Title: Gaussian Processes for Shuffled Regression
Venue: NeurIPS 2025
Accepted: yes
Keywords: shuffled regression

gaussian processes
Abstract: Shuffled regression is the problem of learning regression functions from shuffled data where the correspondence between the input features and target response is unknown. This paper proposes a probabilistic model for shuffled regression called Gaussian Process Shuffled Regression (GPSR). By introducing Gaussian processes as a prior of regression functions in function space via the kernel function, GPSR can express a wide variety of functions in a nonparametric manner while quantifying the uncertainty of the prediction. By adopting the Bayesian evidence maximization framework and a theoretical analysis of the connection between the marginal likelihood/predictive distribution of GPSR and that of standard Gaussian process regression (GPR), we derive an easy-to-implement inference algorithm for GPSR that iteratively applies GPR and updates the input-output correspondence. To reduce computation costs and obtain closed-form solutions for correspondence updates, we also develop a sparse approximate variant of GPSR using its weight space formulation, which can be seen as Bayesian shuffled linear regression with random Fourier features. Experiments on benchmark datasets confirm the effectiveness of our GPSR proposal.

Gaussian Processes for Shuffled Regression

Abstract

Shuffled regression is the problem of learning regression functions from shuffled data where the correspondence between the input features and target response is unknown. This paper proposes a probabilistic model for shuffled regression called Gaussian Process Shuffled Regression (GPSR). By introducing Gaussian processes as a prior of regression functions in function space via the kernel function, GPSR can express a wide variety of functions in a nonparametric manner while quantifying the uncertainty of the prediction. By adopting the Bayesian evidence maximization framework and a theoretical analysis of the connection between the marginal likelihood/predictive distribution of GPSR and that of standard Gaussian process regression (GPR), we derive an easy-to-implement inference algorithm for GPSR that iteratively applies GPR and updates the input-output correspondence. To reduce computation costs and obtain closed-form solutions for correspondence updates, we also develop a sparse approximate variant of GPSR using its weight space formulation, which can be seen as Bayesian shuffled linear regression with random Fourier features. Experiments on benchmark datasets confirm the effectiveness of our GPSR proposal.

1 Introduction

The purpose of shuffled regression (SR) is to learn regression functions from shuffled data where the correspondence between the input features and the target response is unknown [1, 2, 3, 4]. This situation often arises with data collected independently from multiple devices/viewpoints [4, 5, 6] or when personal information must be anonymized for privacy reasons, such as in cases involving clinical history [7]. Since traditional supervised regression methods, collectively referred to as coupled regression (CR), rely on data with input-output correspondences (the pairs of features and responses) and so cannot deal with shuffled data, SR has become the focus of recent studies [8, 9, 10, 11, 12].

In SR literature, main focus is the use of linear models [1, 2, 4, 9] (an exception is [12] which uses neural networks as explained in §2). Accordingly, the use of Gaussian processes (GPs) [13] has not been well investigated. Considering the known benefits of GPs such as nonparametric flexibility and uncertainty quantification, efforts are needed to realize their benefits in SR.

This study proposes a probabilistic model based on GPs for SR called Gaussian Process Shuffled Regression (GPSR). By introducing GPs as a prior of regression functions in function space via the kernel function, GPSR can express wide variety of functions in a nonparametric manner while quantifying prediction uncertainty as shown in Table 1.

By use of the Bayesian evidence maximization framework and a theoretical analysis of the connection between the marginal likelihood/predictive distribution of GPSR and that of standard Gaussian process regression (GPR), we derive an easy-to-implement inference algorithm for GPSR using GPR as a subroutine; it iteratively applies GPR for the optimization of the kernel's hyperparameters and solves the quadratic assignment problem (QAP) for the updates of the input-output correspondence. Although this algorithm is valid for small-scale data, it seems prohibitive for handling medium/large-

Table 2: Classification of GPSR, GPR, Bayesian Linear Regression (BLR) and Bayesian Shuffled Linear Regression (BSLR). We establish the connection between these four methods.

scale data due to the difficulty of solving QAP (QAP is NP-hard in general) and the computation cost of GPR, which scales cubically with the number of samples. Therefore, we also develop a sparse approximate variant of GPSR called Sparse Spectrum GPSR (SS-GPSR) using the weight space formulation of GPSR. This can be seen as Bayesian Shuffled Linear Regression (BSLR) with the random Fourier features, and allows us to reduce both memory and computation cost and to use sorting operations to obtain closed-form solutions of correspondence updating. Note that both developing an algorithm for BSLR and clarifying the relations among four methods in Table 2 are also our contributions. Experiments on benchmark datasets confirm the effectiveness of our proposals.

2 Related Works

This work is positioned within the literature of shuffled regression and that of Gaussian processes. We describe below prior works in these research lines.

Background of shuffled regression (SR). SR [3, 4] which is also called 'regression without correspondence" [2, 14], 'uncoupled regression" [15, 16], 'permuted and unlinked regression" [9, 11], arises in various fields, including flow cytometry for measuring chemical characteristics of cells [4], image/point cloud registration [5, 6], and linkage of health records [7]. Thus various theoretical aspects and problem settings of SR have been studied. For example, Pananjady et al. show that recovering the correspondence between input and output is NP-hard in general [3]; the conditions under which unique solutions can be achieved have been explored [17]. Some existing works consider the setting with additional data such as pairwise comparison data [10, 15, 16, 18]. We focus on the setting where no additional data is, only the shuffled data.

Models and algorithms for SR. Earlier works on SR mainly use linear models [3, 4, 8, 9] and are categorized as shuffled linear regression (SLR). Only in recent years have approaches that utilize more flexible models, such as the neural-network-based method called shuffled deep regression (SDR) [12], been considered, and our study using GPs follows this research line. For comparison, see Table 1. For parameter estimation, variants of the expectation-maximization (EM) [19] are frequently used, e.g., [4, 9, 12, 20]. In contrast, our inference algorithm is derived from the Bayesian evidence maximization framework [21, 22] which has, up to now, not been used in the context of SR. Since the evidence penalizes overly complex models by integrating over parameters, this algorithm offers the way of hyperparameter tuning without validation data while still avoiding overfitting.

Gaussian processes (GPs). For supervised regression, Gaussian process regression (GPR) provides nonparametric flexibility and the quantification of uncertainty. However, naive GPR has limited practicality because its computation cost scales to the cube of the number of data points. For handling datasets with more than a few thousand points, various scalable approaches such as using inducing points [23, 24, 25, 26], grid points [27, 28], and spectral points or called random Fourier features (RFF) [29, 30] have been proposed. These so-called sparse GPs are still an active part of GP research e.g., [31, 32, 33]. Our study adopts the approach using spectral points/RFF [29, 30] to develop SS-GPSR, not only to reduce computation cost, but also because equations for updating the input-output correspondence can be obtained as closed-form solutions via a sorting operation.

3 Preliminaries

In this section, we provide definitions of shuffled data and its connection to coupled data. We also review Gaussian process regression (GPR) for 'coupled" data used in supervised regression. All symbols are listed in the Appendix A.

Definition of Shuffled Data. Let X ⊆ R D and Y ⊆ R be the input space and response space, respectively. Shuffled data D SD are defined by pairs of input-set X i = { x i 1 , x i 2 , · · · , x iL } and response-set y i = { y i 1 , y i 2 , · · · , y iL } , i.e., D SD = { ( X i , y i ) } i M =1 , where x iℓ ∈ X , y iℓ ∈ Y , L is set size 1 , and M is the number of pairs. Unlike 'coupled" data, the correspondence between input and response is unknown in shuffled data; the response corresponding to input x iℓ is one of the elements in response-set y i = { y i 1 , ..., y iL } , but it is not known which one. For simplicity, we define the total number of inputs/responses in D SD as N ≜ ML and denote all inputs and all responses as X = { X i } i M =1 , and y = { y i } i M =1 , respectively. Also we equate symbol y i with L -dimensional column vector y i = ( y i 1 , · · · , y iL ) ⊤ , and symbol y with N -dimensional column vector y = ( y ⊤ 1 , · · · , y ⊤ M ) ⊤ whose { ( i -1) L + ℓ } -th element is y iℓ .

Connection between Shuffled Data and Coupled Data. Shuffled data D SD are regarded as a general representation of coupled data D CD defined as pairs of input x i ∈ X and response y i ∈ Y , D CD = { ( x i , y i ) } N i =1 . This is because shuffled data with set size L = 1 are equivalent to coupled data. To distinguish it from shuffled data, bold symbols/vectors, etc. related to couple data will henceforth be underlined; we denote all inputs and all responses in D CD as X = { x i } N i =1 and y = ( y 1 , · · · , y n ) ⊤ , respectively. Note that early works on shuffled regression such as [2, 3, 4] focus on shuffled data where the number of pairs is M = 1 . In practical scenarios of data collection, data are often collected multiple times by repeated measurements or by changing the target group of users; even if the input-response correspondence is obscured we can know from which measurement/usergroups the data was collected, so we consider shuffled data where the number of pairs is M > 1 .

Gaussian Process Regression (GPR) for Coupled Data [13]. GPs are collection of random variables { f ( x ) | x ∈ X} for which any finite subset follows a multivariate Gaussian specified by mean function m ( x ) : X → R and kernel/covariance function k ( x , x ′ ) : X × X → R . GPR for coupled data uses (zero-mean) GPs with kernel k θ depending on hyperparameter θ as priors over function f , which is denoted as f ∼ GP (0 , k θ ( x , x ′ )) . Each response y i is assumed to be obtained by adding Gaussian noise to f i = f ( x i ) , i.e., y i = f i + ϵ , where ϵ ∼ N ( ϵ ; 0 , β -1 ) and β is the precision. Introducing function values vector f = (f 1 , · · · , f N ) ⊤ , this data generative process is summarized as p ( f | X ) = N ( f ; 0 , K ) and p ( y | f ) = N ( y ; f , β -1 I N ) , where K is the gram matrix whose ( i, j ) -th element is k θ ( x i , x j ) and I N is the N × N identity matrix. Figure 1a shows the graphical model representation of GPR. We get the (analytically-tractable) marginal likelihood by integrating over latent function values f , p ( y | X ) = ∫ p ( y | f ) p ( f | X ) d f = N ( f ; 0 , C θ ) where C θ = K + β -1 I N . Thus hyperparameter θ is estimated by the evidence framework [21, 22] 2 that maximizes the logarithm of marginal likelihood (evidence) defined as

Similarly, we get the predictive distribution of response y ∗ for test input x ∗ as follows.

where k is an N -dimensional vector defined by k = ( k θ ( x ∗ , x 1 ) , · · · , k θ ( x ∗ , x N )) ⊤ . Thus GPR can make predictions with information on confidence levels.

Spectral Representation of Shift Invariant Kernel. One bottleneck of GPs is that, in a direct implementation, the memory and computation requirements for inverting covariance C θ scale as O ( N 2 ) and O ( N 3 ) , respectively. Accordingly, Lazaro et al. [29] use the spectral representation or the

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so called random Fourier features (RFF) ψ : X → F ψ ⊆ R H that approximate the (shift-invariant) kernel function,

where H ′ = H/ 2 and s h is a D -dimensional vector sampled from the kernel's spectral density, e.g., N ( s ; 0 D , (4 π 2 Λ ) -1 ) for the (ARD) Gaussian kernel defined by k ( x , x ′ ) = σ 2 0 exp ( 1 2 ( x -x ′ ) ⊤ Λ ( x -x ′ ) ) . This, together with the weight-space-view formulation of GPR, enables us to create a sparse-spectrum approximation of GPR through Bayesian linear regression (BLR) on feature space F ψ ⊆ R H . This reduces the memory requirement to O ( NH ) since we no longer store the N × N covariance matrix C θ . As a result, the predictive distribution and the logarithm of marginal distribution for BLR, denoted by p BLR ( y ∗ | x ∗ , D CD ) and L BLR , can be computed with lower memory usage and computation cost. See Appendix B for thorough definitions and detailed explanations. Although any positive definite kernel can be adopted in our method, we assume that kernel k is shift-invariant when deriving the approximate variant of GPSR using RFF.

4 Gaussian Process Shuffled Regression (GPSR)

This section presents our probabilistic model based on GPs for shuffled regression (GPSR).

4.1 Generative Process and Marginal Likelihood of GPSR

GPSR is constructed by the following generative process of shuffled data D SD = { ( X i , y i ) } i M =1 . Similar to GPR [13] in the previous section, we put GPs prior on f , f ∼ GP (0 , k θ ( x , x ′ )) , and denote function values at i -th input-set X i = { x i 1 , · · · , x iL } as f i = ( f ( x i 1 ) , · · · , f ( x iL )) ⊤ . So N (= ML ) -dimensional vector f = ( f ⊤ 1 , · · · , f ⊤ M ) ⊤ is subject to the multivariate Gaussian given by

where K is the N × N gram matrix whose (( i -1) L + ℓ, ( i ′ -1) L + ℓ ′ ) -th-th element is k θ ( x iℓ , x i ′ ℓ ′ ) . For each sample i = 1 · · · M , using L × L latent permutation matrix 3 Π i , response-set y i = ( y i 1 , · · · , y iL ) ⊤ is determined by shuffling latent function values f i and adding Gaussian noise, i.e., y i follows p ( y i | f i ) = N ( y i ; Π i f i , β -1 I L ) . This yields the joint probability distribution of all responses y = ( y ⊤ 1 , · · · , y ⊤ M ) ⊤ given f as

From Eq. (4) and (5), we get the marginal likelihood for GPSR by integrating over function value f ,

Figure 1b shows the graphical model representation of GPSR. Note that the order of generating function values f and shuffling by Π can be exchanged; the identical marginal likelihood is derived even if we first shuffle inputs X and then generate the function values of shuffled inputs.

For further analysis of the marginal likelihood, we denote the logarithm of (6) as symbol L GPSR ,

and define pseudo coupled data (PCD) as follows:

Definition 4.1. (Pseudo Coupled Data) Given shuffled data D SD = { ( { x iℓ } L ℓ =1 , { y iℓ } L ℓ =1 ) } i M =1 and permutation matrices { Π i } i M =1 , we define the inversely permuted response ˜ y ⊤ = (˜ y ⊤ 1 , · · · , ˜ y ⊤ M ) as

We also denote the ℓ -th element of ˜ y i as ˜ y iℓ . Pseudo coupled data D PCD is defined as a set of N = ML pairs of input x iℓ with inversely permuted response ˜ y iℓ , D PCD = { ( x iℓ , ˜ y iℓ ) } M,L i,ℓ =1 .

Marginal likelihood of GPSR is equivalent to that of GPR (1) with pseudo coupled data.

Proposition 4.2. (Equivalence of Marginal Likelihood) L GPSR ( θ ; D SD ) = L GPR ( θ ; D PCD ) holds.

Proof. By the multiplicativity of determinant and the properties of permutation matrix ( Π ⊤ = Π -1 and | Π || Π ⊤ | = 1 ), we get

This result allows us to use GPR as a subroutine of the inference detailed in the next subsection.

4.2 Inference

Similar to GPR [13], we adopt the Bayesian evidence maximization framework [21] for estimating kernel hyperparameter θ and permutation matrix Π . To effectively utilize the Proposition 4.2, we adopt the alternating optimization scheme that iteratively updates θ and Π .

Optimization of θ . Proposition 4.2 allows us to employ exactly same approach as GPR when estimating hyperparameter θ . We update θ by applying an optimization routine such as scaled conjugate gradient (SCG) and L-BFGS using the following partial derivatives of L GPSR w.r.t. the hyperparameters,

Optimization of Π . Considering that permutation matrix Π is a discrete variable, we cast this as a known combinatorial optimization problem. By extracting the terms related to Π from L GPSR (Eq. (9)) with multiplication by -2 for simplicity, we get

where we used the trace trick ( z ⊤ Az = tr( Azz ⊤ ) ). We can regard U QAP as the objective function of the quadratic assignment problem (QAP), and so we can employ one of the existing QAP solvers [36] 4 . In the later experiments, we use simulated annealing , which is easy to implement and closely related to Markov Chain Monte Carlo (MCMC) used in existing shuffled regression methods [4, 20]. For more details on the inference procedure, see Appendix C. Note that QAP is a NP-hard problem and this procedure seems prohibitive when handling medium/large scale data; we derive an approximation algorithm for GPSR that can avoid solving QAP in §5.

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4.3 Making Predictions

Wederive the predictive distribution of y ∗ for input x ∗ in GPSR by considering the joint distribution of the ( N +1) -dimensional vector y ′ = ( y ⊤ , y ∗ ) ⊤ . Note that ( x ∗ , y ∗ ) is a 'coupled" pair not included in D SD as shown in Fig. 1b. We define the latent function value at x ∗ as f ∗ = f ( x ∗ ) and f ′ = ( f ⊤ , f ∗ ) ⊤ . From GPs prior on f , p ( f ′ | X ′ ) = N ( f ′ ; 0 ( N +1) , K ′ ) where K ′ is ( N +1) × ( N +1) gram matrix computed using X ′ = X ∪ x ∗ . From Eq. (5), and assuming p (y ∗ | f ∗ ) = N (y ∗ | f ∗ , β -1 ) similar to GPR, y ′ is determined by following p ( y ′ | f ′ ) = N ( y ′ ; Π ′ f ′ , β -1 I ( N +1) ) where Π ′ = diag( Π , 1) . Then, the marginal distribution of y ′ is given by

Note that S ′ can be cast in the following block form:

where k is the N -dimensional vector whose { ( i -1) L + ℓ } -th element is k θ ( x iℓ , x ∗ ) . The predictive distribution of y ∗ given shuffled data is the conditional Gaussian of (12); it is given by

Note that Π ⊤ = Π -1 is used in the above derivation. It is obvious, similar to Proposition 4.2, that the predictive distribution of GPSR is equivalent to that of GPR (2) given pseudo-coupled data. Proposition 4.3. (Equivalence of Prediction) p GPSR ( y ∗ | x ∗ , D SD ) = p GPR ( y ∗ | x ∗ , D PCD ) holds.

Figure 2 demonstrates the effectiveness of GPSR with the derived inference algorithm and predictive distribution for the simple toy problem. GPSR can output a prediction with its uncertainty and capture the true function as the sample size M is increased. However, the prediction and inference procedure presented in this section is prohibitive for handling medium/large scale data since it has two difficulties: (i) need to compute the inversion of the covariance matrix C θ at the cost of O ( N 3 ) operations (same as GPR), and (ii) need to solve QAP for estimating permutation matrix Π . The next section provides a sparse variant of GPSR that can resolve these two difficulties at once.

5 Sparse Spectrum GPSR (SS-GPSR): Approximation of GPSR using RFF.

The key idea behind our sparse variant of GPSR, sparse spectrum GPSR (SS-GPSR), is to use a weight-space-view of GPSR and random Fourier features (RFF) [29, 30]. Here we start with a model with a finite-dimensional feature map ϕ : X → F ϕ whose inner product corresponds to kernel function k . We then consider using RFF defined in (3) to approximate GPSR with a shift-invariant kernel that may not have a corresponding finite-dimensional feature map.

5.1 Weight Space View of GPSR: Bayesian Shuffled Linear Regression (BSLR)

We derive the predictive distribution and inference algorithm for SS-GPSR from the following Bayesian shuffled linear regression (BSLR) models defined as

where Φ i is an L × H design matrix whose ℓ -th row is ϕ ( x iℓ ) ⊤ . We can see that BSLR is a weightspace-view of GPSR as follows. Using the equivalent expression of (15) y i = Π i Φ i w + ϵ i where ϵ i ∼ N ( ϵ i | 0 L , β -1 I L ) , we have the following mean and covariance of y i and y j :

So the joint distribution of response-set y 1 , · · · , y M is given by zero-mean Gaussian with N × N covariance R = Π ( α -1 ΦΦ ⊤ + β -1 I N ) Π ⊤ ; Φ is N × H design matrix Φ = ( Φ ⊤ 1 , · · · , Φ ⊤ M ) ⊤ . Thus we can confirm that BSLR is a weight-space-view of GPSR since R is equivalent to the covariance S in GPSR (6) if the kernel is defined by inner product of the feature map, k ( x , x ′ ) = α -1 ϕ ( x ) ⊤ ϕ ( x ) .

5.2 Prediction and Bayesian Evidence Maximization Framework for BSLR

We derive the predictive distribution and inference algorithm in BSLR. From Bayes rule, p ( w |D SD ) ∝ p ( w , y | X )= p ( w ) ∏ M i =1 p ( y i | X i , w ) holds and thus a posterior distribution of w is given by

So we obtain the following predictive distribution p BSLR ( y ∗ | x ∗ , D SD )= ∫ p ( y ∗ | x ∗ , w ) p ( w |D SD ) d w and marginal likelihood L BSLR ( w ; D SD )=log ∫ p ( y , w | X ) d w (see Appendix D for derivation):

Similar to GPSR, the predictive distribution and marginal likelihood of BSLR are equivalent to those of BLR given pseudo-coupled data (proofs are given in Appendix E).

holds.

Proposition 5.1. (Equivalence of Prediction) p BSLR ( y ∗ | x ∗ , D SD ) = p BLR ( y ∗ | x ∗ , D PCD ) holds. Proposition 5.2. (Equivalence of Marginal Likelihood) L BSLR ( w ; D SD )= L BLR ( w ; D PCD )

We can also show the predictive distribution of BSLR (18) is equivalent to that of GPSR (14) using the connection between GPSR and GPR, GPR and BLR (Appendix B), BLR and BSLR.

Proposition 5.3. (Equivalence of Prediction) p GPSR ( y ∗ | x ∗ , D SD ) = p BSLR ( y ∗ | x ∗ , D SD ) holds if the kernel function k is given by k ( x , x ′ ) = α -1 ϕ ( x ) ⊤ ϕ ( x ) .

Proof. Applying Proposition 4.3, B.1 and 5.1 sequentially, we get p GPSR ( y ∗ | x ∗ , D SD ) = p GPR ( y ∗ | x ∗ , D PCD ) = p BLR ( y ∗ | x ∗ , D PCD ) = p BSLR ( y ∗ | x ∗ , D SD ) .

Proposition 5.3 states the predictive distribution of GPSR can be obtained via BSLR for kernels that admit finite-dimensional feature maps (regardless of whether the kernel is shift-invariant). For shift-invariant kernels that do not admit finite-dimensional feature maps, an approximate predictive distribution can be constructed using RFF ψ (3) as feature map ϕ . Therefore, we can more efficiently construct the (approximate) predictive distribution using BSLR when the number of samples N exceeds the dimension size H since the expression (18) involves the inverse of H × H matrix A instead of N × N gram matrix C θ .

5.3 Inference

By leveraging the result of Proposition 5.2, a simple inference algorithm that iteratively updates the posterior of w and Π can be derived, similar to GPSR. Thanks to the equivalence of marginal likelihood, we can estimate posterior of w by exactly same way of BLR (update ¯ w and A -1 by (17)). Also we can update Π by minimizing the trace term U LAP in (19) that can be seen as the objective of linear assignment problem (LAP). This is a simple instance of LAP, so the optimal permutation matrix Π i can be found by a sort operation in O ( L log L ) without using tailored algorithms for LAP [37, 38, 39] 5 . This is because the objective function U LAP ( Π ) = ( Π i v i ) ⊤ y i is maximized by setting the permutation matrix such that the ℓ -th largest element of response vector y i and that of the predictive mean vector v i ≜ Φ i ¯ w match each other for all ℓ . For more details on the inference procedure, see Appendix F.

Thus our SS-GPSR using the predictive distribution (18) and the above inference procedure reduces memory and computation costs associated with GPs and avoids solving QAP. Although existing studies also use sort operations [4, 12], to the best of our knowledge, this is the first to show that the permutation matrix can be estimated by sorting, even in the evidence maximization framework.

6 Experiments

6.1 Setting

Data. We evaluated GPSR and SS-GPSR using four publicly available data sets found in the UCI machine learning repository 6 : airfoil data ( Airfoil ), concrete compressive strength data ( Concrete ), Boston housing data ( Housing ), auto-MPG data ( MPG ). We also made Housing-1D with input feature dimension of D = 1 by extracting RM (average number of rooms) features for visualization purposes. We prepared 5 data sets by randomly dividing the data and using 60% for training, 20% for validation, and 20% for testing. We made the shuffled data used for training/validation by randomly dividing the training/validation data into M sets each with L elements (i.e., M ≈ 0 . 6 × original datasize /L ) and shuffling the indices in each set. Note that validation data were used only for neural network-based baselines as explained in next paragraph. The test data in the evaluations were coupled data. We used test mean squared error (test MSE) as the performance metric. We ran 5 trials using 5 sets of training, validation and test data.

Baselines. We compared (SS-)GPSR with linear-model-based method (SLR) [4] and the state-ofthe-art method that uses neural networks (SDR) [12]. Both methods use a (Gaussian-based) MSE loss and were trained by the (stochastic) sparse EM algorithm. SDR adopted early-stopping with validation data (which were also shuffled data) [42]. As oracle baselines, we also examined linearregression (LR), GPR, and deep-regression (DR) using coupled data with data size of N = LM . Both SDR and DR used a one-hidden-layer feedforward network with ReLU activation function.

Hyperparameters. GPSR (and Oracle GPR) used Gaussian kernels as in §3, and SS-GPSR used its RFF approximation with dimension size H = 100 in common for all datasets. The other hyperparameters such as precision α and β in (SS-)GPSR are estimated when applying GPR or BLR in inference procedure (see Alg. 1 and 2 in Appendix). So GPSR and SS-GPSR do not use validation data unlike SDR. Moreover, due to their slow convergence, both GPSR and SDR used warm initialization similar to [12], i.e., the initial value of permutation matrix is set using the estimated result of SLR. For SS-GPSR, ten different runs were performed with various initial permutation matrices, and the solution that maximized the marginal likelihood was selected. For more details of implementations and hyperparameters such as optimization setting for SDR, see Appendix G.

6.2 Results

Table 3 shows the results of the experiments. We observe that SDR, GPSR and SS-GPSR which can express nonlinear functions, outperform SLR in almost all settings. When comparing SDR and

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(SS-)GPSR, either of our proposals (GPSR or SS-GPSR) demonstrates the best performance across all datasets, except for Housing ( L = 16 ), confirming our methods' effectiveness. Note that GPSR and SS-GPSR perform similarly well across a comparable number of datasets, so we cannot judge which is superior in terms of prediction performance. From Fig. 3a, we can see that GPSR well captures the non-linear structure thanks to its nonparametric flexibility, and this contributes to its superior performance comparable to that of oracle methods for Housing-1D . Figure 3b illustrates that GPSR effectively captures the inherent uncertainty in the data, while Fig. 3c reports the negative log-likelihood on the test data, demonstrating the method's effectiveness in uncertainty quantification.

We also investigated the detailed behavior of SS-GPSR as illustrated in Fig. 4. In Fig. 4a, we observe that SS-GPSR with the random initialization scheme can identify a better solution than warm initialization (in terms of marginal likelihood) and converges within about ten iterations. Given that SDR requires a longer time to converge without warm initialization [12] 7 , these findings highlight

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the effectiveness of SS-GPSR. Figure 4b further explores the impact of the dimension size of RFF, H . It is shown that although MSE performance is degraded when H is smaller than 100 , it stabilizes when H exceeds 100 . Thus, SS-GPSR utilizing RFF works well without requiring sensitive tuning of the dimension size. These results support the usefulness of SS-GPSR.

7 Conclusion

In this study, we proposed GPSR and SS-GPSR for learning regression functions from shuffled data. By theoretically analyzing the connection between GPSR, GPR and their weight-space formulations, we derived the easy-to-implement inference algorithms for (SS-)GPSR, and confirmed their effectiveness by experiments on public benchmark datasets.

Limitation and Future Work. The proposed methods are the first to apply GPs for shuffled regression, and a promising direction for future work is to extend their applicability to higherdimensional and more complex settings. This may involve incorporating advanced techniques such as deep kernel learning [43], which combines neural networks to learn task-specific feature representations, as well as sparse GPs described in §2, which offer scalable solutions for large-scale problems. Additionally, exploring fully Bayesian approaches that introduce prior distributions over the kernel parameters and permutation matrix offers another promising avenue for future research, as similar techniques have been successfully applied in related domains [20, 33].

Societal Impact. Shuffled regression methods, including ours, are useful for analyzing e.g., independently collected data as mentioned in §1. However, it's important to note that coupled (test) data with input-output correspondence is necessary to evaluate the regression performance (That's why the experiments in this paper are based on benchmark data). When applying the method to real-world applications without coupled (test) data, it's crucial to carefully assess the performance of the estimated model when making predictions or decisions. Furthermore, the interpretation of input-output correspondences via latent variable representations requires careful consideration, as such usage falls outside our primary scope and may pose risks in scenarios involving privacy concerns.

Acknowledgments and Disclosure of Funding

We would like to thank the anonymous reviewers for their insightful comments and constructive feedback, which have helped improve the quality and clarity of this paper. This research was supported by the Human Informatics Laboratories at NTT, Inc. as part of the authors' regular employment responsibilities. The authors declare no competing interests related to this work.

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A Symbol Definitions

Table 4 shows the symbols used in our paper. Bold symbols/vectors related to couple data are underlined.

B Details of Bayesian Linear Regression for Coupled Data

Bayesian linear regression (BLR) with parameter w ∈ R H and feature map ϕ : X → F ϕ ⊆ R H is derived by considering the following prior distribution and model,

where α and β are hyperparameters. From Bayes theorem, p ( w |D CD ) ∝ p ( y , w | X ) = p ( w ) ∏ N i =1 p ( y i | x i , w ) holds and thus a posterior distribution of w given coupled data D CD is written as

where Φ is the N × H design matrix whose i -th row is ϕ ( x i ) ⊤ . We get the predictive distribution of y ∗ at test point x ∗ by integrating over parameter w :

Thus the memory required for Bayesian linear regression is O ( NH ) to store design matrix Φ . Also, the inverse of H × H matrix A can be computed in O ( H 3 ) by direct computation.

Remark. Note that ¯ w in (21) is both the mean and the mode of the posterior distribution p ( w |D CD ) , so ¯ w is equivalent to the penalized maximum likelihood/maximum a posteriori (MAP) estimator. In non-Bayesian/frequentist schemes, however, the predictive distribution is constructed by plugging the estimator into the model (without integration over a posterior distribution), i.e., p ( y ∗ | x ∗ , ¯ w ) = N ( y ∗ ; ¯ w ⊤ ϕ ( x ∗ ) , β -1 ) , and so the predictive variance is different from (22).

It is known that the predictive distribution of BLR (22) and that of GPR (2) are equivalent when the kernel function k is defined by the dot product of feature map ϕ .

Proposition B.1. (Equivalence of Prediction, e.g., [22]) p GPR ( y ∗ | x ∗ , D CD ) = p BLR ( y ∗ | x ∗ , D CD ) holds if the kernel function k is given by k ( x , x ′ ) = α -1 ϕ ( x ) ⊤ ϕ ( x ) .

We define the logarithm of marginal likelihood for BLR by symbol L BLR as follows:

Note that L BLR is an analytically-tractable quantity computed as

C Inference Procedure of GPSR

We detail the inference procedure of GPSR explained in §4.2. As stated, simulated annealing (SA) is used for solving QAP since it is easy to implement and closely related to Markov Chain Monte Carlo (MCMC) used in existing shuffled regression methods [4, 20]. Although not explicitly mentioned in the main text, we can estimate both the precision parameter β and the kernel parameter θ in the GPR subroutine. The equivalence of marginal likelihoods between GPSR and GPR (shown in Prop. 4.2) enables the use of existing GPR as subroutines, making the algorithm easier to implement.

Algorithm 1 shows the pseudo code that implements the inference procedure of GPSR. At initialization (line 1-3), permutation matrix Π is initialized. Note that, as reported in [12], convergence can be accelerated by warm-initialization, i.e., the initial setting of Π is set by using the result of e.g., SLR [4]. Then, the response vector of pseudo-coupled data (Definition 4.1) D PCD , ˜ y = Π ⊤ y , is set. At each (algorithmic) time step t , GPR-step (line 5-6) and Correspondence Update (CU)-Step (line 7-14) are iteratively conducted. As the GPR-step, the parameters of θ are optimized by applying GPR using pseudo-coupled data D PCD represented by X and ˜ y . Next, as the CU-step, permutation matrix Π is updated by randomly swapping its rows following SA. Let us denote the permutation matrices before and after random swapping as Π (old) and Π (new) , respectively. SA accepts this swapping following probability q :

where T is the temperature parameter. From the definition of q , the random swapping is accepted with probability 1 when the solution is improved. Otherwise, SA accepts the swapping with probability exp( -∆ /T ) . i.e., the solution is updated with a certain probability even when the solution becomes worse, which helps to escape from local optima solutions.

Note that the above algorithm can be extended or modified in various ways, such as using the sparse GP methods described in §2 [23, 24, 25, 26, 28, 33] for the GPR-step and using optimization methods other than SA shown in [36] for the CU-Step.

Algorithm 1 Inference Procedure of GPSR

Input: kernel function k (with hyperparameter θ, β ), shuffled data D SD , optimization parameter (e.g., # of iterations/steps T max , S max , initial temperature T init , and cooling rate γ )

Output: hyperparameter θ, β , response vector of pseudo coupled data ˜ y , permutation matrix Π

1: /* Initialization */

2: Randomly initialize Π (Optional: Π could be initialized by solving SLR).

3: Set the response vector of pseudo coupled data (Definition 4.1) as ˜ y ← Π ⊤ y .

4: for t = 1 to T max do

5: /* GPR-Step (Optimization of hyperparameter using partial derivative Eq. (10)) */

6: Update θ and β by SCG/L-BFGS (This could be done by e.g., GPR.fit( X , ˜ y ) in scikit-learn)

7: /* Correspondence Update-Step (Optimization of Π ) by simulated annealing*/

8: Set temperature T ← T init

9: for s = 1 to S max do

10: Randomly select i ∈ { 1 , · · · , N } and ℓ, ℓ ′ ∈ { 1 , · · · , L } ( ℓ = ℓ ′ ) .

̸

11: Swap { ( i -1) L + ℓ } -th row and { ( i -1) L + ℓ ′ } -th row of Π with probability q (Eq. (25))

12: Update temperature T ← γT .

13: end for

14: Update the response vector of pseudo coupled data, ˜ y ← Π ⊤ y

15: end for

D Details of Bayesian Shuffled Linear Regression (BSLR)

Here we derive BSLR's predictive distribution (18) and marginal likelihood (19). We start with the parameter posterior distribution as follows:

Derivation of posterior distribution p ( w |D SD ) (17) . From the BSLR model (15), the joint distribution p ( w , y | X ) is written as

By the trick of adding 0 = ¯ w ⊤ A -1 ¯ w -¯ w ⊤ A -1 ¯ w and I H = A -1 A , Eq. (26) is expanded into

Thus we get the posterior distribution (17) from p ( w |D SD ) ∝ p ( w , y | X ) ∝ exp {-1 2 ( w -¯ w ) ⊤ A ( w -¯ w ) } .

Derivation of predictive distribution p BSLR ( y ∗ | x ∗ , D SD ) (18) . From the property of marginal and conditional Gaussian distributions, we get

Derivation of logarithm of marginal distribution L BSLR ( w ; D SD ) (= log ∫ p ( y , w | X ) d w ) (19) . By the trick of adding 0 = ¯ w ⊤ A -1 ¯ w -¯ w ⊤ A -1 ¯ w and I N = Π ⊤ Π with the definitions ¯ w = β A -1 Φ ⊤ Π ⊤ y and A = α I H + β Φ ⊤ Φ , we can expand the final term of the argument of the exponential function in (27) as follows.

Substituting (29) into (27), we get the following expression of joint distribution:

From the property used for normalizing constant of Gaussian distributions, ∫ exp {-1 2 ( w -¯ w ) ⊤ A ( w -¯ w ) } d w = (2 π ) H 2 | A | -1 / 2 holds and thus we get

We get L BSLR ( w ; D SD ) (19) by taking the logarithm of (30). The expression of E ( ¯ w ) using trace function is obtained by the trace trick ( z ⊤ z = tr( zz ⊤ ) ).

E Connection Between BLR and BSLR

Here we show the connection between BLR and BSLR. Similar to the relationship between GPSR and GPR discussed in § 4, we can prove the equivalence of their predictive distributions and marginal likelihoods using the definition of PCD. We begin by presenting a proof about predictive distributions.

Proof of Proposition 5.1. We need to show the equivalence of the posterior mean of BSLR ¯ w (17) and that of BLR ¯ w (21) when pseudo coupled data D PCD are given. The design matrices ( Φ , Φ ) and posterior variances ( A , A ) are equivalent from the definition. So we get ¯ w = β A -1 Φ ⊤ Π ⊤ y = β A -1 Φ ⊤ ˜ y = ¯ w .

Marginal likelihood of BSLR (19) is also equivalent to that of BLR (23) with pseudo coupled data.

Proof of Proposition 5.2. Similar to the proof of Prop. 5.1, we can show the equivalence of the design matrices, posterior means and variances. Equivalence of the remaining terms can be shown as follows:

F Inference Procedure for SS-GPSR (BSLR with RFF)

We detail the inference procedure of SS-GPSR explained in §5.2. Although not explicitly mentioned in the main text, we can estimate both the parameter posterior and hyperparameters α, β in the BLR subroutine [22, 44]. The equivalence of marginal likelihoods between BSLR and BLR (shown in Prop. 5.2) enables the use of existing BLR as subroutines, making the algorithm easier to implement.

Algorithm 2 shows the pseudo code for inference procedure. Overall structure is analogous to Alg. 1. At initialization (line 1-5), permutation matrix Π and the response vector of pseudo coupled data (Definition 4.1) ˜ y is set. Also, vector s h used in RFF is sampled from the kernel's spectral density. At each (algorithmic) time step t , BLR-step (line 7-8) and Correspondence Update (CU)-Step (line 9-13) are iteratively conducted. As the BLR-step, posterior of the parameters w and hyperparameter α, β is update by applying BLR using pseudo coupled data represented by X and ˜ y . Next, as CU-step, permutation matrix ˜ y is updated by sort operation.

G Implementation Details of Experiments

This appendix explains the details of our experimental setting and implementation.

Dataset: As stated in the Experiments section, we used four publicly available data sets, Airfoil , Concrete , Housing , and MPG . The description of each dataset and the pre-processing we applied are as follows. Airfoil records 1503 samples obtained from a series of aerodynamic and acoustic tests of two and three-dimensional airfoil blade sections with 5 features including angle, velocity and so on. Concrete records 1030 samples of the compressive strength of concrete with 8 features including the amount of cement, water and so on. Housing records 506 samples of home prices (MEDV: Median value of owner-occupied homes) in Boston with 13 socioeconomic and environmental features such as average number of rooms per dwelling (RM). MPG records 398 samples of the city-cycle fuel consumption in miles per gallon of automobiles with 7 features including weight and horsepower. We excluded samples with missing values and converted the categorical feature into a one-hot vector for MPG . Input features were standardized for all data sets.

Oracle method (LR, GPR and DR): For oracle methods other than DR, we used scikitlearn 8 . Specifically, LR uses sklearn.linear_model.LinearRegression and GPR uses sklearn.gaussian_process.GaussianProcessRegressor with default settings (we adopted Gaussian kernel for GPR). Details of DR are presented in the next paragraph.

Neural network-based methods (DR and SDR): As stated in the Experiments section, we used a one-hidden-layer feedforward neural network with the ReLU activation function for neural networkbased methods (DR and SDR). Hyperparameters of these methods were set following [12]. The number of units was set to 20 for all problems. The parameters of DR were optimized using Adam [45] with a learning rate of 0 . 001 . The parameters of SDR were optimized by the stochastic sparse EM algorithm using Adam with a learning rate of 0 . 001 . The mini-batch size of DR and that of SDR were 32 and 32 /L , respectively. The maximum number of epochs was 2000 in common. We used

early-stopping with validation data (coupled data for DR and shuffled data for SDR) [42]. The above was implemented using PyTorch [46]. Experiments were run on a computer with Apple M1.

Proposed methods (GPSR and SS-GPSR): Here, we provide details of the proposed methods (GPSR and SS-GPSR). Their inference algorithms are summarized in Algorithm 1 (Appendix C) and Algorithm 2 (Appendix F), respectively. As mentioned in the Experiments section, GPSR utilized warm initialization, i.e., the initial value of the permutation matrix is set using the estimated result of SLR. For SS-GPSR, ten different runs were performed with various initial permutation matrices, and the solution that maximized the marginal likelihood was selected. For GPSR, the maximum number of iterations T max was set to 100 . The parameter for simulated annealing was configured with S max = 10 -1 N log 2 ( L ) , an initial temperature of T = 1 . 0 and a cooling rate γ = 0 . 99 . In the case of SS-GPSR, the maximum number of iterations T max was set to 20 . The dimension size of RFF H was fixed at 100 for all datasets. Note that for GPR and BLR steps in Algorithm 1 and 2, we used the scikit-learn analogous to oracle methods (BLR uses sklearn.linear_model.BayesianRidge ). So hyperparameters such as precision α and β in (SS-)GPSR are estimated when applying GPR or BLR. So GPSR and SS-GPSR do not use validation data unlike SDR.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel Gaussian Process-based framework for shuffled regression, termed Gaussian Process Shuffled Regression (GPSR), along with its sparse variant, SS-GPSR. The goal of shuffled regression is to learn a regression function from \"shuffled data\" where the correspondence between input features and target responses is unknown. GPSR leverages Gaussian Processes (GP) as a functional prior, employing kernel functions for nonparametric modeling and providing principled quantification of predictive uncertainty. The authors establish the theoretical equivalence between the marginal likelihood and predictive distribution of GPSR and those of standard Gaussian Process Regression (GPR). Building on this, they develop an alternating optimization algorithm that iteratively updates kernel hyperparameters (via standard GPR subroutines) and input-output correspondences (by solving a Quadratic Assignment Problem, QAP). To improve computational efficiency, the authors further propose SS-GPSR, which approximates GPSR using Random Fourier Features (RFF) to map the model into weight space, resulting in a Bayesian Shuffled Linear Regression (BSLR) formulation. This enables closed-form updates of the correspondence via sorting operations. Empirical results on multiple UCI datasets demonstrate that GPSR and SS-GPSR outperform existing shuffled regression approaches, particularly in modeling nonlinear relationships and quantifying uncertainty.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n(1) The proposed method is rigorously designed with clear theoretical underpinnings. In particular, Propositions 4.2 and 4.3 establish a theoretical equivalence between GPSR and standard GPR, and the experimental evaluation is comprehensive, involving multiple datasets and varying set sizes.\n\n(2) The work is the first to introduce Gaussian Processes into the shuffled regression setting and further develops an efficient sparse variant, SS-GPSR.\n\n(3) Shuffled regression has practical relevance in scenarios involving privacy-preserving data (e.g., medical records) and multi-source data integration. The adoption of nonparametric modeling enhances the expressive capacity of existing approaches in these settings.\n\nWeaknesses:\n\n(1) SS-GPSR updates the input-output correspondence via a sorting operation (§5.2), which is optimal only under the assumption of a strict rank alignment between the predicted means v_i and responses y_i. The paper does not analyze the robustness of this assumption under noise or model misspecification, nor does it quantify the impact of sorting errors on regression performance. Notably, the derivation in Appendix F omits consideration of distributional shifts in the noise.\n\n(2) The experiments are limited to low-dimensional UCI datasets (with a maximum feature dimension D=13), and the method's robustness in high-dimensional settings (e.g., image embeddings) or under complex noise structures (e.g., heteroscedastic noise) remains untested. Moreover, SS-GPSR exhibits noticeable performance variance at L=16 in Table 3 (e.g., MPG dataset shows a standard deviation of 3.03 in MSE), but the source of this instability is not discussed.\n\n(3) Implementation details of the simulated annealing algorithm in Appendix C are incomplete—for example, the temperature schedule T and the acceptance probability q are unspecified. Furthermore, the paper does not explain how the QAP solver avoids convergence to suboptimal local minima (§4.2).\n\n(4) Although SS-GPSR is claimed to reduce computational complexity, the paper does not provide runtime comparisons with existing baselines (e.g., SDR), nor does it analyze the joint impact of the RFF dimension H and the set size L on computational cost (§6).\n\nQUESTIONS:\nThe authors are encouraged to address the following critical issues through additional analysis or experiments, as they directly impact the paper’s overall evaluation:\n\n(1) SS-GPSR updates the correspondence matrix $\\Pi_i$ via sorting of the predicted means $v_i$ and observed responses $y_i$ (\\S5.2). However, when the underlying regression function is non-monotonic or the noise level is high, this procedure may yield incorrect pairings. A theoretical analysis is needed to specify the optimality conditions for sorting---e.g., under what smoothness or monotonicity assumptions Proposition 5.1 holds. In addition, synthetic experiments should be conducted to quantify how the sorting error rate and regression MSE vary with increasing noise levels, similar to Figure 2.\n\n(2) Current experiments are limited to low-dimensional tabular datasets. The effectiveness of GPSR and SS-GPSR in higher-dimensional feature spaces (e.g., $D > 100$) or under non-stationary conditions (e.g., covariate shift) remains unexplored. Please include experiments on at least one high-dimensional dataset (e.g., CIFAR-10 image embeddings) or temporally shuffled sensor data collected from heterogeneous devices. Comparative analysis with SDR in these settings would strengthen the empirical claims.\n\n(3) While SS-GPSR is claimed to be more computationally efficient than GPSR, no quantitative comparisons of runtime or memory usage are provided. Please include such metrics and analyze the computational trade-offs, particularly contrasting the theoretical complexities $\\mathcal{O}(N^3)$ (GPSR) and $\\mathcal{O}(H^3)$ (SS-GPSR).\n\n(4) In GPSR, the QAP subproblem is solved via simulated annealing (\\S4.2), a stochastic optimization approach that lacks direct theoretical grounding in Bayesian inference. This introduces a potential disconnect with the marginal likelihood maximization framework. Please discuss how simulated annealing affects convergence and posterior estimation—possibly in comparison with MCMC methods~\\cite{ref20}—or provide a theoretical or empirical guarantee of the monotonic increase of the marginal likelihood $\\mathcal{L}_{\\text{GPSR}}$ during the iterative process.\n\nLIMITATIONS:\nWhile the authors briefly acknowledge certain limitations and social implications in the conclusion section, several critical aspects require further elaboration:\n\n(1) GPSR assumes the underlying regression function follows a Gaussian Process (GP) prior, yet the paper does not discuss potential failure cases when this assumption is violated—for example, in the presence of heavy-tailed or non-Gaussian data distributions. It is recommended to include a synthetic experiment in the appendix that compares GPSR with quantile-based shuffled regression methods when the ground-truth function f deviates significantly from the GP prior.\n\n(2) SS-GPSR relies on Random Fourier Features (RFF), which inherently assume a stationary (translation-invariant) kernel (§3). However, the applicability of the method to non-stationary kernels is not examined, such as deep kernels [42]. The paper should clarify this kernel restriction and discuss possible extensions or limitations regarding kernel choice.\n\n(3) Although the conclusion mentions that prediction performance should be evaluated with caution, it does not address the concrete risks of incorrect input-output alignment, such as patient-label mismatches in medical records. It would be beneficial to discuss whether a confidence-based rejection mechanism is necessary for privacy-sensitive or high-risk applications, e.g., imposing an uncertainty threshold to discard low-confidence predictions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Gaussian Process Shuffled Regression (GPSR) and its scalable variant SS-GPSR for the problem of learning regression functions from shuffled data, where the input–output correspondence is unknown. The authors formulate a Bayesian evidence maximization framework in which both the regression function and permutation matrix are inferred. GPSR uses a Gaussian process prior, and inference alternates between Gaussian process regression and solving a quadratic assignment problem. To address the scalability challenges of GPSR, the authors develop SS-GPSR, a sparse approximation based on random Fourier features and a weight-space formulation. A key technical insight is the introduction of a latent variable $w$, which yields a marginal likelihood that matches that of GPSR at optimality but depends only linearly on the permutation matrix, enabling efficient updates via sorting. Experimental comparisons against baselines show that SS-GPSR achieves competitive accuracy with improved scalability.\n\nSTRENGTHS AND WEAKNESSES:\n#### **Strengths**\n\n* The paper presents a clear, principled formulation of shuffled regression using Gaussian processes and evidence maximization.\n* The derivation of SS-GPSR is elegant, with the introduction of the auxiliary variable $w$ allowing efficient optimization through a linear dependence on the permutation matrix.\n* The paper is clearly written and accessible, even through dense derivations.\n* The closed-form inference and uncertainty quantification offered by the GP-based models set them apart from purely neural alternatives.\n\n#### **Weaknesses**\n\n* While the mathematical development is thorough, the core approach ultimately reduces to kernel ridge regression with a permutation matrix estimated via alternating optimization. From this perspective, the methodological contribution may feel incremental, though well-executed.\n* Experiments are relatively limited in scale and do not fully evaluate how SS-GPSR behaves as dataset size or set size $L$ increases.\n* SS-GPSR does not optimize kernel hyperparameters ($\\theta$), and the impact of this design choice on performance is not explored.\n* No runtime comparisons are provided, even though efficiency is one of the main selling points of SS-GPSR.\n\nQUESTIONS:\n1. **Interaction of kernel and permutation optimization**\n   In Figure 2, GPSR is applied to a toy sine wave task. Can you provide insight into how optimization of the kernel hyperparameters and the permutation matrix interact? For example, if the data is highly shuffled, will the GP favor a short length scale and overfit? Could this lead to poor local optima?\n\n2. **Lack of kernel optimization in SS-GPSR**\n   SS-GPSR does not optimize the kernel hyperparameters $\\theta$. Is this the main price paid for scalability? In settings where tuning $\\theta$ is important for performance, how does this limitation affect results?\n\n3. **Convergence behavior of alternating updates**\n   Since $w$ and $\\Pi$ are each optimized globally in SS-GPSR, is there any convergence guarantee for the alternating procedure? Could the method oscillate between solutions with the same likelihood? Empirical observations or theoretical discussion here would be helpful.\n\n4. **Runtime and scalability experiments**\n   A central claim is that SS-GPSR is more efficient than GPSR and SDR. To support this, it would be helpful to include runtime-to-convergence plots for varying $L$, and wall-clock comparisons across methods.\n\n5. **Larger-scale experiments**\n   Current experiments are limited to relatively small datasets. If feasible, then I think evaluating performance at larger $N$ and especially larger $L$ would better illustrate the scalability of SS-GPSR and its competitiveness with neural methods like SDR.\n\n\nIf the authors can provide meaningful insight into the optimization dynamics (especially around kernel/permutation interactions), clarify the limitations of not tuning $\\theta$ in SS-GPSR, and supply runtime or larger-scale evaluations, I would be inclined to raise my score.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Gaussian Process Shuffled Regression (GPSR), a probabilistic model for learning regression functions when input-output correspondences are unknown, a setting referred to as *shuffled regression*. GPSR places Gaussian process priors over functions in the input space and iteratively recovers the correspondence using a Bayesian evidence maximisation framework. To address computational scalability, the authors propose a sparse variant (SS-GPSR) using random Fourier features, enabling closed-form correspondence updates via sorting. Theoretical connections to standard GPR and Bayesian linear models are established, leading to a principled and efficient inference scheme. Empirical results on benchmark datasets provide comparisons between (SS-)GPSR, classical shuffled regression methods (linear and deep shuffled regression), and oracle baselines using unshuffled data.\n\nSTRENGTHS AND WEAKNESSES:\nThe article is well-written and didactic, making it easy to follow throughout successive derivations. All involved methods are carefully detailed and re-explained in the Appendix for the most basic ones. The problem being addressed, extending GP regression to the case of shuffled measurements, is clearly positioned within existing literature. As a GP expert, I was unaware of this unusual *shuffled* context, which made it interesting to discover. Although the proposed method mainly builds on existing ideas, they are combined in a sensible way. In particular, Random Fourier Features (RFF) tackles both GP scalability and the quadratic assignment problem (QAP) at the same time, which is an elegant proposal. The experimental setting is convincing and effectively highlights the key properties of the methods (except for two important aspects, which are detailed in the *Questions* section). \n\nOverall, I believe that this article meets the threshold for publication and constitutes an incremental yet valuable contribution to both the GP and the shuffled regression literatures.\n\nQUESTIONS:\nThe paper carefully describes and discusses the key aspects we could expect in this framework. I only have two remaining questions after reading: \n\n- Could you provide an empirical comparison of training/prediction times for all methods involved in the experiments? As SLR is used as initialisation in GPSR, I suspect it's orders of magnitude quicker (which is fine, but it would be valuable to know the exact time/accuracy tradeoff). For problems with large values of L, we can observe that the difference between (SS-)GPSR and SLR decreases; therefore, discussing this aspect may be more fair and would provide more nuanced takeaways to readers. \n\n- Although it is a key desirable property of GP-based frameworks, no evaluation of the uncertainty quantification (UQ) is provided. I understand that competing methods do not provide off-the-shelf uncertainty measures for comparison. However, it would be valuable at least to evaluate whether UQ is well-calibrated for (SS-)GPSR, for instance, by computing the number of testing points effectively lying within a credible interval. If a correct UQ is preserved in the (SS-)GPSR framework, it would be a significant benefit compared with existing approaches, and this feature would deserve to be highlighted accordingly. On the other hand, if UQ is somewhat damaged during the inference procedure, it could be presented as a natural and valuable path for future work, which could be mentioned in the conclusion.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the shuffled regression problem from a Gaussian Process (GP) perspective. By introducing the GP, they additionally provide the uncertainty measure while improving the SOTA performance. Moreover, to make the method scalable, they introduce the spectral representation of the kernel to reduce the time complexity and memory consumption.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper clearly presents the idea of Gaussian Process Shuffled Regression (GPSR) and is overall well-written and easy to follow.\n2. It is the first to introduce Gaussian Processes (GPs) to the shuffled regression setting, addressing a novel and relevant problem.\n3. A scalable variant, BSLR, is proposed to address the computational limitations of GPSR.\n4. The paper demonstrates that GPSR and BSLR produce equivalent results under exact spectral approximation of the kernel, providing theoretical justification for the scalable method.\n5. The proposed method is evaluated on four real-world datasets and consistently outperforms existing state-of-the-art baselines.\n\nWeaknesses:\n1. Although the optimization details of GPSR are clearly described, the derivation or computation of the feature map $\\phi$ in BSLR is omitted, which negatively affects the clarity and completeness of the scalable method. It would also be helpful to include error bounds for the spectral approximation to better understand the conditions under which BSLR remains reliable.\n2. The SOTA baseline SDR is claimed to be a deep method, yet the implementation in the experiments uses only a single hidden layer. This may raise fairness concerns in the experimental comparison.\n3. The paper does not report training or inference time, despite the known scalability challenges of GP-based methods.\n4. There is no empirical scalability analysis. It would be valuable to see how the proposed method scales in terms of runtime, memory, and performance with increasing numbers of samples ($N$) or spectral features ($M$).\n5. All experimental datasets are low-dimensional, where a spectral dimension $H=100$ suffices. However, it remains unclear how $H$ needs to scale with higher-dimensional input, which is important for understanding the general applicability of the method.\n\nI would be inclined to raise my score if the authors address the following key concerns: (1) ensuring a fairer comparison by using deeper architectures for the SDR baseline as claimed; (2) reporting training and inference time as well as memory consumption; and (3) providing a clearer analysis or empirical evidence regarding the scalability of the proposed method, particularly in high-dimensional or large-scale settings.\n\nQUESTIONS:\nThe questions are listed in the weaknesses.\n\nBesides, I wonder what the computational complexity is with the spectral decomposition method. The authors only provided the memory complexity, but the computation complexity is missing.\n\nLastly, it seems that a deep version of the proposed method could be developed by replacing the kernel with a deep kernel, following the approach of Wilson et al. [1]. This extension may further boost performance and help adapt the method to high-dimensional or more complex data distributions. Exploring this direction, or at least discussing its feasibility, would enrich the contribution and position the work more clearly within the broader GP literature.\n\n[1] Wilson, Andrew Gordon, et al. \"Deep kernel learning.\" Artificial intelligence and statistics. PMLR, 2016.\n\nLIMITATIONS:\nThe scalability of the proposed method remains uncertain when applied to complex, high-dimensional datasets. In particular, it is unclear how the number of spectral features ($H$) must grow and how computational cost scales in such settings.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}