Efficient neural representation in the cognitive neuroscience domain: Manifold Capacity in One-vs-rest Recognition Limit
Authors: Nga Yu Lo
SueYeon Chung
Venue: Submitted to ICLR 2023 (2023)
Keywords: computational neuroscience
statistical physics of learning
representation geometry
perceptual manifolds
object recognition

Abstract: The structure in neural representations as manifolds has become a popular approach to study information encoding in neural populations. One particular interest is the connection between object recognition capability and the separability of neural representations for different objects, often called "object manifolds." In learning theory, separability has been studied under the notion of storage capacity, which refers to the number of patterns encoded in a feature dimension. Chung et al (2018) extended the notion of capacity from discrete points to manifolds, where manifold capacity refers to the maximum number of object manifolds that can be linearly separated with high probability given random assignment of labels. Despite the use of manifold capacity in analyzing artificial neural networks (ANNs), its application to neuroscience has been limited. Due to the limited number of "features", such as neurons, available in neural experiments, manifold capacity cannot be verified empirically, unlike in ANNs. Additionally, the usage of random label assignment, while common in learning theory, is of limited relevance to the definition of object recognition tasks in cognitive science. To overcome these limits, we present the Sparse Replica Manifold analysis to study object recognition. Sparse manifold capacity measures how many object manifolds can be separated under one versus the rest classification, a form of task widely used in both in cognitive neuroscience experiments and machine learning applications. We demonstrate the application of sparse manifold capacity allows analysis of a wider class of neural data - in particular, neural data that has a limited number of neurons with empirical measurements. Furthermore, sparse manifold capacity requires less computations to evaluate underlying geometries and enables a connection to a measure of dimension, the participation ratio. We analyze the relationship between capacity and dimension, and demonstrate that both manifold intrinsic dimension and the ambient space dimension play a role in capacity. 

Under review as a conference paper at ICLR 2023
EFFICIENT NEURAL REPRESENTATION IN THE COGNI- TIVE NEUROSCIENCE DOMAIN: MANIFOLD CAPACITY IN ONE- -S-REST RECOGNITION LIMIT
Anonymous authors Paper under double-blind review
ABSTRACT
The structure in neural representations as manifolds has become popular approach to study information encoding in neural populations. One particular interest is the connection between object recognition capability and the separability of neural representations for different objects, often called "object manifolds." " In learning theory, separability has been studied under the notion of storage capacity, which refers to the number of patterns encoded in a feature dimension. [Chung et al. (2018) extended the notion of capacity from discrete points to manifolds, where manifold capacity refers to the maximum number of object manifolds that can be linearly separated with high probability given random assignment of labels. Despite the use of manifold capacity in analyzing artificial neural networks (ANNs), application to neuroscience has been limited. Due the limited number "features", such as neurons, available in neural experiments, manifold capacity cannot be verified empirically, unlike in ANNs. Additionally, the usage of random label assignment, while common in learning theory, is of limited relevance to the definition of object recognition tasks in cognitive science. To overcome these limits, we present the Sparse Replica Manifold analysis study object recognition. Sparse manifold capacity measures how many object manifolds can be separated under versus the rest classification, form of task widely used in both in cognitive neuroscience experiments and machine learning applications. We demonstrate the application of sparse manifold capacity allows analysis of wider class of neural data in particular, neural data that has limited number of neurons with empirical measurements. Furthermore, sparse manifold capacity requires less computations to evaluate underlying geometries and enables connection to a measure of dimension, the participation ratio. We analyze the relationship between capacity and dimension, and demonstrate that both manifold intrinsic dimension and the ambient space dimension play a role in capacity.
INTRODUCTION
The approach to study neural populations as manifolds and their geometry has become a popular method to uncover important structural properties in neural encoding and understand the mechanisms behind the ventral stream, the motor cortex, and cognition (Kriegeskorte & Kievit
et al. 2018) (Gallego et al.) 2017) (Sohn et al. (Ebitzz & Hayden 2021 2 (Kriegsskooe & Wei 2021 (Chung & Abbott 2021). In the ventral stream, the invariant ability for humans and animals to recognize object despite changes in pose, position, and orientation has motivated definition of object manifold as the underlying representation of neural responses to a distinct object class. A long-standing hypothesis in visual neuroscience posits that the visual cortex untangles these object manifolds for invariant object recognition (Dicarlo & Cox 2007), relating object recognition to the separation of manifolds by some linear hyperplane.
There a well developed theory of linear separability given by Gardner (1988) that studies the separation of points by perceptron. The theory quantifies capacity load that describes the maximum number of points that can be linearly separated given a random dichotomy (a random assignment of binary labels to the manifolds). The capacity load also encodes the number of points stored per feature dimension required to have linear separability. This theory of separation, however, does not connect to the geometries of the underlying representations.

---

Under review as a conference paper at ICLR 2023
Dense Labels (Computer Science Theory)
Sparse Labels (Cognitive Science)
Probecilyy Sepesiion
VC Dimension
Cover's Theorem
Hamster
Cet
Truck
Figure I: Dense vs Sparse labels (left) Computer science theories such as VC dimension and Cover's Theorem are interested in capacity under all random dichotomies. Under this regime, manifolds are densely labeled where more than one manifold has positive label. (right) Conversely, neuroscience studies cognitive tasks such as object recognition, where one-vs-rest dichotomies are more relevant. Under this regime, manifolds are sparsely labeled where only one manifold is assigned positive labeling at time. In each dichotomy, (red) is the positive label and (blue) is the negative label -1.
Chung et al. (2018) extended the notion of capacity from points to manifolds and related capacity manifold geometric properties. Manifold capacity, defined as the maximum number of object manifolds that can be linearly separated given random dichotomy, has been used to show how classification emerges along layers of deep neural network (DNN) undertaking visual object recognition Cohen et al. (2020) and speech recognition (Stephenson al. 2019). Furthermore, has been shown that geometric properties of the manifold, defined as manifold radius and dimension, decrease in magnitude while capacity increases. This observation suggests feature representations with low intrinsic dimension allows invariance and robustness in classification.
Despite the use of manifold capacity in analyzing DNN, its applicability to neuroscience has been limited due the way manifold capacity is defined. The current theory following Gardner's framework considers all random dichotomies, similar to other theoretical computer science studies. VC dimension in learning theory returns the largest number of points in a set such that every dichotomy of the set can be learned by a given hypothesis function (Vapnik & Chervonenkis 2015 Abu-Mostafa al.] 2012). Cover's theorem returns the number of linearly separable
given P points and a D dimensional space (Cover) 1965). The regime of object recognition for classification model and for a monkey performing a delayed matching Majaj et al. (2015b) or oddity task, however, is equivalent to the separation of manifolds on
basis (Figure TT. In other words, the only relevant dichotomies are those where only one manifold has a positive labeling. This relates to the notion of sparse labeling in Chung et al. (2018).
Sparse labeling also overcome the technical restrictions of using manifold capacity ana- lyze biological neurons like previous works have for artificial neurons. The current manifold capacity theory for random dichotomies falls outside the regime of most available neural datasets, namely, data with limited number of simultaneously recorded neurons (Gao et al., 2017). Hence, as modern large scaled probing techniques improve and become publicly available (Jia et al.) 2019 Steinmetz et al.) 2021), analyzing current data requires further innovations in theoretical and analysis framework. Under sparse labeling, capacity is greater than in the traditional regime (Chung et al. 2018 Gardner 1988). It follows that, under the sparse label regime, we can verify capacity in datasets with fewer number features, or neurons, which was not previously possible (Froudarakis et al. I). Thus, the sparse labeling regime allows us to apply the theory of manifold capacity to real neural data and use capacity as a measure of recognition and similarity between DNN and the biological brain.
In this paper, we extend the work presented Chung et al. 2018)) to analyze neural data by estimating manifold capacity in the one-vs-rest recognition limit. We define sparse manifold
2

---

Under review as a conference paper at ICLR 2023
capacity the ratio of P/N where P is the number of manifolds in given system, and N is the degree of freedom required such that the probability of linearly separating a one-vs-rest dichotomy is 0.5. Our contribution is as follow:
We present the Sparse Replica Manifold analysis for estimating sparse manifold capacity in the one-vs-rest recognition limit. Our analysis overcomes the limitations in Chung et al. (2018) by taking into account of correlation between manifolds and the effects of a heterogeneous mixture of manifold geometries, thereby making the analysis applicable to real complex data
We show that the application of sparse manifold capacity allows analysis of wider class of neural data that has a limited number of neurons with empirical measurements. We demon- strate, for the first time to our knowledge, the match between the theoretical and empirical manifold capacity in real neural data, and suggest capacity as a reliable measurement of linear separability in the biological brain.
We show that the manifold geometries under the sparse label regime is faster to compute than in the classical regime, allowing efficient analyses via manifold geometries.
We explicitly illustrate the effects of ambient dimension and manifold intrinsic dimension on sparse linear separability. In particular, we show that better linear separability is related to both high ambient dimension and low manifold intrinsic dimension, filling in the gap of knowledge on the role of high and low dimensional representations in neural encoding.
2 REVIEW OF MEAN FIELD THEORY MANIFOLD ANALYSIS
We will first summarize the Mean Field Theory Manifold Analysis (MFTMA) for object manifolds first introduced in Chung et al. (2018). This framework extends the calculation of perceptron capacity of discrete points (Gardner 1988) to capacity of manifolds. Consider a system of P manifolds in a N --dimensional ambient space, where each manifold has an affine dimension of D. We define the capacity load az = P/N, where N₂ is the critical number of feature dimensions required such that 0.5 of all manifold dichotomies number of categories encoded per feature dimension. Chung et al. (2018 8) calculated capacity as an average of aM, for each manifold M. To determine am, equation
separable by hyperplane. 0₂, can also be interpreted as the
involves finding manifold anchor points, S(C, to), uniquely determined by Gaussian vectors T R D+1 whose components T; (0, 1). For the rest of this paper, we denote 7 (o) (I E IRP,4 dichotomy and orientation of the manifolds, the anchor point is point on the manifold or its convex hull that contributes to the separating linear hyperplane. The notion of anchor point emerges from the solution to the KKT interpretation of the mean field equations for capacity, discussed in depth in Chung et al. (2018) (see also appendix[A.3 [A.].. Similar to support vectors in Support Vector Machine, these anchor points change based on the location and labels of other manifolds. The Gaussian vectors represent the variability in orientations, locations, and labels of other manifolds.
R). Given some
| - F. S(E, to)
RM
t() + 1)² E.to
(I)
MFTMA links the connection between this capacity load with the manifold geometric properties. In particular, Chung et al. (018)) defined for each manifold, the manifold anchor radius (RM) squared as the mean squared length of all s(F, to). i.c. R2 (I) to)ll In addition, the manifold anchor dimension (Das) measures the mean angular spread between the unit vector of an anchor point (&) and the Gaussian vector that determines the anchor point, i.e. DM ((F s(f,, ))) Chung let al. (2018) also showed that one can estimate a. using manifold anchor radius and dimension by BBlll (K, RM, DM)) with margin of KG
OBuu (1, R, D) 1o dtxo(t) [Metit
Dto (=60+144 (1+82)
+ ---1 Dto (to (x) ++)]
(2)
where Dt is the Gaussian measure and (t = 2'-8 D- 1-1 2d," t 62
T(P)
1 Upoon publication, the Sparse Replica Manifold Analysis will made publicly available for the use of the neuroscience and machine learning community to analyze artificial neural networks and biological data.
{-) denotes taking the average in this paper.
3

---

Under review as a conference paper at ICLR 2023
2.1 GAUSSIAN RADIUS AND GAUSSIAN DIMENSION
Shown in Chung et al. 201 8). for small manifold sizes, the anchor point (3) depends only on This motivates definition of manifold radius similar to the Gaussian mean width. Chung et al. (2018) additionally defined the Gaussian radius (Ry) and Gaussian dimension (Dg) as
R2 $| (013)) D, -(F-& ())))
(3)
where (t) is the anchor point determined by T, and 6g is its unit vector. These measures play important roles when considering classification tasks with sparse labels (see next section).
3 MANIFOLD CAPACITY FOR ONE-VS-REST OBJECT RECOGNITION
The MFTMA estimates the capacity load when 0.5 all 2 dichotomies in system of manifolds are separable. In neuroscience and learning models, however, a more practical usage of this theory is to estimate capacity where manifolds are separated on one-vs-rest basis. This falls under the notion of sparse labeling, in contrast to dense labeling. In other words, want to consider only the P dichotomies, in each which only one manifold has a positive labeling. Assuming that each manifold represents class of object, estimating capacity one-vs-rest basis is equivalent estimating the capacity for a system undertaking an object recognition task.
In the following sections, we present the Sparse Replica Manifold Analysis that computes manifold in the one-vs-rest recogmition limit, also called sparse manifold capacity. Sparse manifold capacity was previously considered in Chung et al. (2018) in the context of random uncorrelated manifolds with identical geometries (i.e. every manifold has the same manifold radius and dimension). We extend this work by taking into account of heterogeneous geometries and correlations between manifolds. In the following sections, we first review the theory for sparse capacity considered in previous work. Then, provide the theorem for calculating sparse manifold capacity in the general case with a mixture of different manifold geometries. Lastly, we discuss the case where manifold center correlation are taken into account.
3.1 SPARSE MANIFOLD CAPACITY
Define f. the sparsity parameter, as the fraction of positively labeled manifolds in a dichotomy. Then, T-1/P corresponds to the one-vs-rest dichotomies. Under the regime of sparse labeling, capacity depends on maximizing a bias parameter, where the hyperplane is not constrained to pass through the origin. This bias parameter has a positive contribution the margin for the positively labeled manifold and negative contribution for the negatively labeled manifolds. The resulting effect of the bias parameter, shown in Chung al. (2018), puts the manifolds regime where lo becomes negligible to the anchor point of contributing manifold. These observations allow us to use equation i to estimate the sparse manifold capacity assuming every manifold behaves as ball with the same Gaussian radius (Rg) and Dimension (D,) (Chung et al. 2018).
Okc, (HMMO Rg: Dg) = max(ʃ [ BB (s + b, Rg D,) + (1 - aau b, Ry D,)1 (4)
b
3.1.1 SPARSE MANIFOLD CAPACITY WITH MIXTURES OF SHAPES
In this work, to account for the different Gaussian Radii and Dimensions for each manifold in heterogeneous system, we derive and employ Theorem T The derivation this theorem is provided in the appendix I and short version is given in the next section. For each manifold i 1, calculate a from Equation where R is the array of all manifold Gaussian Radii in the system, and D the array of all manifold Gaussian dimensions the system. The sparse manifold capacity is the average of all a.s.
Theorem 1. Given P manifolds. Let D DD.t.... be the array of their corresponding Gaussian dimensions. And let Rg = [Rg be the array of their corresponding Gaussian radii. Then sparse manifold capacity, OC.SPARSE is computed as (a (1.2....P) where
O (K., Rg D2)) max(1// / Ptae/Poopullle azdu (x
Rigi Dgi)
Σ 1/P-aaauue (n b, Rgg. Dgi)) (5)
1#1
4

---

Under review as a conference paper at ICLR 2023
3.1.2 DERIVATION TO THEOREM 1
Given Di and Ry by (Chung et al. 2018). the capacity of such ensemble of manifolds is the average
of a.M, computed for each manifold by equation | or estimated as @BALL (K. R. D) (equation
2). Sparse capacity for a system of manifolds with identical shapes, is computed by equation 4
Observe that equation 4 is averaging across AT.I.. ss each with inputs from manifold with either positive labeling or negative labeling indicated by ++b and -b. Since all and are the same in a homogeneous system, equation 4 sufficiently accounts for all possible assignments of positive and negative labeling. In the case of heterogencous geometries, it follows that we need to consider all possible dichotomies given by the sparse parameter. In order to conserve equation 4 averaging across all such dichotomies suffices. For the one-vs-rest regime discussed in this paper, there are only P dichotomies to consider, within each only one manifold is positively labeled. Thus, for each manifold i, compute equation 5 Then, averaging across all ouys, i.e. across all relevant dichotomies, gives us the sparse manifold capacity.
3.1.3 SPARSE MANIFOLD CAPACITY WITH CENTER CORRELATION
MFTMA assumes that the manifold centers of the system are randomly related. Realistic data, however, tend to exhibit correlation. Cohen et al. 2020) took center correlation into account by projecting the manifolds into a space where they will have low-correlated centers. This low correlation space is determined via the Euclidean centroids of each manifolds. We follow a similar approach, but instead using the Euclidean centroids of each manifold to recover the space of low correlations, our approach for computing sparse manifold capacity with center correlation uses categorical local linear differences (CLLD), which are sampled differences between the local linear centers of the manifolds.
Suppose manifold M E DR We define a manifold's local linear centroid as the D-dimensional projection of its Euclidean centroid in the LLE space. The LLE space is the lower d < D dimensional space that resulted from the local linear embedding (LLE) (Roweis & Saul 2000) of a m We describe the algorithm recover the appropriate low correlations space using CLLD and calculate sparse manifold capacity in figure In appendix A.4. we show our method is better than the technique in Cohen al. (2220) for estimating sparse manifold capacity with correlations.
Computing sparse manifold capacity with center correlation using CLLD.
1: Use LLE to project each manifold into a low dimensional space
2: For each manifold, compute its Euclidean center in the new space and project the center back to the original, high dimensional space to obtain local linear centroids.
3: Sample CLLD by sampling differences between local linear centroids.
4: Use CLLD to recover the space of low center correlation via the technique in Cohen et al. (2020).
5:
Project each manifold into the low center correlation space and compute capacity via Theorem
Figure 2: Pseudocode for sparse manifold capacity with correlation. See appendix A.2 for details.
4 RESULTS
By relaxing the homogeneous and random correlations assumptions in Chung et al. (0018), our method estimates manifold capacity for real data in the one-vs-rest recognition limit, the relevant domain for machine learning and neuroscience. While previous works (Cohen et al. 2020 Stephenson al. 2019) estimate manifold capacity in ANNs under the dense label regime, we demonstrate for the first time the application of manifold capacity in the neuroscience domain, enabled by sparse labeling. We use our Sparse Replica Manifold Analysis to measure the sparse manifold capacity of deep neural network manifolds and neural manifolds collected from the ventral streams of primates presented with visual stimulus in experiments done by Majaj et al. (2013a) and Freeman et al. (2013). Details of these datasets are provided in the appendix A.33 We show that sparse manifold capacity requires fewer number of features, ideal for neural data where limited number (e.g., order of hundreds) of
neurons are recorded. With this, we demonstrate the match between the theoretical and empirical manifold capacity in real neural data for the first time (to our knowledge). We then study the manifold
For the experiments in this paper, we chose d=2 to minimize the reconstruction error of LLE projection
5

---

Under review as a conference paper at ICLR 2023
geometries in relation to sparse manifold capacity and show explicitly that smaller manifold intrinsic dimension is desirable for larger capacity and better linear separability.
4.1 SPARSE MANIFOLD CAPACITY THEORY PREDICTS NUMERICS IN REAL DATA
In this section, we show how well theorem predicts the sparse manifold capacity. Figure demonstrates the match between the theoretical sparse manifold capacity (determined by our theorem) and simulated sparse manifold capacity (the ground truth determined empirically) for responses in layers of a CIFAR- 1-0 trained VGG-16 16 and brain regions of Macaque monkeys to given stimuli. The ground truth sparse manifold capacity is determined by interpolating the critical number of features (Ne) over which the probabilityy one-vs-rest dichotomy is linearly separable drops from to O. In practice, we use bisection search for N. where the probability is nearly 0.5. The linear separability of each dichotomy is determined by finding consistent SVM model. The full algorithm to determine simulated capacity is given in the appendix A.66 Interestingly, it appears that the neural data in figure 3 is better estimated by theorem 1, A possible explanation is that neural data have existing neural noise that smooths out the manifold geometries and makes them more amenable to our analysis.
(b)
Theorectios Capeecty.
Thecre-tical Coppecity
Capacity
V4
Vajajj et al Freemann
Sinubtte CaaulitedCCaosilly
Simuleted Cepiocity
'Simulened Capaciit
Figure 3: Theoretical simulated sparse manifold capacity (a) on smooth manifolds in layers of a VGG16 (b) on pointcloud manifolds in layers of a VGG16 (c) on neural data from the ventral stream of Macaque monkeys engaged in a classification task. Each dot is a layer or brain region.
4.2
BROADER APPLICATION OF MANIFOLD ANALYSIS ENABLED BY SPARSITY
(a) Danse Label
(b) Majaj et al, 2013
(c)
Freeman et al,
Testable
Ne
Dense
Ng
Nc.
Dense
Nc.
Sinedbeel
Regine
Protability Separation
Regime given No
1020
1000
Thoorsicel
150
250
Neurons
P/N
Odense
200
Sparse Label
Sparse
Sparsa
250
50
<
Geparse
PINo
Figure 4: (a) Sparse Replica Manifold Analysis enables cotullaiio-thhoon validation in real neural data with limited number of the neurons. Given No neurons in the data, capacity under dense labels tends to be outside the testable regime (top). Conversely, under sparse labels, the theoretical capacity can be validated empirically (bottom). In (b) and (c), we demonstrate this phenomenon. We estimate critical number of features Nº P/a. from the neural data in V4 and IT regions (Majaj al. 2015a) and the V1 and V2 regions (Freeman et al.) 2013) under dense labels (left) and sparse labels (right). The testable regime (highlighted blue) depends on the number of neurons/features available. Sample images of each dataset are shown on the bottom of their respective panel. Observe that under dense labeling, there is no simulated data available for validation.

---

Under review as a conference paper at ICLR 2023
Sparse labeling allows analysis of capacity wider class of neural data that has limited number of neurons measured. Recall that capacity is the ratio of P/Ne (refer to section 2). Figure 4 (top) demonstrates that under dense labels and limited number of neurons/features, Noo capacity often cannot be validated empirically using the bisection search because N₂ > No. On the other hand, the verification for capacity under sparse labeling requires less number of features (Figure|Ap bottom). For the Majaj dataset (Figure 4p), the estimation for No under dense labeling requires almost thousand neurons in the IT region where only 168 neurons were recorded. Under sparse labeling, the estimation is reduced to 65 neurons. Similarly, for the Freeman dataset (Figure(ec). dense labeling estimates 185 and 159 neurons for the VI and V2 regions while sparse labeling estimates just and 27 neurons, well within the number of neurons available.
4.3 COMPUTATIONAL EFFICIENCY
We observe that the computational time for the relevant manifold geometries is faster under sparse labeling compared to dense labeling by order of magnitude. Table compares the CPU time for computing manifold radius and dimension under dense labeling and sparse labeling for layers of the VGG-16. We note that the total sequential computation time for sparse manifold capacity is dominated by the optimization of the b parameter, and is typically not faster compared to capacity under dense labels (see appendix A.5)). Nonetheless, sparse manifold capacity enables the use of Gaussian radius and Dimension as a convenient measure of manifold geometry during object recognition.
Table I: Sparse Replica Manifold Analysis Computation Time (seconds): Manifold Radius and Dimension computed sequentially for 40 manifolds.
VGG-16 Layers
Input
ReLU-4
ReLU-7
ReLU-14
ReLU-21
ReLU-28
RcLU-37
Sparse Labels
1.01
1.16
1,23
1.14
1.15
0.76
1.07
Dense Labels
51.68
50.00
50.30
48.53
46.34
45.76
45.10
4.4 MANIFOLD GEOMETRY AND SPARSE CAPACITY IN ARTIFICIAL AND NEURAL DATA
In this section, we study the behavior between capacity, manifold Gaussian radius and Dimension, along with the manifold effective dimension and radius. Imposing hierarchy on layers of the DNN and the ventral stream, figure [ shows that capacity generally increases while average manifold Gaussian dimension (D₂) and Gaussian radius decrease (Ry). This follows the similar observations made in Co hen et al. (020)) and suggests that compressing the manifold allows better manifold linear separability
Figure 5 also compares the Gaussian geometries we have introduced to the notion of effec- tive dimension and effective radius. Effective dimension and effective radius correspond to measuring the manifold dimension and radius on an ellipsoid imposed on the manifold, whereas the Gaussian geometries are measured on the convex hull of each manifold. Under sparse labeling, the effective dimension is also known the participation ratio in most literature (Chung al., 2018 Gao et al. 2017 Litwin- et al. 2017 Elmoznino & Bonner 2002 Jozwik et al. [019; [Sorccher et al. 2021 In figure 5 a). mean effective dimension and radius behave similarly to mean Gaussian dimension and radius for layers of the DNN. Hence, the manifold intrinsic dimension, measured by the participation ratio of each manifold, also decreases as capacity increases. We note that figure (jb-c)(ii,iv) shows that mean effective dimension and mean Gaussian dimension do not decrease across the hierarchy imposed on the biological ventral stream as evidently as mean effective radius and mean Gaussian radius. Instead, the manifold Gaussian dimension and effective dimension appear to exhibit the opposite behavior relative to the artificial network. These suggestive differences between the artificial and the biological brain points toward using manifold geometries as metric to finding and fitting an artificial brain model.
4.5 ROLE OF DIFFERENT DIMENSIONS IN CAPACITY AND LINEAR SEPARABILITY
We have shown that capacity increases while manifold Gaussian radius and Dimension decrease. This suggests that compressing the manifolds and reducing their dimensions allow better separation. This may seem to contradict the intuitive belief in deep learning that higher dimensions allow better
7

---

Under review as a conference paper at ICLR 2023
(1)
(i1)
(IR)
(Iv)
(V)
0.
Dat
Pb
Ar
DNN
(a)
0.5
Majaj et
5/1
IT V4 1T
VM 14
Frouman et
(c)
V1
V/2
1/1
1/2 V11
V2
V22 1/2 V.
Figure 5: This figure plots (i) capacity, (ii) mean manifold Gaussian dimension, (iii) mean manifold radius, (iv) mean manifold effective dimension, and (v) mean manifold effective radius for smooth manifolds in ReLU layers of the (a) VGG16, (b) neural data in the V4 and IT regions, and (c) in the VI and V2 regions. Effective dimension computed as || / where À is the eigenspectrum of the covariance matrix of manifold. Effective radius is computed as ||A||2/ ||
classification. In F we give experimental evidence how linear separability is related to both lower manifold intrinsic dimension (D) and larger ambient dimension (N). Figure 6a summarizes our point: ambient dimension allows more expressivity for a separating hyperplane, but when the manifold intrinsic dimension high, the manifolds may not be separable.
For these experiments, we examine the interaction between manifold intrinsic dimension (as gaussian mean width and effective dimension), ambient dimension (N), the sparse manifold capacity, and the margin of linear separability in the SVM (i.c. the distance between the anchor point and separating hyperplane). We sample 20 manifolds, consisting of 1000 examples in the CIFAR 100 dataset. We vary the intrinsic dimension by projecting each manifold to a number of their PCA components. We vary the ambient dimension by random projection. In bc..f.g that at a fixed ambient dimension, increasing intrinsic dimension leads to decreasing capacity and margin. On the other hand, at fixed manifold intrinsic dimension, increasing ambient dimension increases capacity and margin (Figure pp.e). Hence, even though higher ambient dimension may enable better linear separability, the intrinsic dimension still plays role.
illustrate
DISCUSSION
We devised the Sparse Replica Manifold analysis for estimating sparse manifold capacity that is applicable neuroscience experiments. As Gardner's original perceptron capacity demonstrated that the sparse labeling regime increases capacity (Gardner] 1988), our new theory allows applications of the replica mean field theory analysis on neural data with few feature dimensions neurons recorded. Thus, we can begin analyzing the dynamics of the neural system where only sufficient samples of neurons were drawn (Gao et al.) (2017). Our theory can estimate sparse manifold capacity in layers of deep neural network and in real neural data of the ventral stream (Figures 3 We are able to estimate the number of features required to separate a one-vs-rest dichotomy with 0.5 success rate within the regime of available number of features.
In contrast to the capacity used in Cohen al. 2020)). sparse manifold capacity depends only on the Gaussian radius and Dimension of each manifold, which is faster to compute than in the classical regime (Table We note, however, that the time takes to compute sparse manifold capacity is longer, attributed to the optimization of the bias parameter, which warrants future studies. To account for correlation between manifolds, we also differ from Cohen al. (2020) by
8

---

Under review as a conference paper at ICLR 2023
(8)
(b)
Manifold Dimension (D)
Population Capaciy (Pochcc)
Dapacly (oaallyye--Phhie (ec-PPNo
Por Newron
Separable
Non-separable
Amblent Dimenial (N)
No
3.5
(e)
(1)
N1 (g)
Mengin (4)
2000
4003
15
Effective Dimansion(Darl) Gaussion Moan Width
Figure 6: (a) illustrates the relationship between ambient dimension N (shown orange axes), manifold intrinsic dimension D (shown as green axes), and linear separability. Manifolds are most easily separable in the regime of high ambient dimension and low manifold dimension. (b) shows that with fixed a, increasing N₂ (critical number of features) will increase the population capacity, which is the number of separable manifolds, P. Recall that we only need N >>> .o.. Otherwise a is not dependent on N. (b) holds true as long as N > No. Refer to section 2 for notations. (g)-(g empirical results sparse manifold capacity and the mean margin achieved by SVM when linear separability plausible. (c)-(d) demonstrate that the per neuron capacity, which the sparse manifold capacity, decreases effective dimension (participation ratio) and Gaussian mean width (R. / D2)) increase respectively. (e) illustrates that the margin of the SVM increases as the ambicnt dimension increases. We increase ambient dimension using random projections. (I)-(g) on the other hand demonstrate that margin decreases effective dimension and the Gaussian mean width increases.
using categorical local linear differences (CLLD) to compute space of low correlation. Thus, we introduce another method to deal with the center correlation for manifold analysis.
Generally, increasing capacity corresponds to decreasing manifold radius and dimension (Figure [5), and figure/Jshows that linear separability is associated with both low manifold dimension and high ambient dimension. The focus on the effective ambient dimension has been used in the literature describe the encoding capability and generalization performance of neural networks (Elmoznino & Bonner 2022] Jozwik et al. 2019) with claims that higher ambient dimensional measures benefit performance (Elmoznino Bonner 2022). Our work studies the manifold intrinsic dimension and illustrates the two types of dimension may behave inversely. filling in the gap of knowl- edge in the argument between the role of high and low dimensional representations neural encoding.
Sparse manifold capacity can be related other work involving sparseness neural en- coding. Previous work associates sparse neural connectivity with better readout performance (Babadi) & Sompolinsky, 2014; Litwin- Litwin-Kumar et al. 2017). These ideas of sparseness are associated with the number of active neurons, where in our work could relate to the number of critical features needed to achieve the given capacity of manifolds, which extends to the representation' is efficiency. Sparse manifold capacity is used for the regime of one-vs-rest class discrimination, where higher sparse manifold capacity is associated with improvement in manifold separability, equivalent to better readout performance.
Our theory opens many future directions in neuroscience and machine learning. Sparse manifold capacity broadens the operating regime of the replica manifold analysis framework to lower ambient dimensions, enabling analysis of both artificial and biological neural networks occupying low dimensional space. Among other things, we hope this work will (a) enable broader accessibility of the present manifold analysis approach to a wider experimental neuroscience community; (b) motivate the usage of present geometric approach to measure and train a broader size range of machine learning models; and (c) inspire future studies in using geometric approaches like ours as method to compare representations between biological neural data and artificial neural network models of the brain, by providing theoretically grounded metrics representation geometry and task performance.

---

Under review as a conference paper at ICLR 2023
REFERENCES
Yaser S. Abu-Mostafa, Malik Magdon-Ismail, and Hsuan-Tien Lin. Learning From Data. AMLBoook, 2012.
Baktash Babadi and Haim Sompolinsky. Sparseness and expansion sensory representations. Neuron, 83(5):1213- 2014.
Yue Chung.
Statistical Mechanics of Neural Processing of Object Manifolds. PhD thesis, Harvard University, 2017.
Yuo Chung and LF Abbott. Neural population geometry: An approach for understanding biological and artificial neural networks. Current opinion in neurobiology. 70:137-144, 2021.
Yeo Chung, Daniel Lee, and Haim Sompolinsky. Classification and geometry of general perceptual manifolds. Physical Review X, 83))031003, 2018.
Uri Cohen, Sue Yeon Chung, Daniel Lee, and Haim Sompolinsky. Separability and geometry of object manifolds in deep neural networks. Nature communications, 11(1):1-13, 2020.
Thomas M. Cover. Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition. IEEE Transactions on Electronic Computers, EC-14(3): 326-334, 1965. doi: 10. 109///GE....95.22266..
James Dicarlo and David Cox. Untangling invariant object recognition. Trends in cognitive sciences, 11:333-41, 09 2007. doi: 10. 2..101/6.iccs.2007.66..
R Becket Ebitz and Benjamin Y Hayden. The population doctrine in cognitive neuroscience. Neuron, 109(19):35553368. 2021.
Eric Elmoznino and Michael Bonner. nigh-performing neural network models visual cortex benefit from high latent dimensionality. bioRxiv, 2022.
Jeremy Freeman, Corey M. Ziemba, David J. Heeger, Eero P. Simoncelli, and J. Anthony Movshon. functional and perceptual signature of the second visual area in primates. Nature Neuroscience, 16(7):974-981, Jul 2013. ISSN 1546-1726. doi: 10. 10.10388nn.3402. 10.1038/nn..4000
Emmanouil Froudarakis, Uri Cohen, Maria Diamantaki, Edgar Y Walker, Jacob Reimer, Philipp Berens, Haim Sompolinsky, and Andreas S Tolias. Object manifold geometry across the mouse cortical visual hierarchy. BioRxiv, pp. 2020-08, 2021.
Juan A Gallego, Matthew G Perich, Lee E Miller, and Sara A Solla. Neural manifolds for the control of movement. Neuron, 94(5):978-95) 2017.
Peiran Gao, Eric Trautmann, Byron Yu, Gopal Santhanam, Stephen Ryu, Krishna Shenoy, and Surya Ganguli. A theory of multineuronal dimensionality, dynamics and measurement. BioRxiv, pp. 214262, 2017.
Elizabeth Gardner. The space of interactions in neural network models. Journal of physics A: Mathematical and general, 21(1):257, 1988.
Xiaoxuan Jia, Joshua H Siegle, Corbett Bennett, Samuel D Gale, Daniel Denman, Christof Koch, and Shawn Olsen. High-density extracellular probes reveal dendritic backpropagation and facilitate neuron classification. Journal of neurophysiology, 121 1831-188881187,, 2019.
Kamila Maria Jozwik, Hyo-Dong Lee, Nancy Kanwisher, and James DiCarlo. Are topographic deep convolutional neural networks better models of the ventral visual stream? 2019 Conference on Cognitive Computational Neuroscience, 2019.
Nikolaus Kriegeskorte and Rogier A Kievit. Representational geometry: integrating cognition, computation, and the brain. Trends in cognitive sciences, 17(8)401-412, 2013.
Nikolaus Kriegeskorte and Xue-Xin Wei. Neural tuning and representational geometry. Nature Reviews Neuroscience, 22( 1):1::733-718, 2021.
10

---

Under review as a conference paper at ICLR 2023
Ashok Litwin- Kumaa, Kameron Decker Harris, Richard Axel, Haim Sompolinsky, and LF Abbott. Optimal degrees of synaptic connectivity. Neuron, 93(5):1153-116) 2017.
Najib J. Majaj. Ha Hong, Ethan Solomon, and James J. DiCarlo. Simple learned weighted
sums of inferior temporal neuronal firing rates accurately predict human core object recognition performance. Journal of Neuroscience, (113)):33022134111 2015a. ISSN 0270-6474. doi: 10. J0.1523/NEUROSCL5118---4222
URL https: /twp::://wwww.. www btttp::/www..jeerrrsss jwe.rnneuis....... wtg/content.....ooooog.o.o./...e
35/ 35/39/13402)
Najib Majaj, Ha Hong. Ethan Solomon, and James J DiCarlo. Simple learned weighted sums of inferior temporal neuronal firing rates accurately predict human core object recognition performance. Journal of Neuroscience, 35(39); 13400-13188 2015b.
Sam Roweis and Lawrence Saul. Nonlinear dimensionality reduction by locally linear em- bedding. Science, 29065500):2323-222200 2000. doi: 10.1126/sciemce.299. 290.5500..232.. URL https: www. science org/ doi/ abs /00. 12611266622 /tte///ccence..eeeeeeeeeeeeeeeec 290 5500 2323
Martin Schrimpf, Jonas Kubilius, Ha Hong, Najib J. Majaj, Rishi Rajalingham, Elias B. Issa, Kohitij Kar, Pouya Bashivan, Jonathan Prescott-Roy, Franziska Geiger, Kailyn Schmidt, Daniel L. K. Yamins, and James J. DiCarlo. Brain-score: Which artificial neural network for object recognition is most brain-like? bioRxiv preprint, 2018. URL https::///// http://www.biixx.. www.oiiww.org//ooo...
100.101//60 10111100407000/0
Martin Schrimpf, Jonas Kubilius, Michael Lee, N Apurva Ratan Murty, Robert Ajemian, and James J DiCarlo. Integrative benchmarking to advance neurally mechanistic models of human intelligence. Neuron, 2020. URL pottps://vve //ww:..w/wwwww/ cell com wwwm//eeroo///ccoooocooooo ful
wtwpo/////lllttttttooeoooee Iteet 6273 (20)
30605-XX
Anirvan Sengupta, Cengiz Pehlevan, Mariano Tepper, Alexander Genkin, and Dmitri Chklovskii. Manifold- localized receptive fields are optimal in mmillarty-preservinng neural networks. Advances in neural information processing systems, 31, 2018.
Hansem Sohn, Devika Narain, Nicolas Meirhaeghe, and Mehrdad Jazayeri. Bayesian computation through cortical latent dynamics. Neuron, (036)):33-944,2000
Ben Sorscher, Surya Ganguli, and Haim Sompolinsky. The geometry of concept learning. bioRxiv, 2021.
Nicholas A. Steinmetz, Cagatay Aydin, Anna Lebedeva, Michael Okun, Marius Pachitariu, Mar- ius Bauza, Maxime Beau, Jai Bhagat, Claudia Böhm, Martijn Broux, Susu Chen, Jennifer Colonell, Richard Gardner, Bill Karsh, Fabian Kloosterman, Dimitar Kostadinov, Carolina Mora-Lopez, John O'Callaghan, Junchol Park, Jan Putzeys, Britton Sauerbrei, Rik J. J. van Daal, Abraham Z. Vollan, Shiwei Wang, Marleen Welkenhuysen, Zhiwen Ye, Joshua T. Dudman, Barundeb Dutta, Adam W. Hantman, Kenneth D. Harris, Albert K. Lee. Edvard I. Moser. John O'Keefe, Alfonso Renart, Karel Svoboda, Michael Häusser, Sebastian Haesler, Matteo Carandini, and Timothy D. Harris. Neuropixels 2.0: A miniaturized high-density probe for stable, long-term brain recordings. Science, 37845539) 2021. doi: 10.1126/science.ab445558 URL 111.111.111211 wtt.://cennceeeeee wwtpssciinnnnee.a.....ee abf4588
https: h/wps://wwww./ www. wwwwwce.ienccecce
Cory Stephenson, Jenelle Feather, Suchismita Padhy, Oguz Elibol, Hanlin Tang, Josh McDermott, and SueYeon Chung. Untangling in invariant speech recognition. Advances in neural information processing systems, 32, 2019.
Vladimir N Vapnik and A Ya Chervonenkis. On the uniform convergence of relative frequencies of events their probabilities. In Measures of complexity, pp. 11-30. Springer, 2015.
A APPENDIX
A.1 DERIVATION TO THEOREM 1
Suppose we have a heterogencous system of P manifolds, Let D5 = [Dgi....opp) be the array of their corresponding Gaussian dimensions. And let Ry [Rg, Ropl be the array of their corresponding
11

---

Under review as a conference paper at ICLR 2023
Gaussian radii. By Chung al. (2018), the capacity of such ensemble of manifolds is the average of a,,, computed for each manifold by equation I or estimated as O (R, , R. D) (equation 2 where R and D are the manifold's ss respective measure of dimension and radius. Furthermore, by Chung et al. 20188) sparse capacity for homogeneous system of manifolds (i.e.assuming all manifolds have the same Gaussian radius and Gaussian dimension), is computed
OHOMOG = maxif BBa (K + b, Rg, D₂) + (1- f) a BBul((K Rg Dg)1=
(*)
b
where f is the sparse parameter (i.e. the fraction of positively labeled manifolds), and b is the bias parameter that has positive contributions positively labeled manifolds and negative contributions to negatively label manifolds. Observe that equation (+) is thus averaging across @BaLL 's each with inputs from manifold with either positive labeling or negative labeling indicated by +b and -b. Since all R and D are the same in homogeneous system, equation (*) sufficiently accounts for all possible assignments of positive and negative labeling.
In the case heterogeneous geometries, follows that we need to consider all possible di- chotomies given by the sparse parameter. In order to conserve equation (+), averaging across all such dichotomies suffices. For the one-vs-rest regime discussed in this paper, there are only P dichotomies consider, within each only manifold is positively labeled. Thus, for each manifold compute
am (1c, Rg. Dy) = max[1/P a'uu b, Rgi. Dgi) + E1/P Bau (n: b, Rgj. Dgj)]= b
j
Then, averaging across all aMi, i.c. across all relevant dichotomies, gives us the sparse manifold capacity. Observe that this general theorem does not deviate from the case of homogeneous system. In the case of = 1/P and we have a homogeneous system,
OHOMOG = aM
aas =< OM >
ijj
A.2 COMPUTING SPARSE MANIFOLD CAPACITY CLLD
AlgorithmDD 1 gives the detailed pseudocode to calculate sparse manifold capacity with center correla- tion. For the experiments this paper, the number of components chosen for local linear embedding of each manifold was heuristically chosen the default value of 2 to minimize the reconstruction error of the dimensional reduction. See section 3.1.3 for definitions.
A.2.1 ALGORIHTMM FOR LOCALIZING LOCAL LINEAR CENTROID
Discovering the local linear centroid involves projecting each manifold low dimensional center the high dimensional space. Since this problem involves solving an overdetermined system, we use algorithms to localize for the local linear centroid. Here we describe two of these algorithms. Results of the paper use the Nearest Neighbor algorithm.
The Nearest Neighbor algorithm tracks the K nearest points on the manifold closest to the Euclidean centroid in the LLE space. The D-dimensional local linear center is the average these K points in the higher D-dimensional space. On the other hand, the Weighted Average algorithm determines a weighted average in the D-dimensional space based on the distance each example of the manifold from the Euclidean centroid in the LLE space. Points the manifold closer to the LLE space Euclidean centroid will have a greater weight. The algorithms to determine local linear centroids given manifold and its LLE equivalent are provided in Algorithms and 3
A.3 DATASETS
We used three datasets for our experimental analysis. In figure 3 capacity is measured across layers of trained VGG-16 and brain regions of Macaque monkeys. For the DNN, we extract the activation from each layer to form manifolds corresponding class form an object manifold. A pointcloud manifold consists of distinct examples from the same class. A smooth manifold is generated by affine transformations (translation) of image belonging to a distinct class. Neural data were taken from Majaj et al.] (2015a) and Freeman et al. (2013) where images shown to Macaque monkeys, and responses from the primate ventral stream were
given input. Responses to inputs of the same
12

---

Under review as a conference paper at ICLR 2023
Algorithm 1 Calculate sparse manifold capacity with Center Correlation
Function: sparse cateeeccpppayaapae with correlation (.))
Input data: Set of manifolds X = {M' MMM ,M....M% M, "JM' M E IR Dam
Output data: sparse manifold capacity a
1: {ui, (22, .PP) + local linear_centroinne centroids
2: Sample A c {u, غylر * j. i.j i {1...P}}
3: Compute common component matrix, V. using A ((Cohen et al. 2020) Suppl)
4: for manifold M i {1, do
5: Compute residual manifold M; = M VVVTM"
6: Compute DgA Ryi of M, (by equation
7: end for
8: for manifold M', P1 do
9: Compute (K,,, Dg R₂) (by equation
10: end for
11: Compute a, = (a:)s
Function: local_lineat Laa__linerrceenrriiii centroids (X)
Input data: Set of manifolds X = {M M², MM,,. ...M.. M E Drm Output data: Local linear centroids {/1, pp) ). M4 RR1
1: for manifold M E (1, do
2: Perform LLE M and project M to its LLE space.
3: Let L be the projected manifold, and compute the center = (L')
4: Compute μ4, the projection of v4 in the original, higher dimensional space. (See A.2.1
5:
end for
Algorithm 2 Nearest Neighbor Algorithm for Computing Local Linear Centroids
Input data: Manifold M Rem; the projection of M after LLE L' Rddm; the mean of L E Rdd1
Output data: Local linear centroid μ4 E Dx1
Parameters: K E: E
I: Compute the set of {ō, - |1u LjllVv {l,...m)}
2: ..okK minftracs {6,}
3: 14 (M)...
Algorithm 3 Weighted Average Algorithm for Computing Local Linear Centroids
Input data: Manifold M RDem the projection of M' after LLE L' Edam, the mean of L',y; E dd
Output data: Local linear centroid #4 E R Dx1
Parameters: & +
I: Vj E {1, ...m. }, compute 8, = 114- Li
2: Vj E (m,.. compute Wij = 1/8,
3: Vj E (1....m) compute vi = Wj/ Σx
4: 14 = "
recorded. Again, responses to inputs of the same class form an object manifold. The Majaj dataset contains neural responses to 64 different classes from the V4 and IT regions. The Freeman dataset contains neural responses to 30 different classes from the V1 and V2 regions.
A.3.1 DNN TRAINING
regular VGG-16 16 was trained using pytorch on the C0 dataset. The model was trained on stochastic gradient descent optimizer with learning rate of 0.01 and momentum rate of 0.9.
13

---

Under review as a conference paper at ICLR 2023
A learning scheduler used to reduce learning rate by 0.5 every 10 epochs. The final training accuracy was nearly 0.90.
A.3.2 GENERATING SMOOTH AND POINTCLOUD MANIFOLDS
Smooth manifolds generated by affine transformations of image. For a Cifar100 image, 24x24 frame is translated randomly from the center of the image by a maximum of 4 pixels horizontally and vertically, Each pointcloud manifold consists of the top 50 well-trained examples from that class. For Figure
we extract smooth manifolds from the following layers of the VGG-16
('layer_ 00 10__Int_______1
01__oo22d___ 'layer 02 02_ReL''', Stayer_07_ReL''
'layer_03 ________________.__
1 "layer_09_Reil "layer_10_MaxPPoo222 'layer_12_Reiu'
[aayer_1_Coon2dd'
'layer_14_RelJ" ''aaee__6_Reeul
'layer_21_Rell
'layer_23_Rel'' "Layer_24_Maxoolll
2_____'', 'layer_27_Conv22
'layer_28_Rell,
1ayer_30_ReLU' 'layer_37RRelU']
We extract our pointcloud manifolds from the following layers of the VGG-16
I' Vlayer_00, _aayee____Input',, 'layer_00 O0_ReeUU
Llaye__0__Relll 'layer_09_Reil 'layer_14_Reln 'layer_21_Reij 'layer_23_ReWJ Bayer_28_Reell
A.3.3 NEURAL DATA
The data from Majaj et al. (2015a) are made available by Brainscore (Schrimpf et al. 2020) (Schrimpi) ct al. 2018). The data consists of neural responses from the V4 and IT regions four Macaque monkeys. The stimulus set consists of 64 classes, and there are total of 88 neurons recorded in the V4 V4 region and 168 neurons recorded in the IT region. The data from Freeman et al. (2013) are made available by the authors of the work. There and V2 regions of 13 Macaque monkeys respectively. The set of stimuli contains 15 texture classes. For the purpose of the author's experiments, the 15 texture classes used to generate corresponding 15 noise classes. We consider the neural data corresponding to texture and noise images together. Hence, total, we had 30 classes for this dataset.
total of 102 and 103 neurons recorded from the V1
A.4 COMPUTING SPARSE MANIFOLD CAPACITY VIA CLLD VS METHOD COHEN AL
The calculation for sparse manifold capacity assumes that manifolds are uncorrelated. Cohen et al. (2020) takes into account of correlation for the classical notion of manifold capacity by finding space of low correlation. This space is computed using the centroids of each manifold. Compared Cohen et al.) (2020), our method outlined by algorithm [ [sees CLLD, categorical local linear differences, to compute the space of low correlation. Figure 7 shows that the method via CLLD is fact superior to the method in Cohen al. (2020) for predicting sparse manifold capacity in our neural datasets. Using CLLDD allows a better match between theoretical and simulated sparse capacity.
A.S SPARSE REPLICA MANIFOLD ANALYSIS COMPUTATION TIME
Table shows the computational time of Sparse Replica Manifold Analysis for layers of VGG-16 for 40 smooth manifolds. Observe that radius and dimension are computed simultancously as capacity in the dense label regime. Also, observe that in the regime of sparse label, the compute time of capacity dominates the total compute time.
A.6 SIMULATING CAPACITY
The algorithm for simulating sparse manifold capacity uses a bisection search to find Ne, the number of feature dimensions the input data can be reduced to such that half of the one-vs-rest dichotomies linearly separable. The full algorithm is in Algorithm 4
14

---

Under review as a conference paper at ICLR 2023
(a) Sparse Manifold Capacity Computed (b) Sparse Manifold Capacity Computed via CLLD
via Cohen et al
V2
Thoorectical Capacity
08-
V1
06
V4
Majaj
Freeman et
0.5
0.8
1.0
3.2 0.4 0.6
1.0
Simulated Capacity
Simulated Capacity
Figure 7: This figure shows the match between simulated and theoretical sparse manifold capacity neural datasets when different methods are used to take into account manifold correlation. In (a), the method presented in Cohen et al. (2020) is employed. The manifold centroids used to recover the space of low correlation. In (b), CLLD are used to recover the space of low correlation. Refer to appendix A.2
Table 2: Sparse Replica Manifold Analysis Computation Time (seconds) for 40 smooth manifolds in VGG-16 Layers
Dense Labels
Sparse Labels
Vgg 16 Layers
RtD
Capacity
Total Time
Rg.Dg
Capacity
Total Time
Input
51.68
51.68
159.99
1.01
771.75
857.46
ReLU-2
50.21
50.21
111.09
1.24
762.17
851.72
ReLU-4
50.00
50.00
99.41
1.16
846.37
932.27
ReLU-7
50.30
50.30
84.51
1.23
832.91
918.44
ReLU-9
50.92
50.92
82.18
1.20
768.40
856.76
ReLU-14
48.53
48.53
68.31
1.14
715.54
802.85
ReLU-16
48.34
48.34
69.76
1.21
798.93
893.71
ReLU-21
46.34
46.34
69.98
1.15
1336.58
1424.74
ReLU-23
46.34
46.34
61.31
1.28
1153.42
1245.36
ReLU-28
45.76
45.76
62.43
0.76
1290.43
1379.53
ReLU-30
45.97
45.97
81,88
0.82
1086.01
1177.79
ReL.U-37
45.10
45.10
105.36
1.07
1198.15
1284.02
A.7 CAPACITY FOR ELLIPICAL MANIFOLDS
Figure measures simulated capacity while varying manifold effective dimension. Effective dimen- sion is equivalent to the manifold Gaussian dimension if the manifold is an ellipsoid (Chung et al.) 2018). Hence, simulated capacity for figure calculated assuming manifolds have an elliptical structure. As such, we use the Maximum Margin Manifold Machines algorithm described in (Chung 2017
A.8 MEAN FIELD THEORY OF MANIFOLD SEPARATION
In this section, we elucidate the emergence of anchor points the solution for manifold capacity. Following Gardner's 's framework, [Chung et al. (2018) determined that in the max margin solution for homogeneous system of manifolds, inverse manifold capacity is given by a' (F(T)) where
F(T) = min ( gs - x 2 0, Yड E S} and gs (V) = min S E S}.
a' is interpreted for a single manifold. T are sampled Gaussian vectors while V. = yu il, 1.... 1), where it is solution hyperplane, is the manifold's ith basis vector, and y is the
15

---

Under review as a conference paper at ICLR 2023
Algorithm 4 Simulated Capacity
Function: bisection searnnhssschchee (X)
Input data: Set of manifolds X = {M', .M2, ,??,, M E IR Dam
Output data: sparse manifold capacity a
1: Initialize = 0 and Found = False
2: Initialize Nmax D and Nunn = 2
3: Naezt + (Nmar + /2
4: probmax is linearly Sinearrl_separableeX, N
5: probmin f is siinearlyssepaaalee_ Nmin)
6:
7: while not Found and N = Nuent do
8: N + Nnext
9: prob + is Serearly_separableca
11:
if prob 0.5|| < tol then
12:
Found+ True
13:
14:
end
15:
16:
if (probmin 0.5) (prob 0.5) < 0 then
Niaa N
18:
probmax - prob
20:
else
Nmin N
22:
probmin t prob
23:
24:
end if
25: Nnent + (Nmax + Nmwn))
26:
27:
end while
28:
29: if Found then
30: No Nnezt
31:
32: else
33:
Interpolate for No
35: end if
a. P/Ne
manifold's label. is described as the signed fields induced by the solution hyperplane, and S can be thought of as the set of points on the manifold. The KKT interpretation is to find V given by
- 7 AŠ(7)
(6)
such that the following holds
2
9s(V) - k6 1x20
Algs(V) (V) - x] =0
and 5 = arg minges S. The mean field interpretation of equation is that is a decomposition of contributions from other manifolds represented by I and the fixed manifold represented by S. Furthermore, equation implies that the resulting F (T) is precisely [[AS(T)) with 5(T) denoted the anchor point of the manifold.
16

---

Under review as a conference paper at ICLR 2023
Function: is ss_lineerly__eene separable (X,N)
Input data: Set of manifolds X (M,, MM,,..M),, M E RDmm Number feature dimensions N
Output data: Probability of separation given dichotomy, prob
1: Initialize sep = 0
2: Project D-dimensional manifolds to N dimensions using random projection.
3:
4: for i E [I...., P) do
5: Assign the set Y of binary labels for each manifold, where manifold gets the positive label.
6: Find consistent SVM solution for input and label
7:
8:
if' solution exists then
9:
scp sep +
10:
11:
end if
12:
13: end for
14: prob t s
A.9 HOMOGENEOUS HETEROGENEOUS GEOMETRIES
We've made the claim in the previous section, Derivation Theorem 1, that the equation 4 is insufficient for a system of manifolds with varying radii and dimensions. We demonstrate the disparity the match between the theoretical and simulated sparse manifold capacity when different assumptions of the geometries are made in figure g and 9 Using the smooth manifold and the pointcloud manifold data from the VGG-16 dataset used throughout this paper, we compute the capacity using homogeneous geometries by averaging across the Gaussian radii and dimensions of all manifolds the system. Comparing the left and the right plots in each picture, it is evident that assuming homogeneous geometries is insufficient in a system where manifolds have varying Gaussian radii and dimensions.
(a) Estimating sparse capacity with Heterogenerous Geometry
(b) Estimating sparse capacity
with Homogeneous Geometry
Theorectical Capacity
0
1
2
Simulated Capacity
Figure Heterogeneous vs Homogeneous geometries in estimating sparse manifold capacity for smooth manifolds. (a) is the result of incorporating the heterogencous geometries in sparse manifold capacity demonstrated in Fig of the paper (i.c. using theorem 1). (b) is the result of assuming homogeneous geometries to estimate sparse manifold capacity (i.e. equation in the paper). If we simply use the average dimension and radius of all manifolds estimate the sparse capacity, we have the resulting mmic) in (b)
17

---

Under review as a conference paper at ICLR 2023
(a) Estimating sparse capacity with Heterogenerous Geometry
(b) Estimating sparse capacity
with Homogeneous Geometry
3
Theorectical Capacity
0
1
2
3
Simulated Capacity
Figure 9: Heterogeneous Homogeneous geometries in estimating sparse manifold capacity for pointcloud manifolds. (a) the result of incorporating the heterogeneous geometries in sparse manifold capacity as demonstrated in Fig 3 the paper (i.e. using theorem
1). (b) is the result of assuming homogeneous geometries to estimate sparse manifold capacity (i.e. equation i the paper). simply use the average dimension and radius all manifolds estimate the sparse capacity, we have the resulting (mis)match in (b)
A. SPARSE REPLICA MANIFOLD ANAYSIS ON IMAGENET-TRAINED RESNET 101
We used an imagenet-trained Resnet101 and create 20 object smooth manifolds from the following extracted convolutional layers. Results of the match between theoretical and simulated sparse manifold capacity is shown in figure 10
x 'layer_122 123 Conv2d'_ "layer_014 [o1____n -layer_155 Conv_dd_ Bayer_181__omml ( Citee_o22__Cov//2
1layer_202_Convel 1layer_048_Conv22
[ayee_222_Conval
Simulated vs Theoretical Capacity of Imagenet-
trained Resnet101 Convolutional Layers
2.00
1.75
Theorectical Capacity
1.25
1.00
0.50
0.25
0.00
0.00
0.75
1:00
125
1.75 2.00
Simulated Capacity
Figure 10: Theoretical vs simulated sparse manifold capacity measured on 20 manifolds from an imagenet-traiee trained Resnet101.
18

---

Under review as a conference paper at ICLR 2023
A.11 CLLD SENSITIVITY TO NUMBER OF NEURONS
Estimating sparse manifold capacity requires accounting for manifold correlation. Cohen et al. 2020)) successfully account for manifold center correlation in their work that applied dense labeled manifold capacity to artificial neural network. In appendix [..4 and S the method in Cohen et al. (2020) insufficient for sparse manifold capacity. Instead, we show that the method we presented in this paper using CLLD does a better job. In figure we show that using CLLD always outperform Cohen al regardless of the number of neurons present in the data. We vary the number of neurons by random projection. The results show that the estimation for sparse manifold capacity is robust.
(a) Sparse Capacity Computed via CLLD (with varying number of neurons)
(b) Sparse Capacity Computed via Cohen et (with varying number of neurons)
1.2
1.2
Theoretical Capacity
0.8
0.8
0.4
0.4
V4
V1
V2
0.0
0.0
0.0
0.4
0.8
1.2
0.0
0.4
0.8
1.2
Simulated Capacity
(a1) Number of Neurons vs Sparse
(b1) Number of Neurons vs Sparse
capacity computed via CLLD
capacity computed via Cohen et al
1.5
1.5
V4
1.0
1.0
Capacity
0.5
0.5
0.0
0.0
300
100
400
Number of Neurons
Figure 11: In (a) and (b), each color represents the capacity at various layers of the ventral stream. Each point of the same color is the capacity measured from randomly projecting the neural manifold some number of features. (al) and (b1) show explicitly the number of neurons the resulting estimated capacity. We vary the number of neurons by random projection. The resulting theoretical capacity changes little as result. Furthermore, compared the method in Cohen al (bl), the method via CLLD is closer to the ground truth capacity (represented by x).
19


Reviews:
[{"review_id": "vuXrz3dhci", "reviewer": "Reviewer_TxMd", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors of the paper propose Sparse Replica Manifold analysis for estimating sparse manifold capacity that measures how many object manifolds can be separated under sparse classification tasks, which is pervasively used in both neuroscience (real neural data) and deep learning (artificial neural data) domains. Building upon Mean Field Theory Manifold Analysis (MFTMA), the authors derive and prove Theorem 1, which theoretically computes sparse manifold capacity under the assumption of different Gaussian Radii and Dimensions for each manifold. To account for the correlation between manifolds, authors leveraged CLLD to compute a space of low correlation. The authors then perform comprehensive experiments on both real and artificial neural data to demonstrate that their theoretical and empirical analyses are consistent. Overall, the paper is presented very well and has convincing results.", "summary_of_the_review": "This study on neural representations is very intriguing for both discriminatory analyses in statistical learning and to better study computational principles in neuroscience. Therefore, studies like this are (and will be) of interest to venues like ICLR. I suggest acceptance of this work. I believe that if the authors address/comment on the concerns mentioned in the Weaknesses section, the overall strength of the paper will be increased.", "strength_and_weaknesses": "Strengths\n- The paper is overall very complete in itself, especially the inclusion of MFTMA in the main manuscript is very helpful.\n- Thorough empirical study is performed.\n- Nice to see that sparse analyses are significantly faster compared o dense analyses.\n- Overall clarity and organization of the paper are great.\n\n\nWeaknesses\n- 2.0  (t^tilda, t_0) vectors are lacking variable definition. Also, why are these sampled from a Gaussian distribution?\n- Appendix section A.1 DERIVATION TO THEOREM 1 may be included in the main manuscript to improve completeness. I regard this derivation as one of the main technical novelties of this paper.\n- I understand that Eqs 4 and 5 are built upon the assumption that the manifolds are distributed Gaussian with a given radius and Dimension. What happens when they are not? Does this analysis break down?\n- 3.1.2 \"null space of low correlations\" what does that mean? Please give some background.\n- 4.1, \"Observe this means that a manifold needs to have enough feature dimension for the bisection search to operate appropriately.\" It is not only a hard sentence to understand, but why this should be the case for a 0.5 probability that a one-vs-rest dichotomy is separable is unclear. For completeness, the authors could elaborate on this.\n- Figure 3 indicates that manifold capacity is underestimated on artificial data, whereas it is almost exactly estimated in real neural data. A discussion on why this might be the case could be interesting.\n- It would be interesting to see the analysis on some other (possibly larger) datasets, such as ImageNet. Also, analysis with a more recent and widely used deep architecture would be more relevant, such as Residual CNNs and/or transformers.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "sALDlby5tC6", "reviewer": "Reviewer_4A5K", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "I thank the authors for their thoughtful replies as well as the clarifications that they have made in the paper. I would increase my score to 7 if that were an option. In any case, I think that this is a valuable contribution and look forward to seeing the paper published =)\n\nThis paper extends prior work on measuring manifold capacity to the setting of sparse one-vs-all classification. This is well matched to biological data and provides a more optimistic bound on the number of neurons needed to compute the proposed metrics. They show that across artificial and biological processing stages manifold dimension decreases while ambient space size might increase – both factors that contribute to superior classification performance.", "summary_of_the_review": "Overall, I think that this is a cool extension for measuring manifold capacity in sparse settings. The results across processing hierarchies also make sense in artificial and biological systems (maybe a newer architecture like ResNet would have been more interesting than VGG). It would be great if the paper were more self-contained, currently it seems somewhat derivative/incremental on prior work. That being said, I think improving previous bounds to realistic sparse settings is really cool and valuable.", "strength_and_weaknesses": "Strength: The paper is clearly written and presents and interesting extension to prior work that is well matched to realistic settings. The experiments are straightforward and support the claims.\n\nWeaknesses: It would be great if the methods would rely less on prior work. Basically, it is impossible to properly understood the current paper fully without having read the prior work by Chung et al. This is unfortunate and might make the current work seem more incremental.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "U9QzXeiYytT", "reviewer": "Reviewer_yhnc", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The submission presents an extension to the representational analysis framework of Chung et al. (2018) in order to apply the framework to more standard computer vision tasks such such as multiway classification, with the goal of more precise comparisons between humans and neural networks.", "summary_of_the_review": "The submission needs improvement in clarity to better convey its contributions to the neuroscience and computer vision communities.", "strength_and_weaknesses": "##### Strengths:\n\n1. The idea seems valid, although I do not have the domain knowledge to place its significance.\n\n##### Weaknesses:\n\n1. The explanation of and motivation for the method generally assume a lot of background; for example, what is meant by \"separability\", \"system capacity\", \"manifold\" and \"manifold capacity\", \"random dichomoties\". As such, it was not easy to understand all details of the method.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "TR8wziDPC7f", "reviewer": "Reviewer_pKPQ", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper extends Chung et al. (2018), which proposes Sparse Replica Manifold analysis to estimate manifold capacity. The authors show that the application of sparse manifold capacity requires a smaller number of features and is faster compared to dense labeling. The authors also illustrate the effects of ambient dimension and manifold intrinsic dimension on sparse separability.", "summary_of_the_review": "This paper is the following work of Chung et al. (2018) and many conclusions have been proposed before.", "strength_and_weaknesses": "Strengths:\n\n1.The proposed Sparse Replica Manifold analysis overcomes the limitation of size of neuroscience dataset and is close to real tasks in cognitive science.\n\n2.The authors analyze the manifold capacity of both deep neural network manifolds and neural manifolds, making the conclusions more convincing.\n\nWeaknesses:\n\n1.The authors follow the work of Chung et al. (2018), but they don’t clearly claim the contributions of previous work.\n\n2.The datasets this paper uses are relatively old and limited.\n\n3.More comprehensive and thorough experiments are necessary to prove the conclusions.\n\nQuestions for the Authors\n\n1.In chapter 3, the authors claims that sparse manifold capacity has been previously considered in Chung et al. (2018) and that they extend this work k by taking into account of heterogeneous geometries and correlation between manifolds. But in “RESULTS”, they don’t clearly claim the advances of Sparse Replica Manifold compared with previous works and the contributions of taking different Gaussian Radii and Dimensions into account.\n\n2.The dataset this paper uses are from Majaj et al. (2015), Freeman et al. (2013) and VGG-16 trained by a CIFAR-100. More experiments on latest and large-scale datasets are needed.\n\n3.In Figure 5, the change ranges of Dg in both neural dataset and the tendency of Rg and Deff is not so evident. More experiments are necessary to prove the conclusion.\n\n4.The broader application of manifold analysis is meaningful; however, the scale of neuroscience dataset is larger recently. The total sequential computation time of Spares Replica Manifold Analysis has also no advantage. The influence of this method needs to prove again.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "MJkAGe9kSt", "reviewer": "Reviewer_EQzE", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper extends the sparse replica manifold analysis approach in Chung 2018 to incorporate across-manifold correlations and account for non-homogeneous manifolds. They demonstrate the application of sparse manifold capacity allows analysis of a wider class of neural data and show that their theory and algorithms provide a better match to real data.\n\n", "summary_of_the_review": "This paper provides new tools for analyzing the geometry of representations in artificial and biological neural networks. They extend theory from Chung 2018 to a more realistic setting where manifolds can have different underlying geometry and propose an algorithm for computing inter-manifold correlations. \n\nWhile their analysis and theory does indeed provide better matches to real data, the theoretical results appear to be relatively incremental and haven't been adequately dissected in the empirical studies. Is the better match due to using heterogeneous manifold estimates, or is the proposed estimation method key? The role of the dimension of the projection, the dimensionality of the manifold versus the number of neurons or ambient dimension hasn't been clearly dissected in the experiments.", "strength_and_weaknesses": "\nStrengths: \n\n- Moving to case where different manifolds can have different radii and varying correlations is better match to real data.\n\n- Simple algorithms and some theoretical justifications.\n\n- Analysis on two neural datasets show that their approach can better predict properties of neural responses than the homogeneous analysis.\n\nWeaknesses: \n\n- The main contribution of the work is extending previous theory to the heterogeneous manifold case, and providing a new algorithm for computing inter-manifold correlations. Thus, it would be useful to have some ablations to understand how adding different assumptions or constraints impacts the theory and match to practice.\n\nIn particular, \n(i) How much does the relaxation of the homogeneity constraint on the manifolds impact the match to data vs. the sparsity assumption? \n(ii) How does ovr match the assumption about how the brain is performing object recognition? It wasn’t entirely clear how the ovr is important -- or if this analysis tells us something about how neural circuits are trying to classify different objects.\n(iii) How much is the low rank projection step in CLLD impacting the match to data? How do you select the dimension to project to?\n\n- Further synthetic experiments involving homogeneity and heterogeneity would be useful to understand the theory\n\n- The extension of the theory is relatively incremental and hasn’t been dissected in a rigorous manner.\n\n- Instead of measuring across manifold correlations using manifold centers, they propose to use categorical local linear differences (CLLD). Are these metrics sensitive to the size of the dataset and how many neurons are in each? \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject