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Similarity of Neural Network Representations Revisited

Simon KornblithMohammad NorouziHonglak LeeGeoffrey E. Hinton

article2019ICML2,552 citations

Establishes centered kernel alignment (CKA) as a reliable method for comparing hidden representations across differently initialized neural networks while exposing the mathematical limitations of canonical correlation analysis.

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Neural networks often produce internal representations that are difficult to compare directly, raising questions about whether models with the same architecture but different random starts learn similar features, whether layers can be matched across architectures, and how training data or design choices affect those features. These comparisons matter because they can reveal whether networks converge to consistent solutions or develop redundant layers, which in turn affects decisions about model scaling, reuse, and interpretability.

The article set out to identify a reliable scalar index for comparing neural-network representations and to test it on practical questions about layer correspondence, network width, depth, and training data.

The authors first analyzed the invariance properties required of any useful index, showing that full invariance to invertible linear transforms renders many existing measures uninformative once the number of neurons exceeds the number of examples. They then introduced centered kernel alignment (CKA), a normalized measure of similarity between the pairwise similarity matrices of two sets of activations. CKA is invariant to orthogonal transformation and isotropic scaling yet remains sensitive to the relative scale of important directions; it can be computed with either linear or RBF kernels and requires only modest numbers of examples.

When applied to families of convolutional and transformer networks trained on CIFAR-10 and CIFAR-100, CKA recovered the correct layer-to-layer mapping in more than 99 percent of cases across random initializations, whereas canonical correlation analysis, singular-vector CCA, projection-weighted CCA, and linear regression succeeded in fewer than half. CKA further showed that representations become steadily more similar as layer width increases, that similarity saturates earlier in shallow layers than in deep ones, and that early-layer representations remain aligned even when networks are trained on different datasets. In very deep plain networks, CKA detected large blocks of nearly identical representations that coincided with stalled improvements in classification accuracy.

These results indicate that CKA supplies a practical diagnostic for representational consistency, redundancy, and cross-architecture alignment—information that loss curves alone do not provide. Practitioners can therefore use the measure to decide when widening is likely to add value, when depth has become counterproductive, or whether a layer from one model can safely replace the corresponding layer in another.

The main limitations are that the study examined only image-classification and translation models on modest datasets, that the preferred linear kernel may miss structure captured by other kernels, and that CKA, like other summary statistics, collapses rich geometric relationships into a single number. Additional experiments on larger models, different modalities, and alternative kernels would strengthen before the method is treated as a default tool for architectural search or model auditing.

Googlecorrelation analysisdata pointslinear transformationsneural network representationsrepresentational similaritysimilarity matrices
arXiv: 1905.00414
Cover for Similarity of Neural Network Representations Revisited

Abstract

Recent work has sought to understand the behavior of neural networks by comparing representations between layers and between different trained models. We examine methods for comparing neural network representations based on canonical correlation analysis (CCA). We show that CCA belongs to a family of statistics for measuring multivariate similarity, but that neither CCA nor any other statistic that is invariant to invertible linear transformation can measure meaningful similarities between representations of higher dimension than the number of data points. We introduce a similarity index that measures the relationship between representational similarity matrices and does not suffer from this limitation. This similarity index is equivalent to centered kernel alignment (CKA) and is also closely connected to CCA. Unlike CCA, CKA can reliably identify correspondences between representations in networks trained from different initializations.

Topics

correlation analysisdata pointslinear transformationsneural networks

Keywords

neural network representationsrepresentational similaritysimilarity matriceskernel alignmentmultivariate similarityconvolutional networkstransformer encoderslinear kernels

Table of Contents

  • 1 Introduction
  • 2 What Should Similarity Be Invariant To?
  • 2.1 Invariance to Invertible Linear Transformation
  • 2.2 Invariance to Orthogonal Transformation
  • 2.3 Invariance to Isotropic Scaling
  • 3 Comparing Similarity Structures
  • 4 Related Similarity Indexes
  • 5 Linear CKA versus CCA and Regression
  • 6 Results
  • 6.1 A Sanity Check for Similarity Indexes
  • 6.2 Using CKA to Understand Network Architectures
  • 6.3 Similar Representations Across Datasets
  • 6.4 Analysis of the Shared Subspace
  • 7 Conclusion and Future Work
  • References
  • A Proof of Theorem 1
  • B Orthogonalization and Invariance to Invertible Linear Transformation
  • C CCA and Linear Regression
  • C.1 Linear Regression
  • C.2 CCA
  • C.3 Projection-Weighted CCA
  • D Notes on Other Methods
  • D.1 Canonical Ridge
  • D.2 The Orthogonal Procrustes Problem
  • E Architecture Details
  • F Additional Experiments
  • F.1 Sanity Check for Transformer Encoders
  • F.2 SVCCA at Alternative Thresholds
  • F.3 CKA at Initialization
  • F.4 Additional CKA Results
  • F.5 Similarity Between Different Architectures with Other Indexes

Knowls

  1. Knowl 1 — Centered Kernel Alignment for Neural Network Representations

    model/method

    Centered Kernel Alignment (CKA) measures the similarity between two neural network representations by comparing their inter-example similarity structures. Let X∈Rn×p1X \in \mathbb{R}^{n \times p_1} and Y∈Rn×p2Y \in \mathbb{R}^{n \times p_2} denote activation matrices of p1p_1 and p2p_2 neurons across nn examples. Let K,L∈Rn×nK, L \in \mathbb{R}^{n \times n} denote kernel similarity matrices with entries Kij=k(xi,xj)K_{ij} = k(x_i, x_j) and Lij=l(yi,yj)L_{ij} = l(y_i, y_j), where xi,yix_i, y_i are activation vectors for example ii.

    The empirical Hilbert-Schmidt Independence Criterion (HSIC) is defined as: HSIC(K,L)=1(n−1)2tr(KHLH)\text{HSIC}(K, L) = \frac{1}{(n - 1)^2} \text{tr}(KHLH) where H=In−1n1n1nTH = I_n - \frac{1}{n}\mathbf{1}_n\mathbf{1}_n^T is the centering matrix.

    Centered Kernel Alignment normalizes HSIC to achieve invariance to isotropic scaling: CKA(K,L)=HSIC(K,L)HSIC(K,K)HSIC(L,L)\text{CKA}(K, L) = \frac{\text{HSIC}(K, L)}{\sqrt{\text{HSIC}(K, K) \text{HSIC}(L, L)}}

    When using linear kernels K=XXTK = XX^T and L=YYTL = YY^T on column-centered matrices XX and YY, linear CKA simplifies to: CKA(XXT,YYT)=∥YTX∥F2∥XTX∥F∥YTY∥F\text{CKA}(XX^T, YY^T) = \frac{\|Y^T X\|_F^2}{\|X^T X\|_F \|Y^T Y\|_F}

    For the radial basis function (RBF) kernel k(xi,xj)=exp⁡(−∥xi−xj∥22/(2σ2))k(x_i, x_j) = \exp(-\|x_i - x_j\|_2^2 / (2\sigma^2)), setting the bandwidth parameter σ\sigma to a fixed fraction of the median inter-example Euclidean distance preserves invariance to isotropic scaling.

  2. Knowl 2 — Spectral Decomposition and Eigenvalue Weighting of Linear CKA

    theoretical result

    Let centered activation matrices have singular value decompositions X=UXΣXVXTX = U_X \Sigma_X V_X^T and Y=UYΣYVYTY = U_Y \Sigma_Y V_Y^T, where UX∈Rn×p1U_X \in \mathbb{R}^{n \times p_1} and UY∈Rn×p2U_Y \in \mathbb{R}^{n \times p_2} contain the left-singular vectors (eigenvectors of XXTXX^T and YYTYY^T), denoted uiXu_i^X and ujYu_j^Y. Let λiX=(ΣX,ii)2\lambda_i^X = (\Sigma_{X, ii})^2 and λjY=(ΣY,jj)2\lambda_j^Y = (\Sigma_{Y, jj})^2 denote the corresponding eigenvalues (squared singular values).

    Mean squared canonical correlation analysis (RCCA2R^2_{\text{CCA}}) for orthonormalized representations QX,QYQ_X, Q_Y with p1≤p2p_1 \le p_2 is: RCCA2=∥QYTQX∥F2p1=1p1∑i=1p1∑j=1p2⟨uiX,ujY⟩2R^2_{\text{CCA}} = \frac{\|Q_Y^T Q_X\|_F^2}{p_1} = \frac{1}{p_1} \sum_{i=1}^{p_1} \sum_{j=1}^{p_2} \langle u_i^X, u_j^Y \rangle^2

    In contrast, Linear CKA evaluates to: CKA(XXT,YYT)=∑i=1p1∑j=1p2λiXλjY⟨uiX,ujY⟩2∑i=1p1(λiX)2∑j=1p2(λjY)2\text{CKA}(XX^T, YY^T) = \frac{\sum_{i=1}^{p_1} \sum_{j=1}^{p_2} \lambda_i^X \lambda_j^Y \langle u_i^X, u_j^Y \rangle^2}{\sqrt{\sum_{i=1}^{p_1} (\lambda_i^X)^2} \sqrt{\sum_{j=1}^{p_2} (\lambda_j^Y)^2}}

    Linear CKA differs from RCCA2R^2_{\text{CCA}} by weighting each squared dot product between canonical eigenvectors ⟨uiX,ujY⟩2\langle u_i^X, u_j^Y \rangle^2 by the product of their corresponding eigenvalues λiXλjY\lambda_i^X \lambda_j^Y. Consequently, linear CKA emphasizes alignment along high-variance principal components while downweighting directions that explain negligible variance, without requiring explicit matrix truncation or decomposition.

  3. Knowl 3 — Triviality of Invertible Linear Invariant Similarity for Wide Representations

    theoretical result

    Let X,Y∈Rn×pX, Y \in \mathbb{R}^{n \times p} be representation matrices of pp features evaluated over nn data points. Suppose a scalar similarity index s(X,Z)s(X, Z) is invariant to invertible linear transformations in its first argument, such that s(X,Z)=s(XA,Z)s(X, Z) = s(XA, Z) for all arbitrary representations ZZ and all invertible matrices A∈Rp×pA \in \mathbb{R}^{p \times p} with rank(A)=p\text{rank}(A) = p.

    If rank(X)=rank(Y)=n\text{rank}(X) = \text{rank}(Y) = n, then: s(X,Z)=s(Y,Z)s(X, Z) = s(Y, Z)

    This holds because when representations have feature dimensionality p≥np \ge n with full rank nn, there exists an invertible transformation A∈Rp×pA \in \mathbb{R}^{p \times p} such that XA=YXA = Y. Consequently, any similarity index invariant to invertible linear transformations (such as unregularized Canonical Correlation Analysis) fails to distinguish between different representations whenever the number of neurons in a layer is greater than or equal to the dataset size.

  4. Knowl 4 — Layer Identification Sanity Check Accuracy across Similarity Indexes

    data/table

    A sanity check for representational similarity requires that for a pair of architecturally identical networks trained from different random initializations, each hidden layer in one network should have maximum similarity to the corresponding layer in the second network. Ten 10-layer convolutional neural networks based on All-CNN-C were trained independently on CIFAR-10. For every layer of each network, the corresponding layer in each other network was identified by maximum similarity, and accuracy was averaged across all network pairs (excluding logits layers).

    Similarity Index Layer Identification Accuracy (%)
    CCA (ρˉ\bar{\rho}) 1.4
    CCA (RCCA2R^2_{\text{CCA}}) 10.6
    SVCCA (ρˉ\bar{\rho}) 9.9
    SVCCA (RSVCCA2R^2_{\text{SVCCA}}) 15.1
    PWCCA 11.1
    Linear Regression (RLR2R^2_{\text{LR}}) 45.4
    Linear HSIC 22.2
    CKA (Linear) 99.3
    CKA (RBF, σ=0.2×median\sigma = 0.2 \times \text{median}) 80.6
    CKA (RBF, σ=0.4×median\sigma = 0.4 \times \text{median}) 99.1
    CKA (RBF, σ=0.8×median\sigma = 0.8 \times \text{median}) 99.3

    CCA, SVCCA, and PWCCA fail to reliably identify architecturally corresponding layers across random initializations, achieving accuracies ≤15.1%\le 15.1\%, whereas linear and RBF CKA achieve over 99%99\% accuracy.

  5. Knowl 5 — Canonical Ridge Regularization Interpolation between CCA, Linear Regression, and CKA

    theoretical result

    Canonical ridge regularized Canonical Correlation Analysis (CCA) optimizes the canonical correlation coefficients σi\sigma_i under regularization parameters κX,κY≥0\kappa_X, \kappa_Y \ge 0: σi=max⁡wiX,wiY(XwiX)T(YwiY)∥XwiX∥22+κX∥wiX∥22∥YwiY∥22+κY∥wiY∥22\sigma_i = \max_{w_i^X, w_i^Y} \frac{(Xw_i^X)^T (Yw_i^Y)}{\sqrt{\|Xw_i^X\|_2^2 + \kappa_X \|w_i^X\|_2^2} \sqrt{\|Yw_i^Y\|_2^2 + \kappa_Y \|w_i^Y\|_2^2}} subject to mutual orthogonality constraints (wiX)T(XTX+κXI)wjX=0(w_i^X)^T(X^TX + \kappa_X I)w_j^X = 0 and (wiY)T(YTY+κYI)wjY=0(w_i^Y)^T(Y^TY + \kappa_Y I)w_j^Y = 0 for all j<ij < i.

    Defining partially orthogonalized bases Q~X=UXΣX(ΣX2+κXI)−1/2\tilde{Q}_X = U_X \Sigma_X (\Sigma_X^2 + \kappa_X I)^{-1/2} and Q~Y=UYΣY(ΣY2+κYI)−1/2\tilde{Q}_Y = U_Y \Sigma_Y (\Sigma_Y^2 + \kappa_Y I)^{-1/2}, the normalized sum of squared singular values yields: CKA(Q~XQ~XT,Q~YQ~YT)=∑i=1p1∑j=1p2λiXλjY(λiX+κX)(λjY+κY)⟨uiX,ujY⟩2∑i=1p1(λiXλiX+κX)2∑j=1p2(λjYλjY+κY)2\text{CKA}(\tilde{Q}_X\tilde{Q}_X^T, \tilde{Q}_Y\tilde{Q}_Y^T) = \frac{\sum_{i=1}^{p_1} \sum_{j=1}^{p_2} \frac{\lambda_i^X \lambda_j^Y}{(\lambda_i^X + \kappa_X)(\lambda_j^Y + \kappa_Y)} \langle u_i^X, u_j^Y \rangle^2}{\sqrt{\sum_{i=1}^{p_1} \left(\frac{\lambda_i^X}{\lambda_i^X + \kappa_X}\right)^2} \sqrt{\sum_{j=1}^{p_2} \left(\frac{\lambda_j^Y}{\lambda_j^Y + \kappa_Y}\right)^2}}

    This normalized statistic interpolates between standard similarity measures:

    1. Setting κX=κY=0\kappa_X = \kappa_Y = 0 recovers RCCA2R^2_{\text{CCA}}.
    2. Setting κY=0\kappa_Y = 0 and taking κX→∞\kappa_X \to \infty (under von Neumann trace normalization) recovers the fraction of variance explained by linear regression, RLR2=∥QYTX∥F2∥X∥F2R^2_{\text{LR}} = \frac{\|Q_Y^T X\|_F^2}{\|X\|_F^2}.
    3. Setting κX=κY=κ\kappa_X = \kappa_Y = \kappa and taking κ→∞\kappa \to \infty recovers Linear CKA, CKA(XXT,YYT)\text{CKA}(XX^T, YY^T).

  6. Knowl 6 — Equivalence between Squared Projection-Weighted CCA and Linear Regression

    theoretical result

    Projection-weighted canonical correlation (PWCCA) computes similarity between activation matrices X∈Rn×p1X \in \mathbb{R}^{n \times p_1} and Y∈Rn×p2Y \in \mathbb{R}^{n \times p_2} (p1≤p2p_1 \le p_2) as ρˉPW=∑i=1p1αiρi∑i=1p1αi\bar{\rho}_{\text{PW}} = \frac{\sum_{i=1}^{p_1} \alpha_i \rho_i}{\sum_{i=1}^{p_1} \alpha_i} with weights αi=∑j∣⟨hi,xj⟩∣\alpha_i = \sum_j |\langle h_i, x_j \rangle|, where xjx_j is the jj-th column of XX and hi=XwiXh_i = X w_i^X is the ii-th canonical variable vector.

    Modifying PWCCA by squaring both the canonical correlation coefficients ρi\rho_i and the projection weights yields: RMPW2=∑i=1p1αi′ρi2∑i=1p1αi′,where αi′=∑j=1p1⟨hi,xj⟩2R^2_{\text{MPW}} = \frac{\sum_{i=1}^{p_1} \alpha'_i \rho_i^2}{\sum_{i=1}^{p_1} \alpha'_i}, \quad \text{where } \alpha'_i = \sum_{j=1}^{p_1} \langle h_i, x_j \rangle^2

    For column-centered activations XX and YY, this squared projection-weighted index is algebraically identical to the fraction of variance explained by linear regression of XX onto YY: RMPW2=∥QYTX∥F2∥X∥F2=RLR2R^2_{\text{MPW}} = \frac{\|Q_Y^T X\|_F^2}{\|X\|_F^2} = R^2_{\text{LR}} where QYQ_Y is any orthonormal basis for the column space of YY.

  7. Knowl 7 — Invariance Principles for Representational Similarity Indexes

    model/method

    Representational similarity indexes s(X,Y)s(X, Y) for feature activation matrices X∈Rn×p1X \in \mathbb{R}^{n \times p_1} and Y∈Rn×p2Y \in \mathbb{R}^{n \times p_2} require specific invariance properties:

    1. Orthogonal Transformation Invariance: s(XU,YV)=s(X,Y)s(XU, YV) = s(X, Y) for all full-rank orthonormal matrices U∈Rp1×p1U \in \mathbb{R}^{p_1 \times p_1} (UTU=IU^TU = I) and V∈Rp2×p2V \in \mathbb{R}^{p_2 \times p_2} (VTV=IV^TV = I). Orthogonal invariance accommodates neuron permutation symmetries and rotationally symmetric weight initializations while preserving pairwise Euclidean distances and scalar products between examples.
    2. Isotropic Scaling Invariance: s(αX,βY)=s(X,Y)s(\alpha X, \beta Y) = s(X, Y) for any positive scalars α,β>0\alpha, \beta > 0.
    3. Sensitivity to Non-isotropic Linear Transformations: Neural network training via gradient descent is sensitive to feature scaling and aligns representations along dominant eigenvectors. Similarity indexes must not be invariant to arbitrary invertible linear transformations or non-isotropic scaling, as doing so discards feature magnitude information that is consistent and meaningful across trained networks.

    Any orthogonal-invariant similarity index s(X,Y)s(X, Y) can be converted into an invertible linear transformation invariant index s′(X,Y)=s(QX,QY)s'(X, Y) = s(Q_X, Q_Y) by evaluating ss on the orthonormal bases QX,QYQ_X, Q_Y obtained from thin QR decompositions X=QXRXX = Q_X R_X and Y=QYRYY = Q_Y R_Y.

  8. Knowl 8 — Pathology Detection in Overly Deep Plain CNNs versus ResNets

    empirical result

    Linear CKA identifies representation degradation when network depth increases without residual connections. In plain convolutional networks where layer depth is scaled (1×,2×,4×,8×1\times, 2\times, 4\times, 8\times depth):

    • Doubling network depth improves CIFAR-10 test accuracy from 94.1%94.1\% (10 layers) to 95.0%95.0\% (18 layers), but 4×4\times and 8×8\times multipliers degrade performance to 93.2%93.2\% (34 layers) and 91.9%91.9\% (66 layers).
    • In the 8×8\times plain architecture (Plain-66), layer-to-layer linear CKA shows that representations across the entire second half of the network become nearly identical to each other and to the final layer (CKA ≈1.0\approx 1.0), demonstrating representational saturation where additional layers fail to refine features.
    • Layer-wise linear classification probing (training an ℓ2\ell_2-regularized logistic regression classifier on each hidden layer) mirrors CKA: classification accuracy plateaus less than halfway through the 66-layer plain network.
    • In contrast, deep residual networks (ResNet-62) show no representation collapse, exhibiting a characteristic grid pattern where post-residual activations are similar across blocks at the same spatial resolution, while activations within residual blocks remain distinct.

  9. Knowl 9 — Representational Convergence and Layer Saturation with Network Width

    empirical result

    Evaluating convolutional networks with widths ranging from n=4n=4 to n=4096n=4096 channels trained independently from different random initializations on CIFAR-10 demonstrates two phenomena:

    1. Width Increases Representational Similarity: As network width increases, representational similarity between independently initialized networks monotonically increases, with linear CKA approaching 1.01.0 across all layers for very wide networks.
    2. Early Layer Saturation: Earlier convolutional layers (e.g., layers 1 and 2) saturate in similarity at much narrower channel widths (e.g., 64 to 256 channels) than later convolutional layers (layers 6 to 8), which require significantly wider layers to achieve high CKA across seeds.

    Furthermore, networks of a given width are generally more similar to independently initialized networks of the same width than they are to much wider models.

  10. Knowl 10 — Cross-Dataset Layer Similarity and Impact of Initialization

    empirical result

    Comparing 10-layer convolutional networks trained on CIFAR-10 versus networks trained on CIFAR-100 reveals that:

    1. Early layers (layers 1 through 4) learn representations that are highly similar across the two datasets, exhibiting linear CKA values comparable to models trained on the same dataset (CKA >0.8> 0.8).
    2. Later layers diverge markedly between CIFAR-10 and CIFAR-100 models, capturing dataset-specific label structure and higher-level semantics.
    3. This cross-dataset representational similarity is learned rather than an architectural inductive bias: untrained networks at random initialization exhibit substantially lower similarity to trained networks (CKA <0.4< 0.4) and to other random initializations.

  11. Knowl 11 — Transformer Encoder Representational Sublayer Structure

    empirical result

    Evaluating representations across 12-sublayer Transformer encoders trained on the WMT18 English-to-German translation task demonstrates that:

    1. Representational similarity displays an alternating checkerboard pattern: self-attention sublayer representations are more similar to other self-attention sublayers than to feed-forward network (FFN) sublayers, and FFN sublayer representations are similarly more aligned with other FFN sublayers.
    2. CKA reveals this alternating sublayer structure both after the self-attention/FFN transformation and immediately after channel-wise scaling operations.
    3. CCA fails to detect the alternating representational structure present after channel-wise scaling because CCA is invariant to non-isotropic scaling, causing representations before and after channel-wise scaling to appear identical under CCA.

  12. Knowl 12 — Dimensionality and Alignment of the Shared Subspace

    empirical result

    Analyzing the geometric action of the representational similarity matrix (RSM) YYTYY^T of network BB on the eigenvectors uiXu_i^X of RSM XXTXX^T of network AA (both trained from different initializations on CIFAR-10) demonstrates:

    1. The representational similarity captured by CKA is primarily driven by alignment along the eigenvectors corresponding to the largest eigenvalues of XXTXX^T and YYTYY^T.
    2. The rank of the subspace where YYTYY^T and XXTXX^T align (i.e., where YYTuiXYY^T u_i^X exhibits high norm and high cosine similarity with uiXu_i^X) is substantially lower than the total feature dimensionality of the activations.
    3. In the penultimate global average pooling layer (64 total units), the effective dimensionality of the shared aligned subspace between distinct random initializations is approximately 10, corresponding directly to the 10 target classes of CIFAR-10.

Coverage note — None was omitted; all key theoretical derivations, invariance proofs, unified framework bounds, and experimental findings across CNNs, ResNets, and Transformers are represented.

References

  1. 1.Advani, M. S. and Saxe, A. M. High-dimensional dynamics of generalization error in neural networks. arXiv preprint arXiv:1710.03667, 2017.
  2. 2.Amari, S.-i., Ozeki, T., Karakida, R., Yoshida, Y., and Okada, M. Dynamics of learning in MLP: Natural gradient and singularity revisited. Neural Computation, 30(1): 1–33, 2018.
  3. 3.Björck, Å. and Golub, G. H. Numerical methods for computing angles between linear subspaces. Mathematics of Computation, 27(123):579–594, 1973.
  4. 4.Bojar, O., Federmann, C., Fishel, M., Graham, Y., Haddow, B., Huck, M., Koehn, P., and Monz, C. Findings of the 2018 Conference on Machine Translation (WMT18). In EMNLP 2018 Third Conference on Machine Translation (WMT18), 2018.
  5. 5.Chen, A. M., Lu, H.-m., and Hecht-Nielsen, R. On the geometry of feedforward neural network error surfaces. Neural Computation, 5(6):910–927, 1993.
  6. 6.Connolly, A. C., Guntupalli, J. S., Gors, J., Hanke, M., Halchenko, Y. O., Wu, Y.-C., Abdi, H., and Haxby, J. V. The representation of biological classes in the human brain. Journal of Neuroscience, 32(8):2608–2618, 2012.
  7. 7.Cortes, C., Mohri, M., and Rostamizadeh, A. Algorithms for learning kernels based on centered alignment. Journal of Machine Learning Research, 13(Mar):795–828, 2012.
  8. 8.Cristianini, N., Shawe-Taylor, J., Elisseeff, A., and Kandola, J. S. On kernel-target alignment. In Advances in Neural Information Processing Systems, 2002.
  9. 9.Dinh, L., Sohl-Dickstein, J., and Bengio, S. Density estimation using real NVP. In International Conference on Learning Representations, 2017.
  10. 10.Dumoulin, V., Shlens, J., and Kudlur, M. A learned representation for artistic style. In International Conference on Learning Representations, 2017.
  11. 11.Edelman, S. Representation is representation of similarities. Behavioral and Brain Sciences, 21(4):449–467, 1998.
  12. 12.Elsayed, G. F., Lara, A. H., Kaufman, M. T., Churchland, M. M., and Cunningham, J. P. Reorganization between preparatory and movement population responses in motor cortex. Nature Communications, 7:13239, 2016.
  13. 13.Freiwald, W. A. and Tsao, D. Y. Functional compartmentalization and viewpoint generalization within the macaque face-processing system. Science, 330(6005):845–851, 2010.
  14. 14.Garriga-Alonso, A., Rasmussen, C. E., and Aitchison, L. Deep convolutional networks as shallow Gaussian processes. In International Conference on Learning Representations, 2019.
  15. 15.Gatys, L. A., Ecker, A. S., and Bethge, M. Image style transfer using convolutional neural networks. In IEEE Conference on Computer Vision and Pattern Recognition, 2016.
  16. 16.Golub, G. H. and Zha, H. The canonical correlations of matrix pairs and their numerical computation. In Linear Algebra for Signal Processing, pp. 27–49. Springer, 1995.
  17. 17.Gretton, A., Bousquet, O., Smola, A., and Schölkopf, B. Measuring statistical dependence with Hilbert-Schmidt norms. In International Conference on Algorithmic Learning Theory, 2005.
  18. 18.Haxby, J. V., Gobbini, M. I., Furey, M. L., Ishai, A., Schouten, J. L., and Pietrini, P. Distributed and overlapping representations of faces and objects in ventral temporal cortex. Science, 293(5539):2425–2430, 2001.
  19. 19.He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learning for image recognition. In IEEE Conference on Computer Vision and Pattern Recognition, 2016.
  20. 20.Ioffe, S. and Szegedy, C. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning, 2015.
  21. 21.Jacobsen, J.-H., Smeulders, A. W., and Oyallon, E. i-RevNet: Deep invertible networks. In International Conference on Learning Representations, 2018.
  22. 22.Jacot, A., Gabriel, F., and Hongler, C. Neural tangent kernel: Convergence and generalization in neural networks. In Advances in Neural Information Processing Systems, 2018.
  23. 23.Johnson, J., Alahi, A., and Fei-Fei, L. Perceptual losses for real-time style transfer and super-resolution. In European Conference on Computer Vision, 2016.
  24. 24.Khaligh-Razavi, S.-M. and Kriegeskorte, N. Deep supervised, but not unsupervised, models may explain it cortical representation. PLoS Computational Biology, 10(11): e1003915, 2014.
  25. 25.Kriegeskorte, N., Mur, M., and Bandettini, P. A. Representational similarity analysis-connecting the branches of systems neuroscience. Frontiers in Systems Neuroscience, 2:4, 2008a.
  26. 26.Kriegeskorte, N., Mur, M., Ruff, D. A., Kiani, R., Bodurka, J., Esteky, H., Tanaka, K., and Bandettini, P. A. Matching categorical object representations in inferior temporal cortex of man and monkey. Neuron, 60(6):1126–1141, 2008b.
  27. 27.Kuss, M. and Graepel, T. The geometry of kernel canonical correlation analysis. Technical report, Max Planck Institute for Biological Cybernetics, 2003.
  28. 28.Laakso, A. and Cottrell, G. Content and cluster analysis: assessing representational similarity in neural systems. Philosophical Psychology, 13(1):47–76, 2000.
  29. 29.LeCun, Y., Kanter, I., and Solla, S. A. Second order properties of error surfaces: Learning time and generalization. In Advances in Neural Information Processing Systems, 1991.
  30. 30.Lee, J., Sohl-dickstein, J., Pennington, J., Novak, R., Schoenholz, S., and Bahri, Y. Deep neural networks as gaussian processes. In International Conference on Learning Representations, 2018.
  31. 31.Li, Y., Yosinski, J., Clune, J., Lipson, H., and Hopcroft, J. Convergent learning: Do different neural networks learn the same representations? In NIPS 2015 Workshop on Feature Extraction: Modern Questions and Challenges, 2015.
  32. 32.Lorenzo-Seva, U. and Ten Berge, J. M. Tucker’s congruence coefficient as a meaningful index of factor similarity. Methodology, 2(2):57–64, 2006.
  33. 33.Morcos, A., Raghu, M., and Bengio, S. Insights on representational similarity in neural networks with canonical correlation. In Advances in Neural Information Processing Systems, 2018.
  34. 34.Mroueh, Y., Marcheret, E., and Goel, V. Asymmetrically weighted CCA and hierarchical kernel sentence embedding for multimodal retrieval. arXiv preprint arXiv:1511.06267, 2015.
  35. 35.Novak, R., Xiao, L., Bahri, Y., Lee, J., Yang, G., Abolafia, D. A., Pennington, J., and Sohl-dickstein, J. Bayesian deep convolutional networks with many channels are Gaussian processes. In International Conference on Learning Representations, 2019.
  36. 36.Orhan, E. and Pitkow, X. Skip connections eliminate singularities. In International Conference on Learning Representations, 2018.
  37. 37.Press, W. H. Canonical correlation clarified by singular value decomposition, 2011. URL http://numerical.recipes/whp/notes/CanonCorrBySVD.pdf.
  38. 38.Raghu, M., Gilmer, J., Yosinski, J., and Sohl-Dickstein, J. SVCCA: Singular vector canonical correlation analysis for deep learning dynamics and interpretability. In Advances in Neural Information Processing Systems, 2017.
  39. 39.Ramsay, J., ten Berge, J., and Styan, G. Matrix correlation. Psychometrika, 49(3):403–423, 1984.
  40. 40.Robert, P. and Escoufier, Y. A unifying tool for linear multivariate statistical methods: the RV-coefficient. Applied Statistics, 25(3):257–265, 1976.
  41. 41.Romero, A., Ballas, N., Kahou, S. E., Chassang, A., Gatta, C., and Bengio, Y. FitNets: Hints for thin deep nets. In International Conference on Learning Representations, 2015.
  42. 42.SAS Institute. Introduction to Regression Procedures. 2015. URL https://support.sas.com/documentation/onlinedoc/stat/141/introreg.pdf.
  43. 43.Saxe, A. M., McClelland, J. L., and Ganguli, S. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. In International Conference on Learning Representations, 2014.
  44. 44.Sejdinovic, D., Sriperumbudur, B., Gretton, A., and Fukumizu, K. Equivalence of distance-based and RKHS-based statistics in hypothesis testing. The Annals of Statistics, pp. 2263–2291, 2013.
  45. 45.Smith, S. L., Turban, D. H., Hamblin, S., and Hammerla, N. Y. Offline bilingual word vectors, orthogonal transformations and the inverted softmax. In International Conference on Learning Representations, 2017.
  46. 46.Song, L., Smola, A., Gretton, A., Borgwardt, K. M., and Bedo, J. Supervised feature selection via dependence estimation. In International Conference on Machine learning, 2007.
  47. 47.Springenberg, J. T., Dosovitskiy, A., Brox, T., and Riedmiller, M. Striving for simplicity: The all convolutional net. In International Conference on Learning Representations Workshop, 2015.
  48. 48.StataCorp. Stata Multivariate Statistics Reference Manual. 2015. URL https://www.stata.com/manuals14/mv.pdf.
  49. 49.Sussillo, D., Churchland, M. M., Kaufman, M. T., and Shenoy, K. V. A neural network that finds a naturalistic solution for the production of muscle activity. Nature Neuroscience, 18(7):1025, 2015.
  50. 50.Tucker, L. R. A method for synthesis of factor analysis studies. Technical report, Educational Testing Service, Princeton, NJ, 1951.
  51. 51.Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, Ł., and Polosukhin, I. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 5998–6008, 2017.
  52. 52.Vaswani, A., Bengio, S., Brevdo, E., Chollet, F., Gomez, A. N., Gouws, S., Jones, L., Kaiser, Ł., Kalchbrenner, N., Parmar, N., et al. Tensor2tensor for neural machine translation. arXiv preprint arXiv:1803.07416, 2018.
  53. 53.Vinod, H. D. Canonical ridge and econometrics of joint production. Journal of Econometrics, 4(2):147–166, 1976.
  54. 54.Wang, L., Hu, L., Gu, J., Wu, Y., Hu, Z., He, K., and Hopcroft, J. E. Towards understanding learning representations: To what extent do different neural networks learn the same representation. In Advances in Neural Information Processing Systems, 2018.
  55. 55.Yamins, D. L., Hong, H., Cadieu, C. F., Solomon, E. A., Seibert, D., and DiCarlo, J. J. Performance-optimized hierarchical models predict neural responses in higher visual cortex. Proceedings of the National Academy of Sciences, 111(23):8619–8624, 2014.
  56. 56.Zagoruyko, S. and Komodakis, N. Wide residual networks. In British Machine Vision Conference, 2016.

Citation

MLA
Kornblith, S., et al. “Similarity of Neural Network Representations Revisited”. arXiv, 2019, https://doi.org/10.48550/arxiv.1905.00414.
APA
Kornblith, S., Norouzi, M., Lee, H., & Hinton, G. (2019). Similarity of Neural Network Representations Revisited. arXiv. https://doi.org/10.48550/arxiv.1905.00414
Chicago
Kornblith, S., M. Norouzi, H. Lee, and G. Hinton. 2019. “Similarity of Neural Network Representations Revisited”. Preprint, ArXiv. https://doi.org/10.48550/arxiv.1905.00414.
Harvard
Kornblith, S. et al. (2019) “Similarity of Neural Network Representations Revisited”. arXiv. Available at: https://doi.org/10.48550/arxiv.1905.00414.
Vancouver
1. Kornblith S, Norouzi M, Lee H, Hinton G (2019) Similarity of Neural Network Representations Revisited. https://doi.org/10.48550/arxiv.1905.00414

BibTeX

@misc{https://doi.org/10.48550/arxiv.1905.00414,
  doi = {10.48550/ARXIV.1905.00414},
  url = {https://arxiv.org/abs/1905.00414},
  author = {Kornblith, Simon and Norouzi, Mohammad and Lee, Honglak and Hinton, Geoffrey},
  keywords = {Machine Learning (cs.LG), Neurons and Cognition (q-bio.NC), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Biological sciences, FOS: Biological sciences},
  title = {Similarity of Neural Network Representations Revisited},
  publisher = {arXiv},
  year = {2019},
  copyright = {arXiv.org perpetual, non-exclusive license}
}
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