Distance-informed Neural Processes
Authors: Aishwarya Venkataramanan
Joachim Denzler
Venue: NeurIPS 2025 poster ()
Keywords: neural processes
Bi-Lipschitz regularization
Uncertainty estimation
uncertainty calibration
Out-of-distribution (OOD) detection
gaussian processes
stochastic processes
Primary area: probabilistic_methods
TL;DR: We introduce Distance-informed Neural Processes (DNP), a Neural Process variant with bi-Lipschitz regularization that preserves input geometry. DNP yields better-calibrated uncertainty & stronger OOD detection across regression + classification.

Abstract: We propose the Distance-informed Neural Process (DNP), a novel variant of Neural Processes that improves uncertainty estimation by combining global and distance-aware local latent structures. Standard Neural Processes (NPs) often rely on a global latent variable and struggle with uncertainty calibration and capturing local data dependencies. DNP addresses these limitations by introducing a global latent variable to model task-level variations and a local latent variable to capture input similarity within a distance-preserving latent space. This is achieved through bi-Lipschitz regularization, which bounds distortions in input relationships and encourages the preservation of relative distances in the latent space. This modeling approach allows DNP to produce better-calibrated uncertainty estimates and more effectively distinguish in- from out-of-distribution data.  Empirical results demonstrate that DNP achieves strong predictive performance and improved uncertainty calibration across regression and classification tasks.

Distance-informed Neural Processes
Aishwarya Venkataramanan
Joachim Denzler
Computer Vision Group
Computer Vision Group
Friedrich Schiller University Jena, Germany aishwarya. venkataramannnnuui- jenaad
Friedrich Schiller University Jena, Germany joachim denzler@uni jena.
Abstract
We propose the Distance-informee Neural Process (DNP), novel variant of Neural Processes that improves uncertainty estimation by combining global and distance- aware local latent structures. Standard Neural Processes (NPs) often rely on a global latent variable and struggle with uncertainty calibration and capturing lo- cal data dependencies. DNP addresses these limitations by introducing a global latent variable to model task-level variations and local latent variable to capture input similarity within a distance-preserving latent space. This is achieved through bi-Lipschitz regularization, which bounds distortions in input relationships and encourages the preservation of relative distances in the latent space. This modeling approach allows DNP to produce better-calibrated uncertainty estimates and more effectively distinguish in- from
data. Empirical results demon- strate that DNP achieves strong predictive performance and improved uncertainty calibration across regression and classification tasks.
Introduction
Deep neural networks have achieved remarkable success in fields such as computer vision S 60
natural language processing [57] [2], and reinforcement learning (9]. However, they often produce
overconfident and unreliable predictions, particularly when encountering data that fall outside the
Stochastic models such as Gaussian Processes (GPs) (6) address this limitation by defining prior over functions using kernel functions, which encode the assumption that similar inputs produce similar outputs. Kernel functions measure similarity using distance metric, assigning higher scores to inputs closer the training data. Consequently, when the model encounters data points that are far from the training set, the predictive distribution reverts to the prior. In many cases, this prior has high variance, indicating high uncertainty about the output. However, the practicality vanilla GPs is limited by their high computational cost, scaling cubically with the size of the training data and by their lack of flexibility in nigh-dimensional problems.
support of the training set 23
58 17
This paved way the development of Neural Processes (NPs) [2], that leverage meta-learning principles 220 to learn distribution over functions from observed data. Unlike traditional kerne-- based methods, NPs employ neural encoders infer a global latent variable that summarizes the underlying function. The NP then conditions its predictions on the inputs and the global latent variable, facilitating rapid adaptation to new functions. However, the reliance on a single global latent variable limits representational expressiveness, and standard NPs often produce uncertainty estimates that are not well calibrated [4 25 63
To address these limitations, we propose the Distance-informer Neural Process (DNP), which integrates both global and local priors for better uncertainty estimation and generalization. The global prior functions like standard NPs, capturing task-specific patterns. The local prior models distance relationships between observed data and test inputs through a latent space, introducing an inductive bias similar to GPs. However, neural networks used to parameterize these latent spaces can warp the input geometry causing similar points be mapped far apart and unrelated points 39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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to appear close together in the learned representation 36 55 69 To mitigate this, add bi- (b) Vanilla MLP embeddings
(a) Input
(c) Bi-Lipschitz MLP embeddings
1.0
10
1.0
0.5
0.0
0.0
-055
-1.0
-2.5
0.0
2.5
5.0
Figure 1: Visualization of input data and learned representations on a 2D synthetic data. (a) shows the original data. (b) shows representations from a 3-layer vanilla MI.P, where the input structure distorted. (c) shows representations from the MLP with bi-Lipschitz regularization, which better preserves the input structure and relative distances, important for computing reliable input similarity.
Lipschitz regularization on the local latent network's ' weight matrices [68]. This constrains each layer's singular values within fixed range, which minimizes distortions and encourages the latent space to approximately preserve input distances [36]. The distance- distance-preserring latent space is then used define local prior that encodes data similarity. As shown in Fig. the latent representations learned by vanilla neural network distorts input geometry, while bi-Lipschitz regularization helps preserve the input structure by minimizing such distortions_ By conditioning on both global and distance-aware local latent variables, DNP yields better-calibrated uncertainty estimates and improved distinction between in- and -i-or--distribution (OOD) data. We demonstrate its effectiveness in regression and classification through experiments on synthetic and real-world datasets. Code is available: https://gitthi com/ cvj erjena//a wnw.DNP...........a.a.aaaaaaa... git.
Background
Let D = (x;, y:) represent dataset consisting of input-outpul pairs, where E IR is input and RR is the corresponding output. NPs define family of conditional distributions over functions f E F, mapping x to y. The dataset split into a context set (xc, yc) = (x,, y;); where .M S N an arbitrary subset of D, and target set yr) all data points. The model uses the context set to create summary that conditions how it predicts the target outputs yr from the target inputs Xy. Different variants of NPs differ how this context- conditioning is performed and how the predictive distribution is defined.
y:) which includes
The Conditional Neural Process (CNP) IT models the conditional distribution as -Agg((Xe. (XeyNc)eE)) is a permutation -nvvariaa summary (e.g. mean) of the context points. Each (x.yye is first encoded by a neural network. after which Tc and Xy parameterize a factorized Gaussian distribution over yr. The model is trained by maximizing the log-likelihood of the target set conditioned on the context set. addition to the deterministic encoding of the context set, Neural Process (NP) [12] introduces global latent variable zg Rdd to capture task-level uncertainty. A separate neural network encodes (xc. yc) by first aggregating the context set into global summary Sc, which is then used to parameterize Gaussian distribution over zg. The decoder models the conditional distribution over yr using XT. Tc, and zg:
P(yrixc,> VV, p(yr/r(, xr), where
MYYT(T,, 1 p(yT|XT, zG) p(zc)ss) pacc
dz.
The Attentive Neural Process (AttnNP) 25 extends NP by replacing the shared context representa- tion re with a target-speeific representation r, computed using self-attention over the context and cross-attention with each target input x,. The decoder then conditions on Xt, I, and zg to model the target distribution.
For illustrative purposes, we used special case where the bi bi-Lipschitz bounds were set to 1, resulting in an isometric mapping.

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GLOBAL LATENT PATH
Moan
Context set
X
Y1
NN
X₂
NN
DECODER
X₃
NN
2
LOCAL LATENT PATH
Y₁
h(-)
X
U1
X
X2
U2
Cross-
Attention
X
NN
X
Ug
Y2
Y
Y
Yc
Bi-Lipchitz regularization
Figure 2: The encoder consists of global and local latent paths, each learning distribution over its respective latent variables. The local path incorporates bi-Lipschitz regularization to approximately preserve input distances to model similarity relations among the data. The decoder then conditions on the sampled latents and the target embedding predict the output.
While AttnNP uses attention capture input-specific patterns, it does explicitly preserve distance relationships among inputs. Its attention weights, computed over learned embeddings, may distort input relations 69 55), hindering the capture of local dependencies crucial for similarity- predictions. In contrast, while DNP also employs cross-attention to obtain target-specific latent variables, its key contribution is the construction regularization, from which attention weights are computed.
distance-awarr latent space via bi-Lipschitz
3 Distance-informed Neural Process
The Distance-informed Neural Process (DNP) is illustrated in Fig. 2 The encoder consists of both global and local latent paths. The global path models distribution over the global latent variable zG E Rdd conditioned on the context data. To capture local dependencies, the distance-informer local path models distribution over the target-specific latent variable z, Rd conditioned on the target input x, and context data. The decoder then uses both the latent variables for the predictions.
3.1 Global latent path
The global latent variable zα follows the standard formulation NP and AttnNP, where serves as function-level representation that captures global uncertainty. The context set (Xc,Yc) is first encoded into a summary representation Sc using mean aggregation over the encoded context pairs. This is further used to define the prior distribution as:
Pdo (zg Xx. yc) = -N(Ha, Eoc, (sc)),
where pea (-) and Σe. (+) are neural networks that parameterize the mean and diagonal covariance of the Gaussian distribution.
3.2 Local latent path
To derive the local latent variable, we encode the input points from both the context and target sets into low-dimensional latent representations using neural network Each context point x, E x is mapped to embedding u, h(xx), and each target input xT is mapped to h(x:),

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where ue Uy E I These representations are then used to model similarity relationships among the inputs via cross-attention.
Accurately capturing these similarity relations require that the latent embeddings preserve the geometric structure of the original input space. However, neural networks can distort this structure during encoding [55] 19), which compromises meaningful distance relationships in the latent space (see Fig. D. Such distortions can lead to two key issues: (I) over-sensittvity, where small perturbations in the input cause disproportionatel large changes in the latent space; and (2) feature collapse, where distinct inputs are mapped similar embeddings, thereby reducing the discriminability the learned representations.
Definition 1 (Distortion bounded mapping). A function h : X - U is said to be a distortion bounded mapping if there exist positive constants L1 and L2 such that, for all X1, X2 E X, the following bi- bi-Lipschitz condition holds [45 361:
L1 dx [1,, X2) S. (x(1) h(x2)) S L2 122dxx dx(x1,
(3)
Here, dx(x1, X2) denotes the distance between X1 and X2 in the input space X, and du (lx(h((()))) h(x2)) is the corresponding distance in the oulput (latent) space U.
The bi- bi-Lipschitz mapping in Eq. implemented by a neural network ensures that the latent space approximately preserves meaningiul distances from the input manifold. The upper Lipschitz bound limits the sensitivity of the mapping, preventing small perturbations in the input from causing dispro- portionately large changes in the latent space. Conversely, the lower bound enforces sensitivity to meaningful variations by ensuring that distinct inputs remain distinguishable after mapping. Together, these constraints yield an approximately isometric transformation, promoting latent representations that remain faithful to the underlying geometric structure of the data.
Some well-known methods for enforcing bi-Lipschitz constraints in Euclidean space include two- sided gradient penalties 16, orthogonal normalization 4 spectral normalization in architectures with residual connections
While these techniques are used in the context of GANs (39 16 and deep kernel learning (55). applying bi-Lipschitz regularization to NPs is, to our knowledge, novel. Among the existing approaches. spectral normalization offers the best empirical trade-off between stability and computational cost [55]. For architectures with residual connections, bounding the largest singular values of the weight matrices via spectral normalization (39 68) combined with Lipsecitz-continuuui activation functions is sufficient to achieve the -i-shiiccitz property of the network (2). However, for more general architectures without residual links, we extend this approach by explicitly constraining both the smallest and largest singular values of each weight matrix to directly control the network's s contraction and expansion behavior across layers.
36
39, and reversible models
Weight Regularization. A neural network typically consists of a series of L layers, where each layer applies a linear transformation followed by non-linear activation function a as g (x) a( ([x+b,)., To encourage bi- bi-Lipschitz behavior, both the linear transformation and the activation function must together form a bi bi-Lipschitz mapping. For the linear part, this achieved by bounding the maximum and minimum singular values of the weight matrices W. To ensure overall bi- bi-Lipschitz continuity. this bounding combined with bi-Lipsehitz activation functions such as Leaky ReLU 38, Softplus, ELU (6) PReLU [8]. etc which preserve both upper and lower Lipschitz bounds. Note that residual architectures using spectral normalization [2] have similar requirements, as they rely on Lipschitz- continuous activations to ensure the bi-Lipschitz property. Let of min and o denote the smallest and largest singular values of Wi respectively. These values control how much the input can be contracted or stretched by the linear transformation. Accordingly, we define the bi-Lipschitz regularization loss:
Coi-Lip m0 A1 - - o'min)
) + max d x2)
where 0 < A1 S. A are hyperparameters that control the lower and upper bounds of o' and of " respectively. These hyperparameters provide flexibility in managing the effective Lipschitz constants when combined with regularization techniques (e.g., Dropout, Batch Normalization). The overall Lipschitz constants of the function = gr. -T.-1 o 91 are governed by the product of the per-layer singular value bounds.
Note that computing the exact singular value decomposition (SVD) of each weight matrix can be computationally expensive. particularly for deep networks. For matrix W e 18xx the computa-

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tional cost of SVD is (mmnn pp'4)) which can be prohibitive for large and To obtain fast approximation of of max and o² quine we apply the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method [29 511. an iterative algorithm originally designed compute a few extremal eigenvalues and corresponding eigenvectors of large symmetric matrices. In our case, LOBPCG used to approximate the largest and smallest eigenvalues of the symmetric positive semi-definite matrix W,W, or W Wi, depending on which of the two is smaller in dimension. For iiterations, the total computational cost per layer (Tp))) where z 5 10. The approximate values of of and min are then obtained the square roots of these eigenvalues respectively.
Similarity relations. Once the latent space embeddings are obtained, we use cross-attention [57 to model the relationships between the context and target points. The latent embeddings of the context data u, serve as the keys, while the embedding of a target point u, acts as the query. The cross-attention weights are computed using Laplace attention follows:
u..l
exp
Vd..
0j
[|u.-u,r]]
Eeec exp
The attention mechanism assigns relevance score to each context point based on its similarity to the target point. Since the embeddings are constructed preserve the geometric structure of the input space, the resulting attention weights more accurately reflect the true proximity between points.
Local latent variables. The target-specific prior distribution for the local latent variable z, is Gaussian and follows similar construction to 37
paz (2 Xc, =N
Σ aapan (Xes Vc),
Σ explaf (Xe: Yel)
10 (3)) and ()) neural networks used parameterize the Gaussian distribution. The attention weight of control how much each context point contributes to the local latent distribution. When target point lies in an OOD region and far from the context data, the attention weights tend toward zero. As a result, the local prior distribution over Z; approaches standard normal, functioning non-informative prior.
3.3 Generative Model
Once the global and local latent variables are defined, we construct the decoder (likelihood model) that generates outputs Y1:N from inputs X1:N. This decoder network is conditioned on both the global latent zg and the local latent 21:N corresponding to X1:N, and is given by the conditional distribution Py(N1:N zg, Z1:N; XL:N). The complete generative process is given by:
//
Pxi: (y1:N)
polys/za, 2,, pe,, (z,(x,) ) pea (zg) dzi:N dzg i=1
Note that we choose factorized Gaussian priors for zg and each z; for simplicity. This choice does not limit the generality of our approach: any permutation-invariat distribution over the global and local latents could be substituted in its place.
Proposition 1. Eq. defines an exchangeable stochastic process by satisfying both exchangeability and marginal consistency, as required by the Kolmogorov Extension Theorem [41].
The proof is provided in Appendix C Intuitively, exchangeability holds because the likelihood and local priors factorize across data points, so permuting the input-output pairs does not change the joint distribution. Marginal consistency holds because integrating out subset of outputs and their corresponding latents simply removes those terms from the product, yielding a valid joint distribution over the remaining variables.
3.4 Training and Inference
Having defined the generative model, we aim to fit the model parameters to infer the latent variables, and to make predictions for new inputs x". However, exact inference using Eq. 7 is intractable.
5

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To address this, we employ variational inference, which involves maximizing the Evidence Lower Bound (ELBO) on the log marginal likelihood (28.. We introduce the variational distributions Ide (ZG) XT. Yr) and Qd (2, Ye) ()) to approximate the true posterior distributions over the global and local latent variables, respectively. The global variational posterior modeled as Gaussian:
qec (za yT) N (Mec (ST)) Egc (ST));
(8)
where Sy is the mean-aggregated summary of the target set, obtained in the same manner sc. The local variational posterior To, is also modeled as Gaussian with diagonal covariance, parameterized by neural networks:
do (t.) Xc) = (Nuo Ve,.. Xx.
(9)
The resulting ELBO, derived in Appendix is:
CELBO
Bg Rsy pog/plli|xx, 2z, za)l DK (uo (xo|xT, (xcl/xx (zalxc.
(10)
-DKL(96, (zi X4, NI, pe,, (2I
The final training objective of DNP is the sum of the ELBO loss and the bi-L.isschiz regularization loss:
L = LILOO + BLriLiip
(11)
where 3 is a hyperparameter that control the trade-off between data likelihood and weight regulariza- tion. For inference on unseen data points x", we use the posterior predictive distribution:
// pe (y" |za, z², x" "<"))or (z" (z"|x", xc, yc)Po. (xc.cc)po,.ccc// yc) dzg- (12)
p(y"|x", xx.,C)
Note that at test time, the learned prior networks Pe,, and pe, are used to generate the latent variables and for the prediction. The algorithms and computational complexity for training and inference is provided in Appendix. A.I and A.2
4 Related Works
Stochastic Neural Networks (SNNs) have been widely explored for capturing uncertainty in neural networks. Bayesian Neural Networks (BNNs) (3) introduce a prior distribution over weights and use Bayesian inference to estimate posterior distributions, and estimate uncertainty in deep learning models. However, BNNs are computationally expensive. leading the development of approximation techniques such as variational inference [15, Monte Carlo dropout (10) and deep ensembles (32). While BNNs approximate posterior distributions over weights, an alternative Bayesian approach is provided by Gaussian Processes (GPs)]66), which define distributions over functions directly. Unlike BNNs, GPs offer a non-parametric framework for capturing uncertainty, making them particularly appealing for small-date regimes. However, the cubic computational complexity of standard GPs limits their scalability, and they are less flexible in digeesiimniio data, motivating various approximations such sparse GPs 54 49 and deep kernel learning (67]
12 25 have emerged powerful class of models that combine the flexibility of deep learning with the uncertainty quantification of Bayesian inference. In addition to the NPs discussed in See. several other variants have been introduced model different inductive biases. Bootstrapping Neural Process (BNP) (34) improves uncertainty estimation in NPs by incorporating bootstrapping procedure to iteratively improve latent variable estimates. Convolutional CNP (ConvCNP) [14] and ConvNP [ introduce translation equivariance by leveraging convolutional neural networks, which improves generalization in tasks involving spatial dependencies such as time- series. Functional NP 337 leverages graph structures to model data dependencies, thereby obtaining more localized latent representations.
Neural Processes (NPs)
contrast, DSVNP 63 introduces hierarchical structure of global and local latent variables to capture uncertainty at multiple levels of granularity. Transformer Neural Processes 440 replace the standard NP encoder-decode with transformer, improving expressiveness while ensuring context invariance and target equivariance.
[64] extend NPs with

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multiple expert latent variables to better capture multimodal distributions. [52] reformulate context aggregation as probabilistic graphical inference. yielding mixture and robust Bayesian aggregation variants that improve OOD robustness. [62] address NP underfitting via a surrogate objective in an -ntirationmmmxxiizzaaiionn framework, leading to more accurate distribution learning. [33 propose decoupled encoder trained with contrastive objective, enhancing transferability and robustness across tasks. While these works focus on expressiveness and inference, DNP takes a complementary path by enforcing geometry- preserves input structure and improves uncertainty estimation and OOD generalization. Following 63], summary of selected NP variants is provided in Table [
regularization through bi-Lipschizzcoontrrinne latent path. This
Table 1: Comparison of different Neural Process (NP) variants.
NP Family
Recognition Model
Generative Model
Prior Distribution
Latent Variable
= 3().
Global
CNP
(Deterministic)
NP
(Zzc) Xr)
plyr (2,xr)
p(ac|Xe,3e)
Global
(Stochastic)
(x,,
p(y.
Local
ConvCNP
(Deterministic)
ConvNP
2, Gncp(xc.eexx)
p(y./z.)
Implicit via encoder
Local (Stochastic)
AttaNP
((Z XI.XT .r)
21,
p(zoxxc,yc)
Global (Stochastic)
f(xc. Xc)))
Local (Deterministic)
DSVNP
g(zG|xr (yT)
plr/za,22,80 2.)))
pizc XC.YC).
Global (Stuchastic)
o ig: X., Y.)
plz, zG.
Local (Stochastic)
DNP (Ours)
g(2c|XT, YT). (2,
ply:/ac:2, 2,
piz Xc: Ych,
Global (Stochastic)
piz; NC.
Local (Stochastic)
5
Experiments
We evaluate DNP both regression and classification tasks. For regression, consider 1D synthetic examples, as well as multi-dimennsonal tasks on several real-world datasets. Performance is assessed in terms of predictive quality and uncertainty calibration [31]. For classification, we assess the model on CIFAR- -10 and CIFAR-100 [30], evaluating classification accuracy, calibration performance, and OOD detection capabilities. The regression networks use 3-layer MLP, while a VGG-16 architecture 30 is used for feature extraction in classification. The data generation process for the experiments, model architectures used and the hyperparameters are provided in Appendix A
5.1 1D Synthetic Regression
In this experiment, we evaluate the methods on 1D functions generated from GP priors with three different kernels: (i) the Radial Basis Function (RBF) kernel, (ii) the Matérn-5/2 kernel, and (iii) the Periodic kernel. At each training iteration, kernel hyperparameters are randomly sampled to introduce functional diversity. During testing, the methods are evaluated 2000 function realizations, using the log-likelihood (LL) and Expected Calibration Error (ECE) [3 of the target points.
Table shows the results for evaluation the target set, excluding the context. Across all three kernel types, DNP achieves a superior lig-likelihood and lower ECE scores in majority of the cases compared to the other baselines. Evaluation on the context set, inference latency and noisy data is provided in Appendix [B.]] Figure 3 shows predictions from AttnNP and DNP on GP- sampled functions, evaluated both in-distribution ([-2, 2) and in OOD regions. While AttnNp often underestimates uncertainty. DNP generally better calibrated. DNP can occasionally be overconfident due the approximate nature of the bi-Lipschitz constraint. However, DNP achieves more reliable predictions, accurately modeling the target function in-distribution while expressing higher uncertainty in OOD regions.
5.2 Synthetic to World Regression
Following 14, we consider the Lotka-Volterra model 65 which captures predator-prey population dynamics. Models are trained on simulated data generated from the Lotka-Voltert equations and
7

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Table 2: Performance comparison for ID synthetic regression. Results are averaged over 10 runs.
Matém-
Periodic
Method
LL-(1)
ECE (1)
11.(1)
ECF (
T1.(f)
ECE (
CNP 11
0.490=0.000 006
0.122=00.061
0.419:00004
0.088:20.047
0.239=0.001
0.120=0.029
NP 02
0.480+0.000
0.102=0.062
0.318+0.008
0,111=0.053
0.228=0.002
0. 0.145=0.047
ConvCNP
0.496--0.007
0.118-00.04
0.329+00001
0.335+0.002
0.106++0.032
ConvNP
g
0.190.0.0074 0.009
0.31310.002
0.137.10.095
0.43110.003
0.11711.011
AmmNP
0.578=0.003
0.103=0.026
0.49910.009
0.078:10014
0.457=0.001
0.098:10038
DSVNP
66
0,582=0.000 0.002
0.10120.052
0.512=0.006
0.101=0.012
0.412=0.002
0.092=0.042
NP-mBA
0.521+0.005
0.108-00.012
0.324+000007
0.091++0.021
0.375-0.005
0.091+0.010
Ours
0.696.00.00
0.093.10.054
1.62010.002
0.083.10.11
1.635_0.001
0.074.1.002
(a)
(b)
(c)
(d)
1 AttnN
Ours
Figure 3: Predictive distributions for sample realizations from the GP kernel. The model is trained on inputs from the region [2.2]. The orange curve represents the ground truth, and the green points are the observations. The blue curve shows the model's mean prediction. Shaded regions indicate the +:3 standard deviation confidence interval.
evaluated on real-world data from Hudson's Bay hare-lynx data [35]. presenting distribution mismatch scenario. For each training batch, the number of context points is sampled uniformly from the interval [3, 50], while the number of target points fixed 50. During testing, 30 context points are randomly selected from the training set. In addition to NPs, we also include an exact GP with RBF kernel as baseline. Table 3 reports log likeelioloood scores on both simulated and real data. Results indicate that DNP achieves better performance over the other baselines. Interestingly, the exact GP performs worse than the NP variants, which can be attributed the limited context points. In contrast, NP-based methods are able to more efficiently incorporate context information, resulting higher-quality predictions.
5.3 Multi-output Real- World Regression
To assess the performance of DNP more complex and practical scenarios, we follow
[x] and eval-
uate the models on the SARCOS [6]] Water Quality (WQ) ( and the SCM20D (50) datasets. The SARCOS dataset contains 48.933 samples, each with 21 input features and 7 output variables. The Water Quality (WQ) dataset includes 1,060 samples with 16 input features and 14 output variables. The SCM20D dataset comprises 8,966 samples, with 61 inputs and 16 outputs. Performance mea- sured using Mean Squared Error (MSE) and ECE scores. The results are provided in Table 4 which shows that DNP achieves superior predictive performance in most cases and provides consistently improved uncertainty calibration across all three datasets.
Table 3: Log-likelihood values for the predator-prey experiment. Results are averaged over 10 runs.
Method
Gi(Exact)
CNP
NP
ComNN
AttaNP
DSVNP
Ours
Simulated
0.952=0.024
1.293-10.034
1.382-0.036
1.608
1.590-100119
1.50740.027
1.712:10.029
Real
.961240.018
6.43
5.207=
-. 4.103
0.003-10.022
1.897-0.031
0.027

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Table Prediction performance and uncertainty calibration on multi-output real-world regression datasets, averaged over 10 runs.
Method
Metries
GP (exact)
CNP 11
NP 12
AttnNP 25
DSVNP 63
Ours
MSE (
0.998
0.020
1.093:0.029
1.004
0.938:10.018
0.886 0.012
SARCOS
ECB (
0.102+0.012
0.113=0.013
0.105=0.018
0.108=0.015
0.102:+00012
0.013=0.011
MSE (3)
0.721
0.739=0.010
0.724=0.005
0.692=0.005
0.723=00003
0.715=0.006
WQ
ECE (1)
0.089=0.007
0.087=0.009
0.100=0.013
0.126=0.008
0.108:10.012
0.084
MSE (L)
0.932-00.088
0.956-0.004
0.962-00.05
0.936+0.003
0.912-400004
0.887 0.007
SCM20D
ECE (1)
0. 0.109=0.008
0.124=0.005
0.131=0.003
0.122=0.007
0.1 0.110000002
0.088=0.003
5.4 Image Classification
We evaluate the performance of DNP on image classification using the CIFAR-10 10 and CIFAR- 100 datasets. The models are assessed in terms of classification accuracy, ECE, and the predictive entropy. Additionally, we evaluate OOD detection performance using the Area Under the Reecil curve (AUPR) between the IID and the OOD probabilities. For classification using CIFARTO, we use the SVHN [46), CIFAR100 and the TinyImageNet (33] as the OOD datasets. For CIFAR 100, the OOD datasets are SVHN, CIFARIO and Tiny ImageNet. The VGG16 model [48] is used as the feature extractor for all the methods. In addition to the NP-based baselines, we also provide the results for a VGG-16 classifier trained using cross-entropy loss, referred to as "Deterministic".
Table 5: Image classification using CIFARIO (ID) vs SVEN/CIFAARIO///inyIImnnggg (OOD).
Results are averaged over 10 runs.
Dataset
Metries
Deterministic
CNP
NP
AttNPP
DSVNP
Ours
Accuracy (f)
92.08=10.01
89.72=00.2
90.86-+0005
90.37=0.01
90,41=0.03
91.97=+001
CIFARIO
ECE (L)
0.032=0.002
0.034:0.001
0.025=:0.011
0.020:10.000
0.018:0.001
0.011=0.002
Entropy (L)
0.122=00001
0.196::00011
0.179110000
0,237110.000
0,201 10.001
0.209=0.000
Latency (ms) (L)
3.679
6.011+0.113
8.207++0.135
8,432+0.109
6.716+0.122
AUPR (t)
76.34+-0.01
89.68:0.02
35.66+00001
93.69=0002
93.42=0.01
97.05-0.01
SVHN
Entropy (1)
0.134=0.006
1.02==0.006
0.95110.002
1.41::0.001
1816:200002
0.279
AUPR (f)
80.96:10.00
87.23:10.01
87.33
87.11:0.001
8.014=0.01
90.58 0.00
CIFARIOO
Entropy (1)
0.173=00.005
0.790:10.004
0.735:100001
0.8911000001
0.912100001
1.167=0.001
AUPR (f)
74.72:10.01
85.7420.01
86.4811001
86.34=0.01
87.23=0.01
90.04:0001
TinyImageNet
Entropy (1)
0.168=0.005
0 0.800000003
0.761+0.002
0,900+000000
0,912+00001
1.19:=0.001
Table 6: Image classification using CIFAR100 (ID) vs SVHN/CIFAR IVENICIFARRO/TTiniinggeeee (OOD). Results are averaged over 10 runs.
Dataset
Metries
Deterministic
CNP
NP
AttaNP
DSVNP
Ours
Accuracy (t)
70.02:1000
68.74:1005
69.26:10.01
69.30
69.28=0004
69.3 09.00:10.04
ECE (1)
0.136=0.002
0.103=00001
0.119+0.001
0.118+00.09
0,104+0.002
0.082
CIFARI00
Entropy (L)
0.258-0.001
0.834:10001
0.80511..000
0.6631100011
0.672.10.007
0.677:10.11
Latency (ms) (L)
3.662-0.110
3.825+0.116
6.228++.1117
8.424+0.117
8.637400.25
6.876+0.118
AUPR (f)
75.32+10.0
84.62:10:11
85.3410.01
87.01=0.01
86. 86.28=0.01
89.92:10.01
SVHN
Entropy (1)
0.298=00003
L424:000033
0.937110.03
1.221-10.002
1.292:1.001
3.494=0.002
AUPR (1)
70.64+00.02
73.49+00011
76.32+0.01
69.57-00011
76.33-0.01
80.56-+0.01
CIFAR1O
Entropy (1)
0.282=0.003
1.515=0.002
L.533+0.003
1.225++0.003
1.232:+0.004
0.548=0.003
TinylmageNet
AUPR (f)
72.73:10.01
74.84100001
75.46:0001
73.41=0.01
76.82=0001
80.09:10.01
Entropy (1)
0.306=0.003
1431-10.002
1.534:10.003
1.206
1.291 1.0003
2.441 0.003
Table shows the results for classification uisng CIFARIO. From the results, DNP achieves the least ECE score, and closely follows Deterministic in terms of accuracy. DNP incurs higher inference latency per instance compared to CNP and vanilla NP. However, when compared AttnNP and DSVNP, DNP achieves a faster inference by avoiding self- over the context set, instead relying on a bi-Lipschitz regularized latent space to preserve the relational structure. Interestingly, the in-distribution entropy of DNP is slightly higher than that of the vanilla NP. This could be attributed to DNP's s more expressive uncertainty modeling, where the combination of global and local latent variables introduces additional variance to capture the predictive uncertainty. Consequently, DNP also shows increased entropy on the OOD datasets, allowing to more effectively distinguish ID from OOD samples. This leads to improved performance in OOD detection, as reflected in its higher AUPR

---

scores compared to other methods. A similar trend is observed in the classification performance on CIFAR-100 as shown in Table 6 DNP achieves a lower ECE, while also outperforming all baselines in OOD detection.
To evaluate the contribution of dif- ferent regularization methods aimed controlling the Lipschitz constant of the network, we compare DNP variants using the 2-sided gradient penalty [16], orthogomal regulariza- tion and spectral regulariza- tion 68 for CIFAR1O (ID) vs CI- FARTOO (OOD) classification. The results in Table that while all regularization methods improve ECE and AUPR scores compared to the no- regularization baseline, bi-Lipschitz weight regularization consistently outperforms the others. No- tably, both spectral and bi-Lipschitz regularization yield substantial reduction in ECE, with the latter achieving the lowest ECE and highest AUPR scores. contrast, methods like orthogonal regularization are more restrictive, as they impose stronger constraints on the model's 's weight space, which may limit representational flcxibility. Additional ablations on the influence of the number of context points, dimensions of d,, and dz, ) A2 values for the bi-Lipschitz regularization, trade off parameter / and comparison with dot-product attention are provided in Appendix B.2
Table 7: Comparison of regularization methods DNP on accuracy, calibration (ECE), and OOD detection (AUPR).
Results are averaged over 10 runs.
Regularization
Accuracy (1)
BCB (U)
AUPR (t)
DNP w/ reg. 2-sided Gradient Penalty
92.08+0.01
0.042++0001
87.85++0.01
88.92:10.01 0.032+10000 90.90:0.02
Orthogonal reg.
90,96.10.01
0.034.10.000
88 88.64.10.01
Spectral Norm reg. Bi-Lipschitz reg. (ours)
91.42:10.01
0.0151100011
89.7341.01
91.97=0.01
0.011=20.002
90.58:0.01
Limitations and Future Work
The current bi-Lipschitz constraint is applied in Fuclidean space, which may fully capture complex semantic relationships. Extending this constraint to incorporate more flexible distance measures or task-specific similarity metries (69] could improve generalization across diverse domains. Moreover, the use of cross-attention introduces additional computational overhead, which may limit offers a promising
scalability very large datasets. Exploring sparse attention mechanisms 22 53 path to reduce this cost while maintaining expressiveness.
7 Discussion and Conclusion
We introduced the Distanceinffomme Neural Process (DNP), novel variant within the Neural Process family that integrates both global and local latent representations. DNP leverages a distance- aware local latent path, enforced via bi- bi-Lipschitz regularization approximately preserve distance relationships in the input space. This leads to a more faithful modeling of similarity relationships and improves both predictive accuracy and uncertainty calibration across regression and classification tasks. The broader implications of this work lie in its potential applications in safety-critical domains, where calibrated uncertainty estimates and generalization to unseen conditions are essential. Like most neural models, DNP can be sensitive the quality and bias of its training data and dataset shifts. We encourage practitioners to validate the model carefully to ensure safe and reliable deployment.
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A Implementation Details
A.1
Algorithm
The training and inference procedures for DNP is outlined in Algorithms aeeeeesptctivee The model combines a global latent variable that captures task-level patterns and distance-aware local latent variable that models input similarity. During training, both the global and local paths are optimized jointly using a variational objective and bi-Lipschitz regularization term. At test time, the model predicts target outputs by conditioning on both the global and local priors inferred from the context set.
Algorithm Training DNP
Input
:Mini-batch (x, y) from dataset D, trade-off pa- rameter B, minimum and maximum number of con- text data Nmin Nmax..
Output: Updaated parameters 8, Ag. fL, ΦG. ØL
foreach mini-batch do
Randomly draw the number of context points Ne
(NNiin: Nmax))
Algorithm Inference with DNP
Sample context set (x),yc) C (x,y) of size No. Input :Context set (Xc.ec) test in- put x", trained parameters
define the target set as (xT, = yx
Compute global summary: 8C = mean mean (xx,n Ve));
Output: Predictive distribution
Compute prior Pes (2g Xc.
p(y"[x",xc., Xc)yc)
Encode inputs: u. h(xx),
h(x:);
Encode: u, h(xx), u l(xx);
Compute attention weights of using dot-product atten- tion;
Compute attention weights of using dot-product attention;
Compute local prior pe,, (zu Xx, xc, Yc);
Sample zg ~ Pao (zg (zalxc, (xc.yc);
Compute variational posteriors: Asa (zal xy. yr) and
Sample z po, (z 1x" (xx
90 (2,)x,,.)))
Return (y x ,x","", za);
Sample zg Qéa and 24 Jdzi
Compute likelihood (y: Xt, 24, 2G);
Compute LELOO (Fq. 10);
Compute LiLLip (Eq. 4
Total loss: CELBO + L
Update model parameters via gradient descent;
end
A.2
Computational Complexity
CNPs and NPs aggregate the context set using a permutation-invarian operation (e.g., mean pooling), resulting in an ((M) operation. Each of the N target points is then processed independently, yielding a total prediction-time complexity
O (NN. AttnNPs and DSVNP uses self-attention over the M context points, which incurs cost of O(M?). Additionally, they compute cross-attention between each of the N targets and the M context points, resulting in a further O(NM) cost. Thus, the overall complexity during prediction becomes O(M² N NM). Our proposed DNP eliminates the need for self-attention over the context set by constructing a distance- prreservii latent space using a Liipschitz-reuulariii
encoder. At prediction time, each target point performs cross-attention the context representations, resulting in a total complexity of (NN During training, there an added cost from the LOBPCG computation. The cost per layer is (TTpq) where T the number of iterations, and are the dimensions of the weight matrix.
A.3 Regression
A.3.1 Network Architecture
The overall architecture of regression using DNP is follows.
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Global latent encoder. The global latent encoder is a 3-layer MLP, constructed similarly to the structure NP and AtmNN. The imput to the MLP is the concatenation of the input variable Rd and output variable y E Rdd,
Each context and target pair is first encoded independently, and then the resulting representations aggregated via the mean functiin to produce global summary.
The MLP architecture is as follows:
dy + dn + dn - dn da.
Note that unless specified, each - in this text represents linear layer followed by a Leaky ReLU activation, except for the final layer. The output of the final layer provides an embedding of each input pair Y;). These embeddings are then aggregated across the entire context set using a mean operation, resulting in single summary vector. This summary vector is subsequently passed through separate linear layer to produce the mean and log-veriance of Gaussian distribution over the global latent variable Zy.
dp
jgg Eng where Mg Eg Rd
Local latent encoder. To obtain distortion bounnded latent representation, we first construct a shared 2-layer MLP that maps each input into intermediate feature space of dimension d,. The MLP follows the structure:
d2 - dp dp.
The shared backbone is used to extract two components: (i) the -opootdion-buddedd representation u, and (ii) the parameters of the amortized posterior distribution. The inctortion-bonddee latent representation Rdd is obtained by applying linear projection to the backhone output:
da + du
To obtain the approximate posterior and prior distributions over the local latent variable, the hidden representation from the backbone is concatenated with the corresponding target output and passed through an additional linear layer. This layer outputs the mean and log-variance Gaussian distribution:
dn + dy + [m,, En]
where 14, E, E Rdd.
Next, to incorporate context information into each target, we apply cross-attention between the target's embedding u, and all context embeddings u, to obtain the attention weights af These attention weights then aggregate the per-context Gaussian parameters into target-specific local prior.
Decoder. The decoder takes input the concatenation of the sampled global latent Zg Rdd, the sampled local latent z Idd= and the iestortion-boundee embedding e Rd,, This produces vector of dimension 2d, + du. which is passed through a 3-layer MLP to output the parameters of the predictive Gaussian over y Rdy;
2 d. dd, d4 dr
2 dy
Other baselines. The network architecture of CNP consists of a 3-layer MLP with hidden dimension dy and latent representation of dimension d.. Each context pair (x, y) is passed through the encoder MLP, and the resulting representations are mean aggregated to form a global representation. The decoder then takes the concatenation of this global representation and the target input x; and passes through another 3-layer MLP to predict the mean and log- variance of the output y,- Each layer in the MLPs except the output layer is followed by Leaky ReLU activation.
The encoder is composed of two parallel paths: deterministic path and latent path. Both paths utilize 3-layer MLP with hidden layers of dimension
For NP, we use the architecture proposed 25)
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dp and an output dimension of dg. Each input-ouiput pair (x,y) is encoded independently, and the resulting representations are aggregated using a mean operation. In the latent path, the aggregated representation is used to produce the parameters of multivariate Gaussian distribution, namely the mean My and log-variance log E.g of the global latent variable. The decoder receives the sampled latent variable, the aggregated deterministic representation, and the target input x; as input, and outputs the prediction y: through a separate 3-layer MLP to predict the mean and log-variance of the output y:-
For AttnNP and DSVNP, we follow similar architecture as NP, with the key difference that the mean aggregation in the encoder is replaced by multi-head self- atff-attention and the deterministic path uses mullti-head cross- atseettotn network, as proposed [25]. The main distinction between AttNN and DSVNP is that DSVNP models both global and local latent variables by predicting Gaussian distribution parameters for each, where the local latent conditioned on the global latent. All other components, including the decoder, remain identical to those used in NP.
For both ConvCNP and ConvNP, we follow the architecture proposed in [448) using 1D convolu- tional layers. The encoder consists of 3 convolutional layers, each with 64 channels and followed by Leaky ReLU activations. The input data is first interpolated onto a uniform grid using a Gaussian kernel before being passed through the encoder. The decoder also consists of convolutional layers, each with 64 channels, and outputs the mean and log- variance of the predictive distribution for each target point.
The models were implemented using PyTorch 43 and trained with 12 GB of RAM. To obtain the singular values for DNP, we use Py implementation of the LOBPCG method, which employs orthogonal basis selection [51).
Nvidia GeForce GTX 1080
A.3.2 1D Regression
Inputs for both context and target points are drawn uniformly from the interval [-2, The number of context points is sampled uniformly from [3, 50J, while the number of target points is fixed at 50. For all kernels, the lengthscale and the output scale values are sampled uniformly from [0.6, 1.0) and [0.1, 1.0), respectively. Additionally, for the Periodic kernel, the period parameter is sampled uniformly from the interval [0.5, 1.5). A likelihood noise of 0.02 is added for all the cases. A batch size of 50 is used for training. During evaluation, the number of context and target points is sampled from the same distributions used during training. The models used the following hyperparameters: the encoder/decoder hidden dimension
set to d₁ 64, the latent dimension to d= = 64, and the distance-and aware latent dimension to du = 64. All the models are trained using the Adam optimizer [26] a learning rate of 1e-3, 3, for 200 epochs. For DNP, the bi-Lipschitz constraint is enforced with lower and upper singular value bounds \ = 0.1, A2 = 1.0, and the balancing the ELBO and bi-Lipschitz regularization is set 3
A.3.3 Synthetic to Real-World Regression
The data generation process follows [4]. For each training batch, the number of context points is
sampled uniformly from the interval [3, 50), while the number of target points is fixed at 50. At test time, 30 context points are randomly selected from the train data. The following hyperparameters were used for the models: encoder/decoder hidden dimension dy. 64, latent dimension d, = 64, and distance-aware latent dimension d, = 64. All networks were trained with the Adam optimizer [26] at a learning rate of 1e 3, using a batch size of 50 for 200 epochs. For DNP, the lower and upper bounds on the singular values were set to A = 0.1, , X2 = 1.0, with a trade-off weight 3 = 1 balancing the ELBO and bi-Lipschitz regularization.
A.3.4 Multi-output Real- World Regression
We evaluate DNP on three benchmark datasets: SARCOS (1], which models the inverse dynamics oofereedoofffroomm
of a seven-degree-of of- freedom robot arm by predicting seven joint torques from 21 inputs (positions, velocities, accelerations); Water Quality (WQ) 77. which predicts species abundances Slovenian rivers from 16 physical and chemical indicators; and SCM20D (50). supply-chain time series for multi-item demand forecasting. All datasets are standardized to zero mean and unit variance per feature and split 80/20 into training and test sets.
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For every training batch, the number of context points is randomly sampled from a uniform distribution in the range [3. 100), and the number of target points is 100. The network architecture uses hidden and latent dimensions d, 128, dz 128, and the distance-aware latent dimension d, = 128. Models are trained using Adam optimizer [26] with a learning rate of 1e and a batch size of 100, for 100 epochs. At test time, we randomly select 30 context points from the train data. For the bi-Lipschitz regularizer, we use Å1 = 0.1, X2 = 1.0 and set the ELBO/bi-Lipschitz trade-off weight to B 1.
A.4 Image Classification and OOD Detection
For classification tasks on CIFAR- 10 and CIFAR-100 100, we use VGG-16 network 48 with Leaky ReLU activation as the feature extractor.
Global latent encoder: In the global latent encoder, the feature embedding obtained from VGG-16 is concatenated with the one-hot encoded label y form the input representation. The flow of operations is summarized as:
d, VGG-16 dr
(d,, one one_hot(y)] + dg - [gg Eg]
where
Mg Eg E Rdd
Local latent encoder. The local latent encoder also uses the feature embedding extracted by VGG-16. Each input x is passed through VGG-16 to obtain an embedding of size dn:
VGG-16 dn d
This backbone is used to produce two outputs: (1) iesstortion-buundee latent embedding u E R.,,, obtained via a linear projection:
dp + du
(2) The parameters (mean and variance) of the amortized posterior distribution over the local latent variable:
dn + dy [un, Enj where
pn, Ex IR
Cross-attention is then applied between the distortion-bounded target embedding u, and the set of context embeddings u, obtain an attention-based aggregation of local latent parameters.
Decoder. The decoder takes as input the concatenation of the sampled global latent variable 2, E RRd, the sampled local latent variable z, E IR:d, and the distortion-boundee latent embedding u E TR The resulting vector of dimension 2d, + d,, is passed through a 3-layer MLP predict the parameters of a categorical distribution over the output classes.
2d + dg > d, + dr - d,
The other baselines follow a similar architecture to that described in Section A.3.1 with the key difference that a VGG-16 network used feature extractor for the imput images. The extracted feature embedding is concatenated with the one-hot encoded label vector before being passed through the subsequent layers. Similar to DNP, the predictive distribution modeled as a categorical distribution over the output classes.
For the experiments, the networks are trained the training sets of CIFAR-10 and CIFAR-100, each consisting of 60,000 images categorized into 10 and 100 classes, respectively. Evaluation performed on the corresponding test sets, each containing 10,000 images. The images have dimension of 32 x 32, and are normalized to zeros mean and unit variance. For OOD detection, images from the test of the SVHN, CIFARIO//CIFAT For fair comparison, all the OOD images are resized to 32 32 and are normalized the same way as the chosen between from a uniform distribution between 16 and 128. A batch size of 128 is used. The network architecture has the dimesnions d,, d: = 512 and d,, = 256. The models were implemented using PyTorch 43 and trained on an NVIDIA 1100 GPU with 40 GB of memory. Models are trained using the Adam optimizer (26] with a learning rate of 14 for 200 epochs. At test time, 100 context points are randomly selected from the train set. For the bi- bi-Lipschitz regularizer, λ 0.1 and
and the TinyImageNet dataset is used.
images. For training, the number of context points is randomly
25

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Table 8: Performance comparison across different kernel functions for ID synthetic regression for context set. Results are averaged over 10 runs.
RBF
Matém- 2
Periodic
Method
Latency (L)
(t)
BCE (4)
LL (t)
BCE
LL (1)
BCE (L)
(ms)
CNP 11
0.832++0.003
0.072++0.034
0.828++0.003
0.073+0.047
0.332+000044
0.139-0.043
0.046
NP 12]
0.856=#0.022
0.084:0.034
0.692++0.004
0.093=+0.037
0.631=0.003
0.115=0.040
0.095
ConvCNP
14
0.855=10.002 0.088:10.032
0.926:10.002
0.094:10.038
0.438:10.002
0.119=0.030
0.058
CP
8
0.964440.004
0.092:40.052
0.920:40.002
0.062++0.013
0.766:+0.003
0.121=00021
0.117
AttnNP
25
1.313:10.002 0.076:100018
0,931=:0.006
0.077=:00011
0.76411.001
0.072=00012
0.106
DSVNP
1.322++0.006
0.082+00.027
1.018++0.003
0.065++0.018
0.829++0.001
0.102-00003
0.109
Ours
1.499:000044
0.064:10.01
1.229::0.001
0.054:10.012
0.934:10.001
0.068=0.022
0.102
A2 = 1.0, with the ELBO/bi-Lipschizz trade-off weight 3 = 0.5. To obtain the singular values, we use
PyTorch' implementation of the LOBPCG method, which employs orthogonal basis selection [51].
B
Additional Experiments
B.1 ID Synthetic Regression
Table reports the loglliillhoood (LL) and expected calibration error (ECE) for the target points, excluding the context data. In Table 8 following the evaluation protocol of [25 we provide the LL and ECE scores on the context points for the 1D synthetic regression task. Consistent with the results on the target set, DNP achieves higher L.L. and lower ECE scores, indicating better predictive performance and calibration. terms of inference speed, CNP incurs the lowest computational cost due to its fully deterministic architecture. In contrast, ConvNP is more expensive because of its convolutional structure. Meanwhile, AtmNP, DSVNP, and DNP exhibit comparable inference times, as they share similar architectural components such as attention and latent variable sampling.
To assess the robustness of DNP observation noise, we evaluate the performance of the NP models on ID synthetic regression tasks. The functions are generated from the RBF kernel, with noise levels of 5%, 10%, and 15% on the observations. Evaluation on the target set excluding the context data is provided in Table Results show that DNP consistently achieves the highest liglliieddood across all noise settings, while also maintaining strong uncertainty calibration with the lowest ECE in most cases. We also study the role of geometry-awar regularization. Table T0 reports the effect of bi-Lipschitz regularization on ConvCNP and ConvNP across different kernel families in the 1D) synthetic regression task. Incorporating the bi-Lipschitz constraint consistently improves log-likeliin log-likelihood and reduces calibration error compared to the unregularized counterparts, with the strongest gains observed for ConvNP.
Table 9: Performance comparison for ID synthetic regression under noisy data for RBF kernel function. Results are averaged over 10 runs.
5% Noise
10% Noise
15% Noise
Method
LL (f)
BCB (1)
LL
ECB (L)
(L)
BCB (4)
CNP 101
0,462:10.003
0.11710.032
0.421:0.006
0.10410.028
0.32110.003
0.073100031
NP 12
0.47510.000 0.5
0.1124:0043
0.43:10.012
0.094=0.031
0.364:0.004
0.095110021
Conv
0.512-10.005
0.090 048
0.502:10.001
0.083:10029
0.452:10.004
0.054110.027
ConvNP
8
0.532-10.004
0.1111:0.71
0.526:0.003
0.082=0.039
0.479-0.005
0.05110.019
AllnNP
25
0.571::0.004
0.082+40.031
0.515=0.011
0.097
0.485
0.021:10026
DSVNP
63
0.567-10.003
0.094.10.043
0.532-10.008
0.092-10001
0.476:10.005
0.048110.038
Ours
0.625-0.004
0.073+000036
0.003
0.081+0.00
0.522
0.037+000229
B.2 Ablation Study
Table IT presents an ablation study evaluating the impact of the number of context points on CIFAR- 10 (ID) vs. CIFAR- 100 (OOD) classification performance for both AttnNP and DNP. We observe that the accuracy and AUPR scores are robust to the number of context points for both models, remaining approximately constant across different context set sizes, with only minor fluctuations in ECE. Notably, DNP consistently outperforms AttnNP across all metrics and context sizes, achieving
26

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Table 10: Evaluation of bi bi-Lipschitz regularization on ConvNP for 1D synthetic regression. Results are averaged over 10 runs.
Method
RBF
Matem-
Periodic
L.L. (1)
ECE (J)
(1)
ECE (L)
L1)
ECE
CanvCNP
0.496.00.07
0.32910.001 0.118.0..04
0.098
0.335.10.002
0.106_0.032
ConeNP
0.609+00005
0.190+000074
0,313++0002
0.137-0.095
0.431
0.117-0.011
ConvCNP + hi Lipschitz
0.507:1.0.013
0.006-10.013
0.400:10.014
0.081
0.082-0.022 0.39911.0004
ConvNP bi-Lipachitz
0.702+00008
0.073+00018
0.462+00.16
0.093=00019
0.515+0.003
0.098=00018
superior calibration and predictive performance. In line with this, we use 100 context points for the main experiments, as it offers a good balance between performance and computational cost.
Table 1I: Ablation study on the number of context points for CIFAR-IO CIFAR-100 classification. Results are averaged over 10 runs.
AttNN
DNP
Context
Accuracy (*)
ECE (L)
AUPR (1)
Accuracy (1)
ECE (L)
AUPR
50
90.36:10.01
0.021110.01
87.1210.011
91.92.10001
0.01510.002
90.5110.01
100
90.37+0.01
0.020+0.000
87.11+0.01
91.97+0.01
0.011+0.002
90.58-0.01
200
90.39=0.01
0.0211:0.033
87.18=0.02
91.96:10.22
0.012+10.002
90.58=0.02
500
90.40=00.01
0.02340.002
87.17=0.02
91.95:0.01
0.010=0.001
90.58=0.02
Table 12 reports the metrics on the effect of varying the distance-aware latent dimension dy and latent dimension d_ CIFAR-10 (ID) CIFAR-100 (OOD) classification. We observe that increasing both d, and d2 generally improves performance across all metrics, with the best results achieved when d 256 and d_ 512.
Table 12: Ablation study on the impact of varying dy and d2 on classification performance for CIFAR-10 10 vs. CIFAR-100. Results are averaged over 10 runs.
du d2 Accuarcy(*)
ECE(L)
AUPR(1)
512, 512
91.36-0.02
0.011+0.001
90.14+00.01
512, 256
90.47-0.02
0.014+0.000
90.39+0.01
256, 512
91.97=0.01
0.011::.002
90.5810.01
256, 256
90.64=0.01
0.012:10.002
90.36:00001
256, 128
90.82-0.01
0.0144:0.002
89.52++001
128, 256
90.44-0.01
0.014+0.001
90.04+0.01
128, 128
88.83-0.01
0.019+0.001
87.81+0.01
Table 13 presents an ablation study examining the impact of varying the lower (A1) and upper (22) bounds of the bi-Lipschitz constraint on model performance. The configuration x = 0.1 and A = 1.0 yields the best performance, balancing regularization strength and representational flexibility. As A2 increases while keeping X = 0.1, we observe modest trade-off: the accuracy remains competitive, but there is slight degradation
calibration (ECE) and AUPR scores as X2
becomes larger. In contrast, increasing A (e.g., to 0.6 or higher) or setting Åι = X2 results in more substantial performance degradation. This suggests that overly tight constraints on the bi-Lipschitz regularizer hinder the model's ability adapt complex input-output relationships, limiting both its expressiveness and the quality its uncertainty estimates.
Table 14 presents an ablation study on the effect of the trade-of off parameter 3 which balances the ELBO and the bi-Liisehitz regularization loss, on CIFAR-10 10 (ID) vs. CIFAR-100 (OOD) classification. As 3 increases, calibration and uncertainty quantification improve significantly, peaking around 8 = 0.5. Beyond this point, calibration remains stable but accuracy slightly drops, suggesting that excessive regularization limits predictive performance. While 33 was manually tuned in this work to balance predictive performance and uncertainty calibration, automating its selection remains an important direction for future work (24).
27

---

Table 13: Effect of bi-Lipschitz constraint bounds (A1, X2) on CIFARIO CIFARI00 classification. Results are averaged over 10 runs.
A1. A2
Accuarcy(1)
ECE(1)
AUPR(1)
A1.
Accuarcy(1)
BCB())
AUPR(1)
0.1. 0.6
89.61=0011
0.014=0.002
88.98=0.01
0.2, 1.0
91.41=0.01
0.013++0000
89.72=0.011
0.8
91.09+0.01
0.012=0.001
89.46+0.02
0.4, 1.0
90.89+0.011
0.012++0.022
88.43++0.01
0.1,
91.97++001
0.011=0.002
90.58++0.01
0.6, 1.0
89.60+0.01
0.017++0.002
88.92++0.02
0.1,
91.92+0.01
0.011=0.002
89.36=#0.01
89.86=0.01
0.027=+0.003
89.05=00011
0.1, 1.4
92.02+0.01
0.015=0.002
89.52++0.11
1.0, 1.0
89.92+0.01
0.031+0.03
88.73+0001
Table 14: Evaluation of the trade-off parameter between the ELBO and the bi-Lipschitz regularization loss on CIFARIO vs CIFARI00 classification. Results are averaged over 10 runs.
8
Accuracy (1)
ECE (L)
AUPR (?)
0
0.042::0001 9.008=0.01
87.8520.01
92.01=0.01
0.023+10003
88.1410.01
0.2
91.92=0.01
0.019:10001
89.62=0.01
0.5
91.97=0001
0.011=-0.022
90.58+00.11
0.7
90.62=0.01
0.011=00003
90.42:0.02
1.0
90,59
0.0124+0.022
90.43+00.11
Table 15: Comparison of the Laplace and dot-product similarity functions for cross-attention
CIFARIO (ID) vs CIFARI00 100 (OOD). Results are averaged over 10 runs.
Attention Type Accuracy(1)
ECE(L)
AUPR(1)
Laplace Dot-product
91.97110.01 91.12+00.01
0.01110.002
90.5810.01
0.018+0.002
89.62+0.011
We also evaluate the performance when using Laplacian attention (Eq, 5) and dot-product attention, given by
n
exp
0
(13)
Edec exp
u U Vdl,.
The results are presented in Table 15 While both attention mechanisms perform similarly overall, Laplace attention yields improved classification accuracy, better uncertainty calibration, and OOD detection.
Table 16 presents ablation analysis of the global and local latent paths in the DNP model. We observe that using only the local latent path yields the best performance in terms of classification accuracy and calibration (ECE), indicating its strong contribution to modeling instance-specific uncertainty. In contrast, the global latent path provides slightly better AUPR. Incorporating both paths yields the best overall performance (see main results in T)ble
Table 16: Ablation study evaluating the contribution of global and local latent paths in DNP.
Latent path
Accuracy (f)
BCB ())
AUPR (t)
DNP (Global latent only)
90.44:10.01
0.046110.002
90.59110.01 90.56:10.01
DNP (Local latent only)
91.11:10.01
0.016:10.022
C Proof of Proposition 1
To show that the generative model defined in Eq. is stochastic process, it needs to satisfy two conditions of the Kolmogorov Extension Theorem 41:
1. Permutation Invariance (Fxchangeability):: The joint distribution Prix (yi-N) must be invariant under any permutation и of the indices (1,...,N) N); that is,
Pusin (VI.N) PRecuayy (Jx(1.,))
(14)
28

---

2. Consistency (Marginalization): For any N and any M < N, the marginal distribution over any subset (. 3ar) of the outputs must be recoverable by integrating out the remaining variables from the joint distribution Pa (V1:N).
We now prove each of these in turn. For the proof, assume that the joint distribution over targets, latents, and inputs factorizes as
N
/f
I ply: 2a, Zi, xs) p(zz x:) p(zG) dza:N dza
PeLL (YINN) =
(15)
Exchangeability: Let π be any permutation {1, 2,.... .N. Since we use permutation-inee -nvvarian distribution over the global and local latent variables i.e., the prior p(zg) is shared across all inputs and each p(z,|x, xx) is independently defined- -te joint distribution remains invariant under reordering of the inputs and outputs:
PRIN (y1:N) = /f
N
II
plys zc, 2,, p(z, x.) p(za) dz1: dzc
2m1
N
(16)
II p(yn()) za, Rn(i), Xm(i)) p(Zz()) Xx(i)) p(za) day: dzg
Px (V>(1:N))
Thus, the model defines permutation invariiant distribution over the input-output pairs, satisfying the exchangeability condition required by the Kolmogorov Extension Theorem.
Consistency: From Eq.
7 we have
I PXIx (YI:N) dy dyN++IN
II ply: aG, 2i, p(z; x;) p(ZG) d2y: dza] dy
(17)
(since all densities nonnegative and integrable), may interchange the integrals and the product over i, yielding
By Pubini-Tonell 44
f P (YIN) dym+I:N= = //
II (=++1
ply: 2G. 24, x:) p(z; x;) dy dyN+ dzasiiN
II ply: zG. z. p(z: xx) p(zG) dz dzg.
(18)
The inner bracket integrates to (marginalizing out the N M points), leaving exactly
/f
M
II plys zG. 2,, xx) p(z, x;) p(zG) dz dzg.
(19)
1=1
which matches the form Pxix (Ylar). Hence, marginal consistency holds.
D Derivation of ELBO Loss
The generative model in Eq. is intractable, we introduce variational approximation de (z,,,, zG)
to approximate the true posterior. Using the Evidence Lower Bound (ELBO) (27]. we rewrite the log-marginal likelihood by multiplying and dividing by (z. za);
logp(y:|xe, Xc, yc) = log /f ply,, 2t, 20|x,. Nc) 4ab 2t, 20)
Iss (2) zG)
dz, dz.
(20)
Applying Jensen's inequality, we obtain:
plye, Zto zgl Xx, XC.NO
log logp(y, Xc. 2 Rgg log
(21)
no (2,, 20))
29

---

This expectation defines the ELBO.
To make inference tractable, assume structured variational approximation where the posterior factorizes as:
go (0 zc) a (zG , yr) (z Xx, Va, Xc, Nc)-
(22)
Expanding the ELBO by substituting the factorization, obtain:
lo, xc, 2 Egg [(rlg Zr, x.) + logp(zz Xx,, Nc) +
Egg [log qo (z, Yt, Yr, Xc) + log tea (T,
(23)
Rewriting the expectation terms in terms of Kullback-Leibler (KL) divergences, we obtain the final ELBO:
log (yyy(lxx,,x,,8000 Bgg ftg
ply /aa Zz, xx )
-DK (z Yc. yc) pez (2,)x,,,,,, DxL (9o0 (zG xr. pea (zG xc. yC ))
(24)
This ELBO consists of: A reconstruction term that encourages accurate predictions of yt. Two KL divergence regularization terms that enforce approximate posteriors qag and qo, to be close to their respective priors.
30


Reviews:
[{"review_id": "eHlBCZ4zdi", "reviewer": "Reviewer_C6NN", "summary": "This paper introduces the Distance-informed Neural Process (DNP), a new member of the Neural Process family designed to improve uncertainty quantification and generalization. The authors identify a key weakness in standard NPs: their reliance on a single global latent variable often leads to poorly calibrated uncertainties and a failure to capture local data dependencies. To overcome this, DNP employs a dual-path architecture. A global path, similar to standard NPs, captures task-level uncertainty. The main contribution is a local latent path, which models input similarity explicitly. This is achieved by learning a distance-preserving latent space for the inputs, enforced through a novel bi-Lipschitz regularization on the encoder network. The similarity between points in this regularized latent space is then used to compute attention weights, allowing the model to make predictions that are informed by local context. Through extensive experiments on regression and classification tasks, the paper demonstrates that DNP achieves superior predictive performance, better-calibrated uncertainty estimates, and more effective out-of-distribution (OOD) detection compared to several state-of-the-art NP variants.", "questions": "1. **Connection to Classical Methods**: The core idea of using distance-based weights for local modeling is conceptually close to classical non-parametric methods like locally weighted regression (LWR) or kernel regression. Could you elaborate on the relationship between DNP and these methods? Specifically, what are the key advantages of your deep learning approach with bi-Lipschitz regularization compared to a modern variant of LWR or a deep kernel learning model that also aims to preserve local similarity?\n\n2. **Scalability of the Local Path**: The O(NM) complexity of the cross-attention mechanism is a practical limitation for large context sets. You mention exploring sparse attention as future work. Have you considered or run preliminary experiments on simpler, more scalable approximations? For instance, one could attend only to the k-nearest neighbors in the latent space. How does performance degrade under such approximations? A brief analysis would provide valuable insight into the performance-efficiency trade-off.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The paper proposes a well-motivated and technically sound solution to a known limitation in NPs by integrating a distance-aware local latent path.\n- The method's effectiveness is convincingly demonstrated through comprehensive experiments showing strong performance, especially in uncertainty calibration and OOD detection.\n- The paper is exceptionally clear and well-written, with intuitive figures and a thorough appendix that aids understanding and reproducibility.\n\nWeaknesses\n- The work is an incremental advancement over existing NP variants, combining known concepts in a novel way rather than introducing a completely new paradigm.\n- The model's O(NM) computational complexity from cross-attention presents a scalability challenge for tasks with large context sets.\n- The proposed bi-Lipschitz regularization is limited to Euclidean spaces, which may hinder performance on data with non-Euclidean geometries.", "was_revised": false, "final_justification": "", "rebuttal": "We would first of all like to thank you for the positive evaluation of the paper’s motivation, clarity and experimental rigor. We now address the weaknesses and the questions raised.\n\n**W1. Incremental Nature of the Contribution**\n\nWhile DNP builds upon existing Neural Process variants, we emphasize that our work introduces a **novel integration of architectural and regularization techniques** that together yield substantial empirical improvements in uncertainty modeling. We are the **first to incorporate a bi-Lipschitz-constrained local latent path** within the NP framework. This introduces an inductive bias that improves the model’s ability to capture local structure and produce well-calibrated uncertainty estimates. Empirically, DNP consistently outperforms the strong NP baselines such as AttnNP, ConvNP, and DSVNP across a variety of regression and classification tasks (Tables 1-5). These results indicate that DNP offers a principled and practically useful advancement in NP design.\n\n**W2, Q2. Scalability of Local Path**\n\nWe evaluated the kNN-based sparse attention variant on the CIFAR10 (ID) vs. CIFAR100 (OOD) classification task. The results for 3 different values of k are as follows:\n\n| k   | Accuracy (%)     | ECE             | AUPR (%)        |\n|-----|------------------|------------------|------------------|\n| 5   | 87.34 $\\pm$ 0.01     | 0.021 $\\pm$ 0.002    | 87.34 $\\pm$ 0.02     |\n| 10  | 88.19 $\\pm$ 0.01     | 0.019 $\\pm$ 0.003    | 87.92 $\\pm$ 0.01     |\n| 20  | 89.05 $\\pm$ 0.02     | 0.019 $\\pm$ 0.001    | 89.08 $\\pm$ 0.01     |\n\nThese results indicate that increasing k generally improves model performance. However, even small values of k (e.g., k = 5) yield competitive results, suggesting that the model can achieve strong performance even with limited neighborhood information.\n\n**W3. Euclidean Limitation of Bi-Lipschitz Regularization**\n\nWe agree that the current bi-Lipschitz constraint in DNP preserves Euclidean geometry in the latent space. We would like to clarify that our goal in this work is to take a principled first step toward introducing distance awareness into NPs, and to show that even simple distance preservation can significantly improve uncertainty calibration.\nEven though  this constraint may limit applicability in settings with inherently non-Euclidean data structures (e.g., graphs or manifolds), we note that our approach is already effective across a broad range of regression and classification tasks involving high-dimensional and structured inputs. Moving forward, we see this as a foundation for future extensions, including the use of task-specific or non-Euclidean similarity metrics as noted in Section 6. We believe this opens up a promising research direction for incorporating more general geometric priors into the latent variables.\n\n**Q1. Connection to Classical Methods**\n\nWe really appreciate this thoughtful observation. Indeed, the bi-Lipchitz preservation in the local latent path in DNP is conceptually inspired by distance-based non-parametric methods. The classical methods, though excellent in uncertainty calibration, are computationally expensive and lack flexibility when dealing with high-dimensional inputs. The main advantage of our method lies in **scalability and generalization**. Modern methods like Deep Kernel Learning (DKL) combine neural feature extractors with Gaussian Processes (GPs), but they inherit the computational limitations of GPs: $O(N^3)$ training time, where $N$ is the number of training points. Even sparse approximations with inducing points reduce this only to $O(NM^2)$, where $M$ is the number of inducing points, which remains costly for large datasets. In contrast, NPs, including DNP avoids these costs by framing the problem within a meta-learning framework. Once trained, DNP requires only $O(N'M)$ computation, where $M$ is the number of context points and $N'$ is the number of target points. Moreover, DNP can generalize across tasks without retraining. This is unlike DKL which typically requires retraining or fine-tuning per task. In summary, DNP thus offers a scalable and flexible alternative that preserves the inductive bias of distance-based classical non-parametric methods while being better suited for modern large-scale, task-distributed learning scenarios."}, {"review_id": "6Q9tFaWvB7", "reviewer": "Reviewer_n9Sq", "summary": "The paper introduces *distance-informed neural processes* (DNP), which extends attentive NPs (AttnNPs) by encouraging the representations of the context and target points to maintain their relative distances from the input space. The authors achieve this by regularising the encoder through a bi-Lipschitz constraint, which encourages the singular values of the weight matrices to be within some predefined upper and lower bounds. The resulting model is trained through variational inference and evaluated on both regression and classification problems, outperforming baseline models on most tasks.", "questions": "**Suggestions**\n1. As mentioned under weaknesses, I am not convinced that the motivation for introducing the bi-Lipschitz constraint (i.e., to maintain relative distances) is helpful. I would suggest focusing instead on the smoothness properties of the non-linear transformation, which, I realise, is to some extent a different way of saying the same, but it would be clear that you are not trying to learn an identity mapping (which Figure 1c seems to be and which would be counterproductive).\n\n**Questions**\n1. It seems to me that your approach to imposing the bi-Lipschitz constraint is readily applicable to other architectures, e.g., convolutions. Is this correctly understood? If so, did you test bi-Lipschitz ConvNP or similar?\n2. I’m curious to understand the effect of the smoothness constraint on the model performance. As another ablation study, did you try to train the same architecture as the DNP but with some standard regularisation (e.g. weight decay) instead of the bi-Lipschitz constraint?\n3. What do the errors in the tables refer to? They seem quite small.\n4. Did you try to visualise samples from your model, e.g., for the 1D synthetic experiment? AttnNP can suffer from weird kinks in the samples, and I am curious to see if the bi-Lipschitz constraint had an effect here.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n1. The proposed model appears original and generally outperforms the baselines significantly. \n2. The experimental section is extensive. In particular, the authors have performed several ablation studies to understand the effect of the introduced smoothness constraint.\n3. The paper is well-written and well-structured.\n\n**Weaknesses**\n1. While the model seems original, its novelty is limited. The paper seems quite incremental, and given that adding the bi-Lipschitz constraint to the objective function is a fairly small change, I would have expected a more thorough analysis of the effect of the constraint on, e.g., training stability, sample quality, and other NP architectures, say, the ConvNP.\n2. The description of the proposed model and its relation to other models from the literature could be improved, for instance with a side-by-side comparison of their (simplest possible) graphical models and/or key equations.\n3. While the reference list is extensive, there are surprisingly few papers on neural processes or neural network smoothness cited. And even among those that are cited, it is not entirely clear how the current model positions itself in the literature.\n4. I found the motivation for introducing the bi-Lipschitz constraint and Figure 1 confusing and a bit misleading. To me, it sounds like the constraint is imposed to try to enforce that the relative distances of the data are unchanged by the encoding, which Figure 1 seems to support, but that would essentially mean that the encoding is an identity map. Rather, the bi-Lipschitz constraint imposes smoothness in the encoding transformation, and I think this is a more helpful mental picture of what the proposed model does and why it should work.", "was_revised": true, "final_justification": "The authors have mostly addressed my questions and concerns, and while I still see the submission as an incremental contribution, I think it deserves to be accepted.", "rebuttal": "We thank you for the thoughtful and constructive feedback. Below we address the weaknesses and questions.\n\n**W1, Q1. Method Novelty and additional bi-Lipschitz analysis**\n\nWe would like to clarify that our contribution lies in the novel integration of architectural and regularization components that, together, lead to significant empirical gains in uncertainty modeling. In particular, DNP is the first to introduce a **bi-Lipschitz-constrained local latent path** within the NP framework. This promotes the model’s capacity to capture local structure and produce more **calibrated uncertainty estimates**.\n\nPlease note that we have included additional ablation analyses on the impact of bi-Lipschitz bounds and the trade of parameter between the ELBO and the bi-Lipschitz loss in the Appendix B.2 Tables 11 and 12 respectively. Our results show that there is a trade-off between model expressiveness and regularization strength. \n\nWe will include the following additional results when applying bi-Lipschitz regularization to ConvCNP and ConvNP.\n\n\n| Method                  | RBF LL ↑         | RBF ECE ↓       | Matérn-5/2 LL ↑   | Matérn-5/2 ECE ↓ | Periodic LL ↑     | Periodic ECE ↓   |\n|-------------------------|------------------|------------------|-------------------|------------------|-------------------|------------------|\n| ConvCNP                 | 0.496 ± 0.007     | 0.118 ± 0.054     | 0.329 ± 0.001      | 0.098 ± 0.032     | 0.335 ± 0.002      | 0.106 ± 0.032     |\n| ConvNP                  | 0.609 ± 0.005     | 0.190 ± 0.074     | 0.313 ± 0.002      | 0.137 ± 0.095     | 0.431 ± 0.003      | 0.117 ± 0.011     |\n| ConvCNP + bi-Lipschitz  | 0.507 ± 0.012     | 0.086 ± 0.013     | 0.400 ± 0.014      | 0.081 ± 0.014     | 0.399 ± 0.004      | 0.082 ± 0.021     |\n| ConvNP + bi-Lipschitz   | 0.702 ± 0.008     | 0.073 ± 0.018     | 0.362 ± 0.016      | 0.093 ± 0.019     | 0.515 ± 0.003      | 0.098 ± 0.018     |\n\nThe results indicate that adding the bi-Lipschitz constraint improves the log-likelihood and ECE scores of the base ConvCNP and ConvNP models.\n\n**W2. Comparative Model Descriptions**\n\nWe currently provide textual comparisons of the baselines in Sec. 2 and 4. In the revised version, we will include the following table highlighting the architectural and probabilistic differences between the various baselines.\n\n\n| NP Model | Recognition Model | Generative Model | Prior Distribution | Latent Variable |\n|------------|-------------------|------------------|--------------------|------------------|\n| **CNP**    | $\\mathbf{z}_C = f(\\mathbf{x}_C, \\mathbf{y}_C)$ | $p(\\mathbf{y}_T \\mid \\mathbf{z}_C, \\mathbf{x}_T)$ | - | Global (Deterministic) |\n| **NP**     | $q(\\mathbf{z}_G \\mid \\mathbf{x}_T, \\mathbf{y}_T)$ | $p(\\mathbf{y}_T \\mid \\mathbf{z}_G, \\mathbf{x}_T)$ | $p(\\mathbf{z}_G \\mid \\mathbf{x}_C, \\mathbf{y}_C)$ | Global (Stochastic) |                        \n| **ConvCNP**      | $\\mathbf{z}_i = f(\\mathbf{x}_C, \\mathbf{y}_C, \\mathbf{x}_i)$                        | $p(\\mathbf{y}_i \\mid \\mathbf{z}_i)$                            | -                                                                                   | Local (Deterministic)           |\n| **ConvNP**       | $\\mathbf{z}_i \\sim f(\\mathbf{x}_C, \\mathbf{y}_C, \\mathbf{x}_i)$             | $p(\\mathbf{y}_i \\mid \\mathbf{z}_i)$   |  Implicit via f | Local (Stochastic) |\n| **AttnNP** | $q(\\mathbf{z}_G \\mid \\mathbf{x}_T, \\mathbf{y}_T)$, $\\mathbf{z}_i = f(\\mathbf{x}_C, \\mathbf{y}_C, \\mathbf{x}_i)$ | $p(\\mathbf{y}_i \\mid \\mathbf{z}_G, \\mathbf{z}_i, \\mathbf{x}_i)$ | $p(\\mathbf{z}_G \\mid \\mathbf{x}_C, \\mathbf{y}_C)$ | Global (Stochastic) + Local (Deterministic) |\n| **DSVNP**  | $q(\\mathbf{z}_G \\mid \\mathbf{x}_T, \\mathbf{y}_T)$, $q(\\mathbf{z}_i \\mid \\mathbf{z}_G, \\mathbf{x}_i, \\mathbf{y}_i)$ | $p(\\mathbf{y}_i \\mid \\mathbf{z}_G, \\mathbf{z}_i, \\mathbf{x}_i)$ | $p(\\mathbf{z}_G \\mid \\mathbf{x}_C, \\mathbf{y}_C)$, $p(\\mathbf{z}_i \\mid \\mathbf{z}_G, \\mathbf{x}_i)$ | Global (Stochastic) + Local (Stochastic) |\n| **DNP (Ours)**    | $q(\\mathbf{z}_G \\mid \\mathbf{x}_T, \\mathbf{y}_T)$, $q(\\mathbf{z}_i \\mid \\mathbf{x}_C, \\mathbf{y}_C, \\mathbf{x}_i)$ | $p(\\mathbf{y}_i \\mid \\mathbf{z}_G, \\mathbf{z}_i, \\mathbf{x}_i)$ | $p(\\mathbf{z}_G \\mid \\mathbf{x}_C, \\mathbf{y}_C)$, $p(\\mathbf{z}_i \\mid \\mathbf{x}_C, \\mathbf{y}_C, \\mathbf{x}_i)$ | Global (Stochastic) + Local (Stochastic) |\n\nUnlike AttnNP and DSVNP, which also use global and local latent variables, DNP incorporates a bi-Lipschitz constraint on the local latent path. This improves uncertainty calibration and OOD detection.\n\n**W3. Related Work and Literature Positioning**\n\nOur model extends the NP family by introducing a distance-aware inductive bias in the latent path, inspired by non-parametric modeling methods such as Gaussian Processes. The use of bi-Lipschitz regularization aligns with prior efforts to control neural network sensitivity and preserve structure [1, 2], but ours is, to our knowledge, the first to apply it within the NP setting to improve uncertainty calibration and OOD robustness. We will revise the related work section in the final version to better articulate this positioning, along with the description and positioning of the additional references suggested by Reviewer HQey.\n\n**W4 , S1. Motivation and Interpretation of Bi-Lipschitz Regularization and Figure 1**\n\nWe thank the reviewer for pointing out this potential source of confusion. In Figure 1, we used a special case of our method where the bi-Lipschitz bounds were set to $\\lambda_1$ = $\\lambda_2$ = 1, resulting in an isometric mapping. This was done intentionally for visualization purposes, to clearly highlight the issues of over-sensitivity and feature collapse in the latent space, and to show how the bi-Lipschitz constraint mitigates them. However, we acknowledge that not stating this explicitly in the paper was an oversight, and we will clarify this setup and its illustrative intent in the text.\nThat said, we agree that the main use of the bi-Lipschitz constraint is not to enforce exact distance preservation, but rather to promote smoothness and control sensitivity in the encoder. This interpretation provides a more general and helpful understanding of the regularizer’s role. In the figure, the encoder maps from a 2D input to a 2D latent space using a shallow network, which makes the transformation appear close to an identity. However, in all our experiments, we use relaxed bounds (e.g., $\\lambda_1$ = 0.1, $\\lambda_2$ = 1) and project to a latent space of dimension different from the input. In these scenarios, identity mappings are neither desirable nor possible. \n\n**Q2. Effect of smoothness constraint**\n\nThank you for the thoughtful question. The ablation study in Table 6 investigates the effect of the smoothness constraint on DNP's performance by comparing several regularization strategies.\nWe treat DNP without regularization as one extreme case where no explicit smoothness constraint is enforced. On the other extreme, we evaluate gradient penalty and orthogonal regularization, which strongly constrain all singular values to be close to 1, essentially enforcing a near-isometry. Both these scenarios degrade the model’s performance and uncertainty calibration.\nIn contrast, spectral normalization only bounds the largest singular value, promoting Lipschitz continuity but allowing lower singular values to collapse. Our proposed bi-Lipschitz regularization lies between these extremes: it enforces both upper and lower bounds on singular values, promoting smooth yet expressive transformations by avoiding both over-sensitivity and feature collapse.\nWe now include results for CIFAR10 vs CIFAR100 using the standard weight decay value of 1e-4. \n\nAccuracy =   91.92, ECE = 0.038, AUPR = 87.84\n\nCompared to the no-regularization case (ECE = 0.042, AUPR = 87.85), weight decay has a marginal improvement on uncertainty calibration. In contrast, bi-Lipschitz regularization significantly improves both ECE and AUPR. \n\n**Q3. Error in the table**\n\nThe errors in our tables represent the standard deviation of the mean, computed across multiple evaluation tasks (10 runs). As the metrics are averaged over many data points and tasks, the resulting error bars are small and reflect the stability of the metric estimates, not predictive uncertainty. If I misunderstood the question, please let me know.\n\n**Q4. Visualization of sample**\n\nIn our experiments, DNP's predictive samples were smooth, and we did not observe any “kinks”. Additionally, as shown in Figure 3, we observed that AttnNP occasionally produces excessively high uncertainty in certain regions, while simultaneously underestimating uncertainty in some OOD regions. This behavior is consistent with the qualitative findings reported by [3]. We hypothesize that both these issues in AttnNP stem from geometric distortions in the latent space. This is especially problematic because AttnNP relies directly on this latent space for obtaining the attention values during inference. In contrast, the smoothness property of the bi-Lipschitz constraint in DNP helps mitigate this. As a result, we obtain coherent samples and calibrated uncertainty estimates.\n\n\n[1] Jeremiah Liu, Zi Lin, Shreyas Padhy, Dustin Tran, Tania Bedrax Weiss, and Balaji Lakshminarayanan. Simple and principled uncertainty estimation with deterministic deep learning via distance awareness. Advances in neural information processing systems, 33:7498–7512, 2020\n\n[2] Joost Van Amersfoort, Lewis Smith, Andrew Jesson, Oscar Key, and Yarin Gal. On feature collapse and deep kernel learning for single forward pass uncertainty. arXiv preprint arXiv:2102.11409, 2021\n\n[3] Qi Wang and Herke Van Hoof. Doubly stochastic variational inference for neural processes with hierarchical latent variables. International Conference on Machine Learning, pages 10018–10028. PMLR, 2020."}, {"review_id": "WgkdKoEJTb", "reviewer": "Reviewer_HQey", "summary": "The paper introduces Distance-informed Neural Processes (DNP), a novel variant of Neural Processes that improves uncertainty caliration and OOD detection by integrating both global and distance-aware local latent variables. DNP employs bi-Lipschitz regularization on its local latent space to preserve geometric structure, enabling better similarity modeling. Experiments show state-of-the-art results in regression and classification tasks with improved uncertainty calibration.", "questions": "See the weakness part.", "limitations": "See the weakness part.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "I summarize the strength of this work as follows:\n\n(1) This paper is easy to follow and the introduction of bi-Lipschitz constraints to preserve input space geometry in latent embeddings is well-motivated. (2) The developed model improved uncertainty quantification. The dual-pathway (global and local latent) design enhances the expressiveness and reliablity of uncertainty modeling. (3) It conducts extensive evaluation.The paper presents extensive empirical validation across synthetic and real-world tasks, including regression, classification, and OOD detection.\n\n---\n\nAs for the weaknesses, I find the following concerns are waiting to address:\n\n(1) Baselines are a bit outdated, I hardly see new after 2022. It would be necessary to include one of the following ones [1-5] published in NeurIPS/ICML/ICLR with others carefully discussed in the literature work.\n(2) As the distance regularizer is the primary contribution in this work, it is necessary to include one ablation experiment to show Lipshitz term’s influence under varying scales.\n(3) The limitation part should be added in the main paper.\n\n---\n\nReference\n\n[1] Nguyen, Tung, and Aditya Grover. \"Transformer neural processes: Uncertainty-aware meta learning via sequence modeling.\" arXiv preprint arXiv:2207.04179 (2022).\n\n[2] Wang, Qi, and Herke Van Hoof. \"Learning expressive meta-representations with mixture of expert neural processes.\" Advances in neural information processing systems 35 (2022): 26242-26255.\n\n[3] Tailor, Dharmesh, Mohammad Emtiyaz Khan, and Eric Nalisnick. \"Exploiting inferential structure in neural processes.\" Uncertainty in Artificial Intelligence. PMLR, 2023.\n\n[4] Wang, Qi, Marco Federici, and Herke van Hoof. \"Bridge the Inference Gaps of Neural Processes via Expectation Maximization.\" The Eleventh International Conference on Learning Representations.\n\n[5] Gondal, Muhammad Waleed, et al. \"Function contrastive learning of transferable meta-representations.\" International Conference on Machine Learning. PMLR, 2021.", "was_revised": true, "final_justification": "The authors mostly addressed my concerns from more experiments and discussions.", "rebuttal": "We thank you for noting the clarity and thoroughness of the paper. Below, we address the weaknesses: \n\n**W1. Selection of Baselines and Related Work**\n\nWe appreciate the references and will incorporate the following discussion of the works in the related work section. \n\nTransformer Neural Processes [1] uses autoregressive sequence modeling and replaces the traditional encoder-decoder architecture of NPs with a transformer-based approach that improves model expressiveness and ensures context invariance and target equivariance. [2] extends NPs by using multiple expert latent variables and assignment variables to capture multimodal distributions, addressing the expressiveness limitations of single global latent variables in standard NPs. [3] replaces NP's standard context aggregation with inference in probabilistic graphical models, treating context embeddings as neural sufficient statistics. The approach yields mixture and robust Bayesian Aggregation variants that show improved performance on out-of-distribution data and robustness to context corruption. [4] address the under-fitting problem in Neural Processes (NPs) from an optimization perspective. They identify that NPs struggle to learn accurate probability distributions during training as a key limitation, and propose a surrogate objective for the target log-likelihood within an expectation maximization framework to achieve improved performance. [5] propose a decoupled encoder-decoder approach where the encoder is trained with a contrastive objective to learn transferable meta-representations of functions.  This improved transferability across multiple downstream tasks and greater robustness to noise compared to end-to-end trained baselines. While these models focus on improving expressiveness or inference strategies, DNP takes a complementary approach by enforcing geometry-aware regularization through a bi-Lipschitz-constrained latent path. This inductive bias helps preserve input structure, leading to more coherent uncertainty estimates and stronger OOD performance.\n\nBelow, you can find the metrics obtained for [3] ,referred to as NP-mBA on the 1D Synthetic regression task. DNP still achieves the best performance compared to the baselines.\n\n\n\n| Method         | RBF LL ↑         | RBF ECE ↓        | Matérn-5/2 LL ↑  | Matérn-5/2 ECE ↓ | Periodic LL ↑     | Periodic ECE ↓    |\n|----------------|------------------|------------------|------------------|------------------|-------------------|-------------------|\n| CNP            | 0.490 ± 0.006    | 0.122 ± 0.061    | 0.419 ± 0.004    | 0.088 ± 0.047    | 0.239 ± 0.001     | 0.120 ± 0.029     |\n| NP             | 0.480 ± 0.009    | 0.102 ± 0.082    | 0.318 ± 0.008    | 0.111 ± 0.053    | 0.228 ± 0.002     | 0.145 ± 0.047     |\n| ConvCNP        | 0.496 ± 0.007    | 0.118 ± 0.054    | 0.329 ± 0.001    | 0.098 ± 0.032    | 0.335 ± 0.002     | 0.106 ± 0.032     |\n| ConvNP         | 0.609 ± 0.005    | 0.190 ± 0.074    | 0.313 ± 0.002    | 0.137 ± 0.095    | 0.431 ± 0.003     | 0.117 ± 0.011     |\n| AttnNP         | 0.578 ± 0.003    | 0.103 ± 0.026    | 0.499 ± 0.009    | **0.078 ± 0.014**| 0.457 ± 0.001     | 0.098 ± 0.038     |\n| DSVNP          | 0.582 ± 0.002    | 0.101 ± 0.032    | 0.512 ± 0.006    | 0.101 ± 0.012    | 0.412 ± 0.002     | 0.092 ± 0.042     |\n| NP-mBA         | 0.521 ± 0.005    | 0.108 ± 0.012    | 0.324 ± 0.007    | 0.091 ± 0.021    | 0.375 ± 0.005     | 0.091 ± 0.010     |\n| **Ours**       | **0.696 ± 0.003**| **0.093 ± 0.054**| **0.620 ± 0.002**| 0.083 ± 0.010    | **0.635 ± 0.001** | **0.074 ± 0.022** |\n\n**W2. Ablation for the Bi-Lipschitz Term**\n\nIn Appendix B.2, we already include ablation analyses over the lower and upper Lipschitz bounds ($\\lambda_1$, $\\lambda_2$) (Table 11), and over the regularization weight $\\beta$ (Table 12). These results show that there is a trade-off between model expressiveness and regularization strength. When the bounds are too tight, the encoder becomes overly constrained, limiting the model's expressiveness. Conversely, when the bounds are too loose or the regularization weight is too small, the latent space can become distorted, leading to degraded uncertainty calibration.\n\n**W3. Including Limitations in the Main Paper**\n\nWe discuss the limitations (scalability and Euclidean assumptions) and potential future directions to address them in Section 6 of the main paper. We will create a dedicated “Limitations” subsection and move the relevant content there."}, {"review_id": "DdYrzEQ6Bt", "reviewer": "Reviewer_jh3q", "summary": "The paper introduces so called Distance-informed Neural Processes (DNP). DNPs build on Neural Processes but additionally include a local latent path and variable, with the main objective of an improved uncertainty calibration. To this end, bi-Lipschitz regularization is used to preserve distances from input to output. DNPs are evaluated on different rasks ranging from onedimensional regression to image classification, with the novel method typically showing strong accuracy and improved uncertainty calibration compared to competing methods such as Attentive NPs.", "questions": "1. The paper on ConvNP (Foong et al. 2020) states to propose a new maximum-likelihood objective replacing ELBO, how does the objective in the paper (cf. Appendix D) compare to it?\n2. It was surprising to see DNP being so overconfident in some settings (cf. Figure 3). What are the drivers behind this?\n3. Is there an alternative to the LOBPCG method applied?\nIf these questions are addressed satisfactorily, I may increase my evaluation score.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Quality: I was not able to identify a technical issue in the paper. A proof for Proposition 1 is provided in Appendix C. A strong point is that a variety of applications/experiments from different contexts is shown, including an illustration of predictive distributions/confidence intervals. In addition, substantial additional material is contained in the Appendix, and the code is available in the supplementary material. Overall, I believe the paper to be a thoughtful and sound work.\n\nClarity: The paper was quite clear and relatively straightforward to read. The motivation for the research is presented well, as is the justification of the bi-Lipschitz regularization. Most of the technicals are also presented in a straightforward manner. Much detailed information is provided. I have only smaller suggestions for improvement. What was most confusing to me was the description of Table 4/5 and the corresponding results. The differences between using the two different datasets as ID should be discussed properly. Also, why is latency only shown for the CIFAR10-case? Moreover, I find it suboptimal that some quite closely related work is discussed in Section 2 “Background”, while some is discussed in Section 4 “Related Works”. For example, I was aware of the existence of ConvNP (Foong et al. 2020) and was thus surprised to not see it being mentioned in Section 2, but then later found it to be mentioned in Section 4. Another smaller point is that I did not get the rationale behind sampling the number of context points in Section 5.2.\n\nSignificance: Machine learning methods which provide well-calibrated uncertainty estimates are, without doubt, highly valuable. I thus believe that the paper aims at an important contribution and it improves upon existing NPs. That being said, Figure 3, especially the cases (c) and (d), show that the suggested method DNP still can suffer from substantially underestimating uncertainty and thus yield estimations that are way too confident, even slightly OOD. This would warrant further discussion.\n\nOriginality: To the best of my knowledge, while bi-Lipschitz regularization has been applied to many variants of neural networks, applying bi-Lipschitz regularization to NPs is novel and promising. I also found the reasoning for originality to be convincing overall. \n\nIn sum, I found the paper to be an interesting read and wish the authors all the best for their work on uncertainty calibration.\n\nReference: Foong, A., Bruinsma, W., Gordon, J., Dubois, Y., Requeima, J., & Turner, R. (2020). Meta-learning stationary stochastic process prediction with convolutional neural processes. Advances in Neural Information Processing Systems, 33, 8284-8295.", "was_revised": true, "final_justification": "This is a paper of high quality with high significance. It showcases strong results across a variety of experiments from different contexts. The clarity of the paper was further improved during the discussion, and the questions have been answered satisfactorily. I thus recommend to accept the paper.", "rebuttal": "We thank you for the positive and constructive feedback. We are glad that the clarity, technical soundness, and significance of our work were recognized. Below, we address the specific questions and suggestions raised.\n\n**Clarifications on Tables 4 and 5**\n\nThank you for pointing this out. We will add the distinction between ID and OOD datasets in Table 4 and Table 5. Additionally, latency was reported only for the CIFAR10-ID setting as a representative case. The CIFAR100 experiment uses the same model architecture, with the only difference being the number of output classes (10 vs. 100), which had a marginal impact on inference speed. We will also include the following latency results for the CIFAR100 in Table 5 setting for completeness.\n\n| Model         | Latency (ms)        |\n|---------------|---------------------|\n| Deterministic | 3.662 ± 0.110       |\n| CNP           | 3.825 ± 0.116       |\n| NP            | 6.228 ± 0.117       |\n| AttnNP        | 8.424 ± 0.117       |\n| DSVNP         | 8.637 ± 0.125       |\n| Ours          | 6.876 ± 0.118       |\n\n**Organization of Related Work**\n\nSection 2 was primarily intended to provide background for readers less familiar with NPs, focusing on the core models CNP, NP, and AttnNP that our method builds upon. Due to space constraints, we chose to defer the discussion of more recent or orthogonal variants such as ConvNP and DSVNP to Section 4.\n\n**Sampling the Number of Context Points**\n\nWe sample the number of context points per task to reflect the meta-learning setting, where the model must generalize to tasks with varying context sizes. This sampling strategy improves robustness and better reflects the uncertainty and variability encountered in real-world tasks. Moreover, this is a standard procedure followed in most NP literature, including NP, AttnNP, and ConvNP, to ensure fair comparison and realistic evaluation.\n\n**Q1. ConvNP Objective function**\n\nThe ML-based objective in ConvNP involves sampling a latent variable $z \\sim p(z|D_c)$ from a ConvCNP encoder and marginalizing over it, without learning a posterior conditioned on the target set. This approach is effective for models focused purely on predictive performance and is simpler to train. However, it does not provide the explicit control over the latent space structure that our distance-aware design requires.\nIn our case, the use of a VI framework is essential due to the way we introduce inductive bias into the model. The local latent variable is designed to encode the geometry of the input structure. To effectively model and learn from this structure, it is crucial to maintain a posterior distribution over the latent variable that adapts to the task. The VI framework allows us to learn this posterior in an amortized manner and is therefore better aligned with the goals of the model than the ML-based objective of ConvNP.\n\n**Q2. Underestimation of Uncertainty**\n\nThis is a good observation. The underlying reason is that the bi-Lipschitz constraint does not strictly preserve input geometry; it only enforces approximate distance preservation within a bounded range. As a result, the model may sometimes underestimate distances between some inputs in the latent space, which can lead to overconfident predictions in certain regions.\nThis reflects an inherent trade-off: enforcing geometry too rigidly may compromise the model’s expressiveness, making it less able to adapt to complex or non-uniform data distributions. To manage this, we allow bounded flexibility in the Lipschitz constants, which provides a balance between geometric regularization and functional capacity. This trade-off and its impact on uncertainty are further explored in our ablations (see Tables 6 and 11).\n\n**Q3. Alternatives to LOBPCG**\n\nAn alternative to LOBPCG is the Lanczos algorithm, which is another iterative method for estimating the extreme eigenvalues of symmetric matrices. We considered this option. However, in our experiments, we found LOBPCG to offer more stable and accurate estimates and decided to use LOBPCG."}]

Author rebuttal:

Meta-review: The paper introduces the Distance-informed Neural Process (DNP) designed to improve uncertainty quantification and generalization in NPs. In addition to a global latent path as implemented in standard NPs the model employs a local latent path, which models input similarity explicitly. This is implemented by distance-preserving bi-Lipschitz regularization on the encoder network used to compute attention weights to local context. 

All reviewers agreed that this paper is well motivated, technically sound and well written. 
There were some concerns regarding the incremental nature of the work, and more specific concerns about comparisons and ablations, however the authors have addressed most of them in the rebuttal.
Decision: Accept (poster)