.12 written explicitly as the th backward difference of a sequence , and the implication arrow is the binomial inversion of sequences. This is the representation of the 's in terms of the states .
Remember that if there are -many nodes per layer, then each maps to , and so these matrices are block matrices. For example, the entry is the matrix with the number along all of the diagonals, for and . Similarly, the matrix is the matrix of all zeros.
Equation 2.14 and, equivalently, equation 2.15, is the state-space representation of the additive dense network for general . The schematic of the additively dense network architecture, with its equivalent state-space representation, is given in Figure 2. By introducing -many lags into the dense network, the dimension of the state space increases by a multiple of for an equivalent first-order system, since we are concatenating all of the 's to define the complete state of the system as , which maps to .
3 Network Capacity and Skip Connections
The objective of this section is to partially explain why imposing high-order skip connections on the network architecture is likely to be beneficial. A first-order system has one state variable (e.g., position), while a second-order system has two state variables, (e.g., position and velocity). In general, a th order system has -many state variables, for .
Recall that when maps to , the equivalent first-order system maps to , for a th-order system. This holds since each of the -many functions mapping to operates independent of each other through their independently learned weight matrices, and so their concatenation spans .
This immediately implies that the weight matrix for transforming the th order system is , while the weight matrix for transforming the equivalent first-order system is . Therefore, by imposing -many skip connections on the network architecture, from a dynamical systems perspective, we only need to learn up to as many parameters to maintain the same embedding dimension when compared to the equivalent zeroth or first-order system. Also notice that the weight matrix for transforming the 's to the state vectors 's is a lower block diagonal matrix, and so it is full rank, and so state variables defined by this transformation matrix do not introduce degeneracies.
4 Numerical Experiments
4.1 Visualizing Implicit Dimensions
An experiment was conducted to visualize and understand these implicit dimensions induced from the higher-order dynamical system. The one-dimensional data were constructed such that of the data are the red class and the other are the blue class, and the blue data are separated with half to the left of the red data and half to the right. It might seem that there is no sequence of single-neuron transformations that would put these data into a form that can be linearly separated by hyperplane, and at best one could achieve an accuracy of . This is the case with the standard residual network (see Figure 3a). The architecture has only one state variable, position, and therefore cannot place a hyperplane to linearly separate the data along the positional dimension.
In contrast, the architecture has two state variables, position and velocity , and therefore its equivalent first-order system is two-dimensional. When visualizing both state variables, one sees that the data in fact get shaped such that a hyperplane only along the positional dimension can correctly separate the data with accuracy. If one were looking only at the positional state variable, that is, the output of the single node, it would seem as if the red and blue curves were passing through each other; however, in the entire two-dimensional space, we see that is not the case. Although this network has only a single node per layer and the weight matrices are just single scalars, the equivalent first-order dynamical system has two dimensions and therefore the one-dimensional data can be twisted in this two-dimensional phase space into a form such that it is linearly separable in only the one positional dimension.
4.2 Estimating the Magnitude of the Perturbations
This section seeks to quantify the magnitude of the perturbation, and therefore validate the perturbation approximations being made. In order for to be a valid perturbation expansion from the transformation , we require . This implies that the magnitude of should be such that . In addition, assuming the image is traveling a constant distance from input to output, one would expect the average size of the perturbation to be roughly . That is, as one increases the number of layers, the average size of each partition region (mesh size) should get smaller as . Experiments were conducted on MNIST, measuring the size of the perturbation term for a network with two sections of residual blocks of sizes and , with the number of blocks in each section being . The results are shown in Figure 4. Details of this experiment are given in the appendix. Several conclusions are drawn from this experiment:
• The magnitude of the perturbation term, for sufficiently large , is in fact much less than one. At least in this setting, this experimentally validates the intuition that residual networks are learning perturbations from the identity function.
• With increasing the number of layers , the magnitude of the perturbation goes as , suggesting that there exists a total distance the image travels as it passes through the network. This implies that the image can be interpreted as moving along a trajectory from input to output, in which case the network is a finite difference approximation to the differential equation governing this trajectory. Performing a linear regression on yields that the image travels a ”computational distance” of through the first section and through the second section. This may suggest that the first section is more important when processing the image than the second section. If taken literally, it would imply that the average MNIST image is traveling a total ”computational distance” of from the low-level input representation to the high-level abstract output representation. This measure is a depth-invariant computational distance the data travel through the network.
• This analytical approach suggests a systematic way of determining the depth of a network when designing new network architectures. If one requires a certain minimum mesh size, after estimating the 's, one can then calculate the minimum number of layers required to achieve a mesh of this size. For example, on this MNIST experiment, if one requires a minimum average mesh size of , then the first section should have about 74 layers while the second needs only 36 layers.
4.3 Comparison of Various Order Network Architectures
This section experimentally compares the classification performance of various order architectures that are described in this letter. The architectures that are tested are the networks for , as well as the additive dense network for ; note that the additive dense network is the same as the network. In all of the experiments, first the ResNet architecture was designed to work well; then, using these exact conditions, the described skip connections were introduced, changing nothing else. Further details of the experiments are in the appendix.
It is seen in Table 1 that in both CIFAR10 and SVHN, the , , and architectures all perform similarly well, the architecture performs much more poorly, and the three additive dense networks perform fairly well. On CIFAR10, the architecture achieved the lowest test error, while on SVHN, this was achieved by the architecture. A likely reason that the architecture is performing significantly worse than the rest could be that this architecture imposes significant restrictions on how data flow through the network. Thus, the network does not have sufficient flexibility in how it can process the data.
Notes: All networks had three sections where the data are transformed to sizes , and (denoted by height width number of channels), and each section having five residual blocks. Training procedures were kept constant for all experiments; only the skip connections were changed. The numbers in bold highlight the lowest test errors achieved per data set among the different architectures being compared. The network achieves the lowest test error of 9.46% on the CIFARIO detaset, and the network achieves the lowest test error of 2.66% on the SVHN data set.
5 Conclusion and Future Work
This letter has developed a theory of skip connections in neural networks in the state-space setting of dynamical systems with appropriate algebraic structures. This theory was then applied to find closed-form solutions for the state-space representations of both networks as well as dense networks. This immediately shows that these th-order network architectures are equivalent, from a dynamical systems perspective, to defining -many first-order systems. In the design, this reduces the number of parameters needed to learn by a factor of while retaining the same state-space embedding dimension for the equivalent and networks.
Three experiments were conducted to validate and understand the proposed theory. The first had a carefully designed data set such that restricted to a certain number of nodes, the neural network is able to properly separate the classes only by using the implicit state variables in addition to its position, such as velocity. The second experiment on MNIST was used to measure the magnitude of the perturbation term with varying levels of layers, resulting in a depth-invariant computational distance the data travel, from low-level input representation to high-level output representation. The third experiment compared various-order architectures on benchmark image classification tasks. This letter explains in part why skip connections have been so successful and motivates the development of architectures of these types.
While there are many possible directions for further theoretical and experimental research, we suggest the following topics of future work:
Rigorous design of network architectures from the algebraic properties of the space-space model, as opposed to engineering intuitions.
Analysis of the topologies of data manifolds to determine relationships between data manifolds and minimum embedding dimension, in a similar manner to the Whitney embedding theorems.
Investigations of the computational distance for different and more complex data sets. This invariant measure could be potentially used to systematically define the depth of the network, as well as to characterize the complexity of the data.
Appendix: Description of Numerical Experiments
For the experiment of section 4.2, no data augmentation was used and a constant batch size of 256 was used. In the network, each block has the form , where the is the convolution operation, and are the learned filters, and and are the leaky-ReLU activation and batch-normalization functions. For specifying image sizes, we use the notation . The first section of the network of constant feature map size operates on images, and a stride of 2 is then applied and mapped to . After section 4.2, global average pooling was performed to reduce the size to 64 length vectors and fed into a fully connected layer for softmax classification.
For section 4.3, the batch size was updated automatically from , when a trailing window of the validation error stopped decreasing. In CIFAR10, 5000 of the 60,000 training samples were used for validation, while in SVHN, a random collection of of the training and extra data was used for training, while the remaining was used for validation. The only data augmentation used during training was that the images were flipped left-right and padded with four zeros and randomly cropped to .
S. S. has been supported by the Walker Fellowship from the Applied Research Laboratory at the Pennsylvania State University. The work reported here has been supported in part by the U.S. Air Force Office of Scientific Research under grants FA9550-15-1-0400 and FA9550-18-1-0135 in the area of dynamic data-driven application systems. Any opinions, findings, and conclusions or recommendations expressed in this letter are our own and do not necessarily reflect the views of the sponsoring agencies.