Question:
The paper presents two algorithms, namely the one based on matrix completion and the one based on the Nyström method. Regarding the following description of the Wasserstein distance matrix approximation algorithm and its theoretical basis, which of the following statements are incorrect?
A. The W₂-Nyström algorithm approximates the matrix by calculating certain columns (such as the O(d log d) columns) and using the Nyström method. Its theoretical stability depends on the high rank and coherence conditions of the matrix.
B. The W₂-MC algorithm is based on a random sampling matrix completion method. It requires optimization using the Gram matrix representation of the Euclidean distance matrix and solves the low-rank constraint problem through the Lagrange method.
C. The experimental results show that, under the same sampling budget, the error of the W₂-MC algorithm is always lower than that of the W₂-Nyström algorithm, and the classification task accuracy is also higher.
D. The theoretical proof of the paper indicates that when there is no noise and the number of sampling columns satisfies O((d + 2)log(d + 2)), the Nyström method can accurately recover the distance matrix required for MDS embedding.

Answer: ABC

2.3
Nystr¨om method
Consider a symmetric matrix in block form D =
 A
B
B⊤
C

, where A ∈Rm×m and B ∈Rm×n. We
are interested in the regime m ≪n, where the aim is to estimate the block C using only the smaller
blocks A and B. Suppose that rank(A) = rank(D). Then, one can show that the columns of the
submatrix
B
C

lie in the span of the columns of
 A
B⊤

. Specifically, there exists an X ∈Rm×n
such that B = AX and C = B⊤X. If A is invertible, then X = A−1B, which yields the exact
relation C = B⊤A−1B. In the more general case where A is not invertible, we may instead use
the Moore–Penrose pseudoinverse to obtain X = A†B leading to C = B⊤A†B. This procedure is
known as the Nystr¨om method [15].
While it is not a completion algorithm per se, the Nystr¨om method can be used to complete a
distance matrix. In particular, if only the columns containing A and B⊤are computed, then the
entries of C can be recovered, or in the general case approximated, by B⊤A†B.
Notations For a distance matrix D, owing to symmetry and zero diagonal entries, we only consider
samples from the strictly upper-triangular part.
We denote by Ω⊂{(i, j) : i ∈[N], j > i}
the set of sampled index pairs, with cardinality |Ω| = m. Given a matrix D and an index set
I ⊂[N] := {1, . . . , N}, we write D(:, I) for the submatrix of D consisting of the columns indexed
by I. Analogously, D(I, :) denotes the submatrix formed by the rows indexed by I. PΩ(D)ij = Dij
if (i, j) ∈Ωand is 0 otherwise. If rank(D) = r and D = VrΣrV ⊤
r , then the incoherence ν of D is
p
n/r maxi ∥|Vr(i, :)∥.
3
Algorithms
Here we describe two algorithms for estimating the Wasserstein distance matrix to compute em-
bedings of images. Algorithm 1, W2-MC, randomly computes a given number of entries of the
distance matrix, and then uses a matrix completion algorithm to estimate the full squared distance
matrix. We leave the completion algorithm as a generic plug-in for existing algorithms, and consider
one designed for distance matrices in our experiments. We sample from the strict upper triangle,
include the diagonal, and reflect the entries across the diagonal to obtain the observed distance
matrix. We apply the same procedure after completion to ensure that the resulting matrix retains
the properties of a distance matrix.
Algorithm 2, W2-Nystr¨om, uses the Nystr¨om method to complete the distance matrix.
In
particular, we compute only a set of columns of the distance matrix Cobs = D(:, I), and estimate
Dest = CobsUC⊤
obs where U = Cobs(I, :) = D(I, I). As in W2-MC, we also enforce the distance
matrix structure by enforcing the 0 diagonal.
Note that for large N, the computational cost of both algorithms is dominated by the cost of
computing the W2 distances, and the cost of the matrix completion subroutine MC is often negligible
by comparison. Thus, algorithms that limit sampling complexity of the matrix are more desirable.
For this reason, we do not truncate the rank of the Nystr¨om approximation to D in W2-Nystr¨om,
which is usually done to save computation time.
3
Algorithm 1: W2-MC
1 Input: Data measures {µi}N
i=1, sample size m, matrix completion subroutine MC
2 Randomly sample m entries Ω
3 Compute observed distance matrix Dobs = PΩ(D), where Dij = W2(µi, µj)2
4 diag(Dobs) = 0
▷enforce 0 diagonal
5 Dobs = Dobs + D⊤
obs
▷reflect across diagonal
6 Dest = MC(Dobs)
7 Dest = 1
2(Dest + D⊤
est)
▷enforce symmetry
8 Output: Estimated distance matrix Dest
Algorithm 2: W2-Nystr¨om
1 Input: Data measures{µi}N
i=1, number of columns c
2 Randomly sample c columns I ⊂[N]
3 Compute columns of distance matrix Cobs =
(
W2(µi, µj)2,
i ∈[N], j ∈I
0,
otherwise.
4 Dest = CobsUC⊤
obs, where U = Cobs(I, :)
5 diag(Dest) = 0
▷enforce 0 diagonal
6 Output: Estimated distance matrix Dest
The Nystr¨om method is well-studied, and using some existing bounds, we are able to prove a
stability result for MDS embeddings on Wasserstein distance matrices. We assume that the data
measures in W2(Rn) are isometric to a set of Euclidean vectors {yi}N
i=1 ⊂Rd. Examples of such sets
have been studied and include common image articulations like translations, scalings, and shears
of a fixed base measure [16, 7]. We assume noise (or error) in the computation of the W2 distances,
and consider how the MDS embedding {zi}N
i=1 ⊂Rd coming from Dest of W2-Nystr¨om compares
to {yi}. MDS embeddings are centered and equivalent up to orthogonal transformation, so we
measure error by a Procrustes distance, defined as minQ∈O(d) ∥Z −QY ∥2, where O(d) is the set
of orthogonal d × d matrices. Note that Z and Y are matrices whose columns are the embedded
vectors.
Theorem 3.1. Suppose {µi}N
i=1 ⊂W2(Rn), and Dij = W2(µi, µj)2 has incoherence ν.
Sup-
pose there exist {yi}n
i=1 ⊂Rd such that Dij = ∥yi −yj∥2. Let λ > 0, δ ∈(0, 1), and compute
m ≳λν2(d + 2) max{log(d + 2), log(1/δ)} columns uniformly randomly in W2-Nystr¨om with er-
ror Cobsi,j = W2(µi, µj)2 + Eij such that ∥E∥2 ≤σd+2(D)/3.
If ∥Y †∥2∥E∥1/2
2
≤1/
√
2, then
Dest = CobsUC⊤
obs yields an MDS embedding {zi}N
i=1 ⊂Rd such that
min
Q∈O(d) ∥Z −QY ∥2 ≤(1 +
√
2)∥Y †∥2∥E∥2 ×

5
N
(d + 2)(1 −δ) + 3
s
N
(d + 2)(1 −δ) + 2
!
with probability at least 1 −max{re−λ(δ+(1−δ) log(1−δ)), δ}.
Proof. A simple modification of the proof of [17, Theorem 3.3] implies the upper bound with
(1 +
√
2)∥Y †∥2∥D −Dest∥2. Then [18, Theorem 3] implies that the columns selected would yield
4
an exact Nystr¨om decomposition of D without the presence of noise. Thus, the hypotheses of
[19, Remark 4.7] are satisfied, and yield that ∥D −Dest∥2 ≤5∥Vd+2(I, :)†∥2
2 + 3∥Vd+2(I, :)†∥2 + 2
where I is the set of columns selected by Algorithm 2 and Vd+2 are the first d + 2 left singular
vectors of D. Finally, [20, Lemma 3.4] shows that with the given probability, ∥Vd+2(I, :)†∥2 ≤
p
N/((d + 2)(1 −δ)) for all δ ∈(0, 1).
A simple corollary is that if we compute O((d + 2) log(d + 2)) columns of D without error, then
the MDS embedding is exact up to orthogonal transformation. Exact MDS embeddings of W2 data
were studied in [7].
4
Numerical Experiments
4.1
Fixed sample budget
We first do a fixed sample budget comparison of W2-MC and W2-Nystr¨om for completing the
Wasserstein distance matrix of 2, 000 points from the OrganCMNIST dataset from the MedMNIST
benchmark [21], comprising coronal plane CT scans of 11 different organs downsampled to size 28×
28. For uniform sampling, we subsample the matrix at different rates γ ∈{3%, 5%, 10%, 20%, 25%}.
We then match the number of observed entries between uniform sampling and the Nystr¨om method
by solving the equation c(c−1)
2
+ c(n −c) = γ · n(n−1)
2
for c (the number of columns selected by
Nystr¨om). This equation equates the number of off-diagonal entries used by both methods. For
MC in W2-MC, we use the factorized Gram matrix formulation from [14] based on the relationship
between the Gram matrix and the square distance matrix:
Find
P ∈Rn×q,
s.t.
(PP ⊤)ii + (PP ⊤)jj −2(PP ⊤)ij = Dij,
P ⊤1 = 0,
(2)
where q denotes the estimate for the rank of the underlying points. The centering constraint can
be removed by reparameterization. Defining A : Rn×n →R|Ω| as
A(X)α = Xii + Xjj −2Xij,
α = (i, j) ∈Ω,
the augmented Lagrangian is
L(Q; Λ) = 1
2

A(QQ⊤) −b + Λ

2
2 ,
(3)
with b is the vector of observed distances and Λ ∈R|Ω|.
The algorithm starts with an initial
estimate of Q (randomly initialized in our experiments) and then alternates between two steps:
updating Q via Barzilai–Borwein descent, and updating the Lagrange multipliers. For the dataset
in consideration, we set the rank estimate to be 200.
Table 1 reports the results averaged over 10 random trials. We observe that for fixed sample
budget, the Nystr¨om method consistently yields better accuracy compared to matrix completion.
While both Nystr¨om and typical matrix completion algorithms scale similarly up to logarithmic
factors, the Nystr¨om method is a single-step procedure, whereas the cost of the matrix completion
algorithm depends on the target accuracy, with the number of iterations required to reach a given
error often dominating the runtime. In our experiments, we run W2-MC for a sufficient number
of iterations, as we consistently observe stable convergence across trials. In practice, choosing a

Recovering Wasserstein Distance Matrices from Few Measurements
Muhammad Rana1, Abiy Tasissa3, HanQin Cai4, Yakov Gavriyelov1, and Keaton Hamm1,2
1Department of Mathematics, University of Texas at Arlington
2Division of Data Science, University of Texas at Arlington
3Department of Mathematics, Tufts University, Medford, MA 02155, USA.
4Department of Statistics and Data Science and Department of Computer Science,
University of Central Florida
Abstract
This paper proposes two algorithms for estimating square Wasserstein distance matrices from
a small number of entries. These matrices are used to compute manifold learning embeddings
like multidimensional scaling (MDS) or Isomap, but contrary to Euclidean distance matrices, are
extremely costly to compute. We analyze matrix completion from upper triangular samples and
Nystr¨om completion in which O(d log(d)) columns of the distance matrices are computed where
d is the desired embedding dimension, prove stability of MDS under Nystr¨om completion, and
show that it can outperform matrix completion for a fixed budget of sample distances. Finally,
we show that classification of the OrganCMNIST dataset from the MedMNIST benchmark is
stable on data embedded from the Nystr¨om estimation of the distance matrix even when only
10% of the columns are computed.
1
Introduction
The bare representation of data in many applications is high-dimensional [1]. In today’s era of
big data, this high dimensionality poses significant challenges and necessitates specialized methods.
Manifold learning refers to a class of techniques that compress data representations into lower-
dimensional forms, making them more suitable for analysis and computation. A widely used linear
technique is principal component analysis (PCA). However, linear approaches can be restrictive
when the data lie on or near a nonlinear manifold. To address this, nonlinear manifold learning
techniques aim to obtain compact representations while preserving local geometric structures [2].
We note that, in the standard way they are used, all of the aforementioned methods typically treat
data as vectors in a Euclidean space.
While many applications yield vector data, there are a number of domains that produce data
that may be best modeled as distributions or measures. For example, gene expression and natural
language processing data can be represented as distributions over gene networks or dictionaries,
respectively, and image data can be considered as discrete distributions over a 2-dimensional pixel
grid. Recent works have used this notion to improve understanding of data geometry and per-
formance on tasks such as classification or clustering [3, 4, 5]. One of the advantages of treating
images, for example, as measures, is that spatial relationships are not obscured as they are under
vectorization. Moreover, the natural notion of a metric structure on the space of measures is that
of the Wasserstein metric from optimal transport (OT) [6].
1
arXiv:2509.19250v1  [stat.ML]  23 Sep 2025
Several works have explored manifold learning using probability measures and the Wasserstein
metric, e.g., [5, 7]. This approach has shown improvement in task performance and data modeling,
but it comes with the significant drawback of slow computation time. In particular, computing
one Wasserstein distance exactly for discrete measures supported on m points costs Ω(m3 log(m))
flops. To form neighbor embeddings for N data measures requires computing the square distance
matrix Dij = W2(µi, µj)2, carrying total complexity O(N2m3 log(m)).
Mitigating this high computational cost can be approached in two ways: 1) speeding up a single
Wasserstein distance computation, and 2) computing fewer entries of D to form an approximation
of it. Many methods from computational OT explore the first approach, e.g., through entropic
regularization, multiscale methods, or Nystr¨om approximations to the cost matrix in the discrete
formulation of the Kantorovich problem (see [6] for a survey). In contrast, relatively few works have
explored the problem of estimating an entire Wasserstein distance matrix directly from samples.
The focus of this paper is to propose algorithms for estimating square Wasserstein distance
matrices for use in neighbor embeddings by computing as few entries as possible. Our approach
is based on exploiting the approximate low-rank structure of Wasserstein squared distances: if the
matrix D is approximately rank r, we propose two algorithms based on matrix completion and
the Nystr¨om method which allow us to estimate D from O(r log(r)) fully observed columns and
maintain stability of the MDS embedding.
2
Technical background
2.1
Optimal transport and MDS
The Kantorovich formulation of the quadratic Wasserstein metric on discrete probability measures
µ = Pn
i=1 αiδxi and ν = Pm
i=1 βiδyi is
W2(µ, ν) :=
min
P∈Γ(µ,ν) ⟨C, P⟩
(1)
where Γ(µ, ν) = {P ∈Rm×n
+
: P1m = α, P ⊤1n = β} are couplings with marginals µ and ν,
and Cij = ∥xi −yi∥2 [6].
This problem computes an optimal matching of mass between two
discrete measures. Given measures {µi}N
i=1 ⊂W2(Rn), the Wassmap algorithm [5, 7] computes
Dij = W2(µi, µj)2, a double centering of it B = −1
2HDH where H = I −1
N 11⊤, with truncated
SVD Bd = VdΣdV ⊤
d . The embedding vectors in Rd are then rows of VdΣd.
2.2
Matrix completion
Given a low-rank matrix with missing entries, the problem of estimating the unknown entries
is known as the matrix completion problem. This problem has been studied extensively due to
its broad applications, most notably in recommendation systems [8]. A naive formulation seeks
to minimize the rank of the matrix subject to the constraints imposed by the observed entries;
however, rank minimization is computationally hard in general [9]. A common convex relaxation is
to minimize the nuclear norm, which serves as a surrogate for rank. In this convex setting, seminal
theoretical results show that, under random sampling, an N × N rank-r matrix can be exactly
recovered from O(νrN log N) observed entries, where ν is an incoherence parameter capturing
the conditioning of the underlying matrix [10].
Given the computational challenges of convex
methods, non-convex algorithms have also been explored, and similar recovery guarantees have
been established in this setting [11]. In this paper, we focus on the completion of distance matrices.
In the case of Euclidean distance matrices, it can be shown that the squared distance matrix
2
has rank at most d + 2, where d is the embedding dimension of the underlying points [12]. This
structural property enables the use of low-rank optimization techniques for Euclidean distance
matrix completion [13, 14].
2.3
Nystr¨om method
Consider a symmetric matrix in block form D =
 A
B
B⊤
C

, where A ∈Rm×m and B ∈Rm×n. We
are interested in the regime m ≪n, where the aim is to estimate the block C using only the smaller
blocks A and B. Suppose that rank(A) = rank(D). Then, one can show that the columns of the
submatrix
B
C

lie in the span of the columns of
 A
B⊤

. Specifically, there exists an X ∈Rm×n
such that B = AX and C = B⊤X. If A is invertible, then X = A−1B, which yields the exact
relation C = B⊤A−1B. In the more general case where A is not invertible, we may instead use
the Moore–Penrose pseudoinverse to obtain X = A†B leading to C = B⊤A†B. This procedure is
known as the Nystr¨om method [15].
While it is not a completion algorithm per se, the Nystr¨om method can be used to complete a
distance matrix. In particular, if only the columns containing A and B⊤are computed, then the
entries of C can be recovered, or in the general case approximated, by B⊤A†B.
Notations For a distance matrix D, owing to symmetry and zero diagonal entries, we only consider
samples from the strictly upper-triangular part.
We denote by Ω⊂{(i, j) : i ∈[N], j > i}
the set of sampled index pairs, with cardinality |Ω| = m. Given a matrix D and an index set
I ⊂[N] := {1, . . . , N}, we write D(:, I) for the submatrix of D consisting of the columns indexed
by I. Analogously, D(I, :) denotes the submatrix formed by the rows indexed by I. PΩ(D)ij = Dij
if (i, j) ∈Ωand is 0 otherwise. If rank(D) = r and D = VrΣrV ⊤
r , then the incoherence ν of D is
p
n/r maxi ∥|Vr(i, :)∥.
3
Algorithms
Here we describe two algorithms for estimating the Wasserstein distance matrix to compute em-
beddings of images. Algorithm 1, W2-MC, randomly computes a given number of entries of the
distance matrix, and then uses a matrix completion algorithm to estimate the full squared distance
matrix. We leave the completion algorithm as a generic plug-in for existing algorithms, and consider
one designed for distance matrices in our experiments. We sample from the strict upper triangle,
include the diagonal, and reflect the entries across the diagonal to obtain the observed distance
matrix. We apply the same procedure after completion to ensure that the resulting matrix retains
the properties of a distance matrix.
Algorithm 2, W2-Nystr¨om, uses the Nystr¨om method to complete the distance matrix.
In
particular, we compute only a set of columns of the distance matrix Cobs = D(:, I), and estimate
Dest = CobsUC⊤
obs where U = Cobs(I, :) = D(I, I). As in W2-MC, we also enforce the distance
matrix structure by enforcing the 0 diagonal.
Note that for large N, the computational cost of both algorithms is dominated by the cost of
computing the W2 distances, and the cost of the matrix completion subroutine MC is often negligible
by comparison. Thus, algorithms that limit sampling complexity of the matrix are more desirable.
For this reason, we do not truncate the rank of the Nystr¨om approximation to D in W2-Nystr¨om,
which is usually done to save computation time.
3
Algorithm 1: W2-MC
1 Input: Data measures {µi}N
i=1, sample size m, matrix completion subroutine MC
2 Randomly sample m entries Ω
3 Compute observed distance matrix Dobs = PΩ(D), where Dij = W2(µi, µj)2
4 diag(Dobs) = 0
▷enforce 0 diagonal
5 Dobs = Dobs + D⊤
obs
▷reflect across diagonal
6 Dest = MC(Dobs)
7 Dest = 1
2(Dest + D⊤
est)
▷enforce symmetry
8 Output: Estimated distance matrix Dest
Algorithm 2: W2-Nystr¨om
1 Input: Data measures{µi}N
i=1, number of columns c
2 Randomly sample c columns I ⊂[N]
3 Compute columns of distance matrix Cobs =
(
W2(µi, µj)2,
i ∈[N], j ∈I
0,
otherwise.
4 Dest = CobsUC⊤
obs, where U = Cobs(I, :)
5 diag(Dest) = 0
▷enforce 0 diagonal
6 Output: Estimated distance matrix Dest
The Nystr¨om method is well-studied, and using some existing bounds, we are able to prove a
stability result for MDS embeddings on Wasserstein distance matrices. We assume that the data
measures in W2(Rn) are isometric to a set of Euclidean vectors {yi}N
i=1 ⊂Rd. Examples of such sets
have been studied and include common image articulations like translations, scalings, and shears
of a fixed base measure [16, 7]. We assume noise (or error) in the computation of the W2 distances,
and consider how the MDS embedding {zi}N
i=1 ⊂Rd coming from Dest of W2-Nystr¨om compares
to {yi}. MDS embeddings are centered and equivalent up to orthogonal transformation, so we
measure error by a Procrustes distance, defined as minQ∈O(d) ∥Z −QY ∥2, where O(d) is the set
of orthogonal d × d matrices. Note that Z and Y are matrices whose columns are the embedded
vectors.
Theorem 3.1. Suppose {µi}N
i=1 ⊂W2(Rn), and Dij = W2(µi, µj)2 has incoherence ν.
Sup-
pose there exist {yi}n
i=1 ⊂Rd such that Dij = ∥yi −yj∥2. Let λ > 0, δ ∈(0, 1), and compute
m ≳λν2(d + 2) max{log(d + 2), log(1/δ)} columns uniformly randomly in W2-Nystr¨om with er-
ror Cobsi,j = W2(µi, µj)2 + Eij such that ∥E∥2 ≤σd+2(D)/3.
If ∥Y †∥2∥E∥1/2
2
≤1/
√
2, then
Dest = CobsUC⊤
obs yields an MDS embedding {zi}N
i=1 ⊂Rd such that
min
Q∈O(d) ∥Z −QY ∥2 ≤(1 +
√
2)∥Y †∥2∥E∥2 ×

5
N
(d + 2)(1 −δ) + 3
s
N
(d + 2)(1 −δ) + 2
!
with probability at least 1 −max{re−λ(δ+(1−δ) log(1−δ)), δ}.
Proof. A simple modification of the proof of [17, Theorem 3.3] implies the upper bound with
(1 +
√
2)∥Y †∥2∥D −Dest∥2. Then [18, Theorem 3] implies that the columns selected would yield
4
an exact Nystr¨om decomposition of D without the presence of noise. Thus, the hypotheses of
[19, Remark 4.7] are satisfied, and yield that ∥D −Dest∥2 ≤5∥Vd+2(I, :)†∥2
2 + 3∥Vd+2(I, :)†∥2 + 2
where I is the set of columns selected by Algorithm 2 and Vd+2 are the first d + 2 left singular
vectors of D. Finally, [20, Lemma 3.4] shows that with the given probability, ∥Vd+2(I, :)†∥2 ≤
p
N/((d + 2)(1 −δ)) for all δ ∈(0, 1).
A simple corollary is that if we compute O((d + 2) log(d + 2)) columns of D without error, then
the MDS embedding is exact up to orthogonal transformation. Exact MDS embeddings of W2 data
were studied in [7].
4
Numerical Experiments
4.1
Fixed sample budget
We first do a fixed sample budget comparison of W2-MC and W2-Nystr¨om for completing the
Wasserstein distance matrix of 2, 000 points from the OrganCMNIST dataset from the MedMNIST
benchmark [21], comprising coronal plane CT scans of 11 different organs downsampled to size 28×
28. For uniform sampling, we subsample the matrix at different rates γ ∈{3%, 5%, 10%, 20%, 25%}.
We then match the number of observed entries between uniform sampling and the Nystr¨om method
by solving the equation c(c−1)
2
+ c(n −c) = γ · n(n−1)
2
for c (the number of columns selected by
Nystr¨om). This equation equates the number of off-diagonal entries used by both methods. For
MC in W2-MC, we use the factorized Gram matrix formulation from [14] based on the relationship
between the Gram matrix and the square distance matrix:
Find
P ∈Rn×q,
s.t.
(PP ⊤)ii + (PP ⊤)jj −2(PP ⊤)ij = Dij,
P ⊤1 = 0,
(2)
where q denotes the estimate for the rank of the underlying points. The centering constraint can
be removed by reparameterization. Defining A : Rn×n →R|Ω| as
A(X)α = Xii + Xjj −2Xij,
α = (i, j) ∈Ω,
the augmented Lagrangian is
L(Q; Λ) = 1
2

A(QQ⊤) −b + Λ

2
2 ,
(3)
with b is the vector of observed distances and Λ ∈R|Ω|.
The algorithm starts with an initial
estimate of Q (randomly initialized in our experiments) and then alternates between two steps:
updating Q via Barzilai–Borwein descent, and updating the Lagrange multipliers. For the dataset
in consideration, we set the rank estimate to be 200.
Table 1 reports the results averaged over 10 random trials. We observe that for fixed sample
budget, the Nystr¨om method consistently yields better accuracy compared to matrix completion.
While both Nystr¨om and typical matrix completion algorithms scale similarly up to logarithmic
factors, the Nystr¨om method is a single-step procedure, whereas the cost of the matrix completion
algorithm depends on the target accuracy, with the number of iterations required to reach a given
error often dominating the runtime. In our experiments, we run W2-MC for a sufficient number
of iterations, as we consistently observe stable convergence across trials. In practice, choosing a
5
Rate / # of cols.
W2-Nystr¨om
W2-MC
25% / 268
1.58 × 10−2
5.20 × 10−2
±1.96 × 10−3
±1.91 × 10−2
20% / 211
1.85 × 10−2
3.42 × 10−2
±2.2 × 10−3
±8.26 × 10−5
10% / 103
2.85 × 10−2
5.65 × 10−2
±2.60 × 10−3
±6.01 × 10−3
5% / 51
4.82 × 10−2
1.01 × 10−1
±8.59 × 10−3
±2.50 × 10−2
3% / 30
7.21 × 10−2
1.32 × 10−1
±1.20 × 10−2
±9.77 × 10−3
Table 1: Relative error (mean ± standard deviation) of Nystr¨om and matrix completion under
different sampling rates of a 2000 × 2000 OrganCMNIST W2 distance matrix. For the Nystr¨om
method, the numbers of columns associated with the sample rates are also reported.
suitable stopping criterion for matrix completion problems could also be challenging. Therefore, this
fixed sample budget experiment leads us to conclude that computing full columns of W2 distance
measurements may be preferred for approximating Wasserstein distance matrix. Studying how far
this generalizes is the subject of future work.
4.2
Classification stability
We next consider the stability of classification performance under the Nystr¨om approximation as
the number of sampled columns varies. Specifically, we compute 5%, 10%, 20%, . . . , 100% of the
columns of the 2000 × 2000 OrganCMNIST distance matrix using W2-Nystr¨om. The resulting
approximation Dest is embedded via MDS, followed by a 90 −10 train-test split. For the MDS
embedding, we set the dimension to 120, chosen such that the retained singular values account
for 97% of the total sum of singular values. Classification is then performed on the embeddings
using linear discriminant analysis (LDA), 1-nearest neighbor (KNN1), kernel SVM with Gaussian
kernel (SVM RBF), and random forest. We note that the dataset has 11 underlying classes, and we
evaluate the quality of the classification using classification accuracy, defined as the proportion of
correctly identified data points. Figure 1 shows classification accuracy as a function of the number
of columns sampled.
The main conclusion we draw from Figure 1 is that the classification performance is quite stable
after computing about 10–20% of the columns of the distance matrix. This result and Theorem 3.1
suggest that the Nystr¨om Wassmap algorithm (MDS on the output of Algorithm 2) is stable after
sampling O((d + 2) log(d + 2)) columns of D.
5
Conclusion
We showed that the Nystr¨om method yields a good approximation of W2 distance matrices, leading
to stable MDS embeddings. On the OrganCMNIST dataset, we showed that classification accuracy
is stable even when only sampling a small fraction of columns, and for a fixed sampling budget, the
Nystr¨om method outperforms a distance matrix completion approach. Unlike classical Euclidean
distances, where the tight link between Gram and distance matrices grounds the theory of multidi-
mensional scaling, the structural properties of W2 distance matrices are far less understood. This
6
t
Figure 1: Classification accuracy (mean ± 1 standard deviation of 100 trials) vs. number of columns
computed in Algorithm 2 for 2000 OrganCMNIST data points.
gap makes it especially important to develop principled methods that reduce the cost of computing
W2 distances.
In future work, we will explore theoretical conditions under which W2 distance
matrices admit approximate low-rank structure or fast spectral decay, as well as explore hybrid
strategies that combine Nystr¨om-selected columns with partial entry sampling similar following
ideas similar to [22].
Acknowledgements
Abiy Tasissa acknowledges partial support from the National Science Foundation through grant
DMS-2208392. HanQin Cai acknowledges partial support from the National Science Foundation
through grant DMS-2304489.
Muhammad Rana and Keaton Hamm were partially supported by a Research Enhancement
Program grant from the College of Science at the University of Texas at Arlington and by the
Army Research Office under Grant Number W911NF-23-1-0213. The views and conclusions con-
tained in this document are those of the authors and should not be interpreted as representing the
official policies, either expressed or implied, of the Army Research Office or the U.S. Government.
The U.S. Government is authorized to reproduce and distribute reprints for Government purposes
notwithstanding any copyright notation herein.
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9


Question:
Regarding the calculation, properties, and applications of the Wasserstein distance matrix in manifold learning, which of the following statements are incorrect:
A. Calculating the exact Wasserstein distance matrix between M discrete metrics at n points has a time complexity proportional to n³, which constitutes the main computational bottleneck of this method.
B. The square Wasserstein distance matrix D has the same eigenvalues and eigenvectors as the matrix B obtained after double centering (B = -1/2 H D H, where H is the centering matrix). Therefore, the eigen decomposition of D can be directly performed to execute MDS.
C. The experiments conducted on the OrganCMNIST dataset show that, for the distance matrix estimated by W₂-Nyström, even if only 10% of the columns are calculated, the classification accuracy after MDS dimensionality reduction is similar to that when using the full matrix.
D. The matrix inversion method used in the W₂-MC algorithm in the paper is specifically designed for distance matrices. It generates the distance matrix by finding a low-rank Gram matrix G, and satisfies the constraint condition that the sum of the rows or columns of G is zero.

Answer: ABD

Several works have explored manifold learning using probability measures and the Wasserstein
metric, e.g., [5, 7]. This approach has shown improvement in task performance and data modeling,
but it comes with the significant drawback of slow computation time. In particular, computing
one Wasserstein distance exactly for discrete measures supported on m points costs Ω(m3 log(m))
flops. To form neighbor embeddings for N data measures requires computing the square distance
matrix Dij = W2(µi, µj)2, carrying total complexity O(N2m3 log(m)).

2.1 Optimal transport and MDS
The Kantorovich formulation of the quadratic Wasserstein metric on discrete probability measures
µ = Pn
i=1 αiδxi and ν = Pm
i=1 βiδyi is
W2(µ, ν) :=
min
P∈Γ(µ,ν) ⟨C, P⟩
(1)
where Γ(µ, ν) = {P ∈Rm×n
+
: P1m = α, P ⊤1n = β} are couplings with marginals µ and ν,
and Cij = ∥xi −yi∥2 [6].
This problem computes an optimal matching of mass between two
discrete measures. Given measures {µi}N
i=1 ⊂W2(Rn), the Wassmap algorithm [5, 7] computes
Dij = W2(µi, µj)2, a double centering of it B = −1
2HDH where H = I −1
N 11⊤, with truncated
SVD Bd = VdΣdV ⊤
d . The embedding vectors in Rd are then rows of VdΣd.

Recovering Wasserstein Distance Matrices from Few Measurements
Muhammad Rana1, Abiy Tasissa3, HanQin Cai4, Yakov Gavriyelov1, and Keaton Hamm1,2
1Department of Mathematics, University of Texas at Arlington
2Division of Data Science, University of Texas at Arlington
3Department of Mathematics, Tufts University, Medford, MA 02155, USA.
4Department of Statistics and Data Science and Department of Computer Science,
University of Central Florida
Abstract
This paper proposes two algorithms for estimating square Wasserstein distance matrices from
a small number of entries. These matrices are used to compute manifold learning embeddings
like multidimensional scaling (MDS) or Isomap, but contrary to Euclidean distance matrices, are
extremely costly to compute. We analyze matrix completion from upper triangular samples and
Nystr¨om completion in which O(d log(d)) columns of the distance matrices are computed where
d is the desired embedding dimension, prove stability of MDS under Nystr¨om completion, and
show that it can outperform matrix completion for a fixed budget of sample distances. Finally,
we show that classification of the OrganCMNIST dataset from the MedMNIST benchmark is
stable on data embedded from the Nystr¨om estimation of the distance matrix even when only
10% of the columns are computed.
1
Introduction
The bare representation of data in many applications is high-dimensional [1]. In today’s era of
big data, this high dimensionality poses significant challenges and necessitates specialized methods.
Manifold learning refers to a class of techniques that compress data representations into lower-
dimensional forms, making them more suitable for analysis and computation. A widely used linear
technique is principal component analysis (PCA). However, linear approaches can be restrictive
when the data lie on or near a nonlinear manifold. To address this, nonlinear manifold learning
techniques aim to obtain compact representations while preserving local geometric structures [2].
We note that, in the standard way they are used, all of the aforementioned methods typically treat
data as vectors in a Euclidean space.
While many applications yield vector data, there are a number of domains that produce data
that may be best modeled as distributions or measures. For example, gene expression and natural
language processing data can be represented as distributions over gene networks or dictionaries,
respectively, and image data can be considered as discrete distributions over a 2-dimensional pixel
grid. Recent works have used this notion to improve understanding of data geometry and per-
formance on tasks such as classification or clustering [3, 4, 5]. One of the advantages of treating
images, for example, as measures, is that spatial relationships are not obscured as they are under
vectorization. Moreover, the natural notion of a metric structure on the space of measures is that
of the Wasserstein metric from optimal transport (OT) [6].
1
arXiv:2509.19250v1  [stat.ML]  23 Sep 2025
Several works have explored manifold learning using probability measures and the Wasserstein
metric, e.g., [5, 7]. This approach has shown improvement in task performance and data modeling,
but it comes with the significant drawback of slow computation time. In particular, computing
one Wasserstein distance exactly for discrete measures supported on m points costs Ω(m3 log(m))
flops. To form neighbor embeddings for N data measures requires computing the square distance
matrix Dij = W2(µi, µj)2, carrying total complexity O(N2m3 log(m)).
Mitigating this high computational cost can be approached in two ways: 1) speeding up a single
Wasserstein distance computation, and 2) computing fewer entries of D to form an approximation
of it. Many methods from computational OT explore the first approach, e.g., through entropic
regularization, multiscale methods, or Nystr¨om approximations to the cost matrix in the discrete
formulation of the Kantorovich problem (see [6] for a survey). In contrast, relatively few works have
explored the problem of estimating an entire Wasserstein distance matrix directly from samples.
The focus of this paper is to propose algorithms for estimating square Wasserstein distance
matrices for use in neighbor embeddings by computing as few entries as possible. Our approach
is based on exploiting the approximate low-rank structure of Wasserstein squared distances: if the
matrix D is approximately rank r, we propose two algorithms based on matrix completion and
the Nystr¨om method which allow us to estimate D from O(r log(r)) fully observed columns and
maintain stability of the MDS embedding.
2
Technical background
2.1
Optimal transport and MDS
The Kantorovich formulation of the quadratic Wasserstein metric on discrete probability measures
µ = Pn
i=1 αiδxi and ν = Pm
i=1 βiδyi is
W2(µ, ν) :=
min
P∈Γ(µ,ν) ⟨C, P⟩
(1)
where Γ(µ, ν) = {P ∈Rm×n
+
: P1m = α, P ⊤1n = β} are couplings with marginals µ and ν,
and Cij = ∥xi −yi∥2 [6].
This problem computes an optimal matching of mass between two
discrete measures. Given measures {µi}N
i=1 ⊂W2(Rn), the Wassmap algorithm [5, 7] computes
Dij = W2(µi, µj)2, a double centering of it B = −1
2HDH where H = I −1
N 11⊤, with truncated
SVD Bd = VdΣdV ⊤
d . The embedding vectors in Rd are then rows of VdΣd.
2.2
Matrix completion
Given a low-rank matrix with missing entries, the problem of estimating the unknown entries
is known as the matrix completion problem. This problem has been studied extensively due to
its broad applications, most notably in recommendation systems [8]. A naive formulation seeks
to minimize the rank of the matrix subject to the constraints imposed by the observed entries;
however, rank minimization is computationally hard in general [9]. A common convex relaxation is
to minimize the nuclear norm, which serves as a surrogate for rank. In this convex setting, seminal
theoretical results show that, under random sampling, an N × N rank-r matrix can be exactly
recovered from O(νrN log N) observed entries, where ν is an incoherence parameter capturing
the conditioning of the underlying matrix [10].
Given the computational challenges of convex
methods, non-convex algorithms have also been explored, and similar recovery guarantees have
been established in this setting [11]. In this paper, we focus on the completion of distance matrices.
In the case of Euclidean distance matrices, it can be shown that the squared distance matrix
2
has rank at most d + 2, where d is the embedding dimension of the underlying points [12]. This
structural property enables the use of low-rank optimization techniques for Euclidean distance
matrix completion [13, 14].
2.3
Nystr¨om method
Consider a symmetric matrix in block form D =
 A
B
B⊤
C

, where A ∈Rm×m and B ∈Rm×n. We
are interested in the regime m ≪n, where the aim is to estimate the block C using only the smaller
blocks A and B. Suppose that rank(A) = rank(D). Then, one can show that the columns of the
submatrix
B
C

lie in the span of the columns of
 A
B⊤

. Specifically, there exists an X ∈Rm×n
such that B = AX and C = B⊤X. If A is invertible, then X = A−1B, which yields the exact
relation C = B⊤A−1B. In the more general case where A is not invertible, we may instead use
the Moore–Penrose pseudoinverse to obtain X = A†B leading to C = B⊤A†B. This procedure is
known as the Nystr¨om method [15].
While it is not a completion algorithm per se, the Nystr¨om method can be used to complete a
distance matrix. In particular, if only the columns containing A and B⊤are computed, then the
entries of C can be recovered, or in the general case approximated, by B⊤A†B.
Notations For a distance matrix D, owing to symmetry and zero diagonal entries, we only consider
samples from the strictly upper-triangular part.
We denote by Ω⊂{(i, j) : i ∈[N], j > i}
the set of sampled index pairs, with cardinality |Ω| = m. Given a matrix D and an index set
I ⊂[N] := {1, . . . , N}, we write D(:, I) for the submatrix of D consisting of the columns indexed
by I. Analogously, D(I, :) denotes the submatrix formed by the rows indexed by I. PΩ(D)ij = Dij
if (i, j) ∈Ωand is 0 otherwise. If rank(D) = r and D = VrΣrV ⊤
r , then the incoherence ν of D is
p
n/r maxi ∥|Vr(i, :)∥.
3
Algorithms
Here we describe two algorithms for estimating the Wasserstein distance matrix to compute em-
beddings of images. Algorithm 1, W2-MC, randomly computes a given number of entries of the
distance matrix, and then uses a matrix completion algorithm to estimate the full squared distance
matrix. We leave the completion algorithm as a generic plug-in for existing algorithms, and consider
one designed for distance matrices in our experiments. We sample from the strict upper triangle,
include the diagonal, and reflect the entries across the diagonal to obtain the observed distance
matrix. We apply the same procedure after completion to ensure that the resulting matrix retains
the properties of a distance matrix.
Algorithm 2, W2-Nystr¨om, uses the Nystr¨om method to complete the distance matrix.
In
particular, we compute only a set of columns of the distance matrix Cobs = D(:, I), and estimate
Dest = CobsUC⊤
obs where U = Cobs(I, :) = D(I, I). As in W2-MC, we also enforce the distance
matrix structure by enforcing the 0 diagonal.
Note that for large N, the computational cost of both algorithms is dominated by the cost of
computing the W2 distances, and the cost of the matrix completion subroutine MC is often negligible
by comparison. Thus, algorithms that limit sampling complexity of the matrix are more desirable.
For this reason, we do not truncate the rank of the Nystr¨om approximation to D in W2-Nystr¨om,
which is usually done to save computation time.
3
Algorithm 1: W2-MC
1 Input: Data measures {µi}N
i=1, sample size m, matrix completion subroutine MC
2 Randomly sample m entries Ω
3 Compute observed distance matrix Dobs = PΩ(D), where Dij = W2(µi, µj)2
4 diag(Dobs) = 0
▷enforce 0 diagonal
5 Dobs = Dobs + D⊤
obs
▷reflect across diagonal
6 Dest = MC(Dobs)
7 Dest = 1
2(Dest + D⊤
est)
▷enforce symmetry
8 Output: Estimated distance matrix Dest
Algorithm 2: W2-Nystr¨om
1 Input: Data measures{µi}N
i=1, number of columns c
2 Randomly sample c columns I ⊂[N]
3 Compute columns of distance matrix Cobs =
(
W2(µi, µj)2,
i ∈[N], j ∈I
0,
otherwise.
4 Dest = CobsUC⊤
obs, where U = Cobs(I, :)
5 diag(Dest) = 0
▷enforce 0 diagonal
6 Output: Estimated distance matrix Dest
The Nystr¨om method is well-studied, and using some existing bounds, we are able to prove a
stability result for MDS embeddings on Wasserstein distance matrices. We assume that the data
measures in W2(Rn) are isometric to a set of Euclidean vectors {yi}N
i=1 ⊂Rd. Examples of such sets
have been studied and include common image articulations like translations, scalings, and shears
of a fixed base measure [16, 7]. We assume noise (or error) in the computation of the W2 distances,
and consider how the MDS embedding {zi}N
i=1 ⊂Rd coming from Dest of W2-Nystr¨om compares
to {yi}. MDS embeddings are centered and equivalent up to orthogonal transformation, so we
measure error by a Procrustes distance, defined as minQ∈O(d) ∥Z −QY ∥2, where O(d) is the set
of orthogonal d × d matrices. Note that Z and Y are matrices whose columns are the embedded
vectors.
Theorem 3.1. Suppose {µi}N
i=1 ⊂W2(Rn), and Dij = W2(µi, µj)2 has incoherence ν.
Sup-
pose there exist {yi}n
i=1 ⊂Rd such that Dij = ∥yi −yj∥2. Let λ > 0, δ ∈(0, 1), and compute
m ≳λν2(d + 2) max{log(d + 2), log(1/δ)} columns uniformly randomly in W2-Nystr¨om with er-
ror Cobsi,j = W2(µi, µj)2 + Eij such that ∥E∥2 ≤σd+2(D)/3.
If ∥Y †∥2∥E∥1/2
2
≤1/
√
2, then
Dest = CobsUC⊤
obs yields an MDS embedding {zi}N
i=1 ⊂Rd such that
min
Q∈O(d) ∥Z −QY ∥2 ≤(1 +
√
2)∥Y †∥2∥E∥2 ×

5
N
(d + 2)(1 −δ) + 3
s
N
(d + 2)(1 −δ) + 2
!
with probability at least 1 −max{re−λ(δ+(1−δ) log(1−δ)), δ}.
Proof. A simple modification of the proof of [17, Theorem 3.3] implies the upper bound with
(1 +
√
2)∥Y †∥2∥D −Dest∥2. Then [18, Theorem 3] implies that the columns selected would yield
4
an exact Nystr¨om decomposition of D without the presence of noise. Thus, the hypotheses of
[19, Remark 4.7] are satisfied, and yield that ∥D −Dest∥2 ≤5∥Vd+2(I, :)†∥2
2 + 3∥Vd+2(I, :)†∥2 + 2
where I is the set of columns selected by Algorithm 2 and Vd+2 are the first d + 2 left singular
vectors of D. Finally, [20, Lemma 3.4] shows that with the given probability, ∥Vd+2(I, :)†∥2 ≤
p
N/((d + 2)(1 −δ)) for all δ ∈(0, 1).
A simple corollary is that if we compute O((d + 2) log(d + 2)) columns of D without error, then
the MDS embedding is exact up to orthogonal transformation. Exact MDS embeddings of W2 data
were studied in [7].
4
Numerical Experiments
4.1
Fixed sample budget
We first do a fixed sample budget comparison of W2-MC and W2-Nystr¨om for completing the
Wasserstein distance matrix of 2, 000 points from the OrganCMNIST dataset from the MedMNIST
benchmark [21], comprising coronal plane CT scans of 11 different organs downsampled to size 28×
28. For uniform sampling, we subsample the matrix at different rates γ ∈{3%, 5%, 10%, 20%, 25%}.
We then match the number of observed entries between uniform sampling and the Nystr¨om method
by solving the equation c(c−1)
2
+ c(n −c) = γ · n(n−1)
2
for c (the number of columns selected by
Nystr¨om). This equation equates the number of off-diagonal entries used by both methods. For
MC in W2-MC, we use the factorized Gram matrix formulation from [14] based on the relationship
between the Gram matrix and the square distance matrix:
Find
P ∈Rn×q,
s.t.
(PP ⊤)ii + (PP ⊤)jj −2(PP ⊤)ij = Dij,
P ⊤1 = 0,
(2)
where q denotes the estimate for the rank of the underlying points. The centering constraint can
be removed by reparameterization. Defining A : Rn×n →R|Ω| as
A(X)α = Xii + Xjj −2Xij,
α = (i, j) ∈Ω,
the augmented Lagrangian is
L(Q; Λ) = 1
2

A(QQ⊤) −b + Λ

2
2 ,
(3)
with b is the vector of observed distances and Λ ∈R|Ω|.
The algorithm starts with an initial
estimate of Q (randomly initialized in our experiments) and then alternates between two steps:
updating Q via Barzilai–Borwein descent, and updating the Lagrange multipliers. For the dataset
in consideration, we set the rank estimate to be 200.
Table 1 reports the results averaged over 10 random trials. We observe that for fixed sample
budget, the Nystr¨om method consistently yields better accuracy compared to matrix completion.
While both Nystr¨om and typical matrix completion algorithms scale similarly up to logarithmic
factors, the Nystr¨om method is a single-step procedure, whereas the cost of the matrix completion
algorithm depends on the target accuracy, with the number of iterations required to reach a given
error often dominating the runtime. In our experiments, we run W2-MC for a sufficient number
of iterations, as we consistently observe stable convergence across trials. In practice, choosing a
5
Rate / # of cols.
W2-Nystr¨om
W2-MC
25% / 268
1.58 × 10−2
5.20 × 10−2
±1.96 × 10−3
±1.91 × 10−2
20% / 211
1.85 × 10−2
3.42 × 10−2
±2.2 × 10−3
±8.26 × 10−5
10% / 103
2.85 × 10−2
5.65 × 10−2
±2.60 × 10−3
±6.01 × 10−3
5% / 51
4.82 × 10−2
1.01 × 10−1
±8.59 × 10−3
±2.50 × 10−2
3% / 30
7.21 × 10−2
1.32 × 10−1
±1.20 × 10−2
±9.77 × 10−3
Table 1: Relative error (mean ± standard deviation) of Nystr¨om and matrix completion under
different sampling rates of a 2000 × 2000 OrganCMNIST W2 distance matrix. For the Nystr¨om
method, the numbers of columns associated with the sample rates are also reported.
suitable stopping criterion for matrix completion problems could also be challenging. Therefore, this
fixed sample budget experiment leads us to conclude that computing full columns of W2 distance
measurements may be preferred for approximating Wasserstein distance matrix. Studying how far
this generalizes is the subject of future work.
4.2
Classification stability
We next consider the stability of classification performance under the Nystr¨om approximation as
the number of sampled columns varies. Specifically, we compute 5%, 10%, 20%, . . . , 100% of the
columns of the 2000 × 2000 OrganCMNIST distance matrix using W2-Nystr¨om. The resulting
approximation Dest is embedded via MDS, followed by a 90 −10 train-test split. For the MDS
embedding, we set the dimension to 120, chosen such that the retained singular values account
for 97% of the total sum of singular values. Classification is then performed on the embeddings
using linear discriminant analysis (LDA), 1-nearest neighbor (KNN1), kernel SVM with Gaussian
kernel (SVM RBF), and random forest. We note that the dataset has 11 underlying classes, and we
evaluate the quality of the classification using classification accuracy, defined as the proportion of
correctly identified data points. Figure 1 shows classification accuracy as a function of the number
of columns sampled.
The main conclusion we draw from Figure 1 is that the classification performance is quite stable
after computing about 10–20% of the columns of the distance matrix. This result and Theorem 3.1
suggest that the Nystr¨om Wassmap algorithm (MDS on the output of Algorithm 2) is stable after
sampling O((d + 2) log(d + 2)) columns of D.
5
Conclusion
We showed that the Nystr¨om method yields a good approximation of W2 distance matrices, leading
to stable MDS embeddings. On the OrganCMNIST dataset, we showed that classification accuracy
is stable even when only sampling a small fraction of columns, and for a fixed sampling budget, the
Nystr¨om method outperforms a distance matrix completion approach. Unlike classical Euclidean
distances, where the tight link between Gram and distance matrices grounds the theory of multidi-
mensional scaling, the structural properties of W2 distance matrices are far less understood. This
6
t
Figure 1: Classification accuracy (mean ± 1 standard deviation of 100 trials) vs. number of columns
computed in Algorithm 2 for 2000 OrganCMNIST data points.
gap makes it especially important to develop principled methods that reduce the cost of computing
W2 distances.
In future work, we will explore theoretical conditions under which W2 distance
matrices admit approximate low-rank structure or fast spectral decay, as well as explore hybrid
strategies that combine Nystr¨om-selected columns with partial entry sampling similar following
ideas similar to [22].
Acknowledgements
Abiy Tasissa acknowledges partial support from the National Science Foundation through grant
DMS-2208392. HanQin Cai acknowledges partial support from the National Science Foundation
through grant DMS-2304489.
Muhammad Rana and Keaton Hamm were partially supported by a Research Enhancement
Program grant from the College of Science at the University of Texas at Arlington and by the
Army Research Office under Grant Number W911NF-23-1-0213. The views and conclusions con-
tained in this document are those of the authors and should not be interpreted as representing the
official policies, either expressed or implied, of the Army Research Office or the U.S. Government.
The U.S. Government is authorized to reproduce and distribute reprints for Government purposes
notwithstanding any copyright notation herein.
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