Diffusion Curvature for Estimating Local Curvature in High Dimensional Data
Authors: Dhananjay Bhaskar
Kincaid MacDonald
Oluwadamilola Fasina
Dawson S Thomas
Bastian Rieck
Ian Adelstein
Smita Krishnaswamy
Venue: NeurIPS 2022 Accept (2022)
Keywords: local curvature estimation
neural network loss landscape
diffusion

Abstract: We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.

Diffusion Curvature for Estimating Local Curvature in High Dimensional Data
Dhananjay Bhaskar
Kincaid MacDonald Department of Mathematics Yale University
Department of Genetics Yale University
jhananjay. bhaskar@yale. edu
kincaid. www.mmaldddall.....mmmmmm edu
Oluwadamilola Fasina
Dawson Thomas
Applied Mathematics Program
Department of Mathematics
Yale University dami. wasinnnyylll.e edu
Department of Physics
Yale University
dawson. thomas@yale edu
Bastian Rieck
Ian Adelstein
Institute of AI for Health Helmholtz Pioneer Campus, Helmholtz Munich bastian, rieckShelmholtee tz www.mmhheemeennmmmmm de
Department of Mathematics
Yale University
ian adw.stein@yalleeeeeeeeeeoeeemeee edu
Smita Krishnaswamy Department of Computer Science
Department of Genetics
Applied Mathematics Program
Program for Computational Biology and Bioinformatics
Yale University
amita. haitaa.krsshnswwamyyylll.l.. edu
Abstract
We introduce intrinsic measure of local curvature point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, structure point cloud data and define local curvature based on the laziness of random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data and estimating local Hessian matrices of neural network loss landscapes.
Introduction
With the advent of high-throughput high- dighedimensional point-cloud data in many fields including biomedicine, social science, physics, and finance, there is increasing need for methods extract structure and patterns. key framework for exploring such data is the manifold hypothesis, which states that high-dimensional data arise samples from underlying lower dimensional manifold. The diffusion geometry and diffusion map framework first introduced by Coifman et al. [1] has proven to be useful framework for the shape and structure of the data. Diffusion geometry involves
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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converting data into an affinity graph which is then Markov normalized to form data diffusion operator. This diffusion operator allows for the understanding of geometric features of the data such as manifold-intrinsic distances, and intrinsic manifold eigendimensions (also known as diffusion maps). However, curvature, salient feature of Riemannian geometry, has received less attention. Being an inherently smooth quantity, its extension to the discrete case is not straightforward. The most authoritative definitions of curvature in the discrete setting have been that of Ollivier-Ricci [2] and of Forman [3. which compute curvature on edges of a graph. By contrast, here we propose method of estimating curvature at points or regions in point-cloud data, where there are no inherent edges to consider- one of the first such formulations to our knowledge.
To achieve this, we utilize the connection between data diffusion and volume to define a new formulation of curvature known as diffusion curvature based on the volume growth of graph diffusion starting from a vertex. Starting from unit mass placed at one or several points, we show that in cases of positive curvature the volume growth tends to be slower with more of the mass remaining in the original point, whereas in cases of negative curvature the mass tends to disperse faster to other points. Therefore the ratio of mass in the original point, compared to the remainder of the cloud gives a scalar value of curvature. We note that this method avoids the complexity of having to consider all (potentially exponentially many) edges that would arise by simply attempting to directly translate notions of edge curvature to the data affinity graph. In addition to the scalar notion of curvature, we also show that network training based on diffusion probabilities as the input.
able to compute quadratic form of arbitrarily high dimensions using neural
We validate our notion of curvature on toy datasets as well as real single-cell point cloud data of very high dimensionality. Then, we study both the scalar diffusion as well as the diffusion based quadratic form in studying the area surrounding found minima in neural network loss landscapes. The Hessian of the parameter space is generally difficult to compute as contains O(n²) entries for a neural network with n parameters. However, restricting ourselves to the area around local minima yields a lower dimensional space, where only subset parameter combinations change in the vicinity. We sample in this lower dimensional space and compute a diffusion operator which allows us to assess the curvature of the minima. First, we can assess if the point of convergence is indeed minimum potentially saddle. Further, studies have shown that flat minima, i.c., minima situated in a neighborhood of the loss landscape with roughly similar error, tend to generalize better than sharp minima. These qualities can be assessed from curvature estimations of minima regions.
Our key contributions include:
The formulation of diffusion curvature, an intrinsic scalar valued curvature for regions of high dimensional data sampled from an underlying manifold, computed from data diffusion. Proof that our notion of curvature will result in higher values in positive curvature based on
volume comparison results from Riemannian geometry.
A neural network version that computes a quadratic form starting from a diffusion map.
Validation of our framework on toy and single cell datasets.
Application to Hessian estimation in neural networks.
2 Background
2.1 High Dimensional Point Clouds and the Manifold Assumption
A useful assumption in representation learning is that high-dimensional data is sampled from an intrinsic low.dimensional manifold that is mapped via nonlinear functions to observable high- dimensional measurements; this is commonly referred to as the manifold assumption. Formally, let M be hidden d dimensional manifold that is only observable via collection of >>> d nonlinear functions fi,...: fi.----ni: fn: Md + IR that enable its immersion in a high-dimensional ambient space as F(M")= = (f(z) Conversely, given a dataset X = (2). fn())) = E MM) C R
" from which data is collected. 2 C In" of high-dimensional observations, manifold learning methods assume data points originate from a sampling Z = { C -Md of the underlying manifold via v, f(2),, and aim to learn low dimensional intrinsic representation that approximates the manifold geometry of Ma.

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2.1.1 Diffusion Maps
To learn a manifold geometry from collected point cloud data, we use the popular diffusion maps construction M. This construction starts by considering local similarities defined via a kernel K(x, y). with y E F(M"), that captures local neighborhoods in the data. While the Gaussian kernel is a popular choice for IC, it encodes sampling density information in its computation. Hence, to construct a diffusion geometry that is robust to sampling density variations, we use an anisotropic kernel
(x v)
K(x,, y) = [[[(z, )ll (V,,
with || || the 1-norm (or row sum) and g(z, y) = - as proposed in [. where 0 S a S 1 controls the separation of geometry from density, with
yielding the classic Gaussian kernel, and a = 1 completely removing density and providing a geometric equivalent to uniform sampling of the underlying manifold. In [1]. it is shown that in the limit of infinitely many points this becomes equivalent to a Laplace-Beltrami Beltrami operator, which encodes geometric properties of the underlying manifold. Next, the similarities encoded by K. p(z, y) = /(<=,))//KK( that are organized an N x N row stochastic matrix Pú = p( xry) that describes a Markovian diffusion process over the intrinsic geometry of the data. Finally, diffusion map is defined by taking the cigenvalues 1 A 2 X2 2 ... 2 An and (corresponding) cigenvectors { J.y of P. and mapping each data point T; E X an N dimensional vector:
normalized to define transition probabilities
T (x;)) (x:). .... (wen:)))....,,e(a (x,)(
Here represents diffusion-time. In general, as t increases, most of the cigenvalues become negligible; truncated diffusion map coordinates can thus be used for dimensionality reduction, and Euclidean distances in this space eigenvector, Φ1 (with corresponding eigenvalue A = 1), is often omitted since it represents the stationary distribution of the Markov random walk and does not contain information about the data. Previous work has demonstrated robustness to noise and outlined the best practices for choosing kernel parameters for various applications, including diffusion-based dimensionality reduction and manifold learning of single-cell data [5] (7].
manifold-intrinsic distance [1] 4 Note that the trivial
The eigenvalues of the Laplace- Laplace-Beltrami operator, well the eigenvalues of the diffusion map, encode geometry information about a compact Riemannian manifold M. as can be seen via the asymptotic expansion of the trace of the heat kernel [8. Dimension, volume, and total scalar curvature are spectrally determined properties of the manifold. Given the relationship between diffusion coordinates and the eigenvectors of the Bellace-Beltrami operator, one surmise that the diffusion operator also encodes geometric information. These relations motivate our use of diffusion measure curvature on a data manifold. Section A5 in the Appendix contains additional details in the Riemannian setting.
2.2 Discrete Curvature in Riemannian Geometry
There a long history in Riemannian geometry of using curvature to study geometric and topological properties of a Riemannian manifold. In particular, lower bounds on Ricci curvature have been related to diameter (Bonnet-Myers (9). volume (Bishop-Grome Gromov [00), the Laplacian (Cheng- Yau [T), the
isoperimetric constant (Buser (12), and topological properties of the manifold (Hamilton's 's Ricci flow 13) Although somewhat paradoxical, there has recently been work to extend these smooth Riemannian ideas to the discrete setting, in particular to graphs and Markov chains. Especially notable in this regard is the definition of discrete Ricci curvature due to Ollivier which makes use of the transport distance between probability distributions.
Ollivier's Ricci curvature 2 starts with a metric space X equipped with a random walk (a probability measure (.) for each xr E X) and assigns the edge-wise scalar curvature kon(I, y) Wi(m,,,))/dda, where -.(..) is the L' transportation distance (also known as earth mover's distance or Wasserstein distance). In the Riemannian setting, if one defines the random walk to be
dm = dool()l/wolB((.r)) then Ollivier demonstrates that koR(2, y) = + Rsc(v,))/2442 + O(r3 g))") where is unit tangent vector, and is a point on the geodesic issuing from with d(x, y) small enough. Ollivier (and others 14) (5)) have used lower bounds on this notion of curvature to study global properties of the space (i.e. diameter, volume growth, spectral gap).
3

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The main downsides of Ollivier's Ricci curvature are that it is an edge-wise wise notion and that it requires knowledge of the transport distance. A priori point cloud data does not come with a graph structure. We therefore propose a diffusion based method for prescribing a point-wise scalar curvature to point cloud data. Like Ollivier's s notion, this diffussoon curvature is based on the idea that the spread of geodesics is influenced by Riemannian curvature. We prove theoretical bounds on the diffusion curvature by appealing to the Bishop-Gromov volume comparison theorem.
2.3 The Loss Landscape and Hessian of Neural Networks
In the context of neural networks, second-order information about the loss function contains useful information about the generalizability of minima, the heuristic being that flat minima generalize better than sharp minima. Previous studies have sought to directly examine the loss landscape of neural network. Li et al. [66 used random samples around found minima to visualize and quantify sharpness of minima, Horoi et al. [77 utilized jump-and-retran
sampling visualize and classify "good" and "bad" minima. However, computing the full Hessian of the parameter space is only possible in special cases such as variational quantum classifiers [18] and it is generally intractable for nigh-dimensional systems (such as the parameter space of a neural network). Consequently, previous work has sought to approximate the largest (and smallest) eigenvalues of the Hessian [99 20] yielding an incomplete picture. To circumvent this, we estimate a low dimensional Hessian of the loss function around its optimum using a quadratic approximation. Restricting ourselves to the area around the optimum yields a lower dimensional space, where only a subset of parameter combinations change in the vicinity. This Hessian approximation is achieved by sampling around the optimum, constructing a diffusion operator based on these sampled points, and using neural network to learn the coefficients of the quadratic approximation.
Neural networks are trained such that their output function f 8) matches a given function y(X) on training data X by performing, e.g,, stochastic gradient descent loss function comparing y and f with respect to the parameters 0. A typical example of a loss function is mean squared error L(X. 0) = Erex x (x)) 122) The loss function defines loss landscape for fixed training data X. Here, we use data diffusion to directly estimate the Hessian of the loss landscape, with respect to the parameters of neural network, around found minima, providing a quantitative measure of the region in terms of its spectrum and condition number. In general, the Hessian of a scalar-valued function, (() is symmetric matrix of partial derivatives with the quadratic form:
82 C
82 C
30,00m
VI
HC(v)= -01
vn)
: 0²
0²c C
89, 09,
89,, 09.
Since the gradient is zero at critical points, i.e., VC(0o) = O. the quadratic approximation to the function around its critical points is given by: L()) ~ [(0o) + 1/2H 0o)- We see that the signature of the Hessian (the signs of its eigenvalues) precisely classifies the critical points (V)(()) = 0) as a local maximum (all cigenvalues are negative), minimum (all cigenvalues are positive) saddle (eigenvalues have mixed signs). Computing local Hessian yields rich information about the local curvature around the optimal point in the neural network parameter space.
3 Methods
3.1 Diffusion Curvature
We now illustrate how diffusion can be used to measure the relative spreading of geodesics under the influence of Riemannian curvature. To build intuition, we first discuss the case of surfaces (dimension 2). Three canonical surfaces are the sphere, the cylinder, and the saddle. The Gaussian curvature of these surfaces are positive, zero, and negative, respectively. Imagine taking a sticker and trying to adhere it to one of these surfaces. The sticker, having been printed on a flat piece of paper, has zero Gaussian curvature. will adhere perfectly to the cylinder, will bunch up (there will be too much sticker material) when trying to adhere it to the sphere, and will rip (there will be too little sticker material) when trying to adhere to the saddle. This example showeases the area comparison definition [21 pp. 225-226 of Gaussian curvature, k where one computes the limiting difference

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between the area A(r) of a geodesic disk on the manifold and standard Euclidean disk,
lim 12 TT² A(r) -
Tr
For example, Gaussian curvature will be positive when Ny > A(r), i.e., when the area of the sticker exceeds the area of the corresponding geodesic disk on the sphere. The Bishop-Gromov volume comparison theorem formalizes how extend this into dimensions to M the complete n-dimensional Sumply-connected Riemannian manifold of constant sectional
2. Here we will compare
curvature k. i. the sphere of radius 1/v0 when k > O, Euclidean n-space when A: = 0, and the appropriately scaled version of hyperbolic n-space when k <
Theorem 1 (Bishop-Gromov)). Let (M" :9) be a complete Riemannian manifold with Ricci curvature bounded below by (n 1)k. Let r) denote the ball of radius about point p E M and B(ps,") r) denote a ball of radius r about a point Pk E ME (as defined above). Then v(r) = Vol(B(p-r)) B(p..))r) is non non-increasing function on (0, oo) which tends Las 0. In particular. Vol(B(p,r) r)) I (p)))(
Theorem I captures the sticker phenomenon: curvature increases, the volume of comparable geodesic balls decreases. Positive curvature corresponds to geodesic convergence and smaller volumes, whereas negative curvature corresponds to geodesic divergence (spread) and larger volumes. The discrete nature of the data manifold makes it impossible to compare volumes as the distance scale (radius) goes to zero. Theorem 2 gives us access the intrinsic distance between data points, i.e., the diffusion distance, which we then use to define metric balls on the Riemannian manifold M. (x:)) embeds
Theorem 2 (Coifman et al. 122). The diffiusion map (x.))) = (Z)) data into Euclidean space where the Euclidean distance is equal to the diffussonn distance Dm, i.e. D2 (z. y) = (x) - 4 (1) (())))))
We can now define B,, (x, {v E M: (z,y) Sr) C M to the ball centered at x E M with diffusion radius Let r denote the set of sampled points from Bn (2, and let [B(x, r)] denote the cardinality of this set. To define diffusion curvature, we generate random walk from a point x by diffusing Dirac based æ, i... m. S,Pº where 8₂ is the one-hot vector, so that mz (y) = P' y) is the transition probability from x to y-
Definition 1. The pointwise diffusion curvature C(z) is the average probability that a random walk starting from point ends within B r) after t steps of data diffusion, i.e.,
E Eve my(y)
C(x)
[B(rr)) r)
We extend this definition contiguous region U of the manifold M consisting of neighboring points U = {x,) ( by defining my (-) = 6 where By is the indicator vector on the set U.
We have C(x) E [0, 1/N] where = [B(x,)) r) with larger values indicating higher curvature relative to lower values. The idea is to use diffusion probabilities to capture the relative spreading of geodesics. Intuitively, random walker more likely to return to their starting point in a region of positive curvature (where paths converge) than negative curvature (where paths diverge).
More precisely, in negatively curved regions, the random walker can get lost in the various divergent (disconnected) branches, whereas in positively curved regions the paths of the random walker exhibit more intercoonnectivity. so that the return probability of a walk is higher. We make this idea more formal in the following:
Theorem 3. If we sample uniformly from a Riemannian manifold M with Ricci curvature bounded below by k(n 1) then C(x)) Z C(æR), where M and Hg ME (as defined above).
Proof. From Theorem I we have vol (m r)) < vol (M r)) and together with uniform sampling this yields BB(rrr) r) < [B(x,,)) r)). We then have B/1 r) Z 1/1 so that even with uniform transition probability measures we achieve the desired inequality. Moreover, as the probability measure m.x()) is constructed by diffusing a Dirac according to affinities based on a
Gaussian kernel, the measure decays with relative diffusion distance from x. Hence for sets centered
5

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B]
Seep
Radiss
Step 2
Figure 1: Diffusion recovers Gaussian curvature information (up to a scaling factor and offset) on test manifolds, c.g. sphere, torus, ellipsoid and hyperboloid.
at 2, larger cardinality implies lower average transition probability. We may therefore conclude that C(x)= EyeB(u.r) m=( EyeB(uz.r) Mz (v)
C(rx).
[B(x,r)
[BIk:T]]
While diffusion curvature can be computed easily, it also necessitates an accurate sampling procedure, which can problematic for difficult spaces. Moreover, due to formulation, diffusion curvature only provides relative (non-negative) values.
Limitations.
3.2 Neural Network Implementation of Quadratic Forms
In the previous section, have shown that the diffusion operator and associated diffusion proba- bilities contain information for computation of curvature from point cloud data. In this section, we extend this measure from scalar quantity an entire quadratic form by training a neural network on the diffusion map embedding of the point cloud. A schematic of our neural network, called CurveNet, shown in Figure 2
CurveNet takes as input points {X; = (x,, 3 sampled near local minimum or saddle point (x,: ys). Here x; E IR* are input parameters for an objective function loss, f, with y; = fx; and fx(x)) = O. Points sampled from the parameter space of neural network are mapped to an N dimensional diffusion vector, (X,).) = o1 d1 (X.)..... X on (X.))" Recall from Equation I that the diffusion map is an eigendecomposition of the diffusion operator P Hence, without diffusion operator, and contains N dimensions for sampling of N points. Here utilize this to estimate & quadratic form of the sampled point cloud, y/ = x? Qx,, using neural network. Note that when the point cloud is sample of local region in the manifold, this gives a localized quadratic form describing 2nd order information the region sampled.
reduction dimensionality, it encodes all information present
Since our focus is on local quadratic approximation, we train CurveNet on samples from idealized surfaces of the form f(x) x" Qx where Q is a symmetric matrix with dimensionality dependent on the cardinality of the point cloud; see Figure 3 for some examples of generated 2-D surfaces. We generate series of such surfaces by varying parameters within Q and sampling points. We then provide the diffusion map embedding of the sampled points to the neural network and train it to predict Q using mean-squared error loss. We also include an L, regularization term to promote sparsity in the coefficients, which estimates the dimensionality of the input data, leading to
C(Q
(Qy Q1x)) a
(3)

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A) Training
3/
Generato randam quaddics x x
Semple around minimattaddee (xg, y: 1-11
Train MLP to prodict hessian
()
B) Test
(w
II
Points sampled landscape around minima/taddle (w,, 4.3.
Trained predicts
Figure 2: The neural network is trained points (x;, yr) sampled from randomly generated quadries and tested on weights (W,) and loss values (1,) sampled from the loss landscape neural networks. Objective function values, Yi, and loss values, 4, are treated special axes containing curvature information. Points sampled from the loss landscape, X,, are embedded using the diffusion map 4..
Curvature and the Hessian. In the classical setting of surfaces, a relevant notion of curvature is the Gaussian curvature In the special case that M is realized the graph of function f(x, y). the Gaussian curvature can be calculated explicitly from the Hessian matrix H of f by the formula
(
<(x, v)
(1 + (22))
Note that at local optimum of (where =/= the Gaussian curvature is completely characterized by the eigenvalues of the Hessian. In higher dimensions, by contrast, single number is longer sufficient to capture the full curvature information of a Riemannian manifold, and one instead appeals the Riemann curvature tensor. In the special case that M is realized the graph of function f: Iº + IR, the spectrum of the Hessian of f again contains important (albeit not complete) curvature information. As are interested in measuring curvature in the setting of loss landscapes, we apply CurveNet described above to directly compute the Hessian of the parameters with respect the loss function of a neural network in question.
4 Results
We validate both diffusion curvature and CurveNet using (a) synthetic test cases, and (b) single-cell RNA sequencing data (scRNA-seq). Moreover, we analyze the quality of our curvature estimation methods a data sampling scenario, where we sample points in the vicinity of an optimum. All test cases are chosen to assess the efficiency and accuracy of our methods.
We trained CurveNet N 1000 samples from idealized quadratic surfaces. Later, we sampled 1000 points in the local neighborhood of the model parameters to estimate curvature of the loss landscape via the diffusion map embedding. As the parameter space of neural networks can be large, sampling points around the minima is method of reducing dimensionality and restricting the estimation less complex space. In these local neighborhoods, the intrinsic dimension of the loss manifold is low, since trainable parameters are unlikely to change near the minimum. We used intrinsic dimensions of CurveNet can approximate the Hessian within reasonable range of intrinsic dimensions. We trained on 5000 different randomly generated quadrics, sampled using N = 1000 points, for each intrinsic dimension. CurveNet then outputs quadratic form with K(K 1)/2 entries, where (K max(k) 20), which we compare with the known ground truth using mean squared error (Table I). Training and testing were done on 8 core Tesla K80 GPUs with 24 GB memory/chip.
(2.3... .0] to generate idealized surfaces ensure that

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Table 1: MSE (u o) of neural network Hessian estimation, All values scaled by 10 lower is better.
Test Data
CurveNet epochs 1000)
CurveNet (# epochs 100)
Bascline (random)
Intrinsic dim.
nonzero coeffs.)
2 (4)
2.115 1.250
14.071 8.664
612.145 + 2. .36
5 (16)
0.964 + 0.144
4.865 + 3.382
554.652 = 1.704
10 (56)
0.752 = 0.013
0.798 = 0.029
530.203 = 0.645
15 (121)
0.766 + 0.022
0.085 + 0.007
521.050 + 0.485
20 (211)
0.711 0.034
0.082 0.004
515.739 + 0.397
Diffusion Curvature
Gausilan Curvature
Mean Curvature
Olivier Ricci Curvature
Correlation
(1e) Z
H - 1/2(k) xg)
Aonio k)
Mastual Information
[
Ste, Sterk, M0
Seet 1100 Mi.D4
Figure 3: Curvature estimation toy dataset of quadric surfaces.
4.1
Toy test cases for curvature estimation
First, we calculated diffusion curvature for simple 2-manifolds embedded in R3, We performed t-step random walks on points sampled from test surfaces like the torus, sphere, ellipsoid and hyperboloid. shown in Figure The resulting diffusion curvature measurement is strongly correlated Gaussian curvature, well known intrinsic notion of curvature. Next, we generated series of synthetic datasets representing minima and saddles where the primary objective is to estimate curvature at central points (which are not affected by edge effects). Figure 3 shows series of artificially generated 3D surfaces whose curvature varies in two principal directions from positive negative. In the left column we show the curvature estimate given by diffusion curvature (Eqn. Z}. The second, third and fourth columns show comparisons to Gaussian curvature (defined in Section 3.1 mean curvature and Ollivier's s Ricci curvature (defined in Section 2.2) respectively. The last column contains biaxial plots showing the correlation between Gaussian and diffusion curvature.
We observe that diffusion curvature captures Gaussian Curvature on these data. Despite being scaled differently. diffusion curvature highlights essentially the same structures as Gaussian Curvature. This qualitative observation quantified by the correlation. Overall, we obtain high correlations for the different surfaces. The last surface is slightly different, as high values measured by diffusion curvature are more concentrated within the "cusp", whereas Gaussian curvature spreads out such values over a larger part of the surface. This example demonstrates the benefits of diffusion curvature, though: the context of loss landscape analysis, diffusion curvature is much more sensitive such sharper minima, thus facilitating their detection.

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Diffusion Curvature
0.52
0.50
0.48
0.46
PHATE
Figure 4: Diffusion curvature of the iPSC data.
4.2 Curvature estimation for single-cell data
We also estimated the curvature of a publicly available single-cell point cloud dataset obtained using mass cytometry of mouse fibroblasts. The dataset does not contain any identifiable or human information. Mass cytometry is used to quantitatively measure 2005 mouse fibroblast cells induced to undergo reprogramming into stem cell state using using 33 channels representing various protein biomarkers. Such system is often called induced pluripotent stem cell (iPSC) reprogramming [23] This dataset shows progression of the fibroblasts to point of divergence where two lineages emerge, one lineage which successfully reprograms and another lineage that undergoes apoptosis [4). We note that our model correctly identifies the initial branching point (with cells that don't survive) as having low values of diffusion curvature indicating relatively negative curvature due to divergent paths out of the point (resulting in divergent random walks, see Figure 4). On the other hand it shows higher values indicating flat curvature along the horizontal branch. Diffusion curvature of another single-cell dataset, measuring embryonic stem cell differentiation, is provided in Section Al of the Appendix.
4.3 Hessian estimation and sampling
To obtain second-order information around critical point, Za. we estimate a local Hessian: we first (for both toy test cases and neural networks) sample points, 21, 22 Tn. at random in the neighborhood of the critical point. Here x; E R* where k the full dimensionality of the domain of the toy functions the neural network: f: IR + E and the value of the objective function or loss, I R can be obtained by evaluating f(x;). In both settings, this allows us to obtain an (k++)) matrix. Each row represents sampled point 2; E I2 with its associated loss or objective function in the last column. We consider this column as a special loss axis and use it as an input to the neural network to learn the coefficients along with the diffusion axes, which are obtained from the sampled points.
The diffusion axes were obtained by constructing a diffusion map in RR based on the sampled points around the optimum, i.e., the minimum. We uniformly sampled 1000 points k-dimensional hypersphere which was then scaled by the parameter space the optimum or saddle, as well as the gradient that point. Special at the relative difference between the evaluated loss at locally sampled points around the optimum or saddle and the actual loss at the optimum or saddle. We then use these diffusion coordinates &xx) and the value at the sampled point f(x) E IR as an input pair (#(x), f()) to the neural network, which we ultimately use to estimate the Hessian. Figure shows the eigenspectrum of the Hessian estimated using CurveNet for feed-forward neural network trained to classify MNIST digits. We observe that the number of negative eigenvalues of the Hessian matrix (indicative of maximum in the loss landscape) decrease over training epochs. In Figure 5 when comparing epoch 25 and epoch 200, we observe a marked shift the (cumulative) density of eigenvalues towards positive eigenvalues, showing that the feed-forward neural network is approaching minima in the parameter space. Hessian eigenspectrums estimated using CurveNet for convolutional neural network and ResNet- 18 are provided in the Appendix (Sections A2 and A3).
taken ensure the points were sampled locally by evaluating

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080
Figure 5: Density (left) and cumulative density (right) of the eigenspectrum when sampling around an optimum. At epoch 200, we observe substantial decrease in the number of negative eigenvalues, showing that CurveNet is better capable of approximating the minimum.
Conclusion
We proposed diffiusion curvature, measure of intrinsic local curvature of point clouds. Our measure leverages the laziness of random walks, obtained using the diffusion maps framework. We link diffusion curvature to existing volume comparison results from Riemannian geometry. In contrast to these notions, diffusion curvature can be computed effectively via neural networks even for high- dimensional data. While we demonstrated the effectiveness of such curvature measure by analyzing numerous datasets of varying complexities, our formulation also leads to new research directions. Of particular interest will be proving additional results about our measure, relating it to existing quantities such as the Laplace-Beltrann Beltrami operator, as well as formally proving its stability properties. We also want to develop new methods that use diffusion curvature to compare different datasets; being a quantity that is invariant under transformations (such as rotations of point cloud), we consider diffusion curvature be suitable candidate for assessing the similarity of high-dimensional complex point clouds.
Negative Impact: We cannot envision any negative societal impact of this work.
Acknowledgments and Disclosure of Funding
D.B. is funded by a Yale BleeBBoerringe Ingelheim Biomedical Data Science Fellowship. S.K. re- ceived funding from the NIH (R01GM135929, RO1GM130847, R01HD100035), NSF Career Grant (2047856), and the Sloan Fellowship (FG-2021-15883). D.B., K.M., D.T., B.R., LA.,, and S.K. gratefully acknowledge funding from NSF DMM-2050398 which enabled this project.
References
[1] Ronald R. Coifman and Mauro Maggioni. Diffusion wavelets. Applied and Computational Harmonic Analysis, 21(1):53 144433994444-1 July 2006.
[2] Jayson Sia, Edmond Jonekheere, and Paul Bogdan. Ollivier-Ricii Curvature- Based Method to Community Detection in Complex Networks. Scientific Reports, 91):98800 July 2019, Number: Publisher: Nature Publishing Group.
[3] Robin Forman. Bochner's method for cell complexes and combinatorial Ricci curvature. Discrete & Computational Geometry, 29(3):323-374, Feb 2003.
[4] Kevin R. Moon, David van Dijk, Zheng Wang, Scott Gigante, Daniel B. Burkhardt, William Chen, Kristina Yim, Antonia van den Elzen, Matthew J. Hir, Ronald R. Coifman, Natalia B. Ivanova, Guy Wolf, and Smita Krishnaswamy. Visualizing Structure and Transitions in High- Dimensional Biological Data. Nature Biotechnology. 37( 14:2-1144 December 2019.
[5] Scott Gigante, Adam S. Charles, Smita Krishnaswamy, and Gal Mishne. Visualizing the PHATE of Neural Networks, August 2019. 1eXiv:1908.. 008802831111100000 [cs, stat].
[6] David van Dijk, Roshan Sharma, Juozas Nainys, Kristina Yim, Pooja Kathail, Ambrose Carr, Cassandra Burdziak, Kevin R. Moon, Christine L. Chaffer, Diwakar Pattabiraman, Brian Bierie,
10

---

Linas Mazutis, Guy Wolf, Smita Krishnaswamy, and Dana er. Recovering Gene Interactions from Single-Cell Data Using Data Diffusion. Cell, 174(3):16-772.ee) July 2018.
[7] Daniel B. Burkhardt, Jay S. Stanley, Alexander Tong, Ana Luisa Perdigoto, Scott A. Gigante, Kevan C. Herold, Guy Wolf, Antonio J. Giraldez, David van Dijk, and Smita Krishnaswamy, Quantifying the effect of experimental perturbations at single-cell resolution. Nature Biotech- nology, 39(5):619-62 May 2021. Number: 5 Publisher: Nature Publishing Group.
[8] Carolyn S. Gordon. Survey of isospectral manifolds. In Franki J.E. Dillen and Leopold C.A. Verstraelen, editors, Handbook of Differential Geometry, pages 747-778. North-Holland, 1st edition, 2000.
[9] S.B. Myers. Riemannian manifolds with positive mean curvature. Duke Mathematical Journal, 8(2):401-404, 1941.
[10] R.L. Bishop. A relation between volume, mean curvature, and diameter. Annales Scientifiques de I' l'École Normale Supérieure, 10(312), 1963.
[11] Y. Cheng and S. T. Yau. Differential equations on Riemannian manifolds and their geometric applications. Communications on Pure and Applied Mathematics, 28(3):333-33) 354, 1975.
[12] Peter Buser. A note on the isoperimetric constant. Annales Scientifiques de l'École Normale Supérieure, 15(2):213-230, 1982.
[13] Richard S. Hamilton. mhree-manifolds with positive Ricci curvature. Journal of Differential Geometry, 17(2):255-366, 1982.
[14] Scong-Hun Paeng. Volume and diameter of graph and ollivier's ricci curvature. European Journal of Combinatorics, 33:1808-1819, 2012.
[15] Yong Lin, Linyuan Lu, and Shing-Tung Yau. Ricci curvature of graphs. Tohoku Mathematical Journal, 63(4):605-627, 20
[16] Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, and Tom Goldstein. Visualizing the Loss Landscape of Neural Nets. In Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018.
[17) Stefan Horoi, Jessie Huang, Bastian Rieck, Guillaume Lajoie, Guy Wolf, and Smita Krish- naswamy. Exploring the Geometry and Topology of Neural Network Loss Landscapes. Techni- cal Report arXiv:2102.0048: arXiv, January 2022. S.Ki:2102.00048 [es, stat] type: article.
[18] Pinaki Sen, Amandeep Singh Bhatia, Kamalpreet Singh Bhangu, and Ahmed Elbeltagi. Vari- ational Quantum Classifiers Through the Lens of the Hessian. PLOS ONE, 17())e0222344, January 2022. arXiv:2105. 10162 [quant-ph].
[19] Guillaume Alain, Nicolas Le Roux, and Pitore-Anoiine Manzagol. Negative eigenvalues of the Hessian in deep neural networks. ICLR, 2018.
[20] Levent Sagun, Utku Evci, V. Ugur Guney, Yann Dauphin, and Leon Bottou. Empirical Analysis of the Hessian of Ever-Parametrized Neural Networks, May 2018. arXiv:1706.04454 [es].
[21] Marcel Berger. A Panoramic View of Riemannian Geometry. Springer, Heidelberg, Germany, 2003.
[22] R. R. Coifman, S. Lafon, A. B. Lee, M. Maggioni, Nadler, F. Warner, and S. W. Zucker. Geometric diffusions tool for harmonic analysis and structure definition of data: Diffusion maps. Proceedings of the National Academy of Sciences, 102(21):7426-777) May 2005.
[23] Eli R. Zunder, Ernesto Lujan, Yury Goltsev, Marius Wernig, and Garry P. Nolan. A continuous molecular roadmap to iPSC reprogramming through progression analysis of single-cell mass cytometry. Cell Stem Cell, 16(3):323-37, March 2015.

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
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(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? (Yes) See Section additional details are provided in the Supplementary Materials.
(c) Did you report error bars (e.g., with respect to the random seed after running exper- iments multiple times)? [Yes] Mean and standard deviation of model predictions is reported T
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] We specify number and type of GPUs in Section 4 training time is provided in Supplementary Materials.
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12


Reviews:
[{"review_id": "q3HtYb4UmIl", "reviewer": "Reviewer_39uS", "summary": "The paper presents a technique to estimate local curvatures on point clouds using diffusion maps. The proposed metric, diffusion curvature, is quadratic in nature and resembles the gaussian curvature, in terms of discriminatively. The authors demonstrate its usability for estimating local curvatures on high (and low)-dimensional data and hessians of a neural network loss. The underlying idea is simple: the diffusion curvature measures the divergence in a local neighbourhood wrt to a ball of a constant radius. It can be computed on a point cloud of any dimension. ", "questions": "Please address the points raised in the weakness section.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n1. The proposed diffusion curvature is simple yet sufficiently novel, to my best knowledge.\n2. In general, the paper reads well. The main idea is made sufficiently clear.\n\n\nWeakness:\n1. As seen in figure 3, the proposed  diffusion curvature is more discriminative than gaussian one. Perhaps, it should also be compared with mean curvature as it is more discriminative than the gaussian one, especially in the flat regions. Is there any reason why comparison with mean curvature was ignored?\n2. It is unclear how the diffusion curvature handle the noisy data. Given its discriminative power, it seems that diffusion curvature will be more affected by the noise in data as compared to the gaussian curvature. Repeating the hessian estimation (figure 3) with varied levels of noise in the data is important to understand how unstable the proposed metric is.\n3. The experimental section is missing a practical use-case. A possible use-case could be the use of this metric to correct noisy 3D data obtained through various 3D sensors such as Kinect.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "fGpQpE9Hxmf", "reviewer": "Reviewer_PnD7", "summary": "This paper introduces a new measure of the local curvature of a point cloud sampled from an underlying manifold: the so-called diffusion curvature. This measure uses the framework of diffusion maps and random walks. More precisely, the idea is to use random walks to capture the relative spreading of geodesics. Actually, a random walker is more likely to return to its starting point in a region where geodesic paths converge, i.e. in a region of positive curvature than negative curvature. The authors demonstrate by experiments that their notion of curvature scattering can be related to the Gaussian one.\n\nUsing a diffusion map, the authors also propose a way to locally approximate the Hessian associated with the loss function of a neural network for a given data set. This second-order approximation allows quantifying the (local) minima and thus quantifying their generalizability. Intuitively, the authors claim that flat minima generalize better than sharp minima. This estimation is done using a neural network: CurveNet.\n\nThe experiments are conducted on simulated data, which allows, in particular, the comparison with Gaussian curvature, and single-cell data.", "questions": "Here are some minor comments and suggestions:\n* _l41:_ “the minima” Which one?  Which function?  Uniqueness?\n* _l69:_ $\\lVert.\\rVert_1$ is not defined.\n* _l77:_ Maybe add a sentence to justify why $\\lambda_1=1$?\n* _l88 (and the next times):_ Which annex?\n* _Section 2.3_: Revise this paragraph to better motivate the introduction of the Hessian that arrives without context. Maybe write the paragraph the other way around and start talking about the generalizability of minima? Just a suggestion!\n* _l129:_ No need to redefine the sphere.\n* _l130:_ $K$ is not defined.\n* _Theorem 1:_ The same notation ($B(p,r)$) is used to define two different objects.\n* _Figure 1:_ The article does not refer to Figure 1.\n* _l254:_ “sample $n$ points” According to which law?", "limitations": "This work is theoretical and has no short-term societal impact.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I appreciated the efforts of the authors to give geometric insights, especially in dimension 2 (see section 3.1), throughout their paper. In general, the article is pleasant to read and well constructed. \n\nIn my opinion, one of the main limitations of this work concerns the sampling of points (also by the authors' admission, line 175). This point is not discussed here, and I wonder how this notion of diffusion curvature behaves in very high dimensions.\n\nOn the other hand, as much as the article is well worked, there is a lot of redrafting of supplementary material to be done. Several times in the article, the authors refer to it, but without specifying which part of the appendix. Moreover, the different sections are never put in context, and their titles are not explicit. I have the feeling that the authors have cut and pasted pieces of their articles to fit into the 9-pages limit without trying to recontextualize the different paragraphs. In the end, it makes this supplementary material difficult to follow.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "dJ9Fzm6R93q", "reviewer": "Reviewer_4SQF", "summary": "The paper formulates a discretized scalar curvature for graph and point cloud data, using the diffusion rate of a random walk started at the query point. This is tested on toy cases, and on single-cell data, showing its basic efficacy and correlation with Gaussian curvature (in the toy cases).\n\nAdditionally, it considers a procedure for Hessian estimation using diffusion maps to perform dimensionality reduction. In particular, they first train a neural network to estimate Hessians from quadrics of various degrees. Next, given a local optimum, they tack on the label dimension and put it through a diffusion map for the embedding, and apply the neural network. This was tested on synthetic data from a high-dimensional hypersphere, giving again, some evidence of efficacy.", "questions": "1. How do you deal with the problem that the dimensions of your input data far outweigh your output label? This would seem to suggest that it would be easy for this information to get washed out in the embedding, and you might simply end up measuring curvature of the input.\n2. It's repeatedly mentioned that sampling around the minima is a method of dimensionality reduction. I'm not sure I understand why this is the case. It seems to me that if you randomly perturb about a point in $R^{|W|}$ where $|W|$ is the dimension of the weight space, then you are still generating a random sample from a full-dimensional ball.\n3. In practice, many diffusion maps methods are quite sensitive to the kernel choice when building a graph from a dataset. Did you find that your method was robust to changes in some of those parameters (e.g. $\\alpha$)?\n4. Did you test whether your estimated Hessians and their spectra actually gave information on how robust a local optimum was?", "limitations": "I think the method presents an interesting idea, but the exploration is lacking, and with a lack of strong experimental evidence, I'm not clear that its assumptions are valid. Most of my questions above reference my doubts and potential pitfalls I see for the method.\n\nI would recommend that further work be done on experimental validation be done to address these concerns, if the paper is not accepted.", "rating": 3, "confidence": 5, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The method is certainly novel, and the Hessian optimization method tackles a problem of importance.\n- I quite liked Figure 2, which succinctly summarized their Hessian estimation method.\n\nWeaknesses:\n\n- The scalar curvature quantity and the Hessian estimation method are only loosely related. Either one is not really necessary for the other. The linking concept is that of diffusion maps.\n- The effectiveness of the Hessian optimization method is never tested on a real dataset, in the proposed application, as some measure of how well a local minimum might generalize (question 4 below). \n- I think the method overlooks the challenges presented by the potential high dimensionality of the weight space. See question 1 below.\n- The experimental comparisons are relatively weak, with no competing methods considered for either concept.\n- There was no test for robustness to kernel parameters (see question 3).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "3Cgd0B0DPK2", "reviewer": "Reviewer_r7PE", "summary": "This paper proposes a new local descriptor on sampled data, which quantifies the local recurrence properties of a random walk. This corresponds to a notion of curvature: random walks diffuse rapidly in regions that are negatively curved (such as branching trees), and stay more concentrated in regions of positive curvature (such as boundaries).\n", "questions": "Please refer to the questions above on implementation choices.", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper is original and well-written. It addresses an interesting topic in data sciences: what are the local descriptors that we can define (and understand!) on high-dimensional datasets.\n\nUp to and including Section 3.1, this paper is truly excellent: the authors introduce their ideas clearly, with an appropriate level of details. I appreciate the fact that the authors included relevant references to concepts and theorems in the mathematical literature. An unfortunate trend at NeurIPS/ICML is to provide truncated overviews of the literature in pure mathematics in order to present classical ideas as “novel”: the diligent work of the authors should be rewarded.\n\nOverall, I believe that this paper builds a very nice  bridge between modern geometry and machine learning applications. It introduces inspiring ideas in a way that is easy to digest by the NeurIPS community.\n\nPlease note, however, that I have some reservations about the content of pages 6 to 9:\n\n- I do not understand why the authors decided to rely on a black-box neural network to estimate a local quadratic fit to the input point cloud. This method is intrinsically less reliable and interpretable than a simpler estimator, and goes against the clean mathematical definition of the proposed diffusion curvature. Couldn’t the authors have simply relied on a least-square formulation? Even if we add an L1 regularization term as in Eq. (3), it seems clear that we could compute a decent quadratic fit to the input point cloud by solving a small convex optimization problem with respect to Q. In 2022, this is easy to do using e.g. the “cvxpy\" library.\n\n- One of the main motivations that is proposed by the authors for their method is that it allows one to estimate the Hessian of a neural network. A discussion of alternative methods and approximations would have been appreciated.\n\n- As far as I can tell, the authors do not discuss the scalability of their method. Since the proposed method relies on the computation of the first eigenvectors of a large N-by-N (normalized) kernel matrix, this is an important point. Could the authors explain why Eq. (1) is necessary? Couldn’t the authors simply iterate the application of a (normalized) Gaussian kernel matrix to simulate diffusion?\n  I assume that the authors could be interested by recent papers and codes on diffusion or kernel matrices, such as \"Fast geometric learning with symbolic matrices\", Feydy et al., that was presented at NeurIPS 2020 (with a relevant example on eigendecompositions at: http://kernel-operations.io/keops/_auto_tutorials/backends/plot_scipy.html) or the work of Nicholas Sharp (https://nmwsharp.com/) on the vector Heat method and Diffusion networks.\n\n- I would have appreciated some more details on the proposed experiments (e.g. better legends for the figures). Currently, it is hard to see them as anything more than toy examples: this could be fine for a theoretical paper, but there is certainly room for improvement here. For instance, the authors could illustrate the difference between their notion of curvature and other alternatives (discussed in Section 2.2) to highlight different behaviours, or discuss the influence of the choice of their kernel matrix (i.e. of the underlying random walk). This would be relevant for practitioners, in a context where the “best” local metric for a given dataset is often not known.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept