Spatially Informed Autoencoders for Interpretable Visual Representation Learning
Authors: Dominik Sturm
Hiba Bensalem
Ivo F. Sbalzarini
Venue: ICLR 2026 Poster (2026)
Keywords: autoencoder
visual representation
point process
conditional simulation
interpretable machine learning
self supervision
spatial statistics
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We present spatially informed variational autoencoders that use stochastic point processes to learn interpretable spatial patterns from images.

Abstract: We introduce spatially informed variational autoencoders (SI-VAE) as self-supervised deep-learning models that use stochastic point processes to predict spatial organization patterns from images.  Existing approaches to learning visual representations based on variational autoencoders (VAE) struggle to capture spatial correlations between objects or events, focusing instead on pixel intensities. We address this limitation by incorporating a point-process likelihood, derived from the Papangelou conditional intensity, as a self-supervision target. This results in a hybrid model that learns statistically interpretable representations of spatial localization patterns and enables zero-shot conditional simulation directly from images. Experiments with synthetic images show that SI-VAE improve the classification accuracy of attractive, repulsive, and uncorrelated point patterns from 48% (VAE) to over 80% in the worst case and 90% in the best case, while generalizing to unseen data. We apply SI-VAE to a real-world microscopy data set, demonstrating its use for studying the spatial organization of proteins in human cells and for using the representations in downstream statistical analysis.

Published as a conference paper at ICLR 2026
SPATIALLY INFORMED AUTOENCODERS FOR INTER- PRETABLE VISUAL REPRESENTATION LEARNING
Dominik Sturm 1,2.3,4,- Hiba Bensalem 122.3,4* & Ivo F. Sbalzarini 12,3,4
Dresden University of Technology. Faculty of Computer Science, Dresden, Germany
2 Max Planek Institute of Molecular Cell Biology and Genetics, Dresden, Germany
CCenee for Systems Biology Dresden, Dresden, Germany
*Center for Scalable Data Analytics and Artificial Intelligence ScaDS.AI, Dresden/Leipzig, Germany corre rorreseendece: sbalzari wewaariniee ni@mpiiiiiii (mmpimmbp -cbg de
ABSTRACT
We introduce spatially informed variational autoencoders (SI-VAE) as self- supervised deep-learning models that use stochastic point processes to predict spa- tial organization patterns from images. Existing approaches to learning visual rep- resentations based on variational autoencoders (VAE) struggle to capture spatial correlations between objects or events, focusing instead on pixel intensities. We address this limitation by incorporating a point-process likelihood, derived from the Papangelou conditional intensity, as a self-supervision target. This results in hybrid model that learns statistically interpretable representations of spatial localization patterns and enables zero-shot conditional simulation directly from images. Experiments with synthetic images show that SI- SI-VAE improve the clas- sification accuracy of attractive, repulsive, and uncorrelated point patterns from 48% (VAE)
80% in the worst and 90% in the best case, while gen- eralizing to unseen data. We apply SI-VAE to a real-world microscopy data set, demonstrating its use for studying the spatial organization of proteins in human cells and for using the representations in downstream statistical analysis.
INTRODUCTION
The spatial distribution of objects or events in images is an important readout in many applications. Examples include distributions of forest fires (Kato al., 2020) or species abundance (Gillespie et al. 2024) in satellite imagery, or the distributions of viruses in biological cells observed by fluo- rescence microscopy (Helmuth al,, 2010). In these examples, semantic categories are not defined by appearance or texture of the imaged objects or events. The goal then is to infer representations of the observed spatial patterns that determine or explain them.
Deep learning has been particularly powerful to infer visual representations (Moen et al. 2019) and identify major sources of variation (Bengio et al. 2013). Microscopy images, for example, encode spatially structured patterns of discrete objects, such cells in tissues molecules in cells, that are causal for biological function and its dysregulation in discase (Hung & Link 2011).
Several un- and self-supervised approaches have been proposed to extract information about spatial distributions in images. This includes contrastive learning by comparing augmented views of the same image (Chen al. 2020). Contrastive learning, however, relies pixel similarity and might therefore fail to capture spatial correlations. This was addressed by Cytoself in a domain-aware approach that used classification pretext task to predict ground-truth protein labels (Kobayashi) et al. 2022), constituting a semi-supervised approach (Kingma et al.) 2014). Alternatively, protein sequences have been used to predict cellular localization (Khwaja et al. [2023] Kilgore al. 2025).
Such hybrid approaches model the joint density of the feature and label space, leading to useful
representations (Nalisnick et 2019) leveraging ideas from predictive coding (Oord al.
2019
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learni ng/s -vve

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In the absence of annotated ground truth, self- self-supe autoencoders have shown promise for tracting biological features from microscopy images (Kraus et al. 2024). Usually, self--upervision
is based pixel intensities using masked autoencoders (He al. 2022) image transformations
(Gatopoulos & Tomczak 2021). Gaussian processes (GP) have been used to encode correlations in the data through structured priors (Casale al.] 2018 Pearce 2020 Jazhec al. (022). This allows modeling correlations between images but not spatial correlations among objects within im- ages. Recently, Vasan et al. (2025) proposed point clouds for representation learning of shapes and spatial distributions. The prediction likelihoods of point-cloud models structure the latent space to become discriminative about the spatial distribution. This is in contrast to class labels encouraging categorical latent space and augmentations encouraging pixel-level similarity.
Such learned representations can be used to study biological associations and perturbations (Celik et al. 2022). They do, however, not provide mechanistic insight into spatial organization, and they lack statistical framework for rigorous downstream analysis, which is prerequisite for scientific applications. It has been shown that VAE can learn accurate surrogate models of GP priors for fast sampling (Semenova et al. 2022). While this accelerates spatial Bayesian inference, it does not model interactions between objects within image. Recent work also revealed fundamental limi- tations of un- and supervupe settings, such as the Clever Hans effect preferring bogus cues to true features (Kauffmann al. 2025). Transformer-based masked autoencoders have been argued to primarily learn a representation based on the unmasked patches, ignoring the spatial arrangement of masked tokens in the decoder (Fu al., 2025). This limits their ability to reason about spatial correlations between objects within an image. In addition to inherent limitations of the architec- tures, Gunawan et al. (025) and Abbaryan et al. (2025) have recently shown that image metrics commonly used in loss functions focus on image appearance rather than spatial content. Together, these observations suggest that learning interpretable spatial representations from images requires additional priors.
Here, we propose spatial point processes as self-supervision prior for visual representation learn- ing. Spatial point processes are discrete stochastic processes from spatial statistics. As we show, they enable spatially informed variational autoencoders (SI-VAE) that learn statistically interpretable rep- resentations of spatial distributions of point-like objects in images. Spatial statisties has long been a powerful tool for analyzing localization patterns in images (Helmuth et al. 2010; Lagache et al.) 2015 Summers al. 2022). It provides interpretable and generative models, aligning with the growing interest in explainable models for biology (Chen et al. 2024), Rotem & Zaritsky] 2024). SI- VAE combine the mathematical rigor and statistical interpretability of spatial point processes with the approximation power of deep learning, where the latent representation acts as a predictor for the probability density of spatial distributions.
2
METHODS
SI-VAE learn spatially referenced representations from images by augmenting VAE with spatial point processes. We describe the VAE framework and introduce spatial point processes before com- bining them to the SI-VAE architecture in common probabilistic framework. There, the VAE approximates the density of a spatial point process for modeling spatial distributions. This pro- vides statistically interpretable models, well as zero-shot generative models to sample from the estimated distribution.
2.1 VISUAL REPRESENTATION LEARNING USING VARIATIONAL AUTOENCODERS
We use VAE learn representations from an unlabeled set of images x {..) Ty E C A VAB is a generative model working under the assumption that the data x can be reconstructed from latent vector = E R (Kingma & Welling 2014). The aim is to maximize the likelihood of the data under the latent representation 2, pe (z(
Po po (z(2))((2 dz. Since this likelihood is intractable, VAE approximate the posterior distribution by a variational form, (2(2)) p(z]z). The variational form is commonly chosen as go (2|2x) = N (n((),, o (0)))) Here, A-(-) is a Gaussian with mean (() and standard deviation (0), predicted by an encoding neural network, samples of which can be used to approximate the evidence lower bound (ELBO) (Kingma & Welling,
log po 2 F.y (x) (go ( II)())),
2

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where p(2) is prior over the latent vector 2, commonly chosen (0, It), and 3 weighs between the reconstruction and the prior (Higgins et al. 2017). The standard ELBO is obtained for 3 = 1. Maximimzing the ELBO approximately maximizes (z) and minimizes the Kullback-Leible (KL)
divergence between qo (2(2z) and the true posterior p(z] (Kingma & Welling 2019). Therefore, samples from the variational distribution
(0(2.x) (or quantities derived from them) can be
used as representations for downstream analysis (Zhang et al. (022). This, however, is limited to information contained in the posterior, possibly ignoring the second-order correlation structure between objects in the data.
2.2 SPATIAL POINT PROCESS MODELS
A spatial point process is a discrete stochastic process on W S R d 2 1, where X C W is an finite unordered set of points. Their distribution can be characterized by ((B) [ n BJ, i.e., the number of points in some subregion B C W (Møller & Waagepetersen 2003). Such distributions are difficult to characterize, except for the uncorrelated Poisson point process. It is
therefore common to define the density of a point process relative to the unit-rate Poisson process
(see Appendix A)). This defines Gibbs point processes, which model second-order correlations as interactions between points, specified in terms of an energy-based density
(x) x exp {
xexp
Σ WEX
c -
de v)
(2)
Here, &5 Rdd R and v/r Rd x Rd + R are the first- and second-order potentials. In SI-VAE, they are represented by two separate shallow neural networks with parameters { = (vo E E. These potentials control the a-priori propensity of observing point and the pair-wise interaction between points, respectively. Depending on the (", (u,v), v), points can be attractive, neutral, or repulsive.
This defines general class of point processes, which we infer from data.
likelhool estimation (MLE) of Gibbs processes from data is intractable due to the un- known normalizing partition function of the density Pe (.)). This is common to many energy-
models (Bengio et al. 2013 Gao et al., [021 Tomezak 2024). Therefore, we instead model the
Papangelou conditional intensity (Ba & Coeurjolly] 2023
Ae u) = [Pe(X U (x))) u u X
(3)
(x)(X)/pe \ (u)) WEX.
In an infinitesimal volume du around u, Ae (X, u) du can be interpreted as the probability of ob- serving point at u given all other points in X. For densities following equation 2 the Papangalou conditional intensity XX,, u) exp { Σ Evex (u, u)}. This has been used derive pseudo-liked likellhod estimators that provide unbiased estimating equations without the normalizing approximates the MLE under a conditional independence assumption (Baddeley) While this can limit the statistical efficiency of the estimator, we still use -iuudl-ikelihood estimation here, as it provides a computationally ef- ficient and statistically consistent objective for learning spatial point-process models from data. For a Gibbs process with conditional intensity (X, u), the dog-pseudo-lig secudo-lieelhood is
constant (Møller & Waagepetersen (007). The pseudo-likelihone
log PL(()) = Σ log Ae u) -f (X(, u) du,
(4)
"EXnD
where = W R is an erosion of the domain W by an interaction distance R to avoid edge effects (B & Coeurjolly 2023). We directly use equation 4 as loss function for learning the Papangelou conditional intensity xg X, u) in a VAE.
2.3 SPATIALLY INFORMED VARIATIONAL AUTOENCODERS
We derive a VAE architecture that learns a latent image representation from which the Papangelou conditional intensity of a Gibbs point process, defined in equation 2 can be predicted that explains the observed point pattern in the image. While this self-supervision target is not limited to VAE architectures, VAE enables uncertainty quantification of the model by resampling (2(
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The pretext task for the VAE exploits point-proces statisties with realizations X obtained from images x through, e.g., spot detection. Such weak labeling is commonly used for biomedical data Yakimovich et al. 2021). The loss enforces the joint parameters of the model go (2|z) to approxi- mate the marginal (c) j and the conditional probabilities p& (X|=) of a point process by maximizing the likelihood of the observed points X with the latent vector = as predictor:
E) = Blog (x]=)] + EEg Pl (
(X | - (KL(g) p()))
(5)
=log (L(g]=)
This is illustrated in Fig. The inference model (0(2/z) only relies on æ. such that X is only necessary during training, as usual in self-supervised VAE (Kobayashi et al.) (022). This estima- tion procedure generalizes the Bayesian variational formulation proposed by Zhou et al.] (2022) for parametric models to non non-parametricll Therefore, SIE do not require (learning) an inference function from the complete point pattern to the model parameters, which is usually unavailable in practice. We approximate log Pş (X]=) by the bog-pseudo-lkeelle likelihood in equation f represented by two-layer neural networks with input 2. We choose ub v)
estimating model of the point process from images.
using (x, ul2). The Gibbs potentials dg and 1. are
4 ( vl[2) to be a symmetric, isotropic function. This ensures that the interactions between points are invariant under translation and rotation of the whole point pattern. As shown in Appendix B models with more degrees of freedom for de tend to converge to trivial solutions that do not account for the local interactions between points. We further constrain the model with distance-decaying weights Wae (equation 14) to learn local interactions of range L with (he u) = Wun the - v1l2). Since long- range interactions are indistinguishable from density inhomogeneity, this regularization is required for identifiability of inhomogeneous point processes. The interaction range L is hyperparameter (see Appendix C.2) Conceptually, this framework extends to anisotropic interactions by consider- ing the (signed) difference between point positions, i.e., (le(u, v) 4/s (u- -) v) if sufficiently diverse directional data are available. The input to the interaction network then becomes a vector.
The loss in equation admits probabilistic in- terpretation as a hybrid model of =(X, = p(X[=)p(z). Hybrid models p(X,z) have been shown to learn richer and more outlier- robust representations than purely discrimina- tive models (Nalisnick et al. 2019: Tomczak 2024]. We assume that captures all information, and that the vari- ational posterior qo (2)2)) provides good ap- proximation to Mf= .... The first assumption common in multimodal VAE (Wu & Good- man, 2018). It is not limiting in our setting, since the point pattern X is deterministically obtained from the image x. Therefore, cap- tures all relevant information about both modal- ities, trivially rendering them conditionally (on 2) independent. The second assumption implies that the model should only rely images for inference, which is fulfilled by design when learning visual representations. Under these assump- tions, the loss in equation 5 the ELBO of a joint generative latent-variable model of the point pro- cess, see Appendix D for a proof" This model has joint density p(X, xx 2) p(X 2)p(z 22 <))(2),, where p(x)) = p(æ, 2) dæ is a standard VAE. The aseudo-likelihodd in equation 4 can more- over be interpreted as the limit of Bernoulli random variables over partitions (or pixels) C W. i EEE (I,...,/) /} (Møller & Waagepetersen) (007) with each partition/pixel conditioned on X:
x(X[2)
A[()) = exp bb(
Σ 1,
p(x)
Prediction Model
X|z. such that
2 m (=lar)
Inference Model
X
Figure I: Schematic of the proposed SI-VAE ar- chitecture consisting of an inference model go and a prediction model Ag- The inference model uses input image z to sample the latent code z. The latent sample is used by the prediction model to predict the Papangelou conditional in- tensity >g u) of the point process X.
PL( = lim II (>e(X, wxlz))u,)) No (1- >x ucl=)|ul))+ 1u,1-00 1 15i51
(6)
The indicator N;- ((N > 0) denotes the presence of a point in partition i. Therefore, the prediction model Ae (X ulzz can be interpreted as the limit of a pixel-wise classifier for the presence
NNte that the assumptions only needed for the interpretation of the loss as BLBO. They are not needed for SI-VAE predict valid features of the Papangelou conditional intensity of Gibbs process.
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Table I: Accuracy (Acc) and Fi score for linear classification of point-pattern types in the latent
space of the models (SI-VAE, VAE, mask VAE trained with perfect location knowledge) trained on images of different quality (SNR). We compare two different leak-labeling methods (Spotiflow, Thresholding) with ground truth (GT) baseline using perfect knowledge of point locations. For the
weak-labeling methods we also report the sensitivity Sf, = (FI.cr F //m (1- quantify the robustness of the Fi score to spot-detection errors.
F-spol
F1.
T.method/ to
GT Knowledge
Spotiflow
Thresholding
Model
SNR
Acc (t)
(1)
Ace (1)
Sp, (4)
Acc (t)
F1 (1)
SF, (1)
mask VAE
0.63
0.62
x
VAE
12.8
0.48
0.47
X
x
X
VAE
9.6
0.48
0.47
X
SI-VAE
12.8
0.90
0.90
0.90
0.90
0.0
0.80
0.80
0.56
SI-VAE
9.6
0.88
0.88
0.83
0.83
0.42
0.81
0.81
0.22
of a point given the data X. From Jensen's inequality, the second term in equation 5 provides lower bound on the approximate conditional log-likelihood log Pt.e e (x) se log E, [pe [2]
Eg llog Pg (x [2]]. This connects SI-VAE with other hybrid models, such as DIGLM (Nalisnick et al. 2019 VAE, however, estimate z using amortized variational inference instead of flow models.
Therefore, SI-VAE learn latent that approximates (x(x( while providing model for the
features po(r) (r). The details of the SI-VAr architecture used here, and of training, are given in
Appendix
EXPERIMENTS AND RESULTS
We benchmark SI-VAE on synthetic data, comparing them VAE with the same architecture. The only difference between the VAE baseline and the SI-VAE is the presence of spatial supervision in the latter. This permits relative comparison. Then, we illustrate the workflow of applying SI- VAE to learning interpretable representations of protein localization patterns in human cells. We highlight introspection and interpretation of the latent space and show how the probabilistic framework of VAE enables conditional simulation and downstream statistical analysis over learned representa- tions.
3.1 LEARNING SPATIAL INTERACTIONS ON SYNTHETIC DATA
We first show that SI-VAE are able to learn representations that disentangle clustering due to attrac- tive correlations from clustering due to inhomogeneous intensity functions. For this, we show that the latent space of an SI- VAE linearly separates homogeneous from inhomogeneous point processes and is able to classify their correlation structure.
Data Generation We generate noisy synthetic images with known ground-trute point locations. Point locations are sampled from attractive (Thomas), repulsive (Strauss) and uncorrelated (Poisson) point processes with either homogeneous or inhomogeneous intensity functions. Appendix A details the benchmark point processes, Appendix E the synthetic image generation. It is important to note that Cox processes, such the Thomas process, are not Gibbs processes, since their distribution is not defined via an energy function. Nevertheless, Gibbs processes constitute a flexible modeling framework, and aim is to identify Gibbs process that provides suitable approximation their distribution. We generate 5000 images for each the six cases. All images have the same expected number of points (N((W) 52. This prevents the encoding network from simply discriminating between the processes by total pixel intensity.
Experiment Design We compare the eppresentation-leamiin performance of SI-VAE against a baseline VAE with the same architecture (see Appendix C.I) but without the point-procest supervi- sion target. Both models use the same encoder and decoder networks. The Papangelou conditional intensity dg (X, u/2) is modeled using separate two-layer neural networks for Φe and thg (see
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Appendix C.2) to predict the spatial distribution of X from given latent vector Both models are trained using the Adam optimizer (Kingma & Ba 2015) over their respective ELBO until con- vergence the validation set. We set 3 = 0.1, identified in grid search over three orders of magnitude (0.001 to 1.0), and distance weight war with = 0.25 determined from prior knowledge (see Appendix C.2) During evaluation, a representation is obtained the mean of the posterior qe(2 qe(z|z). We evaluate the representations obtained by both models using a linear evaluation protocol following Chen al. (2020). This tests if contains enough information to distinguish the six dif- ferent point processes in the data, thus linearly disentangling correlation from inhomogeneity. Since the performance of the proposed self supervierriio task depends on the accuracy of the point patterns X provided during training. we repeat the experiment for two different signal-to-noise ratios (SNR) using two different spot detectors (simple Otsu thresholding and the recent deep-learning method Spotillow (Dominguez Mantes et al. 2025)). We compare them to a baseline with ground-truth (GT) knowledge of X.
Results The results in Table show that the SI-VAE consistently outperforms the standard VAE across all experiments. They also show that simple spot-detection methods, here Otsu thresholding, suffice to provide weak labels for the self-supervision task, especially at low SNR. When using state-of-the-art spot detection (Spotiflow, Dominguez Mantes et al. (2025)), the SI-VAE performs close to the GT-knowledge baseline. The SI-VAE predicts meaningful representations even at high noise levels (low SNR). We also report the degradation of the F1 classification score w.r.t. the Fi score the spot detector, yielding the error sensitivity index Sp,- In all cases, SI- VAE predictions degrade less quickly than spot-detection performance (Sf, I 1), indicating that SI- VAE dampen spot-detection errors during training. As errors mainly occur by missing points in dense regions (see Table 3 however, they bias the learned correlations (Kuronen et al. 202T
Since SI-VAR incorporates structured knowledge in the form of (detected) point locations, we also compare it against a VAE trained on ground- truth binary object-location masks (mask VAE). This performs better than the VAE trained on whole im- ages but does not reach the perfor- mance of SI-VAE. The UMAP pro- jections of the latent spaces in Fig. 2 reveal that the standard VAE mainly learns global pixel-intensity patterns. This clearly manifests in the lin- ear separation between homogeneous and inhomogeneous point processes, while not separating different pro- cess types (Strauss, Thomas, Pois- son). While (Semenova et al. (2022) have shown that VAE can learn spa-
VAE
SI-VAE
655
8.0
0.0
2.0
H7
0.5
115
-10
-25 7.5 10.0 125
Figure 2: UMAP projections of the latent representations obtained by VAE (left) and SI-VAE (right) trained on im- ages with an SNR of 12.8 and perfect GT knowledge of the point locations. Cold colors denote homogeneous pro- cesses (Poisson (HP), Strauss (HS), Thomas (HT)), warm colors inhomogeneous processes (Poisson (IP), Strauss (IS), Thomas (IT)).
tial correlations for GP priors, our results indicate that this is insufficient for explaining spatial point patterns. Only the SI- SI-VAE distinguishes between process types, explaining the downstream classi- fication accuracy. The latent space of the SI-VAE also maintains interpretability in the sense that mathematically more similar processes, like Poisson and Thomas, are mapped closer together than dissimilar processes (Strauss). This confirms that the proposed self-supervision task leads more meaningful representations of the underlying spatial organization.
3.2 GENERALIZATION TO UNSEEN PROCESSES
In addition to testing the generalization of SI- VAE on a holdout data set, we quantify generalization to point processes of types not seen during training.
Experiment Design We generate images from homogeneous log- Gaussian Cox process (LGCP), which admits clustering through a Gaussian process (see Appendix [A.4). Because of this clustering and definition based on background process, it should behave similar to homogeneous Thomas

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process, which is special case of a Cox process. We generate 150 synthetic images of point patterns from a LGCP with the same expected number of points as the other processes. We then use the linear classifier trained in the previous subsection the (SI-)VAE representations from Table M to predict the class labels for the LGCP images. (SI-)VAE are provided with images only, without (extracted ground truth) point locations.
The results shown in Fig. The VAE baseline fails to generalize to the LGCP, as mainly captures global pixel-intensity patterns. Even when trained on weak labels (colors, inset legends), the SI- VAE correctly classifies the LGCP belong the same class of processes as the Thomas process most similar to a homogeneous Thomas process (HT). The SI-VAE trained in Section [.1] with complete knowl- edge of point locations classifies perfectly. Since the SI-VAE was not trained on LGCP data, this shows its capacity to generalize to unseen point processes, even when trained weak labels.
Results
1.8
SE
SiMGE [Thveshos)
SHARE (Spotifico)
SIARE (Domplete)
2.6
Clasiftccion
6.4
6.8
H7
17
6P
Figure 3: Class frequencies of the LGCP images predicted by the linear classifier the representations trained in Table for the higher SNR.
3.3 INTERPRETING
THE LEARNED REPRESENTATIONS
SI-VAE provide representations that are interpretable in the framework of spatial statistics. The SI-VAE prediction )x(X, u/2) approximates the Papangelou conditional intensity of a point process from an image. According to equation 3 the inferred point process is repulsive for (x, ul2) 2 AY,Y, z). X C Y, and attractive otherwise. The predicted conditional intensities also be visualized over the domain D. Finally, sampling multiple latent vectors z from the posterior (=|2r) enables uncertainty quantification of the estimates.
Palsson
Streuss
Thomas
40
150
962
0..5
-0.4
0.0
0.4
0.6
-.08
-d.4
0.0
24
0.5
-08
0.8
-0.4
0.4
0.8
100
-0.4
-0A-
0.8
0.5
-0.4
0.9
0.6
0.5
-0.8
2.8
-0.4
0.9
24
0.5
0.8
0.8
-0.4
0.0
0.4
0.5
Figure 4: Visualization of the predicted conditional intensities Ag u|2) for the latent codes obtained from different images from all considered point processes with homogeneous (top row) and inhomogeneous (bottom row) distributions over the eroded domain = W e R. Orange circles show the ground-truth point set X for image x, which not used during inference.
trained with GT knowledge of the point locations at the higher image SNR. We randomly select images from the test set for each of types. For each image, without knowledge of the point locations, the SI- VAE predicts the conditional intensity Àg (X, u/2).
Experiment Design
We consider the SI-VAE from Section 3.1
the six
Results Figure 4 shows the predicted conditional intensities de (X, u/2) for different latent codes 2 (additional plots in Fig. [T]. Differences between homogeneous and inhomogeneous distributions clearly visible. Moreover, the learned representations correctly capture local repulsion (Strauss) or clustering (Thomas) between the points, providing insight into the spatial organization directly from an image. Since SI-VAE model Gibbs processes, the learned potentials &e and the can directly
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Table 2: Relative intensity errors (RIE) of intensity estimates and rejection rates (RR) of second- order models for zero-shot conditional simulation of Xx, u/z) from the SI- VAE trained in Section 3.1 Average scores are computed over 100 random samples from the test set for each class.
Poisson
Strauss
Thomas
RIE (L)
RR (L)
RIE (4)
RR (L)
RIE (L)
RR (4)
Homogeneous
0.34
0.08
0.23
0.13
0.62
0.20
Inhomogencous
0.27
0.03
0.14
0.14
0.49
0.15
be interpreted as the first- and second-order interaction structure of the process. For Poisson point process, Xę(X, u) expf (x))= such that ( v) 0 for all u, U. This means that the SI-VAE should not predict any interactions between points. In Fig. 4 the SI-VAE predicts both attractive and repulsive interactions for the Poisson processes, depending on the latent code z. This suggests that the representations z are sensitive toward the observed point configurations and unable to disentangle first- and second-order characteristics of the observed point pattern, which a difficult task in general. Nevertheless, SI-VAE provide interpretable, mechanistic insight into learned representations. The interpretations, however, may be biased by specific point configurations observed in the image.
3.4 ZERO-SHOP SHOT CONDITIONAL SIMULATION FROM IMAGES
As shown in the previous experiment, SI-VAE learn Papangelou conditional intensitics of point processes. Since (X, u/zz) parameterizes the distribution of X, it can be incorporated into an MCMC sampler Moller & Waagepetersen 2003), enabling conditional simulation. This provides conditional (on query image) generative model of the point process in addition to the generative model Po for the images. Once trained, SI- VAE can thus explore the distribution X directly from an image x in a zero-shot fashion. To our knowledge, this is the first instance of image-based conditional simulation for point processes.
Experiment Design We perform conditional simulation (see Appendix A.6) using the SI-VAE from Section 3.1. trained with GT point locations at high image SNR. For each point-process class, we select 100 random test images. We evaluate the simulations by comparing the predicted process Xg to the observed X under Ag. Accuracy is assessed by nonparametrie estimates of first- and second-order quantities: kernel density estimation (KDE) for intensity, and the K-function (Diggle) 2013) for interactions (see Appendix [F). We compute the relative intensity error (RIF) between the KDE of observed and simulated point patterns, measuring over-/underpreediction relative to the observed pattern. For interactions, we use a Monte Carlo test for point processes (Baddeley al., 2014) and measure its rejection rate (RR), i.e., the frequency with which the K-Tunction of the observed pattern significantly (5% significance level) deviates from the mean K K-function.
Results Table shows the RIE and RR obtained from the simulations. They confirm that the SI- VAE is able to generate point patterns close to the observed ones. The lowest RIE are obtained for the Strauss process, which is correctly predicted to be repulsive. In this case approximating Xę ul2) works best and does not require model saturation (see Appendix A.6). As the envelopes in Fig. [2] show, the majority of hom. Poisson samples are also predicted to be repulsive. The higher RIE in this case is due to samples that appear clustering, producing excess points in the simulation. For estimates with non-clustering potentials we obtain good simulation results (Fig. L4p. The RIE further increases for the Thomas process, which is always clustering. Here, model saturation (Appendix A.6 crucial to obtain stable simulations. The error indicates that the model predicts too many points, which can also be seen in Figs. [4] and [.5] This suggests that the saturation parameter is too large for these cases, motivating further investigation. The RR computed from the K-functions (Figs. 12 and 13) are overall low, indicating that the interaction structure is well captured. The lowest RR are obtained for the Poisson process, where the true interaction is always close to the conditioning sample when corrected by the normalization through the KDE (Fig, [3). The Strauss and Thomas processes have higher RR, which is expected due to the more complex interactions (repulsive, attractive). For these, Figs. T2 and 13 show that simulated point patterns exhibit weaker
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interactions than the conditioning sample. Overall, however, the results show that the SI-VAE can be used for zero-shot conditional simulation of point processes, qualitatively capturing the correct correlation structure of the process from single query image.
3.5 APPLICATION TO PROTEIN LOCALIZATION HUMAN CELLS
Having benchmarked SI-VAE synthetic data, next illustrate workflow applying them to protein localization patterns in human cells. Specifically, we show that SI-VAE representations of cellular microscopy images distinguish between different localization patterns and that, thanks to their statistical interpretability, they allow for a meaningful biological interpretation.
STAM
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Figure 5: First-order (contour plots with detected point locations in orange) and second-order (top- right plot) Gibbs potentials of the SI-VAE representations on the test set. Insets show the nucleus channel for each sample reference. The two clusters (blue, orange) emerging in the latent space separate Doscclelocallzing proteins from nuclear proteins with cluster-assinmmet frequencies for the six proteins given at the bottom right.
Experiment Design We train an SI- SI-VAE on the OpenCell data set (Cho et al. 2022). comprising fluorescence microscopy images of the subcellular localization of over 1000 human proteins. We focus on six proteins from three families: four that localize to vesicles (SNXI, SNXI2, STAM, STAM2) and two that form nuclear punctac (POLR3E, POLR3F). For SI-VAE training, generate weak labels using Spotiflow (Dominguez Mantes et al. 2025). The SI-VAE is trained on 5499 images of the fluorescence signal of the tagged proteins, the nucleus channel, and channel for the signed distance to the nucleus as previously described (Kobayashi et al. 2022). The full details are provided in Appendix G We set the interaction length scale L 0.25, corresponding to the mean half-radius of the nuclei. We assess the learned representations on disjoint test set containing 687 images of the same six proteins by clustering them with Gaussian mixture model (GMM), selecting K = 2 clusters based on the Akaike information criterion (AIC) (see Appendix G We compare the SI-VAE results with two baselines - VAE with the same architecture without spatial supervision and the specialized state-of-tth-aar Cytoself model (Kobayashi et al. 2022) with a substantially deeper architecture two VQ- VAE (Oord al. 2017) and classification head.
(u)} and second-order (r) Gibbs po- tentials learned by the SI-VAE for the two GMM-identified clusters (cluster-assigmmen frequencies the bottom right). The first cluster (blue) contains the four vesicle-associated proteins, while the second (orange) contains the two nucleus-localizin proteins. The GMM clusters obtained by the VAE and by Cytoself show the same result (Fig. [0). For the VAE, however, the clusters are diffuse with Silhouette score of 0.04 (Table 4.. SI-VAE and Cytoself obtain tighter clusters with Silhou- ette scores of 0.29 and 0.33, respectively. indicating that the learned representations more effectively capture differences in the images. This is confirmed by the UMAPs in Fig, 6 where SI- VAE obtains a good visual representation the two protein families. The UMAP for VAE similar to that in Fig. 2 suggesting that the VAE again mainly separates global intensity patterns. The tighter cluster-
Results Figure shows the first-order P: (u)

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VAE
SI-VAE
Cytoself
STM2
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Figure 6: UMAP projections of the latent representations obtained by VAE (left). SI-V SI-VAE (middle), and Cytoself (right) trained on the OpenCell (Cho et al., (022) human-protein data set. Cold colors denote vesicle-localizing proteins (SNX1, SNX12, STAM, STAM2), warm colors nuclear proteins (POLR3E, POLR3F).
ing of Cytoself is likely due to its deeper architecture and discrete latent space (vector quantization and classification labels as supervision target). SI-VAE achieves comparable clustering quality at much lower model complexity, while ensuring interpretability as spatial point processes. It also does not require ground-truth classification labels during training. enabling unbiased discovery.
The first-order potentials correctly reflect known localization patterns: proteins in vesicles are more homogeneously distributed, leading the VAE group them together, while nuclear proteins show inhomogeneous distributions localized to the nuclei. The two baseline models identify the same classes but do not explain them by an underlying probability distribution. The second-order SI- VAE potentials further distinguish the clusters according to the spatial interactions. They reveal short-range repulsion and long-range attraction. The repulsion has smaller range for nuclear than for vesicular proteins, explaining the tighter molecular packing in the nucleus. The nuclear proteins further show stronger long-range attraction, which facilitates efficient coverage of the nucleus, ac- celerating biochemical reactions in the diffusion- -imited regime (Subic & Sbalzarini 2024). This is corroborated by the conditional intensities )e(X, u/|z) predicted by the SI-VAE (Fig. 16) show- ing high event probabilities in the nuclei. We did model or impose nuclear confinement. The SI- VAE reveals it from the data, providing mechanistic explanation for the observed patterns.
4 CONCLUSION
We introduced self-supervision target grounded in point- process statistics. The loss function of the resulting VAE architecture can be understood as the ELBO of a joint model over images and point processes. This showed effective in capturing spatial interactions and generalizing to unseen data. The learned representations can directly be interpreted in the framework of spatial statistics. Since SI-VAE constitute hybrid models, also demonstrated zero-shot conditional (on query image) simulation of point processes. Unlike purely generative models (Lüdke et al.] 20255 Zhou et al.) 2022). SI-VAE remain interpretable. We highlighted the practical utility of SI-VAE by applying them to localization patterns of proteins in human cells. The SI- SI-VAE correctly identified protein localization classes and provided mechanistic explanation for their differences. This demonstrates that SI-VAE learn interpretable visual representations of spatial localization patterns.
In the future, SI-VAE could be extended to non-pairwise interactions by estimating the full density using, e.g., score matching (Hyvärinen 2005 Cao et al. 2024) or Deep Sets (Zaheer et al. 2017). SI-VAE could also be extended to anisotropic interactions, if directionality is present in the data, and to marked point processes to model interactions across different types of points. Additionally, SI-VAE could be combined with more expressive priors on the latent variables, such as GP-VAR
Casale et al. 2018 Pearce 2020 Jazbec et al. 2021) or VampPrior (Tomczak & Welling 2018).
SI-VAE are straightforward to implement and train; they only require a small prediction model to be added to VAE. The main additional cost during training is the evaluation of the Papangelou conditional intensity, which requires computing pairwise interactions between points and numerical quadrature. During inference, SI-VAE behave like regular VAE, the point-process component is only used for training. This makes SI- SI-VAE a drop-in replacement for VAE in applications where spatial localization patterns important feature.
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ACKNOWLEDGMENTS
We thank Dr. Nandu Gopan and Dr. Abhishek Behera (both Sbalzarini group) for helpful discus- sions. We acknowledge financial support by the Federal Ministry of Research, Technology and Space Germany and by the Saxon State Ministry of Science, Culture, and Tourism in the Cen- ter of Excellence for Al-research "Center for Scalable Data Analytics and Artificial Intelligence (ScaDS.AI)". This work was supported by the German Research Foundation (DFG) under Ger- many's Excellence Strategy, Cluster of Excellence "Physics of Life" (EXC2068).
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A POINT PROCESS MODELS
We provide details of the point processes used in the synthetic benchmarks. All parameters were chosen such that all samples, independent of the process model and the homogeneity, have an ex- pected number of EN(W) inference, whereas sampling was performed in extended domains Wexi depending on the process, as discussed below, avoid edge effects. For the inhomogeneous processes we used intensity trend functions of the form p(u) = (xul/=]uli///a), For each process we generated 5000 independent samples. More information on spatial point processes can be found the popular monographs by Baddeley (007). Møller & Waagepetersen (2007). and Møller & Waagepetersen (2003).
52 points. We used the observation window W [-1, 1|a for
A.1 POISSON PROCESS
The Poisson point process is the simplest model, in which spatial variations in the point density arise solely from intensity inhomogeneitites as there are no interactions between points. Since points in a Poisson process are pairwise uncorrelated, this provides baseline that is fully characterized by the intensity function. The intensity function p(u) of a point process is related to the first moment of the point distribution as EN(W) Jw p(u) du. X is a Poisson point process with intensity function p(u) if and only if (BB 10 ((EN((B) for all B C W, and all E X are i.i.d. in W with
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density p(u) p(u)//N((W) Here, P denotes the Poisson probability distribution. The density of a Poisson point process with respect to the unit-rate Poisson process (p(u) = 1) is
p(x) exp {-/,, (p(u) 1) du II p(u)- wEX
Therefore, the Papangelou conditional intensity of Poisson process is A(X, u) = p(u), such that the probability of observing point at u does not depend on X.
We sample realizations of a homogeneous Poisson process with p(u) 13 and N ((EN(W)) points uniformly
W. For the inhomogeneous Poisson case, choose 0.5. The inhomogeneous samples are obtained by first sampling a homogeneous Poisson process with intensity Pmax max p(u), followed by independently thinning it using the thinning probability (u) = p(u)/ This results in an inhomogeneous Poisson process with the desired intensity (Møller & Waagepetersen 2003).
67 and &
A.2 THOMAS PROCESS
A Thomas process is clustering process and a special case of Cox process (Moller & Waagepetersen/ 2003). It is described by parent Poisson process Y with intensity function p(u). Conditional on each y E Y, there is a child Poisson process Xy with intensity
(y WW y. d²Ia).
(8)
where (-) is the d-dimensional normal distribution and À > 0 controls the number child points. The standard deviation controls the spatial range (length scale) of the clustering. A Thomas point process is then defined by the superposition of all child point patterns Ugey This also provides straightforward way of sampling from a Thomas process by two nested realizations of Poisson processes. Due to the dependence on the parent process Y, Thomas processes are said to be driven by random field (()) EyEy Po (u). Following Coeurjolly et al. 2017), we write their density as
xXX) = exp { {-/ (Z(w) - 1)
} II
7(w) REX
Ep(X
W
where p(X is the density of X conditioned on Z. The Papangelou conditional intensity follows X(X, u) Ep(X (u]]2)/Bp(X12)) which is expected increase in the vicinity of a point in X.
For our synthetic samples, choose 0.1 as the clustering range. We sample the Thomas process in a domain that is extended by 7a in all directions in order to capture all interactions. For the homogeneous Thomas case, we use the parent intensity p(u) inhomogeneous case, we use K 23, a 0.3. and À = 3 and obtain the parent Poisson process through thinning as described in Appendix A. The number of child points for each cluster center E Y follows Ny ~ ((A) sampled i.i.d. in W following u ~ (Ny.
4 and X 3. In the
points
A.3 STRAUSS PROCESS
The Strauss process a repulsive point process where points repel each other within a certain radius R. Unlike Poisson and Thomas processes, is directly defined in terms of its Papangelou conditional intensity
X(X, u) = -(u) I(vEB(u.RR)
(10)
where p(u) = ex)(- corresponds to equation 2 Bu, R)) is the ball with radius R centered at u and < 7 < controls the strength of repulsion. For y = 0 no points are allowed within distance R. from any other point (hard-core process), whereas ? 1 results in Poisson point process where points do not interact at all.
Point processes directly defined by conditional intensity are best sampled using MCMC (Møller Hastings sampler from the spatstat
& Waagepetersen, 2003). We here used the package available in (Baddeley et al. 2016).
For all samples, we chose R = 0.2 and y = 0.4, which constitutes a strongly repulsive process.
For the homogeneous case, chose p(u) 40, and for the inhomogeneous case 400 and
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0.5. We rum each MCMC chain for 10 iterations and use 200 uniformly random points to define the initial condition. The sample is obtained after the last iteration.
A.4 LOG-GAUSSIAN Cox PROCESS
log- Gaussian Cox process (LGCP) is clustering Cox process similar a Thomas process. How- ever, instead of using parent Poisson process to define the driving random field Z, a LGCP uses Gaussian process. Therefore, Z exp(Y), where Y GP(p, K) is a Gaussian process with mean function y and covariance function K (Møller & Waagepetersen 2003). The density is the same as that of Thomas process, albeit with this Z.
We sample LGCP with μ 1 and Gaussian covariance function. The variance of the Gaussian covariance function set to o² = 3.31 and the scale parameter to 0.4. This LGCP is sampled using the spatstar package in R (Baddeley et al. 2016).
A.5 SATURATION PROCESSES
The Strauss process defined in Appendix A.3 only possesses a valid density for 0 S. 7 S. 1. For ? 1, the density is integrable with respect to the unit-rate Poisson process. This implies that Strauss process cannot be used to model clustering point patterns. as the parametric model might suggest. fact, this property holds for all Gibbs processes, where purely attractive potentials lead to ill-defined densities. To avoid an unbounded number of events during simulation in such cases, we introduce saturation parameter similar to Geyer saturation process (Geyer) 1999 Rajala al., 2018 Ba & Coeurjolly 2023). Then, the density with respect to the unit-rate Poisson process is
P(X) exp
Σ X
(u)
Σ max BEX
Σ tu v)
(11)
DEX uyte
where 2 is a saturation parameter capping the interaction strength. The one-half prefactor originates from iterating over all non-unique point pairs. This recovers the density in equation 2 in the limit s co, since va is symmetric. The saturation s renders the density integrable, defining valid point process that can also account for clustering. The Papangelou conditional intensity of this process is
X(X, u) exp exp { -o((u) -0).(_((x))) (v((,,))+
Eu Xu v.(o.))]
(12)
veX
with Gibbs interaction term V (u, max max 8, Lvex (u, v) 12.
In practice, needs to be estimated from the data. Since SI-VAE do not account for saturation, we estimate the saturation parameter post hoc by minimizing the MSE between the saturated and unsaturated models over X n D. This ensures that the true conditional intensity is approximately equal to the estimated one under saturation.
A.6 ZERO-SHOT CONDITIONAL SIMULATION
To simulate from the learned conditional intensity (X, w we use a death Metropolis Hastings Algorithm (Alg. 7.4 in Moller & Waagepetersen (2003). The sampler iteratively proposes to add or remove point in the current configuration X (with probability p 0.5) using uniform proposal and death probabilities, i.e., (b( = D| and pa(u) = 1/ with acceptance deter- mined based on de u/2). We perform simulations in the croded domain D W e R, with R = 0.2, to avoid bias outside the estimation domain. Each point is obtained after 30,000 iterations of the sampler. The obtained points then eroded by R again avoid edge effects. We initialize the Markov chain with 52 uniformly distributed points D.
In Gibbs point process, the density become ill-defined for attractive potentials, it may become unbounded for increasing numbers of points. This can lead to instabilities during simulation. We avoid this by introducing saturation parameter previously suggested (Geyer 1999 Rajala
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et al. 2018 (see Appendix A.5 This caps Xę ul22 beyond a certain number of observed points
around u, ensuring that the density remains bounded and hence valid for simulation. The saturation parameter influences the model. We determine using grid search over E [10-3, 3.5] with 50 log-equidistant points such that the MSE between the saturated and unsaturated models over X K is always below 10-3 This ensures that the saturated model is close to the original model, while avoiding the instability. We find this to work reliably in our benchmarks.
We simulate the Strauss process directly from the learned conditional intensity for the Poisson and Thomas processes we use the saturated model to mitigate instabilities due to clustering.
2), while
B APPROXIMATING CONDITIONAL INTENSITIES BY NEURAL NETWORKS
SI- VAE represent the Papangelou conditional intensity of a point process in Gibbs form using first- and second-order potential functions Φ : IR + R and & F x RR + R (see equation 2). We restrict 1/ R IR. be isotropic and symmetric in its arguments. This ensures that interactions between points are invariant under translation and rotation. While this choice reasonable and widely used in spatial statistics (Møller & Waagepetersen 2003). does limit the space of functions an SI-VAE model can approximate.
In the following we will motivate this choice by validation study considering different choices to model v). Specifically, we compare (u, v), (b (() and tb bi[]l We use 150 samples from the homogeneous Strauss process as described Appendix A.3 We train the three different models by minimizing the negative log eseudo-likelihoo size 32 and the Adam optimizer (Kingma & Ba 2015) with learning rate 5 10-4, For these tests, we use deeper neural networks than those used in the main text (see Appendix
from equation 4 ovver 500 iterations using batches of
Specifically, we here use 4 layers with 64 neurons each for both the first- and second order potential networks de and thg. This increased depth accounts for the missing information about the latent variable of the VAE inference model, which is absent in the present test. Overall then, the models have a similar number of parameters to the SI-VA VAE conditional intensities in the main text.
Figure 7 shows the predicted conditional intensities (CI) for a fixed point pattern X for the different models. It can be seen that only the isotropic and symmetric model d/[lu. ell) is able to approx- imate the Papangelou conditional intensity of the observed point pattern (which, in this case, is a collection of overlapping disks). The other two models (1,u, v). (uu v)) fail to capture the locally repulsive behavior and instead predict an almost homogeneous conditional intensity. The magnitude of these conditional intensities corresponds to the intensity of Poisson process (Appendix I.I). We think that this identifiability issue arises due to the conditional independence assumption of the pseudo-iikelhhood likelihood in equation 4 which does not sufficiently account for the local interactions be- tween points. This causes the network to favor trivial solutions when given too much freedom in the form of the function to be learned. Constraining the second-order potential be symmetric and isotropic sufficiently regularizes the second-orde interaction structure in SI-VAR This rationalizes the choice of an isotropic interaction function di| u - -ll) in the main text.
C ARCHITECTURE DETAILS
We detail our VAI architecture and the training procedure. The code, implemented in pytorch Paszke et al. 2019). is available at https:/ aet..pp.. cbg. wt/mmeeacc////c//mmmmoo
C.I VAE ARCHITECTURE FOR THE INFERENCE MODEL
Both VAE and SI-VAE use the same convolutional variational autoencoder architecture to encode images into a 64-dimensional latent space and reconstruct them from there. This constitutes the inference model from Section 2.3 The encoder consists of three convolutional layers with kernel size 4, stride 2, and padding I, progressively downsampling the images from 256 x 256 pixel to 32 32 while increasing the number of channels as 1 + 16 32 - 64. The output is flattened into a 512-dimensional vector and passed through fully connected layer. Two linear layers then map this 51 2-dimensionnl vector the mean y and log-variance log の² of the 6i-dimmesionnl latent
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Strauss
Leamas slu.
0.4
54
0.5
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Figure 7: Comparison of different neural network architectures model the conditional intensity (CT) X(X, u) of homogeneous Strauss process. We compare the ground truth Papangelou condi- tional intensity of the Strauss process (top left) to predictions obtained from neural networks making no assumptions about the structure the second-order potential (u, v), top right), assuming it to be stationary (1/u v),, bottom left), and assuming it to be symmetric and isotropic (vb(1 u|]), bottom right).
distribution (512 - 64). Latent vectors z are sampled using the reparameterization trick (Kingma & &Welling 2014 To reduce the variance, we average 100 samples per image to obtain z, effectively
changing the latent distribution to ((μ, o² /100) with the Bullback-Leiblee divergence in the loss adjusted accordingly. We observed that this stabilizes training and improves convergence.
The decoder reconstructs images via symmetric sequence of three transposed convolutional layers, progressively upsampling the original image size, with channels decreasing as 64 - 32 >> 16 + 1. All layers except the output layer use ReLU activations. The output layer generates the standardized image & E RW x xC' without activations. As outlined Appendix [E. we treat pixel intensities as continuous and standardized. The autoencoders are thus trained by minimizing the mean squared error (MSE) between the input and reconstructed images in the reconstruction term of equation| I Alternatively, pixel intensity levels could be treated as discrete categories, and the data- fitting term in the loss replaced with categorical cross-entropy loss. All VAE models use 3 = 0.1, identified in grid search over [0.001, 1.0]-
C.2 ARCHITECTURE OF THE CONDITIONAL INTENSITY PREDICTION MODEL
In addition to the common inference model, SI-VAE additionally contain a prediction model as described in Section 2.3 The prediction model uses two neural networks, Φe and vhg. predict the conditional intensity lunction of Gibbs process (cf. equation] 2 as:
(X( u|== exp exp -E( z)
Σ Wwn the (1 u112, =) vEX
(13)
where 2 the latent representation of the image x, X is the corresponding point pattern, and u e W is the location at which the conditional intensity is to be predicted. The weights war emphasize local interactions between points to ensure model identifiability as discussed Section 2.3 The network &e predicts the first-order potential (,, 2) of the point process at location u from the latent representation z. The network t/e models the second-order potential the (u, U, z) for the pairwise interaction between two points u, X. Its inputs are latent vector from the inference model and the Euclidean distances ||u 2112- These pairwise interactions are aggregated X. Both &e and dee are implemented as separate two-layer neural networks, where the input is projected to 128 dimensions using a fully connected layer with ReLU activation, followed by another fully connected layer predicting the sealar potential value.
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To ensure model identifiability discussed in Section 2.3 the network dee emphasizes local inter- actions through exponential weights
Wuo exp
vll2
(14)
21
with the hyperparameter L informed by the interaction range in the data. For training. we compute the pssudo-likellhoon for each pattern as in equation The integral is numerically evaluated using the trapezoidal rule over a fixed grid of size 100 x 100. Edge correction is applied via domain erosion D W e with == 0.2.
For each model variant, the three neural networks (VAE, Φg. whe are trained jointly, each with its own Adam optimizer (Kingma & Ba 2015) using fixed learning rate of 10-5 and a weight decay of 10-5 All SI-VAE variants were trained for 100 epochs, while the baseline VAE was trained for 150 epochs to achieve convergence. Early stopping was permitted to prevent overfitting.
D DERIVATION OF THE ELBO
For paired set {Zx,, X.)] of unlabeled images I, R and corresponding point patterns X; = [u, U W C ]. we aim to learn latent representations fz using VAE with joint probability density p(X, 2) = p(X|z,))p<rpp =))((=)))(() We assume that X L. xz i.e., that and are conditionally independent on z. We assume furthermore that p(=|X,,a0 a:) can be approximated by the inference model (0(2(z). While X could hold additional information in practice, such as labels the points, require that during inference only is required. This is architectural design choice. Under these assumptions, the evidence lower bound (ELBO) of the log- likeliidd log ax) can be derived as
logp( (xxx) = F. [log p( Xx,
p(X, 2; 2)
= EEy log
p(=[X, x))
p(X, 2)
qo (2(x)
E,
log
(0(2(x))
p(z x))
p(X, x. =)
go
F.g
log
+ Byg log
go (2(1))
p(=|X,
=:ELBO
Since the KL divergence in the last term is always positive, we find a lower bound on the log- evidence logp(X X, x). Moreover, maximizing the first expectation, which corresponds the ELBO, minimizes the KL divergence between the variational distribution go(2[<r) and the true posterior p(z|X, z). The quality (z(rx) thus determines how well the ELBO approximates the true log- likelihood. This implies that the inference model (=|2²) has to be sufficiently expressive to ap- proximate the true posterior well. This approximation could be improved by also using the point locations X during inference. Here, however, aim to learn visual representations, for which in- ference should only rely on images x. Focusing on the bound, the expression above can be further simplified obtain equation with B
log p( X,x) 2 F. [log p( x, =) log (=(x))
E. [log (p(X, x12)p(2)) - log go (2[x)]
F.g (oop((xx(22) + E, (() - log (=[[))
F., [ogg(XX=(1)) KT. (==))()))))
E. [[ogи[>|=)] E. [logp(X (xxpp(X))) [])] (L(((=))))
The last line relies the conditional independence assumption.
The supervupervi target E, (log p(X provides a lower bound on the conditional log- likelihodo log p( (|x)). This follows from the joint model under the conditional independence assumption with
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p(X(2,2) p(X)2). It then follows that
log (ogp(X(2) = log / dz
= log / p(X(2))(()z) de
1ogp((]2)p((lzxdd
2
[ogp(( dz = E, flog [(|])]]
This shows that g0(2[z) is sufficient for predicting the point process X conditioned on an image æ.
The conditional independence assumption X ø/2 is an architectural design choice to simplify the model and the derivation of the ELBO. It postulates that captures the relevant information from both and x, such that once is known, knowing in addition to x does not provide additional information about ær and vice versa. As consequence, p(X [2r,,) = p( p(]]=), i.e., the point process X can be predicted from the latent representation 2 alone. This is desired, since we want 2 to act the predictor for the spatial distribution. Without this assumption, we would have to model (xx 2) directly, which would require the image as direct input to the prediction model, similar to conditional VAE (Sohn al.] 2015). The same conditional independence assumption is also used in multimodal VAE (Wu & Goodman 2018). In practical applications of SI-VAB the assumption should be fulfilled by construction, such that it is not limiting. Whenever the point locations X are deterministically obtained from the image I, e.g., by spot detection (see Appendix (..)). or the images are synthetically generated from point locations (Appendix()), this trivially the case. Both data modalities then capture the same underlying point process, and the conditional independence assumption is fulfilled.
E SYNTHETIC BENCHMARK DATA SET
We describe the procedure used to generate the synthetic benchmark data set used in the main text and how was prepared for model training. The data sets are available at htttps://git. mpi-cbg de/mosaace wtw.aa////ootwa////////////////e
E.1 SYNTHETIC IMAGE GENERATION
We generate synthetic images that mimic the appearance of biological ffuorescence-microseee im- ages. Images are generated from point process realizations by rendering each point disk-shaped region on a 300 x 300 pixel grid. Disk radii are sampled uniformly between 2 and 6 pixels to reflect the natural size variability of subcellular structures. All disks have a foreground intensity of 1.0 and the background intensity is 0.0 at the stage. The resulting binary disk image is convolved with pixels, modeling the point-spread function of
Gaussian kernel with standard deviation Oblur fluorescence microscope (Zhang et al. 2007). A constant background intensity of 0.2 is added to simulate autofluorescence and ambient light. To realistically model photon shot noise, the blurred image with background is scaled by gain factor of 100, converted to photon counts, and corrupted by Poisson noise with a rate proportional to the pixel intensity. This models detector shot noise. Subsequently, additive Gaussian noise with zero mean and standard deviation Fread is superim- posed to model electronic amplification and readout noise. The final noisy image is normalized to J0, intensity range. An example of this process is shown in Fig. g The images were saved in PNG format with bit depth uint8. 8. While this effectively discretizes the intensity levels, the discretiza- tion is fine enough resolve the simulated photon-count levels. Therefore, mathematically treat pixel intensities as continuous, thus images x E R xHxC
To evaluate the impact of noise on representation learning, also generated data set with higher background intensity (log = 0.4). This changes the signal-to-noise ratio (SNR) of the images,
Iuax
Ibg
SNR
(15)
Vimax
where Imax is the maximum pixel intensity and Ang is the background intensity. The numerator represents the maximum signal contrast, while the denominator measures the Poisson shot noise
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Binary mask of disis
Gaussian blur simuaaingg PSF
3. Addedd uniform bad background (0.2)
06
Scaled to photon counts
5. Applied po sson shot noise
Added Gaussian read notse
100
03
Figure 8: Example of the synthetic microscopy image generation pipeline for spatial point pattern sampled from a homogeneous Poisson point process.
the brightest pixel. The SNR of the images with the standard background (Tug 0.2) is 12.8, whereas it is 9.6 for the higher background (Tog = 0.4).
E.2 DATA SET PREPARATION
The data set consists of 5000 simulated images for each of the six point processes considered, paired with their respective ground-truth point clouds (see Section E. We split the data set into training (80%), validation (10%), and test (10%) sets. All images are scaled to 256 x 256 pixels to match the model input layer and normalized using the mean and standard deviation computed from the training set, with the same normalization applied to all splits to prevent data leakage. Batches of size 64 were used for training, with point coordinates zero-padee zero-padded the maximum number of points in the batch and an additional channel indicating the number of points.
E.3 WEAK LABEL GENERATION
In a synthetic benchmark setting, the ground-trute point locations shown in an image are known. In practical applications, however, points first need to be detected in the images. This is only required for training an SI-VAE model. Inference can then be done on raw images.
We evaluate the robustness of SI-VAE different point-detection methods and imperfect point detection. In addition to the ground-truth baseline, we therefore generate imperfect point locations as weak labels using two methods: (1) a simple global Otsu thresholding using OpenCV (Bradski] 2000) followed by contour centroid extraction, (2) the state-of-the-art deep-learning spot-detection method Spotiflow (Dominguez Mantes et al. 2025
Before applying either spot-detection method, images were denoised using an isotropic Gaussian filter with standard deviation o 3.0 pixels. This value of provided the best results. Smaller values left residual noise, whereas larger values caused too many misdetections.
Global Otsu thresholding was performed using OpenCV (rrddski 2000) to obtain a binary mask. From this mask, we extracted the contour pixels of each connected component and retained all con- tours with nonzero enclosed area. For each retained contour we computed the geometric centroid (in R²). The centroid coordinates were then mapped to the normalized domain 1|1 using standard Cartesian coordinates. This provides simple and fast spot detector, which, however, tends to miss nearby spots and have low detection accuracy. It serves
low-quality baseline for weak labeling.
To obtain high-quality weak labels of point positions, we used Spotiflow (Dominguez Mantes et al., 2025), deep-learning spot-detection method specifically trained for fluorescence microscopy. It predicts the locations of bright spots in the image with subpixel resolution IR?, For each denoised image, we ran the pretrained Spotiflow model in subpixel mode, lowering the probability threshold to 0.40 to include candidates that might otherwise be rejected. This choice was informed by visual inspection of stereographic flow plots, where some spots were clearly visible but excluded under the
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Table Spot-detection performance for the two weak labeling methods (Spotiflow and Otsu thresh- olding) for images of different quality (SNR). We report the absolute difference (AD) between the ground truth and detected numbers of points, the True Positive Rate (TPR, recall), the Positive Pre- dictive Value (PPV, precision), and the Fi score. The expected number of points for all generated point patterns is (W)) R 52.
Sponylow
Otau thresholding
Point process
SNR
AD (L)
(1)
(1)
Fi (1)
AD (U)
TPR
PPV
Ft. (f)
Homsggencous Paisson
6,68
B874
0.999
19.03
0.677
1.00
0.777
Homogeeenus Poisson
12.8
7.16
0.865
1.000
0.937
9.31
0.855
0.999
0.902
Homogencuus Sirauss
9.6
4.32
0,918
0.999
0.956
15.60
0.701
1.000
0.824
[lomogeneuus Strauss
12.8
4.65
0.912
0.999
0.953
6.18
0.885
0.997
0.936
Homogeeens Thomas
9.6
14.99
0,732
0.990
0.843
29.38
0.466
1.000
0.635
Honoggreems Thomss
12.8
15.64
0.719
1,000
0.835
20.35
0.635
0.999976
0.772
Inbomogemewas Poisson
9.6
14,47
0.727
1.000
0.840
29.68
0.437
1.000
0
labomoggreews Pousson
12.8
15.28
0.712
1.000
20.06
0.622
0.999969
0,762
Iekoeeegennvs Stranss
9.6
6.7
0,870
0.999
0.929
18.81
0.634
1,000
0.775
Ialounogeeeews Stranes
12.8
7.3
0.859
0.824
0.999168
0:901
Iehomogmmewx Thomas
9.6
19.6
0.645
0.999
0.781
35.73
0.342
1.000
0.503
Inbomogeneuu Thomas
12.8
20.57
0.627
0.767
27.10
0.508
0,999752
0.665
default adaptive probability threshold. The resulting coordinates were normalized to [-1 and mapped to standard Cartesian coordinates.
We quantify the quality of the two weak labeling methods- Otsu thresholding and Spotiflow-fo the synthetic images for both image quality levels (SNR). For each image, we compare the detected points against the known ground truth. For this, we first use one-to-one Hungarian matching to compute a globally optimal correspondence between the two point clouds that minimizes the sum of squared distances between predicted and ground- truth points. A spatial cutoff of 3 pixels is applied to determine valid matches (Dominguez Mantes et al. 2025). This ensures that each predicted point is matched to most GT point, and vice versa. A matched pair is counted if their Euclidean distance is below the cutoff. Unmatched predictions are counted as false positives (FP), and unmatched ground-truth points as false negatives (FN), while true negatives are undefined in this continuous-spac setting.
frue positive (TP)
For each point process type, we report the absolute difference (AD) between the ground truth num- ber of points and the detected number of points, the True Positive Rate (TPR, recall), the Positive Predictive Value (PPV, precision), and the Fi score across all 5000 images. The results are given in Table 3, As expected, Spotiflow consistently outperforms Otsu thresholding and is more robust to noise in the images (low SNR).
F SUMMARY STATISTICS AND SIMULATION METRICS
To validate the accuracy of the learned conditional intensity models, conditioned on a latent variable compare nonparametric estimates of the first two moments of the predicted process and the ob- served point pattern. The first moment is the intensity function, defined as EN(W) = Jw p(u) du. It can be interpreted the expected number of points per unit area. We obtain a nonparametric estimator using kernel density estimation (Møller & Waagepetersen (003). In particular, for an observed point pattern X C W. the intensity at location u E W is estimated as
p(u) = Σ (, - (ll-u)/wi w)/w(v),
(16)
DEX
where (u) k(u/n)/1² with density function k. We choose A as the multivariate standard Gaussian and the weights wvv) fy (u, v) du to account for edge effects in the finite domain W (Møller & Waagepetersen 2003). We choose the bandwidth 77 0.5 empirically to yield good estimates for all considered point-process types. This is close to Scott's 's rule of thumb for EN(W) x 52 points in W [-1, 1]2, which is Mscoott 0.518 (Scott 2015
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The accuracy of an intensity estimate is then quantified by the relative intensity error ( X) n-1 Ei= Sw lĀ(u) Xx,g) Á(u) X) (X) du for different samples Xię obtained from the condi- tional intensity A 2/22) predicted by the SI-VAE for a given input image x.
To assess the quality of the estimated interaction structure, we leverage ideas from Monte Carlo goodness-of-fft tests for spatial point processes (Diggle 2013 Baddeley et al., 2014). While such tests are strictly invalid and conservative for any significance level when parameters need to be esti- mated from the data, they provide useful proxy to measure the quality of the learned model. We say that a model is not able to capture the interaction structure in the observed pattern if the observation X lies outside the maximum envelope of the samples X1, from the predicted conditional intensity Ag(X, u| x..
The sample envelope considers the minimum and maximum values of a functional summary statistic, such as the K-function, over multiple distances r. The K-function counts the normalized number of points at distance r. It is commonly used to characterize interactions in spatial point processes (Diggle 2013). Following Baddeley et al. (2000), we estimate it as
1(0 E B(u,r)) *
(r))= =
Σ (u,DeeX ((u)p(v)ew
(17)
where j(u) is the estimated intensity at ut, and Cuv is an edge correction weight account for the do- main boundaries. We use translation-base edge correction as described Moller & Waagepetersen (2003). The intensity function p(u) is estimated using equation 16 This shows that it can be difficult to disentangle errors in the first and second moments of the distribution, since both depend on p(u). This is especially true for clustering processes, where it is often unclear whether the clustering is due to a spatially varying intensity or due to attractive interaction between the points.
Since the functional summary statistic function of the r, we reduce 4, Following |Badeeley et al. (2014), we perform maximum absolute deviation (MAD) test, which
single test statistic
considers the maximum deviation from the mean K(r) n+I (Ri(r) + ........ Rn(r) + Konn (r)).
where Robs (r) is the K-function of the conditioning sample and K2(r) is the K-function of the i-th simulated sample. The test statistic then is
4 = max OSr5R
(18)
which fulfills the necessary symmetry property under the null hypothesis Ho for Monte Carlo test. We choose Rt = 0.24, which corresponds to the rule of thumb proposed by Diggle (2013). The rate of rejection of the test is used as metric assess the quality of the learned model. It quantifies how frequently the observed pattern X achieves a larger maximum deviation from the mean than the samples X from the learned model. Formally,
1 M Σ 1 (maxtis S tobs.))
RR
(19)
which measures the number times the test rejects Ho. high rejection rate (RR) indicates that the predicted conditional intensity significantly deviates from the observation and therefore insufficiently captures the interaction structure in the image. This is in line with the post-hoc model evaluation typically done for spatial point processes (Møller & Waagepetersen 2007). We use M 100 random images for each class from the test set and n = 19. This would correspond to a test significance level of 5% for one-sided test, if the parameters of the null model were known.
G APPLICATION TO SUBCELLULAR PROTEIN LOCALIZATION
We describe the data preparation, model architecture, and training procedure for the protein localization experiment of Section 3.5
We used the publicly available OpenCell data set (Cho et al. 2022). which contains fluorescence microscopy images of over 1000 human proteins. The present VAE model was trained on images of size 100x 100 pixels, each centered around nucleus. Three channels were used for training: the
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25
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POLISE
Figure 9: Representative standardized images from the test set for the six proteins (rows) considered in Scction 3.5 Each image consists of three channels: the protein fluorescence signal (first columm), the nucleus fluorescence signal (second column), and the signed distance function to the nucleus (third column). The same example images are also used Figs. and 16
protein fluorescence signal, the nucleus fluorescence signal, and the signed distance function to the nucleus (see Fig.9) Centering the observation window around nucleus ensures that at least one cell is completely contained within an image. Since the same data preparation was used for Cyto- self (Kobayashi et al., 2022). directly took the images from the Cytoself repository (Royer Lab) 2025 We generated weak labels for protein spot locations using Spotiflow (Dominguez Mantes al. 2025) on the protein fluorescence channel, with a low probability threshold of 0.35 to include candidate points that might otherwise be rejected. No Gaussian smoothing was applied to the im- ages prior spot detection. This yielded better results upon visual inspection. The resulting point
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coordinates were normalized to [-1, 1]2 and mapped to standard Cartesian coordinates. These point coordinates were then used as self-supervision target when training the SI- VAE model.
For the application case presented in the main text, we chose six proteins from three families. For each protein, about 1000 images were available (POLR3E: 1 162. POLR3F: 1068, SNXI: 1001, SNX12: 1260, STAM: 1361, STAM2: 1021). This resulted in data set of 6873 images, which was split into disjoint training (80%=5499 images), validation (10%=687 images), and test (10%=687 images) sets. Each channel in the training data set was standardized zero mean and unit variance. The same standardization was then also applied to the validation and test sets to prevent data leak- age. Representative images for each of the six proteins are shown in Fig. 9 These are the same representative examples as in Figs. [ and 16.
C.I was adjusted to the image shape of this application (100x x3). The input channel size was set to 3, while the rest of the convolutional layers re- tained the same parameters as for the synthetic benchmarks, progressively downsampling the images to 12x 12 while increasing the number of channels as 3-11--32 -64. All convolutional layers used ReLU activations. The latent space dimension remained 64. The decoder reconstructed images via a symmetric sequence of three transposed convolutional layers, decreasing the number of channels as 64-332-11--3 All transposed convolutional layers used the same parameters as in Appendix CC.I, with ReLU activations in all layers except the output, to predict the standardized images. The recon- struction error in equation | was minimized using the mean squared error, with the KI. divergence scaled by B 0.1 as in the synthetic benchmarks.
The VAE architecture from Appendix
The conditional intensity prediction model remained unchanged from the synthetic benchmarks, as described in Appendix C.2 We set the range of the interactions half the average nucleus radius,
L = 0.25. This radius was identified from the central nuclei in all cropped images by taking the largest connected component in the nucleus channel and computing its equivalent diameter (diameter of a circle with the same area). This resulted in an average (across the entire data set) half-nucleus radius of 12.59 pixels. Transformed to the coordinate space [-1, 1]2 used by the model, this yields == 0.25. As in the synthetic benchmark, we used an erosion of R = 0.2 to avoid edge effects and discretized integrals over the domain using trapezoidal quadrature on a fixed grid of size 100x 100.
Like in Appendix C all three neural networks (VAE, de. W/) were trained jointly on the training set, each with its own Adam optimizer (Kingma & Bal (013), using batches of size 16 and learning rate 10-4. Training stopped upon convergence of the validation loss.
We compared SI-VAE with two baselines: the VAE without spatial supervision and Cytoself (Kobayashi et al.) (022). semi-supervised model based vector-quamtized VAE (VQ-VAB) that leverages ground- tround-truth protein labels as self-supervision target. Cytoself constitutes the state of the art in human protein localization modeling, and it is significantly more complex than the present SI-VAE.
For the VAE baseline, we retained the SI-VAE's VAE component alone with the same parameters (64 latent dimensions, 33 = 0.1). Training was stopped upon convergence of the validation loss to ensure comparability.
For Cytoself, we used the published implementation (Royer Lab 2025) but retrained the model from scratch our data set. To correct for the smaller size of our data set, we reduced both VQ- VAE codebooks to 128 entries (from the original 512). The two VQ-VAE learn representations on different scales: a local representation (VQ1) at 25x25 pixel capturing texture detail and global representation (VQ2) as a 4x4 pixel feature map with 64 features per pixel aimed at capturing long- range patterns. Following the original Cytoself paper, downstream clustering was performed on the global representation (VQ2). Training was performed with an initial learning rate of 10-4, which was reduced ten-fold whenever the validation loss did not improve for four consecutive epochs. Early stopping was applied with 10 epochs patience. Training terminated after 14 epochs.
All learned representations were analyzed using images from the test set. We determined the latent representation of each test image using the trained SI-VAE, VAE, and Cytoself encoders. For SI-VAE and VAE, we used the mean prediction of the posterior (o(2[z). We then standardized the latent representations across the test set zero mean and unit variance for each latent dimension and performed clustering using a Gaussian Mixture Model (GMM) with full covariance matrix (Hastic et al. 2009). The number of clusters K was determined using the Akaike Information Criterion
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(AIC) (Akaike 1974), AIC -2 2K, where is the maximized likelihood of the model. We fitted GMM for K = 1..... , and selected the model with the lowest AIC, which was 2. We used the scikit-lears implementation of GMM (Pedregosa et al.) 2011) with default settings. Silhouette scores (Rousseeuw) 1987) were computed using Fuclidean distance on the serialized (into vector) 64-dimensional embeddings (1024-dimensional for Cytoself). This the average distance between points within the same cluster relative to the points in the nearest neighboring cluster. Higher Silhouette scores indicate tighter and more separated clusters. The Silhouette score for single sample is defined as
quantifies
S. max(a, b)
(20)
where a is the mean intra-cluster distance, and is the mean nearest-cluster distance. Scores range from -1 to 1.
Cluster Protain Distribution VAE (GMM,
Cluster. Protein Distribution CyMMse
0.00
0.06
Cluster
1.00
0.94
Protain
Figure 10: Cluster assignment frequencies for all proteins in the VAE (left) and Cytoself (right) embeddings. Each square shows the fraction of images of given protein that was assigned to the corresponding cluster. Clusters were determined by Gaussian Mixture Models (GMM) with K 2.
Table 4: Silhouette scores on the test set embeddings of each model. Higher values indicate more compact and better-separated clusters. The table reports the average Silhouette score for each cluster (Nuclear, Vesicular) as well as the overall score across all samples.
Model
Overall
Vesicular cluster
Nuclear cluster
Cytoself
0.331
0.227
0,550
SI-VAE
0.294
0.286
0.310
VAE
0.040
0.042
0.035
All three models consistently identified the same clusters (see Fig.5ifor S1- SA- and Fig, [0 VAE and Cytoself). SI-VAE and Cytoself, however, achieved significantly higher Silhouette scores than the VAE baseline (Table 4. Since the cluster indices assigned by the GMM differ across models, each cluster in the table is labeled according to its dominant protein localization: nuclear (POLR3E, POLR3F) vesicular (SNXI, SNX12, STAM. STAM2).
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H ADDITIONAL FIGURES
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Figure 14: Examples of conditional simulation results from the trained conditional intensity model different latent codes = in the homogeneous cases over the evaluation domain [-0.6, 0.6]2. The trained conditional intensity model is from the SI-VAE from Section 3.1 trained at high SNR with ground-truth truth knowledge. Ten i..d. samples are shown for each point process type (Poisson, Strauss, Thomas).
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Published as a conference paper at ICLR 2026
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Figure 15: Examples of conditional simulation results from the trained conditional intensity model different latent codes in the inhomogeneous cases over the evaluation domain [-0.6, 0. The trained conditional intensity model is from the SI-VAE from Section 3.1 trained at high SNR with ground-trutt -runh knowledge. Ten i..d. samples are shown for each point process type (Poisson, Strauss, Thomas).
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Published as a conference paper at ICLR 2026
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Figure 16: Conditional intensity predictions (contour plots) over the eroded domain D = W e R for representative samples from the set of the OpenCell data with detected points in orange. The same samples as in Fig. aare shown with insets showing the corresponding nucleus channel for reference.
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Reviews:
[{"review_id": "aah6oDvMks", "reviewer": "Reviewer_zSUz", "summary": "The paper developed a self-supervised deep-learning model  that use stochastic point processes to predict spatial organization patterns from images, coined as Spatially Informed Variational Autoencoders (SI-VAE). The self-supervision mechanism is modeled by the Papangelou conditional intensity. Extensive experiments were presented to illustrate the effectiveness of the SI-VAE model.", "questions": "What are the connections between the present work with existing Gaussian process variational autoencoders? What is new in the SI-VAE? Does the current model promote easy implementations?\n\nIf Gaussian process variational autoencoders can also be used to capture spatial information in the images, can the SI-VAE model still outperform its Guassian process counterparts?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper provided a comprehensive illustration of the idea of Spatially Informed Variational Autoencoders (SI-VAE), which leverages the Papangelou conditional intensity as the self-supervision target for measuring the spatial information of images. Extensive experiments provided convincing results to showcase the effectiveness of the SI-VAE model in capturing spatial interactions and its generalization to unseen data in terms of zero-shot learning. The impact of the SI-VAE was also demonstrated on a challenging real-world application of protein localization in human cells.", "weaknesses": "Despite the strength as mentioned above, the paper only compares the proposed SI-VAE to the original VAE, while ignoring the existence of similar techniques, such as \n\n1) Semenova, et al., PriorVAE: encoding spatial priors with variational autoencoders for small-area estimation, J R Soc Interface, 2022\n2) Jazbec, et al., Scalable gaussian process variational autoencoders, AISTATS 2021.\n\nSuch an incompleteness of refereeing and comparisons weakens the overall quality of the paper.", "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": "iaaD2UHH3v", "reviewer": "Reviewer_B6yH", "summary": "This paper proposes a spatially informed variational autoencoder - SI-VAE, which consists of a VAE augmented with a spatial point process.  The latent representation z from the VAE is used as input to a neural network that then predicts the Gibbs potentials that define the point process.  Experiments are performed with synthetic data, showing a comparison to standard VAE, generalization to unseen processes, and zero-shot conditional simulation.  Lastly, an application to protein localization patterns is given, where it is demonstrated that the learned potentials agree with domain knowledge (eg proteins in vesicles being homogeneously distributed versus nucleus proteins being inhomogeneously distributed within the nuclei).", "questions": "see weaknesses above", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Interesting proposed model, sufficient technical contribution/novelty.  Validation on synthetic data.", "weaknesses": "Although the proposed model is interesting, majority of the validation is on synthetic data in some sense tuned to the specifics of the model.  Demonstration of applicability to a real problem is somewhat limited - consisting of only one specific test application where the final evaluation is a check that the learned model potentials agree at a high level with what is expected from domain knowledge.  Within this particular task, there is also no notion of a baseline for comparison.  This is an area where the impact could be much improved, by showing broader applicability to other domains, or giving a more qualitative analysis against baseline methods, or showing some unexpected/novel finding instead of confirming existing domain knowledge.", "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": "Kx7KXvnROr", "reviewer": "Reviewer_vPwX", "summary": "The authors propose using spatial point processes as a self-supervision prior that explicitly models spatial distributions of objects to address the gap in previous un- and self-supervised methods that miss the spatial correlations.  Thus, the paper proposes spatially informed variational autoencoders (SI-VAE) to predict spatial organization patterns from images. The authors apply SI-VAE to a real world microscopy dataset, OpenCell, and correctly identify the protein localization classes.", "questions": "- Biological tissues can have direction dependent spatial structure (e.g.  muscle fibers, neuronal axons) which would lead to directionally correlated spatial patterns. Since SI-VAE assumes pairwise interaction potential to be symmetric, isotropic, from an implementation standpoint, how difficult would it be to extend SI-VAE to include anisotropic or direction aware interactions?\n\n- How sensitive are the SI-VAE representations to errors in spot detection, and could the model be extended to be fully end-to-end (e.g. learning jointly the point locations and spatial interactions)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Improving VAEs by fusing them with spatial point processes, thus learning statistically interpretable representations of spatial distributions of objects in images can improve biological data analysis.\n\n- Furthermore, the decomposition into interpretable potentials is also a non-trivial contribution that is based on well-established statistical frameworks.\n\n- The paper introduces a principled way of interpreting the learnt representations within spatial statistics. The results on the biological dataset are promising.", "weaknesses": "- The real world applications were limited in the paper. Given the novelty of the proposed method, I would have wanted to see more results on biological data and also interpretations. While the results are promising, it is unclear to me whether SI-VAE generalizes to more complex localization patterns or proteins with overlapping spatial distributions.\n\n- SI-VAE assumes that given z, the image and point pattern would be independent.\nBut since X is deterministically obtained from x, wouldn't this assumption be false? Could the authors clarify this?", "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)