A T ENSOR A NALYSIS ON D ENSE C ONNECTIVITY VIA C ONVOLUTIONAL A RITHMETIC C IRCUITS
Anonymous authors
Paper under double-blind review
A BSTRACT
Several state of the art convolutional networks rely on inter-connecting different layers to ease the ﬂow of information and gradient between their input and output layers.
These techniques have enabled practitioners to successfully train deep convolutional networks with hundreds of layers.
Particularly, a novel way of interconnecting layers was introduced as the Dense Convolutional Network (DenseNet) and has achieved state of the art performance on relevant image recognition tasks.
Despite their notable empirical success, their theoretical understanding is still limited.
In this work, we address this problem by analyzing the effect of layer interconnection on the overall expressive power of a convolutional network.
In particular, the connections used in DenseNet are compared with other types of inter- layer connectivity
We carry out a tensor analysis on the expressive power inter-connections on convolutional arithmetic circuits (ConvACs) and relate our results to standard convolutional networks.
The analysis leads to performance bounds and practical guidelines for design of ConvACs.
The generalization of these results are discussed for other kinds of convolutional networks via generalized tensor decompositions.
I NTRODUCTION
Recently, densely connected networks such as FractalNet (Larsson et al., 2016), ResNet (He et al., 2016), and DenseNet (Huang et al., 2016), have obtained state of the art performance on large problems where highly deep network conﬁgurations are used.
Adding dense connections between different layers of a network virtually shortens its depth, thus allowing a better ﬂow of information and gradient through the network.
This makes possible the training of highly deep models.
Models with these types of connections have been successfully trained with hundreds of layers.
More speciﬁcally, DenseNets have achieved state of the art performance on the CIFAR-10, CIFAR-100,
SVHN, and ImageNet datasets, using models of up to 1 thousand layers in depth.
Nevertheless, whether these connections provide a fundamental enhancement on the expressive power of a network, or just improve the training of the model, is still an open question.
In Huang et al.
DenseNet models with 3 times less parameters than its counterpart (ResNets) were able to achieve the same performance on the ImageNet challenge.
Moreover, a theoretical understanding of why the connections used by DenseNets lead to better performance compared with FractalNets or ResNets is still pending.
Despite the popularity of these models, there are few theoretical frameworks explaining the power of these models and providing insights to their performance.
In Cohen et al.
(2016a)
, the authors considered convolutional networks with linear activations and product pooling layers, called convolutional arithmetic circuits (ConvACs), and argued for the expressiveness of deep networks using a tensor based analysis.
This analysis has been extended to rectiﬁer based convolutional networks via generalization of the tensor product Cohen & Shashua (2016a).
In Cohen & Shashua (2016a), it was shown that ConvACs enjoy a greater expressive power than rectiﬁer based models despite the popularity of rectiﬁer based networks in practice.
Indeed the empirical relevance of ConvAC was demonstrated through an architecture called SimNets Cohen et al. (2016b).
In addition, the generative ConvAC of Sharir et al.
(2016) achieved state of the art performance in classiﬁcation of images with missing pixels.
These results served as motivation for the works of Cohen & Shashua (2016b);
Cohen et al.
; Levine et al.
Sharir & Shashua (2017), where different aspects of ConvACs were studied from a theoretical perspective.
In Cohen & Shashua (2016b) the inductive bias introduced by pooling geometries was studied.
Later, Levine et al.
(2017) makes use of the quantum entanglement measure to analyze the inductive bias introduced by the correlations among the channels of ConvACs.
Moreover, Sharir & Shashua (2017) generalizes the convolutional layer of ConvACs by allowing overlapping receptive ﬁelds, in other words permitting stride values lower than the convolution patch size.
These locally overlapping connections led to an enhancement on the expressive capacity of ConvACs.
The notion of inter-layer connectivity for ConvACs was addressed by Cohen et al.
(2017) in the context of sequential data processing, such as audio and text related tasks.
In that work, the expressive capabilities of interconnecting processing blocks from a sequence was studied.
Nevertheless, these types of interconnections are related to the sequential nature of the problem and different from the ones used in ResNet, FractalNet and DenseNet.
In this work, we extend the tensor analysis framework of Cohen et al.
(2016a) to obtain insightful knowledge about the effect of dense connections, from the kind used in DenseNets, FractalNet and
ResNet, on the expressiveness of deep ConvACs.
We study the expressive capabilities provided by different types of dense connections.
Moreover, from these results we derive performance bounds and practical guidelines for selection of the hyperparameters of a deep ConvAC, such as layer widths and the topology of dense connections.
These results serve as the ﬁrst step into understanding dense connectivity in rectiﬁer networks as well, since they can be further extended to include rectiﬁer linear units, in the same spirit as the generalization of the tensor product s done by Cohen & Shashua (2016a).
The remainder of this paper is organized as follows.
In Section 2, we introduce the notation and basic concepts from tensor algebra.
In Section 3, we present the tensor representation of ConvACs as introduced by Cohen et al.
(2016a), and later in Section 4, we obtain tensor representations for densely connected ConvACs.
In Section 5, performance bounds and design guidelines are derived for densely connected ConvACs.
P RELIMINARIES
The term tensor refers to a multi-dimensional array, where the order of the tensor corresponds to the number of indexes required to access one of its entries.
For instance, a vector is a tensor of order 1 while a matrix is a tensor of order 2 .
In general a tensor A of order N requires N indexes ( d 1 , . . .
, d N ) to access one of its elements.
For the sake of notation, given I ∈ N , we use the expression [ I ] to denote the set { 1 , 2 , . .
, I } .
In addition, the ( d 1 , . . .
d N ) -th entry of a given tensor of order N and size M 1 × M 2 × · · · × M N is denoted as A ( d 1 ,...,d N ) , where d i ∈ [ M i ] for all i ∈
[ N ] .
Moreover, for the particular case of tensors of order N with symmetric sizes
M 1 = M 2 = · · · = M N = M , we use ( R M ) ⊗ N as shorthand notation for R M ×···× M .
A crucial operator in tensor analysis is the tensor product ⊗ , since it is necessary for deﬁning the rank of a tensor.
For two tensors B ∈ R M 1 ×···× M p and C ∈ R M p +1 ×···× M p + q , the tensor product is deﬁned such that B⊗C ∈ R M 1 ×···× M q and ( B⊗C ) d 1 ,...,d p + q = B d 1 ,...,d p C d p +1 ,...,d p + q for all ( d 1 , . . .
, d p + q ) .
In tensor algebra, a tensor A ∈ R M 1 × M 2 ×···× M N is said to have rank 1 if it can be expressed as A = v (1) ⊗ · · · ⊗ v ( N ) , where v ( i ) ∈
R M i for all i ∈
[ N ] .
Moreover, any tensor A ∈ R M 1 × M 2 ×···× M N can be expressed as a sum of rank-1 tensors, that is
[Equation]
where Z ∈ N is sufﬁciently large and v ( i )
z ∈ R M i for i ∈
[ N ] .
Note that this statement is trivial for Z = (cid:81)
Ni =1 M i .
On the other hand, when Z is the minimum number such that (1) is satisﬁed, the rank of the tensor is deﬁned to be rank( A )
=
Z and (1) becomes equivalent to the well known CANDECOMP/PARAFAC (CP) decomposition of A .
Another operator, that is on the core of the former works of Cohen & Shashua (2016a); Cohen et al.
(2016a)
; Levine et al.
(2017), is the matricization operator.
The operator [ A ] denotes the matricization of a tensor
A ∈ R M 1 ×···× M N of order N .
This matricization of the tensor A re-orders its elements into a matrix [ A ] ∈ R M 1 · M 3 ··· M N − 1 × M 2 · M 4 ··· M N with A yd 1 ,...,d N in the row 1+ (cid:80) N/ 2 i =1 ( d 2 i − 1 − 1) (cid:81) N/ 2 j = i +1 M 2 j −and column 1 + (cid:80) N/ 2 i =1 ( d 2 i − 1 − 1) (cid:81) N/ 2 j = i +1 M 2 j .
This is operator is of great use since it enjoys properties such as [ A ⊗ B ] =
[ A ] (cid:12) [ B ] and rank( A ) ≥ rank([ A ]) , where (cid:12) denotes the Kronecker product of two matrices.
Note that, since the Kronecker product is multiplicative in the rank, we have that rank( A ⊗ B ) ≥ rank([ A ⊗ B ])
= rank([ A ])rank([ B ]) which is a central property of this theoretical analysis framework.
C ONVOLUTIONAL A RITHMETIC C IRCUITS AS T ENSOR D ECOMPOSITIONS
[Equation]
convpool
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
[Equation]
[Figure]
A ConvAC is a convolutional neural network that utilizes linear activation functions with product pooling, unlike most popular convolutional networks which make use of rectiﬁer activations with max or average pooling.
Moreover, the input of the network is modeled by X = ( x 1 , . . .
, x N ) ∈ ( R s ) N , where x i ∈ R s denotes the vectorization of the i -th patch of the input image.
For this analysis, it is assumed that a set of M features is obtained from every patch, that is f θ d ( x i
) ∈ R for all i ∈
[ N ] , d ∈ [ M ] .
These features are selected from a given parametric family F = { f θ : R s → R : θ ∈ Θ } , such as Gaussian kernels, wavelet functions, or learned features.
Then, to determine whether an input X belongs to a class belonging to the set Y , the network evaluates the some score functions h y ( X ) ∈ R and decides for the class y ∈ Y such that
[Equation]
Using this formulation, in Figure 1 (a) we observe an example of a single hidden layer ConvAC, while in Figure 1 (b) we observe the general case of a deep arithmetic circuit of L layers.
As shown by Cohen et al. (2016a), any score function of a ConvAC can be expressed as an homogeneous polynomial with degree N on the input features of the form
[Equation]
where A yd 1 ,...,d N ∈ R are the polynomial coefﬁcients stored in the grid-tensor A y ∈ ( R M ) ⊗ N .
In other words, a score function h y ( X ) is a polynomial of MN variables f θ d ( x i ) ∈ R for all i ∈
[ N ] , d ∈ [ M ] , degree N , and M N polynomial coefﬁcients stored in the grid-tensor A y .
For the special case of a shallow ConvAC with 1 × 1 convolutions and Z hidden units 1 , shown in
Figure 1 (a) , the score functions are computed from the weight vectors a z,i (cid:44) [ a z,i 1 , . .
a z,iM ] T ∈ R M and a y (cid:44) [ a y 1 , . .
. , a yZ ] T ∈ R Z for all i ∈ [ N ] and z ∈ [ Z ] .
This leads to the score function
[Equation]
The ﬁrst step of the tensor analysis framework is to obtain an expression (in terms of the network parameters a yz and a z,id ) of the grid-tensor A y that represents this concrete network architecture.
In other words, obtaining the expression for A y that transforms (2) into (3).
This expression was already obtained in Cohen & Shashua (2016a) as
[Equation]
where ⊗ denotes the tensor product.
Note that (4) is the form of a standard CP decomposition of the grid tensor A y .
This implies that the rank of A y is bounded by rank( A y ) ≤ Z .
Moreover, the obtained results where generalized in Cohen et al.
(2016a)
for the case of a deep ConvAC with sizepooling windows 2 , thus L = log 2 N hidden layers as shown in Figure 1 (b) , leading to a grid-tensor given by the hierarchical tensor decomposition
[Equation]
,α ... φ l,j,γ = r l − 1 (cid:88)
α = a l,j,γα φ l − 1 , 2 j − 1 ,α ⊗ φ l − 1 , 2 j,α ...
A y =
φ L, 1 , 1 =
r L − 1 (cid:88)
α = a L, 1 ,y
α φ L − 1 , 1 ,α ⊗ φ L − 1 , 2 ,α , (5) where r 0 , . . .
, r L − 1 ∈ N are the number of channels in the hidden layers, { a 0 ,j,γ ∈ R M } j ∈ [ N ] ,γ ∈
[ r 0 ] are the weights in the ﬁrst hidden convolutions, { a l,j,γ ∈ R M } j ∈ [ N/ 2 l ] ,γ ∈ [ r l ] are the weights of the hidden layers, and a L, 1 ,
y ∈ R r L − 1 stores the weights corresponding to the output y in the output layer.
D ENSELY C ONNECTED A RITHMETIC C IRCUITS
The recent empirical success of densely connected networks (DenseNets), presented by Huang et al.
(2016), has served as motivation for our theoretical analysis on dense connectivity.
Dense connectivity in a convolutional neural network refers to the case when a number k ∈ N (known as growth rate) of previous layers serve as input of the forthcoming layer.
More precisely, in Huang et al.
(2016), a DenseNet performs this via concatenation along the feature dimension of the current layer inputs with the preceding layer features.
Note that these feature must have compatible sizes along the spatial dimension for the concatenation to be possible.
To address this issue, Huang et al.
(2016) proposed to group blocks of the same spatial dimensions into a dense block , as shown in Figure 2.
These dense blocks do not contain operations such as pooling, that alter the spatial dimensions of the input features.
Moreover, in the DenseNet architecture the layers that perform the pooling operation are called transition layers , since they serve as transition between dense blocks.
For example, in
Figure 2 we depict a dense block of 4 layers with growth rate k = 2 , followed by a transition layer.
[Figure]
In the original DenseNet these transition layers included one convolution layer before the pooling operation.
Nevertheless, for this work we consider transition layers composed of only pooling operations.
Note that this does not affect the generality of the model, since avoiding dense connections on the convolutional layer preceding the transition layer is equivalent to including a convolution in that transition layer 3 .
In the case of ConvACs, any dense block of size greater than 1 can be represented as a dense block of size 1 , since the activation function is the linear function (the non-linearity comes from the product pooling operator in the transition layer).
Therefore, for ConvACs, it is only reasonable to analyze dense blocks of size 1 .
Note that, if we only allow dense connections between hidden layers within a dense block, a ConvAC is limited to a maximum growth rate of k = 1 .
In order to analyze the effect of broader connectivity we extend the concept of growth rate by allowing dense connections between dense blocks.
With proper pooling, outputs of hidden layers belonging to different dense blocks can also be concatenated along the feature dimension.
In the reminder of this paper we refer to the dense connections between hidden layers of the same block as intra-block connections , while the connections between hidden layers of different blocks as inter-block connections .
4. D ENSE I NTRA - BLOCK C ONNECTIONS
In this section we analyze the effect of intra-block connections.
We ﬁrst start by constructing a densely connected version of a single hidden layer ConvAC.
The resulting network with growth rate k = 1 is shown in Figure 3 (a) .
In the same manner as in (3), this architecture leads to the score function
[Equation]
Then, we present the following proposition regarding shallow ConvACs with dense connections of growth rate k = 1 .
Proposition 1 The network’s function of a densely connected shallow ConvAC shown in (6) corresponds to the grid tensor
[Equation]
where Sdiag N (cid:8) a yz + Z (cid:9) M z =1 ∈
( R M ) ⊗ N denotes the super-diagonal tensor of order N with a yZ +1 , . .
. , a yZ + M in its diagonal.
Proof See appendix B.1.
Note that the rank of this tensor is now bounded by rank( A y ) ≤ Z + M instead of Z , but adding these dense connections increases the number of parameters of the network from MNZ + Z to
M
Z
Z f
[Equation]
d ( x i ) conv
[Equation]
a
[Equation]
,i ,
[Equation]
δ i pool (
[Equation]
(cid:81) Ni =1 conv
[Equation]
) h y ( X )
[Equation]
(cid:104) a y , pool (:) (cid:105)
M
M conv poolconcat (a)
M rr 2 r 2 rrr
M M rrr 1 r(
[Figure]
M rr
[Figure]
r
[Figure]
rrr
M M
[Figure]
MNZ + Z + M .
Then, for large values of N , there is no clear advantage on using dense connections on a shallow ConvAC.
Nevertheless, in Section 5 we show that dense connections are capable of increasing the expressive power of deep ConvACs, specially for large values of N .
We now generalize the obtained results for the case of a L -layered dense arithmetic circuit, with growth rare k = 1 , as the one in Figure 3 (b) .
Similarly to (5), the obtained grid tensor has the hierarchical decomposition given by
[Equation]
From this result we observe that inter block connections account for virtually increasing the width of the network’s hidden layers from r l to ˜ r l (cid:44) r l + r l − 1 for all l = 0 , 1 , . . .
, L − 1 , where r − 1 (cid:44)
M .
Note that this increased width comes at the expense of increasing the network’s parameters.
Moreover, in Section 5 we discuss whether increasing the network’s width via intra block dense connections leads to an enhancement in its overall expressive power.
4. D ENSE I NTER - BLOCK C ONNECTIONS
In this section we study broader connectivity via dense inter-block connections.
As discussed in Section 4, proper pooling of the preceding features must take place before the concatenating them into the current layer.
Since this type of connections have not been considered in the former DenseNets, we propose 3 possible ways of realizing such connections (via product, average, or max pooling).
For a ConvAC with pooling window size w pool , an inter block connection that connects block l ∈
[ L ] with block p ∈
[ L ] is said to be of jump length L jump ∈
[ L − 1] if p = l + L jump .
An example of an inter block connection of jump length L jump = 1 can be seen in Figure 3 (c) .
To perform this inter block connections, the sizes along the spatial dimensions of preceding features must be reduced by
L jump w pool , before concatenating them along the feature dimension of layer l .
This spatial size reduction may be realized via pooling of the preceding features with window size L jump w pool .
When using a pooling layer the size along the feature dimension remains unchanged.
Moreover, the type of pooling employed (product, average, or maximum) affects the expressive potential of the resulting
ConvAC.
Furthermore, the following proposition addresses the effect that adding dense inter block connections, via average pooling, has on the network function of a ConvAC.
Proposition 2 Adding inter block connections via average pooling of jump length L jump ≥ 1 to a standard ConvAC with grid-tensor A y ∈ ( R M ) ⊗ N leads to a network function of the form
[Equation]
where g ( X ) contains polynomial terms on f θ d ( x i ) for d ∈ [ M ] ,
i ∈
[ N ] of degree lower than N .
Remark
This result is also valid when the connections are done by addition instead of concatenation, as it is done in ResNet and FractalNet.
Proof See appendix B.2.
From this proposition we conclude that adding inter block connections average pooling does not alter the grid tensor
A y , instead these connections account for extra polynomial terms of degree strictly less than N .
Note that, for the special case where the input features belong to an exponential kernel family, such as F = { f θ ( x ) =
e θ T x : R s → R : θ ∈ Θ } or F = { f θ ( x ) =
e (cid:107) θ − x (cid:107) p : R s → R : θ ∈ Θ } where (cid:107) · (cid:107) p denotes the (cid:96) p norm with p ∈ N , the number of polynomial terms is equivalent to the number of exponential basis that the network function can realize.
Therefore, the another valid measure of expressiveness is the number of polynomial terms a ConvAC is able to realize.
Given a certain ConvAC topology, the number of polynomial terms can be computed inductively by expanding the polynomial products of every layer via generalized binomial expansions.
Such an analysis is left for future contributions.
Moreover, if we perform this connections via product poling, the features to be concatenated correspond to polynomial terms of the same order.
This leads to a generalization of the intra-block connections from 4.1, leading to virtually increased widths ˜
r l (cid:44) r l + (cid:80) L jump q =1 r l − q .
Finally, we leave the analysis of inter-block connections via maximum pooling for future work and consider only product pooling inter-block connections in the remainder of this paper.
P RACTICAL I MPLICATIONS
For the sake of comparison, let us assume networks with hidden layer widths r l decaying (or increasing) at an exponential rate of λ ∈ R .
Formally, this is r l = λr l − 1 ∈ N , thus r l = ( λ ) l r for all l = 0 , 1 , . . .
, L − 1 , where r (cid:44) r 0 .
To shorten the notation, we denote as ( L, r, λ, k ) to a ConvAC with of exponential width decay λ ∈ R , length L ∈ N , initial with r ∈ N and growth-rate k ∈ N .
A growth-rate of k = 0 refers to a standard ConvAC with no dense connections.
Deﬁnition 1 Suppose that the weights of a ( L, r, λ, k ) ConvAC, with L, k ∈ N and r, λ ∈ R , are randomly drawn according to some continuous non-vanishing distribution.
Then, this ( L, r, λ, k )
ConvAC is said to have weak dense gain G w ∈ R if, with probability p > 0 , we obtain score functions that cannot be realized by a ( L, r (cid:48) , λ, 0) ConvAC with r (cid:48) <
G w r .
When p = 1 , this ( L, r, λ, k ) ConvAC is said to have a strong dense gain G s =
G w ∈ R .
Using this deﬁnition we present a bound for the weak dense gain G w in the following theorem.
Theorem 5.1 Given M ∈ N , any a ( L, r, λ, k ) ConvAC with L > 1 , r ≤ M, λ ≤ 1 , k > 0 has a dense gain is bounden by G w ≤ Mλr .
Proof See appendix B.3.
This general bound may serve as guideline for tayloring M and the widths r 0 , . .
, r L − 1 such that we exploit the expressiveness added by dense connections.
Theorem 5.2 For the particular case of theorem 5.1 when k = 1 , the weak dense gain is bounded by G w ≤ min (cid:0) 1 + 1 λ , Mλr (cid:1) .
Proof See appendix B.3.
Using this result, we able able to quantify the expressive gain provided by dense inter block connections.
If a ConvAC has a dense gain G w = (1 + 1 λ ) that is already close to the general bound from
Theorem 5.1 it is less encouraging to include broader dense connections, since it would increase the number of parameters of the model while there is no room for a signiﬁcant expressive gain increase.
In this scenario, connections as the ones in ResNet and FractalNet may result more beneﬁcial since they do not increase the size of the model, while at the same time enhancing its trainability.
Theorem 5.3 For the particular case of theorem 5.1 when k
= 1 , if r ≤ 11+ λ √ M , then the bound of theorem 5.2 is achieved with equality and strong dense gain G s = 1 + 1 λ .
Proof See appendix B.3.
This last theorem shows that there exist a regime where this bounds can be achieved with strong dense gain.
Whether this is true outside this regime is still an open question, since further knowledge about the rank of random tensors is limited.
Moreover, these theorems does not consider the additional amount of parameters added by dense connections.
We complete our analysis by addressing this issue in the following proposition.
Proposition 3 Let ∆P dense ∈
N be the additional number of parameters that are added to a ( L, r, λ, 0) ConvAC when we introduce dense connections of growth-rate k > 0 .
In the same manner, let ∆P stand ∈ N be the number of parameters that are added to a ( L, r, λ, 0) ConvAC when we increase its initial width r by a factor G ∈ R .
Then the ratio between ∆P dense and ∆P stand is greater than
[Equation]
Proof See appendix B.4.
The factor G from this proposition directly relates to the dense gain of a ConvAC, thus this ratio may be used to decide whether is interesting to add dense connections to a model (we want this ratio to be as large as possible).
Finally Theorems 5.1 and 5.2 directly bound this ratio, which give the practitioner a guideline to decide which connections (if any) should be added to a given model.
A P RELIMINARY L EMMAS
Lemma
Given Z ∈ N , let A ∈ ( R M ) ⊗ P be a random tensor of even order P ≥
such that
[Equation]
where a ( k ) z ∈ R M are randomly drawn from a non-vanishing continuous distribution for all k ∈
[ P ] and z ∈
[ Z ] .
Then, if Z ≤ M P/ 2 we have that rank( A )
= rank
[ A ])
=
Z with probability 1.
This lemma also holds when for a subset Z ⊆
[ Z ] we have that a ( k ) z
= a z e z ∈ R M for all z ∈ Z , where a z ∈ R are randomly drawn from a non-vanishing continuous distribution.
Proof Using the deﬁnition of the matricization operator, we get that the matricization A is [
[Equation]
Note that, from this expression, it is straight forward to see that the rank of [ A ] is always less or equal than Z .
Let ˜ Z ⊆
[ M P/ 2 ] be the subset ˜ Z
= { ˜ z : ˜ z =
M ( P/ 2) − 1 M −
[Equation]
z −
[Equation]
+
[Equation]
: z ∈ [ Z ] } and U ∈
R M ( P/ 2) × M ( P/ 2) be a permuted version of [ A ] such that the ﬁrst Z rows of U correspond to the rows
˜ z ∈ ˜
Z of [ A ] , and the ﬁrst Z columns of U correspond to the columns ˜
z ∈ ˜
Z of [ A ] .
Since permuting the rows and the columns of a matrix does not alter its rank, we have that U has the same rank as [ A ] .
Now, let us partition U into blocks as
[Equation]
where P is of size Z -by- Z , and Q , W , Z have matching dimensions.
Note that, if rank( P ) =
Z then rank( U ) ≥ Z , which leads to Z ≤ rank( U ) =
rank([ A ]) ≤ Z , thus rank([ A ])
=
Z .
Therefore, it is sufﬁcient to show that rank( P ) =
Z with probability 1 to conclude this proof.
To that end, let us deﬁne the mapping from x ∈ R MPZ to P = P ( x ) as
[Equation]
Note that this deﬁnition of x implies that a ( i )
z
= a ( i ) z
( x ) for all
z ∈ [ Z ] and i ∈
[ P ] .
Therefore, since [ A ] is computed as in (10), we have that
[ A ] =
[ A ]( x ) , thus Q = Q ( x ) and P = P ( x ) .
Now, det P ( x ) is a polynomial on x , then it either vanishes in a set of measure zero or its the zero-polynomial (see Caron & Traynor (2005)).
If we set x to be equal to some x 0 ∈ R MPZ such that a ( i ) z
= e z for all
z ∈ [ Z ] and i ∈
[ P ] , we have that ˜ a ( odd ) z
= ˜ a ( even ) z
=
e ˜ z ∈ R M P/ 2 with
˜ z
(cid:44) ( (cid:80) P/ 2 − 1 n =
M n
[Equation]
( z − 1) + 1 =
M ( P/ 2) − 1 M −( z − 1) + 1 .
Therefore, since [ A ] is calculated as in (10) and ˜ a ( odd ) z
˜ a ( even ) T z is now a matrix with 1 on the entry (˜ z, ˜ z ) and zero elsewhere, we have that [ A ]( x 0 ) is a diagonal matrix with ones on the diagonal elements ˜
z ∈ ˜ Z and zero elsewhere.
This leads to P ( x 0 ) =
I Z which has a determinant det P ( x 0 ) (cid:54) = 0 .
Finally, since there exist x 0 such that the polynomial det P ( x 0 ) is not zero, we conclude that det P ( x ) is not the zero-polynomial, which means that det P ( x ) (cid:54) = 0 with probability 1, thus proving this lemma.
Lemma 2 Let A ∈ ( R M ) ⊗ P and B ∈ ( R M ) ⊗ P be random tensors of even order P ≥ 2 and Z ∈ N be tensors such that
[Equation]
where a ( i ) z ∈ R M and b ( i ) z ∈ R M are randomly drawn from a non-vanishing continuous distribution.
Then, if Z 1 ≤ M P/ 2 and Z 2 ≤ M P/ 2 , we have that rank ( A ⊗ B ) =
Z 1 Z 2 with probability 1.
Proof Let C ∈
( R M ) ⊗ 2 P be a random tensor deﬁned as C =
A ⊗ B .
Therefore, we may express C as
[Equation]
Then, we deﬁne rank-1 tensors C ( q,z ) to be C ( q,z )
= a (1) q ⊗ · · · ⊗ a ( P ) q ⊗ b (1) z ⊗ · · · ⊗ b ( P )
z to get
C =
Z (cid:88) q,z =
C ( q,z ) .
Since C is now expressed as a sum of Z 1 Z 2 rank-1 tensors, we have that rank( C ) ≤ Z 1 Z 2 .
Since Z 1 ≤ M P/ 2 and Z 2 ≤ M P/ 2 we may use Lemma 1, this leads to rank([ A ]) =
Z 1 and rank([ B ]) =
Z 2 with probability 1.
Finally we, use the properties of the Kronecker product to obtain the rank of the matricization C as rank([ C ])
= rank([ A ⊗ B ]) =
rank([ A ])rank([ B ]) , leading to rank([ C ])
=
Z 1 Z 2 ⇒ Z 1 Z 2 = rank
([ C ]) ≤ rank( C ) ≤ Z 1 Z 2 ⇒ rank( C ) =
Z 1 Zwith probability 1, thus proving the Lemma.
Lemma 3 Let A ∈ ( R M ) ⊗ P and B ∈ ( R M ) ⊗ P be tensors of order P > 2 and Z ∈ N be tensors such that
[Equation]
where a ( i ) z ∈ R M and b ( i ) z ∈ R M are randomly drawn from a non-vanishing continuous distribution.
Then, if Z 1 + Z 2 ≤ M P/ 2 , we have that rank ( A + B ) =
Z 1 + Z 2 with probability 1.
Proof Let C ∈ R M ×···× M be a tensor of order P deﬁned as C
=
A + B .
Therefore, we may express
C as
[Equation]
where
[Equation]
Since Z 1 + Z 2 ≤ M P/ 2 we may use Lemma 1, leading to rank ( C )
=
Z 1 + Z 2 with probability 1, thus proving this Lemma.
Corollary 1 Let A , B , C be tensors of the same size with ranks Z A (cid:44) rank ( A ) , Z B (cid:44) rank ( B ) , and Z C (cid:44) rank ( C ) .
Then, the following statements hold true.
[Equation]
B D EFERRED P ROOFS B. P ROOF P ROPOSITION
Proof We reformulate this (6) to have the same form as (3).
To that end we deﬁne a d,iz , for z =
Z + 1 , . .
, Z + M , to be a d,iz
= 1 if z − Z = d and zero otherwise.
This deﬁnition of a d,iz leads to (cid:80)
Md =1 a d,iz f θ d ( x i ) =
f θ z − Z ( x i ) for z
= Z + 1 , . . .
, Z + M .
Using this relation we get
[Equation]
z =
Z
+ a
[Equation]
N
[Equation]
i =
M (cid:88) d = a
[Equation]
f
[Equation]
d
[Equation]
x
[Equation]
) =
[Equation]
(cid:88)
[Equation]
a
[Equation]
N (cid:89) i =
M (cid:88) d
[Equation]
a
[Equation]
f
[Equation]
d ( x
[Equation]
) , which has the same form as (3).
Therefore, as done in (4), we obtain the grid tensor for this architecture as
[Equation]
thus proving this proposition.
B. P ROOF P ROPOSITION
Proof Given x (cid:44) [ f θ 1 ( x 1 ) , · · · , f θ M ( x 1 ) , f θ 2 ( x 1 ) , · · · , f θ M ( x N )] T ∈ R MN , the output of the l -th layer of a ( L, r, λ, 0) ConvAC can be stored into the vectors of mappings δ l,j ( x ) (cid:44) [ δ l,j 1 ( x ) , . . .
, δ l,jr l ( x )] T ∈ R r l for j ∈ [ N/ 2 l ] and l ∈ [ L ] .
Moreover, since the entries of these vectors are the result of l − 1 convolution-pooling layers with product pooling of window size 2 , all the mappings δ l,j 1 ( x ) can be expressed as a sum of polynomial terms on x of degree 2 l .
Now, let the coefﬁcient vectors a l,j,γ (cid:44) [ a l,j,γ, . .
. , a l,j,γr l ] T ∈ R r l for j ∈ [ N/ 2 l ] and γ ∈ [ r l +1 ] , be the weight vectors for the convolution of the l -th layer.
To shorten the notation we use (cid:104) a l,j,γ , δ l,j (cid:105) = (cid:80) r l d =1 a l,j,γd δ l,jd as shorthand for the convolution between these vectors.
Then, the outputs the the layer l of this ConvAC are given by δ l +1 ,j ∈ R r l +1 with δ l +1 ,j γ = (cid:104)
a
[Equation] [Equation]
, δ
[Equation] [Equation]
− 1 (cid:105)(cid:104) a
[Equation] [Equation]
,γ , δ l, 2 j (cid:105) for j ∈ [ N/ 2 l +1 ] , γ ∈ [ r l +1 ] .
If we recursively calculate these out vectors up to the L -th layer we obtain the score functions h y stand ( x ) (cid:44) δ L, 1 ( x ) =
δ L, 1 1 ( x ) ∈ R .
We now consider the effecct of adding dense connections via average pooling from some k ∈ N preceding layers l − 1 , . . .
, l − k .
To this end, let ˜ r l = (cid:80) kq =1 r l − q be the total size along the feature dimesnion of the vectors to be concatenated.
In addition, let ω l,j ( x ) (cid:44) [ ω l,j 1 ( x ) , . .
, ω l,j ˜ r l ( x )] T ∈
R ˜ r l be the vectors of mappings of the corresponding preceeding features at the layer l for j ∈ [ N/r l ] .
In order to compute the convolutions of this layer, an additional vector of coefﬁcients is required as b l,j,γ (cid:44) [ b l,j,γ, . . .
, b l,j,γ ˜ r l ] T ∈ R ˜ r l .
Then, the outputs of the l -th layer of this ( L, r, λ, k ) ConvAC are the denoted as the vectors ˜
δ l,j ( x ) (cid:44) [˜ δ l,j 1 ( x ) , . . . , ˜
δ l,jr l +1 ( x )]
T ∈ R r l +1 for j ∈ [ N/ 2 l +1 ] where ˜ δ l +1 ,
[Equation]
= (cid:104) (cid:20) a l, 2 j − 1 ,γ b l, 2 j − 1 ,γ (cid:21) , (cid:20) δ l, 2 j −ω l, 2 j −(cid:21) (cid:105)(cid:104) (cid:20) a l, 2 j,γ b l, 2 j,γ (cid:21) , (cid:20) δ l, 2 j ω l, 2 j (cid:21) (cid:105) .
From this expression it follows
[Equation]
where
[Equation]
Note that the entries of ω l,j ( x ) are assumed to come from preceding layers with an appropiate average pooling.
Since performing avergae pooling does not increase the degree of the polynomial terms involved (only product pooling does) and the jump length L jump is at least 1 , the entries of ω l,j ( x ) have at most polynomial degree 2 l − 1 , which is strictly less than the degree of the entries of δ l,j ( x ) (i.e., 2 l ).
Therefore, from the obtained expression of ω l +1 ,j we observe that it has polynomials withb degree no greater than 2 l + 2 l − 1 , while the entries of δ l +1 ,j have a strictly higher degree of 2 l + 2 l = 2 l +1 .
Moreover, since (cid:104) a l,j,γ , δ l,j + ω l,j (cid:105) can be expressed as
[Equation] [Equation]
we can make use of the obtained results in an unductive manner up to the L -th layer, thus leading to
[Equation]
where g ( x ) contains polynomial terms of x of order strictly less than N , thus proving this theorem.
Note that this result also applies to additive and resudial connections, as de ones used in ResNet and
FractalNet, since they can be expressed as in (11).
B. P ROOF OF T HEOREMS 5.1 TO 5.
Proof Given M ∈ N , a ( L, r, λ, 0) ConvAC with L > 1 , r ≤ M, λ ≤ 1 has a grid tensor
A y stand ∈ ( R M ) ⊗ N .
For the forthcoming analysis let us assume r 0 ≤ M .
This assumption is done, so that we can write min { r 0 , M } = r 0 , merely for notation purposes since we show that this does not affect the generality of the results.
Using this assumption, we upper bound the rank of the grid tensor as rank (cid:0) φ 1 ,j,
[Equation]
rank (cid:0)
[Equation]
rank (cid:0)
[Equation]
It was shown in Cohen & Shashua (2016a)
that, when the weights are independently generated from some continuous distribution, we have that rank (cid:0) φ 1 ,j,γ (cid:1)
=
min { r 0 , M } with probability 1.
Note that, the bounds obtained for r 0 values greater than M is the same as for r 0 =
M , thus implying that the assumption of r 0 ≤ M does not affect the generality of the results.
Finally, by induction up to the L -th layer, we obtain a bound for the grid tensor rank as
[Equation]
Since we assumed networks with hidden layer widths r l decaying (or increasing) at an exponential rate of λ ∈ R .
Formally, this is r l = λr l − 1 ∈ N , thus r l = ( λ ) l r for all l = 0 , 1 , . . .
, L − 1 , where r (cid:44) r 0 .
Therefore, we may simplify the obtained bound to
[Equation]
We this analysis by proving Theorem 5.1.
To that end let A y dense be the grid tensor of a dense ( L, r, λ, k ) ConvAC with k > 0 , while A y stand is the grid tensor of a ( L, r (cid:48) , λ, 0) ConvAC with r (cid:48) ∈ R .
As discussed in Section 4, this dense version of the former ( L, r, λ, 0) ConvAC is equivalent to virtually increasing the widths of the ConvAC, which translates extra additive terms in the expressions from 12.
Moreover, using corollary 1 we observe that, if the ranks of the tensors φ l,j,γ are additive and multiplicative up to rank( A y dense ) > rank( A y stand ) , so they are up to rank( A y stand ) .
A weak dense gain value G w ∈ R is achieved when there is a set of functions realized by the ( L, r, λ, k )
ConvAC that cannot be realized by ( L, r (cid:48) , λ, 0) ConvAC unless r (cid:48) =
G w r .
To bound this gain, let us assume the best case scenario where rank( A y dense ) reaches the maximum possible rank, from the size of A y dense this can be at most rank( A y dense )
=
M 2 L − 1 .
As discussed, would imply that the ranks of the tensors φ l,j,γ are additive up to M 2 L − 1 for both ConvACs.
Therefore, A y stand achieves its maximum rank given by rank ( A
[Equation]
Then, a ( L, r (cid:48) , λ, 0) ConvAC is able to realize the functions of a ( L, r, λ, k ) ConvAC when rank ( A y stand )
= rank ( A y dense ) , thus ( λ )
[Equation]
L ( r
[Equation]
L
[Equation]
1 = M
[Equation]
L − 1 .
Finally, since r (cid:48)
=
G w r , this leads to a maximum value of
[Equation]
which proves Theorem 5.1.
For Theorem 5.2 we use consider the particular case of k = 1 , which yields a core tensor given by the hierarchical tensor decomposition from (9).
We use the same assumption of r 0 ≤ M and deﬁne the virtually increased widths ˜
r l (cid:44) r l + r l − 1 ∈ N for l = 1 , . . .
, L − 1 and ˜ r 0 (cid:44) M .
This leads to rank (cid:0) φ
[Equation]
= rank
[Equation]
rank (cid:0)
[Equation]
and
[Equation]
Note that for r l = λr l − 1 ∈ N ( λ ∈ R ), we get virtually increased widths ˜
r l = (1 + λ ) l r
=
(cid:0) λ (cid:0)
+ 1 λ (cid:1)(cid:1) l r , for all l = 0 , 1 , . . .
, L − 1 , leading to rank ( A y dens ) ≤
[Equation]
As in for the proof of Theorem 5.1, the maximum dense gain G w is obtained when rank( A y dense ) reaches the maximum possible rank.
In this particular case, this corresponds to rank( A y dense ) =
min (cid:16) ( λ (1 + 1 /λ ))
2 L − 1 − L r 2 L − 1 , M 2 L − 1 (cid:17) .
Furthermore, for obtaining rank ( A y stand ) = rank ( A y dense ) a gain of
G w ≤ min (cid:18) 1 + 1 λ , M λr (cid:19) is required, thus proving Theorem 5.2.
Finally, for proving Theorem 5.3 we show that this bound can be attained with probability 1.
Note that this bound is achieved when the inequalities from (12) hold with equality for all l ∈
[ L ] .
The ﬁrst inequality of (12) holds when the tensors
[Equation]
are additive on the rank.
This can be proven to be true with probability 1 when
[Equation]
by applying Lemma 3.
In the same manner, the second inequality of (12) holds when the tensor pairs (cid:0) φ l − 1 ,
[Equation]
j − 1 ,α , φ l − 1 , 2 j,α (cid:1) are multiplicative in the tensor rank.
We may use Lemma 2 to prove this is the case with probability 1 if we can bound
[Equation]
In summary, if equations (16) and (17) hold, we may use Lemmas 3 and 2 to prove that (12) reaches equality with probability 1, thus implying that (13) also reaches equality everywhere outside a set
Lebesgue measure zero.
It is straight forward to see that, if (16) holds, so does (17).
For the case of a network with exponential width decay λ and r ≤ 1 λ √ M we have that
[Equation]
thus (16) holds.
Therefore, within this regime of r ≤ 1 λ √ M , we may ensure that the tensor rank is additive and multiplicative with probability 1.
Now we apply the same reasoning for a densely connected arithmetic circuit of L layers and width decay λ such that r ≤ 11+ λ √ M = 1 λ (1+1 /λ ) √ M .
In the same manner as for the standard ConvAC we bound
[Equation]
which enables us to make use of Lemmas 3 and 2 to prove that (15) holds with equality everywhere except from a set of Lebesgue measure zero.
Note that, for r ≤ 11+ λ √ M < 1 λ √ M , we have that both equations (13) and (15) reach equality with probability 1, thus proving Theorem 5.3.
B. P ROOF P ROPOSITION
Proof Let P ( L, r, λ, k ) ∈ N be the number of parameters of a ( L, r, λ, k ) ConvAC.
A standard ( L, r, λ, 0) ConvAC is composed of the weights { a 0 ,j,γ ∈ R M } j ∈ [ N ] ,γ ∈
[ r 0 ] in the ﬁrst hidden convolutions, { a l,j,γ ∈ R M } j ∈ [ N/ 2 l ] ,γ ∈ [ r l ] in the hidden layers, and a L, 1 ,y ∈ R r L − 1 in the weights corresponding to the output y in the output layer.
Therefore, this ConvAC has a number of weights
[Equation]
When adding dense connections of growth-rate k , we need additional weights for the convolution between the preceding layers and the current layer.
Therefore, at the l -th layer we have an extra (cid:80)
[Equation]
r l r l − 1 − q weights, which leads to
[Equation]
By deﬁnition, we have that ∆P std =
P ( L, Gr, λ, 0) − P ( L, Gr, λ, 0) and ∆P dense =
P ( L, Gr, λ, k ) − P ( L, r, λ, 0) , thus yielding
[Equation]
and
[Equation]
Finnaly
[Equation]
we use these expressions to compute the ratio of interest as
[Equation]
which proves this proposition.