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Nonparametric Distribution Matching for Self-Supervised Whole-Slide Image Condensation

Duong M. Nguyen, Trong Nghia Hoang, Hang Thi Nguyen, Thanh Trung Huynh, Phi Le Nguyen, Minh N. Do

arXiv:2610.00678v1eess.IVcs.LG

TL;DR

WSI condensation must reduce substantial computational costs, but existing methods lack a principled account of feature distributions under a fixed encoder. NICER formulates condensation as distribution matching and uses a nonparametric MAP-based approximation. Across histopathology benchmarks, it reports consistent improvements in efficiency-accuracy trade-offs, supported by pathological evaluation.

  • Problem

    WSI condensation methods often rely on heuristic prototypes and lack a principled formulation for preserving feature distributions under a fixed representation lens.

    Source: §1 · pp. 1–2

  • Method

    NICER matches feature distributions under a fixed pretrained encoder and uses a Poisson point process MAP formulation with a tractable approximation.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    Across cancer-subtyping and survival-prediction benchmarks, NICER consistently outperforms baselines with improved performance-computation trade-offs and pathological assessment support.

    Source: §1 · p. 2, §6 · p. 10

  • Takeaways & Limitations

    The results underscore the benefits of a theoretically grounded approach to scalable WSI representation learning.

    Source: §6 · p. 10

    NICER uses bag-of-features and fixed pretrained representations, so spatial or multi-scale dependencies are not explicitly modeled and broader encoder evaluation is needed.

    Source: §6 · p. 10

Abstract

from arXiv · show

Histological whole-slide images (WSIs) are central to computational pathology but pose severe computational challenges due to their extremely high resolution, often spanning several gigabytes per slide. To enable scalable learning, existing methods apply self-supervised data condensation to reduce computational cost, but typically rely on heuristic prototype learning and do not explicitly preserve learning-relevant feature distributions for downstream tasks. In response, we introduce a principled reformulation of WSI condensation as a distribution-matching problem under a fixed representational lens, and develop NICER, a tractable approximation framework based on a nonparametric prior with slide-adaptive capacity. Experiments on five histopathology datasets, together with clinical evaluation from a board-certified pathologist, show that NICER consistently outperforms prior methods, achieving an average accuracy improvement of 7.44% while offering improved efficiency-accuracy trade-offs, highlighting the benefits of principled, distribution-aware condensation for scalable histological representation learning. Source codes are available in https://github.com/nmduonggg/NICER.

1 Introduction

WSIs create substantial training and storage costs, while existing condensation methods lack a principled account of feature distributions under a fixed encoder. NICER reframes condensation as distribution matching and approximates it with a nonparametric framework.

1 Introduction

  • WSIs can span several gigabytes, so pipelines divide them into thousands of patches, creating substantial training and storage costs.Patches are embedded with a pretrained encoder and aggregated into slide-level representations.Source: §1 · p. 1
  • Roughly 19% better computation-accuracy tradeoff is reported for NICER than PANTHER at similar compression; across benchmarks, NICER consistently outperforms baselines.The comparison reports higher downstream F1 and a smaller compression rate for NICER; evaluation also includes cancer subtyping, survival prediction, and pathological assessment.Source: Figure 2 · p. 2, §1 · p. 2
  • Existing condensation methods lack a principled formulation for preserving condensed information under a fixed representation lens and often rely on heuristic prototypes.Task-agnostic clustering and prototype methods may overlook the pretrained feature extractor's fixed inductive bias.Source: §1 · p. 2
  • NICER formalizes WSI condensation as distribution matching under a fixed representation lens and connects the objective to MAP inference under a point-process prior.This yields a nonparametric formulation that characterizes an equivalent solution class.Source: §1 · p. 2
  • NICER approximates the objective by decoupling local pattern coverage from global concept abstraction, enabling adaptive-capacity condensation.Its empirical evaluation reports improved performance-computation trade-offs while preserving clinically meaningful histological structure.Source: §1 · p. 2

2 Theoretical Framework

The framework formulates WSI condensation as matching real and condensed feature distributions under a fixed pretrained encoder. It reduces the global objective to slide-level problems and interprets their solutions through a Poisson point process prior.

2 Theoretical Framework

  • The objective matches real and condensed WSI representation distributions under the same pretrained encoder, using squared 2-Wasserstein distance.The paper leaves other discrepancy measures to future work.Source: §2 · p. 3
  • A tractable surrogate upper-bounds global Wasserstein discrepancy with per-slide discrepancies, enabling independent inference for each slide.The reduction preserves slide-specific traits; the paper favors slide-specific prototypes for MIL, where each WSI is processed independently at inference.Source: §2 · p. 3
  • Theorem 2.2 characterizes slide-level solutions as MAP estimates under a Poisson point process prior, equivalent to minimizing a cardinality-penalized optimal-transport objective.The prior models finite unordered prototype sets with unknown cardinality and encourages sparsity through a linear size penalty.Source: §2 · p. 4
  • NICER makes the intractable inference tractable by introducing a reduced, data-adaptive pattern set and alternating between pattern summarization and prototype optimization.This approximation assumes conditional independence of the empirical embeddings from concepts given the pattern set.Source: §2 · p. 4

3 A Tractable Condensation Approximation

NICER approximates nonparametric WSI condensation in two stages: it first summarizes patch embeddings as local patterns, then condenses those patterns into adaptive slide-specific concepts. Alternating updates make the concept-learning objective tractable.

3 A Tractable Condensation Approximation

  • NICER decouples pattern exploration from concept condensation, using a nonparametric prior to merge redundant concepts and adapt model complexity to slide variability.The overview frames these stages as a learning-based approximation to distribution matching.Source: §3 · p. 4, Figure 3 · p. 5

3.1 Local Pattern Exploration

  • Local pattern exploration selects a compact set Z that summarizes empirical embeddings H through similarity-based retrieval.Each feature distributes its mass over its top-κ nearest patterns, yielding a compact intermediate representation.Source: §3.1 · p. 5

3.2 Global Concept Condensation

  • Global concept condensation models patterns as generated from a latent concept set with unknown capacity and assigns each pattern to exactly one concept.Learnable assignment logits permit many-to-one aggregation, while the pattern-concept likelihood uses a squared-distance geometry with learnable covariance.Source: §3.2 · pp. 5–6

3.3 Condensation Learning

  • Condensation learning alternates continuous updates of concepts and model parameters with discrete pattern assignments.The active-set update removes concepts with no assignments, implementing the PPP preference for smaller configurations rather than separate heuristic pruning.Source: §3.3 · p. 6

4 Experiments

NICER is evaluated on five datasets across cancer subtyping and survival prediction, with additional morphological-prototyping analysis and pathologist assessment. Results report gains across tasks, alongside trade-offs and sensitivity analyses for pattern capacity and top-κ.

4.1 Experiment Settings

  • The experiments cover five datasets and cancer subtyping and survival prediction, using task-specific metrics and both condensation-ability and morphological-prototyping evaluation settings.Cancer subtyping uses Kappa, weighted F1, accuracy for PANDA, and balanced accuracy for NSCLC; survival prediction uses C-Index.Source: §4 · p. 6, §4.1 · pp. 6–8

4.2 Performance on Condensation Ability

  • NICER outperforms competing methods across cancer subtyping and survival settings, with gains on PANDA and NSCLC, in-cohort BRCA and NSCLC, and cross-cohort CPTAC evaluation.Reported comparisons include PANDA gains of 2.28–15.04% with DSMIL and ILRA and up to 33.72% with ABMIL; survival gains reach 79.73%, and cross-cohort gains are approximately 5%.Source: Table 1 · p. 7, §4.2 · p. 8

4.3 Performance on Morphological Prototyping

  • Across morphological-prototyping scenarios, NICER outperforms other unsupervised methods and shows minimal performance drops across tasks and datasets.The comparison uses a nonlinear transformer layer; the supplied passage describes the result pattern but does not provide individual table values.Source: Table 2 · p. 7, §4.3 · p. 8

4.4 Additional Analysis & Ablation Studies

  • Pathologist-annotated prototypes show intra-pattern morphological consistency and distinguish diagnostically distinct patterns, including viable tumor and adjacent necrosis.The assessment is based on a representative TCGA-BRCA WSI.Source: Figure 6 · p. 8, §4.4 · p. 9
  • Both NICER and PANTHER reduce storage from 194GB to 3.2–5.7GB, while NICER has higher per-iteration GPU and runtime costs during condensation.The passage states that condensation is performed once per WSI, making the additional cost negligible relative to the reported downstream performance gains.Source: Table 4 · p. 8, §4.4 · p. 9
  • On PANDA with ABMIL, increasing initial pattern count improves performance but gains begin to saturate beyond roughly 200 patterns.The reported explanation is that larger sets increase distillation capacity, while condensation removes redundancy and approaches stable representations.Source: Figure 4 · p. 7, §4.4 · p. 9
  • Performance improves as top-κ increases but plateaus or slightly declines beyond κ = 3, where patch information diffuses across patterns and weakens distillation.The sensitivity analysis covers Condensation Ability and Morphological Prototyping.Source: Table 3 · p. 7, §4.4 · p. 9
  • On NSCLC, NICER concept embeddings become more dispersed over the first five iterations, reflecting increasingly diverse and specific information.The t-SNE visualization is described as showing reduced overlap and redundancy while preserving diversity.Source: Figure 5 · p. 7, §4.4 · p. 9

5 Related Work

Prior work spans dataset condensation, multiple-instance learning, and self-supervised pathology methods, but existing WSI condensation approaches rely largely on heuristics. The section motivates a principled formulation aligned with task-agnostic representation space.

Multiple Instance Learning

  • MIL treats each WSI as an unordered patch set, motivating condensation; NICER is framed as MIL-agnostic and prioritizes computational and storage efficiency.Source: §5 · p. 9

Dataset Condensation

  • Dataset condensation synthesizes compact training datasets to reduce storage and training costs, with methods including gradient matching, trajectory matching, and factorization-based approaches.The passage notes that many such methods rely on expensive supervision, limiting applicability to histological imaging.Source: §5 · p. 9

Self-supervised Data Condensation and Computation Pathology Applications

  • In computational pathology, self-supervised condensation targets transferable task-agnostic information, while existing WSI methods typically use handcrafted features or unsupervised clustering heuristics.The section calls for a principled formulation aligned with the task-agnostic representation space.Source: §5 · pp. 9–10

6 Conclusions and Limitations

The conclusion presents NICER as a nonparametric, slide-adaptive distribution-matching approach and reports improved efficiency-accuracy trade-offs across histopathology benchmarks.

6 Conclusions and Limitations

  • NICER’s nonparametric distribution-matching formulation adapts prototype capacity to slide complexity and improves efficiency-accuracy trade-offs across histopathology benchmarks.The conclusion also cites pathological evaluation.Source: §6 · p. 10

A Broader Statement of Impact

  • NICER is presented as an effective nonparametric WSI condensation algorithm for learning from histological data with varying complexity; experiments use publicly available datasets.The paper describes potential ethical implications for real-patient use as indirect and outside the study’s focus.Source: A Broader Statement of Impact · p. 17

B Pseudocode for NICER

  • NICER repeatedly assigns patches to top-κ patterns, distills them, condenses patterns into concepts, and reinitializes the pattern set from the resulting prototypes.The CONDENSE routine alternates concept and assignment updates, then removes unused concepts.Source: B Pseudocode for NICER · p. 17

C Extended Related Work

  • Prior work motivates NICER’s self-supervised setting: WSI methods use manual features or clustering, while many condensation methods rely on supervision and costly full training.MIL treats each slide as a set of patches; NICER prioritizes computing and storage efficiency across MIL frameworks.Source: C Extended Related Work · pp. 17–18

D.1 Datasets

The evaluation spans five histopathology datasets for cancer subtyping and survival prediction, with complementary condensation and prototyping settings and specified baselines and training procedures.

D.1 Datasets

  • The five datasets support prostate cancer grading, lung adenocarcinoma-versus-squamous subtyping, breast and lung survival prediction, and external-cohort survival evaluation on CPTAC LUAD.PANDA contains 10,616 biopsies; the NSCLC cohort has 1,041 slides (529 LUAD, 512 LUSC). TCGA-BRCA includes 1,133 slides from 1,062 patients, and TCGA-LUAD includes 529 slides from 478 patients. CPTAC LUAD provides 185 patients and 486 WSIs for external evaluation without further adaptation.Source: §D.1 · pp. 18–19

D.2 Evaluation Setting Details

  • Evaluation separates preservation of predictive information from the original feature bag from the usefulness of condensed prototypes as transferable morphological representations.The prototyping setting applies methods to training and test sets, then uses prototypes directly for downstream prediction.Source: §D.2 · p. 19

D.3 Baseline Details.

  • Baselines include unsupervised prototyping methods and supervised MIL predictors, spanning clustering, nonparametric factor analysis, soft assignments, and attention-based models.The compared methods include DeepSets, ProtoCounts, H2T, InfiniteGPFA, OT, PANTHER, ABMIL, DSMIL, and ILRA.Source: §D.3 · p. 19

D.4 Training Details and Computational Resources

  • Experiments use UNI patch embeddings and an NVIDIA RTX A6000; NICER uses 20 iterations, κ = 3, and an initial pattern set size of M0 = 1000.The patch inputs are non-overlapping 256 × 256 regions at 20× magnification.Source: §D.4 · pp. 19–20

E Space Complexity and Scalability

The appendix estimates NICER’s memory scalability and explains how its Poisson point process prior yields a variable number of active concepts through assignment-based pruning.

E Space Complexity and Scalability

  • O(Nd + Md + NM) space is dominated by patch-pattern similarities; an RTX A6000 estimate supports about 5.9 × 10^6 patches, and larger bags can be batched without changing the objective.Typical extreme clinical WSIs at 0.5 MPP contain fewer than 10^6 tissue patches after background filtering.Source: E Space Complexity and Scalability · p. 20

F Nonparametric Prior in NICER

  • The Poisson point process prior contributes a cardinality penalty, so NICER removes concepts with no assigned patterns instead of fixing the effective concept count in advance.The active concept set adapts to slide complexity.Source: F Nonparametric Prior in NICER · pp. 20–21
  • The MAP objective combines patch-distribution preservation by local patterns, assignment of patterns to latent concepts, and a prior over the concept configuration.The passage identifies the empirical patch-embedding bag, learned local patterns, and latent concepts as the objective’s central quantities.Source: F Nonparametric Prior in NICER · p. 20

G Derivation of P(H | Z)

The derivation gives a probabilistic basis for feature-pattern similarity and motivates top-κ retrieval as a compact approximation for pattern selection.

G Derivation of P(H | Z)

  • Maximizing the Gaussian likelihood is approximately equivalent to maximizing feature-prototype similarity, providing a probabilistic justification for the design.Pattern selection uses cosine similarity and distributes each feature’s mass over its top-κ nearest patterns.Source: G Derivation of P(H | Z) · p. 22
  • Top-κ assignment avoids spending computation distilling patches into irrelevant prototypes, and reported performance improves from κ = 1 to κ = 3 before plateauing beyond κ = 5.The passage attributes this plateau to a negligible residual approximation gap once patterns are well-separated.Source: G Derivation of P(H | Z) · p. 22
  • A rigorous theoretical analysis of the anticipated absence of performance degradation remains future work.The paper bases its expectation on conceptual and empirical validation.Source: G Derivation of P(H | Z) · p. 22

H Proof of Theorem 2.2

This section links the per-slide distribution-matching objective to MAP inference under a Poisson point process (PPP) prior. The prior makes prototype-set size a sparsity penalty, yielding a cardinality-penalized optimal-transport objective under a uniform parameter prior.

H.1 Problem Setup

  • Per-slide distribution matching is equivalent to MAP inference over prototypes and model parameters under a PPP-induced Gibbs posterior.The correspondence motivates alternating MAP-based optimization.Source: §H.1 · p. 22

H.2 Poisson Point Process prior and MAP equivalence

  • The PPP prior induces a sparsity-promoting penalty proportional to the number of prototypes.Its configurations are finite, unordered prototype sets with unknown cardinality.Source: §H.2 · p. 23
  • With a uniform prior on θ, MAP inference coincides with minimizing the cardinality-penalized OT objective.This identifies the PPP-based inference objective with the penalized OT problem.Source: §H.2 · p. 24

I Proof of Proposition 2.1.

The proof uses joint convexity of optimal transport (OT) and Jensen’s inequality to relate dataset-level transport to per-slide costs. Applying the same argument to the penalized objective yields the corresponding inequality.

I Proof of Proposition 2.1.

J Lemmas and Derivations

The appendix derives linearity properties for assignment-based objectives and reports additional evaluations on Camelyon16 and PANDA with CONCH features. NICER leads the reported comparisons in both settings.

J Lemmas and Derivations

  • The lemmas establish that the assignment-dependent objectives R1(b) and R2(b) are linear in the assignment parameter b.The derivations use the stated assignment likelihoods and a binary vector with exactly one nonzero component.Source: J Lemmas and Derivations · pp. 24–25

K.1 Performance on a Needle-In-A-Hay-Stack Dataset

  • NICER achieves the best performance across metrics and predictors on Camelyon16, outperforming baselines by up to approximately 43% and 39% in accuracy with ABMIL and ILRA, respectively.The evaluation uses UNI as the foundation encoder.Source: §K.1 · p. 25, Table 6 · p. 26

K.2 Extended Results with a Different Pretrained Encoder

  • NICER achieves the best performance across metrics and predictors on PANDA with CONCH, outperforming the most competitive baselines by around 23% and 13% in accuracy with ABMIL and DSMIL, respectively.This evaluation tests a different pretrained encoder.Source: §K.2 · pp. 25–26, Table 7 · p. 26

K.3 Prototype Analysis

The appendix examines prototype dynamics, sensitivity to capacity and pattern-selection settings, and controlled-budget comparisons. It also reports pathologist-assessed local patterns and more coherent global maps for NICER than for PANTHER.

K.3 Prototype Analysis

  • NICER prototypes diverge and specialize across iterations, while PANTHER’s fixed 16-prototype comparison collapses heterogeneous regions; NICER also preserves diverse patterns across t-SNE, UMAP, and PCA projections.Prototype trajectories vary with slide morphology and feature distribution, and the prototype count stabilizes after a few iterations.Source: §K.3 · pp. 26–27, Figure 7 · p. 27, Figure 8 · p. 28
  • Larger, more complex WSIs receive more NICER prototypes, and the reported average prototype count varies across datasets.The table reports average final concept-prototype counts across datasets.Source: §K.3 · p. 27, Table 8 · p. 29

K.4 Sensitivity Analysis

  • Increasing initial pattern count M improves performance until gains plateau, while increasing top-κ helps up to a moderate value such as κ = 3 before gains plateau or slightly decline.At larger κ, information is dispersed across patterns, weakening the distilled representation.Source: Table 9 · p. 28, §K.4 · p. 29, Table 11 · p. 30
  • At comparable condensation levels, NICER surpasses PANTHER by nearly 19% in F1 when using 11.32% Input versus PANTHER’s 10.39% Input.Both methods show an upward F1 trend as % Input increases.Source: Table 10 · p. 29, §K.4 · p. 30
  • The average number of final concepts increases with κ and then plateaus as condensation merges overlapping patterns; the figure tracks prototype counts over the first eight iterations.Source: Figure 9 · p. 28, Figure 10 · p. 29, §K.4 · p. 30

K.5 Performance under Controlled Compute Settings

  • NICER outperforms PANTHER and OT under both mean- and maximum-prototype matched settings, supporting gains beyond its prototype-budget advantage.The maximum-budget setting gives baselines a budget larger than NICER’s.Source: §K.5 · pp. 30–31, Table 12 · p. 31, Table 13 · p. 31

L Pathological Interpretation Details

  • The pathology assessment propagates K-means assignments on condensed features back to original patches and evaluates local annotations and global WSI maps with a board-certified pathologist.The local clusters correspond to recognizable histopathological entities and separate diagnostically distinct patterns, including viable tumor versus adjacent necrosis.Source: L Pathological Interpretation Details · p. 31, §L.1 · p. 34

L.2 Global Pattern Visualization

  • NICER’s global pattern maps align more closely with WSI histology than PANTHER’s, with stronger tumor-region agreement and more coherent necrotic and non-tumor groupings.PANTHER’s assignments are described as more fragmented and spatially noisy.Source: Figure 11 · p. 32, §L.2 · p. 34

M Limitations

NICER’s limitations concern omitted spatial structure, the PPP independence assumption, and reliance on fixed pretrained representations. The checklist also records where experimental details are provided and how results and code are reported.

M Limitations

  • NICER treats patches independently, so it does not directly capture fine-grained tissue architecture or spatial and multi-scale relationships.The authors note that this may matter for clinically sensitive tasks.Source: M Limitations · p. 34
  • The PPP prior assumes independent concept occurrence, which may limit representation of highly complex or correlated morphological structures in extreme cases.The authors describe independence as reasonable when concepts represent distinguishable prototypical patterns.Source: M Limitations · p. 34
  • NICER uses a fixed pretrained representation; evaluation with two foundation models leaves broader robustness and applicability across encoders to be clarified.Source: M Limitations · p. 34

NeurIPS Paper Checklist

  • Checklist responses refer to Sec. 4.1 and App. D for experimental details, report error bars, and state that source code will be released upon acceptance.Source: §2 · p. 37

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