FedMC: Federated Manifold Calibration
Authors: Yanbiao Ma
Wei Dai
Gaoyang Jiang
wanyi Chen
Chenyue Zhou
Yiwei Zhang
Fei Luo
Junhao Wang
Andi Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Federated Learning
Distribution Calibrations
Geometric Knowledge
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: Data heterogeneity in Federated Learning (FL) leads to significant bias in local training. While recent efforts to introduce distributional statistics as priors have shown progress, they universally rely on a flawed global linearity assumption, failing to capture the nonlinear manifold structures prevalent in real-world data. This model-reality mismatch causes the calibration process to generate out-of-distribution (OOD) samples, which fundamentally misleads the model. To address this, we introduce a paradigm shift. We propose Federated Manifold Calibration (FedMC), a novel framework that learns and leverages the local, nonlinear geometry of data. FedMC employs local kernel PCA on the client side to learn fine-grained local geometries, and constructs a global "geometry dictionary" on the server side to aggregate and distribute this knowledge. Clients then utilize this dictionary to perform context-aware, on-manifold calibration. We validate our proposed method by integrating it with a wide range of existing FL algorithms. Experimental results show that by explicitly modeling nonlinear manifolds, FedMC consistently and significantly enhances the performance of these state-of-the-art methods across multiple benchmarks.

Published as a conference paper at ICLR 2026
FEDMC: FEDERATED MANIFOLD CALIBRATION
Yanbiao Ma', Wei Dai', Gaoyang Jiang', Wanyi Chen', Chenyue Zhou', Yiwei Zhang', Fei Luo", Junhao Wang's & Andi Zhang² GGooling School of Artificial Intelligence, Renmin University of China University of Warwick, Coventry, CV4 7AL, United Kingdom ybmal 9988 ruc. edu. cn, az381 cantab. ac.uk
ABSTRACT
Data heterogeneity in Federated Learning (FL) leads to significant bias in local training. While recent efforts to leverage distributional statistics show promise, they universally underpinned by flawed global linearity assumption, fail- ing to capture the nonlinear manifold structures prevalent in real-world data. This model-reality mismatch causes the calibration process generate out-of- distribution (OOD) samples, which fundamentally misleads the model. To address this, we introduce a paradigm shift. We propose Federated Manifold Calibra- tion (FedMC), novel framework that learns and leverages the local, nonlinear geometry of data. FedMC employs local kernel PCA on the client side to learn fine-grained local geometries, and constructs global "geometry dictionary" on the server side to aggregate and distribute this knowledge. Clients then utilize this dictionary to perform context-aware, on-manifold calibration. We validate our proposed method by integrating it with a wide range of existing FL algo- rithms. Experimental results show that by explicitly modeling nonlinear mani- folds, FedMC consistently and significantly enhances the performance of these state-of-the-ane methods across multiple benchmarks.
INTRODUCTION
Federated Learning (FL) has become a key paradigm for collaboratively training machine learning models while preserving privacy Konečny (016);; [Sun et al. (2020); Huang et al. (2024b); Qi et al. 2023). However, the non-Independent and Identically Distributed (non-llI) nature of client data, that is, data heterogeneity, remains the core obstacle to its success Li et al. (2020b); Tiang et al.
(022b); Ma et al. (2025a). To address this challenge, researchers have explored multiple technical avenues, from constraining model parameter updates Li et al. (2000a;; Acar et al. (2021); Lee et al. (002) to optimizing server-side aggregation strategies Xia et al. (2021); Sun et al. (2023); ) Chen et al. (022)). Among these efforts, promising direction is to leverage global statistical priors to guide local training Huang et al. (2023) /hou & Konukoglu (2023); Ma et al. (2025b), with the goal of providing isolated clients with a picture of the "global data landscape." Early attempts focused on sharing first-order moment information, for instance, by communicating class prototypes Mu et al. (2023) Huang et al. (2024a); Meng et al. (2024): Qi al. (2025b) give clients a sense of the global class centers. While these methods partially alleviate biases caused by missing classes, the information they provide is overly simplistic, depicting only the "position" of the data distribution
while ignoring its "shape."
To obtain richer description of the distribution, recent studies have begun to explore the use of second-order moments (e.g., covariance) by constructing global linear geometric model (i.e., hyper-ellipsoid) to characterize the "shape" of the data distribution and guide calibration Diino et al. (202a)); Ma et al. (2025a). Such methods have taken an important conceptual step, proving that explicitly modeling data geometry is an effective path to resolving heterogeneity. Nevertheless, these methods based global statistical priors, whether first- second-order, all share reliance on a fundamental yet flawed global linearity assumption. They attempt to summarize a complex data distribution with single, globally consistent, simple bodel-be point ellipsoid.
This assumption fails catastrophically when confronted with the data manifolds that are more rep- resentative of real-world scenarios TTnenbaum al. (2000); Ma al. (2025c). High-dimensional

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data is typically concentrated near low-dimensional, curved manifold M, rather than be- ing uniformly distributed throughout Euclidean space Lei et al. (2020); Ma et al. (2024); Qi et al. global
(025a). As illustrated in Figure for an S-shaped data manifold, any method based linear assumption will produce a severe geometric misinterpretation. For instance, global PCA will erroneously identify the Fuclidean "shortcut" between the manifold's endpoints as its principal component. Calibrating along this direction, which completely deviates from the data's 's true intrinsic path, will systematically generate oul-of-distribution (OOD) pseudo-samples that have zero proba- bility under the true data generating distribution, fundamentally misleading the model's learning process.
To overcome this fundamental deficiency, we argue that paradigm shift is necessary: from rely- ing on flawed, global linear models to new framework capable of understanding and leveraging the local, nonlinear geometry of data. To this end, we propose Federated Manifold Calibration (FedMC). Our method discards the global linearity assumption, instead adopting more flexible, localized strategy inspired by manifold learning. The core innovations of our framework are:
At the client side, we employ local kernel PCA to extract fine-grained, nonlinear geometric features that respect the local curvature of the manifold.
At the server side, we aggregate these diverse local geometries into a globally shared ge- ometry dictionary, which serves as an atlas of the manifold's 's morphology.
During calibration, clients dynamically query this dictionary for each of their data points to obtain a context-aware geometric prior, ensuring that the calibration process unfolds on the manifold. This entire process is realized within the distributed and privacy-constrained framework of federated learning.
Although the manifold calibration principle we propose is general, we implement and validate within the framework of Federated Prompt Learning (FPL), as FedMC is particularly suited to ad- dress the biased prompts learned on heterogeneous data. We choose FPL as our validation platform because it is an advanced, communication--fficcenn FL paradigm with particularly acute need for precise calibration. The main contributions of this paper can be summarized as follows:
We move beyond the limitations of the implicit global linearity assumption prevalent in federated learning by introducing a nonlinear manifold hypothesis. This initiative estab- lishes a more profound and realistic theoretical foundation for how the federated learning field can understand and utilize the intrinsic structure of data distributions.
We propose FedMC, a novel and general federated learning framework. It is capa- ble of learning, aggregating, and leveraging the local, nonlinear geometry of data a privacy-preserving manner, featuring core components such as the Global Anonymous Ba- sis (GAB), Secure Geometry Descriptors (LGDss, and the Geometry Dictionary.
We conduct extensive experiments across scenarios with label skew, domain skew, and a combination of both. Results show that FedMC significantly enhances the performance of multiple advanced FPL methods, demonstrating its effectiveness and broad applicability.
2 RELATED WORK
2.1 FEDERATED LEARNING WITH DATA HETEROGENEITY
The challenge of data heterogeneity in Federated Learning (FL), stemming from the non- Independent and Identically Distributed (non-IID) nature of client data Ma et al. (2025b); Li et al. (2024)). has catalyzed a vast body of research Ma et al. (202a)); Huang et al. (2024b). These works can be broadly categorized into several mainstream directions Huang et al.) (2022). The first cate- gory focuses on client-side regularization to constrain local updates Li et al. (2023)); Miao et al.) 20023). For instance, FedProx Li et al. (2020a) introduces a proximal term to limit the divergence between local and global models, while SCAFFOLD Karimireddy et al. (2020) employs control variates to estimate and correct for the "client drift" in local updates. HPL Huang et al. (2023) su- pervises the learning of local class prototypes by aggregating and sharing class prototypes across clients. second line of work centers on server-side aggregation strategies Chen et al.] (2023); Huang et al. 2022), such FedAvg Hsu al. (2019) which incorporates momentum smooth
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the update process, and FedOPT Reddi et al. (2021), which applies adaptive optimizers to the global model aggregation. A third category leverages globally shared information to guide local train- ing Ma et al. (2025a); Li et al. (2024b); Xiong al. (202)). This includes sharing distilled public datasets Shi et al. (2024), knowledge from the global model Lee et al. 2002]; ; Ma et al. (025a)) or,
more directly, sharing class prototypes Mu et al. (2023); Tan al. (2022) Huang et al. (2024a) to
provide clients with sense of the global class centers.
Despite their effectiveness, these methods are typically agnostic to the intrinsic data structure. That is, they either operate directly on the model's parameters or address the first-order moment (mean) deviations caused by imbalanced sample sizes or missing classes, but they seldom delve into the second -odde moment or the higher-order order intrinsic geometry of the data distribution.
2.2 FEDERATED PROMPT LEARNING
With the rise of Vision Foundation Models Radford al.] (021); Oquab et al. (2023) Parameter- Efficient Fine-Tuning (PEFT) techniques, particularly prompt learning Zhou et al. (2022); ; Gao et al.)
(2024), have been integrated into the federated framework to reduce communication overhead, giv- ing rise to Federated Prompt Learning (FPI.) [Su et al. (2024); Lu et al. (2023). Early research in FPL primarily revolved around the modality and sharing strategies of prompts. In the domain of vi- sual prompts, for example. FedPR Feng ct al. (2023) creatively proposed learning federated visual prompts in the null space of the model to minimize interference with pre-trained knowledge. Other studies have also noted that visual prompts can improve the privacy-utility trade-off with minimal expenditure of the privacy budget Li et al. (2023c). Research in textual prompts is equally rich. PromptFL Guo et al. (2023) is a representative work on learning single text prompt shared among all clients, while Fed Qiu et al. (2024) takes a step further by designing a unified prompt gen- eration network across clients achieve better generalization. Furthermore. an important branch of research is personalized prompt design, which aims to learn unique prompt for each client that adapted its local data distribution ei (2023a): Su et al. (2024).
Although these methods have significantly advanced the field of FPL., they mostly frame the core challenge as "how to better design and aggregate prompt parameters." They often overlook a more fundamental problem: during the local optimization process on heterogeneous data, the prompts learned by clients are inherently biased due to a lack awareness of the global distribution.
THE FAILURE OF THE GLOBAL LINEARITY ASSUMPTION
Before detailing our new framework, this section theoretically deconstructs the limitations of exist- ing geometry- based calibration methods Ma et al. (202aa) 2023). The core of these approaches is to compute global covariance matrix, Eglobal, and use its eigendecomposition,
A). to define "global geometric prior" for guiding local calibration: x' = x + Σm=1 C V Um- This formu- lation implicitly makes fundamental global linearity assumption: that the intrinsic structure of the data can be effectively approximated by a single linear subspace, span(U), passing through the global data center.
However, this assumption fails when confronted with the data manifolds that are more representa- tive of real-world. High-dimensional data is typically concentrated on or near a low-dimensional manifold, M, rather than being uniformly distributed throughout Euclidean space. On a manifold, the meaningful distance is the geodesic distance along its surface, not the Euclidean distance upon which PCA relies. For an S-shaped manifold, global PCA will erroneously identify the Euclidean "shorteut" between its endpoints as the principal component u1 This path traverses a void where no data exists, thus severely distorting the true geometry of the data.
This geometric misinterpretation has disastrous consequences. An ideal, manifold-aware calibration should operate within the local tangent space, T.M, at given data point z. Yet, the global subspace span(U) (U) found by PCA is almost never aligned with the local tangent space T..M for an arbitrary point on a curved manifold; the former represents flawed global average, while the latter captures the true, immediate geometry. Therefore, in a calibration step x' = x + d, the perturbation vector d SU) will almost certainly contain normal component, d_, that pushes the point off the manifold M. This means that calibration based on the global linearity assumption systematically generates duttof-distribution (OOD) pseudo-samples that have zero probability under the true data
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generating distribution. Training a model with these fundamentally flawed samples forces it to learn a spurious correlation between class labels and features. To achieve a truly precise calibration, must move from relying on flawed global linear
paradigm shift is therefore necessary: model to a new framework capable of understanding and leveraging the local, nonlinear geometry of the data. This provides the direct motivation for our proposed FedMC framework.
4 METHODOLOGY: THE FEDMC FRAMEWORK
Building upon the analysis in the previous section, we propose the Federated Manifold Calibration (FedMC) framework. This framework aims to learn and leverage the intrinsic manifold geometry of data within the strict privacy constraints of federated learning. It fundamentally departs from prior work by abandoning the global linear assumption in favor of localized, nonlinear approach. The framework's s architecture is structured into a preparatory, by two core iterative phases for federated training: (I) Federated Aggregation of Local Geometry and (II) Client-side Manifold-Aware Calibration.
setup phase, followed
4.1 SETTING: FEDMC FOR FEDERATED PROMPT LEARNING
We instantiate our FedMC framework within the context of Federated Prompt Learning (FPL). In a typical FPL setting with vision-language model like CLIP, a powerful vision encoder Evision (-) and text encoder Etext (-) are shared and frozen across all clients. The goal is to collaboratively learn prompt vector, P. which is a set of learnable parameters. The core challenge addressed by our work stems from the input to the learning process. Each client & feeds its local images through the frozen encoder obtain set of image embeddings, D2 {2g]10 = Fvissoon (image,)) ]. Due to data heterogeneity, this local embedding set Dj represents only a small, biased patch of the global data manifold M. Training a local prompt P& solely on D3 leads to significant bias.
FedMC directly tackles this issue by operating in the image embedding space. Our framework does not alter the raw images or the frozen encoder. Instead, it calibrates the biased local embedding set DE to generate an enhanced set Dr that better reflects the global manifold structure. The local prompt is then trained on this calibrated set, effectively debiasing the learning process at its source. In the following sections, any reference to "data point x" pertains to these image embeddings.
4.2 PREPARATORY PIASE: GLOBAL ANONYMOUS BASIS CONSTRUCTION
The foundational challenge for any collaborative geometric learning in federated setting to establish common language, or coordinate system, for describing geometric structures without violating client privacy. naive approach where clients share geometric information defined by their own local data points would constitute a severe privacy leak. This preparatory phase is designed to solve this problem by constructing a Global Anonymous Basis (GAB)-a shared set of reference points that is collectively generated and decoupled from any single client's raw data.
Local Prototype Generation and A nonyymization: Each client first identifies representative land- marks on its local data manifold by performing K-Means clustering its embeddings Dj to obtain a set of prototypes { To ensure these landmarks can be shared without revealing precise infor- mation about the client's 's data distribution, we employ Differential Privacy. Each client perturbs its prototypes with calibrated Gaussian noise to satisfy (c, D).P:
ريج = Ck. + (0,
Clients upload only these anonymized prototypes (Ōk,) ) which carry statistical information about local data density while providing formal privacy guarantees.
The server aggregates the thousands of anonymized proto- types from all clients. This pooled set represents noisy, collective snapshot of the entire global data manifold. To distill a stable and representative basis from this set, the server performs a global clus- tering (e.g., K-Means) on it, identifying the Nusse most significant centroids. This resulting set of Nuuse points forms the Global Anonymous Basis (GAB), denoted B, {bi.... .bi...> J. Crucially, the GAB is inherently anonymous because is derived from a mixed and noisy pool of prototypes from all clients, making it impossible to trace any basis point back to specific client's original data. The server then distributes this common basis B, to all clients, equipping them with a shared, privacy-safe coordinate system for all subsequent geometric communications.
GAB Formulation and Distribution:
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4.3 PHASE I: FEDERATED AGGREGATION OF LOCAL GEOMETRY
With the GAB in place as a common language, we now describe the iterative process within each federated learning round. The goal is to build global understanding of the manifold's geometry by aggregating insights from all clients. This phase details the secure, two-part process to achieve this: (1) how clients can extract rich, nonlinear geometric information from their local data and represent it in secure, standardized format, and (2) how the server can then fuse these individual representations into a coherent global geometry dictionary that benefits all participants.
4.3.1 CLIENT-SIDE: SECURE LOCAL GEOMETRY REPRESENTATION
To capture the manifold's s local curvature, we first approx- imate local regions where the geometry is assumed to be consistent. The client partitions its local data D3 into clusters { using K-Means. For each cluster C₁. which represents an approximate local patch of the manifold, the client employs Kernel PCA to model its nonlinear structure. The process begins by constructing the Gram matrix K, using the Gaussian (RBF) kernel:
Local Geometry Extraction via KPCA:
k(x, v) = exp(~>1l= | yl1),
(2)
where the hyperparameter controls the kernel's bandwidth and thus the perceived scale of locality. After centering the Gram matrix to obtain K necessary step to compute variance in the high- dimensional feature space H-an eigendecomposition is performed:
Kjaj, Jj,
(3)
This procedure yields the eigenvalues Ajas quantifying the variance along each principal direction, and the corresponding coefficient vectors Czj.. Crucially, these outputs implicitly define the nonlin- ear principal components in H:
U3S = Σ (ags)a
(4)
a=1
These components form an orthonormal basis that effectively describes the geometry of the local manifold patch by capturing its directions of maximum variance.
Projecting Geometry for Secure Representation: Having defined the local principal components, the client faces the central privacy challenge: how to share this geometric information without re- vealing the underlying data {za] used its construction. Directly transmitting the coefficients a along with the basis points { would constitute a severe data leak. To overcome this, we introduce a crucial projection step that acts as a secure informmtiio-encooiinn mechanism. The client leverages the shared Global Anonymous Basis (GAB), By. as universal language for describing geometry. represents each its local principal components Vj,i by projecting it onto this global basis. The resulting representation is an Nhase -ime-dimmnii coefficient vector Bi.,. where each element is the projection of Us. onto the corresponding basis vector (bb.):
(Br,), (UJ,.. ((b))n
Σ (ass)a
(5)
The brilliance of this step lies in converting geometric information that depends on private data {za} into a standardized coordinate vector By, expressed purely in the public, anonymous basis B,,. thus achieving secure and communicable representation. The vector By. is a secure and standardized rep- resentation of the local geometry. It is secure because is expressed solely in terms of the public, anonymous basis B,, effectively decoupling it from the client's private data {x,,, J. It is standardized because all clients will now use the same basis for their reports, making their geometric findings di- rectly comparable and aggregatable. The client then uploads Local Geometry Descriptor (LGD) containing only this secure information:
LGD = (e,, ny, ((,),,1,,,))1
where df is the number of principal components retained. To make the process of generating a Secure Local Geometry Descriptor concrete, we provide Python-style pseudocode in Listing ??
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4.3.2 SERVER-SIDE FUSING GEOMETRIES INTO GLOBAL DICTIONARY
The server's role is to act as curator, intelligently fusing the multitude of received LGDs from all clients into compact, globally consistent Geometry Dictionary. This dictionary is not a simple average, but rather a structured atlas that maps different regions of the manifold to their unique, consensus-based geometric shapes.
Grouping by Region Sieta-Clustering) Upon receiving LGDs, the server first groups them by performing meta-clustering the anonymous prototypes {č,} thereby identifying which LGDs describe the same macroscopic region the global manifold.
Fusion of Geometrie Templates: For each macroscopic region (meta-cluster) I, the server must now fuse the geometric information from all constituent LGDs into a single, representative tem- plate. Because all the geometric vectors B have been standardized onto the common GAB coordi- nate system, this complex fusion task simplifies to a secure and robust weighted averaging process. The server computes fused coefficient vector Bi, and fused eigenvalue Mi, for each principal component
EjennddgaBe Eye: nzAga
Ejer Ni, Ej
Bi,
The weighting scheme, incorporating both the number of local samples n, and the local variance captured by the eigenvalue XI.i. intuitively ensures that LGDs from data-denser and more geomet- rically pronounced regions contribute more significantly to the final template. This approach is not merely heuristic; we demonstrate in Appendix C.2 that this weighted average is the optimal solu- tion weighted least squares problem, which seeks geometric consensus that best represents the collection of local geometries. This provides a strong theoretical justification for our fusion strategy, resulting in single, robust geometric template for each macroscopic region of the manifold.
Dictionary Assembly and Distribution: Finally, the server assembles these fused templates into the global Geometry Dictionary D. Each entry in the dictionary is a key-value pair, mapping a macro- macro-prototype prototype gI (the centroid of the meta-cluster) to its corresponding fused geometric template:
Entry; = (gi, ((M,, ML.)
This completed dictionary, a comprehensive and privacy-preserving map of the global data mani- fold's geometry, is then distributed to all clients to guide the next phase of calibration.
4.4 PIIAE II: CLIENT-SIDE MANIFOLD-AWARE CALIBRATION
Armed with the geometry dictionary, the client performs a sophisticated local calibration. The pro- cess can intuitively understood as a "map-based navigation": for any given data point, the client first looks up the "local map" (the most relevant geometric template) from the globally-provider "atlas" (the dictionary). It then calculates its own position on this map, moves to a new plausible position the same map, and finally translates this new map coordinate back into real-world data point. This process for a given local data point x unfolds in a sequence. First, the client per- forms Dynamic Geometry Query, finding the most relevant template Entry, from the dictionary by identifying the nearest macro-prototyy gt-:
1" = arg min
||
(9)
It worth noting that the Fuclidean distance is not used here to define the intrinsic shape of the man- ifold. Euclidean distance serves to efficiently localize the data point x within the global "geometry dictionary." On local scale, Euclidean distance acts computationally efficient and reasonable proxy for geodesic distance, which enables rapid identification of the most relevant geometric tem- plate. The actual manifold-awar calibration is then carried out using the nonlinear structure of the selected template, not the distance metric itself. Next, the calibration itself is performed within a d-dimensional subspace of H spanned by the retrieved principal components, which serves as an approximation of the manifold's 's local tangent space. This involves three steps:
Projection onto Principal Components: We first need to compute the projection of ((x) onto each principal component ... Using the kernel trick, this projection value p; can be

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computed without explicitly forming (x)) or of
Nuse
P: (0(x), 28 dn
Σ (B). ()) b.).
(10)
2. Perturbation in the Principal Component Space: We perturb the projection vector p [p...... Pel in the d-dimensional principal component space to get p':
ps = P Mr
where
& ~0,11,
(11)
3. Reconstruction of the Perturbed Feature Vector: The perturbed nigh-dimensional fea- ture vector (xr) can be expressed as ((x Σd P/
The perturbed vector (x)) resides in the high-dimensional feature space and cannot be used di- rectly. We need find its pre-image x' the original input space RP. We solve this classic inverse problem using an iterative optimization approach. The goal is to find an x' that minimizes the distance between (2) and the target vector 4 (xr) in the feature space. This can be formulated as:
x'
arg min [ (z))
(12)
=ER
This objective can be solved with gradient descent. The target inner product between (a) and any basis point bş is pre-computed
Nus Σ [=1
(Bia)) ek (be. (brelbe) (hueltss)
T, ((b))) = Σ M (vi.. (bb))
pl
(13)
The optimization then minimizes the loss function:
Nuw
C(x') Σ (k('', -T,)².
(14)
It is important to clarify that while the true data manifold MM is unknown, our calibration is designed to produce an on-manifold approximation. Specifically, the perturbation is performed in the local tangent space approximated by kernel PCA, ensuring the update direction aligns with the mani- fold's intrinsic geometry rather than a global Euclidean shortcut. The pre-image optimization then seeks point whose feature- space embedding matches this on-manifold target, thereby avoiding the systematic generation of ouito--distributiin samples that plagues global lincar methods. Starting with initialization (c.g., 2 2),, we iteratively update x' until convergence using gradient-based methods. The detailed derivation of the gradient for the loss function Eq. 14 is provided in Ap- pendix CR to facilitate implementation. While iterative, this optimization is performed locally the client and its computational cost is manageable. The resulting x' is our calibrated data point. This calibrated embedding x' along with its original label y forms new training pair (x', y). The client's local prompt PA is then updated by minimizing loss function (e.g.. cross-entropy) on this augmented dataset of calibrated embeddings. The full calibration pipeline for single data point is illustrated in the Python-stye style pseudocode in Listing 2 (Appendix C.1.2). The entire process of our FedMC framework summarized in Algorithm [ (Appendix C.I.1). We also provide scalability experiments for FedMC in scenarios where both the number of local data points and the number of clients may be large in Appendix B.3.
5 EXPERIMENTS
To comprehensively and rigorously evaluate the effectiveness and versatility of our proposed Feder- ated Manifold Calibration (FedMC) framework, we conduct series of extensive experiments. This section aims to address two fundamental research questions:
Can FedMC significantly enhance the performance of -tate-of-the-ar Federated Prompt Learning (FPL) methods across various data heterogeneity scenarios?

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Table 1: Performance comparison on Label Skew Scenarios. Our proposed FedMC (FedVTP) is benchmarked against SOTA methods and global linear geometry baseline, GGEUR (Fe) Best results are in bold and second-best are underlined.
CIFAR-100
Tiny-ImageNet
Methods
B = 0.5
8 = 0.3
B 001
3 0.5
8 = 0.3
8 0.1
ZS-CLIP
64.87
63.67
State-of-ther Art Baselines
FedVPT
83.53
83.18
80.99
75.91
75.67
74.30
FedCoOp
78.42
77.89
74.67
76.53
75.61
72.58
FedVTP (Base)
84.90
84.26
81.01
80.97
80.26
77.58
PromptFL
75.05
75.36
72.19
72.66
71.51
67.85
FedTPG
71.40
70.95
68.63
67.63
66.72
64.71
FedPR
81.17
80.44
78.91
72.59
72.22
70.80
FedCLIP
72.03
71.20
70.64
70.41
70.37
69.50
Our Proposed Framework and Key Baseline
GGEUR (FedVTP) 85.21
84.55
82.55
81.15 81.53
80.35 80.85
78.02
FedMC (FedVTP) 86.72
85.90
85.08
80.12
Is the core principle of FedMC generalizable, extending beyond the FPI. paradigm to em- power broader range of Federated Learning (FI.) algorithms?
To answer these questions, experiments structured into main parts. In the first part (Section 5.2). we enhance FedVTP Jia et al. (2022) with FedMC and compare it against other ad- vanced FPL methods. the second part (Section we apply FedMC to several classic, non-FPL algorithms to validate its capability as a general-purpose framework.
5.1 EXPERIMENTAL SETUP
Datasets. We evaluate our methods across comprehensive suite of six benchmark datasets covering three types of data heterogeneity. For label skew, we use CIFAR-100 Krizhevsky Krizheevkky et al. (2009) and Tmay-ImageNet Deng al. (2009). For domain skew, (017),, Office-31 Gong et al. (2012), and DomainNet Peng et al. (2019). For the combined label and domain skew scenario, employ Office-Home--.DS Ma et al. (2025a). Detailed descriptions
Office-Home /enkateswara et al.
of these datasets and the specific heterogeneity settings are provided in Appendix B.
Compared Methods. In Part (FPL), enhance FedVTP Jia al. (2022) with FedMC and
implemented linear baseline, GGEUR Ma et al. (2025a). We compare against SOTA FPL methods including FedCoOp Zhou al. (2022), PromptFLL Guo al. (2023), FedCLIP Lu al. (003))
FedPR Feng et al. (2023), and Fed Quu et al. (2024). In Part (General FL.) we apply FedMC
and GGEUR FedAvg McMahan et (2017). SCAFFOLD Karimireddy al. (2019). Li
(2021), FedDyn Acar et al. (2021), FedOPT Reddi et al. (2021), FedNTD Lee et al. 2022 and
FedProto Tan et al. (2022).
Implementation Details. All FPL experiments use the pre-trained CLIP ViT-B/16 model as backbone. To ensure fair comparisons, all experiments are conducted with unified federated hy- perparameters (e.g.. T = 50 communication rounds, E 1 local epoch). The performance of all methods measured by the average classification accuracy. complete breakdown of network architectures, federated hyperparameters, and our FedMC parameters can be found in Appendix(B.2
5.2 PART FEDMC EMPOWERS FEDERATED PROMPT LEARNING
In this section, demonstrate that FedMC boost the performance of an advanced FPL method, FedVTP. We present the main results in Table and Table 2 comparing our enhanced method against all baselines.
as plug-and-play module significantly
5.2. PERFORMANCE ON LABEL SKEW SCENARIOS
presenns the results on the CIFAR- 100 and Tiny- ImageNet datasets, which are characterized by severe label distribution skew. The results clearly show that proposed FedMC (FedVTT)
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Table 2: Performance comparison Domain Skew Scenarios. As in the label skew setting, our method consistently outperforms all baselines, demonstrating its robustness handling complex domain shifts. Best results are in bold and second-bess underlined.
Office311
Office-Home
DomainNet
Methods
3 0.5
B - 0.3
3 0.1
3 0.5
3 0.3
8 0.1
8
0.5
3 0.3
0.1
ZS-CLIP
75.76
82.35
73.68
State-of-the-Art Baselines
FedVPT
91.76
90.64
91.35
87.83
87.89
87.81
82.57
82.12
81.56
FedCoOp
95.17
94.48
94.96
88 18.14
88.00
87.77
82.26
82.20
81.34
FedVTP (Basc)
94.93
95.32
94.58
89.11
89.09
88.92
84.54
84.11
83.82
PromptFL
93.38
94.47
94.19
87,41
87.15
87.52
79.16
79.00
78.59
FedTPG
92.38
91.06
91.19
86.73
86.43
85.91
77.71
77.65
77.34
FedPR
91.65
92.30
91.03
87.33
87.00
86.40
80.77
80.75
80.11
FedCLIP
89.69
89.66
90.27
85.29
85,32
85,23
78.76
78.71
78.61
Our Proposed Framework and Key Baseline
GGEUR (FedVTP)
95.24
95.45
94.71
89.35
89.42
89.15
84.65
84,30
83.85
FedMC (FedVTP)
96.88
96.95
96.12
91.26
91.45
91.03
86.51
86,25
85.93
consistently and significantly outperforms all state-of-the-art baselines, including the strong FedVTP method, across all degrees of data heterogeneity. For instance, in the most challenging setting Tiny-ImageNet with 3 = 0. our method achieves an accuracy of 80.12% %, securing a significant 1.74% improvement over FedVTP. Notably, the label skew becomes more severe (i.e., as 3 decreases from 0.5 to 0.1), the performance of methods degrades. However, the performance drop of FedMC (FedVTP) is markedly less pronounced. This widens the performance gap between our method and the baselines, highlighting its superior robustness. This stems from FedMC pro- vides much richer geometric prior by capturing the manifold's curvature. This allows the client to generate high-fidelity, en-manifold pseudo-samples that offer a more accurate representation of the global distribution, thereby mitigating the bias from local data more effectively.
The direct comparison with GGEUR (FedVTP) further validates our core hypothesis. Across all settings, the performance of FedMC (FedVTP) far superior that of GGEUR (FedVTP), leading by margin of 2.53 % on CIFAR-100 (B = 0.1). This provides powerful empirical evidence that modeling the nonlinear manifold geometry fundamentally more effective than relying on flawed global linear approximation.
5.2.2 PERFORMANCE ON DOMAIN SKEW SCENARIOS
The superiority of FedMC is even more pronounced in the more challenging domain skew scenar- ios, achieving a remarkable 2.05% lead on the difficult DomainNet dataset (3 = 0.1), as detailed in Table This is because methods based on prototype averaging or linear assumptions fail when confronted with distinct client manifolds (e.g., photos vs. sketches). In contrast, FedMC's "ge- ometry dictionary" learns these domain-specinf geometric signatures, enabling a far more effective calibration and leading to more generalizable prompt.
5.3 PART FEDMC AS GENERAL FRAMEWORK
To validate its universality, we apply FedMC a diverse suite of seven foundational FL meth- ods. The results, summarized in Table 3 show that FedMC provides consistent and significant perfor- mance gains across all algorithms, confirming its value general-purpose enhancement module.
Table 3: Generality of FedMC. Performance (% accuracy) on Office-Home-LDS (B
0.1) across broad suite of FL. algorithms.
FL Algorithm
Base
+GGEUR
+FedMC
FedAvg
70.14
83.99
85.11
SCAFFOLD
74.82
83.96
85.25
The most critical evidence comes from the compar- ison with FedProto. FedProto is a strong baseline that already mitigates heterogeneity by sharing first- order statistics (prototypes). The significant addi- tional gain provided by FedMC powerfully demon- strates that sharing statistics representing data posi-
MOON
76.83
78.08
80.73
FedDyn
65.99
84.09
86.32
FedOPT
65.59
84.20
86.58
FedNTD
75.46
82.46
84.91
FedProto
69.40
83.35
85.92
tion is insufficient. By capturing and leveraging higher-ordee geometric information that defines the manifold's shape, FedMC provides more fundamental calibration signal, Across all methods,

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Published as a conference paper at ICLR 2026
the performance uplift from FedMC is consistently greater than that from GGEUR, reaffirming the universal superiority of the nonlinear manifold hypothesis.
CONCLUSION
This paper identifies and rectifies flaw in existing geometry-aware FL methods: their liance on flawed global linearity assumption. We propose FedMC, a novel framework that instead embraces more realistic nonlinear manifold hypothesis, learning local data geometry and per- forming privacy-preserving, on-manifold calibration via a global geometry dictionary. Extensive experiments demonstrate that FedMC provides significant and universal performance gains across both state-of-ther FPL and wide range of classic FL algorithms, establishing more faithful geometric foundation for tackling data heterogeneity in federated learning.
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A APPENDIX
Here we provide summary of the key notations used throughout the paper for easy reference.
Table Summary of key notations.
Symbol
Description
Total number clients.
Dy
The set of local data embeddings for client k.
x, y
data embedding and its corresponding label.
x'
A calibrated (augmented) data embedding.
M
The underlying low-dimensional data manifold.
Emision (-)
The frozen vision encoder (e.g., from CLIP).
Pr.
The global prompt model at communication round r.
FcdMC Framework Components
Bg
The Global Anonymous Basis (GAB), set of shared reference points {ba}.
Nuuse
The number of basis vectors in the GAB.
k(r,y)
The Gaussian (RBF) kernel funetion.
(())
The mapping to the high-dimensional feature space H.
Coefficient vector for the i-th principal component of local cluster j.
The secure representation of principal component, projected onto the GAB.
LGD
Local Geometry Descriptor, uploaded by the client.
D,
The global Geometry Dictionary at round
B EXPERIMENTAL DETAILS
This appendix provides a detailed description of the datasets and implementation settings used in experiments.
B. DATASET DETAILS
Our dataset selection is consistent with recent state-of-the-art benchmarks Qiu et al. 2024 and covers three primary types of data heterogeneity.
Label Skew. We evaluate performance on two single-domain image classification datasets where clients have imbalanced class distributions:
CIFAR-100 Krizhevsky et al. 2009): Contains 100 classes with 50,000 training and 10,000 validation images.
Tiny-ImageNet [eng et al. 2009 A subset of ImageNet with 200 classes and 100,000 training images.
Heterogeneity Setting: Following Qiu al. (024). we partition the data among 10 clients using Dirichlet distribution, Dir(B). We experiment with / E {0.5, 0.3, 0.1}. where smaller / value indicates higher degree of label skew.
Domain Skew. We use three standard multi-domain datasets where each client's data comes from a different feature distribution (domain):
Office-Home Venkateswara et al. (017): Comprises 65 classes across 4 domains: Art (A), Clipart (C), Product (P), and Real World (R).
Office-31 Gong al. (2012): Consists of 31 classes from 3 domains: Amazon (A), DSLR
(D), and Webcam (W)
DomainNet Peng et al. (2019): A large-scale dataset with 345 classes across domains: Sketch, Real, Painting, Clipart, Infograph, and Quickdraw.
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Heterogeneity Setting: Consistent with Qiu et al. (2024), each domain is assigned to distinct set of clients. The total number of clients is set to be twice the number of domains (e.g., 8 for Office-Home).
Label and Domain Skew. To evaluate performance in a more realistic and challenging setting where both label and domain shifts coexist, we employ the Office-omee-LDS Ma al. (2025a). This dataset is constructed from Office-Homa While ensuring that each client is assigned data from only one domain, the number of samples per class on each client determined by coefficient matrix generated via a Dirichler distribution with parameter f3. This setup creates complex scenario with simultaneous domain and label heterogeneity.
B.2 IMPLEMENTATION DETAILS
Network Architecture. For all FPL-related experiments (Part 1), we use the pre-trained CLIP VIT-B/16 model as the backbone, consistent with Qiu et al. (2024). For the experiments in Part 2, we use standard architectures appropriate for the datasets (e.g., ResNet-18 for CIFAR-100).
Federated Settings. To ensure fair comparison, all experiments are conducted with the following unified federated hyperparameters: communication rounds T = 50, local epochs E = 1, SGD optimizer with momentum 0.9, batch size 32, and initial learning rate of 0.001.
Mc Parameters. For our FedMC framework, the size of the Global Anonymous Basis (GAB) set to Nase 512. The number of local clusters per client is m 3. For Kernel PCA (KPCA), we employ the Gaussian (RBF) kernel. Its bandwidth parameter was set to standard and effective default, ? 1/d, where is the dimensionality of the feature embeddings (e.g.. d 512 for VIT-B/16).
Evaluation Metric. The performance of methods is measured by the average classification accuracy on global, unified test set, evaluated after the final communication round.
B.3 SCALABILITY AND COMPUTATIONAL OVERIEAD ANALYSIS
Kernel-based manifold learning methods are often criticized for poor scalability with large datasets. However, in the context of Federated Learning (FL), FedMC is carefully designed to mitigate this issue through localized computation and structural constraints, ensuring its practicality even in large- scale settings.
Local Computation is Bounded and Amortized. All kernel-dependent operations (K-Means clus- tering, Kernel PCA, and pre-image reconstruction) are performed locally on each client and only per communication round, not per local training step. Crucially, instead of applying Kernel PCA to the entire local dataset Dj. we first partition into m small clusters (e.g., m experiments). Since Kernel PCA has O(r²) complexity in the number of samples n, this clustering step effectively caps the input size for each KPCA instance, rendering the per-client cost manageable [Dx] grows.
in our
Empirical Time Overhead is Modest. To quantify the practical overhead, we measure the average training time per communication round for FedVTP with and without FedMC on three large domain- skew datasets. As shown in Table 5) FedMC introduces only small time increase- from 2.8s 4. 4. per round while delivering substantial accuracy gains (cf. Table 2.
Table 5: Average training time (seconds) per communication round.
Method
Office-31
Office-Hoone DomainNet
FedVTP
38,4
40.8
34.5
FedMC (FedVTP)
41.2
44.5
38.6
Scalability with Respect to Number of Clients. The server-side operations are highly scalable. The Global Anonymous Basis (GAB) is constructed once offline. In each round, geometry fusion
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involves only weighted averaging of vectors (Eq. 7. which scales linearly with the number of partic- ipating clients. To validate robustness under massive client settings, we evaluate FedMC on CIFAR- 100 (3 0.1) with K 100, 300, and 500 clients. As shown in Table 66 FedMC consistently outperforms the base Fe across all scales, confirming its resilience to client growth.
Table 6: Accuracy (%) on CIFAR-100 (3 0.1) with varying numbers of clients K.
Method
K 100
K -00
K 500
FedVTP
75.73
73.51 76.44
69.85
FedMC (FedVTP)
77.95
73.26
Computation Communication Trade-off. In FL, communication is typically the dominant bottleneck. By investing modest additional local computation to generate high-fidelity, on-manifold samples, FedMC enables faster convergence and higher final accuracy-often reducing the total number rounds needed to reach a target performance. Thus, the slight increase in local compute not only acceptable but strategically advantageous in Bandwidth-constrainee federated environ- ments.
C DETAILED ALGORITHM AND PSEUDOCODE WALKTHROUGH
In this section, we provide the main algorithm for FedMC and detailed, commented pseudocode for its key components. This is intended to clarify the implementation steps and rationale behind our framework. First, we present the overall workflow of FedMC as applied to Federated Prompt Learning. Algorithm [ shows the complete process, including the preparatory GAB construction and the iterative training loop.
C.I WALKTHROUGH KEY CLIENT-SIDE FUNCTIONS
To further demystify the core mechanisms of FedMC, we now provide a step-by-step walkthrough of the two most critical client-side functions, accompanied by Python-style pseudocode.
C.1.1 GENERATING SECURE LOCAL GEOMETRY DESCRIPTOR (LGD)
The primary challenge in sharing geometric information is privacy. Our solution, detailed in Section 4.2.1, is to transform locally computed geometry into a secure, standardized format using the shared GAB. This process involves three main steps:
1. Local Geometry Extraction: The client first partitions its local data embeddings into small clusters, each assumed be relatively flat "patch" on the manifold. On each patch, it performs Kernel PCA to find the principal directions of variance in a high-dimensional feature space. This captures the local, non-linear structure.
2. Secure Projection:
This is the crucial components (which depend on private data), the client projects them onto the public GAB. As shown in Eq. 5 and the code below, this results in a coefficient vector 'betas' which describes the local geometry purely terms of the anonymous basis vectors. This representation now safe to share.
step. Instead of sharing the principal
3. Assembling the LGD: Finally, the client packages this secure geometric information (the 'be- tas'), along with variance information ('lambdas') and other metadata, into a Local Geometry De- scriptor (LGD) for upload.
The entire process is illustrated the pseudocode below.
C.1.2 PERFORMING MANIFOLD-AWARE CALIBRATION
Once client receives the global Geometry Dictionary, it perform on-manifold calibration for its local embeddings, as described Section 4.3. This process effectively generates high-quality, synthetic data points that lie on the learned global manifold.
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Algorithm FedMC for Federated Prompt Learning
1: Server Initializes: Pre-trained frozen encoders Evision: Etent-
2: // Preparatory Phase: GAB Construction
3: Server requests initial prototypes from all K clients.
4: for each client E {1..... K) in parallel do
5: Extract embeddings D Bessioon (img)
Looallmages
6: Compute local prototypes {ck.j) } from Dk-
7: Anonymize and send prototypes {Čt.j} } to server.
8: end for
9: Server aggregates prototypes and computes GAB, Bg-
10: Server initializes global prompt Po and distributes it with By to all clients.
Initialize Geometry Dictionary Du + 0.
// Federated Training Loop 13: for round r = 0, 1, . R 1--1 do
14: Server selects a subset of clients Sp.
15: Server broadcasts current prompt P. and dictionary D, to clients in S,..
16: Let LGDaI + 0.
17:
for each client & E S, in parallel do 1 LGD, + CliendPropttpplate(ft P,, Dr. Bg).
18:
19:
Client k sends updated prompt pk and LGD; to the server.
LGD + LGD LLD
21: end for
Server-Side Aggregation
23:
e.g., FedAvg Fuse geometries: D,+1 +- FuseGeeometries((GG....)) Meta-cluster weighted avg (Eq. 7)
Update global prompt: Pr+1 +- Aggregate( 1
24:
25:
end for
26:
function CLIENTRRMPPUPPATTEKK.PP
27:
Let local prompt Pa < P,-
28:
Extract embeddings from local images: Dj (Evision (img)]img LocalImages,
29:
Phase I: Manifold-Aware Calibration
30:
DR 0.
31:
for each embedding label pair (x, y) E D2 do
Generate calibrated embedding x' from x using D, and By.
Eqs. 9-13
33:
D + Dr (((2,l()))))))))
34:
end for
35:
// Local Prompt Training
36:
Train local prompt P& on calibrated dataset Dr for E epochs to get pk Phase I: Local Geometry Extraction Extract and securely represent local geometry from D3 to get LGD,. return P LGD,,
r+1
38:
>> Eq.
40:
end function
1. Dynamic Geometry Query: For any given embedding 'x', the client first "looks up" the most relevant geometric map in the dictionary by finding the closest prototype. This ensures the calibra- tion is context-aware and uses the appropriate local geometry.
On-Manifold Perturbation: The client projects 'x' onto the principal components of the re- trieved geometric template. It then adds a scaled random noise to these projection coordinates. This analogous to taking small, random step, but crucially, this step constrained the directions that are "valid" on the manifold (i.e., the tangent space).
The perturbed coordinates now exist in the sigh-dimessional fea- ture space. The final, challenging step is find the corresponding point 'x" back in the original embedding space. This is classic inverse problem which we solve via optimization, seeking an "x" whose feature-space representation closest perturbed target.
Pre-image Reconstruction:
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Published as a conference paper at ICLR 2026
Fython-style Pseudocode for Secure Geometry Representation
def serract_secure__oalaggeeeee n_compenents):
CAB, n_clusters,
Client-side function to extract local geometry and represent it securely.
Corresponds to Section 4.2.1.
local__ _embeddings:: Emoeddings from the client's s private data CAB: Global Anonymous Basis received from the server
Partition data into local patches
clusters KMeans (n_clusters) fit_predict [oocal_mbeddiiggs)
14
client LGDs = []
for cluster_ic id in range (n (n_clusterss
cluster_points local_embeddings [clusters cluster_id]
# 1. Local Geometry Extraction via Kernel PCA
19
kpca = KernelPCA www.eaocccccnetetecccpoooooooeee kpca. fit (cluster_points)
kernel="rbf")
30
22
Lambdas are the eigenvalues (variance)
15
$ alphas are the coefficients in the dual space lambdas = kpca. alphas apca.dual_coef_
36
77
$ 2. Secure Projection ento GAB (The key privacy step Eq. 5) # This converts private gecnetry (defined by alphas and cluster_peints:
28
into public representation (betas) using the GAB. kerncl_matrix røf_kernel (cluster_points, CAB) k(xa,, b_s) betas T 8 kernel_matrix
30
31
Shape: (t_components, b_sas))
32
# 3. Assemble the Local Gecmetry Descriptor (LGD) descriptor
"center_prototype" np. mp.mean (cl ptss, axis-0), "nun samples"; Len cocluster_pointtee "geonetry"; (1am, beta) for 1am, beta in zip (lambdas, betas)
6
88
client_LGDs. append (descripter)
return client_IGDS
Listing 1: Python-style pseudocode for generating Local Geometry Descriptors (LGDs) on the client side. This illustrates the core process of local KPCA followed by secure projection the GAB.
This calibration pipeline is shown in the pseudocode below.
C.2 JUSTIFICATION FOR SERVER-SIDE GEOMETRIC FUSION
Section 4.2.2 (Eq, 7 of the main paper, we proposed weighted averaging scheme for the server to fuse local geometric information from multiple clients into single, representative template. Here, we provide a theoretical justification for this scheme by demonstrating that it is the optimal solution to well-posed optimization problem.
Problem Formulation. Consider meta-cluster the server, which groups together set of Local Geometry Descriptors (LGDs) deseribing the same region of the manifold. For each principal component index the server receives set of secure representations (Bjabeet from the clients in
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Published as a conference paper at ICLR 2026
Fython-style Pseudocode for Manifold-Aware Calibration
def calibrate__mbeddiing (x, geometry_dictionary, GAB)
Client-side function perform on-manifold calibration for embedding x. Corresponds to Section 4.3.
A single embedding vector to be calibrated geometry_aictional The global atlas received from. the server
0
0
1. Dynamic Geometry Query (Fq. 8)
Find the most relevant geometric template from the dictionary prototypes
for entry in geometry_dictionaryy best entry idx find nearest (x, prototypes)
55
template = geometry ditp:ryyyddtiioorybeeeeeee htwpr//nnrryyeettttetreee.e entrr_iitryyy
fased_lambdas template ("lambdas""""
fused_betas template ["botas"] # Shape: (n_components, N_baso)
9
20
# 2. Projection onto Principal Components (Eq. 9) Project onto the local tangent space defined by the template kernel_vec rbf_kernel rx.reshape (1, -1), GAB) # k b_s) projections peses kernel_vec.7 p_í sum (beta ( b_s))
23
3. Perturbation in the Principal Component Space (Eq. 10) noise np. random. randn (len (fused_ perturted__rojeetiion projections + noise np. (fused_
66
88
# 4. Reconstruction of the Perturbed Feature Vector (Pre-image problem)
30
need find x' such that its projection matches the perturbed one.
This stop is an optimization problem (Eq. 11-13). x_calibrated solve_pre_image image ceerturbe__proecctte fused_betas, GAB, init-x)
return x_calibrated
Listing sython-style pseudocode for the client-side manifold-awand calibration process. details how client uses the global geometry dictionary to calibrate single embedding.
this group. The goal is to find single consensus vector Br, that best represents this collection of vectors.
We can formalize this as finding a vector & that minimizes the weighted sum of squared Euclidean distances all crient-provided vectors. This be expressed weighted least squares problem:
B arg min
Σ w B3,x112
(15)
BERMaa:
Justification of the Weights. The choice of weights wy is critical and should reflect the "confi- dence" "importance" of the geometric information provided by client j. We argue that suitable weight should be proportional to two factors:
Sample Size (n,): LGDs derived from a larger number local data points (ny) are statis- tically more stable and less prone to sampling noise.
Variance ()..):: :) The eigenvalue λy.ı represents the variance captured by the i-th principal component. A larger variance indicates that this direction is more significant for describing the local data structure.
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Published as a conference paper at ICLR 2026
Therefore, natural and effective choice for the weight is wy nzAj,.-
Derivation of the Solution. The objective function Eq. 15 let's call L(3), is convex and differentiable with respect to B. To find the minimum, we set its gradient to zero:
VOL(3)
VB
wy wy(B
(B-
(16)
jEI
Σ JEL
2m0 (B- 33,))
(17)
Setting the gradient to zero to find the optimal Bi.
Σ
2w, (3t - -3,,)= = 0
(18)
jEL
Σ wi jEl
Bi, Σ wjBpir
(19)
Ejel
WzB3s
37
(20)
Ejes wy
By substituting our chosen weight wy = n Aj.s. we arrive at the exact fusion formula for the coeffi- cient vectors as presented in Eq.
ng
Bi,
(21)
Ejer "g
This derivation shows that our intuitive weighted averaging scheme is not ad-hoc, but fact the optimal solution for finding geometric consensus under weighted least squares criterion. The fusion of eigenvalues Xi, follows similar logie, averaging them based on the number of samples nj that contributed to each estimate.
C.3 GRADIENT DERIVATION FOR PRE-IMAGE RECONSTRUCTION
In Section 4.3 and Appendix B.2.2, we described the process of finding a pre-image x' by minimiz- ing a loss function (Eq. 13). To solve this optimization problem using gradient-based methods (e.g.. Gradient Descent or Adam), we must compute the gradient of the loss function L(x') with respect to the variable z'. We provide the detailed derivation below.
The Loss Function. The objective is to find an x' whose kernel evaluations with the GAB vectors {b.} match the target values (T.) closely possible. The loss function is:
Now
L(x')
Σ
(Ri(2' b.) -T,)²
(22)
where k..,,) is the Gaussian (RBF) kernel.
Step Applying the Chain Rule. We first compute the gradient of L(x') with respect to x' using
the chain rule:
Novs
V L(x') = Σ
2 (k(a', b.) T2) Vark(a' ba)
(23)
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Published as a conference paper at ICLR 2026
Step 2:
Gradient of the RBF Kernel. Next, need the gradient of the RBF kernel function k(2,c) = exp(->il= <l12) with respect to its first argument 2.
Vek(z, c) = V, exp(->il= cl12) -xp(-->|<= cl V --(-vll=-cp
(24)
(25)
-k(,,) (-7) V Σ (Ed - ca)?
(26)
= =((2,)) (-7) (2(z - -c)
(27)
-22 c))(2.))
(28)
Applying this to case, have V.'k(x', b,) -2( bw Ra(' by).
Step 3: Final Gradient Expression.
Finally, by substituting the kernel gradient (Eq. we obtain the complete gradient for the loss function:
28 back
into the chain rule expression (Eq. 23).
Nuee
Vx-((x))
Σ s=1
2 (kia', (k)2),b)))-)) TT) (-27(x' b,))(('(bb)))
(29)
Nuse
4y - ba)( - b.)
(30)
This final gradient expression can be directly implemented and used in any standard iterative opti- mization algorithm solve for the pre-image x'. For instance, the update rule for gradient descent would be 4'41 x V x (xx)), where 1 is the learning rate.
21


Reviews:
[{"review_id": "kdtwmLrlJ9", "reviewer": "Reviewer_TzgE", "summary": "This paper proposed a new FL framework, FedMC, to address data heterogeneity by leveraging the nonlinear manifold structures of client data to describe the true local geometries and then securely aggregating to form a global geometry dictionary. This global dictionary enables on-manifold calibration of local embeddings. The authors conducted extensive experiments across multiple benchmarks to validate the improvement achieved by FedMC for both Federated Prompt Learning (FPL) and conventional FL algorithms.", "questions": "1. Can the authors justify how to ensure the reconstructed x′ lies on the true data manifold?\n\n2. In section 4.4, the proposed “dynamic geometry query” finds the most relevant template entry based on the Euclidean distance (Eq. 9). Given that the concerns raised by the authors in section 3 about the issues regarding the simple Euclidean shortcuts, why is geodesic distance or a manifold-aware metric not considered here?\n\n3. Those kernel-based operations typically scale poorly with large datasets or many clients. Can the authors justify the scalability of FedMC in the settings where both the number of local data points and the number of clients may be large?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "1. The motivation is clearly presented, and the paper provides an intuitive analysis of the limitations of the global linearity assumption, which emphasizes the importance of on-manifold calibration in federated learning.\n\n2. This paper provided a formal derivation in the appendix to justify the server's geometric fusion strategy.\n\n3. This paper explicitly considers privacy preservation. Specifically, the authors proposed two mechanisms: (1) adding differential privacy to the anonymized prototypes and (2) applying secure projection to the extracted local geometry.", "weaknesses": "1. While section 4.3.1 is clear to me, the notations and equations in sections 4.3.2 and 4.4 are too dense. Incorporating one diagram to help illustrate the key steps could significantly improve readability.\n\n2. FedMC seems to introduce too much computational overhead on the client side, requiring many operations such as K-Means clustering, Kernel PCA, and pre-image reconstruction.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "P4Aphl7IDz", "reviewer": "Reviewer_wJpu", "summary": "This paper introduces FedMC (Federated Manifold Calibration), a novel federated learning framework that explicitly models the nonlinear manifold geometry underlying client data. The key idea is to move beyond the conventional global linearity assumption (e.g., global PCA-based calibration) by capturing local nonlinear geometries through kernel PCA at each client. The server then aggregates these local representations via a Global Anonymous Basis (GAB) and a Geometry Dictionary, enabling privacy-preserving, on-manifold calibration during local training. Extensive experiments demonstrate that FedMC consistently improves the performance of both Federated Prompt Learning (FPL) and general FL algorithms across diverse heterogeneity scenarios (label skew, domain skew, and combined skew). The framework is theoretically motivated, methodologically complete, and empirically validated.", "questions": "1. The pre-image optimization in the calibration step (Eq. 12–14) involves iterative updates. Could you share how stable and efficient this process is in practice?\n2. The Global Anonymous Basis (GAB) is initialized once. Would updating it periodically during training further enhance adaptability to shifting data geometry?\n3. Have you explored alternative kernels (e.g., polynomial or Laplacian) for local KPCA, and if so, do they affect performance notably?\n4. Regarding privacy, how does the added geometric sharing interact with differential privacy guarantees? Does it influence the overall privacy–utility balance?\n5. Could FedMC potentially extend to multimodal or cross-modal FL settings, where clients hold different modalities (e.g., image–text)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "1. The paper challenges the often implicit global linearity assumption in geometry-based FL and introduces a principled nonlinear manifold perspective.\n2. The design—from local KPCA to GAB construction and geometry dictionary fusion—is coherent and well justified under privacy constraints.\n3. FedMC yields consistent improvements across six benchmarks and multiple FL paradigms, demonstrating generality and robustness.\n4. The paper is clearly written, well structured, and includes detailed appendices and pseudocode that enhance reproducibility.\n5.  The idea of modeling federated data manifolds could inspire future work on geometric and representation-level calibration in distributed learning.", "weaknesses": "1. It would be useful to briefly comment on the **runtime and communication overhead** of FedMC compared to standard FL baselines, even qualitatively.\n2. Since the framework involves hyperparameters (e.g., kernel bandwidth γ, cluster number m, basis size N_base), a short note on how these were chosen in practice would improve clarity.\n3. The experiments use up to 10 clients. A short discussion on how FedMC might scale with more clients or unbalanced participation would be valuable.\n4. In a few places (e.g., Eq. 12–14), the correspondence between Φ(x), β*, and v* could be made slightly clearer for readers less familiar with kernel methods.\n5. If space permits, a brief ablation on the effect of each key component (GAB, geometry fusion, calibration) would further highlight their individual contributions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OQ49ZSIT8X", "reviewer": "Reviewer_RFAP", "summary": "This paper makes a significant conceptual and practical contribution to federated learning by shifting the paradigm from global linear to local nonlinear geometric modeling. The proposed FedMC framework is innovative, theoretically grounded, and empirically effective across diverse settings. It opens a new direction for geometry-aware FL and has the potential to influence future work on distribution calibration in decentralized learning.", "questions": "1. In Section 4.3.1, you mention using a Gaussian kernel with γ = 1/d for KPCA. Could you clarify what \"d\" refers to here? Is it the dimension of the image embeddings? Also, was this value chosen empirically, or is there a theoretical reason behind it?\n\n2. I’m curious about how often the Geometry Dictionary is updated. Is it rebuilt from scratch in every communication round, or is it initialized once and then incrementally updated? If it’s updated every round, does the dictionary size grow over time, or is there a mechanism to keep it fixed? \n\n3. In Table 1 and 2, GGEUR(FedVTP) serves as a strong linear baseline. Could you briefly explain how GGEUR works in practice? Specifically, how is its “global geometric prior” computed and applied during calibration? Is the main difference from FedMC simply that GGEUR uses standard PCA instead of kernel PCA?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper identifies a key flaw in FL calibration methods, namely the global linearity assumption. This insight is both theoretically sound and empirically validated, and it reframes how we should think about distributional priors in heterogeneous FL.\n\n2. FedMC is a well-motivated, technically sophisticated framework that bridges manifold learning and federated optimization. The use of local kernel PCA, secure projection onto a shared anonymous basis, and dynamic geometry querying constitutes a coherent and elegant solution to the on-manifold calibration problem under privacy constraints.\n\n3. The experiments are comprehensive. Multiple datasets and heterogeneity types (label, domain, mixed). Comparison against SOTA FPL methods and a tailored linear baseline.\n\n4. The construction of the Global Anonymous Basis (GAB) with differential privacy and the secure representation of local geometry via projection onto GAB are thoughtful and align well with FL’s privacy requirements.", "weaknesses": "1. The pre-image reconstruction step (solving for \\(x'\\) in the original space) involves iterative optimization per sample, which may introduce computational overhead. However, the authors note this is manageable and performed locally, and the empirical results justify the cost.\n\n2. While the focus on FPL is well-motivated (communication efficiency, bias in prompts), a brief discussion on applicability to other parameter-efficient FL settings (e.g., LoRA, adapters) could further broaden impact.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)