[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Abstract. In this paper, we first present an explicit expression for the inverse of a type of matrices. As special applications, the inverse of some matrices arising from implicit time integration techniques, such as the well-known implicit Runge-Kutta schemes and block implicit methods, can also be explicitly determined. Additionally, we introduce three fast algorithms for computing the elements of the inverse of these matrices in O(n²) arithmetic operations, i.e., the first one is based on Traub algorithm for fast inversion of Vandermonde matrices, while the other two utilize the special structure of the matrices. Finally, some symbolic and numerical results are presented to show that our algorithms are both highly efficient and accurate.

Key words. Matrix inversion, fast algorithm, Runge-Kutta matrix, implicit time integration

AMS subject classifications. 15A09, 65F05, 65D30, 68Q25, 65Y20

SHISHUN LI* AND HUILE WEI†

1. Introduction. General speaking, it is unnecessary and inadvisable to actually compute matrix inverse in the vast majority of practical computational problems. However, there exist some matrices and their inverse play a very important role in practical applications. In this paper, we investigate a type of matrices with special structure and design some fast algorithms to compute their inverse. Let c₀, c₁,···, cₙ be distinct real or complex numbers, we consider the explicit expression of the elements of the matrix

(1.1)      W = V⁻¹HV ≡ ⎛ w₁₁ w₁ᵀ ⎞
            ⎜              ⎟
            ⎝ w₂  Wₙ   ⎠

where w₁₁ ∈ ℝ, w₁, w₂ ∈ ℝⁿ, Wₙ ∈ ℝⁿˣⁿ,

H = ⎛ 0                ⎞
  ⎜ 1  ⋱            ⎟
  ⎜   ⋱ ⋱        ⎟
  ⎝    n  0        ⎠  and  V = ⎛ 1  1  1  ⋯ 1      ⎞
               ⎜ c₀ c₁ c₂ ⋯ cₙ  ⎟
               ⎜ c₀² c₁² c₂² ⋯ cₙ²⎟
              ⎜ ⋮  ⋮  ⋮   ⋱ ⋮ ⎟
              ⎝ c₀ⁿ c₁ⁿ c₂ⁿ ⋯ cₙⁿ ⎠

Here V ∈ ℝ^(n+1)×(n+1) is a Vandermonde matrix. It is well-known that the inverse of V can be given explicitly, that is, there exists an explicit formula for the element of V⁻¹. By generalizing the scheme for computing the inverse of a Vandermonde matrix described in [26], we give a explicit expression of the matrix W, i.e., the elements of W can be computed by the basic four arithmetic operations (+, −, *, /) of cᵢ.

The implicit Runge-Kutta (IRK) methods and block implicit methods (BIM) have been widely used for the time discretization of ordinary and partial differential equations [6, 4, 10, 17, 16, 18, 21, 22]. The inverse of IRK matrix and BIM matrix are necessary for stability analysis, and plays an important role in the construction of new time discretization methods with desirable properties [2, 17]. In this paper, we

prove that the inverse of the IRK matrix and BIM matrix is the submatrix of \( W^T \),
i.e., the inverse of IRK matrix and BIM matrix is exactly \( W_n^T \) if \( c_i \) (\( 1 \leq i \leq n \)) are
the zeros of the Legendre polynomials and \( c_i = i \) (\( 1 \leq i \leq n \)), respectively. Consider
a parabolic equation of the form

(1.2) \[ u_t - \nabla \cdot (\alpha(x)\nabla u) + \beta(x) \cdot \nabla u = f(x, t), \quad \text{in}~\Omega \times (0,T] \]

equipped with appropriate initial and boundary conditions, where \( \Omega \subset \mathbb{R}^d \) (\( d = 2 \) or 3) is a bounded, open polygonal (or polyhedral) domain, \( f(x, t) \in L^2(\Omega \times [0, T]) \),
\( 0 < \alpha_0 \leq \alpha(x) \leq \alpha_1 < +\infty \), and all the coefficients are sufficiently smooth. The
discretization of (1.2) by IRK in time and finite element in space produces a system
of equations with the coefficient matrix

(1.3) \[ I \otimes M + \tau A \otimes K, \]

where \( \otimes \) denotes the Kronecker product, \( \tau \) is the time step, \( I \in \mathbb{R}^{r \times r} \), \( A \in \mathbb{R}^{r \times r} \),
\( M \in \mathbb{R}^{N \times N} \) and \( K \in \mathbb{R}^{N \times N} \) denote identity matrix, the coefficient matrix of \( r \)-
stage IRK, the mass matrix and the stiffness matrix, respectively. In practice, (1.3)
is often large and highly ill-conditioned, and parallel preconditioners are necessary to
construct for solving the system with coefficient matrix (1.3). Usually, it is preferable
to solve an equivalent system with the coefficient matrix

(1.4) \[ A^{-1} \otimes M + \tau I \otimes K \]

since it can significantly reduce the computational cost [1, 3, 5, 10, 19]. Moreover, it
requires less memory since \( A^{-1} \otimes M \) is in general sparser than \( A \otimes K \), and the stiff
part of the matrix is block diagonal so that the analysis and solution more tractable
[24]. Various preconditioners based on \( A^{-1} \) are constructed, such as, ILU(0)-based
stage-coupled and stage-parallel preconditioners [20], linear and nonlinear parallel
preconditioner [25, 24], and the lower triangular preconditioners [1]. Hence, the
computation of \( A^{-1} \) is very important for solving the system of equations arising from
the discretization of time-dependent PDEs, and we will present an explicit formula to
compute \( A^{-1} \).

The explicit formula for the elements of \( W \) is well suited for hand or machine
calculation. However, the computation of the \( (n + 1)^2 \) elements requires \( O(n^3) \) arithmetic
operations, and it is clearly too expensive in compute time for large \( n \). In
this paper, we present three fast algorithms to compute all the elements of \( W \) in
only \( O(n^2) \) floating point operations (flops). The explicit formula for the Vandermonde
matrix was first presented by Traub in [26], and an algorithm with the complexity
\( 6n^2 \) flops was also proposed by using its special structure. This algorithm has been
generalized to invert the Vandermonde-like matrices [7, 8, 9, 12, 23, 27]. Following
the idea in [26], we first present a fast algorithm to compute all the elements of \( W \)
in only \( O(n^2) \) flops. Similar to Traub algorithm for Vandermonde matrix, this algorithm is numerically unstable and inaccurate due to the rapid propagation of roundoff
errors. Moreover, the solution components may soon exceed the representable machine number and produce overflow as \( n \) increases. Actually, we observe that \( W \) has
better structure than Vandermonde matrix although it is singular. By using a trick
to avoid overflow and underflow in multiplication and division, we derive another two
algorithms with \( O(n^2) \) complexity. It is shown that our algorithms are both highly
efficient and accurate for computing all the elements of \( W \).

The remaining part of the paper is organized as follows. In Section 2, we first give
an explicit formula for the matrix \( W \), then prove that the inverse of IRK matrix and

Here are the corrected texts for each crop:

---

**Crop 1:**
where \(\sigma_{n+1-k}\) denotes the \(k\)th elementary symmetric function of \(c_0, c_1, \cdots, c_n\), i.e., \(\sigma_{n+1-k}(c_0,c_1, \cdots, c_n)\) is the sum of all distinct products of \(k\) of the arguments \(c_0,c_1, \cdots, c_n\) [12, 26]. The inverse of the Vandermonde matrix can be given by the matrix whose elements are

---

**Crop 2:**
BIM matrix is exactly the submatrix of \(W\). In Section 3, three fast algorithms with the complexity \(O(n^2)\) flops are presented to compute the elements of \(W\). In Section 4, some symbolic and numerical results are presented to demonstrate the performance of the proposed algorithms. Finally, some concluding remarks are given in Section 5.

---

**Crop 3:**
\[
(2.5)\qquad \psi(x)=\sum_{k=0}^{n} b_k x^k = \sum_{k=0}^{n} (-1)^{n-k} \sigma_{n-k}(c_0, c_1, \cdots, c_{i-1}, c_{i+1}, \cdots, c_n)x^k
\]

---

**Crop 4:**
\[
(2.2)\qquad \varphi(x) = \sum_{k=0}^{n+1} a_k x^k = \sum_{k=0}^{n+1} (-1)^{n+1-k} \sigma_{n+1-k}(c_0, c_1, \cdots, c_n)x^k
\]

---

**Crop 5:**
\[
(2.3)\qquad (V^{-1})_{ij} = \frac{(-1)^{n-j} \sigma_{n-j}(c_0, c_1, \cdots, c_{i-1}, c_{i+1}, \cdots, c_n)}{\displaystyle\prod_{\substack{l=0 \\ l\ne i}}^{n} (c_i - c_l)}
\]

---

**Crop 6:**
\[
(2.1)\qquad \varphi(x) = \prod_{k=0}^{n} (x - c_k)
\]

---

**Crop 7:**
\[
\sum_{k=0}^{n} (-1)^{n-k} k c_j^{k-1} \sigma_{n-k}(c_0, c_1, \cdots, c_{i-1}, c_{i+1}, \cdots, c_n) = \begin{cases}
\prod_{\substack{k=0 \\ k\ne j}}^{n} (c_j - c_k), & i\ne j, \\[2ex]
\sum_{\substack{m=0 \\ m\ne i}}^{n}\prod_{\substack{k=0 \\ k\ne i,m}}^{n} (c_j - c_k), & i = j.
\end{cases}
\]

---

**Crop 8:**
\[
(2.4)\qquad \psi'(x) = \sum_{\substack{m=0 \\ m \ne i}}^{n} \prod_{\substack{k=0 \\ k \ne i, m}}^{n} (x - c_k)
\]

---

**Crop 9:**
Similar to (2.2), we obtain

---

**Crop 10:**
**LEMMA 2.1.** Let \(\sigma_{n-k}\) be the \(k\)th elementary symmetric function of \(c_0, c_1, \cdots, c_{i-1}, c_{i+1}, \cdots, c_n\), we have

---

**Crop 11:**
From the well-known Vieta's formula, we have

---

**Crop 12:**
Now we introduce the following lemma before giving the explicit formula for \(W\)

---

**Crop 13:**
Proof. Define \(\psi(x) = \prod_{\substack{k=0 \\ k\ne i}}^{n} (x-c_k)\), the derivative of \(\psi(x)\)

---

Here is the corrected text from the OCR:

---

From (2.5), we see that the derivative of \( \psi(x) \) is

\[
(2.6) \quad \psi'(x) = \sum_{k=0}^n (-1)^{n-k} k \sigma_{n-k}(c_0, c_1, \cdots, c_{i-1}, c_{i+1}, \cdots, c_n) x^{k-1}.
\]

Combining (2.4) and (2.6) implies

\[
(2.7) \quad \sum_{k=0}^n (-1)^{n-k} k \sigma_{n-k}(c_0, c_1, \cdots, c_{i-1}, c_{i+1}, \cdots, c_n) x^{k-1} = \sum_{\substack{m=0 \\ m \neq i}}^n \prod_{\substack{k=0 \\ k \neq i,m}}^n (x - c_k).
\]

Setting \( x = c_j \) in (2.7), we have

\[
\sum_{k=0}^n (-1)^{n-k} k c_j^{k-1} \sigma_{n-k}(c_0, c_1, \cdots, c_{i-1}, c_{i+1}, \cdots, c_n)
\]

\[
= \sum_{\substack{m=0 \\ m \neq i}}^n \prod_{\substack{k=0 \\ k \neq i,m}}^n (c_j - c_k) =
\begin{cases}
\prod_{\substack{k=0 \\ k \neq i,j}}^n (c_j - c_k), & i \neq j, \\
\sum_{\substack{m=0 \\ m \neq i}}^n \prod_{\substack{k=0 \\ k \neq i,m}}^n (c_i - c_k), & i = j.
\end{cases}
\]

Which completes the proof.

Define the generalized binomial coefficient \( \binom{c_n}{c_i} \) (\(0 < i < n\)) such that

\[
\binom{c_n}{c_i} = \frac{(c_n - c_0)(c_n - c_1) \cdots (c_n - c_{n-1})}
{[(c_i - c_0)(c_i - c_1)\cdots (c_i - c_{i-1})][(c_n - c_i)(c_{n-1} - c_i)\cdots (c_{i+1} - c_i)]}
\]
\[
= \frac{\prod_{k=0}^{n-1} (c_n - c_k)}
{\prod_{k=0}^{i-1} (c_i - c_k) \prod_{k=0}^{n-i-1} (c_{n-k} - c_i)}
\]

\[
(2.8) \quad = \frac{\prod_{k=0}^{n-1} (c_n - c_k)}
{(-1)^{n-i} \prod_{\substack{k=0 \\ k \neq i}}^n (c_i - c_k)}.
\]

Further, define

\[
(2.9) \quad \binom{c_n}{c_n} = 1 \quad \text{and} \quad \binom{c_n}{c_0} = \frac{(-1)^n \prod_{k=0}^{n-1} (c_n - c_k)}
{\prod_{k=0}^{n-1} (c_{n-k} - c_0)}.
\]

Setting \( c_n = n \) and \( c_i = i \), it is clear that \( \binom{c_n}{c_i} \) is the binomial coefficient

\[
\binom{n}{i} = \frac{n!}{i!(n-i)!},
\]

where \( n! \) denotes the factorial of \( n \).

Next we present the main theorem of this paper.

Theorem 2.2. Let \( c_i \ (0 \leq i \leq n) \) be distinct real or complex numbers, the elements of \( W \) in (1.1) are given by

\[
W_{ij} =
\begin{cases}
\frac{1}{c_j - c_i} \displaystyle\prod_{\substack{k=0 \\ k \neq j}}^n (c_j - c_k) \bigg/ \prod_{\substack{k=0 \\ k \neq i}}^n (c_i - c_k), & i \neq j, \\
\displaystyle\sum_{\substack{k=0 \\ k \neq i}}^n \frac{1}{c_i - c_k}, & i = j.
\end{cases}
=
\begin{cases}
\frac{(-1)^{j-i} \binom{c_n}{c_i}}{c_j - c_i} \binom{c_n}{c_j}^{-1}, & i \neq j, \\
\displaystyle\sum_{\substack{k=0 \\ k \neq i}}^n \frac{1}{c_i - c_k}, & i = j.
\end{cases}
\]

Proof. From the definition of \( H \) and \( V \) in (1.1), it is clear that the elements of \( HV \) are

\[
(HV)_{ij} = i c_j^{i-1}.
\]

It follows from (1.1), (2.3), (2.11) and Lemma 2.1 that

\[
W_{ij} = \sum_{k=0}^n (V^{-1})_{ik} (HV)_{kj}
= \sum_{k=0}^n \frac{(-1)^{n-k} k c_j^{k-1} \sigma_{n-k}(c_0, c_1, \ldots, c_{i-1}, c_{i+1}, \ldots, c_n)}{\prod_{\substack{l=0 \\ l\neq i}}^n (c_i - c_l)}.
\]

Combining (2.8), (2.9) and (2.12) implies

\[
W_{ij} =
\begin{cases}
\frac{1}{c_j - c_i} \displaystyle\prod_{\substack{k=0 \\ k \neq j}}^n (c_j - c_k) \bigg/ \prod_{\substack{k=0 \\ k \neq i}}^n (c_i - c_k), & i \neq j, \\
\displaystyle\sum_{\substack{k=0 \\ k \neq i}}^n \frac{1}{c_i - c_k}, & i = j,
\end{cases}
=
\begin{cases}
\frac{(-1)^{j-i} \binom{c_n}{c_i}}{c_j - c_i} \binom{c_n}{c_j}^{-1}, & i \neq j, \\
\displaystyle\sum_{\substack{k=0 \\ k \neq i}}^n \frac{1}{c_i - c_k}, & i = j.
\end{cases}
\]

Which completes the proof.

Corollary 2.3. Let \( c_i = i \ (0 \leq i \leq n) \), the elements of the matrix \( W \) in (1.1) are

\[
W_{ij} =
\begin{cases}
\displaystyle \sum_{p=1}^i \frac{1}{p} - \sum_{p=1}^{n-i} \frac{1}{p}, & i = j, \\
\frac{(-1)^{j-i} \binom{n}{i}}{j-i} \binom{n}{j}^{-1}, & i \neq j, \quad i,j = 0,1,\ldots,n.
\end{cases}
\]

REMARK 1. For the case \( i = j \), setting \( p = i - k \), we have

\[
\sum_{k=0, k \neq i}^n \frac{1}{i - k} = \sum_{k=0}^{i-1} \frac{1}{i - k} + \sum_{k=i+1}^n \frac{1}{i - k} = \sum_{p=1}^i \frac{1}{p} - \sum_{p=1}^{n-i} \frac{1}{p}
\]

Corollary 2.3 was presented in [2]. However, the techniques used in the proof can not be extended for the proof of Theorem 2.2.

## 2.2. The explicit formula for the inverse of IRK matrix and BIM matrix

The IRK methods based on Gaussian and Radau quadrature formulae possess favorable stability properties, offer a high order of accuracy, and have no start-up issues. In this section, we present the explicit formula for the coefficient matrix of Gauss and Radau IIA methods. The elements of these matrices are given by

\[
(2.14) \qquad a_{ij} = \int_0^{c_i} l_j(\theta) d\theta,
\]

where \( l_i(\theta) \) is the Lagrange polynomial \( \prod_{k\neq i}(\theta - c_k)/(c_i - c_k) \) and \( c_i \) \((1 \leq i \leq n)\) are the zeros of

\[
\frac{d^n}{dx^n}\left(x^n(x-1)^n\right), \quad \text{and} \quad \frac{d^n}{dx^{n-1}}\left(x^{n-1}(x-1)^n\right),
\]

respectively. Due to \( \theta^{k-1} = \sum_{j=1}^n c_j^{k-1} l_j(\theta) \) for \( k = 1,2,\ldots,n \), (2.14) is equivalent to the following systems [11]

\[
(2.15) \qquad \sum_{j=1}^n a_{ij} c_j^{k-1} = \frac{c_i^k}{k}, \quad k = 1,2,\ldots,n, \quad i = 1,2,\ldots,n.
\]

From (2.15), we have

\[
(2.16) \qquad A = C_n V_n D_n^{-1} V_n^{-1},
\]

where

\[
C_n = \begin{pmatrix}
c_1 \\
 & c_2 \\
 & & \ddots \\
 & & & c_n
\end{pmatrix}, \quad
D_n = \begin{pmatrix}
1 \\
 & 2 \\
 & & \ddots \\
 & & & n
\end{pmatrix}, \quad
V_n = \begin{pmatrix}
1 & c_1 & \cdots & c_1^{n-1} \\
1 & c_2 & \cdots & c_2^{n-1} \\
\vdots & \vdots & \ddots & \vdots \\
1 & c_n & \cdots & c_n^{n-1}
\end{pmatrix}.
\]

Define \( e_1 = (1,0,\cdots,0)^T \in \mathbb{R}^n \), \( e = (1,1,\cdots,1)^T \in \mathbb{R}^n \), and \( c_0 = (c_0, c_0^2, \cdots, c_0^n)^T \), we have the following theorem.

THEOREM 2.4. Let \( c_0 = 0 \), \( c_i \neq 0\) \((1 \leq i \leq n)\) be distinct real numbers, we have

\[
\begin{aligned}
\left( e_1^T V_n^{-1} C_n^{-1} \right)_i &= \frac{(-1)^i}{-c_i} \begin{pmatrix} c_n \\ c_i \\ c_0 \end{pmatrix}, \quad
(-A^{-1}e)_i = \frac{(-1)^i}{c_i} \begin{pmatrix} c_n \\ c_0 \\ c_i \end{pmatrix} \text{ and} \\[2ex]
(A^{-1})_{ij} &=
\begin{cases}
\frac{(-1)^{i-j}}{c_i - c_j} \begin{pmatrix} c_n \\ c_j \\ c_i \end{pmatrix}, & i \neq j, \\[2ex]
\sum_{k=0, k\neq i}^n \frac{1}{c_i - c_k}, & i = j.
\end{cases}
\end{aligned}
\]

Proof. It is clear that \( c_0 = (0,0,\cdots,0)^T \in \mathbb{R}^n \) since \( c_0 = 0 \). From the definition of the Vandermonde matrix \( V \), we have

\[
(2.17) \quad V^{-T} = \begin{pmatrix} 1 & c_0^T \\ e & C_n V_n \end{pmatrix}^{-1} = \begin{pmatrix} 1 & c_0^T \\ -V_n^{-1}C_n^{-1}e & V_n^{-1}C_n^{-1} \end{pmatrix},
\]

Set

\[
F = \begin{pmatrix}
0 & 1 & & \\
& \ddots & \ddots & \\
& & \ddots & 1 \\
& & & 0
\end{pmatrix}_{(n+1)\times(n+1)}
\]

A straightforward computation implies

\[
(2.18) \quad V^T F = \begin{pmatrix} 0 & e_1^T \\ c_0 & V_n \end{pmatrix} \quad \text{and} \quad H^T = F \begin{pmatrix} 0 & c_0^T \\ c_0 & C_n \end{pmatrix}.
\]

It follows from (2.17) and (2.18) that

\[
\begin{aligned}
W^T &= V^T H^T V^{-T} = V^T F \begin{pmatrix} 0 & c_0^T \\ c_0 & D_n \end{pmatrix} \begin{pmatrix} 1 & c_0^T \\ -V_n^{-1}C_n^{-1}e & V_n^{-1}C_n^{-1} \end{pmatrix} \\
&= \begin{pmatrix} 0 & e_1^T \\ c_0 & V_n \end{pmatrix} \begin{pmatrix} 0 & c_0^T \\ c_0 & D_n \end{pmatrix} \begin{pmatrix} 1 & c_0^T \\ -V_n^{-1}C_n^{-1}e & V_n^{-1}C_n^{-1} \end{pmatrix} \\
&= \begin{pmatrix} 0 & e_1^T D_n \\ c_0 & V_n D_n \end{pmatrix} \begin{pmatrix} 1 & c_0^T \\ -V_n^{-1}C_n^{-1}e & V_n^{-1}C_n^{-1} \end{pmatrix} \\
(2.19) \quad &= \begin{pmatrix}
-e_1^T D_n V_n^{-1} C_n^{-1} e & e_1^T D_n V_n^{-1} C_n^{-1} \\
-V_n D_n V_n^{-1} C_n^{-1} e & V_n D_n V_n^{-1} C_n^{-1}
\end{pmatrix}.
\end{aligned}
\]

Combining (2.16), (2.19), and Theorem 2.2 completes the proof.

REMARK 2. From Theorem 2.4, we see that the inverse of IRK matrix can be given explicitly by choosing \( c_i \) as the zeros of the shifted Legendre polynomial of degree \( n \).

Recently, a family of A-stable BIM for solving parabolic equations has been introduced in [17]. BIM have the same desirable properties as IRK methods and are time-parallel in the sense that they compute the solutions at several time steps simultaneously. It shows that the inverse of the BIM matrix can be also given explicitly, i.e.,

\[
N = D_n (V_n D_n V_n^{-1} D_n^{-1}) D_n^{-1} + D_n^{-1},
\]

where \( D_n \) and \( V_n \) are defined in (2.16) and \( c_i = i \). The explicit expression of \( N \) plays an important role in constructing new BIM with positive definite BIM matrices. Then the traditional finite element theory for parabolic equations discretized by the backward Euler or Crank-Nicolson schemes can also be extended to BIM.

3. The fast algorithms for the inverse of the matrix. The explicit formula for \( W \) is suitable for symbolic operations using a symbolic manipulation package. However, it may be very expensive for large \( n \). In this section, we present three fast numerical algorithms with \( O(n^2) \) complexity to compute \( W \).

Following the idea of Traub' algorithm for the inverse of Vandermonde matrix \( V \) [26], we introduce the first algorithm to compute the elements of \( W \). Firstly, we calculate the coefficients \( a_k \) of the monic polynomial \( \varphi(x) \) in (2.2) recursively by

\[
\left\{
\begin{aligned}
&\beta_{m+1, m+1} = 1, \quad \beta_{m+1, i} = \beta_{m, i-1} - c_m \beta_{m, i}, \\
& a_k = \beta_{n+1, k},
\end{aligned}
\right.
\]
for \( 1 \leq m \leq n, \; 0 \leq i \leq m, \; 0 \leq k \leq n+1, \)

where \( \beta_{1,0} = -c_0, \; \beta_{1,1} = 1 \) and \( \beta_{m,-1} = 0 \). Then, we calculate the coefficients \( b_k \) of \( \psi(x) \) in (2.5) by
\[
b_k = a_{k+1} + c_i b_{k+1}, \quad n \geq k \geq 0,
\]
where \( b_{n+1} = 0 \). Finally, we calculate the scalars \( \psi(c_i) \) by Horner's rule and obtain
\[
(3.1) \quad
W_{ij} =
\begin{cases}
\frac{1}{c_j - c_i} \frac{\psi(c_j)}{\psi(c_i)}, & i \neq j, \\[2ex]
\displaystyle\sum_{k=0, k \neq i}^{n} \frac{1}{c_i - c_k}, & i = j,
\end{cases}
\]

The above scheme is implemented thoroughly in Algorithm 3.1. We see that the computational complexity of Algorithm 3.1 is \( 11n^2 + 12n \) flops. However, it may soon exceed the representable machine number and produce overflow (or underflow) when computing the coefficients \( a_k \) and \( b_k \) for large \( n \), this issue also occurs in computing \( \psi(c_i) \) since it involving a product of \( n \) factors.

In order to avoid the above issue and improve accuracy, we present another fast algorithm by using the special structure of \( W \). Firstly, we introduce the matrix \( G \) which is given by
\[
(3.2) \quad
G = \begin{pmatrix}
1 & c_0 - c_1 & c_0 - c_2 & \cdots & c_0 - c_{n-1} & c_0 - c_n \\
c_1 - c_0 & 1 & c_1 - c_2 & \cdots & c_1 - c_{n-1} & c_1 - c_n \\
c_2 - c_0 & c_2 - c_1 & 1 & \cdots & c_2 - c_{n-1} & c_2 - c_n \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
c_{n-1} - c_0 & c_{n-1} - c_1 & c_{n-1} - c_2 & \cdots & 1 & c_{n-1} - c_n \\
c_n - c_0 & c_n - c_1 & c_n - c_2 & \cdots & c_n - c_{n-1} & 1
\end{pmatrix}
\]

It is clear the computational complexity for computing \( G \) is \( (n+1)^2 \). From the definition of \( \psi(x) \), we have

\[
(3.3) \quad \psi(c_i) = \prod_{\substack{k=0 \\ k \neq i}}^n (c_i - c_k) = \prod_{k=0}^n G_{ik}.
\]

Define

\[
(3.4) \quad
\widetilde{W} = \begin{pmatrix}
1 & \frac{\psi(c_1)}{\psi(c_0)} & \frac{\psi(c_2)}{\psi(c_0)} & \cdots & \frac{\psi(c_{n-1})}{\psi(c_0)} & \frac{\psi(c_n)}{\psi(c_0)} \\
\frac{\psi(c_0)}{\psi(c_1)} & 1 & \frac{\psi(c_2)}{\psi(c_1)} & \cdots & \frac{\psi(c_{n-1})}{\psi(c_1)} & \frac{\psi(c_n)}{\psi(c_1)} \\
\frac{\psi(c_0)}{\psi(c_2)} & \frac{\psi(c_1)}{\psi(c_2)} & 1 & \cdots & \frac{\psi(c_{n-1})}{\psi(c_2)} & \frac{\psi(c_n)}{\psi(c_2)} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
\frac{\psi(c_0)}{\psi(c_{n-1})} & \frac{\psi(c_1)}{\psi(c_{n-1})} & \frac{\psi(c_2)}{\psi(c_{n-1})} & \cdots & 1 & \frac{\psi(c_n)}{\psi(c_{n-1})} \\
\frac{\psi(c_0)}{\psi(c_n)} & \frac{\psi(c_1)}{\psi(c_n)} & \frac{\psi(c_2)}{\psi(c_n)} & \cdots & \frac{\psi(c_{n-1})}{\psi(c_n)} & 1
\end{pmatrix}
\]

Algorithm 3.1 Given \( c(0) < c(1) < \cdots < c(n) \) this algorithm computes \( W = V^{-1}HV \)

From (3.1) and (3.4), we have

\[
(3.5) \qquad W_{ij} =
\begin{cases}
\frac{1}{c_j - c_i} \widetilde{W}_{ij}, & i \neq j, \\
\sum_{k=0,\,k \neq i}^n \frac{1}{G_{ik}}, & i = j.
\end{cases}
\]

If \( \widetilde{W}_{ij} \) have been derived, the complexity for computing \( W \) is \( 3n^2 + 7n \) by using formula (3.5). Next we mainly study how to compute the elements of \( \widetilde{W} \).

We first compute the first row of \( \widetilde{W} \) by

\[
(3.6) \qquad \widetilde{W}_{0j} = \frac{\psi(c_j)}{\psi(c_0)} = \frac{\prod_{k=0}^n G_{jk}}{\prod_{k=0}^n G_{0k}} = \prod_{k=0}^n \frac{G_{jk}}{G_{0k}}, \quad 0 \leq j \leq n.
\]

Instead of computing the numerator \( \psi(c_j) \) and the denominator \( \psi(c_0) \) for each \( j \), we compute \( \widetilde{W}_{0j} \) by a product of \( n+1 \) factors \( G_{jk}/G_{0k} \). This trick may avoid overflow.

(3.7)                 g = \left( \frac{1}{\widetilde{W}_{01}}, \frac{1}{\widetilde{W}_{02}}, \cdots, \frac{1}{\widetilde{W}_{0n}} \right)^T,

or underflow and improve accuracy. It is clear that the complexity for computing \(\widetilde{W}_{0j}\) (\(0 \leq j \leq n\)) is \(2n^2 + 4n\). Set

the elements \(\widetilde{W}\) can be compute with \(n^2 + n\) operations by

(3.8)                    \(\widetilde{W}_{ij} = \frac{\psi(c_j)}{\psi(c_i)} = \frac{\psi(c_j)}{\psi(c_{0})} \frac{\psi(c_{0})}{\psi(c_{i})} = \widetilde{W}_{0j} g_i, \qquad 0 \leq j \leq n,\, 1 \leq i \leq n.\)

From the above analysis, we observe that it requires \(7n^2 + 15n\) operations to compute \(W\). The details of this scheme is presented in Algorithm 3.2.

From (3.6) and (3.7), we see that the overflow and underflow may also occur if

Algorithm 3.2 Given \(c(0) < c(1) < \cdots < c(n)\) this algorithm computes \(W = V^{-1}HV\)

Algorithm 3.3 Given \( c(0) < c(1) < \cdots < c(n) \) this algorithm computes \( W = V^{-1}HV \)

\( G_{ik} \) is far greater or less than \( G_{0k} \). Now we introduce vector \( \tilde{\mathbf{g}} \) which is given by

\[
\tilde{\mathbf{g}} = \left( \frac{\psi(c_1)}{\psi(c_0)}, \frac{\psi(c_2)}{\psi(c_1)}, \cdots, \frac{\psi(c_n)}{\psi(c_{n-1})} \right)^T,
\]

where $\tilde{g}_i$ ($0 \leq i \leq n-1$) are computed by

\[
\tilde{g}_i = \frac{\psi(c_{i+1})}{\psi(c_i)}
\]
\[
= \frac{(c_{i+1} - c_0)(c_{i+1} - c_1)\cdots(c_{i+1} - c_{i-1})(c_{i+1} - c_{i})(c_{i+1} - c_{i+2})\cdots(c_{i+1} - c_{n})}{(c_i - c_0)(c_i - c_1)\cdots(c_i - c_{i-1})(c_i - c_{i+1})(c_i - c_{i+2})\cdots(c_i - c_n)}
\]
\[
= -\frac{(c_{i+1} - c_0)(c_{i+1} - c_1)\cdots(c_{i+1} - c_{i-1})(c_{i+1} - c_{i+2})\cdots(c_{i+1} - c_n)}{(c_i - c_0)(c_i - c_1)\cdots(c_i - c_{i-1})(c_i - c_{i+2})\cdots(c_i - c_n)}
\]
\[
\tag{3.10}
= -\prod_{\substack{k=0 \\ k\neq i,i+1}}^n \frac{(c_{i+1} - c_k)}{(c_i - c_k)} = -\prod_{\substack{k=0 \\ k\neq i,i+1}}^n \frac{G_{(i+1)k}}{G_{ik}}
\]

If $c_0 < c_1 < \cdots < c_n$, it is clear that (3.10) may be more accurate than (3.6). From (3.4), we observe that the upper triangular part of $\tilde{W}$ can be computed recursively by

\[
\tag{3.11}
\tilde{W}_{ij} = \frac{\psi(c_j)}{\psi(c_i)} = \frac{\psi(c_{j-1})}{\psi(c_i)} \frac{\psi(c_j)}{\psi(c_{j-1})} = \tilde{W}_{i(j-1)} \tilde{g}_{j-1}, \quad i+1 \leq j \leq n,\, 0 \leq i \leq n,
\]

where $\tilde{W}_{ii} = 1$. Then the strictly lower triangular part of $\tilde{W}$ is given by

\[
\tag{3.12}
\tilde{W}_{ij} = \frac{1}{\tilde{W}_{ji}}, \quad 0 \leq j \leq i-1,\, 1 \leq i \leq n,
\]

A straightforward calculation shows that complexity for computing $\tilde{g}$ and $\tilde{W}$ are $2n^2$ and $n^2 + n$, respectively. Finally, we obtain Algorithm 3.3 with the complexity $7n^2 + 9n$ flops.

Both Algorithm 3.2 and Algorithm 3.3 include three steps, i.e., computing the elements of the matrices $G$, $\tilde{W}$ and $W$ sequentially. The difference between these algorithms lies in the computation of $\tilde{W}$ in the second step. In each step, both Algorithm 3.2 and Algorithm 3.3 can be implemented in parallel since the computations of each row of $G$, $\tilde{W}$ and $W$ are independent.

## 4. Numerical experiments.
In this section, we present some symbolic and numerical results to demonstrate the performance of the proposed formula and algorithms. In the experiments, we calculate $W$ by choosing $c_i$ ($i = 1,2,\cdots, n$) from the following four examples.

**Example 1.** (Gauss nodes in (0,1)): $c_0 = 0$ and $c_i$ ($i=1,2,\cdots,n$) are the zeros of $\frac{d^n}{dx^n}(x^n(x-1)^n)$.

**Example 2.** (Radau IIA nodes in (0,1]): $c_0 = 0$ and $c_i$ ($i=1,2,\cdots,n$) are the zeros of $\frac{d^n}{dx^{n-1}}(x^{n-1}(x-1)^n)$.

**Example 3.** (Chebyshev nodes in (0,1)): $c_i = \frac{1}{2} + \frac{1}{2}\cos\left(\frac{2i+1}{2n+1}\pi \right)$ ($i=0,1,2,\cdots,n$).

**Example 4.** (Equidistant nodes in [0, n]): $c_i = i$ ($i=0,1,2,\cdots,n$).

### 4.1. Results of the symbolic operations.
Since $W$ is given explicitly by (3.1), we can compute it exactly by Mathematica. From Theorem 2.4, we see that the submatrix $W_n^T$ is exactly the inverse of the IRK matrix $A$ when choosing $c_i$ as in Examples 1 and 2. Note that the coefficient matrix of Gauss and Radau IIA methods can

be calculated using the Mathematica command ImplicitRungeKuttaGaussCoefficients and ImplicitRungeKuttaRadauIIACoefficients, which allow software-implemented precision of arbitrary numbers of digits. The number of digits was set to 1000 in the experiment. To test the performance of the explicit formula (3.1), we define

(4.1)    Err = ||WₙᵗA − I||_∞

and list the results in Tables 1 and 2, respectively. It is clear that our formula (3.1) is correct, and the compute time of A and Wₙ grows fast with the increase of n. However, the calculation of Wₙ is much faster than that of A. The calculation of IRK matrix using Mathematica command takes significantly more time (seconds) compared to computing its inverse using the formula (3.1).

4.2. Results of the fast algorithms. Symbolic operations are time-consuming, although they allow for arbitrary precision. Now we will test the performance of Algorithms 3.2 and 3.3 on Examples 1–4. In the experiments, the matrices computed by using long double (80 bits) and quadruple precision (128 bits) are denoted by Wᵈ and Wᵠ, respectively. The compute time is denoted by “T_d” and “T_q”. We consider Wᵠ as exact and define the infinite-norm and componentwise relative errors as follows:

(4.2)  Err₁ = ||Wᵈ − Wᵠ||_∞ / ||Wᵠ||_∞  and  Err₂ = max_{0 ≤ i, j ≤ n} {|Wᵈ_{ij} − Wᵠ_{ij}| / |Wᵠ_{ij}|}.

The numerical results for Example 1 are presented in Table 3, which shows that both Algorithms 3.2 and 3.3 are very fast although the compute time increases as n

grows. We observe that the time for computing \( W^d \) is no more than one second in the case \( n = 5000 \). It is clear that the errors in the computation of \( W \) grow with \( n \), and the accuracy of off-diagonal elements is slightly better than diagonal elements. From (3.1), we see that the computation of the diagonal elements involving recursive summation which may suffer cancellation. Additionally, we observe that Algorithm 3.2 fails for the case \( n = 10000 \). The reason is that the overflow occurs in the computation of formula (3.6) since \( G_{jk} \) is far greater than \( G_{0k} \) when \( j \) is large. However, Algorithm 3.3 works well since the computation of formula (3.10) can avoid overflow.

Similarly to Example 1, Tables 4 and 5 show that the compute time grows mildly with the increase of \( n \) and both Algorithms 3.2 and 3.3 yield very high accuracy in

the infinite-norm. However, the accuracy of the diagonal elements is much worse than the off-diagonal elements if \( n \) is large. In the experiments, we observe that there is heavy cancellation in the sum, that is,
\[
\sum_{k=0, k \ne i}^n \left| \frac{1}{c_i - c_k} \right| \gg \sum_{k=0, k \ne i}^n \frac{1}{c_i - c_k}.
\]
Moreover, cancellation also occurs in the subtraction of two nearly equal numbers. Hence, there exists a severe loss of significant digits and produces a large relative error in this particular situation. In Figure 1, we present 3D heatmap of the matrix elements for Examples 1-4 by using \( \ln|W_{ij}| \) instead of \( W_{ij} \). From the subfigures (a)-(c), we observe that the matrices exhibit a significant difference in element size, with the diagonal elements being small and approaching 1 in subfigure (b), moreover, the term \( \frac{1}{c_i - c_k} \) in recursive summation may be very large and is either positive or negative. So the algorithms suffer severe cancellation. Fortunately, we can achieve desired high accuracy for this special example by using sufficiently high precision, such as quadruple precision.

Finally, we test the fourth example and present the numerical results in Table 6, which illustrates that both Algorithms 3.2 and 3.3 are very accurate in both types of errors. From Remark 1, we see that the heavy cancellation does not occur in the sum since there exist no nearly equal numbers are subtracted from each other. Subfigure (d) shows that the values of the matrix entries varies widely, making underflow and overflow unavoidable consequences when working in finite precision arithmetic for

very large \( n \).

5. Conclusion. We proposed an explicit formula to compute the inverse of a type of matrices which include the well-known fully IRK and BIM matrices as special cases. So the inverse of these matrices arising from implicit time integration can be computed by the basic four arithmetic operations of the quadrature nodes, and we can obtain their inverse exactly by using a symbolic manipulation package. It’s shown that the inverse of these matrices is much easier to compute than themselves.

by Mathematica. We also presented three fast algorithms with \(O(n^2)\) complexity to compute the inverse of these matrices. The numerical results demonstrate that Algorithm 3.3 is very fast and it only requires several seconds to obtain the inverse of the matrices with size \(n = 10000\). Although encountering substantial cancellation in the computation of diagonal elements with Radau IIA nodes and Chebyshev nodes, we can also obtain desirable high accuracy by using higher precision, such as quadruple precision, and the compute time is considerably less than that required by using the symbolic manipulation package.

18                                 Shishun Li and Huile Wei

[23] M.-R. Skrzipek, Inversion of Vandermonde-like matrices, BIT, 44 (2004), pp. 291–306.

[24] B. S. Southworth, O. Krzysik, W. Pazner, and H. D. Sterck, Fast Solution of Fully Implicit Runge-Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part I: the Linear Setting, SIAM J. Sci. Comput. 44 (2022), pp. A416–A443.

[25] B. S. Southworth, O. Krzysik, and W. Pazner, Fast Solution of Fully Implicit Runge-Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs, SIAM J. Sci. Comput. 44 (2022), pp. A636–A663.

[26] J. Traub, Associated polynomials and uniform methods for the solution of linear problems, SIAM Rev. 8 (1966), pp. 277–301.

[27] Z. Xu and Z. You, A fast inversion of confluent Vandermonde-like matrices involving polynomials that satisfy a three-term recurrence relation, SIAM J. Matrix Anal. Appl. 19 (1998), pp. 797–806.


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
The notation and terminology in this note are mostly standard and follow [5, 7]. A cardinal is an initial ordinal and an ordinal is the set of smaller ordinals. In particular, 𝔠 denotes 2^{ℵ₀}. By a Polish group we mean a topological group with a Polish (that is, a separable and completely metrizable) group topology. Given a topological group G, a topological automorphism of G is a map φ: G → G that is simultaneously a group automorphism and a self-homeomorphism.

Outside the class of Polish groups, there are topological groups whose only topological automorphism is the identity. A remarkable example is van Mill's construction of a Baire separable metric connected and locally connected group having no homeomorphisms other than group translations [11]. In the opposite direction, William Barit and Peter Renaud proved that every Hausdorff locally compact group with more than two elements has a non-trivial topological automorphism [3].

Since Hausdorff locally compact groups are Polish if they are second-countable, the question of whether the result due to Barit and Renaud can be generalized to any Polish group naturally arises. We will prove that this is indeed the case. This answers a question posed by Forte Shinko during the Thematic Program on Set Theoretic Methods in Algebra, Dynamics and Geometry (Fields Institute, January–June, 2023).

Note that by van Mill's example, we cannot drop the complete metrizability of the groups considered in Theorem 1.1; even if we assume that such groups are Baire, metrizable and separable.

ABSTRACT. We prove that every Polish group admits a non-trivial topological group automorphism. This answers a question posed by Forte Shinko. As a consequence, we prove that there are no uniquely homogeneous Polish groups.

Theorem 1.1. Every Polish group with more than two elements admits a non-trivial topological automorphism.

CARLOS PÉREZ ESTRADA AND ULISES ARIET RAMOS-GARCÍA

A natural question motivated by Theorem 1.1 is how complicated the non-trivial topological automorphisms of a Polish group can be. For example, at the lowest difficulty level we have that for extended mapping class groups of (connected)

metrizablesurfacesandtheirfiniteindexsubgroups,everyalgebraicautomorphism
isfaithfullyrepresentedbyaconjugationwithamappingclass[4].Inparticular,
everyextendedmappingclassgroupisnaturallyisomorphic(as an abstract group)
to its automorphism group.

Our first observation is that for non-Boolean topological groups there is always
a non-trivial topological automorphism.

**Proposition 2.1.** Let \( G \) be a topological group. If \( G \) is not abelian or it has an
element of order greater than 2, then \( G \) has a non-trivial topological automorphism.

*Proof.* If \( G \) is not abelian, then there exist two non-trivial elements \( g, h \in G \) for
which \( h \neq ghg^{-1} \). Consequently, by conjugating with \( g \) or \( h \) we get a non-trivial
topological automorphism that does not fix \( h \) or \( g \), respectively. On the other hand,
if \( G \) is abelian but it has a non-trivial element \( x \) of order greater than 2, then the
inversion of elements is a non-trivial topological automorphism of \( G \) because it does
not fix \( x \). □

In light of Proposition 2.1, we restrict our attention to non-trivial topological
automorphisms on topological Boolean groups. Given an abstract Boolean group
\( G \), we endow it with its canonical structure as a vector space over the field of
two elements \( \mathbb{Z}_2 \). This vector space structure gives rise to a linear isomorphism
between \( G \) and a direct sum of \( \kappa \) copies of \( \mathbb{Z}_2 \) for some cardinal \( \kappa \). If additionally
\( G \) admits a Polish group topology, either \( G \) is countable, and so \( \kappa \leq \omega \), or \( G \) has
continuum cardinality, in which case \( \kappa = \mathfrak{c} \). In the former case, \( G \) is discrete and
when \( \kappa = n < \omega \), the automorphism group of \( G \) is precisely the general linear
group \( \mathrm{GL}_n(\mathbb{Z}_2) \) because every group automorphism of a Boolean group is a linear
\( \mathbb{Z}_2 \)-automorphism. Such general linear group \( \mathrm{GL}_n(\mathbb{Z}_2) \) has order \( (2^n-1)(2^n-2)\cdots(2^n-2^{n-1}) \). When \( \kappa = \omega \), the automorphism group \( \mathrm{Aut}(G) \) carries a non-
discrete Polish group topology under the pointwise-convergence topology, and thus
its cardinality is \( \mathfrak{c} \).

It remains to consider non-trivial topological automorphisms for Boolean Polish
groups \( G \) of continuum cardinality. In order to do this, we will construct a non-
trivial topological automorphism on a dense subgroup of \( G \) and then we will extend
it to the latter using the following lemma.

**Lemma 2.2.** Let \( G \) be a Polish group and \( H \) a dense subgroup of \( G \). Then every
topological automorphism of \( H \) extends uniquely to a topological automorphism of
\( G \).

*Proof.* Let \( \varphi \) be a topological automorphism of \( H \). If we consider \( \varphi \) to be a con-
tinuous homomorphism from \( H \) to \( G \), then there exists a unique continuous ex-
tension \( \overline{\varphi} : G \to G \) of \( \varphi \) by the density of \( H \) in \( G \) ([5, Page 6]). Extending anal-
ogously \( \varphi^{-1}: H \to G \) to a continuous homomorphism \( \overline{\varphi^{-1}} : G \to G \), we see that
\( \overline{\varphi} \circ \overline{\varphi^{-1}} = \mathrm{Id}_G \) and \( \overline{\varphi^{-1}} \circ \overline{\varphi} = \mathrm{Id}_G \) since the identity map of \( G \) as well as these
compositions extend the identity map of \( H \). It follows that the desired topological
automorphism of \( G \) extending \( \varphi \) is \( \overline{\varphi} \). □

The next key lemma plays an important role in our construction.

Lemma 2.3. Let (G, +, τ) be a Boolean Hausdorff topological group. For every finite collection {xi}i<n ⊆ G of Z₂-linearly independent elements, there exists an identity neighbourhood U such that

(1) for every two distinct elements x, x' ∈ L := span{xi}i<n, (x + U) ∩ (x' + U) = ∅; and
(2) FL := G \ (L + U) has empty interior.

Proof. Consider the family

𝒫 = {U ∈ τ : 0_G ∈ U ∧ ∀x, y ∈ L (x ≠ y ⇒ (x + U) ∩ (y + U) = ∅)}

ordered under inclusion. We will use the Kuratowski–Zorn lemma to show that (𝒫, ⊆) has a maximal element and then we will prove that any such maximal element fulfills items (1) and (2) above.

To verify that 𝒫 is not empty, first note that, since L is a finite set and G is a Hausdorff space, there exists a disjoint family of open sets {Ux}x∈L such that x ∈ Ux for every x ∈ L. Take any such family and notice that the identity neighbourhood U := ⋂_{x∈L} x + Ux ∈ 𝒫. Indeed, given any two different x, y ∈ L, (x + U) ∩ (y + U) ⊆ (x + x + Ux) ∩ (y + y + Uy) = Ux ∩ Uy = ∅.

Now, if {Uα}α∈I ⊆ 𝒫 is a linearly ordered subset, then UI := ⋃_{α∈I} Uα is an upper bound of it since for any two different x, y ∈ L, if

∅ ≠ (x + UI) ∩ (y + UI)
 = (x + ⋃ Uα) ∩ (y + ⋃ Uα)
    α∈I      α∈I
 = (⋃ (x + Uα)) ∩ (⋃ (y + Uα)),
  α∈I     α∈I

then certainly there would exist two α, β ∈ I for which (x + Uα) ∩ (y + Uβ) ≠ ∅, and thus by considering α ≥ β we would get that (x + Uα) ∩ (y + Uα) ≠ ∅, which would be a contradiction.

Let U be a ⊆-maximal element of 𝒫. To see that FL := G \ (L + U) has empty interior, we will proceed by contradiction. First note that if V := int(FL) ≠ ∅, then V + L ⊆ FL since if there were v ∈ V, u ∈ U and x₁, x₂ ∈ L such that v + x₁ = u + x₂, then v = (x₁ + x₂) + u ∈ (L + U) ∩ V, which is not possible. A consequence of this fact is that for any x ∈ L, x + V = V. Indeed, as V is the greatest open set contained in FL, certainly x + V ⊆ V for every x ∈ L. On the other hand, since any v ∈ V can be written as v = x + (x + v) and x + v ∈ x + V ⊆ V for every x ∈ L, we conclude that V ⊆ x + V for all x ∈ L.

Now we construct a non-empty open subset V′ ⊆ V for which (x + V′) + (y + V′) = ∅ if x, y ∈ L are distinct. In order to do this, for any v ∈ V consider a disjoint family of open sets {Ux}x∈L in V such that v + x ∈ Ux for any x ∈ L. Then we can consider V′ as ⋂_{x∈L} x + Ux. Note that v ∈ V′ because for any x ∈ L, v = x + (x + v) ∈ x + Ux. Moreover, V′ ⊆ U_{0_G} ⊆ V and certainly for each two distinct x, y ∈ L, (x + V′) + (y + V′) ⊆ (x + x + Ux) ∩ (y + y + Uy) = ∅.

We claim that the identity neighbourhood \( W := U \cup V' \in \mathbb{P} \). Indeed, note that for any two different \( x, y \in L \),

\[
(x+W) \cap (y+W) = (x + (U \cup V')) \cap (y + (U \cup V')) \\
= ((x+U) \cup (x+V')) \cap ((y+U) \cup (y+V')) \\
= ((x+U) \cap (y+U)) \cup ((x+U) \cap (y+V')) \cup \\
\qquad ((x+V') \cap (y+U)) \cup ((x+V') \cap (y+V')) \\
= \varnothing.
\]

Since \( V' \neq \varnothing \), \( W \) is an element of \( \mathbb{P} \) that strictly contains the maximal element \( U \). Hence we reach a contradiction by assuming that \( F_L \) has non-empty interior. \(\Box\)

Proof of Theorem 1.1. Let \( G \) be a Polish group with more than two elements. By the preceding remarks, the only case left to consider is when \( G \) is a Boolean group of continuum cardinality.

Let \( \{x_i\}_{i<n} \subset G \) be a finite collection with more than two elements of \(\mathbb{Z}_2\)-linearly independent elements. Consider an identity neighbourhood \( U \subset G \) as in Lemma 2.3 associated to \( \{x_i\}_{i<n} \). As \( G \) is necessarily non-discrete, the neighbourhood \( U \) is infinite. Since \( U \) does not contain non-trivial linear combinations of \( \{x_i\}_{i<n} \), every \( t \in U \setminus \{0_G\} \) is \(\mathbb{Z}_2\)-linearly independent from \( \{x_i\}_{i<n} \).

With the Kuratowski–Zorn lemma we can construct a maximal \(\mathbb{Z}_2\)-linearly independent subset \( Y \subseteq U \). Note that \( U \subseteq \text{span}(Y) \) since if there were a \( u \in U \setminus \text{span}(Y) \), then certainly \( Y \cup \{u\} \) would be a linearly independent subset of \( U \) strictly larger than \( Y \), contradicting the maximality of the latter. Consequently, we can consider the subgroup \( H \subseteq G \) generated by \( Y \) and \( \{x_i\}_{i<n} \). By item (2) of Lemma 2.3, \( H \) is dense in \( G \) since it contains the dense subset \( U + \text{span}\{x_i\}_{i<n} \).

As a result, to construct the desired topological group automorphism we take any non-trivial automorphism \( \varphi \) of \( \text{span}\{x_i\}_{i<n} \) and extend it to a \(\mathbb{Z}_2\)-automorphism of \( H \) by setting \( \varphi(y) = y \) for \( y \in Y \). Note that \( \varphi \) is a homeomorphism since for any neighbourhood identity \( V \) of \( 0_H \), the identity neighbourhood \( U \cap V \) is such that \( \varphi(U \cap V) = U \cap V \subseteq V \) and certainly the same happens with \( \varphi^{-1} \). Consequently, by Lemma 2.2 we can extend \( \varphi \) to a non-trivial topological automorphism of \( G \). \(\Box\)

The main motivation of [3] was to fully answer a question posed by Edmund Burgess\(^1\) at the 1955 Wisconsin topology conference ([1]) about the existence of uniquely homogeneous continua, i.e., about the existence of a compact connected metrizable space \( X \) such that for any two points \( p, q \in X \) there is a unique homeomorphism carrying \( p \) to \( q \).

In [10] Gerald Ungar used the renowned work of Edward Effros [6] on Polish transformation groups; i.e., pairs \( (G, X) \) where \( G \) is a Polish group acting continuously on the Polish space \( X \), to negatively answer Burgess’s question in the case of finite dimensional continua. We will roughly sketch Ungar's idea for general locally compact Polish spaces. Given a Polish transformation group \( (G, X) \), the space \( X \) is a quotient of \( G \) if \( X \) is homogeneous ([6, Theorem 2.1]). Therefore, as the homeomorphism group of a locally compact Polish space is a Polish group under

\(^1\)Although this question is often attributed to Burgess, he mentions in the MathSciNet review of [10] that this question was raised by another member of the conference.

the g-topology² and the canonical action of such group on the respective space is continuous [2, Theorems 1 and 3], any homogeneous locally compact Polish space is necessarily a quotient of its homeomorphism group. In particular, uniquely homogeneous locally compact Polish spaces are homeomorphic to their homeomorphism groups and thus carry the structure of boolean locally compact Polish groups with no non-trivial topological automorphisms ([10, Theorems 3.15 and 3.16]). Finally, Ungar used structural results of locally compact groups ([8, Theorems 4.9.3 and 4.10.1]) to remark that finite dimensional Polish groups that are either compact and connected or locally compact and locally connected have lots of non-trivial topological automorphisms.

Based on the idea of Ungar, Barit and Renaud used the same structural theory of locally compact groups and (non-commutative) Pontryagin duality to construct non-trivial topological automorphisms on any locally compact group with more than two elements. This fully answered Burguess' question in the negative for general locally compact Polish spaces.

We must point out that our proof of Theorem 1.1 only uses elementary theory of completely metrizable spaces (in particular, [7, Theorem 3.11] and [5, Page 6]) and thus we consider it to be more elementary than the one given by Barit and Renaud. It also negatively solves Burguess' question for the class of Polish spaces admitting a topological group structure.

Theorem 3.1. A non-trivial Polish group cannot be uniquely homogeneous.

It would be desirable to improve Theorem 3.1 to the class of all Polish spaces [11, Question 5.1]. One strategy for it would be to use Ungar's idea of giving a Polish group topology to the homeomorphism group of a uniquely homogeneous Polish space for which the canonical action were continuous and then use Theorem 3.1 to have a contradiction. However, it is known that the homeomorphism group of certain homogeneous Polish spaces, e.g., the Baire space ω^ω, cannot carry a Polish group topology ([9, Corollary 3]). A weaker statement, but still sufficient for our purposes, would be to regard a uniquely homogeneous Polish space as a quotient of a Polish group. But as before, there are homogeneous Polish spaces that cannot be a quotient of a Polish group ([12]). Bearing in mind that the previous examples are not uniquely homogeneous, there is still hope to use the unique homogeneity of a Polish space to make it the quotient of a Polish group.

Question 3.2. Is a uniquely homogeneous Polish space necessarily the quotient of a Polish group?

As Barit and Renaud, we required the axiom of choice to prove our main result. In particular, we use its Kuratowski-Zorn version in Lemma 2.3 to construct the identity neighbourhood U and the axiom of choice itself when finding a maximal Z₂-linearly independent subset Y ⊆ U. It seems plausible that the maximal Z₂-linearly independent subset can be constructed in ZF alone by using metrics; but it is not clear for us if the use of choice is superfluous in the construction of the neighbourhood U.

²The g-topology on the homeomorphism group of a topological space X has as subbasis the family of neighbourhoods of the form

V(K, U) := {f ∈ Homeo(X) | f(K) ⊆ U}

for every closed K ⊆ X and open U ⊆ X such that either K or X \ U is compact.

Question 3.3. Is the Axiom of Choice really necessary to construct non-trivial topological automorphisms on every Polish group?

Acknowledgments. This research was motivated while the authors were part of the Thematic Program on Set Theoretic Methods in Algebra, Dynamics and Geometry at the Fields Institute. We would like to thank the Institute for its warm hospitality. We would also like to thank Michael Hrušák for reading a preliminary version of our manuscript.

POSGRADO CONJUNTO EN CIENCIAS MATEMÁTICAS UNAM-UMSNH, Morelia, Michoacán, México 58089

Email address: cperez@matmor.unam.mx

CENTRO DE CIENCIAS MATEMÁTICAS, UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO, CAMPUS MORELIA, Morelia, Michoacán, México 58089


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
**Long Li¹, Nian Liu³,*, Dingwen Zhang¹, Zhongyu Li⁴, Salman Khan³,⁵,
Rao Anwer³, Hisham Cholakkal³, Junwei Han¹,²,*, and
Fahad Shahbaz Khan³,⁶**

¹ Northwestern Polytechnical University
² Institute of Artificial Intelligence, Hefei Comprehensive National Science Center
³ Mohamed bin Zayed University of Artificial Intelligence
⁴ Xi'an Jiaotong University
⁵ Australian National University
⁶ CVL, Linköping University

**Abstract.** Inter-image association modeling is crucial for co-salient object detection. Despite satisfactory performance, previous methods still have limitations on sufficient inter-image association modeling. Because most of them focus on image feature optimization under the guidance of heuristically calculated raw inter-image associations. They directly rely on raw associations which are not reliable in complex scenarios, and their image feature optimization approach is not explicit for inter-image association modeling. To alleviate these limitations, this paper proposes a deep association learning strategy that deploys deep networks on raw associations to explicitly transform them into deep association features. Specifically, we first create hyperassociations to collect dense pixel-pair-wise raw associations and then deploy deep aggregation networks on them. We design a progressive association generation module for this purpose with additional enhancement of the hyperassociation calculation. More importantly, we propose a correspondence-induced association condensation module that introduces a pretext task, *i.e.* semantic correspondence estimation, to condense the hyperassociations for computational burden reduction and noise elimination. We also design an object-aware cycle consistency loss for high-quality correspondence estimations. Experimental results in three benchmark datasets demonstrate the remarkable effectiveness of our proposed method with various training settings. The code is available at: https://github.com/dragonlee258079/CONDA.

**Keywords:** Co-salient Object Detection · Deep Association Learning

Co-Salient Object Detection (CoSOD) aims to segment salient objects that appear commonly across a group of related images. Compared to traditional Salient

*Corresponding authors: {liunian228, junweihan2010}@gmail.com*

Figure 1: Difference of raw-association-based image feature optimization strategy (a) and our proposed deep association learning strategy (b)

Fig. 1: Difference of raw-association-based image feature optimization strategy (a) and our proposed deep association learning strategy (b). Our deep association learning deploys deep learning networks on raw associations to achieve deep association features. We also present visual samples of our calculated raw associations (Raw Asso.), optimized image feature (Opt. Fea.), and our generated deep association features (Asso. Fea.) in (c).

Object Detection (SOD) [19–23, 26, 49, 50, 52], CoSOD is a more challenging task because it requires sufficient inter-image association modeling [9].

Recently, many advanced works [8, 13, 16, 33, 38, 41, 46, 47, 51] have emerged and achieved impressive performance. These methods first use related image features to acquire raw inter-image associations (also called consensus representations), and then leverage them as guidance to optimize each image feature, as shown in Figure 1 (a). This approach enables the final image feature to implicitly capture inter-image cues, thereby achieving the purpose of inter-image association modeling. However, we find this raw-association-based feature optimization strategy still has two limitations: i) they directly rely on raw associations, which are acquired in heuristic manners, such as pixel-wise [8, 33, 41, 51], region-wise [16, 46], or image-wise [13] similarity measurements. Although high-quality raw associations can be derived from high-level semantic information in image features, their revelation of common saliency regions still relies on similarity measures, which are unreliable when encountering complex scenarios, such as significant differences between co-salient objects or high foreground-background similarity. ii) the primary focus of building their deep models is on optimizing image features. Compared to directly modeling association relationship, image feature optimization is not an explicit approach for inter-image association modeling and will increase the learning difficulty.

To alleviate these limitations, we propose a deep association learning strategy for CoSOD, as shown in Figure 1 (b). Instead of directly using raw associations to optimize image features, we deploy deep networks on raw associations to learn deep association features. This is a more explicit strategy for inter-image association modeling. Moreover, our deep association features can capture high-level inter-image association knowledge, making them more robust in complex scenarios than raw associations, as shown in Figure 1 (c). Technically, we start by collecting all pixel-wise raw associations across the entire image group as hyperassociations. Then, we propose a Progressive Association Generation (PAG)

module to transform hyperassociations into deep association features. PAG progressively generates association features on varying scales, allowing us to use previous association features to enhance the hyperassociation calculation at the next scale, thereby improving the association quality from the very beginning.

Although deep association learning strategy allows more sufficient inter-image association modeling, it significantly increases the computational burden and reduces the practicality of this approach. Additionally, this study finds that it is not necessary to utilize all pixel associations to generate deep association features. In fact, there are even some noisy pixels that negatively impact the quality of the deep association features. Therefore, we propose a method based on Correspondence-induced Association Condensation (CAC) to condense the original full-pixel hyperassociations. This not only alleviates the computational burden but also further enhances the quality of deep association features.

Specifically, CAC performs the condensation operation by selectively associating pixels that have semantic correspondence in other images, as well as their surrounding contextual pixels, thereby creating lightweight yet more accurate hyperassociations. Here, we introduce a pretext task, i.e. semantic correspondence estimation, into the CoSOD, not only improving the model performance but also delving deeper into the essence of CoSOD. Co-salient objects inherently possess object-level semantic correspondence. However, in this paper, we aim to further explore the finer pixel-level correspondence. Although highly accurate correspondence estimation remains a challenge, we believe it will pave a new way for CoSOD research. We also provide an object-aware cycle consistency (OCC) loss to aid in learning correspondences within co-salient pixels.

In summary, the contributions of this paper are as follows:

Recently, there has been a surge of excellent methods for CoSOD [8,10,13,16,33,38,41–43,45–47,51]. These methods initially acquire raw inter-image associations

with related image features and then utilize them to optimize each image feature. Most methods generate pixel-wise [8,33,41,51], region-wise [16,46], and image-wise [13] raw associations through similarity-based manners, e.g. inner product calculations between image features. Even for transformer-based methods [16,33], they also rely on inner product calculation to produce attention maps [36] as raw associations. The image feature optimization manners include spatial or channel calibration [8,13,51], dynamic convolution [41,46], feature fusion [16,41], and cross-attention [16,33], etc. However, they lack the learning of high-level association knowledge and heavily focus on optimized image features. Unlike them, this paper proposes a new research direction that deploys deep networks on associations to achieve deep association features for CoSOD.

Apart from CoSOD, other tasks necessitating the consideration of inter-image relationships, such as few-shot segmentation [25,27,28,39], stereo matching [2,40], video semantic segmentation [34], etc. These tasks have made significant advancements recently by effectively modeling inter-image relations. Most of these methods [2,4,12,27,39,40] initially create cost volumes to capture dense inter-image pixel-wise similarities and subsequently use various modules to convert these hyper volumes into task-specific features. Our approach distinguishes itself from prior methods in three aspects. Firstly, most of them create 4D cost volumes between two images while we create 6D hyperassociations between all related images. Secondly, they calculate hyper volumes based on original image features, whereas we propose PAG to progressively enhance image features for better hyperassociation calculation. Last and most importantly, they rely on the full-pixel cost volume. We consider the condensation of hyperassociations using semantic correspondences to eliminate noisy pixel associations.

Semantic correspondence estimation [24] aims to establish reliable pixel correspondences between different instances of the same object category. Most works performed this task using fully supervised training [14,24]. Some recent works have utilized unsupervised learning with photometric, forward-backward consistency, and warp-supervision losses [30,35,48]. However, they implement these losses on the entire image, where background pixels may affect the performance. In this paper, we introduce this task to condense hyperassociations for Co-SOD and tailor the cycle consistency loss by only applying it to co-salient pixels, hence effectively avoiding the influence of background and extraneous objects.

As shown in Figure 2, CONDA integrates the deep association learning process into an FPN framework. Specifically, given a group of related images \(\{I_i\}_{i=1}^N\), we

first input them into a VGG-16 [32] backbone to collect its intermediate features for PAG and FPN decoding. In detail, we collect all features of the last three stages for PAG and the last feature of each stage for the FPN decoder as follows:

\[
\mathbb{F}^P = \{F_{s,l} \mid s \in \{3, \cdots, 5\}, l \in \{1, \cdots, L_s\}\},
\]
\[
\mathbb{F}^D = \{F_{s,L_s} \mid s \in \{1, \cdots, 5\}\},
\]

where \(\mathbb{F}^P\) and \(\mathbb{F}^D\) are the feature collections for PAG and FPN decoder, respectively. \(F_{s,l} \in \mathbb{R}^{N \times H_s \times W_s \times C_s}\) represents the VGG feature of the \(l\)-th layer in the \(s\)-th stage. There are in total \(L_s\) layers in the \(s\)-th stage. \(H_s\), \(W_s\) and \(C_s\) represent the height, width, and channel of the \(s\)-th stage, respectively.

Then, we input \(\mathbb{F}^P\) into PAG to calculate hyperassociations and generate deep association features \(\{F_s^A \in \mathbb{R}^{N \times H_s \times W_s \times C_s}\}_{s=3}^5\). Finally, these association features will be fused with \(\mathbb{F}^D\) for the FPN decoder process, formulated as:

\[
F_{s,L_s} = F_{s,L_s} + \phi(F_s^A), \quad s = 3, \cdots, 5,
\]
\[
F = \text{Decoder}(\{F_{s,L_s}\}_{s=1}^5),
\]

where \(\phi\) is a convolution layer. \(F \in \mathbb{R}^{N \times H \times W \times C}\) is the final feature for the final co-saliency prediction. We adopt BCE and IoU losses for supervision.

The rest of this section will introduce PAG with full-pixel hyperassociation and the condensation of hyperassociations by plugging the Correspondence-induced Association Condensation (CAC) module into PAG.

Our deep association learning involves two steps: 1) acquiring the raw hyper-associations \(\{A_s\}_{s=3}^5\) from three stages; 2) employing aggregation networks on \(\{A_s\}_{s=3}^5\) to obtain association features \(\{F_s^A\}_{s=3}^5\).

Earlier methods [4, 11, 31] directly utilize the original backbone features, i.e. \(F^P\), to calculate inter-image interactions, such as the so-called cost volume. We argue that hyperassociations derived straight from backbone features might obstruct further improvement of deep association learning, given that the current backbone is pre-trained without any consideration for inter-image associations.

To alleviate this problem, we propose the PAG module to progressively generate pyramid association features so that we can utilize the high-level association feature, e.g. \(F_{s+1}^A\) from the \((s+1)\)-th stage, to enhance the neighbouring low-level VGG features in \(F^P\), e.g. \(F_{s,l}\), for attaining association-enhanced features \(\hat{F}_{s,l}\), based on which we can calculate high-quality hyperassociations \(A_s\) in the \(s\)-th stage. After that, we execute the subsequent aggregation networks on \(A_s\) to achieve association features \(F_s^A\), which will continue to enhance the VGG features of the next stage and carry out progressive association generation. The whole process of our PAG can be formulated as follows:

\[
A_s = \mathrm{HAC}\left(\{\hat{F}_{s,l}\}_{l=1}^{L_s}\right),
\]
\[
F_s^A = \mathrm{Agg}(A_s), \tag{3}
\]
\[
\{\hat{F}_{s-1,l}\}_{l=1}^{L_{s-1}} = \mathrm{Enh}\left(\{F_{s-1,l}\}_{l=1}^{L_{s-1}}; F_s^A\right),
\]

where \(s\) ranges from 5 to 3 and \(\{\hat{F}_{5,l}\}_{l=1}^{L_5} = \{F_{5,l}\}_{l=1}^{L_5}\). The HAC, Agg, and Enh represent the hyperassociation calculation, aggregation network, and association-induced feature enhancement, respectively. Next, we explain them in detail.

**Hyperassociation Calculation.** For each stage, we first compute the raw associations at each layer using the inner product between \(l\)-2 normalized association-enhanced features of \(N\) related images. After that, we stack the raw associations of all layers to form the final hyperassociation of this stage. The hyperassociation for the s-th stage, i.e. \(A_s \in \mathbb{R}^{N \times H_s \times W_s \times N \times H_s \times W_s \times L_s}\), can be calculated via:

\[
A_s = \mathrm{HAC}\left(\{\hat{F}_{s,l}\}_{l=1}^{L_s}\right),
\]
\[
= \mathrm{Stack}\left(\left\{\mathrm{ReLU}\left(\frac{\hat{F}_{s,l} \cdot \hat{F}_{s,l}^\top}{\|\hat{F}_{s,l}\|\|\hat{F}_{s,l}^{\top}\|}\right)\right\}_{l=1}^{L_s}\right), \tag{4}
\]

where \(\hat{F}_{s,l} \cdot \hat{F}_{s,l}^{\top} \in \mathbb{R}^{N \times H_s \times W_s \times N \times H_s \times W_s}\). The \(\top\) indicates transposing the last dimension and the first three dimensions. \(\|\cdot\|\) represents the \(l\)-2 norm. We employ ReLU to suppress noisy association values.

**Aggregation Network.** The raw hyperassociation \(A_s \in \mathbb{R}^{N \times H_s \times W_s \times N \times H_s \times W_s \times L_s}\) is a hypercube with a nested structure, where each pixel position is characterized by a 4D tensor \((N \times H_s \times W_s \times L_s)\). Each 4D tensor documents the associations of the respective pixel with all other pixels in \(N\) related images.

clarity, we designate the first and second \( N \times H_s \times W_s \) dimensions in \( A_s \) as the source and target dimensions, respectively. Although these 4D tensors are crucial for exploring the consensus information for co-saliency detection, they essentially comprise pixel-to-pixel similarity values, seen in (4), which may be suboptimal and unreliable in complex scenarios. Therefore, we propose using deep networks to transform these pixel-wise similarities into deep association features with contextual and high-order association knowledge. This has never been explored in previous CoSOD methods. This is implemented via context aggregations on \( A_s \) to squeeze these 4D tensors as \( C_s \)-dimensional vectors, formulated as:

\[
N \times H_s \times W_s \times N \times H_s \times W_s \times L_s \rightarrow N \times H_s \times W_s \times C_s.
\]
(5)

In detail, we first deploy several association aggregation layers on \( A_s \) to progressively aggregate context information, enlarging \( L_s \) as \( C_s \), and eliminate the target \( H_s \times W_s \) dimension in 4D tensors. Each aggregation layer consists of 2D convolution layers and a downsampling operation. Specifically, focusing on the first aggregation layer for technical explanation, we first aggregate context information in the target \( H_s \times W_s \) dimension by applying a 2D convolution layer on all 4D tensors. The operations on the 4D tensor at pixel position \( (h_i, w_i) \) in image \( I_i \) can be formulated as:

\[
A_s^1(i, h_i, w_i, j, :, :, :) = C_1(A_s(i, h_i, w_i, j, :, :, :)), \quad j = 1, \cdots, N,
\]
(6)

where \( C_1 \) is a \( 3 \times 3 \) 2D convolution layer. Here, \( j \) is the index of other related images in the 4D tensor, and \( A_s(i, h_i, w_i, j, :, :, :) \in \mathbb{R}^{H_s \times W_s \times L_s} \) illustrates the associations between the pixel \( (h_i, w_i) \) in \( I_i \) and all pixels in image \( I_j \). This interpretation applies to other similar symbols.

Then, a downsampling operation \( D \), i.e. bilinear interpolation, is applied on \( A_s^1 \) to reduce the spatial dimension of the 4D tensor by a scaling factor:

\[
A_s^2(i, h_i, w_i, j, :, :, :) = D(A_s^1(i, h_i, w_i, j, :, :, :)),
\]
(7)

where \( A_s^2(i, h_i, w_i, j, :, :, :) \in \mathbb{R}^{H'_s \times W'_s \times C'_s}\), \( H'_s \) and \( W'_s \) are downsampled height and width. \( C'_s \) is the channel number after convolution \( C_1 \).

Finally, we also aggregate context information in the source \( H_s \times W_s \) dimension. Specifically, we extract the values along the source dimension and channel dimension in \( A_s^2 \) to form 4D tensors and apply a 2D convolution layer on them. For instance, \( A_s^2(:, :, :, j, h_j, w_j, :) \in \mathbb{R}^{N \times H_s \times W_s \times C'_s} \) is such a 4D tensor, where \( (h_j, w_j) \) is a pixel position in the target dimension. This can be formulated as:

\[
A_s^3(i, :, :, j, h_j, w_j, :) = C_2(A_s^2(i, :, :, j, h_j, w_j, :)), \quad i = 1, \cdots, N,
\]
(8)

where \( C_2 \) is a \( 3 \times 3 \) 2D convolution layer. \( i \) is the related image index.

After several association aggregation layers, as shown in (6)-(8), we can achieve the aggregated association features with the target \( H_s \times W_s \) dimension eliminated, denoted as \( F_{s}^{A'} \in \mathbb{R}^{N \times H_s \times W_s \times N \times C} \). Subsequently, we average \( F_{s}^{A'} \) on its second \( N \) dimension and obtain the final association features.

Association-induced Feature Enhancement. Once we have obtained association feature $F_s^A$ of the s-th stage, we will use it to enhance the VGG feature of the $(s-1)$-th stage, i.e. $\{F_{s-1, l}\}_{l=1}^{L_{s-1}}$. Specifically, we upsample $F_s^A$ to align the spatial size of features in the $(s-1)$-th stage, and then add it to $\{F_{s-1, l}\}_{l=1}^{L_{s-1}}$ followed by a convolution layer, formulated as:

$$F_s^A \in \mathbb{R}^{N \times H_s \times W_s \times C},$$
formulated as:

$$F_s^A = \frac{1}{N} \sum_{j=1}^N F_s^{A'}(:,:,:,j,:). \tag{9}$$

$$\hat{F}_{s-1, l} = C_3(F_{s-1, l} + U(F_s^A)),$$
\tag{10}

where $C_3$ and $U$ represent a 2D convolution layer and the bilinear upsampling operation, respectively.

Although PAG based on full-pixel hyperassociations can deliver satisfactory performance in CoSOD, it also introduces substantial computational overhead. Additionally, we argue that for each pixel in an image, it is unnecessary to gather its associations with all pixels of other related images to form hyperassociations. Some pixel associations may even impair the final performance, such as those between ambiguous regions. To this end, this subsection try to condense the original full-pixel hyperassociations to retain only informative pixel associations.

This subsection will focus on explaining the condensation of pixel associations of a pixel (e.g. $(h_i, w_i)$ in $I_i$) to ones of the other images (e.g. image $I_j$), i.e. $A_s(i, h_i, w_i, j, :,:,:)\in \mathbb{R}^{H_s \times W_s \times L_s}$, as shown in Figure 3. We will simplify the symbol $A_s(i, h_i, w_i, j, :,:,:)$ as $a_s^j$ in the subsequent text for convenience.

Specifically, CAC opts to select $K \times K$ ($K < H_s, K < W_s$) informative pixel associations from $a_s^j$ to form its condensed representation, i.e. $\tilde{a}_s^j \in \mathbb{R}^{K \times K \times L_s}$. Thus, the entire condensed hyperassociation can be symbolized as $\tilde{A}_s \in \mathbb{R}^{N \times H_s \times W_s \times N \times K \times K \times L_s}$.

K×K×Ls. To ensure the proper selection of K×K pixels, we introduce a pre-text task, i.e. semantic correspondence estimation [14,24]. This allows us to first locate the corresponding pixel of (hi, wi) in image Ij, i.e. (hj, wj), and then combine (hj, wj) with its surrounding pixels as the K×K pixel set. We design this approach based on the observation that the co-salient objects across N related images belong to the same semantic category, and the pixels within them should have semantic correspondences to each other, as shown in Figure 4. Therefore, the introduction of semantic correspondence in CAC not only improves the CoSOD performance but also delves deeper into the core nature of the CoSOD task. As far as we know, this is the first work to use semantic correspondence in the CoSOD task.

Correspondence Estimation. To estimate the correspondence pixel (hj, wj) in Ij for (hi, wi), we first identify an initial pixel (h^0_j, w^0_j) via a heuristic approach. Subsequently, we produce a spatial offset to refine (h^0_j, w^0_j) into (hj, wj). To achieve this purpose, all initial correspondence pixels should be utilized to form the initial condensed hyperassociations, with which the initial deep association features can be generated for spatial offset prediction.

Specifically, we pick out (h^0_j, w^0_j) that has the largest feature similarity value with (hi, wi). Since we have calculated the feature similarities in a^j_s, the (h^0_j, w^0_j) can be obtained as follows:

$$
\bar{a}^j_s = \sum_{l=1}^{L_s} a^j_s(:,:,l),
$$

$$
(h^0_j, w^0_j) = \text{argmax}_{x,y}(\bar{a}^j_s(x,y)),
$$

where $\bar{a}^j_s \in \mathbb{R}^{H_s \times W_s}$ is a similarity matrix obtained by eliminating the last dimension of $a^j_s \in \mathbb{R}^{H_s \times W_s \times L_s}$ through summation. argmax returns the coordinate of the maximum value.

Next, we select K × K pixels within the square region centered around the initial pixels, e.g. (h^0_j, w^0_j), to construct the initial condensed hyperassociation $\tilde{A}'_s$. Then, we can feed it into the aggregation network, described in Sec 3.1, and achieve the initial aggregated association features $F^{A'}_s \in \mathbb{R}^{N \times H_s \times W_s \times N \times C_s}$. It can be regarded as the association features of each pixel to N other related images. We utilize the feature of (hi, wi) to Ij, i.e. $F^{A'}_s(i, h_i, w_i, j, :) \in \mathbb{R}^{C_s}$, to predict the offsets for (hj, wj) via a linear layer, formulated as:

$$
\Delta^j_s = O(F^{A'}_s(i, h_i, w_i, j, :)),
$$

where O is a linear layer for offset generation. $\Delta^j_s \in \mathbb{R}^{K \times K \times 2}$ consists of K × K offsets, besides the center offset $\Delta^j_s(k_c, k_c, :)$ for correspondence estimation, i.e. refining (h^0_j, w^0_j) as (hj, wj), we also generate other offsets for surrounding pixel selection. Thus, the corresponding pixel (hj, wj) can be obtained by adding the offset to the initial pixel (h^0_j, w^0_j), formulated as:

$$
(h_j, w_j) = (h^0_j, w^0_j) + \Delta^j_s(k_c, k_c, :),
$$

Condensation Operation. Given the estimated correspondence pixel $(h_j, w_j)$ and other offsets in $\Delta_s^j$, we can obtain the surrounding pixels and combine them with $(h_j, w_j)$ to form $K \times K$ pixel set. Their coordinates are stored in $n_s^j \in \mathbb{R}^{K\times K\times 2}$. This process can be formulated as:

\[
n_s^j(x, y, :) = (h_j, w_j) + \Delta_s^j(x, y, :), \quad x, y \in \{1, \cdots, K\};\ x, y \neq k_c,
\]
\[
n_s^j(k_c, k_c, :) = (h_j, w_j),
\tag{14}
\]

where $(k_c, k_c)$ is the center position of $K \times K$ square.

Finally, we can perform the condensation operation via the index selection on $a_s^j$, formulated as:

\[
\tilde{a}_s^j = a_s^j(n_s^j),
\tag{15}
\]

where $\tilde{a}_s^j$ is the condensed representation of $a_s^j$, i.e. pixel associations of $(h_i, w_i)$ to image $I_j$. Furthermore, we also illustrate the condensation for pixel associations of $(h_i, w_i)$ to all images, i.e. $a_s \in \mathbb{R}^{N\times H_s \times W_s \times L_s}$, in Figure 2. By applying such a condensation process to all pixel associations, we can obtain the final condensed hyperassociation $\tilde{A}_s$.

To achieve accurate correspondence estimations, applying effective supervisions on them is necessary. As explicit semantic correspondence annotations are not available, we can only rely on unsupervised losses by imposing correspondence-related constraints on estimated correspondences. Previous unsupervised approaches apply constraints to all pixels [30, 35, 48], including those on the background and extraneous objects that don’t have mutual correspondences, hence harming the model effectiveness. To avoid this problem, we propose an object-aware constraint to only access losses on the co-salient pixels.

We propose an Object-aware Cycle Consistency (OCC) loss for the supervision of correspondence estimations in CoSOD. The cycle consistency can be explained as: if a co-salient pixel $(h_i, w_i)$ of image $I_i$ corresponds to the pixel $(h_j, w_j)$ in image $I_j$, then the pixel $(h_j, w_j)$ should also semantically correspond to pixel $(h_i, w_i)$.

Based on this cycle consistency constraint, we adopt image warping operations to conduct the OCC loss. Specifically, we first warp the image $I_i^s$ (a resized $I_i$ to align the scales in the s-th stage) as $I_{i \rightarrow j}^s$ using the $I_i^s \rightarrow I_j^s$ correspondence estimations. Next, we warp $I_{i \rightarrow j}^s$ backward to $I_{i \rightarrow j \rightarrow i}^s$ using $I_j^s \rightarrow I_i^s$ correspondence estimations. Finally, we can utilize the SSIM loss between $I_i^s$ and $I_{i \rightarrow j \rightarrow i}^s$ to measure the cycle consistency for mutual correspondence pixels in $I_i^s$ and $I_j^s$. Moreover, to ensure the constraints are only conducted on co-salient objects, we mask the images with their ground truth masks, formulated as:

\[
\mathcal{L}_s^C = \frac{1}{N^2} \sum_{i=1}^N \sum_{j=1}^N \mathcal{L}_{SSIM}(I_i^s \cdot G_i^s, I_{i \rightarrow j \rightarrow i}^s \cdot G_i^s),
\tag{16}
\]
where $(k_c, k_c)$ is the center position of $K \times K$ square.

To construct the training data, we follow [51] to use different combinations of three commonly used training datasets, i.e. DUTS class [47] (8250 images of 291 groups), COCO-9k [18] (9213 images of 65 groups), and COCO-SEG [37] (200,000 images of 78 groups), for a fair comparison with other state-of-the-art (SOTA) works. We also implement the synthesis strategy for the DUTS class dataset following [46].

We follow [8] to evaluate our model on three benchmark datasets, i.e. CoCA [47] (1295 images of 80 groups), CoSal2015 [44] (2015 images of 50 groups), and CoSOD3k [7] (3316 images of 160 groups). We adopt four widely-used metrics for the quantitative evaluation, i.e. Structure-measure ($S_m$) [5], Maximum enhanced-alignment measure ($E_\xi$) [6], Maximum F-measure ($F_\beta$) [1], and Mean Absolute Error ($M$) [3].

For training specifics, we employ the data augmentation strategy in [21] and use 256 × 256 as the input size for the network. We employ the Adam optimizer [15] with $\beta_1 = 0.9$ and $\beta_2 = 0.99$ to optimize the network. We train our CONDA model for 300 epochs, starting with an initial learning rate of $1e-4$, which is divided by 10 at the 60,000$^{th}$ iteration. Our experiments are implemented based on PyTorch [29] on a single Tesla A40 GPU, with the batchsize set to $N = 6$. The hyperparameter $K$ in CAC is set to 9.

where $G_i^s$ is the resized ground truth of image $I_i$. The total OCC loss $\mathcal{L}^C$ is the sum of three stages, formulated as: $\mathcal{L}^C = \mathcal{L}_3^C + \mathcal{L}_4^C + \mathcal{L}_5^C$. More details can be found in the supplementary materials.

We conduct ablation studies on the most challenging CoCA [47] dataset. To construct the baseline, we use the FPN [17] (with VGG-16 as encoder) as the

foundational segmentation network and enhance it with the Region-to-Region correlation module (R2R) [16] to simply capture inter-image connections. Then, as shown in Table 1, we incrementally incorporate PAG, CAC, and OCC into the baseline for effectiveness analysis. We trained all ablation models with the DUTS class and COCO9k datasets.

Effectiveness of PAG. PAG first uses intermediate image features from the FPN encoder to calculate full-pixel hyperassociations. Then, aggregation networks are applied to generate deep association features for the decoder process. PAG effectively utilizes deep learning to model pair-wise pixel associations, achieving higher-level association knowledge compared to previous image feature optimization strategies. As shown in the 3rd line of Table 1, PAG shows significant performance boosts compared to the baseline model, with respective gains of 3.72%, 4.80%, 7.30%, and 1.31% in $S_m$, $E_\xi$, $F_\beta$, and $\mathcal{M}$.

Furthermore, to validate our approach of progressively enhancing image features with the previously generated association feature for improving hyperassociation calculation, we conduct an ablation experiment called Separate Association Generation (SAG) where association features are generated for three stages without image feature enhancements. As shown in the 2nd & 3rd lines in Table 1, PAG outperforms SAG, indicating that our progressive enhancement design can obtain better hyperassociation.

Effectiveness of CAC. CAC aims to condense full-pixel hyperassociations in PAG by selecting corresponding pixels and their surrounding contexts. Results in the 3rd & 5th lines of Table 1 show that introducing the CAC module improves the performance. Moreover, it reduces the multiply-accumulate operations (MACs) of aggregation networks from 91.38G in the full-pixel PAG to 77.19G¹. This indicates that utilizing correspondence estimation to condense the hyperassociations not only effectively reduces the computational burden but also helps obtain more accurate pixel associations.

We also analyze CAC in detail. As deep association features are necessary for reliable correspondence estimation, CAC first pre-condenses the hyperasso-

¹ We input a group of 6 related 256x256 images to measure MACs.

ciations using a maximum similarity approach to obtain initial deep association features and then performs further condensation based on the correspondences predicted by these initial deep association features. The hyperassociation condensation process with only the pre-condensation operation is called Similarity-induced Association Condensation (SAC). In Table 1, SAC only brings slight performance improvements due to the heuristic nature of the correspondence estimation. Nevertheless, SAC can provide the initial association feature for CAC to predict reliable correspondence.

Effectiveness of OCC. OCC provides self-supervision for CAC to enable more precise correspondence estimation. Results in the 5th&7th lines of Table 1 show OCC further improves the performance by leveraging more precise correspondence estimations to condense hyperassociations effectively. In addition, we conducted an ablation experiment to validate our object-aware design in CAC by replacing OCC with full-pixel cycle consistency (FCC) loss. Comparing the 5th&6th lines of Table 1, FCC leads to a notable performance decrease due to background pixels disrupting the correspondence learning.

Visualization of Correspondence Estimations. We present some visual samples of correspondence estimations for some co-salient pixels in Figure 4. Our semantic correspondence estimations are meaningful and can effectively depict the common attributes of co-salient objects at the pixel level.

We compare our model with eight recent SOTA methods, i.e. GICD [47], ICNet [13], GCoNet [8], CADC [46], DCFM [41], DMT [16], UGEM [38], and GCoNet+ [51]. We directly utilize their officially released saliency maps for comparison. To ensure fairness, we trained our model with different combinations of three training datasets, following [51], to align with other compared methods. We

denote three training datasets, i.e. DUTS class [47], COCO9k [18], and COCO-SEG [37], as DC, C9, and CS, respectively, for convenience. Our training sets include DC, C9, DC+C9, DC+CS. As shown in Table 2, we can observe that our model achieves the best performance with each training set in most benchmark datasets. What is even more exciting is that we achieve excellent results in the most challenging CoCA dataset, surpassing the second-best models by large margins, e.g. 2.5% $S_m$, 2.5% $E_\xi$, and 4.8% $F_\beta$ with the DC+CS training set.

We also present visual comparisons in Figure 5. Our model can accurately detect co-salient objects in complex scenarios, such as irregularly shaped accordions accompanied by extraneous objects (people). However, other models easily fail to accurately segment co-salient objects.

This paper proposes a deep association learning strategy for CoSOD that directly embeds hyperassociations into deep association features. Correspondence estimation is also introduced to condense hyperassociations, enabling the exploration of pixel-level correspondences for CoSOD. We also utilize an object-aware cycle consistency loss to further refine correspondence estimations. Extensive experiments have verified the effectiveness of our method.

Acknowledgments. This work was supported in part by Key-Area Research and Development Program of Guangdong Province under Grant 2021B0101200001, the National Natural Science Foundation of China under Grant 62136007, 62036011, U20B2065, 6202781, 62036005, the Key R&D Program of Shaanxi Province under Grant 2021ZDLGY01-08, and the MBZUAI-WIS Joint Program for AI Research under Grants WIS P008 and P009.

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