Deep Grassmannian multiview subspace clustering with contrastive learning

Rui Wang; Haiqiang Li; Chen Hu; Xiao-Jun Wu; Yingfang Bao; Rui Wang; Haiqiang Li; Chen Hu; Xiao-Jun Wu; Yingfang Bao

doi:10.3934/era.2024252

Electronic Research Archive

2024, Volume 32, Issue 9: 5424-5450. doi: 10.3934/era.2024252

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Research article

Deep Grassmannian multiview subspace clustering with contrastive learning

1.
School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi 214122, China
2.
Jiangsu Provincial Engineering Laboratory of Pattern Recognition and Computational Intelligence, Jiangnan University, Wuxi 214122, China
3.
Affiliated Wuxi Fifth Hospital of Jiangnan University, Wuxi 214007, China
4.
The Fifth People's Hospital of Wuxi, Wuxi 214007, China

Received: 25 June 2024 Revised: 13 September 2024 Accepted: 20 September 2024 Published: 26 September 2024

This paper investigated the problem of multiview subspace clustering, focusing on feature learning with submanifold structure and exploring the invariant representations of multiple views. A novel approach was proposed in this study, termed deep Grassmannian multiview subspace clustering with contrastive learning (DGMVCL). The proposed algorithm initially utilized a feature extraction module (FEM) to map the original input samples into a feature subspace. Subsequently, the manifold modeling module (MMM) was employed to map the aforementioned subspace features onto a Grassmannian manifold. Afterward, the designed Grassmannian manifold network was utilized for deep subspace learning. Finally, discriminative cluster assignments were achieved utilizing a contrastive learning mechanism. Extensive experiments conducted on five benchmarking datasets demonstrate the effectiveness of the proposed method. The source code is available at https://github.com/Zoo-LLi/DGMVCL.

Keywords:

Citation: Rui Wang, Haiqiang Li, Chen Hu, Xiao-Jun Wu, Yingfang Bao. Deep Grassmannian multiview subspace clustering with contrastive learning[J]. Electronic Research Archive, 2024, 32(9): 5424-5450. doi: 10.3934/era.2024252

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Abstract

1. Introduction

Nonlinear evolution equations (NLEEs) have gained a significant importance in engineering and mathematical sciences, thereby influencing diverse fields such as nuclear physics, plasma physics, kinematics, chemistry, biology, bioengineering, mechanics, electromagnetism, nonlinear optics, mathematical biosciences, and so on. These equations are essential to describe complex scientific phenomena, enabling the modeling and analysis of dynamical systems with differential equations. In particular, mathematicians have made substantial progress in understanding the dynamics of nonlinear partial differential equations (NPDEs), particularly the soliton aspect, thus contributing valuable methods and insights. The exploration of physical behaviors within this framework not only enhances the understanding, but also opens doors to innovative applications, potentially leading to practical breakthroughs in various scientific domains.

A specific class of NLEEs includes the shallow water equations, a set of hyperbolic partial differential equations (PDEs), which have been widely used to model various real-world phenomena governed by PDEs of an arbitrary order. One prominent example of such an equation is the Korteweg-de Vries (KdV) equation, which can be expressed as follows:

$\begin{equation} \eta_{\tau}+\eta \eta_{\xi}+\eta_{\xi\xi\xi} = 0. \end{equation}$

(1.1)

This equation was introduced by Korteweg and de Vries in ^[1], which describes the evolution of small amplitude, long waves in shallow water, where factors such as depth and bottom topography influence the wave speed, thus leading to changes in wave profiles. The KdV equation, precisely solvable with suitable initial data, has captivated widespread interest among researchers for its significance in understanding phenomena like wave breaking and dispersion.

In contrast to the KdV equation, Geyer et al. ^[2] introduced a nonlinear equation to analyze single-layer oceanic fluids in equatorial regions by incorporating the effects of the rotatory motion of Earth on the fluid, commonly referred to as the Coriolis effect. This formulation, termed the geophysical KdV (GeoKdV) equation, contributes to a more comprehensive understanding of oceanic phenomena, and can be written as follows:

$\begin{equation} \eta_{\tau}-\omega \eta_{\xi}+\frac{3}{2}\eta \eta_{\xi}+\frac{1}{6}\eta_{\xi\xi\xi} = 0. \end{equation}$

(1.2)

Here, $\eta$ is a function of $(\xi, \tau)$ and represents the free surface elevation, and $\omega$ is the Coriolis parameter.

In recent years, several authors have explored various aspects of the GeoKdV equation. For instance, Turgut et al. ^[3] investigated the Coriolis effect on oceanic flows in the equatorial region using Lie symmetries and implemented the finite element method for numerical solutions. Rizvi et al. ^[4] employed the Hirota bilinear method to obtain lump-kink and lump-periodic solutions. The solitons and complexions of the GeoKdV equation were derived using Kudryashov and Hirota methods, and the impact of the Coriolis parameter on nonlinear waves was studied by Hosseini et al. ^[5]. Additionally, Alharbi and Almatraf ^[6] investigated the GeoKdV equation by utilizing an improved exp-expansion method, and obtained various solitary solutions. Additionally, they applied shooting and adaptive moving methods to compute numerical solutions and used Fourier analysis to evaluate the accuracy and stability of these numerical schemes. In recent years, Sayed et al. ^[7] explored the modified and generalized forms of the GeoKdV equation. They utilized the extended Tanh method and the generalized Kudryashov method with the Riccati equation to derive sine and cosine hyperbolic solutions.

To address the limitations of integer order operators, which are localized and incapable of predicting complex behaviors, various mathematicians and scientists have introduced non-integer order operators. In recent years, scientific research has increasingly focused on exploring more complex changes through fractional calculus due to its non-local properties and memory-retaining characteristics. This shift has led to significant growth in its applications across various disciplines, including mathematics, physics, engineering, biotechnology, electrochemistry, and more; see ^{[8,9,10,11,12,13,14,15,16]}. However, despite considerable progress, obtaining precise or numerical solutions for models that involve fractional calculus remains a significant challenge. Addressing these complexities requires innovative approaches, as many existing methodologies in the literature employ nonlinear fractional differential equations (FDEs) to capture the behavior of physical phenomena within dynamic systems. In this regard, Karunakar and Chakraverty ^[17] utilized the homotopy perturbation method (HPM) to solve time-fractional third- and fifth-order KdV equations. Shah et al. ^[18] utilized the Mohand decomposition method with Caputo derivatives to tackle fractional and integer-order KdV equations, demonstrating that fractional-order solutions gradually converge to their integer-order counterparts. Similarly, Jena and Chakraverty ^[19] efficiently obtained solitary wave solutions for generalized Hirota-Satsuma coupled KdV and MKdV equations. Al-Sawalha et al. ^[20] made further advancements by applying a natural decomposition method alongside Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to obtain approximate solutions for the Kersten-Krasil'shchik coupled KdV and modified KdV systems. Cao et al. ^[21] explored variational principles for two types of nonlinear GeoKdV equations with fractal derivatives. In recent times, Shehzada et al. ^[22] delved into a modified form of the fractional GeoKdV equation with a power-law kernel. These studies collectively highlight the growing importance of fractional calculus to solve various types of KdV equations, thus demonstrating its potential to provide deeper insights into dynamic systems.

On the contrary, the information about the parameters involved in shallow water wave equations (SWWEs) is often incomplete. To address this, fuzzy parameters can be incorporated into the SWWEs. Zadeh ^[23] introduced the fuzzy set theory in 1965, offers a mathematical framework to handle uncertainty and imprecision. Unlike the classical set theory, where an element either fully belongs or does not belong to a set, fuzzy sets allow for partial membership, with elements belonging to a set to varying degrees, typically represented by values between 0 and 1. This approach is especially useful in cases where boundaries between the categories are not clearly defined, such as when dealing with linguistic variables or ambiguous data. In our study, we apply the fuzzy set theory to address the uncertainty present either in system parameters or initial conditions.

By representing uncertain variables as fuzzy numbers, we capture a range of possible values rather than relying on a single deterministic value. This approach increases the robustness of our model and provides a more comprehensive understanding of the system behavior, particularly in situations where precise information is unavailable. In this regard, several authors have also applied the fuzzy set theory to water wave equations, thus demonstrating its effectiveness in capturing uncertainties in such complex systems. As such, the behavior of two-dimensional SWWEs with fuzzy parameters was investigated by Karunakar and Chakraverty ^[24] using the HPM, treating water depth as uncertain and expressed in terms of fuzzy parameters. Sahoo and Chakraverty ^[25] introduced the Sawi transform-based HPM as a hybrid approach to approximate solutions to one-dimensional SWWEs in tsunami predictions. Additionally, they applied HPM to solve coupled SWWEs, considering the wave height as fuzzy ^[26]. Using this approach, the impact of the coastal slope and the sea depth on tsunami wave characteristics, particularly wave velocity and run-up height, was analyzed in uncertain environments. Extending the study of fuzzy uncertainty in these fields, they further derived an approximate solution for the GeoKdV equation by using the Adomian decomposition method within the fuzzy environment ^[27].

To the best of our knowledge, the proposed model has not been explored within the domain of fractional calculus and fuzzy environment. Drawing inspiration from valuable contributions in fractional calculus and fuzzy uncertainty, we aim to investigate the uncertain GeoKdV equation in the fractional order of Caputo's sense by using the fractional reduced differential transform method (FRDTM). Consequently, we consider the GeoKdV Eq (1.2) in the fuzzy fractional form as:

$\begin{equation} \tilde{\eta}_{\tau}^{\alpha}-\omega \tilde{\eta}_{\xi}+\frac{3}{2}\tilde{\eta} \tilde{\eta}_{\xi}+\frac{1}{6}\tilde{\eta}_{\xi\xi\xi} = 0. \end{equation}$

(1.3)

This study is motivated by the desire to understand the intricate behaviors exhibited by the nonlinear GeoKdV equation in the fuzzy environment and to investigate the effect of the fractional order on its dynamics. Unlike traditional methods, the FRDTM simplifies the complex dynamics of the GeoKdV equation under uncertain and fractional-order conditions, offering an improved accuracy and an ease of implementation. This study not only demonstrates the effectiveness of the FRDTM through multiple case analyses, but also establishes strong agreements between simulated results and provided data, highlighting the robustness of the method. These results confirm that the FRDTM technique is a reliable method for addressing fractional-order nonlinear problems.

The subsequent part of this article is structured as follows: Section 2 covers the fundamental concepts of fuzzy set theory and fractional calculus; in Section 3, the methodology is discussed, with a detailed analysis of the FRDTM; Section 4 aims to provide a comprehensive understanding of application of the FRDTM to solve the time fractional fuzzified GeoKdV equation; in Section 5, results and discussion are presented, where the research outcomes are thoroughly analyzed and interpreted; and finally, Section 6 offers a conclusive summary of the findings and insights for future research directions.

2. Basic concepts of fuzzy sets and fractional calculus

In our subsequent discussions, we will look at some fundamental concepts related to fuzzy and fractional calculus.

Definition 2.1. ^[23,28,29] A fuzzy number denoted as $\tilde{S}$ , is a type of fuzzy set defined on the real line $\mathbb{R}$ . adhering to the conditions of convexity and normality. It is characterized by a piecewise continuous membership function $\mu_{\tilde{S}}(u)$ that satisfies the following:

$\{\mu_{\tilde{S}}(u): \mathbb{R} \rightarrow [0, 1], \forall u \in \mathbb{R}\}.$

Definition 2.2. ^[23,28,29] A triangular fuzzy number (TFN) $\tilde{R}$ can be represented as a triplet

$\tilde{R} = (r_1, r_2, r_3), \ \ \ r_1 \leq r_2 \leq r_3 .$

It falls within the classification of triangular-shaped fuzzy numbers, as shown in . The membership function $\mu_{\tilde{R}}(u)$ of a TFN $\tilde{R}$ may be stated as follows:

$\begin{equation} \mu_{\tilde{R}}(u) = \begin{cases} 0, u < r_1, \\ \frac{u-r_1}{r_2-r_1}, r_1 \leq u \leq r_2, \\ \frac{r_3-u}{r_3-r_2}, r_2 \leq u \leq r_3, \\ 0, u > r_3. \end{cases} \end{equation}$

(2.1)

Figure 1. Schematic diagram of TFN.

DownLoad: Full-Size Img PowerPoint

Definition 2.3. ^[23,28,29] Consider the following TFN

$\tilde{R} = (r_1, r_2, r_3).$

Utilizing the $p$ -cut approach, it may be transformed into a fuzzy interval representation as follows:

$\begin{equation} \tilde{R}_p = [\tilde{R}^l, \tilde{R}^u] = [r_1+p(r_2-r_1), r_3-p(r_3-r_2)], \end{equation}$

(2.2)

where $p \in [0, 1].$

Definition 2.4. ^[23,28,29] By applying the $(p, r)$ -cut technique, the above Eq (2.2) may be expressed in a double parametric form (DPF) as follows:

$\begin{equation} \tilde{R}_{(p, r)} = r[\tilde{R}^u-\tilde{R}^l]+\tilde{R}^l, \ \ \ p, r \in [0, 1]. \end{equation}$

(2.3)

Definition 2.5. ^[23,28,29] For a function $\tilde{f}(t; p, r)$ , with fuzzy values, the fuzzy fractional Riemann-Liouville operator of order $\alpha$ (i.e. ${J_t}^{\alpha}$ ) is defined as follows:

$\begin{equation} {J_t}^{\alpha}\tilde{f}(t;p, r) = \frac{1}{\Gamma(\alpha)} \int_{0}^{t}(t-x)^{\alpha-1}\tilde{f}(x;p, r)dx, \end{equation}$

(2.4)

where $t > 0, \ \alpha > 0.$

Definition 2.6. ^[23,28,29] For a fuzzy valued function $\tilde{f}(t; p, r)$ based on its DPF, the fuzzy fractional differential operator ${D_t}^{\alpha}$ in the Riemann-Liouville sense is interpreted as follows:

$\begin{equation} {D_t}^{\alpha}\tilde{f}(t;p, r) = \begin{cases} \frac{1}{\Gamma(m-\alpha)}\frac{d^m}{dt^m}\int_{0}^{t}\frac{\tilde{f}(x)}{(t-x)^{\alpha-m+1}}dx, \\ m-1 < \alpha \leq m, m \in N, \\ \frac{d^m}{dt^m}\tilde{f}(t;p, r), \ \ \ \alpha = m, m\in N. \end{cases} \end{equation}$

(2.5)

In Caputo's sense, the fuzzy fractional differential operator ${D_t}^{\alpha}$ can be interpreted as follows:

$\begin{equation} {D_t}^{\alpha}\tilde{f}(t;p, r) = \begin{cases} \frac{1}{\Gamma(m-\alpha)}\int_{0}^{t}\frac{\tilde{f}^{(m)}(x;p, r)}{(t-x)^{\alpha-m+1}}dx, \\ m-1 < \alpha \leq m, m \in N, \\ \frac{d^m}{dt^m}\tilde{f}(t;p, r), \ \ \ \alpha = m, m\in N. \end{cases} \end{equation}$

(2.6)

3. Method description

To enhance the understanding of the FRDTM, we will start by looking over the local fractional Taylor's theorem. Moreover, this theorem has been extended to incorporate fractional derivatives, thus establishing a thorough foundation for applying the FRDTM.

3.1. Local fractional Taylor's expansion ^[30,31]

Assuming that

$\psi^{(k+1)\alpha} \in C_{\alpha}(s, v)$

for $k = 0, 1, \ldots, n$ and $0 < \alpha \leq 1$ , we may represent the relationship in the following manner:

$\begin{equation} \psi(\xi) = \sum\limits_{k = 0}^{\infty} \psi^{(k\alpha)}(\xi_0)\frac{{(\xi-\xi_0)}^{k\alpha}}{\Gamma(1+k\alpha)}, \end{equation}$

(3.1)

where

$s < \xi_0 < \xi < v, \forall \xi \in (s, v)$

and

$\psi^{(k+1)\alpha}(\xi) = {D_{\xi}}^{\alpha}{D_{\xi}}^{\alpha}\ldots(k+1 \text{ times }) \psi(\xi).$

Definition 3.1. ^[30,31] For an analytic function $\psi(\xi, \tau)$ , the fractional reduced differential transform denoted as $\psi_k(\xi)$ can be expressed as follows:

$\begin{equation} \psi_k(\xi) = \frac{1}{\Gamma(1+k\alpha)}\Bigg( \frac{\partial^{\alpha k}\psi(\xi, \tau)}{\partial \tau ^{\alpha k}}\Bigg)_{\tau = \tau_0}, \end{equation}$

(3.2)

where $k = 0, 1, \ldots n.$

Definition 3.2. ^[30,31] The fractional inverse differential transform of $\psi_k(\xi)$ can be expressed as follows:

$\begin{equation} \psi(\xi, \tau) = \sum\limits_{r = 0}^{\infty}\psi_k(\xi)(\tau-\tau_0)^{\alpha k}. \end{equation}$

(3.3)

The following results can be drawn from Definitions 3.2 and 3.3:

$\begin{equation} \psi(\xi, \tau) = \sum\limits_{r = 0}^{\infty}\frac{1}{\Gamma(1+k\alpha)} \Bigg( \frac{\partial^{\alpha k}\psi(\xi, \tau)}{\partial \tau ^{\alpha k}}\Bigg)_{\tau = \tau_0} (\tau-\tau_0)^{\alpha k}. \end{equation}$

(3.4)

For a specific time $\tau_0 = 0$ , we have

$\begin{align} \begin{split} \psi(\xi, \tau)& = \sum\limits_{r = 0}^{\infty}\psi_k(\xi)\tau^{\alpha k}\\& = \sum\limits_{r = 0}^{\infty}\frac{1}{\Gamma(1+k\alpha)} \Bigg( \frac{\partial^{\alpha k}\psi(\xi, \tau)}{\partial \tau ^{\alpha k}}\Bigg)_{\tau = 0} \tau^{\alpha k}.\end{split} \end{align}$

(3.5)

3.2. Properties of FRDTM

Let $f(\xi, \tau)$ , $g(\xi, \tau)$ , and $h(\xi, \tau)$ be three analytical functions such that

$\begin{equation} \begin{aligned} f(\xi, \tau)& = {R_D}^{-1}[f_k(\xi)], g(\xi, \tau) = {R_D}^{-1}[g_k(\xi)], \\ h(\xi, \tau)& = {R_D}^{-1}[h_k(\xi)]. \end{aligned} \end{equation}$

(3.6)

Here, ${R_D}^{-1}$ is the inverse reduced differential transform operator. In this instance, we might have the following:

(1) If

$f(\xi, \tau) = g(\xi, \tau)\pm h(\xi, \tau),$

then

$f_k(\xi) = g_k(\xi)\pm h_k(\xi).$

(2) If

$f(\xi, \tau) = ag(\xi, \tau),$

then

$f_k(\xi) = ag_k(\xi),$

where $a$ is a constant.

(3) If

$f(\xi, \tau) = g(\xi, \tau)h(\xi, \tau),$

then

$f_k(\xi) = \sum\limits_{i = 0}^{j}g_i(\xi)h_{j-i}(\xi).$

(4) If

$f(\xi, \tau) = \frac{\partial^n}{\partial \xi^n}h(\xi, \tau),$

then

$f_k(\xi) = \frac{\partial^n}{\partial \xi^n}h_k(\xi).$

(5) If

$f(\xi, \tau) = \frac{\partial^{n\alpha}}{\partial \tau^{n\alpha}}h(\xi, \tau),$

then

$f_k(\xi) = \frac{\Gamma(1+(k+n)\alpha)}{\Gamma(1+k\alpha)}h_{k+n}(\xi).$

3.3. Methodology of FRDTM ^[30,31]

Here, we have discussed the step-by-step explanation of the FRDTM in order to solve a non-linear fractional order PDE.

Consider the following fractional PDE in the operator form:

$\begin{equation} {D_{\tau}}^{m\alpha}g(\xi, \tau)+Lg(\xi, \tau)+Ng(\xi, \tau) = z(\xi, \tau), \end{equation}$

(3.7)

where $m-1 < m\alpha\leq m, m \in \mathbb{N},$ and

● $L$ represents the linear operator;

● $N$ represents the nonlinear operator;

● $z(\xi, \tau)$ is the inhomogenous source term.

Let us consider the initial conditions (ICs) associated with the above equation, i.e., Eq (3.7) is as follows:

$\begin{equation} g^{(i)}(\xi, 0) = l_i(\xi), i = 0, 1, \ldots, m-1. \end{equation}$

(3.8)

Step 1. Implementation of properties of FRDTM

By employing the properties discussed in Section 3.4, Eq (3.7) simplifies to the following:

$\begin{equation} \frac{\Gamma(1+(k+n)\alpha)}{\Gamma(1+k\alpha)}g_{k+m}(\xi) = z_k(\xi)-Rg_k(\xi)-Ng_k(\xi), \end{equation}$

(3.9)

where

$k = 0, 1, 2, \ldots,$

$g_k(\xi)$ and $z_k(\xi)$ are the differential transformed form of $g(\xi, \tau)$ and $z(\xi, \tau)$ , respectively.

Now, applying the FRDTM to the given initial conditions in (3.8), we obtain

$\begin{equation} {g_k(\xi) = l_k(\xi)}, \ \ \text{ for }\ \ k = 0, 1, 2, \ldots, m-1. \end{equation}$

(3.10)

Step 2. Calculation of $g_k(\xi)$

Utilizing Eqs (3.9) and (3.10), the value of $g_k(\xi)$ for $k = 0, 1, 2, \ldots$ can be iteratively obtained.

Step 3. Perform the inverse transformation

After determining the values of $g_k(\xi)$ , let us perform the inverse differential transformation to obtain the approximate solution for $g(\xi, \tau)$ . The inverse transformation allows us to reconstruct the solution from the transformed terms.

The approximate solution for the $n$ term can be expressed as follows:

$\begin{equation} g_n(\xi, \tau) = \sum\limits_{k = 0}^n g_k(\xi)\tau^{\alpha k}. \end{equation}$

(3.11)

Step 4. Final solution

The required solution of Eq (3.7) is given by

$\begin{equation} g(\xi, \tau) = \lim\limits_{n\rightarrow \infty} g_n(\xi, \tau). \end{equation}$

(3.12)

4. Application of FRDTM to the fuzzified GeoKdV equation

By incorporating fuzziness, we may express the time-fractional GeoKdV equation (Eq (1.3)) in an operator form as follows:

$\begin{equation} {D_t}^{\alpha}\tilde{\eta}-\omega\tilde{\eta}_{\xi}+\frac{3}{2}\tilde{\eta}\tilde{\eta}_{\xi}+\frac{1}{6}\tilde{\eta}_{\xi\xi\xi} = \tilde{0}, \ \ \ 0 < \alpha\leq 1, \end{equation}$

(4.1)

with ICs

$\begin{equation} \tilde{\eta}(\xi, 0) = \tilde{\eta}_0(\xi). \end{equation}$

(4.2)

Applying the FRDTM to Eqs (4.1) and (4.2), we arrive at the following recurrence relation:

$\begin{equation} \begin{cases} \begin{aligned} &\frac{\Gamma(1+k\alpha+\alpha)}{\Gamma(1+k\alpha)}\tilde{\eta}_{k+1}(\xi) = \omega\frac{\partial}{\partial \xi}\tilde{\eta}_k(\xi) \\ & -\frac{1}{6}\frac{\partial^3}{\partial \xi^3}\tilde{\eta}_k(\xi)-\frac{3}{2}\sum\limits_{i = 0}^k \tilde{\eta}_i\frac{\partial}{\partial \xi}\tilde{\eta}_{k-i}(\xi), \\ & \tilde{\eta}(\xi, 0) = \tilde{\eta}_0. \end{aligned} \end{cases} \end{equation}$

(4.3)

Now, using the DPF of Eq (4.3), we obtain the following:

$\begin{equation} \begin{cases} \begin{aligned} &\frac{\Gamma(1+k\alpha+\alpha)}{\Gamma(1+k\alpha)}\tilde{\eta}_{k+1}(\xi;p, r) = \omega\frac{\partial}{\partial \xi}\tilde{\eta}_k(\xi;p, r)\\ &-\frac{1}{6}\frac{\partial^3}{\partial \xi^3}\tilde{\eta}_k(\xi;p, r) -\frac{3}{2}\sum\limits_{i = 0}^k \tilde{\eta}_i\frac{\partial}{\partial \xi}\tilde{\eta}_{k-i}(\xi;p, r), \\ &\tilde{\eta}(\xi, 0;p, r) = \tilde{\eta}_0(\xi;p, r). \end{aligned} \end{cases} \end{equation}$

(4.4)

Solving the above for $k = 0, 1, 2, \ldots$ , we obtain the following:

$\begin{align*} \tilde{\eta}_1(\xi;p, r) = &\frac{1}{\Gamma(1+\alpha)}\Bigg[\omega \frac{\partial}{\partial \xi} \tilde{\eta}_0(\xi;p, r)-\frac{1}{6}\frac{\partial^3}{\partial \xi^3}\tilde{\eta}_0(\xi;p, r)\\ & -\frac{3}{2}\tilde{\eta}_0(\xi;p, r)\frac{\partial}{\partial \xi}\tilde{\eta}_{0}(\xi;p, r)\Bigg], \\ \tilde{\eta}_2(\xi;p, r) = &\frac{\Gamma(1+\alpha)}{\Gamma(1+2\alpha)}\Bigg[\omega \frac{\partial}{\partial \xi} \tilde{\eta}_1(\xi;p, r)-\frac{1}{6}\frac{\partial^3}{\partial \xi^3}\tilde{\eta}_1(\xi;p, r)\\ & -\frac{3}{2}\{\tilde{\eta}_0(\xi;p, r) \frac{\partial}{\partial \xi}\tilde{\eta}_{1}(\xi;p, r) \\&+ \tilde{\eta}_1(\xi;p, r) \frac{\partial}{\partial \xi}\tilde{\eta}_{0}(\xi;p, r)\}\Bigg], \\ \tilde{\eta}_3(\xi;p, r) = &\frac{\Gamma(1+2\alpha)}{\Gamma(1+3\alpha)}\Bigg[\omega \frac{\partial}{\partial \xi} \tilde{\eta}_2(\xi;p, r)-\frac{1}{6}\frac{\partial^3}{\partial \xi^3}\tilde{\eta}_2(\xi;p, r)\\ & -\frac{3}{2}\{\tilde{\eta}_0(\xi;p, r) \frac{\partial}{\partial \xi}\tilde{\eta}_{2}(\xi;p, r) \\&+ \tilde{\eta}_1(\xi;p, r) \frac{\partial}{\partial \xi}\tilde{\eta}_{1}(\xi;p, r)\\ & +\tilde{\eta}_2(\xi;p, r) \frac{\partial}{\partial \xi}\tilde{\eta}_{0}(\xi;p, r)\}\Bigg], \\ & \vdots \end{align*}$

Continuing with the process, the remaining components, including $\tilde{\eta}_4(\xi; p, r), \tilde{\eta}_5(\xi; p, r), \tilde{\eta}_6(\xi; p, r)$ and so on, may be evaluated. By combining these results, we derive the series solution of Eq (4.1) as follows:

$\begin{equation} \begin{aligned} \tilde{\eta}(\xi, \tau;p, r) = &[\tilde{\eta}_0(\xi, \tau;p, r)+\tilde{\eta}_1(\xi, \tau;p, r)\\&+\tilde{\eta}_2(\xi, \tau;p, r)+\ldots]\tau^{n\alpha}, \\ &\text{or}\\ \tilde{\eta}(\xi, \tau;p, r) = &\sum\limits_{n = 0}^{\infty}\tilde{\eta}_n(\xi, \tau;p, r)\tau^{n\alpha}. \end{aligned} \end{equation}$

(4.5)

5. Results and conversations

In this work, we obtained approximate solutions for $\tilde{\eta}$ using up to a four-term solution. This was done by considering the initial condition in the context of a TFN. Calculations up to the determination of $\tilde{\eta}_4(\xi, \tau; p, r)$ have been taken due to the complexity associated with presenting more terms. For an improved accuracy, one can calculate an increased number of terms as needed. It is important to note that the obtained solutions can be influenced by adjusting the parameters $p \text{ and } r$ along with the fractional order $\alpha$ . To demonstrate convergence, we produced plots and compared with existing solutions we have assumed the fractional order index with $\alpha = 1$ . Moreover, some graphs are generated with different values of $\alpha$ . Additionally, exploring different values of the parameters linked to the governing equation enables the effective computation of the desired solutions.

To execute our approach, we considered the following initial condition ^[5] in the context of a TFN:

$\begin{equation} \begin{aligned} \tilde{\eta}(\xi, 0) = &\tilde{v}\Bigg\{-\frac{1}{3}b^2+\frac{1}{3}c+\frac{4b^2}{1+da^{\xi-(\frac{c}{6}-\omega)\tau}}\\ &-\frac{1}{3}b^2\Bigg( \frac{1}{1+da^{\xi-(\frac{c}{6}-\omega)\tau}}\Bigg) ^2 \Bigg\}, \end{aligned} \end{equation}$

(5.1)

where

$c = \sqrt{(b^4+108C)}, b = \ln(a),$

with $a, d$ , and $C$ being arbitrary constants, whereas

$\tilde{v} = \Big( \frac{0.4}{3}, \frac{1}{3}, \frac{1.6}{3}\Big)$

is a TFN.

Additionally, the DPF can be applied to the provided initial condition to simplify the calculations. This leads to the following representation:

$\begin{equation} \begin{aligned} \tilde{\eta}(\xi, 0;p, r) = &\tilde{v}(p, r)\Bigg\{-\frac{1}{3}b^2+\frac{1}{3}c+\frac{4b^2}{1+da^{\xi-(\frac{c}{6}-\omega)\tau}}\\ &-\frac{1}{3}b^2\Bigg( \frac{1}{1+da^{\xi-(\frac{c}{6}-\omega)\tau}}\Bigg) ^2 \Bigg\} \end{aligned} \end{equation}$

(5.2)

where $p, r \in [0, 1]$ .

By applying the discussed FRDTM approach, the governing equation (Eq (4.1)) with the provided IC in Eq (5.2) can be solved. The solution is expressed in terms of $\tilde{\eta}(\xi, \tau; p, r)$ , thereby considering up to four terms. To obtain the solution and analyze its physical behavior, the following sets of data are considered:

$\begin{array}{l} \textbf{Set 1.}\ \omega = 1, a = 2, d = 1, C = 0.15. \\ \textbf{Set 2.} \ \omega = 1.2, a = 2, d = 0.5, C = 0.15. \\ \textbf{Set 3.}\ \omega = 1, 1.5, \text{ and } a = 2, d = 0.5, C = 0.15. \end{array}$

Now, when $\alpha = 1$ , the series solution

$\sum\limits_{n = 0}^{\infty}\tilde{\eta}_n(\xi, \tau;p, r) \tau^{n\alpha}$

efficiently converges to $\tilde{\eta}(\xi, \tau; p, r).$

From a graphical point of view, represents the convergence of the obtained fuzzy solutions which are in a triangular-shaped form. It can be observed that as the number of terms increases, the fuzzy approximate solutions demonstrate clear convergence. Some fuzzy solutions are obtained by taking different $\xi$ with a fixed time followed by the set of data 1, and this can be depicted in . To derive the lower and upper bound solutions from the obtained results, it is essential to substitute, $r = 0$ and $1$ , respectively, using $p = 0$ . Furthermore, setting $p = 1$ allows determination of the central/crisp value of the fuzzy solution. By performing these operations, we plotted a 2D figure () to illustrate the lower, middle, and upper solutions of the obtained results. Similarly, represents a 3D plot of the obtained fuzzy solution for $\tau = 0.5$ and $1$ .

Figure 2. Convergence of

$\tilde{\eta}$ for

$\alpha = 1$ at two different time levels for fixed

$\xi = 1.5$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. Plot of

$\tilde{\eta}$ for

$\alpha = 1$ fixed time

$\tau = 1$ and different

$\xi$ .

DownLoad: Full-Size Img PowerPoint

Figure 4. Lower, middle, and upper bound solution of obtained fuzzy solution

$\tilde{\eta}$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. 3D plot of

$\tilde{\eta}$ for two different time levels.

DownLoad: Full-Size Img PowerPoint

The validation phase involves the verification of the accuracy and the reliability of the obtained fuzzy solution by comparing it with established results. A detailed description is given below.

The fuzzy solution is inherently represented as a range of possible values due to the uncertainty modeled by the fuzzy set theory. To facilitate a meaningful comparison, we extract the central values from the fuzzy solution. These central values correspond to specific cases where the uncertainty is minimized, representing a crisp solution derived from the fuzzy framework. This extraction is crucial as it allows us to compare these specific results with existing classical solutions. In this regard, the central values obtained from the fuzzy solution are then compared to an existing solution using dataset 2. The detailed comparison is presented in , which allows for a direct numerical comparison at different time levels. The visual representation of this comparison for different time levels $\tau = 0.25, 0.5, 0.75,$ and $1$ can be observed in Figure 6, which illustrates a good convergence with ^[5].

Table 1. Term-wise solution obtained from the crisp case of the fuzzy solution vs. existing result ^[5].

$\tau=0.5$
$\xi$	Third term	Fourth term	Exact ^[5]
-5	0.356134	0.356137	0.356137
-4	0.364948	0.364954	0.364954
-3	0.387575	0.387588	0.387589
-2	0.44102	0.441032	0.441031
-1	0.542995	0.542953	0.542949
0	0.656964	0.656866	0.656868
1	0.663263	0.663319	0.663324
2	0.553581	0.55366	0.553656
3	0.447823	0.447823	0.447821
4	0.390691	0.390676	0.390676
5	0.366198	0.366189	0.36619
$\tau=1$
-5	0.356986	0.357007	0.357009
-4	0.367185	0.367236	0.36724
-3	0.393177	0.393281	0.393285
-2	0.453388	0.453481	0.453468
-1	0.562592	0.562257	0.5622
0	0.668483	0.667698	0.667739
1	0.650593	0.651041	0.651131
2	0.534035	0.534664	0.534608
3	0.435965	0.435967	0.435942
4	0.385426	0.385307	0.385309
5	0.364114	0.364044	0.364047

| Show Table

DownLoad: CSV

Figure 6. Comparison plot between existing solution and the crisp case of obtained fuzzy solution (i.e.,

$\eta$ ) at different time levels

$\tau = 0.25, 0.5, 0.75$ and

$1$ .

DownLoad: Full-Size Img PowerPoint

Furthermore, to enhance understanding, presents the absolute error plot using $\tau = 0.75$ , $\omega = 1.5$ , and other parameters from dataset 2. The central values of the obtained fuzzy solutions are used to calculate the absolute errors at each term's solution. As seen in the zoomed section of this figure, increasing the number of terms reduces the error. A similar pattern is observed in , where $\tau = 0.5$ and $\omega = 0.5$ are used for different $\xi$ values. For the purpose of this discussion, the fractional order $\alpha$ is taken to be $1$ . The comparison helps to assess how closely the fuzzy-based results align with traditional methods, thereby testing the accuracy and reliability of the fuzzy solution.

Figure 7. Plot of absolute error between obtained crisp solution and exact solution ^[5] for different number of terms.

DownLoad: Full-Size Img PowerPoint

Table 2. Term-wise absolute error at central values of obtained fuzzy solutions at

$\tau = 0.5$ ,

$w = 0.5$ and for various

$\xi$ values.

No.	$\xi$ values
No.	$\xi$ = 0	$\xi$ = 2	$\xi$ = 4	$\xi$ = 6	$\xi$ = 8	$\xi$ = 10
1	$0.5251 \times 10^{-4}$	$0.2224 \times 10^{-2}$	$0.1118 \times 10^{-2}$	$0.3349 \times 10^{-3}$	$0.8766 \times 10^{-4}$	$0.2217 \times 10^{-4}$
2	$0.5251 \times 10^{-4}$	$0.0313 \times 10^{-4}$	$0.1545 \times 10^{-4}$	$0.0572 \times 10^{-4}$	$0.0157 \times 10^{-4}$	$0.0040 \times 10^{-4}$
3	$0.0114 \times 10^{-6}$	$0.0427 \times 10^{-6}$	$0.0836 \times 10^{-6}$	$0.0605 \times 10^{-6}$	$0.0184 \times 10^{-6}$	$0.0048 \times 10^{-6}$
4	$0.2341 \times 10^{-6}$	$0.4636 \times 10^{-6}$	$0.1164 \times 10^{-6}$	$0.0094 \times 10^{-6}$	$0.0027 \times 10^{-7}$	$0.0082 \times 10^{-8}$

| Show Table

DownLoad: CSV

In addition, to investigate the behavior of the solution, a 3D plot is presented in along with 2D and 3D filled contour plots. These visualizations provide a clearer understanding of the solution's evolution over time and space, thus illustrating the dynamic nature of the model. Another key part of the validation phase involves analyzing how external factors, such as the Coriolis parameter, influence the solution. The plot showcased in demonstrates the effect of increasing the Coriolis parameter on the dynamic behavior of $\eta$ . By following the parameter regimes outlined in dataset 3, the observation indicates a decrease in the time required for the free surface elevation to approach zero as the Coriolis parameter increases ^[5].

Figure 8. (a) Surface plot; (b) 2d-filled; (c) 3d-filled contour plots of

$\eta$ for

$\alpha = 1$ .

DownLoad: Full-Size Img PowerPoint

Figure 9. Plot of

$\eta$ for different Coriolis constant

$\omega = 1, 1.5, 2$ .

DownLoad: Full-Size Img PowerPoint

Moreover, in the context of the fractional-order GeoKdV equation, displays a plot of the physical behavior and relationship between the exact solution of the integer-order GeoKdV equation ^[5] and the crisp case of the obtained fuzzy solution for different values of $\alpha$ . It can be observed that as the fractional order increases towards 1, the approximated special case fuzzy solution converges towards the integer-order exact solution, thus highlighting the consistency between the two approaches. Additionally, presents a 3D representation of the obtained solution for various values of the fractional order derivative $\alpha$ under the crisp scenario.

Figure 10. Comparison plot between existing solution and the crisp case of obtained fuzzy solution (

$\eta$ ) for different

$\alpha = 0.5, 0.75, 0.9,$ and 1.

DownLoad: Full-Size Img PowerPoint

Figure 11. Surface plot of

$\tilde{\eta}$ for different

$\alpha = 0.25, 0.5, 0.75,$ and

$1$ .

DownLoad: Full-Size Img PowerPoint

6. Conclusions

This article explored the application of the FRDTM to solve a time-fractional GeoKdV equation under fuzzy initial conditions. Specifically, triangular fuzzy initial conditions are considered, and the study observed that the approximate fuzzy series solution exhibited strong convergence as the number of terms in the series increased, demonstrating the method's robustness and reliability. Under crisp conditions, for the fractional order $\alpha = 1$ , the results produced by the FRDTM closely align with existing precise solutions ^[5], further affirming the accuracy of this approach. This consistency emphasizes the FRDTM's effectiveness in handling fuzzy and crisp cases. The graphical representations provide a visual insight into the solution's behavior, thus showing the impact of different parameter values. A significant advantage observed is that FRDTM avoids the complications of linearization or complex transformations, thus making the method computationally efficient.

Building on the findings of this study, future research could extend the FRDTM to other nonlinear fractional differential equations, particularly those that model complex physical phenomena in higher-dimensional systems. Additionally, incorporating more generalized forms of fuzzy uncertainty, such as trapezoidal or Gaussian fuzzy numbers, could offer a broader perspective on handling imprecision in real-world applications. Furthermore, the implementation of higher dimensional fuzzy numbers, viz., interval type-2 fuzzy numbers and generalised type-2 fuzzy numbers, can be one interesting topic to be considered in the study of various shallow water wave equations.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The first author would like to express his gratitude to the University Grants Commission (UGC), New Delhi, for providing fellowship for conducting research work.

Conflict of interest

The authors declare that they have no conflicts of interest.

References

[1]	M. C. Tsakiris, R. Vidal, Algebraic clustering of affine subspaces, IEEE Trans. Pattern Anal. Mach. Intell., 40 (2017), 482–489. https://doi.org/10.1109/TPAMI.2017.2678477 doi: 10.1109/TPAMI.2017.2678477
[2]	C. You, C. G. Li, D. P. Robinson, R. Vidal, Is an affine constraint needed for affine subspace clustering?, in Proceedings of the IEEE/CVF International Conference on Computer Vision, (2019), 9915–9924.
[3]	P. Ji, M. Salzmann, H. Li, Shape interaction matrix revisited and robustified: Efficient subspace clustering with corrupted and incomplete data, in Proceedings of the IEEE International Conference on computer Vision, (2015), 4687–4695.
[4]	J. Yang, J. Liang, K. Wang, P. L. Rosin, M. H. Yang, Subspace clustering via good neighbors, IEEE Trans. Pattern Anal. Mach. Intell., 42 (2019), 1537–1544. https://doi.org/10.1109/TPAMI.2019.2913863 doi: 10.1109/TPAMI.2019.2913863
[5]	A. Gruber, Y. Weiss, Multibody factorization with uncertainty and missing data using the EM algorithm, in Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (2004), 1–1. https://doi.org/10.1109/CVPR.2004.1315101
[6]	S. R. Rao, R. Tron, R. Vidal, Y. Ma, Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories, in 2008 IEEE Conference on Computer Vision and Pattern Recognition, (2008), 1–8. https://doi.org/10.1109/CVPR.2008.4587437
[7]	E. Elhamifar, R. Vidal, Sparse subspace clustering: Algorithm, theory, and applications, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2013), 2765–2781. https://doi.org/10.1109/TPAMI.2013.57 doi: 10.1109/TPAMI.2013.57
[8]	Z. Kang, G. Shi, S. Huang, W. Chen, X. Pu, J. T. Zhou, et al., Multi-graph fusion for multi-view spectral clustering, Knowl.-Based Syst., 189 (2020), 105102. https://doi.org/10.1016/j.knosys.2019.105102 doi: 10.1016/j.knosys.2019.105102
[9]	G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, Y. Ma, Robust recovery of subspace structures by low-rank representation, IEEE Trans. Pattern Anal. Mach. Intell., 35 (2012), 171–184. https://doi.org/10.1109/TPAMI.2012.88 doi: 10.1109/TPAMI.2012.88
[10]	J. Shi, J. Malik, Normalized cuts and image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 22 (2000), 888–905. https://doi.org/10.1109/34.868688 doi: 10.1109/34.868688
[11]	J. Guo, Y. Sun, J. Gao, Y. Hu, B. Yin, Low rank representation on product grassmann manifolds for multi-view subspace clustering, in 2020 25th International Conference on Pattern Recognition (ICPR), (2021), 907–914. https://doi.org/10.1109/ICPR48806.2021.9412242
[12]	W. B. Hu, X. J. Wu, Multi-geometric sparse subspace clustering, Neural Process. Lett., 52 (2020), 849–867. https://doi.org/10.1007/s11063-020-10274-z doi: 10.1007/s11063-020-10274-z
[13]	D. Wei, X. Shen, Q. Sun, X. Gao, Discrete metric learning for fast image set classification, IEEE Trans. Image Process., 31 (2022), 6471–6486. https://doi.org/10.1109/TIP.2022.3212284 doi: 10.1109/TIP.2022.3212284
[14]	G. Cheng, C. Yang, X. Yao, L. Guo, J. Han, When deep learning meets metric learning: Remote sensing image scene classification via learning discriminative CNNs, IEEE Trans. Geosci. Remote Sens., 56 (2018), 2811–2821. https://doi.org/10.1109/TGRS.2017.2783902 doi: 10.1109/TGRS.2017.2783902
[15]	K. Song, J. Han, G. Cheng, J. Lu, F. Nie, Adaptive neighborhood metric learning, IEEE Trans. Pattern Anal. Mach. Intell., 44 (2021), 4591–4604.
[16]	P. Ji, T. Zhang, H. Li, M. Salzmann, I. Reid, Deep subspace clustering networks, Adv. Neural Inf. Process. Syst., 30 (2017).
[17]	H. Wang, Q. Wang, Q. Miao, X. Ma, Joint learning of data recovering and graph contrastive denoising for incomplete multi-view clustering, Inf. Fusion, 104 (2024), 102155. https://doi.org/10.1016/j.inffus.2023.102155 doi: 10.1016/j.inffus.2023.102155
[18]	W. Wu, X. Ma, Q. Wang, M. Gong, Q. Gao, Learning deep representation and discriminative features for clustering of multi-layer networks, Neural Networks, 170 (2024), 405–416. https://doi.org/10.1016/j.neunet.2023.11.053 doi: 10.1016/j.neunet.2023.11.053
[19]	J. Xu, Y. Ren, G. Li, L. Pan, C. Zhu, Z. Xu, Deep embedded multi-view clustering with collaborative training, Inf. Sci., 573 (2021). 279–290. https://doi.org/10.1016/j.ins.2020.12.073 doi: 10.1016/j.ins.2020.12.073
[20]	H. Wang, W. Zhang, X. Ma, Contrastive and adversarial regularized multi-level representation learning for incomplete multi-view clustering, Neural Networks, 172 (2024), 106102. https://doi.org/10.1016/j.neunet.2024.106102 doi: 10.1016/j.neunet.2024.106102
[21]	Y. Yang, X. Ma, Graph contrastive learning for clustering of multi-layer networks, IEEE Trans. Big Data, 2023 (2023). https://doi.org/10.1109/TBDATA.2023.3343349
[22]	W. Guo, H. Che, M. F. Leung, Z. Yan, Adaptive multi-view subspace learning based on distributed optimization, Int. Things, 26 (2024), 101203. https://doi.org/10.1016/j.iot.2024.101203 doi: 10.1016/j.iot.2024.101203
[23]	Z. Li, Q. Wang, Z. Tao, Q. Gao, Z. Yang, Deep adversarial multi-view clustering network, in Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, 2 (2019), 4.
[24]	H. Ma, W. Wu, A deep clustering framework integrating pairwise constraints and a VMF mixture model, Electron. Res. Arch., 32 (2024), 3952–3972. https://dx.doi.org/10.3934/era.2024177 doi: 10.3934/era.2024177
[25]	Z. Chen, T. Xu, X. J. Wu, R. Wang, Z. Huang, J. Kittler, Riemannian local mechanism for spd neural networks, in Proceedings of the AAAI Conference on Artificial Intelligence, 37 (2023), 7104–7112. https://doi.org/10.1609/aaai.v37i6.25867
[26]	R. Wang, X. J. Wu, Z. Chen, C. Hu, J Kittler Spd manifold deep metric learning for image set classification, IEEE Trans. Neural Networks Learn. Syst., 35 (2024), 8924–8938. https://doi.org/10.1109/TNNLS.2022.3216811 doi: 10.1109/TNNLS.2022.3216811
[27]	T. Liu, Z. Shi, Y. Liu, Visual clustering based on kernel sparse representation on grassmann manifolds, in 2017 IEEE 7th Annual International Conference on CYBER Technology in Automation, Control, and Intelligent Systems (CYBER), (2017), 920–925. https://doi.org/10.1109/CYBER.2017.8446507
[28]	R. Wang, X. J. Wu, Z. Liu, J. Kittler, Geometry-aware graph embedding projection metric learning for image set classification, IEEE Trans. Cognitive Dev. Syst., 14 (2022), 957–970. https://doi.org/10.1109/TCDS.2021.3086814 doi: 10.1109/TCDS.2021.3086814
[29]	R. Wang, X. J. Wu, J. Kittler, Graph embedding multi-kernel metric learning for image set classification with Grassmannian manifold-valued features, IEEE Trans. Multimedia, 23 (2021), 228–242. https://doi.org/10.1109/TMM.2020.2981189 doi: 10.1109/TMM.2020.2981189
[30]	R. Wang, X. J. Wu, T. Xu, C. Hu, J. Kittler, U-SPDNet: An SPD manifold learning-based neural network for visual classification, Neural networks, 161 (2023), 382–396. https://doi.org/10.1016/j.neunet.2022.11.030 doi: 10.1016/j.neunet.2022.11.030
[31]	K. X. Chen, X. J. Wu, R. Wang, J. Kittler, Riemannian kernel based Nyström method for approximate infinite-dimensional covariance descriptors with application to image set classification, in 2018 24th International Conference on Pattern Recognition (ICPR), (2018), 651–656. https://doi.org/10.1109/ICPR.2018.8545822
[32]	T. Bendokat, R. Zimmermann, P. A. Absil, A grassmann manifold handbook: Basic geometry and computational aspects, Adv. Comput. Math., 50 (2024), 1–51. https://doi.org/10.1007/s10444-023-10090-8 doi: 10.1007/s10444-023-10090-8
[33]	D. Wei, X. Shen, Q. Sun, X. Gao, Z. Ren, Sparse representation classifier guided Grassmann reconstruction metric learning with applications to image set analysis, IEEE Trans. Multimedia, 25 (2022), 4307–4322. https://doi.org/10.1109/TMM.2022.3173535 doi: 10.1109/TMM.2022.3173535
[34]	B. Wang, Y. Hu, J. Gao, Y. Sun, B. Yin, Low rank representation on Grassmann manifolds, in Computer Vision–ACCV 2014: 12th Asian Conference on Computer Vision, (2015), 81–96. https://doi.org/10.1007/978-3-319-16865-4_6
[35]	X. Piao, Y. Hu, J. Gao, Y. Sun, B. Yin, Double nuclear norm based low rank representation on Grassmann manifolds for clustering, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2019), 12075–12084.
[36]	C. Zhang, Q. Hu, H. Fu, P. Zhu, X. Cao, Latent multi-view subspace clustering, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2017), 4279–4287.
[37]	R. Li, C. Zhang, Q. Hu, P. Zhu, Z. Wang, Flexible multi-view representation learning for subspace clustering, in Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, (2019), 2916–2922.
[38]	R. Zhou, Y. D. Shen, End-to-end adversarial-attention network for multi-modal clustering, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2020), 14619–14628.
[39]	Z. Kang, Z. Lin, X. Zhu, W. Xu, Structured graph learning for scalable subspace clustering: From single view to multiview, IEEE Trans. Cybern., 52 (2021), 8976–8986. https://doi.org/10.1109/TCYB.2021.3061660 doi: 10.1109/TCYB.2021.3061660
[40]	E. Pan, Z. Kang, High-order multi-view clustering for generic data, Inf. Fusion, 100 (2023), 101947. https://doi.org/10.1016/j.inffus.2023.101947 doi: 10.1016/j.inffus.2023.101947
[41]	J. Chen, S. Yang, X. Peng, D. Peng, Z. Wang, Augmented sparse representation for incomplete multiview clustering, IEEE Trans. Neural Networks Learn. Syst., 35 (2022), 4058–4071. https://doi.org/10.1109/TNNLS.2022.3201699 doi: 10.1109/TNNLS.2022.3201699
[42]	T. Chen, S. Kornblith, M. Norouzi, G. Hinton, A simple framework for contrastive learning of visual representations, in International Conference on Machine Learning, (2020), 1597–1607.
[43]	Y. Tian, C. Sun, B. Poole, D. Krishnan, C. Schmid, P. Isola, What makes for good views for contrastive learning?, Adv. Neural Inf. Process. Syst., 33 (2020), 6827–6839.
[44]	T. Wang, P. Isola, Understanding contrastive representation learning through alignment and uniformity on the hypersphere, in International Conference on Machine Learning, (2020), 9929–9939.
[45]	M. Harandi, C. Sanderson, C. Shen, B. C. Lovell, Dictionary learning and sparse coding on Grassmann manifolds: An extrinsic solution, in Proceedings of the IEEE International Conference on Computer Vision (ICCV), (2013), 3120–3127.
[46]	Z. Huang, J. Wu, L. Van Gool, Building deep networks on Grassmann manifolds, in Proceedings of the AAAI Conference on Artificial Intelligence, (2018), 1137–1145. https://doi.org/10.1609/aaai.v32i1.11725
[47]	A. Edelman, T. A. Arias, S. T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., (1998), 303–353. https://doi.org/10.1137/S0895479895290954
[48]	P. A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2009.
[49]	J. Hamm, D. D. Lee, Grassmann discriminant analysis: A unifying view on subspace-based learning, in Proceedings of the 25th International Conference on Machine Learning, (2008), 376–383. https://doi.org/10.1145/1390156.1390204
[50]	J. Hamm, D. Lee, Extended Grassmann kernels for subspace-based learning, Adv. Neural Inf. Process. Syst., (2008), 21.
[51]	M. T. Harandi, C. Sanderson, S. Shirazi, B. C. Lovell, Graph embedding discriminant analysis on Grassmannian manifolds for improved image set matching, in CVPR 2011, (2011), 2705–2712. https://doi.org/10.1109/CVPR.2011.5995564
[52]	Z. Huang, S. Shan, R. Wang, H. Zhang, S. Lao, A. Kuerban, et al., A benchmark and comparative study of video-based face recognition on cox face database, IEEE Trans. Image Process., 24 (2015), 5967–5981. https://doi.org/10.1109/TIP.2015.2493448 doi: 10.1109/TIP.2015.2493448
[53]	C. Cui, Y. Ren, J. Pu, X. Pu, L. He, Deep multi-view subspace clustering with anchor graph, preprint, arXiv: 2305.06939. https://doi.org/10.48550/arXiv.2305.06939
[54]	P. Xia, L. Zhang, F. Li, Learning similarity with cosine similarity ensemble, Inf. Sci., 307 (2015), 39–52. https://doi.org/10.1016/j.ins.2015.02.024 doi: 10.1016/j.ins.2015.02.024
[55]	J. Chen, H. Mao, W. L. Woo, X. Peng, Deep multiview clustering by contrasting cluster assignments, in Proceedings of the IEEE/CVF International Conference on Computer Vision, (2023), 16752–16761.
[56]	A. Asuncion, D. Newman, UCI Machine Learning Repository, 2007. Available from: https://ergodicity.net/2013/07/
[57]	J. Xu, Y. Ren, H. Tang, X. Pu, X. Zhu, M. Zeng, et al., Multi-VAE: Learning disentangled view-common and view-peculiar visual representations for multi-view clustering, in Proceedings of the IEEE/CVF International Conference on Computer Vision, (2021), 9234–9243.
[58]	H. Xiao, K. Rasul, R. Vollgraf, Fashion-mnist: A novel image dataset for benchmarking machine learning algorithms, preprint, arXiv: 1708.07747. https://doi.org/10.48550/arXiv.1708.07747
[59]	M. D. Addlesee, A. Jones, F. Livesey, F. Samaria, The ORL active floor [sensor system], IEEE Pers. Commun., 4 (1997), 35–41. https://doi.org/10.1109/98.626980 doi: 10.1109/98.626980
[60]	F. F. Li, P. Perona, A bayesian hierarchical model for learning natural scene categories, in 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05), 2 (2005), 524–531. https://doi.org/10.1109/CVPR.2005.16
[61]	U. Von Luxburg, A tutorial on spectral clustering, Stat. Comput., 17 (2007), 395–416. https://doi.org/10.1007/s11222-007-9033-z doi: 10.1007/s11222-007-9033-z
[62]	F. Nie, X. Wang, H. Huang, Clustering and projected clustering with adaptive neighbors, in Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2014), 977–986. https://doi.org/10.1145/2623330.2623726
[63]	H. Tang, Y. Liu, Deep safe incomplete multi-view clustering: Theorem and algorithm, in International Conference on Machine Learning, (2022), 21090–21110.
[64]	Y. Lin, Y. Gou, X. Liu, J. Bai, J. Lv, X. Peng, Dual contrastive prediction for incomplete multi-view representation learning, IEEE Trans. Pattern Anal. Mach. Intell., 45 (2022), 4447–4461. https://doi.org/10.1109/TPAMI.2022.3197238 doi: 10.1109/TPAMI.2022.3197238
[65]	J. Xu, H. Tang, Y. Ren, L. Peng, X. Zhu, L. He, Multi-level feature learning for contrastive multi-view clustering, in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, (2022), 16051–16060.
[66]	L. Van der Maaten, G. Hinton, Visualizing data using t-SNE., J. Mach. Learn. Res., 9 (2008).
[67]	Y. Cai, H. Che, B. Pan, M. F. Leung, C. Liu, S. Wen, Projected cross-view learning for unbalanced incomplete multi-view clustering, Inf. Fusion, 105 (2024), 102245. https://doi.org/10.1016/j.inffus.2024.102245 doi: 10.1016/j.inffus.2024.102245
[68]	K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, inn Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (2016), 770–778.

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