What Representational Similarity Measures Imply about Decodable Information

Authors: sarah e harvey
david  lipshutz
alex h williams
marco  fumero
clementine  domine
zorah  lähner
donato  crisostomi
luca  moschella
kimberly  stachenfeld
Venue: NeurIPS_cc_2024_Workshop_UniReps
Published: 2024-09-24 05:49:57

Abstract: Neural responses encode information that is useful for a variety of downstream tasks. A common approach to understand these systems is to build regression models or “decoders” that reconstruct features of the stimulus from neural responses. Here, we investigate how to leverage this perspective to quantify the similarity of different neural systems. This is distinct from typical motivations behind neural network similarity measures like centered kernel alignment (CKA), canonical correlation analysis (CCA), and Procrustes shape distance, which highlight geometric intuition and invariances to orthogonal or affine transformations. We show that CKA, CCA, and other measures can be equivalently motivated from similarity in decoding patterns. Specifically, these measures quantify the average alignment between optimal linear readouts across a distribution of decoding tasks. We also show that the Procrustes shape distance upper bounds the distance between optimal linear readouts and that the converse holds for representations with low participation ratio. Overall, our work demonstrates a tight link between the geometry of neural representations and the ability to linearly decode information. This perspective suggests new ways of measuring similarity between neural systems and also provides novel, unifying interpretations of existing measures.

Main paper: 
 introduction: 
 The computational neuroscience and machine learning communities have developed a multitude of methods to quantify similarity in population-level activity patterns across neural systems. Indeed, a recent review by Klabunde et al. [20] catalogues over thirty approaches to quantifying representational similarity. Many of these measures quantify similarity in the shape or representational geometry of point clouds. For example, recent papers (e.g. [36, 9] ) leverage the Procrustes distance and other concepts from shape theory-an established body of work that formalizes the notion of shape and ways to measure distance between shapes [18, 10] . Other measures of representational similarity are not quite this explicit but still emphasize desired geometric invariance properties. For example, work by Kornblith et al. [22] popularized centered kernel alignment (CKA) by emphasizing its invariance to isotropic scaling, translation, and orthogonal transformations. These are precisely the same invariances specified by classical shape theory [18, 10] . Earlier work by Raghu et al. [29] argued for a more flexible invariance to affine transformations, and they used canonical correlations analysis (CCA) for this purpose. Contemporary work in neuroscience is replete with similar geometric reasoning and quantitative frameworks [25] .Here we ask: What does the similarity between neural representations, potentially in terms of shape or geometry, imply about the similarity between functions performed by those neural systems? Potentially, not very much. Maheswaranathan et al. [27] showed that recurrent neural networks with different architectures performed the same task with similar dynamical algorithms, but with different representational geometry. More recently, Lampinen et al. [26] showed that representational geometry can be biased by other nuisance variables, such as the order in which multiple tasks are learned. Similar limitations of representational geometry measures are discussed in [11, 8] .On the other hand, several basic observations suggest that geometry and function ought to be related. Consider, for example, the common practice of using linear models to decode task-relevant variables from neural population activity [1, 24] . The premise behind these analyses is that anything linearly decodable from system A is readily accessible to layers or brain regions immediately downstream of A. Therefore, information that is linearly decodable from A may relate to functional role played by A within the overall system [4] . Notably, the invariances of typical representational similarity measurestranslations, isotropic scalings, and orthogonal transformations-closely coincide with the set of transformations that leave decoding accuracy unchanged. For example, translations and rotations of neural population activity will not impact the accuracy of a linear decoder with an intercept term and an L 2 penalty on the weights. Thus, if we accept the premise that decoding accuracy is a proxy-even, perhaps, an imperfect proxy-for neural system function, then representational geometry may be a reasonable framework to capture something about this function [23] .We provide a unifying theoretical framework that connects several existing methods of measuring representational similarity in terms of linear decoding statistics. Specifically, we show that popular methods such as centered kernel alignment (CKA) [22] and similarity based on canonical correlations analysis (CCA) [29] can be interpreted as alignment scores between optimal linear decoders with particular weight regularizations. Additionally, we study how the shape of neural representations is related to decoding by deriving bounds that relate the average decoding distance and the Procrustes distance. We find that the Procrustes distance is a more strict notion of geometric dissimilarity than the expected distance between optimal decoder readouts, in that a small value of the Procrustes distance implies a small expected distance between optimal decoder readouts but the converse is not necessarily true. This formalizes recent observations by Cloos et al. [5] , who found that high Procrustes similarity implied high CKA similarity in practical settings. 
theoretical framework for linear decoding: 
 We use X ∈ R M ×N X and Y ∈ R M ×N Y to denote matrices holding responses to M stimulus conditions measured across neural populations consisting of N X and N Y neurons, respectively. Throughout this paper we will assume that the neural responses have been preprocessed so that the columns of X and Y have zero mean.We first consider the problem of predicting (or "decoding") a target vector z ∈ R M from neural population responses by a linear function X → Xw where w ∈ R N X . One can view this as a simplified neural circuit model where each element of ẑ = Xw + ϵ is the firing rate readout unit the M conditions. Here, we interpret w as a vector of synaptic weights and the random vector ϵ represents potential noise corruptions; for example, ϵ i ∼ N (0, σ 2 ) where σ 2 > 0 specifies the scale of noise. This neural circuit interpretation of linear decoding is fairly standard within the literature [13, 17] .Within this setting, we formalize the problem of decoding z from the population activity X through the following class of optimization problems: where G(•) is a function mapping R M ×N onto symmetric positive definite N × N matrices. The objective is concave, and equating the gradient to zero yields a closed form solution:EQUATIONEQUATIONTo appreciate the relevance of eq. ( 1), note that if G(X) = C X + λI, where C X := 1 M X ⊤ X is the empirical covariance of X, then the solution w * coincides with the optimum of the familiar ridge regression problem: minimizeEQUATIONwith solution w * = 1 M (C X + λI) -1 X ⊤ z, where λ > 0 specifies the strength of regularization on the coefficients. More generally, we can interpret eq. ( 1) as maximizing the inner product between the target, z, and the linear readout, Xw, plus a penalty term on the scale of w as quantified by the function w → w ⊤ G(X)w, which is a norm on R N . Equation ( 1) is a useful generalization of eq. ( 3) because there are situations where the scale of the decoded signal matters. For instance, when the readout unit is corrupted with noise, ẑ = Xw + ϵ where ϵ i ∼ N (0, σ 2 ), we can interpret eq. ( 1) as maximizing the signal-to-noise ratio subject to a cost on the magnitude of the weights w.In most cases, we will consider choices of G(•) that can be written as: G(X) = aC X + bI (4) for some a, b ≥ 0. Under this choice, the penalty on w equals the sum of two terms:EQUATIONThe first term, scaled by a, penalizes the activation level of the readout unit, which can be viewed as an energetic constraint. The second term, scaled by b, penalizes the strength of the synaptic weights, which can be viewed as a resource constraint for building and maintaining strong synapses. We note again that ridge regression is achieved as a special case when a = 1 and b = λ.We now turn our attention to the the problem of quantifying representational similarity between X ∈ R M ×N X and Y ∈ R M ×N Y . As before, we specify a target vector z ∈ R M and optimize w as specified in eq. ( 1). We optimize v accordingly:maximize v 1 M z ⊤ Y v -1 2 v ⊤ G(Y )v (6) yielding the solution v * = 1 M G(Y ) -1 Y ⊤ z.We are interested in quantifying similarity between X and Y by comparing the behavior of these optimal decoders. A straightforward way to quantify the alignment between the decoded signals is to compute their inner product, which we call the decoding similarity (fig. 1 ):EQUATIONwhere we have leveraged eq. ( 2) and the analogous closed form expression for v * while introducing the following normalized kernel similarity matrices:EQUATIONNote that we have suppressed the dependence of K X and K Y on the function G(•) to reduce notational clutter. Equation ( 7) is a reasonable quantification of similarity between X and Y with respect to a fixed decoding task specified by z ∈ R M , and it may be a fruitful approach for hypothesis-driven comparisons of neural systems. We comment further on this possibility in our discussion. On the other hand, many neural representations support a variety of downstream behavioral tasks. For example, features extracted in early stages of visual processing (edges and textures) can be used to support many visual tasks such as object classification, segmentation, image compression, et cetera. How can eq. ( 7) be adapted to quantify the similarity between X and Y across more than one pre-specified decoding task? We explore three simple ideas.Option 1 -best case alignment. An optimistic measure of representational similarity between networks is to find the decoding task z ∈ R M that results in a maximal decoder alignment. Formally,EQUATIONwhere the constraint that ∥z∥ 2 = 1 is needed to keep the maximal value finite.Option 2 -worst case alignment. An alternative is to find the task that minimizes alignment:EQUATIONwhich is obviously the pessimistic counterpart to option 1 above.Option 3 -average case alignment. Instead of maximizing or minimizing over z, we can approach the problem by averaging over a distribution of decoding tasks. Formally, we can quantify alignment in terms of the average decoding similarity:EQUATION)where the expectation is taken with respect to z ∼ P z .In the next subsection, we show that the third option is closely related to several existing neural representational similarity measures. However, all three quantities are potentially of interest and they are easy to compute as we document below in propositions 1 and 2.Proposition 1. The best case and worst case decoding alignment scores, eqs. ( 9) and (10), are respectively given by the largest and smallest eigenvalues of:EQUATIONProposition 2. The average decoding similarity, eq. ( 11), is given by:EQUATIONwhere K z := E[zz ⊤ ] represents a kernel matrix for the targets.Similarity measure a b Linear CKA 0 b GULP 1 λ CCA 1 0 ENSD 0 1 M Tr[C 2 X ] Tr[C X ]Table 1 : Existing similarity measures are equivalent to the decoding score in eq. ( 7) or the decoding distance in eq. ( 16) for different choices of a and b in eq. ( 5).Proof of proposition 1. Let A ∈ R M ×M be any matrix, potentially non-symmetric. We can express A as a sum of a symmetric and skew symmetric matrix, A = B + C, where:EQUATIONFor any z ∈ R M , we havez ⊤ Az = z ⊤ Bz + z ⊤ Cz.It is easy to verify that z ⊤ Cz equals zero. Thus, z ⊤ Az = z ⊤ Bz, and maximizing over z subject to ∥z∥ 2 yields the largest eigenvalue of B (since it is symmetric). Likewise, minimizing yields the smallest eigenvalue of B. Substituting A = K X K Y into this argument proves the proposition.Proof of proposition 2. Using eq. ( 7), we have:E⟨Xw * , Y v * ⟩ = E[z ⊤ K X K Y z].Using the cyclic property of the trace operator to manipulate this further, we get:z ⊤ K X K Y z = Tr(K X zz ⊤ K Y ).Taking the expectation with respect to z and using linearity of trace yields eq. ( 13). 
interpretations of cka and related measures: 
 Intuitively, proposition 2 says that the expected inner product between optimal linear readouts is equal to a weighted matrix inner product between kernel matrices K X and K Y (with respect to a positive semi-definite weighting matrix K z , which of course depends on the distribution of decoding targets z). Such kernel matrices and inner products are important for several methods of quantifying representational similarity. We can therefore leverage this result to provide new interpretations of several distance measures. The most basic idea is to consider a distribution over z which satisfies K z = I. This leads us to measure distances using standard Euclidean geometry, which we state as the following corollary to proposition 2. We explore a relaxation of this assumption and the limit as the number of input samples M → ∞ in appendix C.Corollary 1. For any distribution over decoding targets satisfying K z = I, the Frobenius inner product between normalized kernel matrices K X and K Y is equal to the average decoding similarity:EQUATIONFurther, when K z = I, the (squared) average decoding distance is equal to the (squared) Euclidean distance between K X and K Y :EQUATIONThis corollary can be used to unify several neural representational similarity measures, each corresponding to a different choice of the penalty function G(X), Table 1 . Further, it yields an interpretation of each measure in terms of the expected overlap or difference in decoding readouts between networks.Linear CKA First, for b > 0, the following choice of G(X) yields the linear CKA score [6, 22] :G(X) = bI ⇒ E⟨Xw * , Y v * ⟩ E⟨Xw * , Xw * ⟩E⟨Y v * , Y v * ⟩ = Tr[K X K Y ] ∥K X ∥ F ∥K Y ∥ F = CKA(X, Y )where we have normalized the expected inner product to obtain a similarity score ranging between 0 and 1. Note that the centering step of CKA is taken care of by our assumption, stated at the beginning of section 2, that the columns of X and Y are preprocessed to have mean zero. Thus, linear CKA be interpreted as a normalized average decoding similarity with quadratic weight regularization.The CKA score is closely related to Euclidean distance, so it is not surprising that we can extend our analysis to other methods that utilize this distance. For example, the sample GULP distance, proposed by [3] , is the equal to the average decoding distance with a regularization parameter λ > 0. In particular,G(X) = C X + λI ⇒ E∥Xw * -Y v * ∥ 2 2 = ∥K X -K Y ∥ 2 F = GULP 2 λ (X, Y )yields the plug-in estimate of the squared GULP distance [3, section 3]. We therefore see that the Euclidean distance between normalized kernel matrices can be interpreted as equal to the average decoding distance, or also as an approximation of the population formulation of the GULP distance established in [3] .Canonical correlation analysis (CCA) CCA is another method that has been used to compare neural representations that fits nicely into our framework [29, 36] . When the two networks have the same dimension, i.e.N = N X = N Y , the output of CCA is a sequence of N canonical correlation coefficients 1 ≥ ρ 1 ≥ • • • ≥ ρ N ≥ 0.The average squared canonical coefficients is sometimes used as a measure of similarity, 1 and Kornblith et al. [22] showed this to be related to the linear CKA score on whitened representations. In particular, assuming the covariance matrices C X and C Y are invertible, we have:EQUATION)In practice, it is often useful to incorporate regularization into CCA because the covariance matrices C X and C Y may be ill-conditioned. As mentioned above, the regularization used in GULP distance corresponds to choosing G(X) = C X + λI. A similar approach proposed in [36] is to use G(X) = (1 -α)C X + αI for a hyperparameter 0 ≤ α ≤ 1. This effectively allows for a continuous interpolation between a CCA-based similarity score and linear CKA.Effective Number of Shared Dimensions (ENSD) Finally, we note a connection to recent work by Giaffar et al. [15] , who studied the following quantity, which they call the ENSD:EQUATIONThis is simply a rescaling of the inner product, Tr[XX ⊤ Y Y ⊤ ] and therefore easily fits within our framework using the following choice for G(X):EQUATIONOne of the most interesting features of ENSD is it's connection to the participation ratio, R:EQUATIONwhich is used within physics and computational neuroscience (e.g. [14, 7, 31] ) as a continuous analogue to the rank of a matrix, and is a measure of the effective dimensionality of the data matrix X. In particular, one can show that 1 ≤ R(C X ) ≤ rank(C X ), and that these two inequalities respectively saturate when C X has only one nonzero eigenvalue and when all nonzero eigenvalues are equal. It is also interesting to note that the inverse participation ratio can be captured by average decoding similarity under a very simple choice for G(X):EQUATIONWe reiterate that all of the results enumerated above depend on choosing a distribution over decoding targets which satisfies K z = I. This enforces the decoding targets z 1 , . . . , z M to be uncorrelated random variables with unit variance. The unit variance constraint is analogous to the constraint that ∥z∥ 2 = 1 that we imposed when maximizing or minimizing over z in eqs. ( 9) and (10) . Indeed, scaling the variance of the z i 's simply scales the magnitude of E⟨Xw * , Y v * ⟩. Thus, setting the variance of each sample to one (or some other constant) is reasonable. The constraint that E[z i z j ] = 0 for all i ̸ = j may be relaxed, as we discuss in appendix C. 
relating the shape of neural representations to decoding: 
 In corollary 1 we saw that the Euclidean distance and Frobenius inner product between K X and K Y enjoys a nice interpretation in terms of (average) decoder similarity. This does not provide a completely satisfactory answer of our original question: What does the geometry of a neural representation imply about its function (i.e. decoder performance)? As discussed in section 1, comparing neural representations through linear kernel matrices, K X and K Y , is motivated from geometric principles-namely, their invariance to orthogonal transformations. Indeed, for any orthogonal matrix R, one can verify that:EQUATIONwhenever G(X) is parameterized as in eq. ( 4). Therefore, then the transformation X → XR leaves the whitened linear kernel matrix unchanged. That is, if X ′ = XR, we have:EQUATION)The Procrustes shape distance, P(•, •), is another popular method which is invariant to orthogonal transformations [36, 9] . This approach connects to a broader and decades-old literature on shape analysis [10, 18] whose applications include anatomical comparisons across biological species [12] , and analysis of planar curves such as handwriting data [33] . If the two neural responses are meancentered and have the same dimensionality, then the Procrustes distance is simply the minimal Euclidean distance obtained by rotating and reflecting one set of responses onto the other. That is, when X ∈ R M ×N X and Y ∈ R M ×N Y and we have N = N X = N Y , the Procrustes distance is:EQUATIONWhen N X ̸ = N Y , a straightforward generalization of the Procrustes distance is to zero pad columnwise so that the matrix dimensions match. Geometrically, this can be interpreted as embedding the lower-dimensional point cloud into the higher-dimensional space and optimizing R within this space to align the point clouds.The fact that Procrustes distance is defined in terms of an optimal alignment transformation is appealing both intuitively and because it suggests avenues to generalize the method, such as using permutations (instead of orthogonal transformations) to align representations [19] . Additionally, some empirical results have highlighted cases where Procrustes performs "better" than linear CKA [9, 5] . Rigorously defining and benchmarking what it means to be "better" in this context remains an open discussion [21] , but it is nonetheless of interest to understand what Procrustes distance and associated definitions of shape imply about neural decoding.In appendix A.1 we prove the following result, which states that the Procrustes distance between appropriately normalized representations constrains the average decoding distance, or equivalently, that the average decoding distance bounds the Procrustes distance from above and below. EQUATIONwhereR ∆ = R(K X -K Y )is the participation ratio of the matrix difference K X -K Y . Equivalently, we may rewrite eq. ( 25) to findEQUATIONwhere we have defined normalized transformations X and Ỹ in terms of the transformation G(•):X := 1 √ M XG(X) -1/2 and Ỹ := 1 √ M Y G(Y ) -1/2 .Note that X X⊤ = K X and Ỹ Ỹ ⊤ = K Y .We derive these bounds by leveraging the equivalence between the Procrustes distance between normalized representations P( X, Ỹ ) and a distance metric on positive semi-definite kernel matrices, the Bures distance [16] .We recall that the choice of function G(•) in the decoding problem eq. ( 1) (which determines optimal readout weights w * and v * ) will set the normalization transformation of the representations X and Ỹ . When G(•) is chosen such that Tr K X = Tr K Y = 1,2 we can plot these bounds with α = β = 1 for various values of participation ratio R ∆ (fig. 2 ).Both upper and lower bounds are saturated (appendix A.1), and reveal an allowed region of both Procrustes and average decoding distance as a function of the participation ratio R ∆ . Under the α = β = 1 normalization, the Procrustes distance may range between 0 and √ 2, and the average decoding distance may take values between 0 and 2 √ R∆ , which is reflected in fig. 2 . The bounds in eq. ( 25) and eq. ( 26) reveal an imperfect connection between the Procrustes distance and the average decoding distance, such as the GULP distance. We recall that every choice of readout weight normalization G(X) leads to a measure of representational distance E∥Xw * -Y v * ∥ 2 2 , and that there is an associated Procrustes distance on the normalized representations X and Ỹ , P( X, Ỹ ). These bounds tell us that the average decoding distance E∥Xw * -Y v * ∥ 2 2 and this Procrustes distance constrain each other in a very particular way. Namely, when the Procrustes distance between normalized representations is small, then the average decoding distance must also be small. 3 However, the converse is only necessarily true when the participation ratio of the difference of the kernel matrices K X -K Y is also small (e.g. fig. 2B ). Indeed, as the dimensionality of K X -K Y increases, we see that the Procrustes distance between normalized representations can be large, but the average decoding distance can be relatively small, with the effect becoming more exaggerated as R ∆ becomes large (fig. 2D ). In appendix A.1 we show that for α = β = 1, the minimum possible participation ratio R ∆ is 2. Therefore, fig. 2 also illustrates another interesting phenomenon, namely that the upper left-hand corner of these plots is impossible to populate. That is, in the case of large expected difference in decoded signals, one can never simultaneously measure a small Procrustes distance between the normalized representations. Thus we observe quantitatively, as may be intuitive, that the Procrustes distance offers a more strict notion of geometric dissimilarity than the average decoding distance. 
discussion: 
 Understanding meaningful ways to measure similarities between neural representations is clearly a very complex problem. The literature demonstrates a proliferation of different techniques [20, 34] , but an underdeveloped understanding of how these methods relate to each other. Less still is known about how representational similarity measures interact with functional similarity or the amount and type of information that is decodable from the representation. This paper presents a theoretical framework centered around the linear decoding of information from representations, which allows us to understand some existing popular methods for measuring representational similarity as average decoding similarity or average decoding distance with different choices of weight regularization. These connections relied on averaging the decoding similarity or decoding distance over a distribution of decoding targets z with E[zz ⊤ ] = I. In the future it could be interesting to explore modifying these assumptions. For instance, instead of maximizing, minimizing, or taking the expectation over decoding targets, are there potentially interesting sets of fixed decoding targets that make sense when comparing networks in particular contexts? Furthermore, we focused on linear regression as a decoding method; a potentially interesting line of future work could be to extend this framework to linear classifiers (e.g. support vector machines or multi-class logistic regression). Lastly, in this paper we considered quantifying representational similarity across a finite set of M stimulus conditions. However, these finite-dimensional approaches can be framed as approximations or estimators for a population version of the problem. For example, the framework in [28] for the Procrustes distance, or [3] for the GULP distance (the plugin estimator of which is a special case of our framework as we saw above). We outline a framing of this perspective in appendix C, but a rigorous exploration of the M → ∞ regime and an analysis of the behavior of estimators for similarity scores in the limited sample regime could be an important topic of future work.EQUATIONSo the two eigenvectors are proportional toEQUATIONEQUATIONTo find the eigenvalues, we combine eq. ( 31) and eq. ( 32), arriving atEQUATIONEQUATION)Finally, to calculate ∥M ∥ * we simply sum the magnitudes of these eigenvalues. By inspecting the term under the square root, we see that λ -is always negative. Therefore,|λ + | + |λ -| = λ + -λ -,and we findEQUATIONWe now find a lower bound on the Bures distance between kernel matrices K X and K Y in terms of the Euclidean distance between the same matrices, ∥K X -K Y ∥ 2 F . There are many choices of X and Y multiply to the same positive semi-definite kernel matrices via K X = XX ⊤ and K Y = Y Y ⊤ . Define real positive scalars α = Tr XX ⊤ and β = Tr Y Y ⊤ , and the unit vectorsEQUATIONwhere vec(X) is the vectorization linear transformation converting the M × N X -dimensional matrix X into an M N X -dimensional vector. We will assume that the representations have been zero-padded such that N X = N Y . By Uhlmann's theorem [35] , there exists some choice of X and Y , such thatEQUATIONOne can see this by writing the fidelity as a nuclear norm of X ⊤ Y and considering the variational form of the nuclear norm as a maximization of the Hilbert-Schmidt inner product over unitary transformations (see [16] for a more explicit discussion of this). Now use our identity from earlier with u = x and v = ŷ.EQUATIONThe operation that takes αxx ⊤ and β ŷ ŷ⊤ to K X and K Y , respectively, is called the partial trace. By the monotonicity of the nuclear norm under partial tracing [35] , we have:EQUATIONWith equality when K X and K Y are rank-1, that is K X = αxx ⊤ and K Y = β ŷ ŷ⊤ .Lastly, we rewrite the nuclear norm ∥K X -K Y ∥ * in terms of the Euclidean distance and the participation ratio of the matrix K X -K Y . The participation ratio of a matrix A ∈ R M ×N is defined asEQUATIONso we haveEQUATIONwhereR ∆ = R( X X⊤ -Ỹ Ỹ ⊤ ).We can also use the definition of the Bures distance eq. ( 27) to replaceF(K X , K Y ) = 1 2 (α + β -d B (K X , K Y ) 2 ). Making these substitutions into eq. ( 39), and solving for d B (K X , K Y ) 2 , we find:EQUATION)This bound is saturated when the inequalty in eq. ( 39) is an equality, or when K X and K Y are both rank 1.For a bound that is independent of the participation ratio, we could replace R ∆ with its ostensible minimal value of 1. When K X and K Y are normalized to have trace 1, we would then haveEQUATIONHowever, in this case that Tr K X = Tr K Y = 1, we can actually arrive at a tighter lower bound than eq. ( 43), because we can say something interesting about the minimal participation ratio of matrices of the form K X -K Y . It turns out that under this normalization, the participation ratio is lower bounded by 2, with the minimum of 2 achieved when K X and K Y are rank 1 (also in this case the bound in eq. (42) will be saturated).Lemma 2. For positive semidefinite matrices K X and K Y with Tr KEQUATIONProof. By noting that K X -K Y is Hermitian, we rewrite eq. (40) asEQUATIONwhere λ i is the ith eigenvalue of the matrix difference K X -K Y . Next, we recognize that if Tr K X = Tr K Y , then Tr(K X -K Y ) = 0 and therefore the eigenvalues of K X -K Y sum to 0. The set of eigenvalues {λ 1 , λ 2 , ..., λ M } can then always be partitioned into a set of positive eigenvalues {λ + 1 , λ + 2 , ..., λ + p }, and a set of negative eigenvalues {λ - 1 , λ - 2 , ..., λ - q }, with p + q = M . The negative eigenvalues must balance the positive eigenvalues in magnitude, so we must haveEQUATIONWe can rewrite eq. ( 45) in terms of sums over the positive and negative eigenvalue sets:EQUATIONBy inspection, the term in brackets is ≥ 1 (this can be seen by considering the signs and relative magnitudes of each of the summed terms). So we haveEQUATION)The inequality eq. ( 44) is saturated when K X -K Y is rank 2 and thus only has two equal magnitude and opposite signed eigenvalues, as can be seen from the first line of eq. ( 47). We can also use eq. (34) to see that when K X and K Y are rank 1 and trace 1, we haveEQUATIONFor K X and K Y normalized to have trace 1, we can then state a tighter bound than eq. ( 43) using the minimal participation ratio in eq. ( 44). We haveEQUATIONWe can bound the Bures distance using its variational form [2] ,EQUATION)The Powers-Størmer inequality implies that ∥KEQUATION)Expressing the nuclear norm in terms of the participation ratio of K X -K Y using eq. ( 41), we have an upper bound on the Bures distance:EQUATIONThe inequality in eq. ( 51) is saturated when the optimal orthogonal transformation that aligns K1/2 X and K 1/2Y is the identity. This occurs when K X and K Y are simultaneously diagonalizable. The Powers-Størmer inequality in eq. ( 52) is saturated when K X and K Y have eigenvalues that only take values in {0, 1}. with varying participation ratio R ∆ . We have populated the plots with points representing the respective distances between randomly sampled pairs of positive semi-definite matrices. These positive semi-definite matrices represent normalized kernel matrices X X⊤ and Ỹ Ỹ ⊤ . We have seen in the main text that the squared Euclidean distance between these kernel matrices is equivalent to the expected squared Euclidean distance between decoded signals,E∥Xw * -Y v * ∥ 2 2 = ∥ X X⊤ - Ỹ Ỹ ⊤ ∥ 2F , when the distribution of decoding targets satisfies E[zz ⊤ ] = I. On the other hand, [16] demonstrated how the Procrustes distance P( X, Ỹ ) is equivalent to the Bures distance eq. ( 27), d B ( X X⊤ , Ỹ Ỹ ⊤ ). Therefore, both distances on these plots can be calculated from samples of positive semi-definite matrix pairs. Each color of points in fig. 2 (B-D ) represents a different random ensemble of M × M PSD matrices with M = 50, which are then binned into one of the three plots if the participation ratio R ∆ lies in the respective range indicated above each plot panel.• Pink points are generated by sampling eigenvalues for the two PSD matrices from a Dirichlet distribution with concentration parameters logarithmically spaced between 10 -3 and 10 3 . Matrices of eigenvectors are then randomly sampled from the set of orthogonal matrices. • Green points were generated by sampling a random matrix A of size M × r with each element drawn from the uniform distribution U[0, 1]. A uniform random integer r was selected between 1 and 24. A PSD matrix was generated by multiplying AA ⊤ . Another PSD matrix was then generated by adding a randomly weighted matrix of the same dimension with standard normal distributed entries N to A, soEQUATIONTr BB ⊤ BB ⊤ is then computed using these two distance metrics.• Blue points are seen to only occupy fig. 2 (B) , as these correspond to distances between matrices of rank 1 and trace 1. As we saw earlier in this appendix, in this case the participation ratio of the difference between these two PSD matrices is always 2. The rank 1 matrices were sampled using the same method as the green points above, but setting r = 1. • Purple points were generated by sampling one matrix from a Wishart distribution W M (r, I), with degrees of freedom r chosen as a uniformly random integer between 1 and 50. This matrix was normalized to have trace 1. The second matrix was generated by adding to the first matrix another matrix drawn from the Wishart distribution W M (n, I), with n a randomly selected integer between 1 and 10. Both matrices are normalized to have trace 1. • Yellow points represent distances between PSD matrices that are simultaneously diagonalizable. These were generated by sampling M values uniformly between 0 and 1 for each matrix, but and setting these values to 0 if they fall below the threshold 0.6. The resultant values were then normalized to sum to 1, which became the eigenvalues for the two PSD matrices. A matrix of eigenvectors was then randomly selected from the set of orthogonal matrices.In the main text, we considered quantifying representational similarity across a finite set of M stimulus conditions. Further, when considering the average distance in decoding performance, we have assumed that E[zz ⊤ ] = I. Here, we briefly discuss how to relax both constraints, leaving a full investigation to future work. Readers will need familiarity with the basic principles of kernel methods and gaussian processes in machine learning (see e.g. [32, 37] ) to follow along with certain results in this section.To begin, we must revisit and refine our theoretical framework developed in section 2. Under proposition 2, we treated the neural response matrices, X and Y , as constants while we treated the decoder targets, z, as a random variable. In this section we will treat X, Y and z as joint randomThis integral measures the similarity between two neural systems f and g for a particular choice of decoding task function η. However, following the main text, we would like to take the expectation over some ensemble of decoding task functions η.EQUATIONAssuming the conditions to invoke Fubini's theorem regarding changing the ordering of integration are met, we haveEQUATIONWe now recognize this integral as this as the trace of a composition of integral operators,EQUATIONwhereQ : L 2 (U) → L 2 (U)is the covariance operator associated with the Gaussian process covariance function q. This quantity is finite, since all of the operators involved are Hilbert-Schmidt and trace-class.We now would like to show that the finite-dimensional notion of "decoding similarity" introduced in the main text (appropriately normalized) can be thought of as a plug-in estimator of the quantity eq. ( 64), and that this estimator converges to E η [S] as M → ∞.A 'plug-in' estimator for this could be to use the Gram matrices for each kernel and matrix multiplication to approximate the integrals. First, we sample M examples from the input distribution u ∼ P u and construct the Gram matrices Q ij = q(u i , u j ) andEQUATIONH ij = h(u i , u j ) = g(u i ) ⊤ G(g) -1 g(u j ).(66)Then we have, by the strong law of large numbers, Bounds on the rate of this convergence in operator norm can be obtained using elementary matrix concentration inequalities. We now must argue that G(X) -1 concentrates to G(f ) -1 and similarly G(Y ) -1 concentrates to G(g) -1 . Since the matrices G(X) and G(f ) are symmetric and positive definite, we haveEQUATIONwhere the last inequality on the right hand side is a consequence of the Courant-Fischer theorem and the positive-semidefiniteness of the covariance matrix. Similar relations hold for G(Y ) and G(g). Now we would like to study the quantityEQUATIONMultiplying the quantity inside the norm by identity and invoking the bound in eq. ( 72), we haveEQUATIONPulling a factor of (αE[f (u)f (u) ⊤ ] + βI) -1 out of everything in the norm on the right hand side:∥(G(X) -1 -G(f ) -1 )∥ op ≤ 1 β ∥ (αE[f (u)f (u) ⊤ ] + βI)-( α M X ⊤ X + βI) (αE[f (u)f (u) ⊤ ] + βI) -1 ∥ op and using the submultiplicative property of the operator norm,∥(G(X) -1 -G(f ) -1 )∥ op ≤ |α| β ∥E[f (u)f (u) ⊤ ] - 1 M X ⊤ X∥ op ∥(αE[f (u)f (u) ⊤ ] + βI) -1 ∥ op .Using eq. ( 72) again, we haveEQUATIONorEQUATIONwith an analogous bound holding for G(Y ) and G(g).With this knowledge, we define K and Ĥ as the empirical Gram matrices constructed using the estimates G(X) and G(Y ), and conclude that, for every {i, j},EQUATIONProceedings of the II edition of the Workshop on Unifying Representations in Neural Models (UniReps 2024).In the statistics literature, this summary statistic is sometimes called Yanai's generalized generalized coefficient of determination[30,38].For example G(X) = NX CX or the CKA choice from above with b = Tr[CX ]. This is not a requirement for these bounds to hold, but simply a choice of normalization for the sake of plotting.The scale of "small" and "large" here are determined by α, β, and R∆, as the range of the Procrustes distance is between 0 and √ α + β, while range of the average decoding distance is between 0 and α+β √ R ∆ . 
 
 Appendix:  


Decision: In light of the reviewers' feedback and relevancy of the submission, we are pleased to accept this paper for presentation at UniReps 2024. We kindly ask the authors to incorporate the reviewers' suggestions and feedback in the final camera-ready version of the manuscript.

Reviews: review: title: title: Clarifying and straightforward theoretical contribution
main_review: main_review: This was an nice paper to read, and makes a point that is easy to remember. The authors study a family of linear regression objectives subject to some kind of norm regularization, and show that some popular measures of representation alignment are related to the predictions of linear decoders trained on each representation. My comments are mostly tossing up things I might look forward to in a future instance of this work.

I would be interested in seeing how these results relate to similar ones established previously. In particular, I'm thinking of: 

(1) the result from Cristianini et al. (2001) showing that the (uncentered) kernel alignment is related to generalization of a certain binary classifier. Cortes et al. (2012) also showed this (just empirically I think) for centered alignment.

(2) Gretton et al. (2005) introduced something like the CKA as a test of statistical independence. Specifically, they relate the (numerator of) the CKA to the norm of a "cross-covariance" operator between X and Y, and I think for the choice of a linear kernel (where the RKHS is linear decoders) that might imply some results similar to those of the paper?

I only bring these up in case their frameworks offer an avenue for the kinds of extensions mentioned in the discussion. They could also be good additions to a "related work" section of this paper, but I don't feel strongly about that.

On a more substantive side, it seems like a focus of the paper is on unifying these measures, but I was interested if this framework can help to explain some of their observed discrepancies. I'm thinking, for example, of the recent work by Soni et al. (2024), showing that choice of metric can have a pretty big difference on alignment scores. If geometric alignment is related to functional alignment, it seems important to know why different measures of alignment produce different results. Might be a nice application of this work.

At the moment the numerics are a fairly small part of the paper. A nice proof of principle could be to apply this to artificial networks trained on (simulated) tasks of varying similarity, and to see how much the geometric similarity actually reflects functional similarity, and whether this framework is helpful in understanding those results.

links:

Cristianini et al. (2001): https://papers.nips.cc/paper_files/paper/2001/hash/1f71e393b3809197ed66df836fe833e5-Abstract.html

Gretton et al. (2005): https://www.gatsby.ucl.ac.uk/~gretton/papers/GreBouSmoSch05.pdf

Cortes et al. (2012): https://www.jmlr.org/papers/volume13/cortes12a/cortes12a.pdf

Soni et al. (2024): https://www.biorxiv.org/content/10.1101/2024.08.07.607035v1
title: title: Theoretically well motivated and justified framework but would be more convincing with empirical evaluation.
main_review: main_review: ## Summary of the paper

The paper proposes a novel framework to compare representations from different models on the basis of the functions that they implement — functions that can be decoded from them. They argue that representation geometry should be considered a proxy for model-implemented functions, and comparisons of representation geometries should be done with this in mind. 

They formalize a framework for representational similarity defined based on the algorithmic expressivity of the representations. As a stand-in for the functions implementable by the models, they assume the Linear Representation Hypothesis i.e.  they characterize the representation power with respect to a function as the ability to linearly simulate or “extract” that task from the representations — a practice that is popular in probing for interpretability literature also. 

This allows them to define a notion of the “most decodable” and “least decodable” functions i.e. functions that are the easiest and the hardest to extract from the representations.

Under specific assumptions, the authors demonstrate that their framework becomes equivalent to popular second-order similarity metrics, such as Centered Kernel Alignment (CKA).

## Strengths

- The proposed framework is well motivated — it tries to characterize the relationship of representation geometry to expressivity of representations.
- The framework provides meaningful connections with existing popular and well-motivated similarity metrics.

## Weakness

- There is no empirical validation of their framework. Often, a lot of nice seemingly well motivated properties together lead to unexpected and perhaps undesirable consequences. Therefore, I would have been more convinced if there was empirical demonstration that the metric assigns expected values.
- The meaning of the “least decodable” setting is unclear — why would this not be a function with pseudo-random outputs and if so, why is that a meaningful thing to look at?
- The E[zz^T] = I assumption could use qualification with respect to real data. It seems arbitrary but is important for the relationship to CKA.

## Experiment Suggestion

The metric should consider demonstrating their framework by measuring similarities across models while controlling for the expressive power (in terms of number of parameters or performance on relevant NLP benchmarks)

## Conclusion

While the paper presents a theoretically interesting and well-motivated framework, it would benefit significantly from empirical validation and clearer justification of its assumptions. The relationship between representation geometry and functional expressivity is a valuable contribution to the field, but practical demonstrations would strengthen the paper's impact.
title: title: Interesting framework for unifying existing geometric measures - lacking presentation
main_review: main_review: This work puts forward a framework to unify different representational similarity measures (e.g., CKA, CCA, Procrustes distance) via the concept of linear decodability. The authors show that several measures can be interpreted as “alignment scores” between optimal linear decoders, thus connecting geometric similarity with decoding similarity. This is a key contribution that appears novel and important to me. 

However, it must be said that some of the proven connections are not particularly surprising: linear similarity and linear decodability can be expected to be closely related. I also think that the presentation could be significantly improved as the paper is rather hard to read and appreciate. On this, I would have three specific suggestions. 

1) The main findings of the paper are hard to quickly find and grasp. In this respect, a summary table or summary figure (e.g., outlining the correspondences between the different measures) would **drastically** improve the readability of the paper.
2) Related to 1. I would personally appreciate a presentation focusing on the results and building an intuition around them, with proofs either completely moved to the Appendix or sketched in the main text but isolated in “skippable” sections.
3) Finally, I think the paper would benefit from a discussion of the potential practical implications of the framework, which instead appears completely missing from the current version.

All things considered, I think this is a potentially a clear “accept” (7) in terms of the results, but the presentation is so lacking to significantly lower my rating.
title: title: Interesting paper that aims to link geometric and decoding perspectives in similarity measures
main_review: main_review: The paper studies how representational similarity measures like Centered Kernel Alignment (CKA), Canonical Correlation Analysis (CCA), and Procrustes Shape Distance can be understood through decodable information. Specifically, the authors propose a framework that links these similarity measures to the ability of neural representations to linearly decode information across a distribution of tasks. The paper formalizes these connections by analyzing the alignment of linear decoders and deriving bounds for different similarity measures. 

Strengths:
- The findings on the relationship between similarity measures and linear decoders have the potential to impact a range of fields, including neuroscience and representation learning.
- Solid theory: The authors present detailed mathematical proofs and derive bounds that formalize the relationships between different similarity measures.

Weaknesses:
- The framework is built on the assumption that the distribution of decoding targets satisfies $E[zz^{\top}]=I$, but this condition may not be met in practical settings.
- The paper’s framework is heavily focused on linear decoders.
- Despite the strong theoretical foundations, the paper does not provide empirical evidence to show how the proposed measures perform on actual neural networks or datasets.