Year: 2021

Abstract:
Self-supervised visual representation learning aims to learn useful representations without relying on human annotations. Joint embedding approach bases on maximizing the agreement between embedding vectors from different views of the same image. Various methods have been proposed to solve the collapsing problem where all embedding vectors collapse to a trivial constant solution. Among these methods, contrastive learning prevents collapse via negative sample pairs. It has been shown that non-contrastive methods suffer from a lesser collapse problem of a different nature: dimensional collapse, whereby the embedding vectors end up spanning a lower-dimensional subspace instead of the entire available embedding space. Here, we show that dimensional collapse also happens in contrastive learning. In this paper, we shed light on the dynamics at play in contrastive learning that leads to dimensional collapse. Inspired by our theory, we propose a novel contrastive learning method, called DirectCLR, which directly optimizes the representation space without relying on a trainable projector. Experiments show that DirectCLR outperforms SimCLR with a trainable linear projector on ImageNet.

heading: 1 INTRODUCTION
text: Self-supervised learning aims to learn useful representations of the input data without relying on human annotations. Recent advances in self-supervised visual representation learning based on joint embedding methods (Misra & Maaten, 2020b; He et al., 2020; Chen et al., 2020a; Chen & He, 2020; Grill et al., 2020; Zbontar et al., 2021; Bardes et al., 2021; Chen et al., 2020b; Dwibedi et al., 2021; Li et al., 2021; Misra & Maaten, 2020a; HaoChen et al., 2021; Assran et al., 2021; Caron et al., 2021) show that self-supervised representations have competitive performances compared with supervised ones. These methods generally aim to learn representations invariant to data augmentations by maximizing the agreement between embedding vectors from different distortions of the same images.
As there are trivial solutions where the model maps all input to the same constant vector, known as the collapsing problem, various methods have been proposed to solve this problem that rely on different mechanisms. Contrastive methods like Chen et al. (2020a) and He et al. (2016) define ‘positive’ and ‘negative’ sample pairs which are treated differently in the loss function. Non-contrastive methbods like Grill et al. (2020) and Chen & He (2020) use stop-gradient, and an extra predictor to prevent collapse without negative pairs; Caron et al. (2018; 2020) use an additional clustering step; and Zbontar et al. (2021) minimize the redundant information between two branches.
These self-supervised learning methods are successful in preventing complete collapse whereby all representation vectors shrink into a single point. However, it has been observed empirically in noncontrastive learning methods (Hua et al., 2021; Tian et al., 2021) that while embedding vectors do not completely collapse; they collapse along certain dimensions. This is known as dimensional collapse (Hua et al., 2021), whereby the embedding vectors only span a lower-dimensional subspace.
In contrastive methods that explicitly use positive and negative pairs in the loss function, it seems intuitive to speculate that the repulsive effect of negative examples should prevent this kind of dimensional collapse and make full use of all dimensions. However, contrary to intuition, contrastive learning methods still suffer from dimensional collapse (See Fig. 7). In this work, we theoretically study the dynamics behind this phenomenon. We show there are two different mechanisms that cause collapsing: (1) along the feature direction where the variance caused by the data augmentation is larger than the variance caused by the data distribution, the weight collapses. Moreover, (2)
even if the covariance of data augmentation has a smaller magnitude than the data variance along all dimensions, the weight will still collapse due to the interplay of weight matrices at different layers known as implicit regularization. This kind of collapsing happens only in networks where the network has more than one layer.
Inspired by our theory, we propose a novel contrastive learning method, called DirectCLR, which directly optimizes the encoder (i.e., representation space) without relying on a trainable projector. DirectCLR outperforms SimCLR with a linear trainable projector on ImageNet.
We summarize our contributions as follows:
• We empirically show that contrastive self-supervised learning suffers from dimensional collapse whereby all the embedding vectors fall into a lower-dimensional subspace instead of the entire available embedding space.
• We showed that there are two mechanisms causing the dimensional collapse in contrastive learning: (1) strong augmentation along feature dimensions (2) implicit regularization driving models toward low-rank solutions.
• We propose DirectCLR, a novel contrastive learning method that directly optimizes the representation space without relying on a trainable projector. DirectCLR outperforms SimCLR with a linear trainable projector.

heading: 2 RELATED WORKS

heading: 2.1 SELF-SUPERVISED LEARNING METHODS
text: Joint embedding methods are a promising approach in self-supervised learning, whose principle is to match the embedding vectors of augmented views of a training instance. Contrastive methods (Chen et al., 2020a; He et al., 2016) directly compare training samples by effectively viewing each sample as its own class, typically based on the InfoNCE contrastive loss (van den Oord et al., 2018) which encourages representations from positive pairs of examples to be close in the embedding space while representations from negative pairs are pushed away from each other. In practice, contrastive methods are known to require a large number of negative samples. Non-contrastive methods do not directly rely on explicit negative samples. These include clustering-based methods (Caron et al., 2018; 2020), redundancy reduction methods (Zbontar et al., 2021; Bardes et al., 2021) and methods using special architecture design (Grill et al., 2020; Chen & He, 2020).

heading: 2.2 THEORETICAL UNDERSTANDING OF SELF-SUPERVISED LEARNING
text: Although self-supervised learning models have shown success in learning useful representations and have outperformed their supervised counterpart in several downstream transfer learning benchmarks (Chen et al., 2020a), the underlying dynamics of these methods remains somewhat mysterious and poorly understood. Several theoretical works have attempted to understand it. Arora et al. (2019b); Lee et al. (2020); Tosh et al. (2021) theoretically proved that the learned representations via contrastive learning are useful for downstream tasks. Tian et al. (2021) explained why non-contrastive learning methods like BYOL (Grill et al., 2020) and SimSiam (Chen & He, 2020) work: the dynamics of the alignment of eigenspaces between the predictor and its input correlation matrix play a key role in preventing complete collapse.

heading: 2.3 IMPLICIT REGULARIZATION
text: It has been theoretically explained that gradient descent will drive adjacent matrices aligned in a linear neural network setting (Ji & Telgarsky, 2019). Under the aligned matrix assumption, Gunasekar et al. (2018) prove that gradient descent can derive minimal nuclear norm solution. Arora et al. (2019a) extend this concept to the deep linear network case by theoretically and empirically demonstrating that a deep linear network can derive low-rank solutions. In general, over-parametrized neural networks tend to find flatter local minima (Saxe et al., 2019; Neyshabur et al., 2019; Soudry et al., 2018; Barrett & Dherin, 2021).

heading: 3 DIMENSIONAL COLLAPSE
text: Self-supervised learning methods learn useful representation by minimizing the distances between embedding vectors from augmented images (Figure 1a). On its own, this would result in a collapsed solution where the produced representation becomes constant (Figure 1b). Contrastive methods prevent complete collapse via the negative term that pushes embedding vectors of different input images away from each other. In this section, we show that while they prevent complete collapse, contrastive methods still experience a dimensional collapse in which the embedding vectors occupy a lower-dimensional subspace than their dimension (Figure 1c).
We train a SimCLR model (Chen et al. (2020a)) with a two-layer MLP projector. We followed the standard recipe and trained the model on ImageNet for 100 epoch. We evaluate the dimensionality by collecting the embedding vectors on the validation set. Each embedding vector has a size of d = 128. We compute the covariance matrix C ∈ Rd×d of the embedding layer (here z̄ := ∑N i=1 zi/N and N is the total number of samples):
C = 1
N N∑ i=1 (zi − z̄)(zi − z̄)T (1)
Figure 2 shows singular value decomposition on this matrix (C = USV T , S = diag(σk)). in sorted order and logarithmic scale ({log(σk)}). We observe that a number of singular values collapse to zero, thus representing collapsed dimensions.

heading: 4 DIMENSIONAL COLLAPSE CAUSED BY STRONG AUGMENTATION

heading: 4.1 LINEAR MODEL
text: In this section, we explain one scenario for contrastive learning to have collapsed embedding dimensions, where the augmentation surpasses the input information. We focus on a simple linear network setting. We denote the input vector as x and the augmentation is an additive noise. The network is a single linear layer with weight matrix is W . Hence, the embedding vector is z = Wx. We focus on a typical contrastive loss, InfoNCE (van den Oord et al., 2018):
L = − N∑ i=1 log exp(−|zi − z′i|2/2)∑ j 6=i exp(−|zi − zj |2/2) + exp(−|zi − z′i|2/2) (2)
where zi and z′i are a pair of embedding vectors from the two branches, zj indicates the negative samples within the minibatch. When all zi and z′i are normalized to be unit vector, the negative distance −|zi − z′i|2/2 can be replaced by inner products zTi z′i. The model is trained with a basic stochastic gradient descent without momentum or weight decay.

heading: 4.2 GRADIENT FLOW DYNAMICS
text: We study the dynamics via gradient flow, i.e., gradient descent with an infinitesimally small learning rate. Lemma 1. The weight matrix in a linear contrastive self-supervised learning model evolves by:
Ẇ = −G (3) where G = ∑
i(gzix T i + gz′ix ′T i ), and gzi is the gradient on the embedding vector zi (similarly gz′i ).
This can be easily proven based on the chain rule. See proof in Appendix B.1. For InfoNCE loss defined in Eqn 2, the gradient of the embedding vector for each branch can be written as
gzi = ∑ j 6=i αij(zj − z′i) + ∑ j 6=i αji(zj − zi), gz′i = ∑ j 6=i αij(z′i − zi) (4)
where {αij} are the softmax of similarity of between zi and {zj}, defined by αij = exp(−|zi − zj |2/2)/Zi, αii = exp(−|zi−z′i|2/2)/Zi, andZi = ∑ j 6=i exp(−|zi−zj |2/2)+exp(−|zi−z′i|2/2).
Hence, ∑
j αij = 1. Since zi = Wxi, we have
G = −WX (5)
where
X := − ∑ i ∑ j 6=i αij(x′i − xj) + ∑ j 6=i αji(xi − xj)  xTi −∑ i (1− αii)(x′i − xi)x′i T (6)
Lemma 2. X is a difference of two PSD matrices:
X = Σ̂0 − Σ̂1 (7) Here Σ̂0 = ∑
i,j αij(xi − xj)(xi − xj)T is a weighted data distribution covariance matrix and Σ̂1 = ∑ i(1− αii)(x′i − xi)(x′i − xi)T is a weighted augmentation distribution covariance matrix.
See proof in Appendix B.2. Therefore, the amplitude of augmentation determines whether X is a positive definite matrix.
Next, we look into the dynamics of weight matrix W given property of X . Theorem 1. With fixed matrix X (defined in Eqn 6) and strong augmentation such that X has negative eigenvalues, the weight matrix W has vanishing singular values.
See proof in Appendix B.3. Corollary 1 (Dimensional Collapse Caused by Strong Augmentation). With strong augmentation, the embedding space covariance matrix becomes low-rank.
The embedding space is identified by the singular value spectrum of the covariance matrix on the embedding (Eqn. 1), C = ∑ i(zi− z̄)(zi− z̄)T /N = ∑ iW (xi− x̄)(xi− x̄)TWT /N . Since W has vanishing singular values, C is also low-rank, indicating collapsed dimensions. We experimentally verified the theory in Appendix C.
Our theory in this section is limited to linear network settings. For more complex nonlinear networks, the collapsing condition will still depend on “strong augmentation” but interpreted differently. A strong augmentation will be determined by more complicated properties of the augmentation (higher-order statistics of augmentation, manifold property of augmentation vs. data distribution) conditioned on the capacity of the networks.

heading: 5 DIMENSIONAL COLLAPSE CAUSED BY IMPLICIT REGULARIZATION

heading: 5.1 TWO-LAYER LINEAR MODEL
text: With strong augmentation, a linear model under InfoNCE loss will have dimensional collapse. However, such scenarios rely on the condition that the network has a limited capacity which may not hold for real cases. On the other hand, when there is no strong augmentation (Σ̂1 ≺ Σ̂0) and thus X matrix remains PSD, a single linear model won’t have dimensional collapsing. However, interestingly, for deep networks, dimensional collapsing still happens in practice. In the following, we will show that it stems from a different nature: implicit regularization, where over-parametrized linear networks tend to find low-rank solutions.
To understand this counter-intuitive phenomena, we start with the simplest over-parametrized setting by choosing the network as a two-layer linear MLP without bias. The weight matrices of these two layers are denoted by W1 ∈ Rd×d and W2 ∈ Rd×d. Similar to the setting in Sec 4, the input vector is denoted as x and the augmentation is an additive noise. The embedding vector from each branch is z = W2W1x, hence z ∈ Rn. We do not normalize z. See Figure 3. We use InfoNCE loss defined in Eqn 2. The model is trained with a basic stochastic gradient descent without momentum or weight decay.

heading: 5.2 GRADIENT FLOW DYNAMICS
text: Similar to Lemma 1, we derive the gradient flow on the two weight matrices W1 and W2. Lemma 3. The weight matrices of the two layer linear contrastive self-supervised learning model evolves by (G = ∑ i(gzix T i + gz′ix ′T i ) is defined in Lemma 1):
Ẇ1 = −WT2 G, Ẇ2 = −GWT1 (8)
This can be easily proven based on the chain rule. See proof in Appendix B.4. For the two layer case, similar to Eqn 5, we have the specific form of G:
G = −W2W1X (9) where X is defined in Eqn 6. According to Lemma 2, we know that with small augmentation, X = Σ̂0 − Σ̂1 0 is a positive-definite matrix.

heading: 5.3 WEIGHT ALIGNMENT
text: Since we have two matrices W1 and W2, the first question is how they interact with each other. We apply singular value decomposition on both matrices W1 and W2, i.e., W1 = U1S1V T1 , W2 = U2S2V T 2 and S1 = diag([σ k 1 ]), S2 = diag([σ k 2 ]). The alignment is now governed by the interaction between the adjacent orthonormal matrices V2 := [vk2 ] and U1 = [u k 1 ]. This can be characterized by the alignment matrix A = V T2 U1, whose (k, k ′)-entry represents the alignment between the k-th right singular vector vk2 of W2 and the k ′-th left singular vector uk ′
1 of W1. The following shows that indeed W1 and W2 aligns. Theorem 2 (Weight matrices align). If for all t, W2(t)W1(t) 6= 0, X(t) is positive-definite and W1(+∞), W2(+∞) have distinctive singular values, then the alignment matrix A = V T2 U1 → I .
See proof in Appendix B.5. Here, we also empirically demonstrate that under InfoNCE loss, the absolute value of the alignment matrix A converges to an identity matrix. See Figure 4.
The alignment effect has been studied in other scenarios (Ji & Telgarsky, 2019; Radhakrishnan et al., 2020). In the real case, when some of our assumptions are not satisfied, e.g., there are degenerate singular values in weight matrices, we will not observe a perfect alignment. This can be easily understood by the fact that the singular decomposition is no longer unique given degenerate singular
values. In our toy experiment, we specifically initialize the weight matrices to have non-degenerate singular values. In real scenario, when weight matrices are randomly initialized, we will only observe the alignment matrix to converge to a block-diagonal matrix, with each block representing a group of degenerate singular values.
Given the fact that singular vectors corresponding to the same singular value align, we can now study the dynamics of the singular values of each weight matrix W1 and W2.
Theorem 3. If W2 and W1 are aligned (i.e., V2 = UT1 ), then the singular values of the weight matrices W1 and W2 under InfoNCE loss evolve by:
σ̇k1 = σ k 1 (σ k 2 ) 2(vk1 T Xvk1) (10)
σ̇k2 = σ k 2 (σ k 1 ) 2(vk1 T Xvk1) (11)
See proof in Appendix B.6. According to Eqn. 10, (σk1 ) 2 = (σk2 ) 2 + C. We solve the singular value dynamics analytically: σ̇k1 = σ k 1 ((σ k 1 ) 2 + C)(vk1 T
Xvk1). This shows that a pair of singular values (singular values with same ranking from the other matrix) have gradients proportional to themselves. Notice that X is a positive definite matrix, the term vk1 T Xvk1 is always non-negative. This explains why we observe that the smallest group of singular values grow significantly slower. See demonstrative experiment results in Figure 5a and 5b.
Corollary 2 (Dimensional Collapse Caused by Implicit Regularization). With small augmentation and over-parametrized linear networks, the embedding space covariance matrix becomes low-rank.
The embedding space is identified by the singular value spectrum of the covariance matrix on the embedding vectors, C = ∑ (z − z̄)(z − z̄)T /N = ∑ W2W1(x − x̄)(x − x̄)TWT1 WT2 /N . As W2W1 evolves to be low-rank, C is low-rank, indicating collapsed dimensions. See Figure 5c for experimental verification.

heading: 5.4 EFFECT OF MORE LAYERS AND NONLINEARITY
text: In our toy model, we focused on a two-layer linear MLP setting. Here, we empirically show that our theory extends to multilayer and nonlinear cases, as shown in Figure 6a.
Stronger over-parametrization leads to a stronger collapsing effect, which has been shown theoretically (Arora et al., 2019a; Barrett & Dherin, 2021) and empirically (Jing et al., 2020). This can be explained by the fact that more adjacent matrices getting aligned, and the collapsing in the product matrix gets amplified. Note that for a single-layer case, L = 1, there is no dimensional collapse in the embedding space, which is consistent with our analysis.
We empirically show that the collapsing effect also applies to the nonlinear scenario. We insert ReLU between linear layers and observe a similar singular value collapse compared to the linear case. See Figure 6b.

heading: 6 DIRECTCLR

heading: 6.1 MOTIVATION
text: We now leverage our theoretical finding to design novel algorithms. Here we are targeting the projector component in contrastive learning.
Empirically, adding a projector substantially improves the quality of the learned representation and downstream performance (Chen et al., 2020a). Checking the spectrum of the representation layer also reveals a difference with/without a projector. To see this, we train two SimCLR models with and without a projector. The representation space spectrum are shown in Figure 7b. The dimensional collapse in representation space happens when the model is trained without a projector. Thus, the projector prevents the collapse in the representation space.
The projector in contrastive learning is essential to prevent dimensional collapse in the representation space. We claim the following propositions regarding a linear projector in contrastive learning models.
1. A linear projector weight matrix only needs to be diagonal. 2. A linear projector weight matrix only needs to be low-rank.
Based on our theory on implicit regularization dynamics, we expect to see adjacent layers W1(= U1S1V T 1 ) and W2(= U2S2V T 2 ) to be aligned such that the overall dynamics is only governed by their singular values S1 and S2. And the orthogonal matrices V T2 and U1 are redundant as they will evolve to V T2 U1 = I , given S1 and S2.
Now, let’s consider the linear projector SimCLR model and only focus on the channel dimension. W1 is the last layer in the encoder, and W2 is the projector weight matrix. Our proposition claims that for this projector matrixW2, the orthogonal component V2 can be omitted. Because the previous layer W1 is fully trainable, its orthogonal component (U1) will always evolve to satisfy V T2 U1 = I . Therefore, the final behavior of the projector is only determined by the singular values (S2 ) of the projector weight matrix. This motivates proposition 1: the orthogonal component of the weight matrix doesn’t matter. So we can set the projector matrix as a diagonal matrix.
Also, according to our theory, the weight matrix will always converge to the low-rank. The singular value diagonal matrix naturally becomes low-rank, so why not just set it low-rank directly? This is the motivation of proposition 2.
These propositions are verified via ablation studies in Sec 6.3. Given these two propositions, we propose DirectCLR, which is effectively using a low-rank diagonal projector.
6.2 MAIN IDEA
We propose to remove the projector in contrastive learning by directly sending a sub-vector of the representation vector to the loss function. We call our method DirectCLR. In contrast to all recent state-of-the-art self-supervised learning methods, our method directly optimizes the representation space. See Figure 8, DirectCLR picks a subvector of the representation z = r[0 : d0], where d0 is a hyperparameter. Then, it applies a standard InfoNCE loss on this normalized subvector ẑ = z/|z|, L = ∑ i log exp(ẑi ·̂z′i)∑ j exp(ẑi ·̂zj) .
We train DirectCLR with a standard recipe of SimCLR for 100 epochs on ImageNet. The backbone
encoder is a ResNet50. More implementation details can be found in the Appendix D. DirectCLR demonstrates better performance compared to SimCLR with a trainable linear projector on ImageNet. The linear probe accuracies for each model are listed in Table 1.
We visualize the learnt representation space spectrum in Figure 9. DirectCLR prevents dimensional collapse in the representation space similar to the functionality of a trainable projector in SimCLR.
One may suspect that the contrastive loss in DirectCLR does not apply a gradient on the rest part of the representation vector r[d0 :], then why these dimensions would contain useful information?
Here, we show that the entire representation vector r contains useful information. See Figure 10. First, the gradient backpropagating through the representation vector is low-rank, where only the first d0 channel dimensions are non-zero. When the gradient enters the ResNet backbone and passes through the last nonlinear conv block, it becomes full rank. Therefore, this hidden layer h receives
gradients on all channels. Note that h and r have a same channel dimension of 2048. Next, we consider the forward pass. This hidden layer h is directly fed to the representation vectors via the residual connection. As a result, the rest part of the representation vector r[d0 :] is not trivial. In addition, we run an ablation study in Sec F to test the linear probe accuracy based only on the “directly” optimized vector. This verifies that the whole representation vector is meaningful.

heading: 6.3 ABLATION STUDY
text: Proposition 1 matches the fact that: (a) an orthogonal constrained projector performs the same as the non-projector setting; (b) fixed low-rank projector performs the same as a fixed diagonal projector; (c) trainable linear projector performs the same as a trainable diagonal projector.
Proposition 2 matches the observation that a low-rank projector has the highest accuracy.
Please see more detailed ablation study discuss and additional ablation experiments in Appendix F.

heading: 7 CONCLUSIONS
text: In this work, we showed that contrastive self-supervised learning suffers from dimensional collapse, where the embedding vectors only span a lower-dimensional subspace. We provided the theoretical understanding of this phenomenon and showed that there are two mechanisms causing dimensional collapse: strong augmentation and implicit regularization. Inspired by our theory, we proposed a novel contrastive self-supervised learning method DirectCLR that directly optimizes the representation space without relying on a trainable projector. DirectCLR outperforms SimCLR with a linear projector on ImageNet.

heading: REPRODUCIBILITY STATEMENT
text: We provide detailed proof for every lemma and theorem in the Appendices. We provide an anonymous and executable code for reproducing the results of the experiment section in the supplementary material. We will upload the code to a public Github repo after publication.

heading: A USEFUL LEMMAS
text: We adapt two useful lemmas from Arora et al. (2019a).
Lemma 4. Given a matrix W and the dynamics that W evolves by Ẇ , the singular values of this matrix evolve by:
σ̇k = uk T Ẇ vk (12)
where uk and vk are singular value σk’s corresponding left and right singular vectors. i.e. the k-th column of matrices U and V respectively.
Proof. Given a matrixW and its singular value decompositionW = USV T . We have the dynamics of the matrix
Ẇ = U̇SV T + UṠV T + USV̇ T
Multiplying UT from the left and multiplying V from the right, considering U and V are orthogonal matrices, we have
UT ẆV = UT U̇S + Ṡ + SV̇ TV
Since S = diag(σk) is a diagonal matrix, we have
σ̇k = uk T Ẇvk − ukT u̇kσk − σkv̇k T vk
Again, considering uk and vk have unit-norm, we have ukT u̇k = 0 and v̇k T
vk = 0. Therefore, we derive
σ̇k = uk T Ẇvk
Lemma 5. Given a matrix W and the dynamics that W evolves by Ẇ , the singular vectors of this matrix evolve by:
U̇ = U(H (UT ẆV S + SV T ẆTU)) (13) V̇ = V (H (V T ẆTUS + SUT ẆV )) (14)
where represents Hadamard element-wise multiplication. H is a skew-symmetric matrix
Hk,k ′ =
{ 1/(σk 2 − σk′ 2 ) if k 6= k′
0 if k = k′ (15)
Proof. Same as proof for Lemma 1, we start from the following equation
UT ẆV = UT U̇S + Ṡ + SV̇ TV
Considering the fact that UT U̇ and V̇ TV are skew-symmetric matrices, whose diagonal terms are all zero, we Hadamard-multiply Ī to both sides of the equation. Here, Ī has all diagonal values equal zeros and all off-diagonal values equal to one, we have
Ī UT ẆV = UT U̇S + SV̇ TV (16)
Taking transpose, we have Ī V T ẆU = −SUT U̇ − V̇ TV S (17)
Right-multiplying S to Eqn 16 and left-multiplying S to Eqn 17, then adding them up, we have
UT U̇S2 − S2UT U̇ = Ī (UT ẆV S + SV T ẆU)
Therefore, we have U̇ = U(H (UT ẆV S + SV T ẆTU))
where
Hk,k ′ =
{ 1/(σk 2 − σk′ 2 ) if k 6= k′
0 if k = k′
Similar proof applies to Eqn 14.
Lemma 6 (Alignment matrix dynamics). The alignment matrix A, defined by A = V T2 U1, evolves by: Ȧ = −A(H1 (ATF + FTA)) + (H2 (AFT + FAT ))A (18) where represents Hadamard (element-wise) multiplication. Hl is a skew-symmetric matrix, whose (k, k′)-entry is given by
Hk,k ′
l =
{ 1/(σkl 2 − σk′l 2 ) if k 6= k′
0 if k = k′ (19)
and F is defined by F = S2U T 2 GV1S1 (20)
Proof. According to Lemma. 5, we have
U̇1 = U1(H1 (UT1 Ẇ1V1S1 + S1V T1 ẆT1 U1)) V̇2 = V2(H2 (V T2 ẆT2 U2S2 + S2UT2 Ẇ2V2))
Plugging the above two equations and Eqn 8, the dynamics of the alignment matrix A = V T2 U1 can be written as
Ȧ = V T2 U̇1 + V̇ T 2 U1
= V T2 U1(H1 (UT1 Ẇ1V1S1 + S1V T1 ẆT1 U1)) + (H2 (V T2 ẆT2 U2S2 + S2UT2 Ẇ2V2))TV T2 U1 = −A(H1 (UT1 WT2 GV1S1 + S1V T1 GTW2U1)) + (H2 (S2UT2 GWT1 V2 + V T2 W1GTU2S2))A = −A(H1 (UT1 V2S2UT2 GV1S1 + S1V T1 GTU2S2V T2 U1))
+(H2 (S2UT2 GV1S1UT1 V2 + V T2 U1S1V T1 GTU2S2))A = −A(H1 (ATS2UT2 GV1S1 + S1V T1 GTU2S2A)
+(H2 (S2UT2 GV1S1AT +AS1V T1 GTU2S2))A = −A(H1 (ATF + FTA)) + (H2 (AFT + FAT ))A
where F = S2U T 2 GV1S1
Lemma 7 (Singular value dynamics). The singular values of the weight matricesW1 andW2 evolve by:
σ̇k1 = − ∑ k′ (vk ′ 2 T uk1)σ k′ 2 (u k′ 2 T Gvk1) (21)
σ̇k2 = − ∑ k′ (uk ′ 1 T vk2)σ k′ 1 (u k 2 T Gvk ′ 1 ) (22)
Proof. According to Lemma 4, σ̇r1 = u r 1 T Ẇ1vr1
Plugging in Eqn 8, we have
σ̇k1 = −uk1 T WT2 Gv k 1
= −uk1 T V2S2U T 2 Gv k 1 = − ∑ k′ (vk ′ 2 T uk1)σ k′ 2 (u k′ 2 T Gvk1)
Similar proof applies to Eqn 22.

heading: B DELAYED PROOFS

heading: B.1 PROOF OF LEMMA 1
text: The gradient on matrix W is
dL
dW = ∑ i ( ∂L ∂zi ∂zi ∂W + ∂L ∂z′i ∂z′i ∂W )
We denote the gradient on zi and z′i as gzi and gz′i , respectively. Since ∂zi ∂W = xi and ∂z′i ∂W = x ′ i, we get
Ẇ = −( dL dW )T = − ∑ i (gzix T i + gz′ix ′ i T )

heading: B.2 PROOF OF LEMMA 2
text: Proof. X is defined in Eqn 6. X = ∑ i ( ∑ j 6=i αij(x′i − xj) + ∑ j 6=i αji(xi − xj))xTi − ∑ i (1− αii)(x′i − xi)x′i T
= ∑ i ∑ j 6=i αijx′ix T i − ∑ i ∑ j 6=i αijxjxTi + ∑ i ∑ j 6=i αji(xi − xj)(xi − xj)T
+ ∑ i ∑ j 6=i αji(xi − xj)xTj − ∑ i (1− αii)(x′i − xi)(x′i − xi)T − ∑ i (1− αii)(x′i − xi)xiT
Given the fact that ∑ j 6=i αij = 1 − αii, we have ∑ i ∑ j 6=i αijx ′ ixTi = ∑ i(1 − αii)x′ixTi . Also,
since ∑
i ∑ j 6=i iterates all pairs of i, j, we can replace the index between i and j, we have∑
i ∑ j 6=i αijxjx T i = ∑ i ∑ j 6=i αjixix T j .
Therefore X = ∑ i ∑ j 6=i αji(xi − xj)(xi − xj)T − ∑ i (1− αii)(x′i − xi)(x′i − xi)T

heading: B.3 PROOF OF THEOREM 1
text: Proof. According to Lemma 1, we have
d dt W = WX (23)
For a fixed X , we solve this equation analyically,
W (t) = W (0) exp(Xt)
Apply eigen-decomposition on X , X = UΛUT . Then we have exp(Xt) = U exp(Λt)UT . Therefore,
W (t) = W (0)U exp(Λt)UT
BecauseX has negative eigenvalues, i.e., Λ has negative terms, we have for t→∞, exp(Λt) is rank deficient. Therefore, we know that W (∞) is also rank deficient, the weight matrix W has vanishing singular values.

heading: B.4 PROOF OF LEMMA 3
text: Proof. The gradient on matrix W2 is
dL
dW2 = ∑ i ( ∂L ∂zi ∂zi ∂W2 + ∂L ∂z′i ∂z′i ∂W2 ) (24)
We denote the gradient on zi and z′i as gzi and gz′i , respectively. Since ∂zi ∂W2 = W1xi and ∂z′i ∂W2 = W1x′i, we get
Ẇ2 = −( dL
dW2 )T = − ∑ i (gzix T i + gz′ix ′ i T )WT1 (25)
Similar proof applies to W1.

heading: B.5 PROOF OF THEOREM 2
text: Here, we prove that under the assumption that singular values are non-degenerate, the alignment matrix A = V T2 U1 converges to identity matrix.
Proof. According to Lemma 3, we have d
dt (W1W
T 1 ) = −W1GTW2 −WT2 GWT1
d dt (WT2 W2) = −WT2 GWT1 −W1GTW2
therefore, d
dt (W1W
T 1 −WT2 W2) = 0
or W1W T 1 −WT2 W2 = C
Next, we show that the Frobenius norm of each weight matrix grow to infinitely. d
dt ||W1||2F =
d dt tr(W1W T 1 ) = −tr(WT2 GWT1 )− tr(W1GT1W2)
According to Eqn 9, G = −W2W1X , we have −tr(WT2 GWT1 ) = tr(WT2 W2W1XWT1 )
= tr(W2W1XW T 1 W T 2 )
Because X is a positive definite matrix and for all t, W2(t)W1(t) 6= 0, we know B := W2W1XW T 1 W T 2 is positive semi-definite and B 6= 0. Therefore, tr(B) = ∑ k λk(B) > 0 since not all eigenvalues of B are zero.
Therefore, we know ||W1||2F → +∞ (similarly ||W2||2F → +∞). In the limit t− > +∞, we have W1W T 1 = W T 2 W2
Plug in the singular value decomposition of W1 and W2, we have U1S21U T 1 = V2S 2 2V T 2 . Assuming W1 and W2 have non-degenerate singular values, due to the uniqueness of eigen-decomposition, we have
U1 = V2 therefore,
V T2 U1 = I
Remark. Note that when the non-degenerate singular value assumption does not hold, the corresponding singular vectors are not unique and we will not observe the corresponding dimensions becoming aligned.

heading: B.6 PROOF OF THEOREM 3
text: Proof. According to Theorem 2, for σk1 and σ k 2 with same index, the corresponding singular vector pairs vk2 and uk1 will get aligned, i.e., vk ′ 2 T uk1 → δi,j . Therefore, Eqn 21 and Eqn 22 can be simplified to
σ̇k1 → −σk2 (uk2 T Gvk1) σ̇k2 → −σk1 (uk2 T Gvk1)
Insert Eqn 9 and considering the alignment, we derive
σ̇k1 → σk1 (σk2 )2(vk1 T Xvk1) σ̇k2 → σk2 (σk1 )2(vk1 T Xvk1)

heading: C EXPERIMENTS FOR DIMENSIONAL COLLAPSE CAUSED BY STRONG AUGMENTATIONS
text: Numerical simulation verifies our theory. We choice input data as isotropic Gaussian with covariance matrix ∑ i,j(xi − xj)(xi − xj)T /N = I . We set the augmentation as additive Gaussian with
covariance matrix equal to ∑
i(x ′ i−xi)(x′i−xi)T /N = block diagonal(0, k∗I). We plot the weight
matrix singular value spectrum in Figure 11 with various augmentation amplitude k. This proves that under linear network setting, strong augmentation leads to dimensional collapse in embedding space.
D IMPLEMENTATION DETAIL

heading: D.1 AUGMENTATIONS
text: Each input image is transformed twice to produce the two distorted views for contrastive loss. The image augmentation pipeline includes random cropping, resizing to 224x224, random horizontal flipping, color jittering, grayscale, Gaussian blurring, and solarization.

heading: D.2 NETWORK
text: Throughout the ImageNet experiments in this paper, we use a ResNet-50 (He et al., 2016) as an encoder. This network has an output of dimension 2048, which is called a representation vector.

heading: D.3 OPTIMIZATION
text: We use a LARS optimizer and train all models for 100 epochs. The batch size is 4096, which fits into 32 GPUs during training. The learning rate is 4.8 as in SimCLR (Chen et al., 2020a), which goes through a 10 epoch of warming up and then a cosine decay schedule.
E HYPERPARAMETER TUNING ON d0
Here, we list the ImageNet accuracy with various d0 value. It’s easy to see that when d0 → 0, there’s too little gradient information coming from the loss, the performance drops. When d0 → 2048, the model converges to standard SimCLR without a projector, which we know suffers from dimensional collapse in representation space.

heading: F ABLATION STUDY DETAIL
text: Fixed low-rank projector vs Fixed low-rank diagonal projector: DirectCLR is equivalent to SimCLR with a fixed low-rank diagoanl projector. It performs the same as a SimCLR with fixed low-rank projector, which achieves 62.3% linear probe accuracy. Specifically, the singular values of this low-rank matrix are set to have d0 numbers of 1 and 0 for the rest, then left- and right- multiply
a fixed orthogonal matrix. Therefore, their only difference is that this fixed projector has an extra fixed orthogonal matrix in between.
Trainable projector vs trainable diagonal projector: We trained a SimCLR model with a trainable projector that is constrained be diagonal. The model achieves 60.2% linear probe accuracy on ImageNet, which is close to a SimCLR with a 1-layer linear projector.
Orthogonal projector vs no projector: We train a single layer projector SimCLR model with orthogonal constraint using ExpM parametrization (Casado & Martı́nez-Rubio, 2019). Therefore, the projector weight matrix has all singular values fixed to be 1. This model reaches 52.2% accuracy on ImageNet which is close to a SimCLR without projector.
These ablation studies verify the propostion 1 that the SimCLR projector only needs to be diagonal. Also, according to Table 2, we find that low-rank projector setting consistently improves the performance, which verifies proposition 2.
Linear probe on subvector instead of the entire vector: For DirectCLR, we perform a linear probe only on the sub-vector z and get 47.9% accuracy on ImageNet. This shows that the rest of r still contains useful information even though it does not see gradient directly coming from the loss function.
Random dropout instead of fixed subvector: Since DirectCLR drops out a number of dimensions for the loss function, it would be natural to ask whether random dropping out can reach the same performance. We train a SimCLR model without a projector and randomly feed d0 number of features to InfoNCE loss every iteration. This model reaches only 43.0% accuracy on ImageNet. This demonstrates the importance of applying a fixed subvector, which allows the alignment effect to happen.

Review summaries:
This paper theoretically proves that the contrastive learning results in the dimensional collapse in the feature representation space. By analyzing the covariance matrix of the embedding space, the authors claims that it is specifically arise from the strong augmentation and the implicit regularization. To circumvent this issue, the authors propose a novel method of contrastive learning without using the projector at the end of an encoder.

This paper shows that contrastive methods also suffer from the “dimensionality collapse” phenomenon, a milder version of the “total collapse” problem that originally motivated the development of using explicit negative pairs to prevent learning trivial solutions. Two underlying causes for this problem, i.e too strong data augmentation and implicit low-rank regularization, are proved in simplified settings of shallow, linear networks. The dimensionality collapse problem is then related to the embedding projector component of SimCLR, and theoretical analyses are used to motivate an alternative training technique without the need for the embedding projector. 

This paper investigates the collapsing problem of contrastive learning. It attempts to attribute the collapsing phenomena to strong augmentation and implicit regularization, using simple linear network models. Based on the above analysis, the paper then propose a simple sub-vector based CL method called DirectCLR.

This paper firstly studies the dimensional collapse problem in existing contrastive learning (CL) methods. The authors provided both empirical and theoretical results to reveal that the popular SimCLR method may incur dimensional collapse by showing the low-rank property of the covariance. They thus propose a new framework called DirectCLR to solve the dimensional collapse in CL. Experiments on ImageNet demonstrate the effectiveness of the proposed method.