SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomics
Authors: Qingtian Zhu
Yumin Zheng
Yuling Sang
Yifan Zhan
Ziyan Zhu
Jun Ding
Yinqiang Zheng
Venue: ICML 2025 poster (2025)
Keywords: Spatial Transcriptomics
Implicit Neural Representations
Primary area: applications->health_medicine
TL;DR: SUICA = Spatial Transcriptomics + Implicit Neural Representations → Restored Authenticity + Enriched Bio-conservation

Abstract: Spatial Transcriptomics (ST) is a method that captures gene expression profiles aligned with spatial coordinates. The discrete spatial distribution and the super-high dimensional sequencing results make ST data challenging to be modeled effectively. In this paper, we manage to model ST in a continuous and compact manner by the proposed tool, SUICA, empowered by the great approximation capability of Implicit Neural Representations (INRs) that can enhance both the spatial density and the gene expression. Concretely within the proposed SUICA, we incorporate a graph-augmented Autoencoder to effectively model the context information of the unstructured spots and provide informative embeddings that are structure-aware for spatial mapping. We also tackle the extremely skewed distribution in a regression-by-classification fashion and enforce classification-based loss functions for the optimization of SUICA. By extensive experiments of a wide range of common ST platforms under varying degradations, SUICA outperforms both conventional INR variants and SOTA methods regarding numerical fidelity, statistical correlation, and bio-conservation. The prediction by SUICA also showcases amplified gene signatures that enriches the bio-conservation of the raw data and benefits subsequent analysis.

SUICA:
Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomics
Qingtian Zhu
Yumin Zheng
23
Yuling Sang"
Yifan Zhan
Ziyan Zhu S
Jun Ding 236
Yinqiang Zheng'
Abstract
(a)
(b)
(c)
Spatial Transcriptomics (ST) is a method that cap- tures gene expression profiles aligned with spatial coordinates. The discrete spatial distribution and the super-high dimensional sequencing results make ST data challenging to be modeled effec- tively. In this paper, we manage to model ST in a continuous and compact manner by the proposed tool, SUICA, empowered by the great approxi- mation capability of Implicit Neural Representa- tions (INRs) that enhance both the spatial den- sity and the gene expression. Concretely within the proposed SUICA, we incorporate graph- augmented Autoencoder to effectively model the context information of the unstructured spots and provide informative embeddings that are structure-aware for spatial mapping. We also tackle the extremely skewed distribution in fashion and enforce classification--assed loss functions for the opti- mization of SUICA. By extensive experiments of a wide range of common ST platforms under vary- ing degradations, SUICA outperforms both con- ventional INR variants and SOTA methods regard- ing numerical fidelity, statistical correlation, and bio-conservation. The prediction by SUICA also showcases amplified gene signatures that enriches the bio-conservation of the raw data and bene- fits subsequent analysis. The code is available at https:/ /ggtttb......... cww/ Saam//2ym2la 9/ SUICA.
CMAP2
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Canuil plos ploun Cannssficiinn
bach Kibayy
Mesingo Muscel
Mack Smooth
Sabenaddled
Figure I. Starting with the discretely sampled spots (a) of ST. SUICA performs continuous modeling (b) by aid of the great approximation power of INRs. This approach enables complete profiling cell heterogeneity, as visualized in the UMAP (c), which further facilitating the discovery of new biology.
1. Introduction
Spatial Transcriptomics (ST) enables scientists to quantify gene expression while preserving spatial information in tis- sue sections (Marx, 2021). These platforms use various strategies to capture mRNA transcripts from tissue sections and perform sequencing quantify the gene expressions spatially defined locations (Moses & Pachter, 2022). How- ever, various degradation patterns are being observed due to some limitations throughout the data acquisition pro- cess. For instance, achieving higher resolution essen- tial for accurately modeling and analyzing cellular func- tions (Williams et al., 2022), while for now, ST data can still be rather expensive, e-g., $3,500/em for the capture chip and $800/cm² for the high-density sequencing according to Chen et al. (2022a). Beyond the cost constraints to get high- resolution ST, another key technical challenge lies in the drop-out rates (high sparsity of ST data) that makes the gene expressions less informative for analysis (Qiu, 2020), Com- pounding this challenge, different sequencing techniques and platforms exhibit significant heterogeneity in terms of spatial distribution, sequeneing depth (number of mRNA readouts per spot), and drop-out rates (Moses & Pachter, 2022). As a result, such the difference makes a general backbone to model and analyze data across different tissues by different platforms less feasible. To address these tech- nical challenges, we propose a computational framework
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'Equal contribution 'The University of Tokyo 'MGiil Uni- versity" "MUHC Research Institute "Duke-NUS Medical School CCarnegie Mellon University "Mila Quebec AI Institute. Corre- spondence to: Jun Ding <jun.ding@mmet menng@megill.cam ca?>, Yinqiang Zheng <ygzheng @ai.u tactieng@aiuuutoo...a.c.o.oo top...ac.pp>.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transeriptomies
that transforms discrete ST data into a continuous, compact representation, and thus enables complete reconstruction of ST slides by recovering the degradations in the raw data without the need to know the specific type of degradation.
tion. Moreover, SUICA's imputation ability leads more cell-type informed clustering results of ST. To summarize, our contributions in this paper are three-fold as follows:
We introduce SUICA to model ST data as continu- ous and compact representation while preserving data authenticity;
Recently, Implicit Neural Representations (INRs) have drawn great attention from researchers for their compact, continuous, and differentiable properties as a novel repre- sentation of general coordinate-based signals. INRs map coordinates to corresponding values with a neural network with wide applications in inverse graphics (Mildenhall et al., 2020), geometric modeling (Park et al., 2019), and video compression (Chen et al., 2021). Motivated by INRs' inher- ent smoothness property, we leverage this characteristic to interpolate between ST spots, enabling more comprehensive spatial transcriptomics analysis. However, applying INRs to ST data faces two major challenges. First, scaling INRs to the super-high dimensional space of gene expression is non- trivial, as simply widening or deepening the network cannot effectively overcome the curse of dimensionality. Second, the low mRNA transcript capture rate in ST, combined with varying gene expression patterns across cellular states, re- sults in zero-inflated ST data with high sparsity (Piwecka al., 2023). This sparsity makes it particularly challenging for conventional INRs to accurately capture the underlying complex, non-lineat spatial patterns.
We address the issue that prevent INRs from scaling to the super-high dimensional gene expression of ST by leveraging Graph Autoencoder and classification- based loss function;
Extensive experiments show that SUICA achieves su- perior reconstruction quality and imputation capabil- ities on various ST datasets, facilitating subsequent analyses.
2. Related Work
2.1. Implicit Neural Representations
INRs model signals by mapping input coordinates to cor- responding signal values using neural networks. Unlike conventional discrete grid-based signal representations, e.g., images, videos, and voxels, INRs are known for continuous modeling, allowing queries at arbitrary locations within the definition domain. Typically implemented as Multi-Layer Perceptrons (MLPs), INRs leverage the smoothness bias of MLPs Ps to provide a certain level of interpolability, allowing effective generalization to unseen coordinates.
Building upon this approach, we introduce SUICA, a power- ful variant of INR designed especially for ST data. SUICA fully accounts for the unique properties of ST and the di- versity of current mainstream ST platforms. In SUICA, we employ an Autoencoder (AE) based Graph Convolutional Network (GCN) (Kipf& & Welling, 2017) to bridge the gap between conventional INRs and ST. The GCN enhances the approximation capability for unstructured data by lever- aging contextual information and making the embeddings structure-aware. By the aid of the strong profiling power of Graph Autoencoder (GAE) regarding the sparse and skewed distribution, we perform the fitting of INRs the expressive low low-dimensional embedding space, which has been proven to be compact and more suitable for INRs. To enforce the sparsity within the regression- construct pseudo-prose probbbbiibaiilitiess and adopt a regression-by- classification approach for training.
The idea of neural networks for function approximation dates back to the 1980s and 1990s (Cybenko, 1989; Hornik, 1991) as universal function approximation theories. From relatively modern perspective, SIREN (Sitzmann al., 2020) adopts periodic sine functions as the activation func- tion and learns INRs rich in fine details. Equipped with Random Fourier Features, FFN (Tancik et al., 2020) multi- plies the coordinates with a random feature sampled from a normal distribution. BACON (Lindell et al., 2022) fur- ther explores the spectral bandwidth of the target signal and progressively regresses in coarse-to-fine manner.
scheme of INRs, we
INRs have been applied across various downstream tasks, including inverse graphics and video compression. NeRF (Mildenhall et al.. 2020) parameterizes the 3D scene as an MLP-based radiance field with colors and densities. It applies differential volume rendering to query a series of points in the field and then calculate the weighted sum as the pixel value. Following this parameterization, subse- quent works (Liu et al., 2020; Yu et al., 2021; Sun et al., 2022; Hu et al., 2022; Fridovich-Keil et al., 2022; Müller et al., 2022; Peng et al., 2020; Chan et al., 2022; Chen et al., 2022b) further improve the radiance field for efficient training and rendering. Additionally, the continuity of INR is well-suited for application in 3D geometric reconstruc- tion methods (Park et al., 2019; Wang et al., 2021; Yariv
With SUICA, we are enabled reconstruct diverse ST data from several popular platforms and sequencing protocols using a unified representation, facilitating comprehensive analysis as illustrated in Figure 1. In particular, we apply SUICA to enhance the spatial resolution (spatial imputa- tion), to alleviate the drop-out rates (gene imputation), and to smooth the noisy expressions (denoising), respectively. Extensive experimental results demonstrate that SUICA out- performs existing INR variants as well as the -tate-of-the-art methods in both numerical fidelity and statistical correla-
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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomies
Training-imee
Genes
gt
Cell Graphs
Zgt
Syatiol Imputation
Gene Ingutation
Denoising
Cemba
Sprtt
gt
Agt
Expression Matnx
Encoder 4
P
Decodar
Seci Birurrnation
Yes
Spscat Coordinks
00:0000
1
00:1
1
2
8
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CHAEE LASE
Implicit Neural Represestation Jy
P+
Deooder
Lrocons
Training and Interence
Figure 2. The overall pipeline of SUICA. At training-time, GAE based on cell graphs is trained, with whose pre-trained decoder concatenated to INR. INR then maps spot coordinates the corresponding gene expressions. SUICA capable of performing spatial imputation, gene imputation, and denoising.
et al., 2021; Oechsle et al., 2021). As for video compres- sion, NeRV (Chen et al., 2021) and subsequent variants (Li et al., 2022; Chen et al., 2023) employ compact INRs to reduce data redundancy. applications map from low-dimensional to low-dimensional spaces, e.g., R² -> R for image regression, and R - R4 for inverse graphics. The of INRs for mapping from low dimension (i.e., R²2) super-high dimension (i.e., over 20,000 channels) remains unexplored.
improve resolution. To generate unmeasured spots profiles, STAGE (Li et al., 2024) trains a spatial docation-superviie Autoencoder that enhances resolution by integrating spatial coordinates with gene expression data. In summary, these resolution enhancement methods significantly improve ST quality, improving spatial visualization and facilitating more accurate downstream analyses.
noteworthy that existing INR
3. SUICA
2.2. Deep Learning for Spatial Transcriptomics
3.1. Preliminary
Deep learning methods are increasingly adopted in ST to en- hance spatial resolution, denoise raw data, improve spatial domain clustering, and support more accurate downstream analysis (Zahedi et al., 2024). Given the complexity of cap- turing cell-to-cell and spatial dependencies, graph-base based methods leveraging cell relationships have gained signifi- cant traction. Notable examples include GCN-based meth- ods like SpaGCN (Hu et al., 2021), STAGATE (Dong & Zhang, 2022), GraphST (Long et al., 2023), and SEDR (Xu et al., 2024) as well as graph wansformer-basee model like SiGra (Tang et al., 2023). These graph-based methods pro- gressively refine the modeling of ST data, enabling more nuanced interpretation of complex tissue structure.
3.1.1. 1. SPATIAL TRANSCRIPTOMICS
ST data can viewed as an unordered point set, where each point parameterized by spatial coordinates x. The mRNA readouts are bounded these sampled locations and represented by high dimensional vector, where each channel reflects the numerical abundance of a particular mRNA transcript (0 indicates absence or failed capture).
In terms of the spatial distribution of the sampled spots, dif- ferent ST platforms follow different sampling protocols, e,g., Visium applies a fixed array of probes so the sampled points are distributed uniformly over the whole slice, whereas other common platforms do guarantee such regularity. Thus, to accommodate general use cases, ST data is often modeled as unstructured without being quantized into regular grids to make CNN-based analysis inapplicable.
ST data often require higher resolution to accurately cap- ture fine spatial patterns within tissues. Several methods leverage histological images to enhance the spatial resolu- tion of gene expression data, including Convolution Neural Network (CNN)-based models ST-Net (He et al., 2020) and DeepSpaCE (Monjo et al., 2022), and Vision Transformer- based methods such as Hist2ST (Zeng et al., 2022) and TRIPLEX (Chung et al., 2024). In cases without histological images, other approaches (Zhao et al., 2021; 2023; Li et al., 2024) rely solely on spatial and gene expression data to
1.2. CHALLENGES
The key challenge in modeling ST lies in the super-high di- mensional representation, often exceeding 20,000 channels. Due low mRNA capture rates and varying gene expres- sion patterns across cellular states, ST data are typically highly sparse, with zero values comprising up to 90%. A
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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transeriptomies
satisfying model must maintain the inherent sparsity of ST data, it reflects the true gene expression. However, vanilla INRs are observed to have the tendency to yield normally distributed outputs that are smooth (Lee et al., 2018), 8), rather than the zero-inflated ones of ST, presenting a significant challenge. It is also crucial to ensure the numerical fidelity of non-zero values for accurate cell type identification. It is worth noting that high sparsity complicates the evaluation of reconstruction quality, as an entirely empty prediction may still result in low loss, yet be entirely unacceptable. To this end, the design of the representations and evaluation proto- col must address both sparsity and numerical fidelity-a0 an aspect that has, to our knowledge, not been fully explored in the context of INRs.
GTV: 16.54
GTV: 37.02
Variance: 0.73
Variance: 2.79
(a) AE
(b) GAE
Figure 3. Spectral analysis with the embeddings of AE and GAE. GAE yields structure-aware and disentangled embeddings with high-frequency details. GTV: Graph Total Variation.
is integrated through graph structure, which we model using Graph Convolutional Network (GCN), Importantl, the GCN is incorporated solely in the encoder, as the de- coder soder--repponibleee for reconstructing interpolated embed- dings back into the raw space-lacke lacks the requisite graph structure for such integration.
3.2. Method
3.2.1. OVERVIEW
The overall pipeline is illustrated in Figure 2. To construct a compact and dense embedding domain for subsequent INRs, we first incorporate a graph- based encoder and pre-train Graph Autoencoder (GAB) using the given ST slice in self- regressing manner. With the pre-trained GAE, we obtain the encoded latent representation for all spots, denoted as Zgt- We then initiate INR tuning to optimize the neural mapping from ST coordinates to embeddings, targeting Zg-- Once the INR reaches a stable state, we attach a decoder that is pre-trained in our GAE. Subsequently, we fix the INR and train the decoder to fit the raw dataset readouts Ygt-
As illustrated in Figure 3, we conduct graph-based spectral analysis of embeddings generated by both the AE and the GAE. Specifically, we construct connectivity graph using Fuclidean distance between spots (k- = 5) and calculate the Graph Total Variation (GTV) for embeddings defined on the built graph. Figure is colored based on aggregated edge-wiss wise variations of each vertex, revealing that the GAE produces shaper structural clues. Additionally, we compute the channel-wiss variance for both embeddings. Together with the GTV results, these findings demonstrate that GAE offers a more expressive and informative representation, effectively capturing disentangled embeddings for ST.
Concretely, we perform 3 different tasks with SUICA, namely spatial imputation, gene imputation, and denoising, whose data flows are summarized as follows.
To train GAE, we adopt the conventional Mean Square Error (MSE) for supervision as
Spatial Imputation: Say an ST slice {xx,)); whose data spots are split into training subset and a test subset. With the training subset, we train SUICA (GAE+INR), and infer the test subset for evaluation. Gene Imputation: We randomly mute part of the gene expressions of the data matrix, fit SUICA with all of the data, and infer all of the x.
[( -ye)? My
(I)
|Myl
where My represents the mask over all of the elements y to compute the element-wise average.
Denoising: The data flow is basically the same as gene imputation, but with injected noise as the degradation.
3.2.3. EMBEDDING MAPPING
With the pre- pre-trained GAE and encoded representation Zg: in place, we proceed to train the INR to map x + z. The com- pact, dense embeddings provided by the GAE significantly reduce the workload of the subsequent INR, which must fit an otherwise highly skewed distribution.
3.2.2. GRAPH AUGMENTEE AUTOENCODER
The sequenced readouts of ST are known for an extremely skewed distribution, represented as high-dimensional, sparse data. This sparsity exacerbates the curse of dimen- sionality, rendering data points increasingly dissimilar and challenging to organize efficiently.
For SUICA, we empirically select two baseline architec- tures: FFN (Tancik et al., 2020) and SIREN (Sitzmann et al., 2020), both of which have demonstrated effectiveness across various data types. In our experiments, we apply SIREN for spatially sparse ST and FFN for denser spatial distributions.
To address this, we leverage an AE to transform the high- dimensional raw space into a compact, dense, and infor- mative embedding space. Given the irregular structure, the context information, context or neighborhood information
Following conventional INR approaches, we assume
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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomies
Caussian-distributed error in the predicted and use element-wise mean MSE loss for optimization:
4. Experiments
4.1. Datasets & Metries
Lembd
Σe (2 zz1)²
(2)
For a quantitative benchmarking, we involve a nanoscale resolution Stereo-seq (SpaTial Enhanced REsolution Omies- sequencing) dataset, MOSTA (Chen al., 2022a). MOSTA consists of a total of 53 sagittal sections from C57BL/6 mouse embryos at 8 progressive stages using Stereo-seq. from which we take slice for each stage (from E9.5 to E16.5) for benchmarking. In addition to Stereo-seq, we also leverage ST data by other common platforms, ie., Slide-seq 10x Genomics Visium (see Appendix) and MERFISH (see Appendix), to further demonstrate the gen- eralization of SUICA.
[M2
M
While å can serve as the mapping output for downstream analysis, we consider it as an intermediate result, decoding this latent code back into the raw space to achieve a more comprehensive reconstruction.
3.2.4. DECODING HEAD
A straightforward approach to decoding à back to raw repre- sentations is to directly use the pre-trained GAE decoder for end-to-end INR tuning. However, this method encounters two primary challenges. (1) The embedding mapping of x + 2 is error- prror, and domain shifts can severely im- pair decoding performance, compounding with the inherent imperfections in GAE's reconstruction. (2) pre-trained decoder can become trapped in local minima, impeding the optimization of the embedding mapping if the INR depends on gradients from the decoder. To address (1)(2), we first warm up the INR with Equation (2) alone to stabilize the mapping, then attach the pre-trained decoder to learn the mapping from z + y. leaving the INR fixed. This decoder- only training phase designed to finetune a case-spect sese-specific decoder that compensates for mapping errors and minimizes cumulative errors.
As for evaluation of the fitting performance with super-high dimensional data, we focus on 3 aspects, namely numeri- cal fidelity, statistical correlation, and bio-conservation. For numerical fidelity, we apply MSE, MAE (Mean Absolute Er- ror) and cosine similarity to measure the significant, subtle and directional errors between the predicted and ground- truth values respectively. Note that only measure numer- ical fidelity on non-zero values considering the zero-inflated distribution of ST. To measure the statistical correlation, we employ Pearson Correlation Coefficient and Spearman's Rank Correlation Coefficient (Spearman's 's p), with both of them ranging from -1 to 1. Lastly, to evaluate how well the prediction preserves cellular heterogeneity and spatial coherence within the microenvironment of the slice, we use the Adjusted Rand Index (ARI) metrie for quantifying cioceesservation leveraging the independent hand-crafted cell type annotations.
In INR regression tasks, norm-based loss functions (e.g., f² norm or MSE) are typically used, as they assume nor- mally (Gaussian) distributed errors. However, for zero- inflated ST data, this assumption is invalid. To handle the imbalanced distribution of zero and non-zero values, we apply Dice Loss (Sudre al., 2017), which is sensitive to class imbalance and treats the regression task quasi- classification one. Dice Loss optimizes for Intersection over Union (IoU) between the prediction map and binary ground truth. Specifically, we use the non-negative half of tanh(-) to map network outputs into a range (0, 1), and compute the intersection using element- wise Hadamard products with the ground truth. To avoid division by 0, Dice Loss is computed as:
4.2. Evaluation Protocol
As is the conventional data flow for INRs, SUICA infers gene expressions with coordinates as inputs after fitting the given (x,y) y) pairs. We apply SUICA to perform degradation- agnostic reconstruction upon ST data under various common degradations, including spatial sparsity, gene drop-out, and noise, which we refer as spatial imputation, gene imputa- tion and denoising respectively for clarity. Accordingly, we follow different evaluation protocols: for spatial imputation, we randomly sample 80% of the spots for training, leaving the rest 20% for evaluation; for gene imputation, we ran- domly mute 70% of the elements in the data matrices; for denoising, a standard Gaussian noise is injected to the raw data. Note that, such reconstruction is completely based on the great approximation power and internal smoothness of INRs without any beforehand knowledge of the degradation type in reference free manner.
2 (ttanh(T)) o sgn(y;:)) +
Laice 1
(3)
Σ tanh(y) + > syy +e'
where sgn(-) denotes the sign function returning O when the input is 0 and +1/-1 when the input is positive/negative. The full reconstruction loss function is then defined as:
Lrecons
Σ (y YE) 2 M
Σ 18 My
Yaal +Ddicee
JM
My
For comparison, we compare SUICA with rule-based well-known INR variants (FFN (Tancik et al., 2020) and SIREN (Sitzmann et al., 2020)), and the SOTA learning-
(4)
where M represents the binary mask for yg > O, and A mitigates Dice Loss's numerical instability.
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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomies
Embryo E16.5
hippocamgus
Ground Truth
SUICA
FEN
SIREN
STAGE
"RPI.P expresses vartous parts of emdoryo"
"AFP hughly activited the comfeyunic mice neuromal developmew"
RRTI vital development of sheletal moscle"
"HPCA expresnes mouse hapoccopps
"MALATI abundounly exprerses in mouse brain"
"G4BRBI strongly expresse mouse hipoccmpus
Figure 4. Visual comparisons of predicted marker gene expressions on MOSTA dataset (Chen et al., 2022a) and Slide-seq Mouse hippocampus dataset (Stickels et al.. 2021). with the descriptions markers attached below results.
based reference-free baseline model STAGE (Li al., 2024).
4.4. Spatial Imputation
Stereo-seq MOSTA SUICA outperforms INR meth- ods (Tancik et al., 2020; Sitzmann al., 2020) and STAGE (Li et al., 2024) in predicting the gene expression of unseen spots in the MOSTA dataset (Chen et al., 2022a) as presented in Table SUICA not only achieves the best MAE, MSE and cosine similarity, but also achieves higher correlation. Besides, SUICA surprisingly manages strengthen the biological signals, even 3.9% above the ARI calculated with the ground truth.
4.3. Implemented Details
We implement SUICA with PyTorch. To construct the k- NN graph for GCN, we set k 5. including the given cell itself. During the pre-training phase of GAE, we use Adam with learning rate of - for 200 epochs. After obtaining low-dimensional cell embeddings, we train the INR with Adam at a lcarning rate of le for 1A epochs to learn the embedding mapping. Subsequently, the INR is frozen, with the pre-trained decoder trained for additional 1/ epochs using Adam with the same learning rate. STAGE (Li al., 2024) is trained till convergence for benchmarking. All experiments can be conducted on NVIDIA RTX 4090.
Figure 4 showcases SUICA's ability to closely predict ground-truth gene expression levels, e.g., for RPLPO gene that highly involves in the ribosomal function and con- sistently expresses across various regions mouse em-
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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transeriptomies
Table 1. Quantitative benchmarking results of spatial imputation on MOSTA dataset (Chen al., 2022a) and Slide-seq V2 Mouse hippocampus dataset (Stickels et al.. 2021). Bold figures are best scores and underlined figures are second-best best. The respective reference ARI scores are 0.312 (Stereo-seq MOSTA) and 0.182 (Mouse hippocampus Slide-seqV2), MAE/MSE: x10 for Stereo-seq MOSTA.
Stereo-seq MOSTA
Mouse hippocampus Slide-segV2
Methods
MAE
MSEL
Cosinet
Pearsont
Spearmant
ARI
MAEI
MSEL
Cosinet
Pearsonf
Speurmant
ARI1
FFN (Tancik et al., 2020)
6,51
1.20
0.706
0,718
0.400
0.143
0.378
0.215
0.499
0,442
0.274
0.0523
SIREN (Sizzmann al., 2020)
7.21
1.31
0.661
0.678
0.247
0.289
0.383
0.216
0.494
0.452
0.248
0.110
STAGE (I.i et al., 2024)
6.52
1.11
0.732
0.747
0.365
0.139
0.351
0.198
0.587
0.483
0.0361
SUICA (Ours)
5.66
0.85
0.797
0.792
0.447
0.343
0.265
0.125
0.752
0.473
0,308
0.111
bryo (Taylor & Pikó, 1992; Ozadam et al., 2023), SUICA accurately predicts this uniform expression pattern, while other methods overemphasize it in specific regions. SUICA also accurately localizes AFP expression to the liver in the mouse embryo (Kwon et al., 2006). Beyond accurately pre- dicting gene expression in unobserved regions, SUICA is also capable for imputation, enhancing the data to better reflect true underlying biological signatures. For example, despite its low expression level in the ground-truth, SUICA successfully imputes SEPT3, a gene involved in neuronal development, effectively restoring this signal in the brain region. These results highlight SUICA's capacity not only to interpolate but also to enrich underlying biological sig- natures, making it a useful tool for imputing and enhancing spatial gene expression data with high fidelity preserved.
GT
SUICA
FFN
SIREN
STAGE
sCury Shendigion Co Schurciille paoi
-Monilepiilln Maac
Spiiaal
sSail
les
"Py 14d
Mouse hippocampus Slide-seqV2 V2
Slide-seye V2 allows to
Figure 5. Spatially visualized comparison on bio-conservations of predicted spots on MOSTA mouse embryo E16.5, Slide-seqV2 mouse hippocampus, and Visium-Mouse brain.
sequence RNAs with a spatial resolution of 10 mm. The Mouse hippocampus dataset (Stickels et al., 2021) is ap- plied to evaluate the effectiveness of SUICA. As shown in Table 1, SUICA achieves a substantially lower MAF and a notably higher cosine similarity (marking a 16.5% improve- ment) compared to other methods. Figure 4 illustrates the ground-truth and predicted gene expression from the bench- marking methods. Like other methods, SUICA accurately predicts the expression of hippocampus marker genes, such as HPCA (Hippocalcin) (Park et al., 2017) and MALATI, a gene abundantly expressed in the mouse brain (Park et al., 2017). SUICA also demonstrates its ability to impute the underlying biological signals, as evidenced by GABRB3, which is consistently and strongly expressed in the mouse hippocampus (Tanaka et al., 2012). These findings suggest that SUICA provides reliable predictions that closely reflect ground-truth data, capturing key gene expression patterns in the V2 platforms.
to the ground-truth data (Luecken et al., 2022). A high bio- conservation score indicates that the predicted data aligns closely with biological reality, capturing key aspects of cell identity, functional states, or tissue structure that are essen- tial for accurate analysis.
In addition to quantitative bio-conservation metrics like ARI, Figure 5 provides visual assessment of dio-conservation by coloring predicted spots according to dominant cell types in individual clusters. For fair comparison, ensured that the number of clusters matched the number of cell-type labels in the dataset.
In E16.5, apart from the highest ARI among benchmarking methods, SUICA also offers the closest spatial visualization of cell types to the ground truth. Compared to other meth- ods, SUICA captures fine-grained structures, such as the choroid plexus in the brain, the gastrointestinal (GI) tract, and the muscle cells surrounding the heart. FFN's (Tancik et al., 2020) predictions appear chaotic, with most cell-type predictions misaligned with the ground-truth labels. While SIREN (Sitzmann et al., 2020) correctly captures general cell types patterns, it overlooks many details detected by
4.5. Bio-conservation Analysis Clusters
of Predicted Cell Type
Biological variance conservation (Bio-conservation) refers to the degree to which a computational method or model pre- serves biologically meaningful features, such as cell types, gene expression patterns, or cellular relationships, relative
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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transeriptomies
Table Quantitative benchmarking results of gene imputation and denoising on MOSTA dataset (Chen et al., 2022a). Bold figures are best scores and underlined figures are second- best. MAE/MSE: x x10-3
Gene Imputation
Denoising Cosinet
Methods
MAEL
MSF.
Cosinet
Pearsont
Spearmant
MAE
MSF.
Pearsont
Spearmant
FPN (Tancik al., 2020)
4.88
0.963
0.731
0.610
0.251
7.90
1.95
0.266
0.285
0.0523
SIREN (Sitzmann et al., 2020)
6.44
1.12
0.675
0.652
0.124
7.91
1.97
0.112
0.103
0.0166
STAGE (Li et al., 2024)
4.69
0.738
0.802
0.705
0.264
7.60
1.66
0.606
0.630
0.182
SUICA (Ours)
4.30
0.724
0.798
0.714
0.317
6.03
0.934
0.733
0.737
0.379
SUICA, such as the connective tissue around the cartilage primordium in the mouse's 's foot region and the cavity sur- rounding the heart. STAGE (Li et al., 2024) identifies sev- eral major cell types in the mouse embryo- -such as liver and brain-but misses smaller, yet important, cell populations.
Table 3. Ablation study on model design of SUICA. MSE: x10-² for Embryo E16.5 while x10 for Human Brain.
Embryo E16.5
Human Brain
Settings
MSEJ
Cosinet
Pearson
MSR
Cosinet
Pearson
Vanilla INR
2.35
0.668
0.653
9.33
0.756
0.747
In the Mouse hippocampus dataset, SUICA also demon- strates its ability to capture both global and fin--grained features, accurately identifying endothelial cells, Ependy- mal cells, and inter-neurons. In contrast, FFN and STAGE fail to detect smaller cell populations, while SIREN mis- classifies inter-neurons. Similarly, on the Visium-Mouse Brain dataset, SUICA uniquely identifies specific cell popu- lations, such as Lateral Ventricle cells and Cortex_5, which are absent from the predictions of other methods.
+AE
1.60
0.789
0.751
11.27
0.695
0.691
+Dice
1.48
0.806
0.347
7.05
0.826
0,800
+Graph
1.47
0.807
0.761
5.67
0.860
0.846
the Embryo E16.5 dataset, the primary performance im- provement comes from the AE and the quasi-classification loss. In contrast, in the Human Brain dataset, adding AE causes a performance decrease, while the GCN has a more significant impact on overall performance. Intuitively, we attribute this difference to the varying spatial sparsity of different ST techniques and GAE alleviates such issue by incorporating spatial context.
4.6. Gene Imputation & Denoising
Since INRs are known for its internal smoothness, SUICA is able to perform channel- wise gene imputation and denoising upon muted or contaminated gene expressions. In this way, SUICA can be seen as reference-free degradation-agnostic restoration method for ST.
5. Conclusion
In this paper, we seek to model the super-high dimensional and sparse nature of ST data, enhancing both spatial reso- lution and gene expression with the smooth prior inherent in INRs. To this end, we present SUICA, powerful INR variant tailored to model ST in a continuous and compact manner. Using a Graph Autoencoder, SUICA maps zero- inflated raw data into a lower-dimensional embedding space, preserving frguerryy details and making the complex embedding mapping more feasible for INRs. The INR fitted embeddings are then decoded to the raw expression with the decoding head preventing the prediction error from accu- mulating. To encourage the sparsity in the final predictions, we leverage a quasi-classiffiation loss term as a regular- izer, preserving both the sparsity and numerical fidelity of non-zero values. Extensive experiments on Stereo-seq, 10x Genomics Visium, and Slide-seq datasets demonstrate SUICA's effectiveness, yielding improvements in both in- silico metries and biologically meaningful analyses, with enhanced spatial resolution and biological signatures. It is firmly believed by us that SUICA will be an inspiring attempt from both INRs and ST perspectives. We encourage readers to check the appendix for complementary insights and extended evaluations.
We benchmark the quantitative results in Table 2 upon Stereo-seq MOSTA (Chen et al., 2022a). Note that the pipeline is exactly identical except for the degradation pat- tern SUICA manages to handle the contaminated inputs in almost all metrics. It also shows that GAE manages make it easier for INRs to model the smoothness in the super-high dimensional space of gene expressions.
4.7. Ablation Study
We showcase two examples to illustrate how the pro- posed modules in SUICA contribute to the final model performance, namely the E16.5 embryo of Stereo-seq MOSTA (spatially dense, 121,756 cells, 26, 1559 genes, with FFN (Tancik et al., 2020) as backbone) and the Visium Human Brain (spatially sparse, 4,910 cells, 36,592 genes, with SIREN (Sitzmann al., 2020) as backbone), under the setting of spatial imputation.
As can be indicated from Table 3, the spatial density and sequencing depth may influence SUICA's effectiveness. For
8

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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transeriptomies
Acknowledgements
Chung, Y., Ha, J. H., Im, K. C., and Lee, J. S. Accurate spatial gene expression prediction by integrating multi- resolution features. In Proceedings of the IEEE/CVE Conference on Computer Vision and Pattern Recognition, pp. 11591-11600, 2024.
This work was supported in part by Three Lakes Foundation, the Canadian Institutes of Health Research (CIHR) [PJT- 180505], the Fonds de recherche du Québec- Santé (FRQS) [295298, 295299]. the Meakins-Chrit tory Research, AMED JJP23tm0524003), JSPS KAKENHI [24KK0209, 24K22318, 22H00529], and JST-Mirai Pro- gram [JPMJMI23GI].
Chair in Respira-
Cybenko, G. Approximation by superpositions of a sig- moidal function. Mathematics of control, signals and systems, 2(4):303-314, 1989.
Impact Statement
Dai, B., Wang, Z., and Wipf, D. The usual suspects? re- assessing blame for vae posterior collapse. In Interna- tional conference on machine learning, pp. 2313 2313-2322. PMLR, 2020.
SUICA offers comprehensive solution the challenges of spatial sparsity, gene drop-out, and noise in spatial tran- scriptomics data analysis. It is compatible with a wide range of sequencing platforms, making it broadly accessible to research teams. Given the high cost of obtaining ST data, SUICA empowers the scientific community to enhance data quality and accelerate the discovery of novel biological in- sights.
Dong, K. and Zhang, S. Deciphering spatial domains from spatially resolved transcriptomics with adaptive graph attention auto-encoder. Nature communications, 13(1): 1739, 2022.
Farah, E. N., Hu, K.K Kern, C., Zhang, Q. Lu, T.-Y,, Ma, Q., Tran, S., Zhang, B., Carlin, D., Monell, A., et al. Spa- tially organized cellular communities form the developing human heart. Nature, 627(8005):854 0644,486,,- 2024.
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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomies
A. Background
Spatially indexngg
Spatial Transcriptomics (ST) is a technique that represent- ing the gene expression individual spots within slice of sample, providing a detailed picture of cellular activity and composition in the ihero-environment. ST sequencing is typically performed on fresh frozen tissue samples and is broadly categorized into two main types: sequencing- based ST and imaging-basee based ST. Despite differences in method- ology, both technologies deliver complementary measure- ments of situ gene expression. The sequeneing- ST technologies, including Stereo-seq, 10x Visium, Slide- seqV2, usually rely spatially indexed surfaces to encode spatial information (Figure 6). These methods employ dif- ferent spatial indexing strategies: 10x Visium uses mi- croarryyrr spotting robot, these spots are 50-200 pm in size, to deliver a unique barcode to a fixed, known location on the surface of slide (Tian al., 2023); Slide-seq ap- plies 10 pm diameter nanobeads for spatial barcoding (Tian al., 2023); Stereo-seq uses DNA nanoball-pattereet ar- rays for in situ RNA capture to create high-resolution spatial transcriptomics sequencing (Chen et al., 2022a). Follow- ing barcoding, mRNA diffuses to spatially indexed primers, undergoes reverse transcription, cDNA amplification, and short-read sequencing. This process generates transcript reads paired with barcode sequences, localizing each de- tected transcript specific pixels.
Microarray
Nanoballs
Bead
(b)
Capturing mRNA from each spot
0000
(c)
Sequencing
0000
0000
(d)
Data
coordinate
Expression
x
y
Gene
Gene 2 ...
Gene
For the imaging based-ST methods, RNA molecules are specifically tagged with fluorescent probes by complemen- tary hybridization. These probes are then imaged using fluorescence microscopy. MERFISH, for instance, maps RNA molecules with binary barcodes encoded with error- correcting codes to ensure accuracy. Sequential rounds of fluorescence imaging detect the presence absence of fluo- rescence, decoding the RNA identity and precisely mapping molecules to their spatial locations.
Spot 1
10.1 4.5
10
Spot 2
21.4
9.7
0
5
...
Spot M
8.2 23.1
0
0
Figure 6. Workflow sequencing-based spatial transcriptomies sequencing. (a) Spatially indexing, with microarrayer for 10x Vi- sium, nanobead technology for Slide-seq and nanoball-bas based strategy for Stereo-seq; (b) Placing the slice of tissue on the in- dexed surface and capturing mRNA from each spot; (c) Collecting spots and conducting sequencing: (d) The generated data matrix with gene expression and location information for each spot.
We also want to mention the fact that ST data are not al- ways delivered jointly with histology images as reference. For example, Slide-seq. DBiT-seq. MERFISH offer high- resolution spatial gene expression data but lack correspond- ing histological images. Even in cases where histology is available, challenges in precise alignment between histolog- ical features and ST data can arise due to tissue deformation, sectioning artifacts, or registration inaccuracies. These limi- tations highlight the importance of designing reference-free models that do not rely on histological guidance, making them more broadly applicable and robust across different ST platforms and experimental conditions.
SIREN (Sitzmann et al., 2020) outperforms FFN (Tancik et al., 2020) the backbone of SUICA, while for high- resolution ST slices, FFN prevails. To better showcase such phenomenon, we provide two representative cases of the mouse embryos E9.5 and E1 Table 4. We globally set the angular velocity w of the periodical activation function in SIREN (Sitzmann al., 2020) 30 which is adopted by default and the mapping size of the Random Fourier Features as 256. For more information for these magic numbers, please refer to the respective papers. Note that we do not intend to showcase any superiority among INR variants. Since we adopt a modular design that is agnostic to specific INR implementations, we are able to take advantage of different implementations to meet varying requirements.
B. Model Details & Insights
FEN vs. SIREN As mentioned the main body, it is observed that when applied to ST with low spatial density,
12

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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transeriptomies
abilistic latent space, typically modeled Gaussian distri- butions. While VAEs offer generative capabilities through variational inference and learn smoother, more continuous latent spaces compared to AEs, they often produce blurry outputs (Zhao et al., 2017) and struggle to match AEs in re- construction accuracy (Dai et al., 2020). These issues might lead to ambiguity in gene expression that is not wanted. These limitations stem from the inherent trade-off between the quality of latent representation learning and reconstruc- tion fidelity in VAEs. Although variants like beta-VAE (Hig- gins et al., 2017) allow for adjustable weighting between these objectives, the issue persists. In our setting, where numerical fidelity and reconstruction accuracy take prece- dence over generative and latent representation quality, we select AE over VAE for SUICA's design, leveraging its ability to accurately reconstruct high-dimensional raw gene expression data.
Table Quantitative results of SUICA empowered by different INR backbones. MSE: x10-
Emdeyo F9.5
Eudryoo F76.5
Backbones
MSEL
Covinet
Pearsont
MSEJ
Cosneet
Peanoo"
PPN (Tancik al., 2020)
0.55
0.752
0.799
1.47
0,807
0.761
SIREN (Sizzmano al., 20201
0.52
0.766
0314
162
0.786
0.745
Table 5. Quantitative comparison on Stereo-seq MOSTA dataset (Chen et al., 2022a) between SUICA and NGP (Müller ct al., 2022). MAE/MSE: x10-?.
Methods
MAR
MSEJ
Cosinet
Pearsont
Spearmant
NGP (Mäller et al., 2022)
6.39
0.729
0.742
SUICA
5.66
0.85
0.797
0.792
0.447
Explicit Alternatives Recently, researchers are employ- ing explicit or hybrid alternatives (Müller et al., 2022: Chen al., 2022b) to replace the fully implicit represen- tations (Tancik et al., 2020; Sitzmann et al., 2020; Lindell et al., 2022) for a faster inference speed. For more compre- hensive benchmarking, we here compare SUICA with the trainable hash-based encoding (Müller et al., 2022) (denoted as NGP) on Stereo-seq MOSTA dataset (Chen et al., 2022a). The quantitative results are shown in Table 5. For the imple- mentation, we re-compile the library of t tnn--cuda nn to support float32, to align with other methods.
Reconstruction Loss In SUICA, we train the INR and the decoder separately to minimize the accumulated prediction error, on which we would like to further offer some insights. Obviously, a pre-trained decoder is necessary; otherwise the decoder would become a trivial extension of the INR. It is also easy to understand that such decoder requires finetuning to bridge the gap between the pseudo ground-truth embed- dings Zy: and the predicted embeddings 2. The problem that prevents this two-stage training strategy from merging into one end-to-end stage is that when using Lrecoms through the decoder to supervise the INR, the loss will become unstable. We attribute this observation to the optimizing objectives of femba and Lresoms are somehow contradictory. Though this problem can be alleviated by carefully adjusting the training- relevant hyper-parameters. we opt to adopt the two-stage training strategy for simplicity and stable performance.
CNN It is easy to notice that the network architecture in SUICA is almost fully MLP-based except for the GCN. Such architecture does not explicitly involve any local in- ductive bias that CNNs usually do. The underlying reason for doing so is that along the sequencing depth of ST, it is assumed that the channels (i.e., specific gene segments) are permutation-inee -nvvrriinn even though there might be some statistical correlations. In this way, it is not supposed to rely on the context information along the sequencing depth, but for the correlation among cells, we model such context by the GCN.
C. STAGE
In the main text, we involve STAGE (Li et al., 2024) for both quantitative and qualitative comparisons as a baseline for histology-free spatial imputation (densification) of gene expressions. In addition to solely showing the superiority of SUICA, we would like to shed some light on this matter.
PCA In SUICA, we reduce the dimension of raw gene expressions from thousands to 32 by the encoder of GAE, which is recovered later by the decoder. The function of the GAE could be replaced by PCA (Principal Component Analysis), by a pre-fitted linear projection. However, it obviously challenging for the linear transformation of PCA and inverse PCA to model the complex distribution of ST, leading to a great loss in the reconstructed signal. Empirically, it is observed that when encoding and decoding with PCA and inverse PCA, the IoU between the predicted zero maps and the ground truth decreases to O, indicating a complete failure of recovering the sparsity, while SUICA maintains an IoU of around 0.9.
Regarding the benchmarking task of imputation, we identify two critical sub-tasks, namely the reconstruction of gene expression and the spatial mapping. STAGE models im- putation as a generative task, where an AE is adopted to self-regress the raw representations while enforcing the la- tent as the corresponding 2D coordinates. In this way, the two sub-tasks are coupled as one. Considering the chal- lenges posed, the closely coupled paradigm may not lead to satisfactory results. On the contrary in SUICA, the two sub-tasks are well decoupled, where the spatial mapping is performed by the INR while the description as well as the
VAE Variational Autoencoders (VAEs) are a common al- ternative to AEs, designed to generate samples from a prob-
13

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SUICA: Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomies
Table Quantitative henchmarking results of spatial imputation on two Visium ST cases (Palla al., 2022; Wei al., 2022). Note that for Visium-Huume Brain, there is no annotation of cell type for the evaluation of ARI, while reference ARI is 0.428 for Mouse Brain. Bold figures are best scores and underlined figures are second-bess MAE/MSE: x10 2 for Human Brain while x10* for Mouse Brain.
Visium- Human Brain
Visium-Mouse Brain
Methods
MAEL
MSEL
Cosine"
Pearsonf
Spearmant
MAEL
MSEL
Cosine?
Pearson"
Spearman
ARIT
FEN (Tancik et al., 2020)
5,76
0.881
0.772
0.786
0.402
5.95
5.85
0.832
0.741
0.581
0.000587
SIREN (Sitzmann et al.. 2020)
6,58
0,933
0.756
0.747
0.196
5.35
4.29
0.878
0.804
0.647
0.359
STAGE (Li et al., 2024)
6.19
0,805
0.795
0.772
0.223
4.55
3.20
0.918
0,825
0.666
0.140
TRIPLEX (Chang et al., 2024)
4.75
0.560
0.881
0.850
0,319
9.35
14.0
0,00
-0.00682
-0.00715
0.358
UNIv2 (Chen al., 2024)
7.30
1.41
0.723
0.633
0.129
6.94
7.88
0.790
0.631
0.425
0.228
SUICA (Ours)
4.99
0.567
0.860
0.846
0.445
3.68
2.45
0.932
0.800
0.660
0.393
reconstruction is taken care of the GAE. The progressive training paradigm also guarantees that each module is doing the assigned job, keeping the coupling at minimum.
GT
SUICA
BBC
EPDC
Epicardial
LEC
Neuronal
Pericyte
VEC
VIC
VSMC
WRC
According to the experimental results, SUICA not only achieves better quantitative scores for benchmarking, but also exhibits much richer bio-conservation for downstream applications.
aEadocardial
Fibro
adHibbro
ncCM
vCM
EEnocardial
VFibro
Figure 7. Spatially visualized predicted spots on MERFISH human heart (Farah et al., 2024).
D. Human and Mouse Brains Visium
Table 7. Quantitative results of spatial imputation MERFISH human heart (Farah et al., 2024). MAE/MSE: x x10-1
Different from Stereo-seq datasets, Visium is a lower- resolution but more affordable ST technology, on which we additionally compare the model performance. The exper- iments are conducted under the task of spatial imputation. Due to the lack of independent cell-type annotations, ART is not applicable in Visium-Human Brain dataset (Wei et al., 2022). On the Visium-Mouse Brain dataset (Palla et al., 2022), SUICA demonstrates improved ARI, which indi- cates that the predictions can more accurately describe the underlying cellular heterogeneity. Note that for readers' information, we also involve SOTA histology-aider imputa- tion methods TRIPLEX (Chung al., 2024) and UNI (Chen et al., 2024) for comparison, which additionally has access to extra reference. We have also found that TRIPLEX is rather sensitive to dptasct- characteristics.
Methods
MAEI
MSE
Cosineh
Pearson|
Spearman"
FFN (Tancik et al., 2020)
6.06
5.86
0.8/0
0.717
0.551
SIREN (Sitzmann al., 2020)
5.54
4.98
0.759
0.606
STAGE (1.1 et al., 2024)
5.48
5.10
0.870
0.709
0.558
SUICA
4.65
3.92
0.892
0.718
0.548
annotate the cell types for predicted spots of MERFISH data using dominant cell types in individual clusters. We visu- alize the spatial bio-conservation by coloring the cell types in Figure 7. The results show that the SUICA predictions mimic the ground-truth evidenced the capability of SUICA to predict unseen spots in the MERFISH datasets.
F. Ablation on Resolution
To help with the data-efficieecy analysis of SUICA, we per- form experiments under the task of spatial imputation with E16.5 embryo of Stereo-seq MOSTA (Chen et al., 2022a), the results of which are shown in Table 8. The percentage refers to the the proportion of training samples with regard to the whole dataset, while the test set remains the same (20% of all spots).
E. MERFISH
MERFISH an imaging-based based ST technique that enables the highly multiplexed imaging of RNA molecules in cells while maintaining their spatial context. Compared to the V2,, 10x Visium, and Stereo-seq technologies used in our manuscript, MERFISH data can quantify significantly fewer genes for individual cells. In Table 7, we evaluate the performance of SUICA using a human heart MERFISH dataset (Farah et al., 2024) with 228.635 cells and 238 genes, with the setting of spatial imputation. SUICA achieves sig- nificant lower mean absolute error (MAE) and mean square error (MSE), and the cosine similarity is at least 2.28% higher than other methods, showing its high numerical fi- delity. Following the bio-conservation analysis scheme, we
G. Data Pre-processing
To enable the evaluation of Pearson Correlation Coefficient and Spearman's 's Rank Correlation Coefficient, before the experiments we remove empty rows and lines to make sure each spot is at least expressed by a limited number of genes.
Note that the outputs of conventional INR-based tasks are
14

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SUICA:
Learning Super-high Dimensional Sparse Implicit Neural Representations for Spatial Transcriptomies
Table Ablation study on the data-cfficiency of SUICA, % rep- resents the proportion of the spots used for training while test set remains 20%. MAE/MSE: x10-
%
MAE]
MSEJ
Cosinet
Pearson
80%
8.01
1.47
0.807
0.761
60%
7.96
1.52
0.801
0.752
40%
8.00
1.59
0.790
0.739
20%
8.14
1.62
0.786
0.738
strictly bounded, where the linear layer is usually clamped compressed by sigmoid. For ST data, we do not have such assumption and only employ ReLU as the final activation. Therefore, as a common practice in ST analysis (Melsted et al., 2021; Hwang et al., 2018), we remove the genes whose expressions are overly high and normalize each cell by total counts over all genes of the cell so that every cell has the same total count after normalization while keeping the original sparsity.
H. Limitations
One of the significant limitations SUICA roots in the case-by-case training paradigm of INRs. Similar to NeRF (Mildenhall al., 2020), for each case, i.e., each ST slice, need to training new model from seratch. Considering the overwhelming heterogeneity of different ST data, we temporarily compromise in this issue but ac- knowledge that incorporating domain knowledge to make SUICA generalizable is an interesting future work.
Another potential limitation is that SUICA prefers high- quality ST data, in terms of both spatial density and quencing depth, can be indicated from the experimental results we provide. SUICA exhibits the most performance gain in Sterco-seq compared to other platforms and when the gene expressions no longer meet the super-high dimensional sparse assumption, SUICA will degenerate to vanilla INR, as is consistent with the results of MERFISH Table 7.
15


Reviews:
[{"review_id": "HKotuqD5Sf", "reviewer": "Reviewer_c5Qu", "summary": "The authors introduce SUICA, an implicit neural representation-based ST prediction method, which demonstrates another way of gene/spatial imputation for ST. This provides a different take on ST inference frameworks, which predominantly were based on image-based or neighborhood-cell based without relying on nearby coordinates.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are sincerely thankful for Reviewer C5Qu's comments and suggestions, and we hope the following responses could help to resolve the concerns.\n\n1. **Pipeline of SUICA**: Thank you for raising the issue about the confusion of data flow. We will modify the manuscript to help readers better understand our data flow. Here, we summarize the data flow according to Section 4.2 of the manuscript into the following points: \n\n   - Spatial imputation: Take an ST slice {($x$,$y$)} as an example, whose data spots are split into a training subset {($x_\\textrm{train}$,$y_\\textrm{train}$)} and a test subset {($x_\\textrm{test}$,$y_\\textrm{test}$)}. With the training subset, we train the self-regressing GAE, whose encoder encodes all $y_\\textrm{train}$ as $z_\\textrm{train}$. Then we start fitting the INR to approximate the mapping from $x_\\textrm{train}$ to $y_\\textrm{train}$ (with intermediate supervision of $z_\\textrm{train}$). For test-time inference, SUICA **only needs $x_\\textrm{test}$** to infer corresponding $y_\\textrm{test}$.\n   - Gene imputation: We randomly mute a part of the gene expressions of the data matrix, fit SUICA with all of the data, and infer **all of the $x$**. We expect the prediction $\\hat{y}$ to be imputed. \n   - Denoising: The overall data flow is basically the same as gene imputation, but with injected noise as the degradation.\n\n   As a result, **the required input at test time is only the coordinates** and we do not make any assumption of the spatial distribution (e.g., a uniform distribution as regular grids), which means SUICA is able to be both trained and tested with unstructured coordinates. We will follow the reviewer's suggestion to elaborate this issue more clearly in the revised manuscript.\n\n2. **Image-based super-resolution**: Thank you for suggesting benchmarking against other image-based super-resolution methods. We additionally compare SUICA with TESLA and UNIv2[1] with *Visium-Human Brain* and *Visium-Mouse Brain*. Note that SUICA is reference-free and does not incorporate any image information. We also report IoU of non-zero areas for reference to emphasize the predicted sparsity. MAE/MSE: $\\times 10^{-1}$.\n   | Human Brain | MAE$\\downarrow$ | MSE$\\downarrow$ | Cosine$\\uparrow$ | Pearson$\\uparrow$ | Spearman$\\uparrow$ | IoU$\\uparrow$ |\n   | - | - | - | - | - | - | - |\n   | TESLA | 0.280 | 0.00533 | 0.857 | 0.828 | 0.421 | 0.859 |\n   | UNIv2 | 0.730 | 0.141 | 0.723 | 0.633 | 0.129 | 0.388 |\n   | SUICA | 0.498 | 0.0567 | 0.860 | 0.846 | 0.445 | 0.931 |\n   | **Mouse Brain** | | | | | | |\n   | TESLA | 4.00 | 2.58 | 0.877 | 0.786 | 0.619 | 0.619 |\n   | UNIv2 | 6.94 | 7.88 | 0.790 | 0.631 | 0.425 | 0.0 |\n   | SUICA | 3.68 | 2.45 | 0.932 | 0.800 | 0.660 | 0.655 |\n\n3. **Held-out rates**: Thank you for emphasizing the necessity of ablation study with different training-test ratios. Under the setting of spatial imputation (or \"super resolution\"), we evaluate the performance of SUICA with different held-out rates (the proportion of spots kept for test) with *Embryo E16.5*. According to the results, we find there is no significant performance drop when the held-out rate raises. MAE/MSE: $\\times 10^{-2}$.\n   | Held-out Rates | MAE$\\downarrow$  | MSE$\\downarrow$  | Cosine$\\uparrow$ | Pearson$\\uparrow$ |\n   | - | - | - | - | - |\n   | 20% | 8.01 | 1.47 | 0.807 | 0.761 |\n   | 40% | 7.98 | 1.54 | 0.799 | 0.748 |\n   | 60% | 7.94 | 1.52 | 0.802 | 0.751 |\n   | 80% | 8.13 | 1.66 | 0.781 | 0.735 |\n\n4. **Generalizability**: As is discussed in the limitations of Appendix G (Line 770), INRs are case-by-case optimized. At the current stage, we consider this limitation not fatal since the data heterogeneity across different ST platforms prevents the construction of a unified dataset and many SOTA methods also adopt the case-by-case paradigm, e.g, STAGE and SEDR. But we do acknowledge that incorporating INRs for modeling ST in a generalizable way is a promising direction, and there have been some preliminary attempts for images and 3D assets [2].\n\n5. **Baselines from HEST**: Thank you for suggesting the benchmarking against more image-based imputation methods. Here we have selected UNIv2 [1] and please refer to the previous table for the results. \n\n[1] Chen, Richard J., et al. \"Towards a general-purpose foundation model for computational pathology.\" *Nature Medicine* 30.3 (2024): 850-862.\n\n[2] Ma, Qi, et al. \"Implicit Zoo: A Large-Scale Dataset of Neural Implicit Functions for 2D Images and 3D Scenes.\" *The Thirty-eight Conference on Neural Information Processing Systems Datasets and Benchmarks Track*.", "claims_and_evidence": "I think the authors provide claims that are always backed up by evidence.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "No theoretical claims provided", "experimental_designs_or_analyses": "Yes.", "relation_to_broader_scientific_literature": "I think this reflects a good timely contribution to the field.", "essential_references_not_discussed": "There are few key references on \"super-resolution\" of ST that I felt were not discussed, mostly based on image-based algorithms.\n\n[1] Hu, Jian, et al. \"Deciphering tumor ecosystems at super resolution from spatial transcriptomics with TESLA.\" Cell systems 14.5 (2023): 404-417.\n[2] Zhang, Daiwei, et al. \"Inferring super-resolution tissue architecture by integrating spatial transcriptomics with histology.\" Nature biotechnology 42.9 (2024): 1372-1377.", "other_strengths_and_weaknesses": "I think this is a timely contribution to the field, which has been dominated by image-based prediction or purely cell-based GCN-based approach. However, I think a bit more efforts are required to explain/convince readers who are predominantly familiar with current methods and not so with INR-based. If the authors are able to address few additional suggestions, I am willing to increase the score.\n- Figure 2 is great for explaining the training pipeline, but I think there needs to be an illustrative description (also corresponding detailed explanation in the text) on how the test-time inference is carried out. What are the required inputs? is it just the subset of coordinates? Is it just the subset of the cell-level expressions? Can these be randomly distributed (as in the experiments)? This will help the readers understand the situation in which SUICA is available or not\n- There are a few super-resolution ST approaches, although they are based on image-based inputs (TESLA, ISTAR). I would like to see comparison with them if possible (TESLA should be easier to implement, ISTAR maybe not given the rebuttal timeframe)\n- Ideally I want to see a bit more ablation in the evaluation. The authors experiment with 80% randomly sampled spots for training and remaining 20% for evaluation. What if it's on the lower-end of the training? Such ablation experiments will help readers understand amount of data required to ensure \"decent\" imputation performance.\n- Can the authors comment on whether SUICA needs to be trained everytime for each ST plane? Based on my understanding, it does seem that SUICA needs to be fitted for each plane. However, for the embryo dataset, where it seems to consist of serial sections of ST, maybe only few sections can be used for training and the other purely for evaluation? In other words, I would love to see generalization performance/argument for SUICA.\n- It would be great to see a bit more comparison with image-based ST imputation. TRIPLEX is great, but it uses underpowered vision encoders that might not be as great for downstream imputation performance. Maybe the authors can reference some baselines from recent HEST study [1]?\n\nReferences\n[1] Jaume, Guillaume, et al. \"Hest-1k: A dataset for spatial transcriptomics and histology image analysis.\" Advances in Neural Information Processing Systems 37 (2024): 53798-53833.", "other_comments_or_suggestions": "See above", "questions_for_authors": "See above", "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": "2HXxoHXD28", "reviewer": "Reviewer_Q3xk", "summary": "The paper proposes SUICA, a method for continuous modeling of spatial transcriptomics data by combining a graph-augmented autoencoder (GAE) with implicit neural representations (INRs). The key idea involves compressing high-dimensional, sparse gene expression data into a low-dimensional latent space using a GAE, then mapping spatial coordinates to this latent representation via an INR (using either FFN or SIREN), and finally decoding back to the full gene expression space. To address sparsity in spatial transcriptomics, the authors reformulate the reconstruction loss with a quasi-classification approach based on Dice Loss. Experimental evaluations on multiple datasets (including Stereo-seq MOSTA, Slide-seqV2 mouse hippocampus, and Visium-Mouse brain) demonstrate improvements in spatial and gene imputation as well as bio conservation.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the efforts of Reviewer Q3xk by offering comments and suggestions, which would definitely help us to improve the manuscript for its potential readers. We are also afraid that there is some critical misunderstanding towards SUICA, so we hope this rebuttal could lead to a better alignment and kindly ask Reviewer Q3xk to re-evaluate the technical contribution of SUICA.\n\n1. **SUICA does not use PCA and operates in full gene space**: We apologize for the confusion. To clarify, the pipeline of SUICA does **NOT** use PCA for dimensionality reduction at any stage and both the input and output of the GAE are in full-gene space. During test time, it predicts gene expression profiles of the same dimensionality as the raw input, using only spatial coordinates.\n\n    The discussion about PCA in Appendix B is intended to solve the potential concern of whether the encoder and decoder of GAE could be replaced by PCA and iPCA. However, this is not part of SUICA’s pipeline. The dimension reduction and reconstruction in SUICA are respectively conducted by the encoder and decoder of GAE. Therefore, the notion that input to GAE is dimensionality-reduced input via PCA is **incorrect**. We apologize again for being unclear and will make revisions to Section 4.3 and Appendix B to prevent further misunderstanding. \n\n2. **The use of GAE**: The effectiveness of GAE has been discussed and verified in Section 3.2.2 and Section 4.7. We agree with the reviewer's intuitive idea that the root of performance is by leveraging neighborhood information and exploiting the correlation among data. But we would like to emphasize that the INR in SUICA also plays an important role in the continuous modeling of ST, making it possible to query the gene expressions at arbitrary locations, instead of performing denoising. To this end, we believe that the technical contributions of SUICA are non-trivial.\n\n3. **Regression target for SIREN & FFN**: The regression target for SIREN and FFN is the raw gene expressions without being processed by PCA, as is also the target of SUICA.\n\n4. **Data preprocessing**: As is mentioned in Appendix F, the preprocessing includes cells and gene filtering and count depth scaling. We first filtered spots with less than 200 genes expressed and genes that were detected in less than 3 spots. Then we removed spots whose total number of raw counts is lower than the threshold (set as 200 empirically). Then we performed count depth scaling (``sc.pp.normalize_total()``) following the tutorial of scanpy. \n\n5. **Training-relevant hyperparameters**: SUICA applies staged training to obtain satisfactory results with super-high dimension data, which to our knowledge is acceptable and far from being extremely complex. The training-relevant hyperparameters are set according to empirical results, with which SUICA is able to perform stably across varying ST platforms. It is worth noting that, we use the identical set of hyperparameters for ST data from different platforms.", "claims_and_evidence": "- The authors claim that SUICA outperforms conventional INR variants and existing methods (e.g., STAGE) in terms of numerical fidelity, statistical correlation, and biological conservation. Experimental results are presented over several datasets with metrics such as gene MAE, MSE, cosine similarity, correlations, and ARI of cell type. SUICA achieves new SOTA in most cases. \n- However, while the authors claim their approach uniquely handles super-high dimensionality, the model input to GAE is actually dimensionality-reduced input via PCA  (Appendix B, line 699). This critical detail is omitted from the main text, and Figure 2 is extremely misleading as it depicts identical input and output sizes for the GAE.", "methods_and_evaluation_criteria": "The use of GAE is interesting, but it's unclear whether performance improvements stem from integrating expression from neighboring spots to reduce noise, which is an already well-established approach in the field. For example the original MOSTA paper uses bin 50 (summing 50 spots) for downstream analysis.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "Given that the GAE input is PCA-transformed, the paper would benefit from a detailed ablation study on PCA's contribution to the results.", "relation_to_broader_scientific_literature": "- The paper adequately discusses relevant works in the field.\n- The work presents an interesting use of INR in spatial biology, which is novel to the best of review's knowledge.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "N/A", "questions_for_authors": "- Is the regression target of FFN or SIREN baseline also PCA transformed? \n- How is data preprocessing conducted, including normalization, qc's.\n- Given the extremely complex model with three training stages, how were hyperparameters selected?\n- Can you please share your thoughts on previous sections?", "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": "1X8jun3WzS", "reviewer": "Reviewer_RB2n", "summary": "This paper proposed a method for jointly denoising and improving resolution for spatial transcriptomics data.\n\n## update after rebuttal\n\nI raised score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We want to thank Reviewer RB2n for the valuable comments and suggestions, which have greatly helped us improve the quality of our work.\n\n1. **Definition of imputation**: Thank you for pointing out this issue. In the manuscript, we use \"spatial imputation\" to refer to predict gene expressions at 2D locations not included in the training subset. This setting can also be seen as \"super resolution\" or \"densification\". As for the task of \"gene imputation\", we randomly mute a part of gene expressions and apply SUICA to do the reconstruction, so that such muted values could be restored. This setting is more relevant to \"reconstruction\". Please refer to our response to Reviewer C5Qu for the detailed data flow.\n\n2. **Latent code**: $z_\\textrm{gt}$ refers to the encoded latent code by the GAE's encoder; $\\hat{z}$ represents the fitted $z$ by the INR. The goal of our model is to map spatial coordinates of spots to their corresponding gene expressions. However, as shown in the ablation study (Table 3), a vanilla INR can not perform well due to the dimensional challenge of ST. Therefore, we enforce INRs to learn the mapping between spatial coordinates and lower dimensional embeddings $z_\\textrm{gt}$. Ideally, $\\hat{z}$ should accurately align with $z_\\textrm{gt}$ which allows GAE’s decoder to seamlessly restore the gene expressions. We have visualized the embeddings of $\\hat{z}$ and $z_\\textrm{gt}$ using UMAPs. The results show that the INR can effectively reproduce the embeddings learned by GAE’s encoder, allowing SUICA to infer the embeddings of arbitrary locations for spatial imputation. Please kindly refer to [this anonymous link](https://imgur.com/a/RlYePtD) for the visualization.\n\n3. **Why not use GNN only**: We agree with you that GNN can encode ST data. However, we\n    - apply GAE's encoder to gain low dimensional embeddings to enhance INR (Table 3),\n    - employ INRs to map spatial coordinates of arbitrary spots to their embeddings,\n    - and use GAE's decoder to restore the gene expression from embeddings.\n\n     Without these key factors, using GNN only is difficult to map spatial coordinates to gene expression.\n\n\n4. **Detailed ablation on loss**: For the reviewer's information, we break down the loss terms leveraged in SUICA and offer a more detailed ablation on the identical data in Table 3. To better show the loss combinations' effect on the sparsity, we additionally report IoU of non-zero areas for reference. We make our choice as MSE+MAE+Dice to ensure a relatively stable trade-off between numerical fidelity and statistical correlation.\n\n   | Embryo E16.5        | MSE $\\times 10^{-2}$$\\downarrow$ | Cosine$\\uparrow$ | Pearson$\\uparrow$ | IoU$\\uparrow$   |\n   | ------------ | --------- | ------ | ------- | ----- |\n   | MSE          | 1.23      | 0.843  | 0.522   | 0.558 |\n   | MSE+Dice     | 1.22      | 0.843  | 0.572   | 0.674 |\n   | MSE+MAE      | 1.60      | 0.789  | 0.751   | 0.937 |\n   | MSE+MAE+Dice | 1.48      | 0.806  | 0.747   | 0.928 |\n\n   | Human Brain  | MSE $\\times 10^{-3}$$\\downarrow$ | Cosine$\\uparrow$ | Pearson$\\uparrow$ | IoU$\\uparrow$   |\n   | ------------ | --------- | ------ | ------- | ----- |\n   | MSE          | 10.50     | 0.723  | 0.507   | 0.532 |\n   | MSE+Dice     | 10.72     | 0.719  | 0.574   | 0.696 |\n   | MSE+MAE      | 11.27     | 0.695  | 0.691   | 0.933 |\n   | MSE+MAE+Dice | 7.05      | 0.826  | 0.800   | 0.925 |\n\n5. **Resolution**: We agree with Reviewer RB2n that the performance is relevant to the resolution of data. However, it is somehow difficult to ablate such influence because in practice, the resolution is usually coupled with other factors that will have a potential effect on the performance. For instance, a high spatial resolution usually leads to a high drop-out rate, which is also challenging for modeling. We have performed experiments with different held-out rates in the response to Reviewer Q3xk, and we hope the results of such artificial simulation could be of some insights.\n\n6. **Comparison with SEDR**: As is suggested, we benchmark against SEDR. Since SEDR can not perform spatial imputation, we report its gene imputation and denoising performance. However, when attempting to let SEDR reconstruct the raw gene expressions of *Stereo-seq MOSTA* (the default use case is to adopt PCA-reduced expressions), OOM is observed with RTX 4090, so we instead report the scores on *Visium-Human Brain*. MAE/MSE: $\\times 10^{-2}$.\n   | Gene Imputation | MAE$\\downarrow$  | MSE$\\downarrow$   | Cosine$\\uparrow$ | Pearson$\\uparrow$ | Spearman$\\uparrow$ |\n   | - | - | - | - | - | - |\n   | STAGE | 7.13 | 1.69  | 0.692  | 0.667   | 0.110    |\n   | SEDR | 37.0 | 91.4  | 0.0    | 0.048   | 0.076    |\n   | SUICA | 5.64 | 0.699 | 0.835  | 0.829   | 0.391    |\n   | **Denoising** |  | | | | |\n   | STAGE | 5.49 | 0.757 | 0.814  | 0.740   | 0.140    |\n   | SEDR | 47.5 | 90.8  | 0.0255 | 0.117   | 0.0793   |\n   | SUICA| 4.20 | 0.582 | 0.860  | 0.751   | 0.201    |", "claims_and_evidence": "Yes, their claim is supported.", "methods_and_evaluation_criteria": "I have a question about problem definition, which is discussed in the question section.", "theoretical_claims": "Yes, they look good.", "experimental_designs_or_analyses": "Yes, I have checked the soundness.", "relation_to_broader_scientific_literature": "Yes, their contirbution is important for computational biology.", "essential_references_not_discussed": "Yes, I think they need to include more baselines. Please see the questons.", "other_strengths_and_weaknesses": "I have raised several questions in the Question section.", "other_comments_or_suggestions": "The presentaiton looks good.", "questions_for_authors": "This paper is very interesting, but I have several questions about problem definition, model implementation, and evaluations. I will consider raising my score if the authors fully address my concerns.\n\n1. I am confused about the imputation function. From my understanding, imputation in ST data means predicting unmeasured genes, which always need external resources, such as reference scRNA-seq data, to predict those unseen ones. Could you please explain how to use your pipeline to impute gene expression? Otherwise, is it the same as denoising?\n\n2. What is the difference between z_gt and z^hat in their implementation, could the authors visualize their latent representations and discuss potential differences? I think we can directly use a GNN to encode spatial transcriptomic data with the awareness of spatial context, what is the drawback of this design?\n\n3. The ablation studies seem different from their presentations, as their listed loss functions include L_dice,  L_mae, and L_mse, but in section 4.7, the presented results are different.\n\n4. The authors perform experiments based on both ST data with different resolutions, such as Slide-seq V2 and Visium. It will be interesting to investigate the performance differences across data with different resolutions. Could the authors elaborate on this point?\n\n5. The cited link of STAGATE seems wrong, as the correct one should be: https://www.nature.com/articles/s41467-022-29439-6. I think it is also interesting to include a comparison with recent baselines such as SEDE (https://genomemedicine.biomedcentral.com/articles/10.1186/s13073-024-01283-x)", "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:

Meta-review: This paper proposes SUICA that models spatial transcriptomics data in a continuous and compact manner by using Implicit Neural Representations (INRs). The reviewers had concerns regarding missing key references, unclear explanation of test-time inference, limited evaluations scope, lack of baselines. The authors addressed most of the concerns, and the strengths of the paper outweighs the drawbacks.
Decision: Accept (poster)