GULP: a prediction-based metric between representations
Authors: Enric Boix-Adserà
Hannah Lawrence
George Stepaniants
Philippe Rigollet
Venue: NeurIPS 2022 Accept (2022)
Keywords: representations
CCA
CKA
transfer learning
neural networks

Abstract: Comparing the representations learned by different neural networks has recently emerged as a key tool to understand various architectures and ultimately optimize them. In this work, we introduce GULP, a family of distance measures between representations that is explicitly motivated by  downstream predictive tasks. By construction, GULP provides uniform control over the difference in prediction performance between two representations, with respect to regularized linear prediction tasks. Moreover, it satisfies several desirable structural properties, such as the triangle inequality and invariance under orthogonal transformations, and thus lends itself to data embedding and visualization. We extensively evaluate GULP relative to other methods, and demonstrate that it correctly differentiates between architecture families, converges over the course of training, and captures generalization performance on downstream linear tasks. 

GULP: a prediction-based metric between
representations
Enric Boix-Adserà
Hannah Lawrence MIT hanlaw@mit.ed edu
George Stepaniants MIT gstepan@mit.cm edu
MIT eboix0mit. edu
Philippe Rigollet MIT rigollet@mathh.o mit. edu
Abstract
Comparing the representations learned by different neural networks has recently emerged key tool to understand various architectures and ultimately optimize them. In this work, we introduce GULP, family of distance measures between representations that is explicitly motivated by downstream predictive tasks. By construction, GULP provides uniform control over the difference in prediction performance between representations, with respect regularized linear pre- diction tasks. Moreover, it satisfies several desirable structural properties, such as the triangle inequality and invariance under orthogonal transformations, and thus lends itself to data embedding and visualization. We extensively evaluate GULP relative other methods, and demonstrate that it correctly differentiates between architecture families, converges over the course of training, and captures generalization performance on downstream linear tasks.
Introduction
The spectacular success of deep neural networks (DNN) witnessed over the past decade has been largely attributed to their ability to generate good representations of the data [BCV13]. But what makes representation good? Answering this question theory of DNN design. This fundamental question calls for a metric over representations as a basic primitive. Indeed, embedding representations into metric space enables comparison, modifications and ultimately optimization of DNN architectures [LTQ 18]; see Figure 1.
necessary step towards principled
In light of the practical impact of meaningful metric recently garnered significant attention, leading to a myriad of propositions such as CCA, CKA, and PROCRUSTES. Their relative pros and
representations, this question has
currently the subject of lively debate [DDS21, D2N+ whose resolution calls for a theoretically grounded notion of metric.
Our contributions. this work, we define a new family of metrics called GULP?, over the space of representations. Our construction rests functional notion of what makes two representations similar: namely, that two representations are similar if and only if they are equally useful as inputs to downstream, linear transfer learning tasks. This idca is partially inspired by feature-based transfer learning, in which simple models adapt pretrained representations, such as Inceptionv3 [SVI* 16],
"More specifically, define pseudo-metrics rather than metrics. However, these he readily tumedd into metric using metric identification. This amounts to allowing equivalence classes of representations. *GULP is Uniform Linear Probing.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Figure 1: (-SNE embedding of various pretrained DNN representations of the ImageNet [KSH12] dataset with GULP distance (A = 10-2), colored by architecture type (gray denotes architectures that do not belong to family). The embedding shows good clustering of various architectures (ResNets, EfficientNets etc.), indicating that GULP captures intrinsic aspects of representations shared within an architecture family.
CLIP 21], and ELMo [PNT+18] 18], for specific tasks [RASC14]; indeed, this a key use for pretrained representations. Moreover, our application of linear transfer learning is reminiscent of linear probes, which were introduced by [AB17] as a tool to compare internal layers of DNN in terms of prediction accuracy. Linear probes play a central role in the literature hidden representations. They have been used not only to study the information captured by hidden representations [RBH20), but also to themselves define desiderata of distances between representations [DDS21]. However, previous applications linear probing required hand-selecting the task on which prediction accuracy is measured, whereas our GULP distance provides uniform bound over all norm- corm-bounded tasks.
We establish various theoretical properties of the GULP pseudo-metric, including the triangle inequal- ity (Thm 2), sample complexity (Thm 3), and vanishing cases. In particular, we show that akin to the PROCRUSTES pseudo-metric, GULPP invariant under orthogonal transformations (Thm 1) and vanishes precisely when the two representations are related by orthogonal transformation (Thm 2).
In turn, we use GULP produce low-dimensional embeddings of various DNNs that provide new insights on the relationship between various architectures (Figures 1, 5, and 6). Moreover, in Figure 7, we showcase numerical experiment to demonstrate that the GULP distance between two independent networks decreases during training on the same dataset.
Related work. This contribution is part of a growing body of work that aims at providing tools to understand and quantify the metric space of representations [RGYSD17, MRB18, KNLH19, AB17, ALM17, CLR+ 18, LC00, LV15, LYC 15, LLL 19, Mig19, STHH17, WHG 18, DDS21, DHN+ 22, CKMK22J. Several of these measures, such as SVCCA [RGYSDI7] and PWCCA [MRB18] 18], based classical canonical correlation analysis (CCA) from multivariate analysis [And84]. More recently, centered kernel alignment CKA [CSTEK01, CMR12, KNLH19, DHN has emerged popular measure; see Section 2 for more details these methods. The orthogo- nal procrustes metric (PROCRUSTES) is a classical tool of shape analysis [DM16] to compute the distance between labelled point clouds. Though not conspicuous CaA-based methods in the context of DNN representations, it was recently presented under a favorable light in [DDS21].
Various desirable properties of a similarity measure between representations have been put forward. These include structural propertics such as invariance or cquivariance [LC00, KNLH19], as well as sanity checks such as specificity against random initialization [DDS21], for example. Such desiderata can serve as diagnostics for existing similarity measures, but fall short of providing concrete design guidelines.
Outline The rest of the paper proceeds as follows. Section lays out the derivation of GULP, as well as important theoretical properties: conditions under which it is zero, and limiting cases in terms of the regularization parameter X, demonstrating that interpolates between CCA and a version of CKA.A Section 3 establishes concentration results for the finite-sample version, justifying its use in

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practice. In Section 4 we validate GULP through extensive experiments'. Finally, we conclude in Section 5.
2 The GULP distance
As stated in the introduction, the goal of this paper is develop pseudo-metric over the space of representations of a given dataset. Unlike previous approaches, which work with finite datasets, we take statistical perspective and formulate the population version of our problem. We defer statistical questions arising from finite sample size to Section 3.
Let X E RRd be random input with distribution Px and let f IRd + R denote a representation map, such as a trained DNN. The random vector f(X) R the representation of X by We assume throughout that representation map is centered and normalized, so that [[ 0 and [| =
In particular, this normalization allows us identify (unnormalized) representation maps o, & that are related by (z) = aΦ(z) + b, Px-a.s. for a e R and Rd down to a single representation of X (after normalizing), which is a well-known requirement for distances between representations [KNLH19, See. 2.3].
We are now in a position to define the GULP distance between representations; the terminology "distance" is justified in Theorem 2. To that end, let Φ : Rd - R and v Rd R be two representation maps, where may differ from k. Let (X, Y) E R x IR be a random pair and let n)x)) E æ] denote the regression function of Y onto X. Moreover, for any À > O, let Bx denote the population ridge regression solution given by
Bg arg min E[(3 ((X) y) + A|lB11
8
and similarly for ? with respect to (-()). Since we use squared error, these only depend the distribution of Y through the regression function 1).
Definition 1. Fix X >>>>00 0. The GULP distance between representations ((X) and (X) is given by
dx(ó, = sup (B(3]Ø(X) (X
"
where the supremum is taken over all regression functions y) such that [|nll I 1.
The GULP distance measures the discrepancy between the prediction of an optimal ridge regression estimator based on and its counterpart based on vb, uniformly over all regression tasks. While this notion of distance is intuitive and motivated by a clear regression task, it is unclear how to compute it a priori. The next proposition provides equivalent formulation of GULP, which is amenable accurate and efficient estimation; see Section 3. It is based on the following covariance matrices:
Σ. = co)( [lo(X)
Σx = cov cov(bb(X)) =
(1)
We implicitly used the centering assumption in the above definition, and the normalization condition implies that the covariance matrices have unit trace. Throughout, we assume these matrices invertible, which without loss of generality by projecting onto the image of the representation map. We also define the regularized inverses:
= (Ee + ATa)
Σ=^= = (E. Ale)
as well as the cross-covariance matrices EΦe and Eps as follows:
Egs = =E((())((X) ITo
(2)
Proposition 1. Fix X 2 O. The GULP distance between representations (xX and r (X) satisfies
di v) tr(2 Eg Ee) + tr tr(2 AE - tr - Liors 2i -y:T 2i
(3)
Proof. See Appendix A.I.
OOr code is available hutps//ggiiullcoomssttpaaaaa///
3

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2.1 Structural properties
In this section, we show that GULP invariant under orthogonal transformations and that is valid metric on the space of representations. We begin by establishing a third characterization of GULP that useful for the purposes of this section; the proof can be found Appendix A.I.
Lemma 1. Fix À 2 The GULP distance dx(b, 1b) between the representations ((X) and (x( satisfies
(0,,0) B((()) x (x) "
where X' is an independent copy of X.
We are now in position to state our main structural results. We begin with key invariance result. Theorem 1. Fix x 0. The GULP distance dx v) between the representations 6(X) R and (X) R is invariant under orthogonal transformations: for any orthogonal transformations
U. R + R and V R + R,, it holds
dx(U o Φ, Vo = dd(ó, vb)
Proof. We slightly abuse notation by identifying any orthogonal transformation W to a matrix W such that W(x) W x. Note that for any representation map, we have Σwof WE, WT and
Wof = (WE,W* + -- = + AN)T = WE.
Hence, using Lemma 1, we get that
UO(X) (U o o, Vo ) = -
= =E((() F x(x') E x(')) =
where we used the fact that U = Is and TTV = It-
Next, we show that GULP satisfies the axioms of metric.
Theorem 2. Fix x O. The GULP distance dx(db 1b) satisfies the axioms of pseudometric, namely for all representation maps o, 3/,, sp, it holds
dx(o. Φ) = 0, dx(0, v dx(b, o), and dx(e, vb) S dx(o dx(p,
= 0 if and only if k = l and there exists an orthogonal transformation U such Moreover, dx(Ø, = that o(X) = (X) a.s.
Proof. Lemma provides an isometric embedding of representations + f(x)=>> f(X') into the Hilbert space 1,² (P2))) It readily yields that d, is a pseudometric. It remains to identify for which é, da it holds that d, (,, vb) = 0.
The "easy" direction follows from the invariance property of Theorem I: if de and v satisfy ((X) (X) almost surely, then dd(Ø, v) = (U 3) = 0. We sketch the proof of the other direction, and defer the full proof to Appendix A.2. Define o = (E, + AT) 1/2g and & (E + 1/2-1/2bb. By Lemma 1, the condition that d, (0, v/ = 0 is equivalent to x(X) x(X') (X)) ((x))&hi (X) almost surely over X, X'. So (Ø) v) = 0, then we can leverage classical fact that the Gram matrix of set of vectors determines the vectors up to an isometry [HJ12], to prove that there is an orthogonal
transformation U E R such that x(X) ( almost surely over X. Finally, via analyzing homogeneous Sylvester equation, this implies that ((X) Ub(()) almost surely.
Note that when À = 0, the conclusion of this theorem fails to hold: do still satisfies the axioms of pscudo-distance, but the cases for which do(ó, eb) are different. This point is illustrated in the next section where establish that dg is the CCA distance commonly employed in the literature.
2.2 Comparison with CCA, ridge-CCA, CKA, and PROCRUSTES
Throughout this section, assume that k L for simplicity.

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Ridge-CCA. Our distance is most closely related to ridge-CCA, introduced by [Vin76] as regu- larized version of Canonical Covariance Analysis (CCA) when the covariance matrices Eg or L are close to singular. More specifically, for any 2 0, define the matrix C^ := د 2 Leoi the ridge-CCA similarity measure is defined as Px-CCA = ir(C2). Hence, we readily see from Proposition that GULP and ridge-CCA describing the same geometry over representations. To this, recall that Lemma 1 provides an isometric embedding / + (X)) of representation maps into L (Pxx)) While GULP is the distance on this Hilbert space, ridge-CCA is the inner product.
Ridge-CCA was briefly considered in the seminal work [KNLH19] but discarded because of (i) its lack of interpretability and (ii) the absence of rule to select A. We argue that fact, our prediction-dric derivation of GULP gives a clear and compelling interpretation of this geometry (as well suggests several extensions; see Section 5). Moreover, we show that tunability of X is, in fact, a desirable feature that allows to represent the space of representations at various resolutions, giving various levels of information; for example, in Figure 6, higher À leads to a coarser clustering structure.
CCA. Due to the connection with ridge-CCA, our GULP distance is related to (unregularized) CCA when = 0. Specifically, defining C := Σ-IΣ Egp E-1 Expo. the mean-squared-CCA similarity measure is given by (see [Eat07, Def. 10.2]):
POCx (0, v) = tr(C)
E [
(xx) "
2k
where is an independent copy of X'; the last identity can be checked directly. From Lemma 1 can be seen that our GULP distance do v)) with X = 0 is a linear transformation of Peca-
It can be checked that Plecs takes values in [0, 1], which has led researchers to simply propose 1- PecA as dissimilarity measure. Interestingly, this choice turns out to produce valid (squared) metric, i.e., a dissimilarity measure that satisfies the triangle inequality. Indeed, we get that
1 2k
(Ø, v) = 1 - Paea (Ø, +b)
E [[K ((((()))( -
where K(u, v) u is the linear kernel over I& and Ø == E-1²% (where of and & are the whitened versions of sb and á respectively). These identities have two consequences: (i) we see from Lemma that deca corresponds to the GULP distance with À = 0 up to a scaling factor and (ii) deca is valid pseudometric on the space of representations, since we just exhibited an isometry T K (f(X), f(X') with (P()))). We show in Appendix A.2 that decs (Ø, v) 0 iff (X) = Xx(X) a.s. for some matrix A. Note that the invariance Peca to linear transformations was previously known and criticized in [KNLHI9] as arguably too strong.
CKA. In fact, thanks to the additional structure of the Hilbert space I?(P) the deca distance comes with inner product
(()),, T(D))ca == 2
This observation allows us to connect CCA with CKA, another measure of similarity between distribu- tions that is borrowed from classical literature kernel methods [CSTEK01, CMR12] and that was recently made popular by [KNLHI9]. Under our normalization assumptions, CKA is a measure of similarity given by
PCKA (0)
(E| (XX)))))))))))))XXXXXXX(X)))), (X))
(T(Φ), T(v) )oCA
cos (TT())
[[T()) [c
(
where ||T]|ecA = (T,T))c and L denotes the angle in the geometry induced by (, )ocA. In turn, dexa is chosen as de = 11 PCXA. which does not yield pseudometrie. This observation highlights two major differences between CCA and CKA: the first measures inner products and works with whitened representations, while the second measures angles and works with representations. As illustrated in the experimental section as well as in [DDS21], this additional whitening step appears to be detrimental to the overall qualities of this distance measure.
The fact that GULP with x = 0 recovers deca (i.e. df = 2kdeow) is illustrated in Figure 2. As shown, although GULP has roughly monotone relationship with CKA, they remain quite different.
5

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Figure 2: Empirical relationship between distances. Each point in the scatter-plot corresponds to pair of ImageNet representations; the r-coordinate is the GULP distance, and the y-coordinate is the CCA or CKA distance. Although CCA and GULP for À 0 related, their relationship not linear since the representations' dimensionalities differ. Although CKA and GULP are related for large A, their relationship not linear due the difference in normalization. Appendix B.2 contains more details and comparisons, including surprisingly strong correlation between GULP and PROCRUSTES for some values of X.
PROCRUSTES. The relationship between GULP and PROCRUSTES is not clean in the previous comparisons, but we include for completeness. In the limit of infinite samples, the Procrustes distance as derived by [Sch66] is
dpmennstes ur(Ee) ir(E.)) ir ((Enn) [Te)
Our normalization implies tr(E,)) tr( k. However, the term tr (E Ege) (which is
equal the nuclear norm [>>ممم1>)) is not directly comparable the preceding distances.
3 Plug-in estimation of GULP
In practice, the distribution Px of X is unknown, so cannot compute the population version of GULP exactly. Instead, we have access to sample XI.... Xn, lid Px. In all the experiments of this paper, approximate GULP with the following plug-in in estimator:
Ri v)
tr(E A E00 + ur A bg
2 trr - Eov - )
where
ES = Σ ((x))(()) E. = - (X))(()) and EΦp ((.))((X)) 1=1
1=1
are the empirical covariance and cross-covariance matrices, and
+ x and hー^= (É. + AI)-
the empirical inverse regularized covariance matrices. To justify use of the plug-in estimator, we prove concentration around the population GULP distance as n goes to infinity.
Theorem 3. Assume that [|? (x))2 < almost surely. Then, for any (0, 8> with probability at least 1 the plug in estimator di,,, satisfies
1 A3
log( (k: + 1)/8)
-
s
71
We defer the proof of this theorem to Appendix A.3. At high-level, we first show that the inverse regularized covariance matrices, (E. A/) and (E.g + A/)-1, are well-approximated in operator norm, the expectation of the plug-in estimator is close to the population distance. We then apply

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relative with trece
Saspreon diterce for 1pr1
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average reletvv error
K:00
4-1
D4
6904
Figure 3: Convergence of plug-in es- imator as n + oc. We plot relative error | dz, 1000000 1o00ol/df hapol/d,..oo 10000 aver- aged over pairs of ImageNet DNNs.
Figure 4: GULP captures generalization of linear predictors. We plot Spearman's 's p between the differences in predic- tions by X-regularized linear regression, and the different distances. Results are averaged over 10 trials.
McDiarmid's inequality to show that the plug-in estimator concentrates around its expectation. Note that the boundedness conditions on the representations are here simplify technical arguments by appealing simply to McDiarmid's inequality; these can be presumably be relaxed to weaker tail conditions at the cost of more involved arguments.
Figure 3 supports our theoretical result by showing convergence on pairs of networks on the ImageNet dataset. See Appendix B.3 for more details.
4 Experiments
We evaluate our distance in variety of empirical settings, comparing to CCA, CKA, the classical PROCRUSTES method from shape analysis, and variant of CCA known as projection-weighted CCA (PWCCA); see [DDS21, Sec. 2] for defnittons.
4.1
GULP captures generalization performance by linear predictors
The GULP distance is motivated by how differently linear predictors using the representations o and & generalize. In this section, we demonstrate that GULP indeed captures downstream generalization performance by linear predictors. We consider the representation maps 01,..., Óm given by 37 pretrained image classification architectures on the ImageNet dataset Px (see Appendix B.5). For each pair of representations, estimate the CKA, CCA, PWCCA, and GULP distances, using the plug-in estimators on 10,000 images, sufficient to guarantee good convergence (see Figure 3).
We then draw Yú ~ (0, 1) to each one. For each representation i [m]. we a X-regularized ceast-qqares linear regression to the training data {(X,, Yx) ke))) which gives a coefficient vector 8x.s. Finally, for each I < j S m, estimate the distance Ti Exx~x (
5,000 images from the dataset X XI,..., Xn Px, and assign random label
,x between the predictions with representations Φ; and Φj. by taking the empirical average over 3000 samples in a test set. In Figure 4, we plot Spearman's 's rank correlation between т and each of the distances GULP, CKA, CCA, PWCCA, viewed as vectors with (*") entries, for each pair of networks. Notice that for each X, the distance that attains the best correlation is the GULPP distance with that X. This indicates that while GULP is measure of distance that holds uniformly over prediction tasks, it. retains its meaning in the context of a single prediction task.
4.2 GULP distances cluster together networks with similar architectures
We are interested in how GULP can be used compare networks of different architectures trained on the same task. We begin by comparing fully-connected ReLU networks of widths ranging from 100 to 1,000 and depths ranging from to 10, trained on the MNIST handwritten digit database. Every architecture is retrained four times from different initializations (see Appendix 1. We input all MNIST training set images into each network, save their representations the final hidden layer, and compute CKA, PROCRUSTES, and GULP distances between all pairs of representations.

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OXA
Procrustes
GULP A 16-6
GULP A 1e-2
GULP A
100
Figure 5: Two dimensional MDS embedding plots of fully fully-connected ReLU networks colored by architecture width (top) and depth (bottom). Networks are fully-train fully-trained on MNIST and penultimate layer representations are constructed from 60,000 input train images.
Figure 5 shows Multi-Dimensional Scaling (MDS) embeddings of the distances between all MNIST networks, color coded by width and depth. For GULP with = 10-6, the networks are largely organized according rank of the feature matrix: networks of large width and small depth, ones whose representations have the largest rank, are the most different, as evidenced by the halo of points in the MDS plots. This outcome confirms that CCA simply measures rank [KNLH19]. However, for PROCRUSTES and GULP with À = 10-2, networks become clustered by their depth, as evidenced by the striations in the MDS embeddings (plot colored by depth). Furthermore, networks of the same depth look most similar at large widths, as shown by the red centerline in the MDS embedding (plot colored by width), implying that as width increases networks converge to a shared limiting representation. Finally, GULP with À = 1 closely resembles CKA and roughly organizes networks by depth. takeaway is that GULP with À = 1 resembles PROCRUSTES and CKA, and captures intrinsic characteristics such as width and depth.
Next, we show how distances between penultimate layer representations allow us cluster pretrained networks with more complex architectures and, in turn, draw comparisons between them. To that end, we study 37 state-of-the-an models on the ImageNet Object Localization Challenge, of which the four major groups are ResNets, Efficienteess, ConyNeXts, and MobileNets (see Appendix B.
We compute the baseline distances between every pair of representations using 10,000 training images, and visualize them using a swo-dimensional t-SNE embedding in Figure 6. Below each embedding plot we show the dendogram resulting from a hierarchical clustering of the networks based on their distances. As seen from the embeddings, when À increases, the GULP distance separates the ResNet architectures (blue) from the EfficientNet and ConvNeXt convolutional networks (orange and red). Compared to other distances, GULP with large À is able to more compactly cluster ResNets and convolutional networks separately. In Appendix B.5 further quantify the compactness of clusterings under each distance metric by computing the standard deviation of distances within each cluster.
4.3 Network representations converge in GULP distance during training
So far, we have used GULP to compare static networks taken as a blackbox representation maps. Now GULP to examine how representation maps evolve over the course of training. To that end, we independently train 16 R architectures on the CIFARIO dataset [KH+ 09] for 50 epochs. Figure tracks the distance (averaged over all network pairs) at each epoch.
As shown, other distances change very little or even briefly increase over the course of training. For GULP with small X, the previous sections have demonstrated that our distance captures fine- grained differences between representations; here too, accentuates differences representations mid-training (visible around epoch 25). However, as X increases, the GUL.P distance differentiates less between representations, and smoothly decreases over the course of training, thus indicating that it captures intrinsic properties of the representations rather than artifacts due to random seeds.
8

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PACCA
CGA
PEOCFUSTES
TIE SPRIANE
GULFA
CULP A 1
GULA-=
Figure 6: Embeddings of PWCCA, CKA, PROCRUSTES, and GULP distances between the last hidden layer representations of 36 pretrained ImageNet models (top) along with their hierarchical clusterings (bottom). All distance metrics separate ResNet architectures (brown dendogram leaves) from the rest of the ConvNeXt and EfficientNet architectures (pink dendogram leaves). GULP at À = 1 is the most effective distance at separating ResNets from the remaining architectures.
CCA
PWCCA
Linear CKA
GULP A=0.1
Average Normalizeed Pairwise
GULP 1=10
GULP 2=10.0
Epoch
Figure 7: Empirical distances between representations of 16 independently trained ResNet18 archi- tectures during training, computed using 3,000 samples and averaged over all (10) pairs. Distances are scaled by their average value at iteration 0.

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4.4 Sensitivity versus specificity of GULP
In Appendix B.9, we reproduce the experiments of [DDS21]. Our distance compares favorably to baselines and correlates with measures of a DNN's functional behavior. It achieves the specificity of CCA and PWCCA to random initializations, and improves the sensitivity of CKA to out-of outro--distriuutio performance.
5 Conclusion
In this paper, we have defined family of distances for comparing learned representations in terms of their worst-case performance gap over all X-regularized regression tasks. We proved convergence of the finite-sample estimator of this distance, quantified its relationship to existing notions such as CCA, ridge-CCA, and CKA, and demonstrated promising performance in a variety of empirical settings, including the ability to distinguish between network architectures and to capture performance differences on regression tasks.
Further studying extensions beyond linear transfer learning could provide a rich direction for future work. In fact, preliminary experiments reported in Appendix B.10 indicate that, compared to section 1, GULPP fails to predict generalization performance when the downstream task shifts from linear logistic prediction. This suggests extending GULP uniform bound over other downstream predictive tasks, such as logistic regression, multi-class classification, or kernel ridge regression. Although GULP under kernel ridge regression has closed form using the kernel trick* GULP for logistic regression does not have closed form. This brings additional computational questions of interest that are beyond the scope of this work. Finally, it could be interesting to consider the application of GULP to knowledge distillation, or alternatively to consider adding ridge regularization term to probing methods (inspired by GULP).
Acknowledgments and Disclosure of Funding
EB is supported by Apple AI/ML Fellowship, and the National Science Foundation Graduate Research Fellowship under Grant No. 1745302. HL is supported by the Fannie and John Hertz Foundation and the National Science Foundation Graduate Research Fellowship under Grant No. 1745302. GS is supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1745302. PR supported by NSF awards IIS- 183188 DMS- 2022448, and CCF-2106377.
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13


Reviews:
[{"review_id": "ksxBs3s-C9", "reviewer": "Reviewer_JZhG", "summary": "The submitted paper proposes GULP, a family of dissimilarity measures between representations. GULP is based on the squared error between predictions of a ridge regression solution on the two representations. Each value of the ridge regression parameter $\\lambda$ corresponds to a dissimilarity measure in this family. It is shown that the dissimilarity measures in this family satisfy properties of a pseudometric. Connections and differences to CKA and CCA are discussed in detail. Experiments study if similarity of representations in terms of GULP corresponds to generalization performance of linear regression and logistic regression on these representations, if similar architectures find similar representations, and if networks with the same architecture but different initializations find similar representations after training on the same task.", "questions": "See weaknesses section", "limitations": "See weaknesses section", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The provided metric has desirable properties of a dissimilarity measure of neural networks (invariance to rotation and isotropic scaling and capturing similarities in architecture) and the paper provides an unbiased estimator. The presentation is clear and related work is extensively discussed. I have not checked the proofs in the appendix but the math in the main paper is correct. I am voting for acceptance.\n\nThe following are my major comments:\n\n1. One of the claims is that GULP captures generalization performance of linear regression. In Section 4.1, this claim is tested only on images with random labels. I find this one experiment unconvincing and I recommend either improving this experiment or rescinding the claim. Generalization and other aspects of the model's performance can be completely different if a real-world task is used instead of zero-mean Gaussian labels. Unfortunately there are not many benchmarks for regression compared to classification, which might be why the authors chose this synthetic task. There are some well-known datasets on age estimation that can be used in this section.\n\n2. CKA provides the stated benefits of GULP other than satisfying pseudometric properties. A discussion on why it is important for a similarity measure of neural networks representations to satisfy these properties can help with motivating GULP. Are there currently any theoretical results or algorithms in this context that require these measures to be pseudometric?\n\n3. GULP is a family of dissimilarity measures. How similar two representations are under GULP depends on how the hyperparameter $\\lambda$ is tuned. This is not a critical problem to me. Similarity of representations is a loosely defined notion and different applications would call for different measures of similarity. Even CKA comes in different forms depending on its kernel (linear, RBF, etc.). For GULP, it is possible that a scalar value can be selected or constructed from this family on the specific application. I would suggest including more heuristic or theoretically grounded guidelines for choosing $\\lambda$ in specific applications. \n\nMinor comments:\n- Bringing PROCRUSTES in the main text helps with understanding the results\n- line 217: \"with $\\lambda=1$ resembles\"\n------------------------------------------------------\nUpdate: The revision provides a realistic experiment about generalization with solid results and addresses my main concern. The discussion on the benefits of a pseudometric in the response is also helpful and convincing. I increased the score to 7.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "__un6xUTVc4", "reviewer": "Reviewer_zDqK", "summary": "This paper introduces GULP (a pseudometric) for comparing the representations learned by two different neural networks. The core idea is to use a downstream task on the learned representations to derive the distance function. The performance on downstream tasks generally determines the usefulness of representation. So, I think it is a good idea to rely on a downstream task when comparing the representation space of two different networks.\n\nAuthors use ridge regression as a downstream task. Each sample is assigned a random target label drawn from the Normal distribution, and a linear ridge regression model is trained. The final distance is a difference in the regression estimate of two representations under all regression functions.\n\nThe authors present both theoretical and empirical results. The theoretical results derive the closed-form expression of distance and prove its structural properties. Authors use sample estimates for GULP and theoretically justify that the estimator is concentrated around its expectation. The empirical results show the benefits of GULP, such as it captures generalisation over linear predictors, can be used for comparing architectures, the evolution of distance over training, and the sensitivity analysis of networks.\n\nWhile the work is interesting, I think the critical element missing is the justification behind the choice of distance. The distance is closely related to ridge-CCA. I find it hard to understand the introduced distance's essential theoretical benefits over other alternatives.\n\nI am more concerned about the representation used for computing the distance. After the final layer of neural networks, representation retains class-specific information and throws away the rest. That is to say, in BCE loss, neural networks maximise MI(Z; Y), where Z is the representation and Y is the task. I do not see the usefulness in comparing two networks using a distance on their final layer representation for any different downstream task such as ridge regression. Perhaps comparing the representations learned by self-supervised/unsupervised methods will be more meaningful.\nOverall I think it is a good paper with sound theoretical results. On the practical side, there is a scope to improve the paper. Can GULP be extended for different downstream tasks? Also, some justification for using ridge regression as a downstream task. I give a score of 6. If the authors respond sufficiently to my comments, I will be happy to adjust my score.\n\n***Post Rebuttal***\n\nI appreciate the authors for their effort in replying to my comments. I find it as a good paper that could be more attractive with the right set of empirical results. I prefer to stay with my original assessment.\n", "questions": "In section 2, line 71, the representation map 'f' is centred to 0 mean. Next, the maps phi and xsi are introduced. It is unclear whether phi and xsi are two choices for 'f' also centred to 0 mean. If yes, then I think its wrong to say ‘xsi(x) = a phi(x) + b’ as this would imply E[xsi(x)] = b whereas the normalisation would mean E[xsi(x)] = 0.\n\nCan GULP be used for comparing the representation of networks trained on different datasets? It can be an interesting empirical result.", "limitations": "The authors provide some discussion on the limitations of their work. The main downside, in my view, is that GULP relies on the ridge regression task. If the network is trained with cross-entropy loss, it is not immediately obvious what are the benefits of evaluating it under a ridge regression task. Furthermore, downstream applications have other choices such as classification, clustering, etc. The authors mention that in conclusion, but it appears to be the main issue as much of the downstream tasks are not regression. They also say GULP has a closed-form solution under kernel ridge regression. It would be better if the authors showed empirical results with kernels.\n \nThe empirical evaluations use final layer representation would be interesting to investigate applications of GULP for comparing intermediate representations as well as its suitability for comparing networks trained on different datasets.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nTheoretically sound pseudometric for comparing representations of two architectures. The paper is well written, and the proofs are easy to follow.\nEmpirically demonstrate the use of GULP on applications such as grouping similar architectures, comparing the training dynamics, and the sensitivity analysis of architectures.\n\nWeaknesses:\n             \nThe analysis in section 4.2 relies on MNIST data. Given the complexity of features in MNIST digits, I wonder whether the information content of the features learned by various architectures differs significantly. A little more challenging data, such as cifar10, 100, etc., would be more realistic to consider. Authors later analyse architectures trained on the imagenet. However, I still think MNIST is not a good choice, even as a toy problem for comparing representation.\n\nThe empirical results can be better established. For instance, one can use GULP to compare representations of different layers of the same network to see how downstream performance improves with the depth or width of the network. Can potentially investigate its application to tasks such as knowledge distillation.\n\nI don't understand the motivation and practicality of using GULP for network convergence. In section 4.3, 16 resnet18 models are trained, and their distance is reported under GULP and other distances. It is not apparent what the authors mean by distances capture fine-grained differences. The same model should learn similar features under different initialisation for the same task. One could report mean and std in the loss for this purpose. What is an added gain to looking at \"fine-grained differences in features\". After all, the distance comes at an additional computational cost.\n\nMinor Comment: The choice of notation could have been better. It is easier if random variables are bold, likewise use a standard choice for scalar, vector, matrices, etc. The x-axis labels in dendrogram plots are too small, making them hard to interpret. It would be nice to plot these in higher resolution and move them to the appendix.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "DKDTD3jz-d", "reviewer": "Reviewer_d1FQ", "summary": "Main Points:\n- Provide a distance metric that captures the usefulness of a representation for the downstream prediction task\n- The GULP metric measures the discrepancy between a pair of representations obtained via ridge regression estimators.\n- GULP is invariant to orthogonal transformations, and is zero between two orthogonal representations\n", "questions": "Should you compare your method with probing methods in the NLP literature ? https://nlp.stanford.edu/~johnhew/interpreting-probes.html", "limitations": "Can’t find anything other than the ones mentioned by the authors.\n", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- Proposing metrics for evaluating embedding is an important area for progressing AI\n- Suitable theoretical analysis is provided for the properties of the metric\n- Suitable related works are reviewed and compared\n- Suitable experimental setups and results are provided\n\nWeaknesses\n- Various related works are reported in the introduction, however only 3 of these are compared in the experiments.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "2P8yrc5I5K_", "reviewer": "Reviewer_QQnx", "summary": "This paper proposes Uniform Linear Probing (GULP), a family of distance measures for the representations learned with neural networks, considering the representation performance. GULP satisfies the triangle inequality, shows a bounded sample complexity, and ignores the difference caused by orthogonal transformations. \n\nGULP uses the idea of linear probing. Different from the works in literature, which require hand-selected prediction tasks, GULP naturally considers the prediction of the learned network as the target and measures the difference between predictions of two learned neural networks given the same dataset.\n", "questions": "Please see **Strengths And Weaknesses**. One concern is related to the question above it, so I put them together.", "limitations": "Yes,  the authors addressed the limitations and potential negative societal impact.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper shows several **strengths**:\n\n1. The method proposed is new and contributes to the distance measure between representations; even the representations are trained to have different numbers of dimensions. \n\n2. There are theoretical results provided, suggesting the method is sound. The method shows a bounded sample complexity and stays sound when representations are different because one is an orthogonal transformation of the other, which is a common case in neural network training. \n\n3. There is an analysis of the sensitivity of the newly introduced parameter ($\\lambda$). Both empirical and theoretical evidence are provided.\n\n4. The submission is well organized.\n\nThough the topic studied by this paper is important and theoretical analysis has been provided to make the proposed method sound, I have the following **concerns and questions**.\n\n1. In the introduction, the paper claims that GULP considers the representation's performance on downstream prediction tasks. However, in Definition 1, which defines the GULP distance, the equation uses two linear transformations, $\\beta_{\\lambda}$ and $\\gamma_{\\lambda}$. According to the equation above Definition 1, $\\beta_{\\lambda}=argmin_{\\beta} \\mathbb{E}\\[ (\\beta^T \\phi(X) - Y)^2 \\] + \\lambda\\| \\beta \\|^2$, $\\beta$ is learned on the given dataset, which should come from the source task. Though the paper indicates the label $Y$ is assigned randomly in Section 4.1, the input space remains the same. I assumed $\\gamma$ is learned the same way but could output a representation on a different dimensional space. Downstream tasks could be different from the source task, including the input distribution. Thus the performance could be very different. Is the assumption the source task and downstream tasks remain the same input space? Otherwise, it is not clear why the learned $\\beta_{\\lambda}$ and $\\gamma_{\\lambda}$ can be predictive of the performance on downstream tasks.\n\n2. It is not clear to me how the centralization and normalization conditions are ensured during training in practice. The matrix $\\beta$ and $\\gamma$ are learned with an L2 constraint, which comes with a weight $\\lambda$. With a more strict constraint, i.e., a larger $\\lambda$, it seems like it would be harder to meet the request that the length of the learned representation is normalized to equal to 1. Assuming the ground truth $Y$ to be large and the norm of $\\beta$ is restricted, the value of the learned representation will tend to be large. Or, is the condition of tuning $\\lambda$ that finds the $\\lambda$ that the learned representation is closest to meeting the centralization and normalization assumption?\n\n3. There is also another concern related to the point above. Different $\\lambda$ could lead to very different distance measures (Figure 2). Therefore an inappropriate $\\lambda$ will affect the following evaluation of the distance between representations, thus affecting the empirical result of the proposed method.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept