Clustering by Denoising: Latent plug-and-play diffusion for single-cell embeddings
Authors: Dominik Meier
Shixing Yu
Sagnik Nandy
PROMIT GHOSAL
Kyra Gan
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion
Plug-and-Play (PnP)
Single-Cell Genomics
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Single-cell RNA sequencing (scRNA-seq) enables the study of cellular heterogeneity. Yet, clustering accuracy, and with it downstream analyses based on cell labels, remain challenging due to measurement noise and biological variability. In standard latent spaces (e.g., obtained through PCA), data from different cell types can be projected close together, making accurate clustering difficult.
We introduce a latent plug-and-play diffusion framework that separates the observation and denoising space. 
This separation is operationalized through a novel Gibbs sampling procedure: the learned diffusion prior is applied in a low-dimensional latent space to perform denoising, while to steer this process, noise is reintroduced into the original high-dimensional observation space. 
This unique ``input-space steering'' ensures the denoising trajectory remains faithful to the original data structure. Our approach offers three key advantages:
(1) adaptive noise handling via a tunable balance between prior and observed data; (2) uncertainty quantification through principled uncertainty estimates for downstream analysis; and (3) generalizable denoising by leveraging clean reference data to denoise noisier datasets, and via averaging, improve quality beyond the training set.
We evaluate robustness on both synthetic and real single-cell genomics data. Our method improves clustering accuracy on synthetic data across varied noise levels and dataset shifts. On real-world single-cell data,  our method demonstrates improved biological coherence in the resulting cell clusters, with cluster boundaries that better align with known cell type markers and developmental trajectories.

Published as a conference paper at ICLR 2026
CLUSTERING BY DENOISING: LATENT PLUG-AND- PLAY DIFFUSION FOR SINGLE-CELL DATA
Dominik Meier
Shixing Yu
Sagnik Nandy
Cornell Tech dm9540cornet edu
Cornell Tech
Ohio State University
sy7748cornel 1. edu
nandy 15@osu edd
Promit Ghosal
Kyra Gan
University of Chicago
Cornell Tech
promit@uchicago. edu
kyragan@cornelll.ce edu
ABSTRACT
Single-cell RNA sequencing (scRNA-seq) enables the study of cellular hetero- geneity. Yet, clustering accuracy, and with downstream analyses based on cell labels, remain challenging due to measurement noise and biological variability. In standard latent spaces (e.g., obtained through PCA), data from different cell types be projected close together, making accurate clustering difficult. We introduce a latent plug-and aaig-and-play diffusion framework that separates the observation and de- noising space. This separation is operationalized through novel Gibbs sampling procedure: the learned diffusion prior is applied in a low-dimensional latent space to perform denoising, while to steer this process, noise is reintroduced into the observation space. This unique "input-space steering" ensures the denoising trajectory remains faithful the original data structure. Our approach offers three key advantages: (1) adaptive noise handling via tunable balance between prior and observed data; (2) uncertainty quantification through principled uncertainty estimates for downstream analysis; and (3) generalizable denoising by leveraging clean reference data denoise noisier datasets, and via averaging, improve quality beyond the training set. We evaluate robustness both synthetic and real single-cell genomics data. Our method improves cluster- ing accuracy synthetic data across varied noise levels and dataset shifts. On real-world single-cell data, our method demonstrates improved biological coher- ence in the resulting cell clusters, with cluster boundaries that better align with known cell type markers and developmental trajectories.
original
INTRODUCTION
Single-cell RNA sequencing (scRNA-seq) has revolutionized biomedical research by enabling high- resolution profiling of cellular heterogeneity (Park al., 2020; Miragaia et al., 2019), with large- scale initiatives like the Human Cell Atlas providing foundational references for cell type annotation al., 2021; Elmentaite et al., 2022: Stuart et al., 2019; Lopez
(Regev et al., 2017: Lindeboom et al., 2018at 8a). However, since cells are sequenced without predefined labels, accurate cell type identification must be derived entirely from unsupervised analysis of noisy high-dimensional data. The canonical approach involves reducing dimensionality (e.g., via PCA) followed by clustering and manual annotation based on marker genes-as an iterative and subjective process (Kiselev al., 2019; Stuart et al., 2019). However, this entire paradigm is often undermined by the inherent noise in scRNA-seq data, which arises from technical artifacts like varying capture efficiency (Kharchenko et al., 2014) and biological stochasticity (Wagner et al., 2016), which amplified by standard clustering algorithms and lead to unreliable labels.
We frame single-cell denoising an inverse problem: recovering clean gene expression from noisy measurements without imposing strong generative assumptions. Our method leverages the high- quality signal from reference datasets (e.g., SMART-seq, Picelli et al. 2013; 2014) leam powerful denoising prior, which then applied to enhance noisier target data (c.g.. droplet-based ScRNA-seq, Klein et al. 2015; Macosko et al. 2015) for improved clustering. We operationalize

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this under the plug-anin plug-and-play (PnP) paradigm (Venkatakrishnan et al., 2013a; Zhang et al., 2021; Chan et al., 2016; Ryu et al., 2019), which integrates powerful denoising priors with measurement models. Mainstream PnP diffusion frameworks (Zhu et al., 2023; Go et al., 2023; Wu et al., 2024; Coeurdoux et al., 2024; Xu & Chi, 2024) enable this through combining likelihoods via iterative refinement (e.g., Gibbs sampling), where each denoising step is followed by controlled noise rein- troduction to enforce data consistency. This principled approach enables us to denoise beyond test data quality by transferring patterns from high-signal reference data to noisier technologies, even under mild distribution shifts between the reference and target distribution. However, directly ap- plying image-basee PnP frameworks to single-cell data is challenging. Unlike images where pixel noise is largely independent, gene expression data exhibits intrinsic low-rank structure and com- plex correlations. Moreover, denoising must preserve relational structure between cells for accurate clustering and annotation. Standard dimensionality reduction (e.g., PCA) can collapse distinct cell types, making it impossible to guide denoising accurately.
To address these unique challenges, we introduce a latent plug-and-play diffusion framework that tailors the PnP philosophy to single-cell biology. Unlike prior PnP methods that rely on generic or hand-designed priors, our approach is tailored to biological data and operates through modified two-step procedure, designed to overcome limitations of prior single-cell denoising methods. First, diffusion model is trained in a low-dimensional latent space (analogous PCA) capture the core biological manifold of a high-quality reference dataset. Unlike PCA, this diffusion process learns score functions directly from the data, enabling recovery of complex structures in the latent distribution of cell types. This approach inherits the robustness to prior misspecification and scala- bility to high latent dimensions characteristic of diffusion models (Xu & Chi, 2024). Second, during inference on the noisy dataset, we employ Gibbs sampling procedure that reintroduces noise into the original high-dimensional input space. This critical step directly addresses the catent-space col- lapsing issue inherent in methods like PCA (Burges et al., 2010), where distinct biological states are projected too close together, losing information essential for precise denoising. By operating in the original high-dimensional space, where full geometric relationships are preserved, method steers denoising toward biologically meaningful structures obscured in compressed representations.
Our framework diverges from existing Bayesian approaches for single-cell analysis by removing their need for restrictive generative modeling. While variational autoencoders (VAEs) (Lopez et al., 20; Gayoso et al., 2021; Grønbech et al., 2020) are difficult to train and rely on strong likelihood assumptions, more recent approximate message passing methods with empirical Bayes denoisers (Zhong et al., 2022; Nandy & Ma, 2024) still require parametric noise modeling, operate purely in the latent space, and scale poorly to high-dimensional latent spaces. In contrast, our approach re- quires no explicit generative model or pre-processing for noise structure, instead learning it directly from data.
By combining the adaptability of likelihood-free diffusion with the structure-aware refinement of Gibbs-based input-space guidance, we enhance cluster separation in the denoised low-dimensional embeddings of the target data, even when the biological signal is weak or the target distribution moderately differs from the reference.
Our diffusion based denoising framework offers three key advantages over existing single-cell de- noising methods, and therefore provides a principled, robust, and reproducible pathway for diverse downstream tasks like automated cell type annotation using the denoised embeddings.
Adaptive noise handling through tunable interpolation: We introduce a parameter p that dy- namically balances data-driven information and prior knowledge during denoising. This allows optimal adaptation to varying noise levels and dataset qualities-preseevving data-specific signals when test and training distributions align, while leveraging prior knowledge to stabilize highly noisy inputs. This capability is absent in conventional clustering and imputation methods.
Uncertainty quantification: Unlike standard clustering VAE-based pipelines, our approach provides confidence sets for cell-type predictions, enabling quantitative assessment of annotation relitbility-criticall for downstream analysis and clinical applications.
Generalizable denoising: By training on high-quality reference data, our model learns a robust biological manifold that can denoise even low-quality target datasets, effectively addressing real- world scenarios where data from different labs exhibit substantial quality variations. Further, our averaging-bar approach enables denoising beyond the immediate training distribution, enhanc- ing applicability across diverse experimental conditions.
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Our experimental results demonstrate consistent performance under various mis-specifications of the data- getta-enerating process in synthetic settings (Section 4). In real-world single-cell experiments, method shows strong potential for leveraging high-signal training data to improve denoising in low-signal datasets and to denoise beyond the training distribution by averaging (Section 5). The source code of DICE is publicly available.'
1.1 ADDITIONAL RELATED WORK
Diffusion in Single-Cell Data Diffusion models provide flexible, trainable priors that accommo- date complex noise structures (Sohl-Dickstein et al., 2015; Ho et al., 2020; Song et al., 2021b;a). While initial single-cell applications have focused on transcriptome generation and data imputa- tion (Luo et al., 2024; Wang et al., 2025; 2024; Zhang et al., 2025), their potential for learning denoised dinndimes embeddings, critical for reference atlas construction and robust label transfer, remains largely unexplored beyond restrictive generative modelling assumptions. Our work investigates this gap, using unsupervised clustering and subsequent biological validation as the eval- uation framework.
Single-Cell Preprocessing Analyzing single-cell data challenging because raw UMI counts vary across labs, technologies, and populations (Shaham al., 2017). Several pipelines have been pro- posed aim to mitigate these effects, most prominently the ubiquitous Seurat workflow (Stuart et al., 2019) applying quality control, and normalization. In datasets with pronounced batch effects, such batch-induced variations are typically corrected using techniques like Harmony (Korsunsky et al., 2019), and the batch corrected expressions are used to produce linear embedding for downstream tasks such as clustering or label transfer. An alternative direction incorporates these steps into deep generative models that operate directly on raw counts, such as sc (Lopez et al., 201 8h), which learns latent representations end-to-end.
Instead of creating embeddings for all cells, a complementary line work aggregates similar cells into metacells (e.g., MetaQ (Li et al., 2025)), which are then used for downstream analyses. In contrast, our method produces embeddings for all individual cells, rather than summarizing them.
SINGLE-CELL ATLAS CONSTRUCTION VIA POSTERIOR SAMPLING
This work considers single-cell RNA-seq datasets: (1) reference dataset D)r)
{x(")... ...,,..!!) ( used to train diffusion prior that captures the underlying biological manifold,
and (2) target dataset D {x ......,,, x(,) for denoising. Our framework is designed for single-cell RNA-seq datasets after standard preprocessing steps, such as quality control to filter low-quality cells, library-size normalization, and log Ip transform which renders the residual non-biological variation approximately Gaussian. Within this regime, our method does not rely on stronger assumptions about upstream preprocessing choices. The goal is to construct a single- cell atlas-a reference map of cellular states- learned via denoised low-dimensional embeddings that capture underlying biological structure while removing technical variation arising from non- biological sources.
Data Generating Process Both datasets are modeled using a low-rank factor model (Peng et al., 2021; Weine et al., 2024; Zhong et al., 2022; Nandy & Ma, 2024; Argelaguet et al., 2020), Let X; E Rd represent the gene expression profile for cell i:
X; = VU, Ei, Vi,
where V. E [gd* is factor loading matrix that spans the transcriptional space; U; E g Pprior encodes the low-dimensional biological cell signal drawn i.i.d. from an unknown distribution Pprior's and e; is an independent noise vector, reflecting stochastic variation in the measurement process or biological stochasticity.
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Our method is built generative model that projects both reference and target datasets into shared latent space. This is formalized as:
x unr) (v(') + U R elr) e gmxd
(2)
x = v (v(r) +000, w E BRXK, e()) Raad
The critical modeling choice is the shared factor loading matrix V. learned from the reference dataset. This allows the target data be projected into the same latent space as the reference, enabling knowledge transfer. This approach explicitly accommodates different measurement tech- nologies through distinct noise models for each dataset (e ~ e(T)), with the expectation that our method performs best when the reference dataset D(r) has comparable or lower noise levels than the target dataset D(t).
Given this shared structure, we drop superscripts on V entirely. We retain the (r) and (t) superscripts
only for tataset-specifie terms: Dír), D(O,, X X Utr), e()), and e(f) To further simplify notation, we henceforth drop the superscripts with the convention that all quantities refer to the target dataset D()) unless explicitly marked with (r) for the reference dataset D(rr)
Posterior Sampling for Denoised Embeddings Our goal is to construct a denoised atlas of D(() by computing posterior embeddings B[U;| X:), framed as sampling from the posterior:
TU X) f (x - UV (U) (Un)
where (()) denotes the likelihood associated with the observation model, and Ppicee represents the population prior learned via diffusion on D(r), The main challenge in sampling arises from the joint influence of likelihood and prior. Efficiently sampling from a posterior that couples these two components non-trivial, as requires balancing the local reconstruction constraints encoded by the likelihood with the global manifold structure enforced by the prior. Traditional approaches address this difficulty by imposing restrictive assumptions: conjugate Gaussian priors with Gaussian likelihoods (Gelman et al., 2013) enable tractable inference but fail to capture complex biological distributions, while Merropolii-adjusted Langevin algorithms (Roberts Rosenthal, 1998; Durmus et al., 2018) handle non-Gaussian likelihoods but struggle with implicitly defined diffusion priors.
Remark 2.1 (Design Principle: Peference-Guided Denoising). A core design principle of this method is leverage the high-quality reference dataset learn biologically meaningful prior, P PU), which captures the latent manifold of plausible cell states. This prior is not used to force the target dataset into an identical configuration, but to provide a structural guide for denois- ing. The shared projection matrix V enables this by projecting both datasets into common latent space, allowing the diffusion prior learned from D(r) to provide a structural guide for denoising the target data. By projecting the noisier target data into this well-defined space, our diffusion prior can effectively separate biological signal from datasft-specific noise while accommodating different measurement technologies through distinct noise models.
PnP Framework with Auxiliary Variables By leveraging the PnP diffusion framework, we over- come this challenge using split Gibbs sampling approach (Vono et al., 2019; Xu & Chi, 2024) that introduces auxiliary variables Z, to decouple the likelihood from the diffusion prior. This is achieved by first replacing U/ with Z; in the likelihood generating stage, and then enforce consis- tency between Uj and Z; through a Gaussian penalty, which leads to following augmented joint distribution:
PX,,,, Zi) x exp (sp(-bogg log f(x, V2,)
2pa U. 2,112 log Pprior (U.)
(4)
where / controls the alignment strength. Smaller enforces tighter coupling between U; and its auxiliary counterpart Z, This leads to Gibbs sampler alternating between two conditional updates:
1 2p²
Likelihood Step:
Pp(Zi Xs, U.)
exp(
-log f(X, VZ,)
IU Z,112
(5)
1
Prior Step:
Pp(U/ Z,) exp
IU, - log Ppetuc (Ui)
66)
2p2
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Although resembling Gibbs sampling, the alignment penalty is artificially introduced rather than arising from standard conjugacy. This modification allows us to plug in a diffusion prior at inference time, thereby enabling efficient posterior sampling even when the prior is only implicitly specified.
Evaluating Atlas Quality A key challenge in single-cell atlas construction is the lack of direct evaluation metrics due high-dimensional noise and the curse of dimensionality (Kiselev et al., 2019). We employ a multi-faceted evaluation strategy combining quantitative clustering metrics and qualitative visual assessment:
1. Visual Assessment: We use dimensionality reduction techniques, particularly UMAP (MeInnes al., 2018), to visually examine the quality of denoised embeddings. Well-separated, biologi- cally meaningful structures in 2D visualizations validate the method's performance.
Unsupervised Clustering with Post-Hoc Evaluation: Denoised embeddings {ū:} are clustered without using label information to obtain K clusters. The quality of these clusters is evaluated by comparing them the known cell type labels Lg L..... C using the metries adjusted rand index (ARI) (Hubert & Arabie, 1985), average silhouette score (Rousseeuw, 1987), normalized mutual information (NMI) (Strehl & Ghosh, 2002), and cell rype locally invariant Simpson' index (cLISI) Korsunsky et al. (2019),2 which quantify the agreement between the data-driven clusters and biological annotations.
Our framework is fully unsupervised. relying only on expression data.² We benchmark performance on synthetic data with known ground truth and on real datasets, noting that real-world labels may be imperfect (and thus annotation accuracy provides indirect measure of atlas quality).
3 METHOD
We introduce a latent plug-and-pla diffusion scheme for denoising query cell embeddings under the guidance of diffusion model trained on a large corpus of single-cell gene expression data. Lever- aging the PnP framework, we recover oow-dimensional embeddings U from noisy gene expression profiles (r) by decoupling data d and prior structure: the query cell's expression profile an- chors the denoising process its unique features, while the pretrained diffusion model contributes global information about the structure of the cell population. Our pipeline consists of two stages:
Training stage: jointly estimate the factor loading matrix V and train a diffusion model to learn the prior distribution P Prioor over embeddings D(r).
2. Inference stage: given query expressions x perform posterior sampling using split Gibbs scheme, alternating between iikeifhood-informedd updates and diffusion-guideer updates.
An illustration of the method can be found in Figure A.1. This approach preserves the flexibil- ity of diffusion priors while maintaining tractable posterior inference, providing a scalable and uncertainty-awarc framework for single-cell atlas construction.
We train the diffusion model on the reference dataset D(r), Consider the best rank-k approximation of the reference data produced by singular value decomposition: (r) ss WE¹T, where W E Ra** and V E Rdxk are unitary matrices containing the top 1 left and right singular vectors, respectively. and Σ & is the diagonal matrix of the top k singular values. This decomposition yields our loading matrix estimate in Eq. (1), V. We then compute the transformed observations 0; = V x for i = I...., m, which accurately approximates the latent embeddings U; under a wide range of noise models. These estimated embeddings (Ü: Jms therefore provide approximate training samples from the prior distribution Pprice and are used train the diffusion model.
Diffusion Training
To learn this prior, we adopt the standard harwaad--diffusion framework (Sohl-Dickstein et al., 2015; Ho et al., 2020), which we detail in Appendix A.
Denoising with DICE We now introduce DICE (Diffusion Induced Cell Embeddings, Algo- rithm 1), our split Gibbs sampling procedure for denoising query cell Xg and estimating its la-
For detailed description of these metrics, see Appendix D.
While labels not used in denoising, we explore label-augmented diffusion training in Section 5.
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Algorithm DICE: Diffusion Induced Cell Embeddings
1: Input: query cell Xgi trained diffusion model ég, (+); number of iterations T'; annealing schedule
{ps}x Dayy=is estimated factor loading matrix F
2: Initialize: ( V X₂
3: for 0 to do
4: Likelihood alignment: sample
2 U7(e) & exp cxp(-2)) 2n² 12 - U1 log f(x, - V2,))
5: Prior alignment: Using the reverse diffusion update in Eq. (7), sample
U(++1) (()) exp 2pr 1 12 log P Prior (Ug)
q
6: end for
7: return Ui as the denoised embedding of the query cell.
tent embedding Ug. Given annealing schedule {ps ,,...TTh, the augmented distribu- tion Eq. (4) decomposes posterior sampling into two iterative steps:
T},
1. Likelihood alignment (Line 4) is implemented using either general proximal scheme (Xu & Chi, 2024) or a closed-form Gaussian update when is Gaussian (Proposition 3.1). Unlike the prior alignment step in Eq. (5), Line 4 operates in the original d-dimensional data space, reintroducing noise through the likelihood function log f fX,, VZg).
2. Prior alignment (Line 5) is implemented using the trained diffusion model via the reverse update
1 at Eo(xz) + v1 VI- αz
Ft-1 =
Ft
a4 21,
24
(0,
Vai
rum for t = to. to 1..... with initialization 24, = vas, U!d). that 01, (1 + )²))
The chain length is chosen so
We reduce posterior variability by generating multiple samples and averaging, enabling denoising beyond reference data quality (Figure (first column) and Figure 3). The parameter Ps controls the relative weight of prior and likelihood: larger Ps emphasizes population-leva structure and is suitable for noisy queries, while a smaller Pa emphasizes fidelity to the observed expression profile.
Likelihood Alignment under Gaussian Noise Although framework accommodates general likelihoods, in practice, single-cell data are often log 1p-transformed and modeled with Gaussian noise (Zhong al., 2022; Argelaguet al., 2020). In this case, Proposition 3.1 (proof in Ap- pendix B) establishes that the likelihood update step admits closed-form update:
Proposition 3.1. Assume f is the standard multivariate Gaussian density dimensions. Follow- ing Gaussian conjugacy, for all = O..... T 1, the likelihood update step (Line 4) admits the following update:
Zi1) .o.. (A((AAATT u..)). A). A=
Remark 3.2. The same denoising scheme be applied the training data themselves, yielding refined embeddings that serve as a reference atlas. Notably, atlas construction via DICE does not rely restrictive parametric assumptions for either the likelihood the prior, enabling the method to capture rich and complex population structures as found for example in single-cell data.
Confidence Sets We quantify uncertainty in the embedding query cell X,, by applying DICE multiple times and examining the spread of the resulting denoised embeddings.
EVALUATION ON SYNTHETIC DATA
Setup To evaluate how well DICE recovers clean latent structure from noisy expression profiles, we design controlled setting that mimies two pure cell populations with known labels. In latent
dimension & 15, the training prior is balanced Gaussian mixture Pprior = Nio((O0,,, 1.5 Lι6) +
Wis 15, 1.3 Jss,, with each component corresponding cell type. We sample loading

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Figure 1: UMAP visualizations of the 400 test cells for each of the four configurations. Top row: PCA projections: bottom row: DICE-denoised embeddings. Columns (left to right) correspond to Setups 1-4. Throughout, refer the mixture component centered at 11s with scale 1.3 Iıs as Cluster 1, and to the complementary component as Cluster 2 (Gaussian (15 1.5 I15) where applicable; heavy-tailed (1.3 T15) ) in the latent-prior-shift configuration). Points colored blue for Cluster and orange for Cluster
matrix V E 2000x with entries i.i.d. (N For each cell i, we draw a latent U!!²) ~ P Pprior and measurement noise e(") N2000 (0, I2000), and generate a synthetic expression profile x! vU + = ((r) 22000 = L..... 1600. This yields reference dataset D(r) of size m = 1600
in d 2000 observed dimensions, analogous to high-signal scRNA-seq experiment with two subpopulations present in equal proportion.
Test configurations We examine three train-test shifts covering practical single-cell scenarios, with [D(t) 4000, fixing Ppeie and V as in Section 2 unless noted otherwise:
Setup 1 (matched train-test distribution). We generate D(t) from the same generative distri- bution ((r) to isolate posterior denoising effects. This evaluates whether DICE improves pop- ulation separation in latent space versus PCA, mirroring standard atlas workflows where query cells are mapped via denoised embeddings (Nandy & Ma, 2024).
2. Setup 2 (signal-strength shift). We increase the noise D(t), with (t) ~ N2000 10 I2000): This mirrors single-cell scenarios with lower read depth/fewer UMIs or noisier platforms (c.g., shallow sequencing) that reduce signal-to-noise ratio (SNR) at test time. This tests whether DICE, trained on high-quality data, denoises low-SNR profiles better than PCA.
Setup 3 (noise-model shift). We generate ( with heavy-tailed noise, 2(1)) tv=4 (multivariate t with 4 degrees of freedom and scale I2000), while keeping Gaussian likelihood during denoising. This tests DICE's s robustness to likelihood mis-speciiiicttonn from heavy-tailed residuals (due outliers, doublets, over-dispersion in single-cell applications) against PCA.
4. Setup 4 (latent-prior shift). We change the test latent distribution to heavy-tailed mixture:
(.(0) iid. W1ss 1.5 415) + (1.3 4ıs), with increased noise (!t) ~ 2N000 (0, 10 42000). This tests robustness to prior nissspeccification when deploying novel, heterogeneous subpop- ulations (e.g., new developmental states), where we expect DICE to recover mixture separation better than PCA despite heavier tails and lower SNR.
Training and denoising workflow We train a diffusion model on 15-dimensional latent representa- tions obtained via PCA from the training set D(r) for 2,000 epochs (training details in Appendix C). The resulting model serves the learned prior P Pprior across all four test configurations.
At test time, we denoise the PCA projections of each configuration by running Algorithm 1 (DICE) for T 200 Gibbs iterations with constant annealing level p: 20. We repeat this procedure with 10 independent random iterations and report the mean of the resulting embeddings as Monte Carlo estimate of E.U X] for each test cell.

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Silhouette
cLISI
Semp
PCA
DICE
PCA
DICE
0.25
0.37
1.27
1.17
0.24
0.36
1.27
1.17
1.32
1.18
0.22
0.28
1.35
1.27
Table 1: Comparsion of average silhouette scores (higher is better) and cLISI scores (lower is bet- ter) calculated on the ground-truth cluster labels across four synthetic setups.
Figure 2:
UMAP visualizations of 500 runs of DICE on the same imput point Setup and p = 0.1. Training data are shown in grey. Left: center of Cluster 2; Right: midpoint between Clusters &
Evaluation We assess the cluster separation of DICE-derived embeddings against PCA baseline that projects expression profiles to the same latent dimension used for clustering. Qualitatively, we visualize UMAPs (McInnes et al., 2018) (Fig. 1); quantitatively, we report the average cosine silhouette score (Rousseeuw, 1987), and cLISI (Korsunsky et al., 2019) computed using the true cluster labels the test data (Table 1). Because ground-truth labels are available in this benchmark, both metrics are directly comparable. Across all four settings, DICE yields clearer separation in UMAP and better scores than the PCA baseline, indicating more faithful recovery of the underlying classes.
Uncertainty in the embeddings We visualize our confidence set construction for two fixed points: (i) the center of Cluster and (ii) the midpoint between the two cluster centers. For each input, run DICE 500 times and project the resulting embeddings onto the same UMAP as the training data. When the input lies the center of cluster (Figure left), all embeddings map consistently to Cluster 2, indicating high confidence in the cluster assignment. In contrast, when the input lies be- tween clusters (Figure 2, right), the embeddings are split across both clusters, reflecting uncertainty in the assignment. Such uncertainty could be advantageous for downstream tasks designed to incor- porate soft labels. Additional plots in Appendix E show how the parameter p directly controls the size of the confidence sets. Thus, / must be tuned to balance coverage and performance.
EVALUATION ON SINGLE-CELL DATA
We evaluate the effectiveness of DICE denoising gene expression profiles using two publicly available single-cell RNA-seq datasets: the CITE-seq dataset from Hao et al. (2021) and the human fetal brain development datasets from Polioudakis et al. (2019) and Nowakowski et al. (2017) (see Appendix G for dataset provenance). These datasets originate from distinct tissues and capture diverse cellular populations. They also differ in their relative signal strengths, allowing us to examine the ability of DICE handle both complex cell distributions and varying signal-t-noiis regimes.
Diffusion Training For each dataset, we first select the latent dimension using the elbow of the singular-vad value spectrum, and then project the training data onto this k-dimensional space via PCA. This yields the training embeddings ( and the loading matrix V used by DICE. We train the same diffusion architecture actoss datasets using Adam with cosine-annealed learning- rate schedule. Additional details on the training pipeline, including precise model architecture, are provided in Appendix C. We evaluate the influence of picking the PCA dimension (k) in Appendix F.
For CITE-seq, training takes about 36 minutes and inference takes about 12 minutes on a setup ac- celerated by an NVIDIA RTX PRO 6000 GPU. We provide more details setup in Appendix C.
5.1 ANALYSIS OF THE CITE-SEQ DATASET
Dataset The CITE-seq dataset (Hao et al., 2021) 1) consists of RNA profiles for 20,729 genes across 152,094 PBMCs with paired ontibodd-derrived tags (ADT) measurements. We focus solely the transcriptomic modality and uniformly subsample 10,000 cells for analysis. Ground-truth labels are provided, and adopt the 1.2-level granularity, which distinguishes 300 immune subtypes.
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Embeddings genented by PCA
Embeldling generated by DICE
10
Franery
LOC2
Coloinors
Each
u10000
CRECTT
HSPC
CON
MAIT
CONTOM
PDC
COR
Flonsbadt
-00
48-
COSTOM
0MAP1
URAP1
Figure 3: UMAP of 1,000 held-out PBMCs from the CITE-seq dataset. Left: PCA embeddings in 25-dimensional latent space using V from the training set. Right: embeddings after denoising with DICE using a diffusion model trained on the other 9,000 cells.
CITE-sey
Human Neo-Cortex
Method
ARI
cIJSST
NMI
V Mensure
ART
c1.1S1
NMI
Mensure
DICE
0.813
0.553
0.559
PCA
0,745
1.334
0.689
0.787
0.347
1.496
0.496
0.525
ALRA
0.604
1.361
0.713
0,775
0.310
1.526
0.474
0.512
KNN (S nearest neighhors)
0.713
1.334
0.647
0.756
0.275
1.837
0,349
0.405
KNN (10 nearest neighbors)
0.705
1.385
0.663
0.757
0.285
1,750
0.440
0.451
KNN ncarest neighbors)
0.735
0.766
0.268
1.701
0.442
MAGIC
0.674
1.397
0.648
0.757
0.317
1.451
0,502
0.509
0.556
2.237
scVT (10 latent factors)
0.641
1,361
0.598
0.715
Table 2: Comparison of clustering metrics across single-cell datasets.
Preprocessing and denoising We applied standard QC and pre-processing the raw UMI counts data (see details in Appendix H), and considered the top 3,000 highly variable genes (Seurat v3 criterion) construct the data matrix X E IR 1820,000x3,000 We split into 9,000 training and 1, 1,000 held-out test cells with preserved label proportions. Similar to Setup 1, we test DICE's ability in denoising beyond the quality of the training dataset this setting. The training set was used fit the diffusion model, and the test set was reserved for evaluation.
Using the training pipeline above, we picked = 25 and trained diffusion model on 25d PCA embeddings and applied DICE for T = 100 denoising iterations to the PCA embeddings of the test cells. We used linearly decreasing (equally spaced) annealing schedule {p: from 5 to 0.5 over 100 points. Further details of the denoising procedure are provided in Appendix C.
Evaluation We compare UMAP (McInnes al., 2018) visualizations of the 1,000 held-out test cells denoised by DICE against their PCA projections without denoising (Figure 3). Denoised em- beddings show clearer segregation of immune subtypes, with marked improvement in separating sublineages of CD4 and CD8 T' cells. The DICE atlas also performs better in segregating the MAIT cells from the other cells. These populations are notoriously difficult to resolve from RNA alone (Hao et al., 2021), challenge that originally motivated multimodal approaches combining RNA and protein. In the unimodal setting considered here, however, popular toolkits such as Seurat rely directly on PCA embeddings. Our results indicate that denoising PCA embeddings with priors leamned from high-signal training data substantially improves cluster separation, supporting more reliable annotation of held-out cells. We additionally benchmarked the DICE embeddings against several widely used denoising procedures for RNA-seq analysis-MAGIA (van Dijk et al., 201 8a), ALRA (Linderman et al., 2022a), neiegrr smoothing (used in Seurat 2019)), and non-neeative matrix factorization (NMF)-based embeddings," using the clustering met- rics ARI, NMI, cLISI, and V-measure described in Section D. The results (Table 2) show that DICE consistently outperforms these popular denoising pipelines across most clustering metrics.
(Stuart et al.,
*See Appendix for details the benchmarking procedures and their implementation.

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Embeddings generated by PCA
mbeddings generated by DICE
MG
Picreçu
FIN
DPC
RG
-10-
UNAP1
Figure 4: UMAP of 15,126 cells from the Polioudakis et al. (2019) dataset. Left: PCA embeddings in 15-dimensional latent space using V from the training set. Right: embeddings after denoising with DICE using a diffusion model trained on the Nowakowski et al. (2017) dataset.
5.2 ANALYSIS OF HUMAN FETAL BRAIN DEVELOPMENT DATASETS
Datasets We evaluate the effectiveness of DICE in transferring cell type information learned from high-signal training data to related low-signal test data using scRNA-seq datasets from Nowakowski et al. (2017) and Polioudakis et al. (2019). Both datasets profile human fetal brain tissue during development. The Nowakowski et al. (2017) dataset includes cells from primary cortical and medial ganglionic eminence (MGE) samples across multiple development stages, whereas the Polioudakis et al. (2019) dataset focuses on cells from the neocortex during mid-gestation. While the cell types profiled in the two datasets are related, they are not identical due to differences in the sampled tissue. This contrasts with the CITE-seq dataset considered previously, where training and test cells originated from the same source. This experiment evaluates the robustness of DICE to realistic distributional changes that occur during cross-dataset cell-type label transfer.
Preprocessing and Denoising We analyzed 3,495 cells from Nowakowski et al. (2017) and 15,126 cells from Polioudakis et al. (2019). After performing standard pre-processing on the raw expression data (see Section H for details), we selected 785 highly variable genes shared by both datasets. This yielded matrices Xnow E R and Xpol E R211226788 785 As X Xnow exhibited a stronger signal, we trained the diffusion prior on k 15 latent projection of it (via PCA) and evaluated DICE on 1,000 randomly sampled cells from Xpol to assess denoising and embedding quality. The training and denoising hyperparameters (noise/annealing schedule, number of diffusion denoising steps, and number of Gibbs iterations) matched those used in the CITE-seq example.
Evaluation We compare UMAP visualizations for the embeddings constructed from the test data using DICE and the embeddings obtained via the PCA transform learned from the training data. Denoising yielded substantially improved biological interpretability, producing more coherent and homogeneous clusters of similar cell types. A notable improvement is found in the embeddings of the cells belonging to the canonical excitatory trajectory is visually continuous and easy to follow in the DICE embedding, whereas it appears fragmented and noisy under PCA. Similar to the CITE-seq example, we benchmarked the DICE embeddings against MAGIC, ALRA, k-nearest smoothing, and NMF, using the same set of clustering metrics. The results summarized in 2, show that DICE consistently outperforms all competing denoising methods across all four metrics, validating our claim that diffusion-based denoising sharpens lineage relationships and resolves biologically relevant subpopulations better than existing techniques.
IPC-IEN + EN. This trajectory
Discussion We introduced DICE, a latent plug-and-plas framework for denoising and extract- ing meaningful embeddings from digh-diimesional observational data with an underlying low-rank structure. Both synthetic and real-world experiments demonstrated its ability to denoise beyond the training distribution and the ability to leverage clean reference data for denoising. We further showed that DICE can quantify uncertainty in cluster assignments and remains robust under model misspecification. Future work includes 1) extending DICE beyond linear low-rank structures and the i.i.d. noise assumption, 2) improving the efficiency of the sampling procedure, 3) incorporating multimodal data and spatial context, and 4) evaluating the quality of the produced embeddings on clinically meaningful downstream tasks.
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Reproducibility statement We provide the full codebase supplementary material and will re- lease it publicly upon publication. The appendix details the implementation, including hardware specifications, software packages, and training configurations. For synthetic experiments, we in- clude both a description in the main text and code for data generation and preprocessing in the supplementary files. For real-world datasets, describe dataset acquisition in the appendix, pre- processing steps in both the main text and appendix, and provide scripts to reproduce all experi- ments.
Use of Large Language Models (LLMs) Large Language Models (LLMs) were used to assist in preparing tables and figures and for proofreading.
Acknowledgments This work was supported in part by AWS Cloud Credit Grant from Cor- nell's Center for Data Science for Enterprise and Society.
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A DIFFUSION TRAINING
For given noise schedule {ar Helr), noisy versions of ( are generated
0 Vā. Ü, + vI di E1,b, ECS i.d.
I&),
(8)
where 0g II:=1 o,, A neural network εo, (T i) is trained to predict the injected noise Et from the corrupted sample ()(r) by minimizing the mean mean-squaar squared error
L(() = E Eo :o(T
The fitted network provides an estimate of Eler, | 0,77) which is directly related to the score function of the marginal distribution Pe (U( via
log Pe (0l)
- Co (0,)
Thus, training the model to predict noise is equivalent to estimating the score function, which in turn defines the reverse diffusion process and enables sampling from the prior Pprior- For the detailed procedure, see Algorithm 2.
B PROOF OF PROPOSITION 3.1
To prove Proposition 3.1 let us observe that if f(-) is the density of standard multivariate Gaussian distribution, then
-(X.-V2) - VZ,) = (2T))//2 exp (-211x, - v2 -
Therefore
Pp(Z((4) Xg. Ug) exp
|X, V2
1
|1:b,0) Z
(9)
2p
(1)
(2(0))
1
1 4 "T Zt40 + (2 (vTx,
Uls)
exp
2
2²
(10)
1 (Zg - my )A (Z - my)
& exp
(11)
2
where
A = - 4 + VTT
(12)
mg = A VT X,, + U(s)
(13)
and (2) follows by completing the quadratic form in the power of the exponential in (1) to match the density of a Gaussian distribution. Then the proposition follows by identifying the density on the right-hand side of (2) as that of a Gaussian distibution with mean my and covariance r.
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Algorithm 2 Train diffusion model
Training set {x(")) (xx""err epochs E, noise schedule harte) embedding dimension &
Require:
Ensure: Trained model Eg
1:
Calculate the factor loading matrix V from the training data by projecting X to a dimen- sional latent space using PCA.
2: Compute U, = x
3: for = ,,,,,,, E do
4: Independently sample timestep ((0) ~(),,,
all I,...,m m
5: Draw noises E, -N for all
6: Construct noised imputs
0 Vayao Ü. + V1 ag(0 6,0))
for ,.... n
and 0g(0 II
7:
Compute loss
((0) = 1 Σ
Lt, Ed (0)
==1
8: Take gradient step on 0.
9: end for
10:
return Eg
C MODEL ARCHITECTURE AND IMPLEMENTATION DETAILS FOR TRAINING THE DIFFUSION MODELS USED IN THE SINGLE CELL EXPERIMENTS
denoising network, TabularDiffr www.ffffsiioo onMLP, which predicts Gaussian noise ε,, from the noised sample Üü, R at diffusion step t. The architecture is parameterized by the number of residual MLP blocks M and the hidden dimension D. For synthetic experiments, we use a smaller model with M 2 blocks and hidden dimension D 64 (=> 150,000 trainable parameters), trained for E = 2,000 epochs. For single-cell experiments, we employ larger model with M blocks and hidden dimension D 512 (sz 35,000, 0000 trainable parameters), trained for E 10,000 epochs. We use a batch size of = 4048 across models.
Overview Across all experiments,
the
Input encoders We encode the data and the diffusion time with separate branches: (i) linear projection U. + .R... and (ii) a sinusoidal positional embedding of followed by two U-attiiid linear layers to IR We then concatenate the two D-dimensional features into 2Dddimensional vector.
Backbone The concatenated features are processed by eight residual MLP blocks of constant width 2D. Each block expands to 4D units and contracts back to 2D (Linear 2D + 4D, Batch- Norm1d. SiLU, Linear 4D) + 2D, BatchNormid)), and adds residual skip connection. This design provides sufficient capacity while remaining simple and fast for tabular inputs.
Output head A final projection (Linear 2D + D, SiLU, Linear D++ produces eo c0(0)) ,t) in the same dimensionality as (,(r)
Training objective and schedule We adopt the standard noise-prediction loss L(6) defined in Algorithm with a linear 3 schedule: 8r E [10-4 2x10 linearly spaced over T = 512 steps and a₁ 1 3c. ā, IT a. We optimize with AdamW (learning rate 1 x 10 batch size 4048) for 20000 epochs.
Data augmentation in training High-quality single-cell datasets may contain relatively few cells but provide reliable labels. We leverage these labels for data augmentation during diffusion model training. Inspired by mixup by Zhang al. (2018), interpolate between multiple same-class
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samples rather than mixing across classes. Concretely, given a sample x with label y,O) select four additional same-class points and construct
5 Σ Aaos. (A1...., As) Dirichlet(1). j=l
x
During training, we select the interpolated version ( with probability of p = 0.9. This intra- class mixup enriches the training distribution and improves robustness for the single-cell data. We use no data augmentation for the synthetic experiments.
Layer specification For completeness, Table 3 lists the exact layers and tensor shapes (B denotes batch size).
Stage
Operation / Activation
Output shape
Input projections
(+(")) branch
Linear (k+
(B,
t embedding
Sinusoidal (1-+
(B, 16)
Linear (16 (06+-D) D) SILU
(B,D)
Linear (D 30++10 D) SiLU
(B,D)
CONCAT
(B,4D)
Residual MLP block (repeated x.M)
Hidden
Linear -+D+)D
(B,4D)
BatchNormid Id (4D), SiLU
(B,4D)
Linear 2
(B,2D)
BatchNormld (2D)
(B,2D)
Residual
x - x + (zock(x))
(B,2D)
Output projection Head
Linear (2D SiLU
(B
Linear (D~
(B,
Table 3: Layer specification for TabularDi www.aaiifffuiii.a......... predicting (0( (r) k is the input (latent) dimension, D is the hidden dimension of the network
t). B is batch size,
Practical notes We normalize t to [0, 1] prior to sinusoidal embedding, use SiLU activations throughout, BatchNormid within blocks, and default PyTorel initializations.
Implementation setup We implemented all experiments in Python. Neural network training was performed in PyTorch (Paszke et al., 2019) on GPU-accelerated hardware, using mix of AWS g4dn. x1 arge EC2 instances and a dedicated Ubuntu server with an NVIDIA RTX PRO 6000 GPU. Our whole code is available online at
Runtime The runtime of our method can be divided into training and inference.
Training is dominated by parameter updates for the diffusion model, with speed depending on hard- ware and chosen model size. We report the execution times for the Ubuntu server with NVIDIA RTX PRO 6000 GPU. Using the CITE-seg setup, training on 9,000 cells with 3,000 dimensions for 10,000 epochs required approximately 36 minutes.
Inference proceeds via Gibbs sampling, alternating between the prior and likelihood steps. The prior step, which involves sampling from the diffusion model, is the most computationally expensive. As p determines the starting timestep of the diffusion model t, the duration of the prior sampling is dependent on p, larger values of p result longer runtime. The total runtime for denoising the test set of the CITE-seq dataset with 100 Gibbs iterations and p ranging from 5.0 to 0.5 took 12 minutes.
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D DISCUSSION ON THE CLUSTERING METRICS
To assess the quality of clustering in the learned embeddings, we employ five complementary met- rics: the average cosine silhouette score (Rousseeuw, 1987), the Adjusted Rand Index (ART) (Hubert & Arabie, 1985), the cell-type Locally Invariant Simpson Index (cLISI) (Korsunsky et al., 2019), the Normalized Mutual Information (NMI) (Strehl Ghosh, 2002), and the V-measure (Rosenberg & Hirschberg. 2007). Below, we provide brief description of each metric.
1. Average Cosine Silhouette Score. The silhouette score quantifies how well point is matched to its assigned cluster compared to other clusters, measured here using cosine distance. For point i, let a(i) denote the average intra-cluster cosine distance, and let b(i) denote the minimum average cosine distance between and any other cluster. The silhouette score for i is defined as
s() b(i) a(i) max( b(2)}
We report the mean silhouette across all points global measure of clustering quality, with values closer to indicating more distinct and coherent clusters.
2. Adjusted Rand Index (ARI). The Rand Index evaluates the agreement between two par- titions by counting the proportion of point pairs that are consistently assigned together or apart. The ARI corrects this measure for chance, making it more robust in settings with many clusters. It is defined as
RI E
ARI
max(RII E[R]]
where RI is the raw Rand Index and [[] is the expected value of the number of agreeing pairs under random clustering but fixed cluster sizes. ARI values range from (chance- level agreement)
(perfect alignment with ground truth).
3. Cell-type Locally Invariant Simpson Index (cLISI). The Local Inverse Simpson's 's Index (LISI) was originally introduced to assess batch mixing in single-cell integration tasks. We adapt it to clustering by replacing batch labels with cluster (or cell-type) labels, yielding cLISI. For each cell i, cLISI measures the effective number of distinct clusters represented in its k-nearest neighbor neighborhood:
LISI(i)=
Σ
Pie
where Pie is the fraction neighbors of belonging to cluster c. We report the average cLISI across all cells. Lower values correspond locally purer clusters, while higher val- ues indicate greater mixing. Unlike ARI, cLISI does not require ground truth annotations and provides a local measure that complements the global silhouette score.
4. Normalized Mutual Information (NMT). Normalized Mutual Information is an information-theoreiic measure for computing the agreement between two sets of cluster labels for the same data points. measures the agreement between two partitions of the data, C' and T as follows:
2I(C:T) NMT H(C) + HTT)
where T(C;T) is the mutual information between C and T, and H--) denotes the Shannon entropy (see, Cover & Thomas (2006)).
The NMI ranges from 0 to 1, with larger values indicating stronger agreement between the inferred clusters and the biological ground-truth labels. It provides a global summary of clustering accuracy and is widely used for benchmarking atlas-level annotation quality.
5. V-measure. The V-measure provides cluster labels. In our case, these sets are the cluster labels obtained by clustering the denoised embeddings and the original cell-type metadata. The V-measure computes two
entropy-based evaluation scheme of two sets
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Training stage
Estimate factor loading matrix using PCA
Train diftusion model & over embeddings on D
2
Input:
Reference dataset
Diffusion Model
Utrain
D
DDi>)
V
Iterate
Inference stage
Likellhood step Align samples with observations by
Prior Denoise latent representation with diffusion model
Input:
3
Initialize (
4
Target dataset
introducing noise
D
DD(1)
Z
7
Figure A.1: Visual sketch of the DICE denoising procedure. Training stage: (1) The factor loading matrix V is estimated on the reference dataset. (2) A diffusion model is trained to capture the underlying biological variation. Inference stage: (3) The initial latent representation (7(0) obtained using V. Iteratively: (4) Noise is added to U(s) to produce Z(0), (5) Z(s) is denoised using the learned diffusion model, yielding U(++1).
quantities, namely, homogeneity and completeness c. Let be the true cell labels and C be the predicted clusters. Then
H(K |C) H(K)
and c 1 H(C K)
h = 1
H(C)
where H(-) refers to the Shannon entropy of distribution. Then, the V-measure is defined as
x h x V-measure = h +
Higher values of V-measure indicate that the reconstructed atlas adheres to the cluster struc- ture encoded the metadata.
SENSITIVITY TO
DICE employs split Gibbs sampling scheme that introduces an auxiliary variable Z, to decouple the likelihood and prior updates. The strength of the coupling between Z; and U; is controlled through the Gaussian alignment penalty
2p2 IT. Zi
(14)
which appears in both the likelihood and prior steps (see Equation 5 and Equation 6). Intuitively, controls the distance between the sampling of U, and Z; in both the likelihood and prior step.
To interpret the role of p. we start by reviewing the canonical inverse problem of denoising: we observe a noisy matrix X, and our goal is to recover clean, underlying matrix UVT, where U is unknown. This problem is naturally framed in Bayesian setting. The likelihood ʃ(x - UV U). models the noise process, while the prior Pprior (U) encodes our assumptions about the structure of U. A standard approach to find the Maximum a Posterior (MAP) estimate, which solves the optimization problem:
ÚMAP arg min
log f(x T U) log Pprior (U).
U
((U:X)
R(U)
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where E(U; ) is the data-fidelity term, and R(U) is the regularizer induced by the prior. While this yields a single point estimate, discards all uncertainty about the solution.
To capture the full posterior distribution, turn posterior sampling, targeting:
iTU X)
IU) prior(UU..
This problem can be viewed as a soft relaxation of the MAP problem. The key is to observe that sampling from this posterior can be efficiently achieved using a Split Gibbs Sampler. (Xu & Chi, 2024; Venkatakrishnan et al., 2013b).
The sampler introduces an auxiliary variable Z and iterates between two conditional sampling steps:
1. Data-consistency Step (likelihood step): Sample Z from p( U,X), which typically involves the likelihood f and projects information from the observation X and the current state U.
2. Prior-projection Step (prior step): Sample from p(U 2) x (U) 8(U 2). This critical step draws sample from the prior distribution that is consistent with the auxiliary variable Z.
As we repeat this sampling procedure, the two steps eventually reach a stochastic equilibrium. When p is small, the strong coupling term dominates both steps. In this regime:
The initialization (typically from the data distribution as in Algorithm 1) has a persistent influence because the strong coupling prevents substantial deviation from previous states.
Both the likelihood and prior terms become relatively less influential compared to the align- ment constraint.
The sampler produces outputs that remain close to the initialization throughout the sam- pling process.
When is large, the coupling is weak, allowing:
The prior term to dominate the U-update step, pulling samples toward the prior distribution.
The data likelihood to have more influence in the Z-update step.
Greater exploration away from the initialization.
Overall, P controls the trade-off between fidelity to the observation and adherence to the learned prior. Small enforces tight coupling and rapid convergence near the data distribution, while large p allows freer interaction between the two updates and greater influence of the prior. In the following experiments, we empirically investigate how varying P affects the resulting embeddings when (i) applying DICE repeatedly to the same input observation (subsection E), and (ii) when the clusters present in the test data differ from those seen during training (subsection E.2).
E.1 SPREAD ACROSS RUNS
Setup To assess the sensitivity of DICE to different values of p. we perform an ablation study by denoising a fixed input point multiple times and comparing the spread across runs. By fixing the input observation to single point, we can isolate the effect of on the variability introduced by the stochastic sampling procedure. We use the two-cluster setup (see section 4, Setup 1), and use the same input point for all runs, the center of Cluster 2. We generate 500 independent runs of DICE for p E (0.1, 0.5, 1.0, 5.0, 10.0, 20.0).
Results We visualize the generated embeddings under different values of p with a UMAP in Fig- ure A.2. As p increases, the spread the sampled points grows: for small p. points remain tightly concentrated around the cluster center, while for large p, they disperse more broadly. The spread of the sampled points can be interpreted as the posterior uncertainty under different assumed noise levels. When the assumed noise is low (small p), the posterior is sharply concentrated around the observation, as the likelihood dominates and little deviation from the measurement is plausible. In contrast, for higher noise levels (large p). the posterior becomes broader, reflecting greater uncer- tainty: a wider range of latent points is consistent with the observed data. Thus, varying calibrates how strongly the likelihood influences the denoised embedding.
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Figure A.2: UMAP visualizations of 500 of DICE-denoised embeddings for the Cluster center under Sewp with increasing E [0.1, 1.0, 5.0, 10.0, 20.0] (left to right, top to bottom). Input observation in red; training data in grey.
Figure A.3: UMAP visualizations of PCA projections (left) and DICE-denoised embeddings for test points restricted to single cluster, with increasing values of p E [1.0, 8.0, 20.0] (left to right), As P increases, the embeddings gradually shift from being measuremeen--ligneed prior-aligned. Training data are shown in grey.
8.0,
E.2 MISMATCH BETWEEN TRAINING AND TEST CLUSTERS
In this section, investigate the influence of prior guidance under different p. We create mismatch between the training and the test data, and evaluate whether points stay close to the measurement, or are moved according the prior. We update our training data fixed contain 5 well separated clusters. Then, we vary the input test data, which either comes only from one cluster, or from new cluster which is not present in the training data. Our diffusion model the same as for the other synthetic experiments (see Appendix C). We run DICE for 200 Gibbs iterations without averaging, and report results for different choices of p.
Figure A.4: UMAP visualizations PCA projections (left) and DICE-denoised embeddings for test points from a cluster not observed during training, shown for increasing [0.5, 2.0, 4.0] (left to right). With larger p, the unseen cluster progressively gets mapped the training clusters, reflecting stronger prior alignment. Training data are shown grey. The UMAP projection fitted jointly on the combined training and test data.
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Training data We extend synthetic data generation to include five instead of two cluster centers, while still following the same low-rank latent structure. We chose low noise levels, such that the training data is well separated. In detail, we start by drawing the cluster centers tii ~ (O, /15) in latent dimension & = 15. The training prior is a balanced Gaussian mix- ture. Pprior = Σi Nis(ju, 0.3 I1s) with each component corresponding to one cell type. We sample a loading matrix V E 20000 with entries i.i.d. (0,, 1). For each cell i, we draw latent (r) N200 (0, 0.5 and generate a synthetic expression profile x!") vU e((T) E |2200 I,..., 8000. This yields training dataset D(r) of size m 8,000 in d = 200 observed dimensions. We generate test dataset D(() of size 2,000 by increasing the noise variance ( (0, /200).
u(() Pprior and measurement noise by
E.2.1 FEWER CLUSTERS IN THE TRAINING DATA
Setup We construct test set consisting solely of cells from single cluster. This setup corre- sponds to a mismatch where the model's learned prior encompasses a richer structure than what is present in the observed data: While DICE is trained on uniform distribution across clusters, where each cluster is equally likely, the test data only contains one cluster.
Results We plot the PCA projection (left), and generated denoised embeddings for p {1.0, 8.0, 20.0} (left to right) in Figure A.3. DICE projects all samples into one of the learned clusters, independent of p. However, P controls which of the learned clusters the observations are assigned to. For small p. samples remain close to their noisy measurements, resulting in all em- beddings staying in the original input cluster. For intermediate p. most points concentrate within cluster, but subset diffuses into neighboring clusters. For large p. the denoising process aligns closely with the prior, resulting in approximately uniform assignment across clusters, independent of the original observation.
E.2.2 MORE CLUSTERS THE TRAINING DATA
We now consider the opposite case, where the test data includes a new cluster that was unseen during training. Specifically, we sample an additional cluster center μ4, from the same generative process as in the main setup, ensuring that this cluster lies outside the support of the training data.
Results We show UMAP visualization of PCA projection (left), and generated denoised embed- dings for p E {0.5, 2.0, 4.0) in Figure A.4. To observe the test cluster, the UMAP is fitted jointly on the combined training and test data. The PCA projection (left) shows that test measurements form a new cluster, which is non-overlapping with the initial training clusters plotted in gray. For small p. DICE maintains distinet cluster of points around the novel cluster. As p increases, sam- ples gradually disperse into the nearest training clusters, reflecting the influence of the prior. With p = 4.0, almost all points are assigned to clusters from the training set. Notably, even for large the measurements still exert directional influence on the embedding. Points are not assigned uniformly inside the clusters, but tend to align with parts of the clusters more consistent with the observed input.
E.3 DISCUSSION
The results obtained by varying P highlight the trade-off between data fidelity and prior alignment. A small p keeps denoised samples close to their observations, preserving local variations even when they deviate from the training data. contrast, a large P guides samples toward representations consistent with the learned prior. This ability to steer between these regimes makes a powerful control parameter for balancing embedding quality across datasets of differing noise levels. When the data assumed to be low-noise, small p maintains proximity to the measurements, allowing structure not captured in the training set to emerge. Conversely, for high-noise data, a larger leverages the prior guide the estimates toward more reliable, denoised representations.
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F
SENSITIVITY TO k
Adjusted Rand Index (ARI)
0.7
Neo Cartex DICE
0.6
Neo Cortex PCA
CITEseq DICE
CITEseq PCA
0.1
0.3
10
15
20
25
50
100
200
k
Figure A.S: Analysis of Adjusted Rand Index(ARI) for different values of k, the dimension of the latent space. x-axis is not to scale.
Setup To assess the sensitivity of the DICE embeddings to the choice of latent dimension k in Eq. (2), we conducted an ablation study on both the CTTE-seq and human neocortex datasets using k E (5, 10, 15, 20, 25, 50, 100, 200}. The quality of the resulting embeddings was evaluated using Adjusted Rand Index (ARI).
Setup We summarize our findings in Figure A.5. For the CITE-seq dataset, the ARI values of the DICE embeddings increase up to 100 and decreases thereafter. Furthermore, as the latent dimension grows, the benefit of DICE-based denoising over PCA based embeddings becomes more pronounced. This is expected given the large number of distinct cell types in the dataset, which naturally requires dataset, the ARI values remain relatively stable across different choices of k with DICE outper- forming naive ICA-based embeddings for most values. Based these observations. mend selecting by identifying an elbow in the scree plot of the singular values; choosing a slightly larger & is generally safe, but excessively large values lead noisier embeddings and increased computational cost with minimal marginal gain.
digher-dimensionall latent representation. In contrast, for the human neocortex
recom-
G DATA SOURCES
CITE-seq benchmark. We used the Seurat CITF-seq dataset distributed with scvi-tool s as our atlas building benchmark. The original Seurat object and feature matrices were taken directly from the sevi example and used without modification, except for preprocessing and downsampling steps described in Section 5.
Human fetal brain development. We analyzed two fetal neocortex transcriptomic datasets, Nowakowski et al. (2017) and Polioudakis et al. (2019), obtained via the gEAR (Gene Expression Analysis Resource) portal (Orvis et al., 2). For both studies, we downloaded the author-provided processed count matrices and accompanying metadata from gEAR and restricted analyses to the shared gene set as detailed Section 5.
Provenance and licensing. All datasets were used under the terms specified by their original authors and hosting platforms. We performed only secondary analysis; no new data were generated for this work.
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H PREPROCESSING DETAILS FOR THE SINGLE CELL DATASETS
H. CITE-SEQ DATASET
We adopt the following pre-processing pipeline for the CITE-seq data obtained from scvi-da random stratified sampling was used to subsample 10000 cells from the total pool of cells, main- taining the relative proportion of different cell types.
QC filtering To remove the cells with poor expression quality and sequencing errors, we adopt the standard QC scheme adopted in Seurat Stuart et al. (2019) and scanpy. We exclude cells with fewer than 200 detected genes and filter out genes expressed in fewer than cells. Additionally, to climinate cells that may have been dying or stressed at the time of sequencing, we remove all cells with more than 15% mitochondrial RNA content.
Normalization and Batch Effect Removal. To account for differences sequencing depth across cells, we perform standard library-size normalization: for each cell, we divide the gene counts by the total UMI count and multiply by 10 The resulting values are then transformed using x H log(1 + .:) to stabilize variance and reduce the influence of highly expressed genes. We selected the top 3,000 highly variable genes (Seurat v3 criterion), scaled to unit variance with values capped at 10. Batch effects from sequencing lanes were corrected with Harmony (Korsunsky et al., 2019), yielding an expression matrix X 100000x30000
H.2 HUMAN NEO-CORTEX DATASETS
We analyzed 3,495 cells from Nowakowski et al. (2017) and 15,126 cells from Polioudakis et al. (2019). The pre-processed datasets were obtained from the gEAR (Gene Expression Analysis Re- source) portal (Orvis et al., 2021). The downloaded data had already undergone standard QC proce- dures, as described for the CITE-seq dataset, and the expression values were normalized to a fixed library size and transformed using the log 1p operation.
Because the two studies differed in experimental protocols and developmental stages, only 17,638 genes were shared across both datasets. From these, we selected the top 5,000 highly variable genes (HVGs) within each dataset and retained the intersection of these HVG sets, yielding 785 shared highly variable genes. This resulted in the matrices Nnow
R 3495
and X R 111126x785
Cell-type annotations were available only for the Nowakowski et al. (2017) dataset, so we transferred these labels to the Polioudakis et al. (2019) cells using the Seurat v3 label transfer procedure (Stuart et al., 2019) based on the shared set of genes.
I IMPLEMENTATION DETAILS FOR BENCHMARKING
To benchmark DICE embeddings against those generated by popular techniques for ScRNA-ses analysis, we compared the performance of our method against MAGIC (van Dijk et al., 2018b), ALRA (Linderman et al., 2022b), NMF (Non-negative Matrix Factorization), scVI (Lopez et al., 2018 8a) and k-nearest neighbor based denoising with k 5, 10 and 15. We evaluated clustering performance using Seurat's FindClusters algorithm with the Leiden method (Traag et al., 2019). To select the optimal hyperparameter, ran the algorithm and retained the clustering that achieved the highest ARI. In the following, we summarize the pro- cedures and their implementation inwour analysis.
range of resolution values (0.01-2.0)
MAGIC implements a Markov affinity based diffusion smoothing of cellular expression profiles.In our benchmarking experiment, we computed PCA scores from the expression matrix and supplied them directly to MAGIC for denoising. We used the official imple- mentation available at https: //github. com Kri shnaswamy hww.waaaaLLaa/M/M..C... M.GIIC
ALRA applies an adaptively thresholded low-rank matrix approximation to recover de- noised gene-expression values. We employed the PyALRA Python implementation (Lanau & Waterfall, 2025) for all experiments. The expression profiles were provided to ALRA and a low rank approximation was computed. We used 50 PC scores of the denoised matrix our embeddings for the nechmarking.
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Non-negative Matrix Factorization (NMF) is commonly used alternative PCA for de- noising and latent factor extraction in scRNA-seq analysis (Kotliar et al., 2019; Carbonetto et al., 2026). We used the NMF implementation from scikit- (Pedregosa et al., 201 I), selecting the latent dimension to match the numher of PCs used for the DICE embeddings.
scV/ is a priiational-autoonoodee-----r counting for library-size variation and batch effects. For the CITE-seq dataset, trained se scVI with a 10-dimensional latent spacewand used the estimated posterior expectations of embeddings. We could not apply scVI to the human fetal neo- cortex dataset because the raw UMI counts were not available.
model that denoises raw UMI counts while ac-
the latent variables
k-nearest- k-nearest-neighbor smoothing is classical local-averaging denoising strategy widely used in pipelines such as Seurat v3 (Stuart et al., 2019). We denoised PCA embeddings by replacing each cell's representation with the average of its k nearest neighbors (Euclidean distance), using & E {5, 10, 15).
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Reviews:
[{"review_id": "63RqdG5WYk", "reviewer": "Reviewer_fdSG", "summary": "This paper introduces DICE, a latent plug-and-play diffusion framework for the denoising and clustering of single-cell RNA-seq data. The key innovation is the separation of the observation and denoising space: a diffusion model is trained in a learned low-dimensional latent space, and inference employs a Gibbs sampling algorithm alternating between latent denoising and input-space steering. DICE provides a tunable balance between data fidelity and prior knowledge, enables explicit uncertainty quantification, and leverages high-quality references for improved denoising beyond the training distribution. The paper presents robust evaluations on synthetic and real single-cell datasets.", "questions": "See Weaknesses", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. By using a latent plug-and-play architecture, DICE addresses a core limitation of prior dimensionality reduction strategies and effectively maintains biological relationships lost in classical approaches like PCA or VAEs.\n2.  DICE produces more biologically coherent clusters and quantitative improvements in biological clustering metrics on real data.", "weaknesses": "1. The evaluation compares primarily to a PCA baseline, which is too limited given the recent progress in single-cell denoising and clustering. Including comparisons with methods such as scSiameseClu, scDCCA, or SCDD would provide a more convincing empirical validation and better contextualize the proposed approach within current literature.\n2. One concern about DICE is that the generalizability to more complex data types is unclear. The method focuses solely on scRNA-seq gene expression and does not experimentally explore or theoretically support extension to multimodal or spatially-resolved single-cell data domains where denoising and latent structure recovery are at least as challenging.\n3. Although the plug-and-play approach is theoretically flexible, in practice the latent space is initialized via PCA (Section 3), and the factor loading matrix $\\widehat{V}$ is reused for high-dimensional projections throughout. As a result, the overall performance is partially determined by the initial PCA mapping and its limitations (e.g., axis alignment, linearity)—potentially biasing the outcome, especially when compared to nonlinear baselines (e.g., deep autoencoder, contrastive methods). The choice to always use PCA as a latent basis appears arbitrary and is not justified against recent nonlinear alternatives.\n4. There are some recent related works for clustering and denoising for large-scale single-cell datasets that the authors are encouraged to include in the related work section:\n-> MetaQ: fast, scalable and accurate metacell inference via single-cell quantization", "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": "adazmyjYYX", "reviewer": "Reviewer_NrsQ", "summary": "This paper presents DICE (Diffusion Induced Cell Embeddings), a latent plug-and-play diffusion framework for denoising single-cell RNA-seq data. The core idea is to separate the observation space from the denoising space and perform Gibbs sampling between them, thus maintaining biological structure while reducing noise. The paper targets a topic of increasing interest at the intersection of machine learning and computational biology. Overall, it is clearly written, conceptually coherent, and focuses on a scientifically meaningful problem. The work is solid and promising, though several aspects could be clarified or improved.", "questions": "Please respond to C1, C3, and C4.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. The paper’s motivation is clearly articulated, and the logic throughout is coherent and easy to follow.\n\nS2. The proposed problem single-cell data denoising is important and relevant to scientific data analysis.\n\nS3. The presentation is overall well-structured and the experiments provide reasonable empirical validation.", "weaknesses": "**Concerns**\n\nC1. Any denoising method inevitably relies on certain distributional assumptions. The authors justify their design choices by citing prior work, which is reasonable, but the concern remains. For example, the Gaussian-noise assumption (“single-cell are often modeled with Gaussian noise,” line 251) may not always hold, and Eq. (5)–(6) depend on Gibbs sampling with intuitively motivated regularization terms. The authors may add further discussion or experiments exploring the impact of different likelihood/distributional assumptions (beyond current experimental setups) to demonstrate the robustness and superiority of the chosen modeling strategy.\n\nC2. Although the technical description is detailed, the paper would benefit greatly from a conceptual flowchart summarizing the input–output pipeline and the main computational steps. Such a figure would help readers grasp the end-to-end process—how data flow from raw expression profiles to denoised embeddings—without delving into mathematical detail, and would clarify the intuition behind core approximations or modifications.\n\nC3. In Section 1, the authors emphasize that applying image-based PnP frameworks directly to single-cell data is difficult because gene expression exhibits low-rank and correlated structure. However, the experiments do not compare DICE with straightforward or classical denoising baselines. Given that classic PCA is main baseline in the discussion, including or expanding such baseline comparisons would make the empirical validation more convincing.\n\nC4. It is unclear whether the key sentence \"Our framework is agnostic to specific preprocessing choices and accommodates diverse noise structures\" is supported or demonstrated in different places to prove that this paper is agnostic to preprocessing choices and tolerant to diverse noises. For example, do the noise level settings in the final experimental section reflect \"diverse\"?\n\nC5. The presentation is strong overall, but could still be refined. For example, Section 2 functions largely as preliminaries, yet the text uses first-person language and sometimes blurs whether new contributions are being proposed. Besides, the sentence “The main challenge lies in the likelihood term” should specify what aspect of the likelihood is challenging.", "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": "NcWRzcNzDB", "reviewer": "Reviewer_kuwc", "summary": "This paper introduces DICE (Diffusion Induced Cell Embeddings), a latent plug-and-play diffusion framework for denoising single-cell RNA-seq data. The method trains a diffusion model on low-dimensional PCA embeddings from a high-quality reference dataset, then applies a Gibbs sampling procedure to denoise query cells by alternating between a likelihood alignment step and a prior alignment step. The authors evaluate their approach on synthetic data and two real single-cell datasets, reporting improved cluster separation compared to PCA baselines.", "questions": "1. The paper suggests that DICE addresses batch effects and dataset shifts. If batch effect removal and dataset integration are indeed core objectives, then how does DICE compare quantitatively to Seurat/Harmony/scVI?\n\n2. Can the method scale to 100K+ cells?\n\n3. How do PCA and DICE embeddings compare in terms of direct classification performance (precision/recall) for predicting ground-truth cell type labels?\n\n4. Single-cell data annotation is often performed by biologists and domain experts without extensive computational backgrounds. The method requires selecting multiple hyperparameters (k, ρ, number of Gibbs iterations) and training diffusion models. What steps have been taken to make this approach accessible to practitioners?\n\n5. What is the recommended procedure for selecting ρ in practice?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Principled uncertainty quantification: Unlike standard clustering pipelines, the method provides confidence sets for cell-type assignments through posterior sampling, which is valuable for downstream analyses and could help identify cells with ambiguous identities.\n\n2. Flexibility: The framework accommodates different noise models without requiring explicit parametric assumptions.\n\n3. Comprehensive synthetic evaluation: The four synthetic setups systematically examine different types of distribution shift (signal strength, noise model, latent prior), providing evidence of robustness to model misspecification.", "weaknesses": "1. Unclear problem framing and scope: The paper oscillates between four distinct objectives (atlas generation, batch integration, clustering, and denoising). This makes it difficult to assess the method's contribution.\n\n2. Insufficient quantitative evaluation:\n\n- The real-world experiments rely heavily on visual inspection of UMAP plots, which is subjective.\n\n- The authors claim that “across all four settings, DICE yields clearer separation in UMAP … than the PCA baseline, indicating more faithful recovery of the underlying classes.” However, the meaningfulness of this visual separation is unclear, given that UMAP does not preserve exact metric distances. If DICE truly recovers the underlying class structure better than PCA, then a simple classifier trained on these embeddings should achieve better precision/recall for predicting ground-truth cell type labels compared to the same classifier trained on PCA embeddings.\n\n- The paper suggests the method addresses batch effects and dataset shifts, yet provides no quantitative comparison with established integration methods (Harmony, Seurat integration, scVI).\n\n3. Missing experimental details and comparisons:\n\n- Hyperparameter sensitivity: Figure 5 shows dramatic sensitivity to ρ, but the paper provides no principled approach for selecting ρ in practice.\n\n- Robustness to k: k is set to 15 (synthetic), 25 (CITE-seq), or 15 (fetal brain), with the justification that it corresponds to “the elbow of the singular-value spectrum” for real data. However, no details are provided on how sensitive the results are to this choice. A sensitivity analysis would help assess the impact of different k values on performance.\n\n4. Questionable practical utility:\n\n- The model assumes an identical factor loading matrix V across reference and target datasets, but V represents transcriptional programs that differ across biological contexts. Obtaining reference data that simultaneously matches the target in cell type composition, tissue origin, disease state, and patient population characteristics is infeasible, limiting practical applicability.\n\n- Training takes ~11 hours for 9,000 cells; inference takes ~30 minutes for 1,000 cells with ρ=1. This is prohibitively expensive for typical single-cell datasets with 100K+ cells.\n\n- The cross-dataset experiment shows mixed results, raising doubts about generalization.", "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)