Estimating Shape Distances on Neural Representations with Limited Samples
Authors: Dean A Pospisil
Brett W. Larsen
Sarah E Harvey
Alex H Williams
Venue: ICLR 2024 poster (2024)
Keywords: representational geometry
shape metrics
dissimilarity metrics
Primary area: applications to neuroscience & cognitive science
TL;DR: Novel estimator of geometric similarity with tunable bias-variance tradeoff, outperforms standard estimators in high-dimensional settings.

Abstract: Measuring geometric similarity between high-dimensional network representations is a topic of longstanding interest to neuroscience and deep learning. Although many methods have been proposed, only a few works have rigorously analyzed their statistical efficiency or quantified estimator uncertainty in data-limited regimes. Here, we derive upper and lower bounds on the worst-case convergence
of standard estimators of shape distance—a measure of representational dissimilarity proposed by Williams et al. (2021). These bounds reveal the challenging nature of the problem in high-dimensional feature spaces. To overcome these challenges, we introduce a novel method-of-moments estimator with a tunable bias-variance tradeoff parameterized by an upper bound on bias. We show that this estimator achieves superior performance to standard estimators in simulation and on neural data, particularly in high-dimensional settings. Our theoretical work and estimator thus respectively define and dramatically expand the scope of neural data for which geometric similarity can be accurately measured.

Published as a conference paper at ICLR 2024
ESTIMATING SHAPE DISTANCES ON NEURAL REPRE- SENTATIONS WITH LIMITED SAMPLES
Dean A. Pospisil Brett W. Larsen Sarah E. Harvey2.3 Alex H. Williams Princeton University, Princeton, NJ, 08544; @844riietnooooo edu NNw York University, Center for Neural Science, New York, NY, 10003 FFlatiron Institute, Center for Computational Neuroscience, New York, NY, 10010 "Flatiron Institute, Center for Computational Mathematics, New York, NY, 10010 (brettlarsen, sharvey,
ABSTRACT
Measuring geometric similarity between high-dimensional network representa- tions topic of longstanding interest to neuroscience and deep learning. Al- though many methods have been proposed, only a few works have rigorously ana- lyzed their statistical efficiency or quantified estimator uncertainty in data-limited regimes. Here, we derive upper and lower bounds on the worst-case convergence of standard estimators of shape distance measure of representational dissim- ilarity proposed by Williams al. (2021). These bounds reveal the challenging nature of the problem in high-dimensional feature spaces. To overcome these challenges, we introduce a new method-of-momeen estimator with a tunable bias- variance tradeoff. We show that this estimator achieves substantially lower bias than standard estimators in simulation and on neural data, particularly high- dimensional settings. Thus, lay the foundation for rigorous statistical theory for high-dimensional shape analysis, and we contribute a new estimation method that is well-suited to practical scientific settings.
INTRODUCTION
Many approaches have been proposed to quantify similarity in neural network representations. Some popular methods include canonical correlations analysis (Raghu et al., 2017), centered kernel align- ment (CKA; Kornblith et al., 2019), representational similarity analysis (RSA; Kriegeskorte et al., 2008a), and shape metrics (Williams et al., 2021). Each of these approaches takes in a set of high- dimensional measurements e.g.. hidden layer activations or neurobiological responses- and out- puts (dis) (dis)similarity score. Shape distances additionally satisfy the triangle inequality, thus enabling downstream algorithms for clustering and regression that leverage metric space structure.
and
Here, we take closer look at the estimation of shape distance in high-dimensional, noisy, and sample-limited regimes. While shape distances have numerous applications in the physical sciences (Rohlf & Slice, 1990; Goodall, 1991; Andrade et al., 2004; Kendall et al., 2009; Saito et al., 2015) the use of shape metrics and other measures of neural representational similarity has introduced sta- tistical issues that have not been adequately addressed. Specifically, shape metrics are often applied to low-dimensional noiseless measurements (e.g.. 3D digital scans of anatomy across animals; Rohlf & Slice 1990) whereas in the study of neural networks the applications have been nigh-dimensional (e.g., comparing neural activity between brain regions; Kriegeskorte et al. 2008b).
We demonstrate that the noise and high dimensionality of neural representations pose a substantial challenge to estimating representational similarity in sample-limin limited regimes. Yet, with the note- worthy exceptions of research on RSA (Cai et al., 2016; Walther et al., 2016; Schüit et al., 2023) and CKA (Murphy et al., 2024), there is little work on quantifying accuracy of estimators of repre- sentational similarity (e.g. by developing procedures to compute confidence intervals). This poses a serious obstacle to adoption of these methods, particularly in experimental neuroscience where there a hard limit on the number of conditions that can be feasibly sampled (Shi et al., 2019; Williams & Linderman, 2021).

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Published as a conference paper at ICLR 2024
A
Network i
Network
B
&
Rotationaa Aiigmmett
ky(ss) Ay(kar)
M 2
M
M =
M 16
4g(+)
AC(
hy()
hy()
N.(-)
k2())
Figure 1: (A) Classical shape distances (Kendall al., 2009) be used provide rotation- invariant distance between neural representations (Williams et al., 2021). Given two labelled points clouds in N-dimensional space (left and middle), the distance is computed after optimal orthogo- nal transformation is chosen to align the point clouds (righ(). In this visual example the point clouds trace out low-dimensional manifold.
(B) Heatmap shows the covariances (Z Σgj) and cross- covariance (Egy) of the 3D representations in panel A. Shape distances can be re-expressed in terms of these quantities (see eq. 6. 7). (C) Our ability to estimate the shape distance is related to M, the number of stimuli. As M increases (left to right) the number of sampled points along the underlying manifold increases, and we are better able resolve shape differences between the representations.
To address this challenge, we first obtain high-probability upper and lower bounds on the accuracy of typical "plug-in estimates" of shape distance. These bounds reveal these estimators are biased with low variance. The overall error decays with the number of sampled conditions, M, but the decay rate is inversely related to the number of dimensions, N. To combat these limitations, we propose new method-offmomeen estimator which, while not always strictly optimal compared with the plug-in estimator, provides an explicit and tunable tradeoff between estimator bias and variance.
BACKGROUND AND PROBLEM SETTING
We begin by considering a simple setting where each neural network is deterministic map (for the stochastic setting, see section 3.4). A collection K neural networks can then be viewed as a set of functions, each denoted h, Z RV for {1....,K} Here, Z is feature space and N can be interpreted as the number of neurons each system (e.g. the size of a hidden layer an artificial network, or the number of recorded neurons in biological experiment).
KK.
Let h; and hy denote neural systems which we assume are mean-centered and bounded:
E[h,(=)] (x)
and
(hy())1 < BVN almost surely. (1)
for some constant B > O. Here, the expectations are taken over z ~ P, for some distribution P over network inputs. Our assumption that neural population rates are bounded by BvN can result from assuming each neuron has maximum firing equal B. This assumption is common in the literature and reasonable both artificial networks (since connection weights are finite) and biological networks (since neurons have maximal firing rate).
Motivated by the shape theory literature (Goodall, 1991; Kent & Mardia, 1997; Kendall et al., 2009; Williams al., 2021), consider estimating the Procrustes size-and-shape distance, p. and Rie- mannian shape distance, 0, between neural representations, In our setting, these shape distances can
'The assumption that each layer has the same number of neurons is not essential. A theoretical connection with the Bures distance pointed out by Harvey et al. (2023) allows one to generalize shape distances to networks of dissimilar sizes. Indeed, we will see in eqs. (6) and (7) how shape distances can be expressed in terms of covariance and cross-covariance matrices that are well-defined in unequal dimensions.
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be defined as (see App. D in Williams al., 2021):
p(h;, h1) - min QEO(N)
/Blli(>) Q21(2)11
Qhh,
0(h,, h1)
min cos QEO(N)
(3)
(1))
where O(N) denotes the set of N x N orthogonal matrices. Again, all expectations are taken over P. Note that different notions of distance arise from different choices of input distribution, P.
To simplify our analysis and exposition, we will focus on estimating the squared Procrustes dis- lance, p², and what call the cosine shape similarity, cos O. Thus, we ignore the square root term in eq. (2) and the arccosine term in eq. (3), but it should be kept in mind that one must apply these nonlinear functions achieve proper metric.
Properties of Shape Distance is easy to verify that shape distances are invariant to rotations and reflections: that is, if r I R is an orthogonal transformation, then for any function h: 2 + R representing a neural system we have p(h, r h) = 0(h, r o h) = O, where 'o' denotes function composition. Furthermore, p and 9 are proper metrics, meaning that:
p(h,, h,) p(hj. h.) and p(h,, hy) p(h h2) p(ha. h,) (L..., K}, (4)
and likewise for 0. These properties are fundamental to rigorously establishing downstream analy- ses, such as for clustering networks with similar representations (Williams et al., 2021).
is well-known that the optimal orthogonal alignment appearing in eqs. (2) and (3) can be identified in closed form, allowing us to write the Procrustes and Riemannian shape distances in terms of the covariance and cross-covariance matrices. We define the covariance (En and 23;) and cross- covariance matrices (En)) as
Σ El (z),,
Σ₁ = (1),()); Σ,y E(),())
(5)
and reformulate the squared Procrustes distance and cosine shape similarity:
p² (ha, h2) = Tr Tr|2,s] cos 0 (h,, h2)
21
(6)
| -
(7)
E.
TrE
where EEyli ||+ denotes the nuclear norm (or Shatten 1-norm) of the cross-covariance matrix:
N
Egyk= = Σ Sn (Eyy
(8)
n=1
where (M) 2 ... 2 sN (M) 2 0 denote the singular values of a matrix M. Equations 6 and are derived in Appendix to provide the reader with self-contained narrative.
Plug-in Estimators Suppose we are given M independent and identically distributed network inputs 21...., ZM P. How well can we approximate the shape distances between two networks, as a function of M? The standard approach (Williams et al., 3021), is to use a plug-in estimator in which one computes eqs. (2) and (3) after identifying the optimal Q E O(N). As we show in App. A.2, this is equivalent estimating the squared Procrustes and cosine Riemannian distances by substituting the empirical covariances:
M
M
M
Σn =
1 M
Σ
h5 (4 Σnj = NV
Σ m=1
(zm JT, Σnj = 1 37 Σ m=1
(1(==m)) (25m)) (9)
m==1
to approximate the true covariances appearing in eqs. (6) and (7). Thus, (hi, hz) = Tr Tr(E,, + Tr|E,n] 22 E
(10)
cos (14, hy) = E ll
(11)
Tr E Tr
define plug-in estimators for the squared Procrustes and cosine Riemannian shape distances. The empirical behavior these estimators as a function of M was only briefly characterized by Williams et al. (2021) for pair of artificial networks trained CIFAR-10.
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3 RESULTS
First, we theoretically characterize the accuracy of plug-in estimation as a function of the number of samples, M, and the dimension, N. We show that these estimators are biased and can converge at unfavorably slow rates under certain conditions. To overcome these issues, we introduce a new method--ffmomennt estimator section 3.3 which has lower bias the cost of increased variance.
3.1 NONASYMPTOTIC BOUNDS ON THE PERFORMANCE OF PLUG-IN ESTIMATION
First, it is straightforward to estimate Tr E.] and Tr[E, Their plug-in estimators are unbiased under our assumptions in eq. (I), and they rapidly converge to the correct answer. This is shown in the following lemma, whose proof relies only classical concentration inequalities.
Lemma (App. B.1). Under the assumptions in eq. (1), with probability at least 1 - 8:
Tr - Tr[[Ed] < N 1/2 /B
(12)
In contrast, the plug-in estimator for Eigll+ - is biased upwards (see section 3.2) and turns out to converge more slowly. Using the Matrix Bernstein inequality (see Tropp, 2015), can show: Lemma 2 (App. B.2). Under the assumptions in eq. for any M and N:
E 11 - -E 2B N2 log(2N) 2B2 (lo((2N)
(13)
3M
M1/2
This only upper bounds the expected error. However, the fluctuations around this expectation turn out to be small (see App. B.3), and so we are able to combine lemmas 1 and 2 into the following:
Theorem (App. B.3). Under the assumptions in eq. (1), with probability at least - 5
p² 1²1
A Nlg((N) 4B MENNNoo((2 Mlog(2N)
3B2
6 7
2 2
(14)
N
3M
M1/2
M1/2
Theorem 1 states non-asymptotic upper bound the plug-in estimator's error that holds with high probability. We have expressed this bound on the squared size-and-shape Procrustes distance normalized by 1/N, since the raw error, l pl. will tend to increase linearly with N for an uninter- esting reason massll, since the the Procrustes shape distance is comprised of terms like Tr[Ea] and Tr The choice of normalization in theorem also makes the result more comparable to the cosine shape similiarity (eq. 7), which is normalized by a factor, / Tr Tr[E, of order N.
We can gain intuition for theorem by ignoring logarithmic factors and noticing that the second term dominates. Then, roughly speaking, theorem says that we can guarantee the plug-in error decreases as a function of -1/2 Thus, for any fixed N, we need to increase M by a factor of 4 to decrease estimation error by factor of 2. Further, when comparing higher-dimensional neural representations (i.c. higher N) we need to sample more landmarks if N increases by a factor of 2, then M must be increased by factor of 4 to compensate.
3.2 FAILURE MODES PLUG-IN ESTIMATION AND LOWER BOUND PERFORMANCE
Theorem 1 provides a high probability upper bound on the estimation error. A natural question is whether this upper bound is tight. To investigate, we seek an example where the plug-in estima- tor performs badly. We intuited that the plug-in estimates will have large downward bias when two neural representations are very far apart shape space. This can be understood in two ways. First, from the definitions of and 0 in eqs. (2) and (3), we see that both expressions contain minimization over Q E (NN). J. For large and small M, this high-dimensional orthogonal matrix can be "overfit" to the M observations resulting in underestimate of distance. Second, from the alternative formulations in eqs. (6) and (7), we see that the shape distance is large if the true cross- covariance is "small" as quantified by the nuclear norm. In the extreme case where the singular values of Eg are all zero, the empirical cross-covariance matrix (1/M) Σ., (2, () (xm)² will overestimate the nuclear norm, and therefore underestimate the shape distance. This is more severe
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when M is small, since there are fewer terms in the sum to "average out" spurious correlations, which are particularly problematic in high dimensions (i.e. when N is large).
This intuition led to construct example where plug-in in estimation error approaches the upper bound in theorem 1. This is summarized in the following result.
Theorem 2 (Lower Bound, App. B.4). Under the assumptions in eq. (1), there exist neural networks and a distribution over inputs such that in the limit that N + oo and M > N:
102 p²
16B²
11/2M-1/
(15)
N
3π
In Appendix B.5 we show the validity of this lower bound on simulated data. Although the bound is asymptotic, it gives a highly accurate approximation to the observed plug- in error for reasonable values of M and N (see Fig. 5). Thus, while future work may seek to improve the upper bound in theorem I, we cannot hope to improve beyond the lower bound formulated above. If we ignore constant factors and logarithmic terms to gain intuition, we observe there is (roughly) a gap of / between the upper and lower bounds. Thus, possible that our analysis in section 3.1 may be conservative in terms of the ambient dimension. That is, to compensate for a two-fold increase in N, theorem 2 only shows a case where M needs to increased two-fold, in contrast to the four-fold increase suggested by theorem 1. However, in terms of the number of sampled inputs, the lower and upper bounds match: thus, the rate cannot be improved beyond 1/22
3.3 A NEW ESTIMATOR WITH CONTROLLABLE BIAS
The plug-in estimator Egyl I| has low variance but large and slowly decaying bias (see theorems 1 and 2). Here we develop an alternative estimator that is nearly unbiased.
First, note that the eigenvalues of Σij ST correspond to the squared singular values of Σiy- Thus, Tr((E,)
= ll, and so we can reduce our problem to estimating the trace of (Eig 1/2 which is symmetric. Leveraging ideas from a well-developed literature (Adams al., 2018), we proceed to define the ndh moment of this matrix as:
W TH(
X
(16)
where A1..... AN denote the eigenvalues of Σig ET. Now, for any function f R + R and symmetric matrix S with eigenvalues A1,..., AN. define2 Tr(((S) = E. So long as I is reasonably well-behaved, we can approximate it using a truncated power series with P terms. Thus, with S Eiy Ey and f(x) Vz:
P
E |l- = Tr
Xp
7p W
(17)
P=U
p=0
P=0
where '0..... 7P are scalar coefficients.
In summary, we can estimate || Eijlle by (a) specifying an estimator of the top eigenmoments, W Wi...., Wp, and (b) specifying desired set of scalar coefficients -o To estimate the eigenmoments, we adapt procedures described by Kong & Valiant (2017) to obtain unbiased esti- mates for each moment, WI.... Wp (see App. C). To select the sealar coefficients, we propose an optimization procedure that trades off between bias and variance in the estimate of || |E:g]l.. Our starting point is the usual vtarince decomposition:
E (12:))l- - Ep Tp1))) 2
-IR
[1E
Ep ww + War Ep "ip W
(18)
Since BEWW,) = W, = En Np. the first term above (i.e. the "bias") simplifies and is upper-bounded.
E E B1 Ep 2ip 1o
En, -2 (pp))
(N (z2 Ep 7p200)
max 05
"Tii common convention extend scalar functions (see e.g. Potters & Bouchaud, 2020, sec. 1.2.6).
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The inequality follows from replacing each term in the with the worst case approximation error of the polynomial expansion (given here as the maximization over z). Thus, we seek to:
NN (x)12 Ep Spop)) Epp 7o Cov(W, W.)
minimize TO...PPPP
max 0S251
(19)
We estimate Cov(W,, W by boolstrapping--ee. the empirical covariance these statistics across are sampled with replacement. Given this estimate of
re-sampled datasets where { covariance, eq. (19) can be cast as a convex quadratic program and the maximal bias can be bounded to user defined limit at the expense of variance (see App. C.2). We use the maximal bias (eq. 19, term 1) and variance (eq. 19, term 2) to form approximate confidence intervals (see App. C.3).
3.4 EXTENSION TO STOCHASTIC NETWORKS
Thus far, we have modeled neural networks as deterministic mappings, h; 2 R.N. This as- sumption is not satisfied in biological data and in many artificial networks (e.g. VAEs). Here, we briefly explain how to extend the estimators to the stochastic setting. this setting, the response of network i can be written as (z(2) + ez(z). As before, (z(2) is a deterministic mapping condi- tioned on a random variable 2 ~ P. The "noise" term ex(=) is mean-zero random variable that, in addition to inheriting the randomness of captures the stochastic elements of cach forward pass through the network (i.e. trial-to-trial variability even when the stimulus is fixed). Importantly, noise contributions are independent and identically distributed for each pass through the network.
Given a second stochastic network with same structure, hz(2) + (z)), our goal is to estimate the shape distances eqs. (2) and (3) before, effectively ignoring contributions of the "noise" terms ei(-) and c.). Ignoring these terms is not wholly justified, since it is of great interest to quantify how noise varies across networks (Duong al., 2023). Nonetheless, it is useful to develop metrics that isolate the "signal" component of neural representations, and full development of methods to quantify similarity in noise structure is outside the scope of this paper.
Our basic observation is that it suffices to consider two replicates for each network input. That is, let x' z where P. Then, E,; 1ETh ( )h, (z')[] which can be approximated by the slightly reformulated plug-in estimator: Σnp (1/M) Σ
dent across networks, i.c. ez(z) 1T (z) for all m h4( (2',) Further, since noise indepen- Z, the cross-covariance estimators, including the method-of oof-momentss estimator described section 3.3, do not require any modification.
APPLICATIONS AND EXPERIMENTS
4.1 VALIDATION ON SYNTHETIC DATA
We first validate our method-of-momenn estimator (section 3.3) on simulated responses from multivariate normal distribution. We estimate the cosine shape similarity, cos defined eq. 7. Our estimator of Exy1 || is the principle novelty; thus, it is informative to understand its properties isolation. To achieve this, we use the ground truth covariance of Wp (instead of an estimate from a bootstrap) and use the ground truth values of Tr Tr] and Tr For details see App. D.I.
We first compared the bias of the plug-in estimator to that of the moment-based estimator across range of ground truth shape similarity values (Fig. 2A). As expected from our intuition discussed in section 3. 1, the plug-in estimator (blue line) tends to inflate estimated similarity when ground truth is low (left side of plot). The noment-bbssed estimator (orange line), in contrast, performs well over the full range, at the cost of increases estimator variance (blue vs orange error bars).
Next, we fixed the ground truth similarity at 0.2 and studied the effect of sample size, M (Fig. 2B). The moment estimator (constrained to 5% bias) maintains small bias even with small M, at the cost of high variance (orange error bars). Increasing M quickly reduces the variance of the estimator. A similar story emerges when we fix and vary the dimension N (Fig. 2C). As the dimensionality in- creases, the plug-in estimator bias quickly explodes. In contrast, the moment estimator (constrained to 10% bias) has roughly constant bias; but it's variance grows with N. Thus our estimator bias outperforms the plug-in when the sample size is low and dimensionality is high.

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Finally, important property of the moment- based estimator is ability compute approximate
confidence intervals (CI) (see App. C.3). We demonstrate 95% CIs across simulations in Figure 2D. These CIs are conservative, the true shape score is not within the CI's for only 2.3% of simulations.
A Estimator comparisnn son M-200, N-
B
Effect of semple size
C Effect of dimensionality
D
Confidence intervals
N-100
M-100
M-200, N=100
Estimated simi arity
1.0-
Plag-in un..
C00204000810
600
600
800
1000
50
100
150
200
0
25
50
75
300
True simi arity (1rrt
IRuall
stimuli (M)
# nneurons (N)
Simulation
Figure 2: Validation of estimator on synthetic data. (A) The moment based estimator (orange)
compared to plug-in estimator (blue) in simulation with standard deviation bars calculated across simulations. Estimators are evaluated at 20 linearly spaced ground truth similarity score values. (B) Effect of increasing sample size when moment estimator is constrained to have a bias less than %. (C) Effect of increasing dimensionality. (D) Demonstration of conservative confidence intervals that account for variance and maximal bias of moment estimator. We do not include CIs for the plug in estimator (implied by theorem 1) because for small sample sizes, the theoretical bounds on estimator bias always contain far more than the entire allowable interval (0,, 1))
Control of estimator bias Here we demonstrate the bias-variance tradeoff controlled by the upper- bound on bias defined by the user. The quadratic program in eq. (19) constraings the maximal absolute bias below a chosen constant (Fig. 3A, blue shaded area around true similarity score). The actual maximal bias will then be less than or equal to this user defined bias (cyan shaded area within blue). The expected value of the moment estimator stays within the maximal bias, in this case on its bound (orange trace). The user defined bias bound remains inactive until it is less than the MSE minimizing solution's 's bias (blue completely overlapped by cyan above 0.1). Variance then begins to increase as higher order W terms are weighted more to reduce bias (orange standard deviation bars from simulation increase as cyan region narrows). The mean of the estimator converges to ground truth as constrained by the bias bound (dotted orange line converges to dashed black). The plug-in estimator exceeds the maximal bias of the moment estimator (blue trace above eyan area).
Intuition for the moment estimator can be drawn from plots of solutions to the polynomial approxi- mation (eq. 17, Fig. 3B, orange trace approximates black dashed) of the squared singular values of E1, (black points all overlapping). Here we have re-scaled the the vertical axis so that the deviation between the square root and polynomial approximation is exactly the bias of the moment estimator. In the case where bias is not constrained (associated with left most estimates in panel B) the approx- imation is poor (dashed-dot orange trace does not match dashed black trace). For these eigenvalues the the deviation is near the worst possible bias (distance from black point dashed dot orange line is nearly far any other vertical deviation between the traces), this is why the estimator in panel B sits at the bound of maximal possible bias. When the upper bound on bias is very small (far right of B) the approximation is very good (dashed orange overlaps dashed black) because higher order terms are used. Yet this results in very high variance (Fig. 3B).
4.2 APPLICATION TO BIOLOGICAL DATA
Here we investigate noisy non-Gaussian data where the covariance of the W, p and the denominator of the similarity score must be estimated from data. We do so by applying our estimator to neural data: calcium recordings from mouse primary visual cortex in responses to a set of 2,800 natural images repeated twice (Stringer et al., 2019). Our estimator became highly variable when applied to this data part because of its low SNR (average SNR R 0.1). We therefore restricted our analysis to the neurons with the highest SNR in each recording (80 neurons in each recording).
This dataset contains seven recordings from different animals, but the population responses are not directly comparable since each recording targets a different region of primary visual cortex contain- ing neurons with different receptive field. Thus, even though the same images were shown across recording sessions, the recorded neurons were effectively responding different cropped portions of the image. We therefore only quantified shape similarity between subsampled populations of neurons taken from the same recording session.
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A
B
Similarity
0.75
Cross-covariance singular values normalized
||
0.50
51 MEEEPPE
0.00
0.1
0.01
0.25
0.50
0.75
1.0
User defined bias bound
S7
Truth
Moment expectation
<x
Ground truth s
User defined bias bound
Moment est. simulation
approx. low blas
Moment bound
Plug-in
Poly, approx. high blas low variance
Figure 3: Control of bias-variance tradeoff with user defined bound on bias. (A) Moment based estimator expected value is constrained to be within the user defined bias bound (blue region) while minimizing worst case MSE (eq. (19)). Maximal bias can be less than the user defined bias (cyan region within blue). Lower bias leads to increased variance (orange trace converges to black dashed as SD bars widen). Where simulations become unstable we plot the theoretical expected value (dotted orange). Plug-in estimator is well outside bias bounds of moment estimator thus is more biased than moment estimator (blue trace outside cyan line). (B) Example plots solutions to the quadratic program's approximation (orange traces) square root (black dashed trace) of the eigenvalues of Σ1, (black points). Re-scaling of singular values on vertical axis results in the deviation between the polynomial and the true square root evaluated at the true eigenvalues being exactly the bias of the associated estimates in panel A.
Determining the properties of the bias of our estimator requires comparison to the ground truth value of the similarity score. In the neural data, ground truth is unknown. We thus developed two sampling schemes to set the ground truth similarity in the neural data. To set similarity to 0 we measured similarity between different subpopulations of neurons (N 40 neurons each) shown different stimuli (M=400 stimuli each), thus the two populations responses are independent, thus their cross covariance is so that the similarity score is O. To set the similarity we measured similarity between the same subpopulation of neurons (N = 40 neurons) shown the same stimuli (M 400) but on different trials, thus the only deviation in their responses owing to trial-to-trial variability, thus their tuning similarity is 1.
A
C
Ground truth=0
Ground truth
data
data stimali
1.5
1.0
Estimated similarit
Ground truth
Woment ext
Blas
-0.5
2
3 4
Plgg est
Neural recording
Figure 4: Validation of estimator on neural data (Stringer et al., 2019). (A) Comparison of estimators The estimator applied to three disjoint sets
when ground truth similarity neural data is of random stimuli for each recording (n 7). The estimated maximal bias is plotted in dark orange area and the confidence interval, which includes bias, is plotted in light orange. (B) Same simulation as (A) except ground truth similarity is 1.(C) Same as (B) except estimation of true similarity. (D) Estimation of true similarity on all stimuli. (M Rs 2, 800).
When the ground truth similarity was 0, the moment estimator correctly indicated this outcome (Fig. 4A, orange trace overlaps black dashed) and the confidence intervals always contained the true similarity (light orange contains black dashed). On the other hand the plug-in estimator was upwardly biased (blue above black dashed). Thus the moment based estimator can accurately deter- mine when the similarity is low in noisy neural data whereas the plug-in in estimator cannot.
When ground truth similarity that the plug-in (Fig. 4B, blue overlaps black dashed, orange below). This is consistent with our synthetic simulations (see Fig. 2A far right). The Cls always contained the true value but contained
found the bias of the moment estimator
worse than
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nearly the entire possible range of similarity values. Thus while the average estimate is high our confidence intervals are so wide that we do not have much information about the true similarity.
Finally, we aimed to estimate the true shape similarity between these sub-populations of high SNR neurons (N = 40 neurons each). In Figure 4C, we show the estimated similarity across three independent folds of the stimulus set (M = 400 stimuli each). Across all seven recordings the moment estimator was near 0.5, but confidence intervals were wide so there is little information about similarity even for the highest SNR neurons (light orange extends from 0 to on vertical axis). The plug-in estimator reports a higher degree of similarity, that we heavily discount given its upward bias (documented in Fig. 44A). When we included all stimuli (M ~ 2800) we obtained tighter confidence intervals, learning that the true similarity is most likely between 0.25 and 0.75 (Fig. 4D). Thus small populations of well-tuned neurons in the same brain region have only intermediate levels of representational similarity. Overall, we find noisy data is challenging setting for reducing the bias of shape similarity estimates.
4.3 APPLICATION TO ARTIFICIAL NEURAL NETWORK REPRESENTATIONS
In Appendix E apply the plug-in and noment-based estimator penultimate layer representa- tions between two ResNet-50 architectures (He et al., 2016) trained on ImageNet classification Deng al. (2009). In this setting, we can accurately determine the ground truth by sampling a very large number of images (large M). However, for simulated analyses with small sample sizes (small M) we find that the plug-in estimator of similarity shows a positive bias, in agreement to our observa- tions in Figure 4. In contrast, the moment- based estimator provides, on average, better estimate of the shape similarity (albeit with higher variance across simulated analyses). We also observe that the bias of the plug-in estimator depends on how quickly the eigenvalues of the response covariance decays. ("effective dimensionality": see e-g. Elmoznino Bonner 2022). Thus, analyses of shape distance across large collections of networks risk contamination from confounding variables, such as effective dimensionality, in under-pe ander-powered regimes. Overall, our observations on artificial network representations qualitatively agree with our simulated results (sec. 4.1) and analysis of biological data (sec. 4.2).
5 DISCUSSION
There is a vast literature on measuring representational similarity between neural networks (see Klabunde et al., 2023, for review). Recent works have leveraged distance metrics that satisfy the triangle inequality (Williams et al., 2021; Lange et al., 2022; Duong et al., 2023), yet the statis- tical properties of these shape distances are understudied in high-dimensional settings. Here, we theoretically characterized "plug-in" estimates of shape distance in nigh-dimensional, noisy, and sample-limited regimes. We found that these estimates tend over-estimate representational sim- ilarity when the true similarity is small. Further, they require a large number of samples, M, to overcome this bias in dibbeeiinniii regimes. Theorems and 2 provide precise guarantees on the worst-case performance of plug-in estimators. These bounds can guide the design of biological experiments, including pre-registered statistical power analysis. We strongly suspect that the depen- dence on N in these bounds is improvable (c.g. using techniques in Tropp 2015, the bounds can be refined to depend on the intrinsic dimension). However, in terms of the number of samples, the bounds definitively establish that the plug-in error decays at a rate proportional to 112
An equally important contribution of our work is to provide practical method to (a) reduce the bias of plug-in estimators of shape distance, (b) quantify uncertainty in shape distance estimates, and (c) enable practicioners to explicitly trade off estimator bias and variance. When employed on a biological dataset published by Stringer et al. (2019). we find that shape similarity estimates are highly uncertain, revealing the challenging nature of the problem in high dimensions and with noisy data. Importantly, this degree of uncertainty is not obvious from the procedures and plug-in estimates advertised by existing work on this subject.
In summary, our work characterizes the challenges of estimating shape distances in high- dimensional spaces. While shape distances can be well-behaved in certain settings (e.g. in noiseless artificial networks with many sampled conditions), our results suggest the need for carefully de- signed experiments and estimation procedures in sample-limited regimes.

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ACKNOWLEDGEMENTS
This work was supported by the Center for Computational Neuroscience at the Flatiron Institute of the Simons Foundation. DAP was supported Simons Collaboration on the Global Brain (SCGB AWD543027). We thank Jonathan Pillow for useful feedback. The code for our project is available at htttps:/ //ttt::////hhbbb// com/ dp48440000
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A APPENDIX: BACKGROUND ON GENERALIZED SHAPE METRICS
Here we provide several relevant derivations for generalized shape metrics. For more thorough review, we direct the reader to Williams et al. (2021) for the foundational results on generalized shape metrics and Duong et (2023) for the extension to stochastic neural networks.
We can intuitively think of the Procrustes distance as the Euclidean distance between two vectors remaining when the rotations and reflections have been "removed". Similarly, the Riemannian shape distance can be thought of as the angle between two vectors after these rotations and reflections are removed. These definitions in eq. (2) and eq. (3) also make clear that Procrustes distance, like Euclidean distance, is sensitive to the overall scaling of h; hj, while the Riemannian shape distance, like the angle between vectors, is scale-invariant.
A.1 EQUIVALENCE OF FQS. (2) AND (6); FQS. (3) AND (7)
The squared Procrustes can be reformulated in terms of the covariance and cross-covariance matrices as follows:
p² (2,, h2)
(2) (=)(2) QEO(N)
min
min E
QEO(N)
E [h.(2) h4 (2) + (1, -
max E Qh))))()((z)) QEO(N)
E Tr [h:(2
(=)
max QEO(N)
Tr (2)),
Tr [E (,,
(2 (=)"]]
max QED(N)
Tr
[QE [h,(z)h,(=)]]
[.,] + Tr أزز max QEO(N)
Tr Q
Tr [ + Tr أ - -1
Similarly for the cosine Riemannian distance:
(2,,(T)QQM) (2)
cos 0(t,, h1) = max QEO(N)
(2 h
maXQEO(N)
(T)
m Tr [3)(
(x) Tr (z)h,
maxQEO(N) Tr QE:y]
E ll+
Eg Tr Σ,
/Tr E. Tr E
A.2 REFORMULATIONS OF THE PLUG-IN ESTIMATOR OF PROCRUSTES DISTANCE
Let 21...., ZM denote set of independently and identically distributed samples in the network input space. Then, stack the responses of network i row-wise into a matrix X; IR MxN, Given this set up, a common definition of Procrustes distance is (Gower Dijksterhuis, 2004):
min QEO(N)
X. X,Q1r
(20)
VM
Here, have included multiplying factor of 1/VM for reasons that will become clear shortly. Aside from this factor, the quantity above is how Williams et al. (2021) define the Procrustes dis- tance. Below, show that the square of this quantity is indeed the plug-in estimator we defined in
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eq. (10) in terms of the empirical covariance matrices:
min 1 QEO(N) M
min 1 QEO(N) M
|x - X,Q1
(Tr X Tr - 2TxxXx
Tr
M
+
M
max QEO(N)
Tr
I x
Tr Σ
Tr |
+
max QEOIN)
= -Tr Σ
+
2 Eاأرك
= (ha, by)
B APPENDIX: PLUG-IN ESTIMATOR THEORY
B. PROOF OF LEMMA
Here we show that the plug-in estimate of the total variance Tr [Es] converges to the true variance Tr En] exponentially fast as M increases. We begin with some algebraic manipulations:
Tr Eal
Tr Benrl
,े
Σ
ha
mz 14 (Zm)
m=1
Ezur [TT] AM
Σ
Tr h1 2m )z((m))
M
Ex-P [Tr()s(z) [T<(a(=))<,(<>]]]]
1
Σ
Tr (zm) ) los (2m))
m==1
M hi(#m) he ( hm()m)
ExnP [h:(2)]T,,C # Σ
m=1
where we have used the property Tr[xx]] x x for any column vector in the last two lines.
The main assumption we are going to make is that the neural responses are constrained to an & ball of radius BvN or equivalently h2 hz)=)T,,,2) S B2N for all stimuli in the support of P. Note that this is reasonable assumption in both biological (energy constraints) and artificial neural networks (weight decay common).
Lemma 3 (Bounded Random Variables are Sub- Jub-Gaussinn, Wainwright (2019) Example 2.4). We say that random variable X with mean μ is sub- Sab-Gaussian with parameter if:
E [ < 222/2 for all XER
Intuitively, this means that the tails of X fall off faster than Gaussian. Furthermore, if X is mean zero and supported on the interval [a, b), the is sub-Gaussian with parameter a = (b a)/2.
Thus our assumption implies that each term with 3/ ha Th,(zm) is sub-Gaussian with param- eter B/M We can then immediately apply the Hoeffding bound (Wainwright, 2019, Proposition 2.5) obtain:
2Mt
P TH Eail 21
< 2 exx
(21)
B+N²
Analogously for term (B) we obtain:
2Mt
p
ThE
21 s 2exp
(22)
B'N2
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B.2 PROOF OF LEMMA 2
Our main tool is the matrix Bernstein inequality, given as theorem 6.1.1 in Tropp (2015). We para- phrase version of the theorem here keep our narrative self-contained.
Theorem (Matrix Bernstein). Consider a finite sequence (S SMs of independent, random N N matrices. Assume that:
E [m and ||5m L for each index m
(23)
where = = w ||m||2 S 1 is the matrix operator norm.
Further, define the variance of the sum 2m Sm as:
V max {11 EST. Sm 11 .1 S.EESSS S 1m
(24)
Then:
L E E Sm 1 2V log(2N) log(2N)
(25)
We now turn to the proof of lemma 2. Define:
Sm = M (e (he(zm)hs((e (Zm) Eig)
(26)
for the sequence of network inputs (=1,.... ZM }. Notice that:
1 M
E [Smm]
(E (Zm)) Eij) M
(Ei Enj) 0
(27)
Next, due to triangle inequality, we have:
1 | = M
[h hy (Zm)) Eigloo S M (2mm)) I'1
+ M
|E,
11
(28)
(2)
Terms (1) and (2) are each upper bounded by B² N, since for term (1): [11(2)))) (2(())))
10 [he((Z))ly)(a J ll2
(for any vector ||||| S 1)
(29)
(=m)) J 111/ss (Zm)
(30)
[ (zm lll 111112 (1mg(2m) ll
(Cauchy-Schwal) inequality)
(31)
< BVN 1. BVN= = B
(From assumptions in eq. 1)
(32)
And for term (2):
|
(33)
(2)
(for any vector ||0||22 S 1)
(34)
E (1)x(=)),,))))))
(Jensen's 's inequality)
(35)
< B N
(Repeat the upper bound on term
(36)
To summarize, we have:
2B
|m S M
|| (2(2)) 11.0 +
M 11:0 I
(37)
M
That is, we have shown that the assumptions of eq. (23) are satisfied with = 2B2
Our next task is to determine an expression for the variance V defined in eq. (24). First, we have:
E ST Sm
M El,, (m hr (zm) I hes (zm Mhg Zm T + TT. Eig
Zzm (Zm) - by (z (2 (Zm) 17 Egyl
Elyy mm No, (zm) by (z Ft (z 1
+ ΣJ Eij
ΣT. E[4 (zm (os( Elhr ) (zm) E.g
2
E mm mm) Zm My (Zm) )T
+ ΣT Σij ΣT. Eng ΣT
M
1hh (2m (z.mm)) (zm 1z (zm) I
MTT, Eis
M
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Then, by triangle inequality:
1
[ [ 00 =
[B]h (zm (25(2m)) Mn (hy E/j Dijllo
M
M [ (4 Nel (2m) 1 +
|
Lig
1oc
M
(B)
Terms (A) and (B) are each upper bounded by N2. First, taking term (A):
E (2m))) (zm (. h. (= (=m) I'1l
E [hy (= )a, (2m (= Mh (=m) Jolll
(for || S 1) (Jensen's 's)
hy (Zm (1 h4 (=m h (=m) Noll2
E (2m 1//l/, 112 1 [12)
- [1oll2llh, 112 [/, (=m)
1 B N. N = B*N²
(from eq. 1)
For term (B), we first note that
11 due to the fact that the operator norm Then, term (B) upper bounded by B N2 follows readily from:
E ووg x v || ود
submultiplicatiee
Eggle> bb (z)h, 1o
< | (2) ull2
(for ||l||| S
< E (z))))) o1l2
(Jensen's) 's)
E [1a(2))|2 l2 (=)
(Cauchy-Schwanl
< BvN BvN B
(from eq. 1)
Taking these two bounds together, we have shown |E ST Sm s 2B We are now ready to upper bound the variance term, V. appearing in theorem 3. Specifically, by the triangle inequality and the bounds above, we have:
M
2B N
ST
EMEST S
<
Σ m=1
S S
(38)
M
And it is easy to verify, using the same form of argument, that
213 N2
Σ E S.. S
Σ
JES mo S
(39)
M
Thus, V < 2B
With this, we are equipped to apply the matrix Bernstein inequality to obtain an upper bound on the estimation error of the plug-in in estimator. Specifically, we have:
E ||- EE,
11.
E
E.
Eiyll.
(reverse triangle inequality)
Sm E
Σ
< [E(1) Σ 1molloo]
NL
< N / log 2N)
log (2N)
(theorem 3)
< 2 Tog (2N) + 2B2 N2
log(2N)
3M
Where we have substituted the derived quantities L 2B2 N/M and V s NB*N2/AU /M in the final line.
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B.3 PROOF OF THEOREM
Lemma 2 provides upper bound on the expected value of
which is the error of our plug-in in estimate of covarianve nuclear norm. This bound holds for any true cross- covariance matrix Σij. provided that the constraints eq. (1) are satisfied. However, this tells us nothing about how the estimation error deviates around its expectation.
|ا 1|- 1
|E
Here, we use the bounded differences inequality (Wainwright, 2019, Corollary 2.21), also called McDiarmid's s inequality, to show that deviations around this expectation decrease exponentially fast. Thus, the upper bound on the expected error (lemma 2) provides accurate intuition.
Lemma 4 (Bounded Differences Inequality, Wainwright (2019) Corollary 2.21). Consider a func- tion f In - R. The function is said to have the bounded difference property for the kihh coordinate there exists Lk for which the following holds:
max
[(X - 1(X1=-1, X 1x++ S La
X CR" X
Suppose f satisfies this property with Li,--- La for each coordinate respectively. Then the follow- ing inequality holds:
P - [[(((1
21
21 S exp
(40)
Σラ= 1.3
We start by applying the reverse triangle inequality:
E 10l.-111:g) |ا S Eig Engll. Eig Σ
(z )a (Zm)
We can bound how much this changes we change one coordinate of the function, ie. h4 ((211)(g(22) is replaced by ha(Zi) hy The difference is then bounded by:
M E h4 (Zmm))) (z |. m=l
1 M
||
Σij - -
-(1.
M M m
(2m (y M (1(1))1((11) + " (()
||
Di
1
M (bi(z1)/)(21) - (1, (hg))))((2111
M |1((21)/)((1) - (hz
M (114(z)))y Il. + (1,,(2)) =
M (((x(z)))n<<<))) + (2, h (2:))
Finally, can apply Cauchy-Schwand and assumption about the neural activations being bounded to obtain:
= 1 ((h((t))llb)((())ll) + [1, (21)l12)
M ([h((z1)(ns(<))))
M
2B2N
< M
Thus have EM 12 =
M
EM Diz 4B 4B N2/M, and we can apply the bounded differ-
M
ences inequality to obtain for all 0:
+||
1 - 1
E 2t s
2xxp
Mr
(41)
2B'N2
For the deviation from the expectation to be in the range
[-t, t] with probability 1 1- 6 we require:
Mt
2 exp
< 8
N
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Solving for t gives t 2 B 1/2 22 (2/0), and thus with probability 1 8 the following holds:
E
E E E || |
- BBNMM-122 20gg(2/0)
This also implies that the following holds with probability at least 8:
2 log(2N) 2B2N² (log(2N)
BºN
E
Eigll
2 log
(42)
3M
M1/2
M1/2
where we have plugged in our expectation bound from lemma 2.
To complete the proof we need to combine the above tail bound with lemma inequality we have
By the triangle
10² -
Tr[En]] + 1r 2|
Tr 1r 11
2 |-
TH[E,]] Tr
+
Tr
TT
+ - 2||5, |-
Tr[E.1 Tr
Tr
و Tr
و أ + Eا || وار
For convenience, we replace with 6/3 in our results for these three terms, which yields that the following three inequalities independently hold with probability 8/3:
6 7
Tr [ - Tr [r
2 BNN 1/2
V
log
2
Tr[E,p] - Tr(E,;]
2 B2NM 1/2
log
2Ba N2 log(2.N)
2B² N22 / (2g)
B
6 g
E.gl |1
log
3M
M1/2
M1/2
By applying the union bound, obtain that all three inequalities hold simultaneously with prob- ability < 8/3 + 8/3 + 5/3 The three reverse inequalities then hold simultaneously with probability greater than equal to 1 8. Thus with probability at least - 6, the following holds:
NB2 log(2N) +
N2 [og(2N)
3B2N
6 7
10 --15
2 log
3M
1/22
11/2
as claimed in theorem
B.4 PROOF OF THEOREM (LOWER BOUND ON PLUG-IN ESTIMATOR ERROR)
We derive a lower bound by constructing explicit example where the plug-in estimator performs badly. Specifically, we consider a scenario where two networks have entirely decorrelated, high- variance representations. To do this, is called Rademacher variable if it behaves as follows:
Rademacher random variables. A random variable R
R +1 with probability 1/2
(43)
with probability 1/2
Now, suppose we sample M network inputs, 21,... ZM ~ P. independently. Further, let B be the constant appearing in eq. (1). For m {l..... M define
Xm = B h4 (2m)))
and Ym B h2(2m)
(44)
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Note that Xm and Ym are N-dimensional random vectors. Due eq. (1), have [1], I BvN and [[,z())]l2 BvN almost surely. Thus, ||X S VN and ||Ym vN almost surely.
Define X = (1/B)h,(2) and Y = (1/B)h, (z) for randomly sampled z ~ P. The case we will consider is that X and Y are each composed of N independent Rademacher variables. One trivial way to construct this is to suppose each z ~ P is random vector with 2N elements, all of which independent Rademacher variables scaled by a factor B 0. Then, let h4 R + R be the function which extracts the first N elements of x and let h, RRN + RN be the function which extracts the final N elements.
Thus, we have constructed setting where XI,...,Xay YI,, YI...., Ym are all composed of indepen- dent Rademacher variables. In this setting, the squared Procrustes distance is given by:
p² =T [at T+ 2 2 ||+
(45)
Tr
+
(46)
B². -X 2[
+ T. -
(47)
B². (E[x*) X]
(48)
B² (N +N
(49)
(50)
where we have used the fact that X and are independent, mean zero, random vectors to conclude that the cross covariance is an N x N matrix filled with zeros. Furthermore, note that X Xm. N and VTT Y. = N almost surely for all m E 1,.... M since they are comprised of N Rademacher variables. Thus, the plug-in estimate of the squared Procrustes distance takes the form:
x² = B² (Tr Σm Xm XT ml + Tr M Σm Y - 211 M Σm Xm Xm lmll)
(51)
-Ba- (1 Em XT X m + M Em YTY 211 M Em YT
(52)
B3a (N N 2/ Em Xm l
(53)
2B² N - 2B 1 MM Em Xm I.
(54)
Putting these two results together, we conclude that the absolute error of the plug-in estimator is:
lp² p² 2B Σm Xm XmYll-
(55)
Now, the product of two indepedent Rademacher variables is also a standard Rademacher variable. Thus, each element inside the matrix (1/M) Σm Xm YTT is the empirical average of M indepen- dent Rademacher variables. These matrix elements are asymptotically independent in the limit that M + co. Further, the central limit theorem applies in this limit, and thus the distribution of each matrix element approaches a Gaussian distribution N(, 1/M).
Such random matrices are well-studied under the name of Ginibre ensembles. In the limit that N + oo and the variance of each matrix element taken to be o² P/N,, the density of the singular values takes the following form (see e.g. Potters & Bouchaud, 2020, sec. 3.1.3);
V s2
p(s)
8 E (0, 20)
(56)
This is called the quarter circle law since if we look at the density of s it forms a quarter circle. The nuclear norm of the matrix is N times the expected value of with with respect to the density p(s). Integrating this density, we obtain:
N
f
20
lim
Em X m
402 SV
SV
82
ds
(57)
TT?
N
$2/3/2 20
2402
(58)
2
N
[
N
[8
(402)8822
(59)
TO²
T
3
80
8
N
N3/2
M -1/2 1/2
(60)
3m
3x
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Published as a conference paper at ICLR 2024
Where in the last line we have substituted variance of each matrix element in eq. 56) with 1/M (the variance given by the average Rademacher variables under the central limit theorem). Note that the analysis above holds asymp- totically as M, N - oo and we keep M > N so that the central limit theorem continues to hold.
v N/M, which comes from equating o² /N (the
Plugging eq. (60) into eq. (55) and dividing both sides by N we arrive at the expression appearing theorem 2.
B.5 NUMERICAL VERIFICATION OF LOWER BOUND
Figure below shows in simulation that the plug-in estimator closely agrees with the lower bound established by theorem 2.
N=50
M=200
ground moment est est
||E
plug-in theory
500 1000
0
50 100
sampled inputs (M)
neurons (N)
Figure 5: Simulation of lower bound on plug-in in error. Simulated responses from N neurons were sampled from (0, I) independently for pairs networks over M stimuli. We plot the observed nuclear norm of the empirical cross-covariance matrix in blue, and show close agreement with equa- tion 60 in green. The moment-based estimator is shown yellow for comparison; note that the variance of the moment-based estimator decreases with M and increases with N. Left, simulations from N 50 neurons as M was varied. Right, simulations from 200 stimuli as was varied.
C APPENDIX: METHOD-OF-MOMENT ESTIMATOR
C.I DERIVATION OF METHOD-OF-MOMENT ESTIMATOR
We now turn to constructing our method- of moments estimator of |E,,|- - sn (Ei)) which required our novel estimator the Riemannian shape distance. We can form an unbiased estimator of the matrix Dij by observing single random stimuli in the two networks:
Eiym hi(Zm)h)(Zmm) ERNNN
EJS.gml Egg
Note that here the randomness comes from the selection of the stimuli, i.e. 2m P; the output of the network is deterministic. Assuming m, ml are distinct stimuli drawn independently from the distribution P, we then have:
Eegm Eegm = Ei; E!
E
This means estimate Σ, Σ! by observing pair of stimuli in both networks.
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Published as a conference paper at ICLR 2024
Th (1i))
Σ (c)
2p (iij)
Taylor expansion of f(-)
N
2p (ig)
= Σ
Ip Tr Eis Eij )
Tr
Σ 2p (>s)))
1p
p=0
A=1
p=0
8
P
7pp
Tr
Σ in (20 E0((20)
Substitute unbiased estimator for
p=0
LE
Tr
Σ 12 20 1) (1((20)
Approximate with truncated power series
p=0
Our estimator for the nuclear norm of Eig is thus:
P E Syee II Lo=1
Eigle = Σ
7p TT
E
(61)
p=
Note that for each element of the product we are considering the estimator based stimuli (2a - 1) and (2x), ); in total this estimator will use 2P unique stimuli.
We note that unbiased estimate of the numerator of CKA has been developed and evaluated (Song et al., 2007; Kornblith et al., 2019; Murphy et al., 2024)-but has not been extended to stochastic networks. The numerator of CKA is W1 (eq. 16) thus this extension could be made using the strategy described in Section 3.4.
C.2 DERIVING THE QUADRATIC PROGRAM
The optimization problem in eq. (19) takes the form:
minimize A'A + N2 (max (7,
(62)
where f(7,x) = 1/2 Ep Nno??
Cov(WV, Wi)
Cov(Wi, Wp)
711
= : E R.P.
=
E PxP
(63)
7P
Cov(WP, Wi) Cov(Wp, WP))
Notice that is linear in y, and that A is symmetric, positive-definite
We will reformulate eq. (62) in several steps, and ultimately obtain a quadratic program that can be efficiently solved. First, we introduce bound on f2 x)) for all x E [0, 1]. Thus, the optimal y for the problem:
optimization variable R whose square is upper
minimize 7,u
A + 2
(64)
subject to n² (2)))))) for all x E [0,
coincides to the optimal solving eq. (62). This is essentially an epigraph reformulation of the orig- inal problem (see Boyd Vandenberghe, 2004, equation 4.1 Notice that the objective function is quadratic in this reformulation.
Next, we lay down a fine grid of linearly spaced test points ,1....,,7 Er [0, 1]. We can then obtain good approximation to the solution in eq. (64) by solving:
minimize 7.u
TT Ay + 2
(65)
subject to
2² > f (7,
for all E I....,T
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Of course, increasing T (the number of test points) improves the approximation arbitrarily well.
Finally, the constraints of the problem can be put into form that is jointly linear in 7y and u. First, constraining u² 2 (r, 21) is equivalent to simultancously constraining fz) and u 2 -f(7, 24). Then, plugging the definition of zi), and rearranging we have:
minimize 7:u
Ay 22
subject to
u + Σ
1/2 Ipath 2 xi
for all E ,...,,
(66)
P
u Σ "pat
1/2 for all E El....,? -t
This objective is quadratic and the constraints are linear with respect to the optimized quantities. Thus, a solution (approximated to high accuracy) can be achieved efficiently using off-the-shelf quadratic programming solvers. To enforce the user defined bound the bias final two constraints are be appended to eq. (66): - -N 2 -c -c and Nu 2 -G, where e is the upper bound on the absolute bias.
C.3 CONFIDENCE INTERVALS
To form approximate level confidence intervals around E term 1) and variance (eq. 19, term 2) from the quadratic program's 's solution:
the maximal bias (eq. 19,
E
V Ar Nu, Exyll + V 7 Ay + Nu
where zt is the critical value of the standard normal. For confidence intervals of the similarity score we scale this interval by the denominator the similarity score.
D APPENDIX: EXPERIMENT DETAILS
D.1 SIMULATED EXPERIMENTS
To draw data for simulations, we set the eigenvalues of the En and the singular values of E, to a ground truth nuclear norm and similarity score. To demonstrate the estimators accuracy across the space of orthogonal transformations we apply random orthogonal rotation matrix to each population's 's covariance in each new parameter setting.
D.2 EXPERIMENTAL DATA FROM STRINGER ET AL. (2019)
Neural activity in mouse primary visual cortex recorded using two-photon microscope while mice were free to run on an air-floating ball. Recordings were collected across multiple depth planes frequency of 2.5 3 Hz, with planes 30-35 jmm apart. The field of view of the microscope was selected such that 10,000 neurons could be observed within retinotopic location on the stimulus display.
All stimuli were presented for 0.5s with a random inter-stimulus interval between 0.3 and 1.1s con- sisting of grey-screen. The images used in the experiment were taken from the ImageNet database, which includes categories such birds, cats, and insects. The researchers manually selected im- ages that had a mix of low and high spatial frequencies and that did not consist of more than 50 % uniform background. All images were uniformly nontrastnormalizzd by subtracting the local mean brightness and dividing by the local mean contrast. Each stimulus consisted of different normal- ized image from the ImageNet database, with 2,800 different images used in total. The same image was displayed on three screens, but each screen showed the image at a different rotation. Each of the 2,800 natural image stimuli were displayed twice in recording in two blocks of the same randomized order.
Calcium movie data processed using the Suite2p toolbox to estimate spike rates of neurons. Underlying neural activity was estimated using non-negative spike deconvolu- tion (Frierich al., 2017). These deconvolved traces normalized the mean and
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standard deviation of their activity during 30-minute period of grey-sereen spontaneous activity. For further detail please see the original study Stringer et al. (2019). All anal- yses done in this paper were performed the pre-processed data available figshare (https: /ttts://figgsaaa wttt:/ticccccss/////ccocccccccee in in_visual ceciissaa_cootee__eeeeeeeee worteoortooeeo.ooo wtwpaasseenssseeee_____e__
otternnsssseeeettosssseeeeeeee neuronss__ 800 catooonntuuralla Laapppnssttt_LLLL__naaaaaaaaaaa i
E APPENDIX: APPLICATIONS TO DEEP LEARNING
Here we apply the plug-in and moment based estimator to neural network responses to demonstrate impacts of the differences between these estimators and relevance neural networks. We make the point that the bias of the plug-in estimator, but not the moment estimator, is substantial for small samples. Furthermore, plug-in bias depends on the effective dimensionality of the two populations. Thus, naively using the plug-in can lead to erroneous scientific conclusions because the estimate bias can correlate with irrelevant nuisance variables. Concretely, we find the plug-in estimator bias tends to decrease with the effective dimensionality of neural populations. Thus if similarity between two populations appears to be explained by some manipulation of interest (c.g., training regime) can be confounded through variation in effective dimensionality.
common question in the study of neural network representations is how two networks with the same architecture trained on the same task but with different initializations and training procedures are similar/different from each other. To show how the estimators considered in this paper can be used a tool to study this question, et al., 2016) architectures trained to categorize ImageNet (Deng et al., 2009), specifically two sets of pretrained weights available in Pytorch (ResNet S_tieettt_WWWeat @WWiggtt IMAGENETI K_V1 and ResNet 50_ SeSoWWeighe WWW.ightt............... IMAGENETIK.V. as described here). We then compared a randomly cho- sen subset of the neurons (100 in each network) in the penultimate layer (before the final fully connected layer mapping to the logits) across the two networks using the plug-in estimator and the moment estimator. To compute a ground truth similarity metric we applied the plug-in estimator to the responses of these units across 432,064 images randomly chosen as a subset of the Ima- geNet dataset. To compare finite sample bias for each number of observed stimuli M, we randomly re-sampled across images and calculated the mean and SE of the two estimators as a function of number of images (Fig. 6). We found that the bias of the -nttiimattt was at worst 3-fold and this bias decreased slowly, whereas the moment estimator showed small amount of bias even with the smallest numbers of samples.
considered two ResNet-50 (He
limensionality=100
0.5
Similarity
0.3
μi # SE
0.2
ground truth Moment est. Plug-in est.
0.0
100
200
300
400
stimuli (M)
Figure 6: Comparison of plug-in and moment based estimators bias estimating similarity in the hidden layer of deep neural network as function number of samples. Here we specifically study a subset of the neurons (N = 100) in the penultimate layer of two ResNet-50 architectures trained on ImageNet with different initializations and training procedures.
Finally, considered how the bias of the plug-in estimator would vary with respect to irrele- vant properties of the neural populations chosen. We reasoned that such a dependence could con- found results on similarity between neural populations. It known that the effective dimensionality, (EA1) A.)3/E/ X² of response distribution determines the rate at which its sample covariance and thus singular values can be estimated. To determine if this in turn biased the plug-in estima- tor in a real application we randomly re-sampled with out replacement 100 units of the 2048 from the two neural networks 1000 times. We measured the ground truth similarity for each subset, the geometric mean the effective dimensionality of the 100 units from the two networks (calculated across all images), and the plug-in average estimate across 50 random samplings of images. We
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found that the bias (difference of average plug-in estimate and ground truth), across re-sampling of units had a moderate negative correlation (r=0.331) with the effective dimensionality of those popu- lations. Thus observed differences in the similarity of neural network units may be confounded by their dimensionality and its effects on the plug-in estimator.
r=
0.30
Plug-in estimator bias
0.25
0.20
0.15
0.10
0.05
0.00
1.25
1.50
1.75
2.00
Effective dimensionality
Figure 7: Each dot corresponds to a random selection of 100 neurons from the 2048 neurons in the penultimate layer of a ResNet-50. Across subsets of N 100 neurons, we observe a negative
correlation between the bias of the plug-in estimator with M = 50 stimuli and the effective dimen- sionality of the neural activations.
24


Reviews:
[{"review_id": "TaC8xlkSq7", "reviewer": "Reviewer_ATPd", "summary": "The paper proposes a novel estimator for distances between neural representations from both biological and artificial neural networks. The key technical contribution is a new moment based estimator of the nuclear norm of the cross covariance matrix which has significantly lower bias than the plug in estimator. The authors provide both theoretical and empirical analysis to illustrate the benefits of the proposed estimator.", "questions": "1. In the last equation on page 5, why is $x \\in [0,1]$?\n\n2. In Section 4.2 it is acknowledged that the proposed estimator is highly variable in low SNR regimes and so neurons with the highest SNR are selected. Can you provide the relative proportion of such neurons in the presented scenario and also (if possible) comment on their prevalence in general? This is important because if the relative proportion is small then it means that the proposed estimator can only be applied in very few cases.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The problem is clearly identified and analysed theoretically, and the estimator proposed to solve it is principled and technically sound. The theoretical analysis in Section 3 is fairly detailed and appears to be novel. It clearly explains the flaws in the plugin estimator and the subsequent derivation of the new estimator is also well grounded in theory.\n\n2. Simulations and experiments match the intuition used to derive the estimator (Fig 2A) and clearly highlight the strengths of the estimator over the plug-in baseline.", "weaknesses": "1. The overall motivation of the problem is a bit weak. The experiments don't clearly illustrate the downstream benefits of this work. Further experiments showing tangible gains on at least one real world application would greatly strengthen the paper in my opinion.\n\n2. The main technical weakness of the approach appears to be its effect on the variance. While deriving the estimator the authors seem to primarily focus on reducing the bias compared to the plug-in estimator and acknowledge at the end of Section 3.3 that the bias is being bounded at the expense of the variance. This is also seen in the experiments (Fig 2) where the proposed estimator appears to have a significantly higher variance than the moment estimator. This raises questions about the reliability of the estimator, specifically it is not clear if it can reduce the mean squared error for all distributions. This is further illustrated in Fig 4 where on real neural data it appears that other than the case where similarity is set to 0 (A) the plug-in estimator performs comparable to (C, D) or better than (B) the proposed estimator.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "D2Ea5I7tdK", "reviewer": "Reviewer_sJ3L", "summary": "In this paper the authors study the estimators of the so-called \"shape distance\", more precisely Procrustes size-and-shape distance $\\rho$, and Riemannian shape distance $\\theta$., that are distances defined on data manifold. They measure the uncertainty (bias and variance) of the empirical estimates of these distances, relying on centration inequalities, and design a new estimator whose bias/variaqnce tradeoff can be controlled. Their theoretical results are illustrated on:\n* a synthetic experiment\n* measure of calcium in the neural activity of a mouse", "questions": "### Application to artificial neural networks\n\nWhich results your method yield in the latent space of an *artificial* neural network?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "### Clarity\n\nThe paper is well written, and the and theoretical results are explained in a way that make it accessible to a broad range of readers. The proof techniques are well explained. \n\n### Quality\n\nExperiments are convincing and support", "weaknesses": "### Out of scope: no link with neural networks\n\nThis work has nothing to do with artificial neural networks. In the statement of the problem, the authors assume that thew observations are $X=h_i(Z)$ with $Z$ the input data  and $h_i$ the neural network. No hypothesis is made on $h_i$ nor $Z$. The whole paper could be rewritten with $X$ to get rid of the assumption that the measures are the outputs of a neural network. Even Theorem 2 mentions \"neural\" for no good reason: authors only meant to talk about the existence of some r.v $X$.  \n\nThe only experiment linked with neural network is in Sec 4.2, and the \"neural data\" is actually *calcium measurements* from mouse primary visual cortex, which are typical tabular data.  \n\nOverall, a conference or journal focused on statistics seems to be a better match for the content of the article than ICLR.  \n\n### Novelty\n\nThe novelty is poor: all the proofs are classical bias-variance decomposition and concentration inequalities. \n\nMoreover, there exists previous work on the topic that are not covered in the literature review:   \n   \nKent, J.T. and Mardia, K.V., 1997. Consistency of Procrustes estimators. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 59(1), pp.281-290.\n  \nGoodall, C., 1991. Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society: Series B (Methodological), 53(2), pp.285-321.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "ENPtYDfxR4", "reviewer": "Reviewer_GC8w", "summary": "The paper theoretically quantifies the performance of some typical shape distances between neural representations and the dependence of the performance with respect to the number of samples. The authors find that typical methods have low variance but high variance, and instead propose a new method that enables a tunable bias-variance tradeoff.", "questions": "NA", "limitations": "", "rating": 10, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The paper contains significant insight into the performance of 'plug-in' estimators of shape distance between two neural representations and the performance dependence on the number of samples and the dimensionality of the ambient space.\n- The paper identifies the key component behind the non-ideal performance of these estimators (the $||\\Sigma_{ij}||_\\ast$ term) and proposes a simple but intuitive and effective estimator to allow a tunable bias-variance tradeoff.\n- There are adequate experiments on synthetic data to validate this new estimator.", "weaknesses": "- The paper should contain some more intuition-building sentences so that the mathematical formulation is more easily digestible. However, the authors have done a really nice job of making the mathematical foundations and derivations themselves clear.\n- One small experiment with VAEs (even if it is confined to the appendix) might be useful for quantifying the performance of stochastic networks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}, {"review_id": "Ce2b5iDPFe", "reviewer": "Reviewer_emQE", "summary": "This paper extends some works from shape analysis into the space of neural networks. That is, how does one measure the distance between two NNs where a NN is represented as a mapping h:features ->R^N. The main idea is to think of this mapping into the space of point clouds and hence one can use a Kendall Shape space type distance. The key difference contribution I see is the fact the point clouds have some stochastic properties to them so the shape distance is measuring the distance between expectations. The authors spend a great deal with considering the bounds of these distances.", "questions": "When setting up the background, the authors define $h_i:\\mathcal{Z}\\rightarrow \\mathbb{R}^n$ as a function representing each neuron. I have no problem with this definition. My question is, why do the authors limit their writing to that of NNs while this is true for a much wider class of functions. I may have missed something in an assumption but it seems these functions can be quite generally considered.\n\nFor (5) isn't the last equation enough?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "A key component of this paper is the bounds on the distances when using an estimator for the cross covariance estimator. The authors present Lemma 1, 2 and Theorem 1 to do this (and theorem 2). The authors explain this bound well as it makes sense that the higher the dimension, the more landmarks need to be observed. \n\nThe authors show how under a variety of scenarios their estimator is performing. The variety is suitable.", "weaknesses": "When referencing the shape distances such as (2) and (3) the authors cite Williams 2021 but these are simply common shape distances which have been around for much longer. Unless this specific formulation is novel to Williams 2021 I don't see how it's different than just measuring the distance on a circle.\n\nWhen considering the \"failure modes of plug-in...\" there is the missing component of mentioning scale. If h_i differ in scale their $\\rho$ distance can be large but their $\\theta$ distance can be very small as it rescales in the denominator.\n\nSome of the figure axis are impossible to read.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)