A Resolution-Agnostic Geometric Transformer for Chromosome Modeling Using Inertial Frame
Authors: Yize Zhou
Haorui Li
Shengchao Liu
Venue: ICLR 2026 Poster (2026)
Keywords: Chromosome Modeling
Inertial Frame
Resolution-Agnostic
3D Transformer
AI for Biology
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Chromosomes are the carriers of genetic information. Further understanding their 3D structure can help reveal gene-regulatory mechanisms and cellular functions. However, high-resolution 3D structures are often missing due to the high cost and inherent noise of experimental screening. A standard pipeline for reconstructing the chromosome 3D structure first applies the single-cell Hi-C high-throughput screening method to measure pairwise interactions between DNA fragments at different resolutions; then it adopts computational methods to reconstruct the 3D structures from these contacts. These include traditional numerical methods and deep learning models, which struggle with limited model expressiveness and poor generalization across resolutions. To handle this issue, we propose InertialGenome, a novel transformer-based framework for robust and resolution-agnostic chromosome reconstruction. InertialGenome first adopts the inertial frame for the pose canonicalization. Then, based on such an invariant pose, it proposes a Transformer with geometry-aware positional encoding, leveraging Nyström estimation. To verify the effectiveness of InertialGenome, we conduct experiments on two single-cell 3D reconstruction datasets with four resolutions, reaching superior performance over all four computational baselines. Additionally, we observe that the 3D structure reconstructed by InertialGenome is more in line with the results of real experimental results on two functional verification tasks. Finally, we leverage InertialGenome for cross-resolution transfer learning, yielding up to a 5\% improvement from low to high resolution. The source code is available at https://github.com/yize1203/InertialGenome.

Published as a conference paper at ICLR 2026
A RESOLUTION-AGNOSTI GEOMETRIC TRANS- FORMER FOR CHROMOSOME MODELING USING INER- TIAL FRAME
Yize Zho Haorui Si-2,,hhngehaa Liu
Wave Intelligence Lab
2 Department of Computer Science and Engineering. The Chinese University of Hong Kong
ABSTRACT
Chromosomes are the carriers of genetic information. Further understanding their 3D structure can help reveal gene-regulatory mechanisms and cellular functions. However, high-resolution 3D structures often missing due to the high cost and inherent noise of experimental screening. A standard pipeline for recon- structing the chromosome 3D structure first applies the single-cell Hi-C high- throughput screening method to measure pairwise interactions between DNA fragments at different resolutions; then it adopts computational methods re- construct the 3D structures from these contacts. These include traditional nu- merical methods and deep learning models. which struggle with limited model expressiveness and poor generalization across resolutions. To handle this issue, we propose InertialGenome, a novel transformer-basee framework for robust and resolution-agnostic chromosome reconstruction. InertialGenome first adopts the inertial frame for the pose canonicalization. Then, based on such an invariant pose, it proposes Transformer with geometry-aware positional encoding, leveraging Nyström estimation. To verify the effectiveness of InertialGenome, we conduct experiments
single-cell 3D reconstruction datasets with four resolutions, reaching superior performance over all four computational baselines. Addition- ally, observe that the 3D structure reconstructed by InertialGenome is in line with the results of real experimental results on two functional verification tasks. Finally, we leverage InertialGenome for resssuteso transfer learning. yielding up to 5% improvement from low to high resolution. The source code is available htttpe///ggihubbb..o////eeee2//2 2nee220/////tttee Genoeee
INTRODUCTION
The genome encodes the complete set of genetic information within organism, stored as a full DNA sequence. This information is packaged into chromosomes, which serve as the carriers of ge- netic material and compact the DNA into diree-dimeiiionll structures. These chromosomes adopt complex 3D conformations that play essential roles in gene regulation, cell differentiation, and dis- ease progression (Lieberman-Aidee et al. 2009; Dixon et al., 2012 Rao et al. 2014). Importantly, such an organization cannot be inferred from the linear DNA sequence alone (Consortium et al.) 2024).
Over the past two decades, diverse experimental techniques have emerged to probe the 3D confor- mations. Early approaches such as 3C(Dekker et al., 2002). 4C et al.) (2066, and SC(Dostie et al.) 2006) enable targeted interrogation of chromatin interactions at specific loci. More re- cent genome-widh methods, including ChIA-PET (Fullwoo et al. 2009), SPRITE(Quinddoz et al. 2022). GAM(Bcagric al. 2017). and Hi-C(Belton al.] 2012), provide comprehensive maps of spatial chromatin contacts across the entire genome. Among them, high-throughput chromosome conformation capture (Hi-C) has become the gold standard for 3D modeling because it enables the systematic profiling of interactions between all genomic loci simultaneously. In Hi-C, the genome partitioned into consecutive, non-overlapping segments called bins, with their length determined
Work done during intersship Wave Intelligence Lab.

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(A)
Technology
(B)
Problem formulation
Information Capture and Sequencing
Mapping and Counting
Different Resolitions Contact Matrix Matrin)
Reconstruction Technology
Structure
Sequance Read Parrs
Mapplin
Reference sequence
1M5
MVB
Matheratical Modeling
Sequancing
Categorizing
BOOKE
Distance
Crosslink Chromatin
Probabiity Distribution
Ugation and Reverse
Enrichment and PCR
Courring and Motrix
Shrear
Machine Leariing
long DNA Fragments
Figure I: Overview of chromosome 3D reconstruction using Hi-C technology. (A): Experimental implementation of Hi-C for obtaining contact matrix information. (B): Computational pipeline for 3D structure reconstruction via mathematical modeling or machine learning based on the Hi-C con- tact matrix.
by the chosen resolution (e.g., 1 kb, 10 kb, or 250 kb per bin, with higher resolution correspond- ing to shorter segments). As a result, Hi-C produces a contact matrix that records, for every pair of bins, the frequency of spatial contacts between their underlying DNA segments, as illustrated in Fig. Hi-C is chosen for our task because maps at different resolutions offer complementary structural information: high-resolution maps capture fine local structures but are often sparse and noisy, while low-resolution maps are denser and more robust, reflecting global organization. Driven by this observation, we focus this cross ressuutinoltion task by using low-resolution maps structural priors to guide high-resolution 3D genome reconstruction.
Following the Hi-C matrix, the next step is to computationally reconstruct the 3D conforma- tions (Trieu & Cheng 2017: Oluwwddre et al. 2018 Wang & Cheng 2025). Traditional numerical methods based distance geometry, such ChromSDE (Zhang al. 2013), miniMDS (Rieber & Mahony 2017), and 3DMAX (Oluwadare et al. 2018). attempt to find optimal coordinates by minimizing distance-based objective functions. However, these approaches are computationally in- tensive because the underlying non-convex optimization involves high-dimensional search spaces and quadratic complexity in distance matrix operations. Consequently, they often struggle con- verge on high-resolution genomic data and are largely limited to low-resolution modeling due to data sparsity. To address these issues, deep learning offers an efficient, data-driven alternative. For instance, HiC-GNN (Hovenga et al. 2023) leverages node embeddings and graph neural networks to predict 3D conformation directly from Hi-C contact graphs; HiCEGNN (Wang Cheng, (202)) fur- model constraint. However, these approaches share some key limitations: they rely solely on Hi-C contacts without incorporating explicit geometric priors (e.g., chromatin principal axes or directional chain structure). Moreover, the strong symmetry con- straints of structures such as HiCEGNN limit model expression ability, making it difficult to process asymmetric structures (such anchored loops).
ther incorporates =(3)-qquivariance
Our Contributions. To tackle this problem, we propose InertialGenome, a novel Transformer-based framework for 3D chromosome reconstruction. InertialGenome has two main components. (1) In- ertialGenome performs pose canonicalization. It aligns each chromosome to its inertial frame-a coordinate system defined by the principal axes of its inertia tensor. This tensor computed from the 3D point cloud of the chromosome. The alignment removes arbitrary rotations and translations, re- sulting in a iose-invariant representation. (2) Based on this invariant frame, Inertialeenome employs a geometry-awen aware positional encoding into the Transformer architecture. The core idea is to project the bin-wise positions into an imaginary space and use the Nyström method to estimate the pair- wise distance via inner products. Specifically, Nyström enables efficient low-rank estimation of the radial basis function (RBF) kernel over 3D coordinates, capturing long-range structural dependen- cies without computing the full distance matrix. To verify the effectiveness of InertialGenome, we conduct experiments on two single-cell 3D chromosome reconstruction datasets at four resolutions, where it consistently outperforms four baselines in both two structure metrics and two functional validation tasks. Additionally, InertialGenome excels at cross-resolution transfer tasks, reaching up to 5% performance improvement.
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Related Work. Existing methods for 3D chromosome reconstruction fall into three categories (see
Appendix A): Distance-based methods (e.g., 3DMAX (Oluwadare et al. 2018). LorDG (Trieu &
Cheng 2017 convert contacts to distance constraints; Probabilistic approaches (e.g.. BACH (Hu
et al. 2013), PASTIS (Varoquaux et al. 2014)) model contact matrices as observations from spatial
distributions; Deep learning methods (e.g., HiC-GNN (Hovenga et al. 2023), HiCEGNN (Wang)
& Cheng] 2025)) map interactions to 3D structures via neural networks. However, these meth- ods have some limitations: they rely on simplistic modeling of contact matrices the sole input, lacking deeper structural interpretation, and their model expression ability may be constrained. In- spired by the success of InertialAR (Li et al.] 2025), which leverages inertial frames and geometric encoding for invariant molecule generation, we propose InertialGenome. Our framework extends these geometric principles to chromosome modeling, integrating inertial frame canonicalization and geometry-aware positional encoding to significantly improve robustness across resolutions.
PRELIMINARIES
The 3D chromosome reconstruction methods typically take the Hi-C contact matrix as input. The resolution of Hi-C contact matrix reflects the length of chromosome segments, with higher resolu- tion corresponding to shorter segments. For example, kb resolution, each 1,000 base pairs forms segment. When applied to the human genome with standard masking of unmappable regions, this yields approximately 248,94 947 bins. This underscores the sharp scale differences across resolutions, with high-resolution 3D reconstruction being far more computationally demanding and challenging.
The reconstruction process consists of two key steps, as shown in Figure D8. First, the contact frequencies Fi converted to spatial distances D,j based the inverse relationship between distance and contact frequency (Pombo & Nicodemi, [2014) Barbieri et al., (012)), expressed as
Dij = IFT where y E [0.1, 0.2,.... 2]. Second, the 3D coordinates (I;, V/. 2;) are inferred from these distances using either numerical methods deep learning methods. In the following paper, we denote 4 as the index of the i-th bin within the chromosome. Its canonicalized 3D coordinates are represented s; (S,, S S.) E Ra which are pose-normalized via alignment the chro- mosome's principal axes, as detailed in Section 3.1
Problem formulation. In this work, we are interested in solving the 3D chromosome reconstruction task. Following the existing paradigm (Hovenga et al.) [023 Wang & Cheng 2025), the first step is to apply a numerical method to generate an initial position C' {(2] y 2 Ti=1 where N
L
the number of bins (e.g., nodes), from Hi-C contact matrix. Then, our model takes these initial
3D coordinates C* as the input, and the output is an accurately reconstructured 3D coordinates
Ć ((t;; Di, 2) Rigorously, we are solving the 3D chromosome reconstruction task as Ć S(CC).
3 METHOD INERTIALGENOME
In this section, we introduce InertialGenome, novel Transformer-basse framework for robust and resolution-agnostic chromosome reconstruction. consists of three key components: inertial frame canonicalization, geometry-ave positional encoding, and structure-aware fusion. Figure 2 shows the whole architecture of InertialGenome.
3.1 INERTIAL FRAME CANONICALIZATION
To achieve pose-invariant representation of 3D chromosome structures, we implement an iner- tial frame canonicalization method, with the following steps: (1) Centroid translation: & 1 E C; where c; E C². will adjust position relative to the center c/ = C. (2) Inertia ten- N
sor computation: We estimate the normalized inertia tensor as = N EN (I|c//12I, c)( where I3 is the x identity matrix. (3) Principal axes alignment: We perform eigen- decomposition of the inertia tensor I = LALT, where A = Biag(Au, Ay. >₂) contains the eigenval- with A A 2 >y. The columns of L are the corresponding orthonormal eigenvectors I,, ly l,, which define the principal axes in descending eigenvalue order. (4) Chirality correction: We select the farthest point Gmax arg max; ||c||| and map it into the principal-axis frame: p L Cmax.
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Input Data
Nystram Positional Encoding
Rotory Fostional Encoding
Anchor
2
Hi-C Matrix
Point Po
Anchel
Andhor Point
Point
Bin
Gram
Projection Laveratll Anchors
Raturn Giobal Pairwise Belaiionsiips
X
Initial Coordinates
Output Coordinates
Inertial Frame Canonicalization
Geometry-aware Positional Encoding
The architecture of InertialGenome. The framework takes 3D chromosome coordinates, initialized from Hi-C matrices using numerical methods, input Data. (A) The model first performs inertial frame canonicalization to transform raw coordinates into pose-invariant representation by aligning them with the chromosome's 's principal axes (X', Y' Z'). (B) These canonicalized coor- dinates are processed by the Geometry-aware Positional Encoding mechanism (highlighted by the shaded projection). This component integrates Nyström Positional Encoding for efficient global relationship modeling via learnable anchors, and Rotary Positional Encoding for pairwise distance awareness. (C) Finally, the framework outputs Corrected Coordinates, yielding stable and refined 3D chromosome structures.
Figure 2:
where p = (Pu-Py, Py, is the coordinate the farthest point expressed the principal-axis basis. 1, + sign(p=)1, x Lg. Then, the canonical transform is R [1, ly L,)". Pose-invariant coor- dinates obtained as s; Ref, where s; E S. S A {Sz,, Sy; Ss, 1.V Although chromosome structures naturally occupy 3D space, reconstruction algorithms may yield degenerate cases (e.g., nearly coplanar nodes). We discard such samples before training. After applying the canonical transform, each coordinate is represented
We then adjust the first two system by 1,
sign(py)) my, and enforce right-handed
= Rc,, S (s C R
(1)
3.2 GEOMETRY-AWER AWARE POSITIONAL ENCODING
In 3D chromosome structure modeling. need position embedding method that can maintain both absolute position information and the relative distance when calculating the inner product, e.g., RT
Bey .y1 [8,1 Re., Nyy
Raa, -Brg Byy -Byg 28,1 Bag"
(1) Geometric Position Encoding with RoPE, Inspired by Su al. (2024). we extend Rotary Posi- tion Encoding (RoPE) to 3D Euclidean space. To achieve taining pairwise distance awareness, we decompose 3D spatial encoding into three independent 2D rotary subspaces corresponding the (xr,)) y), (y. 2). and (2, planes. This yields 6-dimensional geometric embedding structure, organized into three pairs, each pair encoding angular information along coordinate axis.
attention while main-
The 3D rotary position encoding is applied to the projected query and key vectors. Let T Rdd denote the query and key vectors after linear projection but before positional encoding, associated
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with 3D coordinate s; (Sx,, Sye $z,
We define the rotation operator Ry..
coss 0p)
)b)
sin S5 Bb)
91
cos S. ())
raw
sin S 0n)
Rs, By 9
cos s,,0o)
T3
sin sy 8y00)
43
cosl (,,
sin 0o)
94
cos (5.00)
45
sin(s;0o) sin (5.00)
95
cos COs (b)
where 2 denotes clement-wisg wise multiplication. The final geometry -ware query and key are given by Rs. TTT and k Rs kr respectively. This formulation ensures that the inner product satisfies
the relative positional property: (R,, ( (Ra,h = (qTTM) Rag - which encodes 3D
spatial relationships directly into attention scores. A detailed derivation is provided in Appendix
We implement three RoPE modes that differ in how the input embedding is processed before apply- ing 3D rotary position encoding:
The input embedding x; E IR is split into two halves. The first half (spatial part) is linearly projected and then transformed by 3D RoPE; the second half (feature part) is kept unchanged and directly concatenated.
Selective:
Separate: The two halves independently linearly projected, but only the first half connected through 3D RoPE.
Full: The entire embedding is linearly projected as vector and fully transformed by 3D RoPE.
Input representation. Each token t; corresponds to genomic bin with known 3D coordinates s, (52, Sy,, 2x.) E R3. The total number of bins for a chromosome a given resolution defines the vocabulary size. We map each bin ID i4 E {O..... vocab.st .iiissiieee 1} to learnable semantic embedding Ex (t;) E RR. via a matrix W iok R Rgvocss.sieexd
The initial token representation is formed by concatenating this semantic embedding with the raw spatial coordinates:
Evoke (fc); s; R2
x
Let x and x ( denote the spatial and feature halves æ,, each of dimension d/2. The 3D rotary position embedding is applied follows:
Ra, (wrope an
(Selective)
3D-RoPE(a;;) =
Rs, (wrope ( Wfant, x (1)
(Separate)
(4)
Rs. Rs, (Wa:)
(Full)
where Wrrope Wfeast W are learnable linear projections, and Rs; Rag, Pa is the 6D rotary transformation defined Equation (
(2) Nyström Positional Encoding for Structure Tokenization. While 3D-RoPE in Equation effectively encodes absolute and relative spatial positions, its axis-wise rotation mechanism in- herently limited in modeling global pairwise distance relationships. such as long-range structural dependencies or non-local geometric patterns. To address this, we incorporate a Nyström-based feature encoder (Williams & Seeger 2000 Yang et al. 2012) that explicitly captures low-rank ap- proximations the radial basis function (RBF) kernel over 3D coordinates.
Formally, let 3 denote the canonicalized coordinate of token i. We define RBF kernel between any two points as:
||s: 5; s;
(s,, 85) = -exp
2n²
where Tg 0 is the bandwidth of the g-th Gaussian kernel, and we consider set of G scales (a, y=1 The Nyström method proceeds as follows:
Step 1: Anchor point selection. We fix set of m anchor points {u 1m C R sampled uniformly from the 3D space. For each scale og. we construct the anchor- anchor Gram matrix Ag E mm with entries:
|lu
u
[Agle Kg (y u²) = exp
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To ensure numerical stability, compute the Cholesky decomposition Ag 0g0g where Og Rmxm is a lower-triangular matrix, and precompute Og for later use.
Step 2: Token-anchor similarity. For each token coordinate S;, we compute its RBF similarities all anchors under scale gg- 4, (Sg u1), Kig (S,, u2), ..... Kig (s,, um)] e R
Step Nyström projection. We project a using the precomputed inverse factor:
kgg. g, O5 IR! This yields a low-rank approximation of the full kernel embedding.
Step 4: Multi-scale fusion and compression. We concatenate the projected features across all scales:
k = [ii k2,5... ...; ka,s E B
and apply learnable linear projection fo Rom R to obtain the final Nyström structure embedding:
Bnystrion (s:) = so (R.)-
(5)
This embedding encodes multi-scale, low-rank geometric information about s; and is fused with token and positional representations in the subsequent transformer layers.
3.3
STRUCTURE-AWARE FUSION
We fuse the geometry-aware positional encoding and Nyström structure features into unified Trans- former input.
Input Representation. For each token i, we construct the initial embedding by concatenating three
geometric components: (1) the base position embedding 2;, (2) the normalized canonical coordinate %; which encodes directional information, and (3) the Nyström structure embedding Emystrm (s;).
[s]]]
This yields:
hg Concat Dits
E Eaystrom(::) (s.) Rd
|ss
Position-Augmeented Transformer Input. We then add the geometry-aware positional encoding 3D-RoPE(Z:) inject relative spatial context, followed by dropout:
Hg Dropout(11 (h 3D-RoPE(m::).
Transformer Backbone. The sequence H, is processed by L stacked Transformer layers:
H TransformerLayer(HH)) 1 0, ... 1.
The final representation H, aggregates multi-scale geometric and structural information, and is used predict the 3D coordinates of chromosome bins.
3.4 LEARNING OBJECTIVE
The overall training objective is a hybrid loss that combines struccuraal-pessrvving term and a value- weighted regression term:
Liotal Lstruct + 3 Lweighted.mse
= 1 111a. a,
(9)
where a E [0, 1] balances the two components.
Structural- loss. Let D { denote the input pairwise distance representation derived from contact frequencies, and let Č = J the predicted 3D coordinates where 6, E R To capture the local topology, define neighborhood selection probabilities for each bin in both the input and output (reconstructed) spaces:
exp(-Dig)
exp( exp(-118:
||8:
Eneo exp(-Dae)
Oil Σ
Exys exp( -1
S
(10)

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To align the neighborhood structures between these spaces, employ bidirectional Kull- (KL) divergence objective (Gong et al. )
Strruct (KI((IIQ) + +1 A)KI(Q)P))
(11)
where (L(P)IQ) E, E Poja log Pr.. The hyperparameter E [0, 1] (defaulted to 0.1) bal-
ances the trade-off between false positives and missing structural features. Detailed derivations provided in Appendix C.I
Weighewe MSE. While Latruct preserves the global topology, precise distance estimation requires an additional supervised term. Given that Hi-C data exhibits higher reliability for smaller spatial distances (corresponding to high-intensity contacts), squared error (MSE) loss (Wang et al.) 2024). This loss assigns adaptive weights based on the rank of true distance values rather than treating all errors uniformly.
introduce value- aalue-weighted mean
For batch containing distinct distance values, we compute weights wo for each value v E V (see Appendix C.2 for details). The weighted MSE is defined as:
Eweighted
Σ VEV
Σ (iJ))I,
(yij Dis)
(12)
N
where I, denotes the set of bin pairs with target distance u, Nr [Iol, Vij is the target distance derived from FF and ūij =||;; $yll represents the predicted Euclidean distance between bins i and j.
3.5 STABILITY OF INERTIAL FRAME ALIGNMENT
During our experiments, we observed that inputs from physics- regularization based reconstruc- tions (e.g., 3DMAX, LorDG) consistently benefited from inertial-frame alignment, whereas contact- matrix eigendecomposition methods (e.g., Gram) showed little or no gain. To explain this contrast, We define its
recall that the input coordinates defined C sample covariance
{(x) VE
H Rx3
Ec.
N CC)
(13)
which admits eigenvalues μi 2 122 2 Ha and orthonormal eigenvectors u, u2, Ua corresponding to principal axes. The spectral gap
8(C*) M1 μ2
(14)
quantifies how well the first principal direction is separated from the remainder.
Given two coordinate sets (1) and (2)2), we measure the angular difference of their leading inertial axes by
OPci (1) Cr(2)) arccos{ | (C*(I) (C (C
(15)
The stability of the principal directions under perturbations is controlled by the Davis-Kahan theo- rem (Davis & Kahan 1970). Let A be a symmetric matrix with leading eigenvector u and spectral
gap 8. If A = A + AA has leading eigenvector ū. then
|
sin Au, 7) <
(16)
8
where ||- ll denotes the spectral norm. Applied to Ec. (Equation (33). Equation (16) shows that when the spectral gap 8(C*) (Equation (14) is small, even minor perturbations in the data can rotate the leading axis significantly. Conversely, large gap yields stable inertial axes and reliable alignment. This analysis and explanation can be found in Appendix
4 EXPERIMENTS
Datasets. We evaluated our method on two single-cell Hi-C datasets: human frontal cortex cell Wang & Cheng 2024) and B-Lymmhhcyyt cell (Oluwadare et al. 2020). Both datasets followed the

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Table 1: Performance comparison of six methods on 3D chromosome structure reconstruction from single-cell Hi-C data (Frontal cortex cell test set). Metrics report distance- based Spearman correla- tion (dSCC t) and root mean square enror (dRMSE 1) four resolutions. Best results in bold.
320КB
160KB
80KB
40KB
Method
dscct
dRMSEL
dScct
dRMSEL
dSCCt
dRMSE,
dSCC1
dRMSEL
3DMAX
0.2780
23.1588 1587
0.2302
23.3439
0.1774
23.9174
0.1754
24.6538
LorDG
0.6681
92.6582
0.6997
102.1392
0.6342
100.0507
0.5841
96, 1048
HiC-GNN
0.2432
0.8366
0.2077
0.9083
0,1370
0.9352
0.0915
0.9456
HiCEGNN
0.5804
0.2744
0.5351
0.3550
0.3288
0.4158
0.2506
0.4317
IG-3DMAX
0.9006
0.1697
0.8577
0.1835
0.7727
0.2192
0.7187
0.2410
IG-LorDG
0,8713
0.1544
0.8056
0.1997
0.6835
0.2398
0.6036
0.2574
chromosome partitioning scheme of Waan & Cheng (2025), with training sets (frontal cortex: chr 13.578.9.1113355..6.7.1199
B-Lymphocyte: adds chr 23), validation sets (chr 2,6,10,12 for both), and test sets (chr 4, 1,,4, for both).
Baselines and Implementation. We compare our method with both classical numerical and deep learnngg baselines for 3D chromosome structure reconstruction. The classical numerical methods include 3DMax (Oluaadare et 2018) and LorDG (Trieu & Cheng (017), while the deep learning baselines include HiC-GNN (Hovenga et al. 2023) and HiCEGNN (Wang & Cheng. 2025). All baselines were run with their default configurations as provided in the respective source codes.
Metrics. We evaluate 3D chromosome reconstruction using two metries: (1) Distance Spearman correlation coefficient (dSCC) (Oluwadare al. 2018) measures rank correlation between pre- dicted and ideal distances (range [-1,1]); higher values indicate better structural quality and scale invariance. (2) Distance root mean square error (dRMSE) (Varoquaux et al.) 2014) quantifies absolute distance errors; lower values denote higher accuracy and similarity to ideal distance map.
4.1 MAIN RESULTS
Reconstruction performance on Frontal cortex cell dataset. Table compares six methods on single-cell Hi-C data. Our two variants, IG-3DMAX and IG-LorDG, consistently outperform all baselines in both dSCC and dRMSE across all resolutions. For example, at 320 kB, IG-3DMAX achieves a dSCC of 0.9006, significantly higher than HiCEGNN (0.5804) and 3DMAX (0.2780), while reducing dRMSE 0.1697 from 0.2744. Similar improvements are observed other reso- lutions, with dSCC gains often exceeding 50% and dRMSE reductions of 30-40%. These results demonstrate that combining inertial-frame canonicalization with our Transformer yields state-of- the-art accuracy and stronger iopoluttoo-agnostie performance compared equivariant and numer- ical baselines. Traditional numerical methods (3DMAX, LorDG) perform substantially worse, with dRMSE values orders of magnitude higher, due to the lack of direct di stance supervision. Gram- matrix inputs also yield poor performance in our pipeline; we analyze this instability in Appendix
Reconstruction performance B-Lymphocyte cell dataset. Table compares six methods single-cell Hi-C data. IG-3DMAX achieves the best dSCC and lowest dRMSE across all four res- olutions, demonstrating robust superiority. For example, at 1MB it attains dSCC of 0.9209 and dRMSE of 0.0822, outperforming all baselines. Similar advantages are observed at finer resolutions (500KB, 250KB, 100KB), with dSCC consistently above 0.87 and dRMSE below 0.08. IG-LorDG shows competitive dSCC at 500KB (0.8367) and 250KB (0.8440), but weaker performance at other resolutions. This variation stems from LorDG's inherent reconstruction limitations on this dataset (e.g., high dRMSE 107.7091 at MBB, which constrain IG-LorDG's input quality. In contrast, IG- 3DMAX's stability highlights our framework's resilience to input variations, delivering state-of-the- art accuracy regardless of baseline method performance.
4.2 ABLATION STUDIES
Learning rate robustness. We evaluate our method's robustness to input quality by varying the learning rates (1, 0.5, 0.1, 0.05, 0.01} of 3DMAX and LorDG on Frontal cortex data. Input coordi-
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Table Performance comparison of six methods on 3D chromosome structure reconstruction from single-cell Hi-C data (B-Lymphocyte cell test set). Metrics report distance-based Spearman correla- tion (dSCC t) and root mean square error (dRMSE 1) at four resolutions. Best results in bold.
IMB
500KB
250KB
100KB
Method
dscct
dRMSEL
dScct
dRMSEL
dScCt
dRMSE.
dSCC1
dRMSEL
3DMAX
0.9131
146.7464
0.8603
139.7606
0.8148
136.9766
0.6548
127.9119
LorDG
0.7462
107.7091
0.8103
117.5711
0.8316
112.0299
0.8395
98,9512
HiC-GNN
0.6778
0.2570
0.6457
0.2373
0.5827
0.1312
0.5334
0.2089
HiCEGNN
0.8847
0.0839
0.8068
0.0838
0.7530
0.0823
0.8017
0.0795
IG-3DMAX
0.9209
0.0822
0.9081
0.0777
0.8861
0.0593
0.8708
0.0790
IG-LorDG 0.8413
0.1114
0.8367
0.0979
0.8440
0.0675
0.7939
0.0867
4i NiTesm 33863
14 Faudlaon 1600S
Feutlaee 33249
20 Duuestann 1008
500
Laarn-ng
25
120
Learning Rate
5.1
Learn
200
LEATT ng
22MAX
40-00M4 C50C
30MAH
d5CC
1040000
(1) 3DMAX vs. IG-3DMAX
(2) LorDG vs. IG-LorDG
Figure 3: Learning rate ablation: performance four resolutions for five learning rates. Red: dSCC
(left axis). Blue: dRMSE (right axis).
nates generated under each setting are evaluated at four resolutions using dSCC (f) and dRMSE (L). Figure]Jshows: (1) 3DMAX vs. IG-3DMAX, (2) LorDG vs. IG-LorDG. Our IG variants maintain consistently high dSCC while reducing dRMSE 10-1 scale across all learning rates. Baselines show significant dRMSE fluctuations (up to 102 scale) and unstable performance. This demonstrates InertialGenome's s superior stability and accuracy regardless of input learning rate and resolution.
Loss Components We analyze the contribution of the structural stability loss (Cstruct) versus the coordinate regression loss (Eweighted.mmee) by varying their weighting ratio a/ (3 the total objective (see Appendix E.I
Component ablation. We assess the impact of key design choices by removing inertial-frant alignment, RoPE-3D, or Nyström encoding from our full model (see Appendix E.2
4.3
CASE STUDIES
TAD domain consistency validation. We assess whether bin within the same TAD are spatially closer than those across TADs (see Appendix F.I). Figure shows intra- vs. inter-TAD distances for IG-3DMAX and HiCEGNN. IG-3DMAX yields consistently shorter intra-TAD distances across chromosomes 4, 14, 18, and 20, with intra/inter ratios of 0.76-0.80 and highly significant p-values (Appendix Table 8 HiCEGNN shows higher ratios (0.91-0.99) and weaker significance (e.g., p 0.159 for chr20), indicating poor domain compaction. These results confirm that IG-3DMAX better captures TAD-level spatial organization than HiCEGNN.
A/B compartment validation. We validate the biological plausibility of our reconstructed struc- tures using A/B compartment analysis (see Appendix F.2). Appendix Figure 6 compares dis- tance distributions for IG-3DMAX and HICEGNN. IG-3DMA- shows significantly shorter intra- compartment (A-A, B-B) than cntrr-oompattment (A-B) distances (pA = 0.0001, Pn = 0.0038), confirming expected compartmental organization. In contrast, HICEGNN shows significant

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(G-30MAX: intra vs Inter TAO distances chromosome
HICFONN: Intra Inter TAD distances chromosorme
2.0
Eucileaan disannce
8.0
0.0
Chromesome
Chromosome
MruiAD
IriesSA
Figure 4: Intra- vs. inter-TAD Euclidean distances for IG-3DMAX (left) HiCEGNN (right). Lower intra-TAD distances indicate stronger spatial clustering.
Table Cross-resolution transfer results. Bold: improvement over same-resolution original model;
underline: HICEGNN transfer improvement.
320kb > 160kb
320kb
80kb
320kb
40kb
Method
dSCC1 f dRMSE |
dSCC f
dRMSE &
dSCC
dRMSE
HICEGNN-Transfer
0.5815
0.2997
0.4080
0,4108
0.2343
0.3840
HICEGNN-Original
0.5351
0.3550
0.3288
0.4158
0.2506
0.4317
IG-3DMAX-Full
0.7455
0.3159
0.6766
0.3464
0.6528
0.3588
D0-3DMAX-Selectivv
0.7399
0.3089
0.6698
0,3447
0.6467
0.3567
IG-3DMAX-Separate 0.7434
0.3057
0.6714
0.3392
0.6480
0.3580
IG-3DMAX-Origina 0.7332
0.3071
0.6451
0.3328
0.6132
0.3374
A-A/A-B separation (pa 0.4360) and weak B-B separation (pn 0.0000), indicating poor compartmentalization. These results demonstrate that IG-3DMAX better captures spatial compart- ment segregation than HICEGNN.
Cross-resolution transfer learning. We evaluate transfer from 320kb to finer resolutions (160kb, 80kb, 40kb). Table 3 shows IG-3DMAX consistently outperforms both its original model and HICEGNN variants in dSCC and dRMSE. At higher resolutions (80kb, 40kb), IG-3DMAX main- tains stable dRMSE 10 and improves dSCC by 5%, while HICEGNN degrades (dSCC drops at 40kb). Our gains stem from RoPE and inertial-frame alignment, which preserve spatial rela- tions across scales. HICEGNN lacks such geometric adaptation, leading to unstable performance. IG-3DMAX demonstrates robust, resolution-agnosti reconstruction, with advantages magnified at finer resolutions.
5 CONCLUSION
We presented InertialGenome, Transformer-based framework for robust, resolution-agnostic 3D chromosome reconstruction. It uses cortial-framt canonicalization for pose invariance and geometry-aware Transformer with Nyström positional encoding to capture long-range interactions efficiently. Experiments two single-cell datasets across four resolutions show that InertialGenome consistently outperforms classical (3DMAX, LorDG) and deep learning (HiC-GNN, HiCEGNN) methods in both structural accuracy and functional validation. Our cross-resolution strategy further boosts high-resolution performance by up to 5%, demonstrating strong gemeralization and biological plausibility.
By decoupling physical constraints from architecture, InertialGenome offers a flexible alternative to (E(3)-equivarian models, paving the way for scalable 3D genome modeling, Future work will in- tegrate multi-modas genomic data to enhance reconstruction robustness across diverse experimental conditions.
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ETHICS STATEMENT
Our study complies strictly with the ICLR Code of Ethics. The research involves no human partici- pants sensitive personal data. All datasets and code employed adhere their respective licensing agreements. This work presents foundational research that does not raise concerns related to fair- ness, privacy, security, potential misuse. We affirm that all ethical aspects have been comprehen- sively considered and addressed.
REPRODUCIBILITY STATEMENT
We are dedicated to facilitating the reproducibility of our work. Full details required to repli- cate our key finding--includan including data access instructions, experimental parameters, model archi- tectures, evaluation metrics, and model checkpoints are provided in our GitHub repository at https://q thub. wtw.b/.co///zeel23333 a Genome Users can utilize our detailed doc- umentation and code scripts to faithfully reproduce the results, ensuring research transparency and methodological rigor.
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THE USE OF LARGE LANGUAGE MODELS
In the preparation of this work, the authors used large language models (LLMs) for two specific purposes: (1) to refine and polish English language expression, and (2) to assist in the formulation and typesetting of mathematical equations in LaTeX. All scientific content, research design, analysis, and conclusions remain solely the responsibility of the authors.
Answer:
A RELATED WORK
A.1 DISTANCE AND CONTACT CONSTRAINED OPTIMIZATION METHODS
Distance-basee based methods rely the assumption that genomic loci closer in the one-dimensional sequence are also spatially proximal in three dimensions. The central idea is to exploit the inverse relationship between Hi-C contact frequency and physical distance, thereby transforming the contact matrix into a distance matrix and reformulating the reconstruction task as inferring 3D coordinates from pairwise distances. variety of algorithms adopt this paradigm, including miniMDS (Rieber & Mahony] (017). 3DMax (Oluwadare et al., 2018). ChromSDE (Zhang et al. 2013). Chromo- some3D (Adhikari et al. 2016). ShNeighl (Li et al. 2020), LorDG (Trieu & Cheng. 2017), and others. These methods aim to faithfully recover the underlying geometric distances. For example, 3DMax (Oluwadare et al., 2018) assumes that Hi-C data follow a Gaussian distribution and that contact counts are independent given the structure, defining a log-likelihood objective to identify the most probable conformation. LorDG (Trieu & Cheng (017]. in contrast, employs nonlinear Lorentzian function to enforce structural consistency while improving robustness to noisy distance constraints. In contrast, contact-based methods (Meluzzi & Arya 2013; Trieu & Cheng 2014, Ab- bas et al., 2019) directly translate interaction frequencies (IFs) into spatial constraints without first converting them into distances. For example, GEM (Abbas et al. 2019) enforces distance thresholds between chromosome segments and incorporates biophysical feasibility to reconstruct 3D structures.
A.2 PROBABILITY BASED METHOSS
Probabilistic methods typically assume that the global 3D structure underlies the observed contact map and formulate chromosome reconstruction as a Bayesian inference or maximum likelihood problem. In this framework, the contact matrix is modeled as data generated from specific prob- ability distributions, with representative approaches including BACH (Hu et al., 2013). MCMCSC (Rousscau et al.] 201I). PASTIS (Varoquaux al. 2014). PGS (Hua et al. 2018), and CHROM- STRUCT 4 (Caudai et al.) 2018). For instance, MCMC5C (Rousseau et al. 2011) employs a Gaus- sian prior and uses Markov Chain Monte Carlo (MCMC) sampling infer spatial coordinates from the posterior distribution of interaction frequencies. BACH (Hu et al.) 2013) assumes Poisson distribution for contact counts and applies MCMC to sample chromosome conformations. PASTIS VVaroquaxx et al., 2014), in contrast, optimizes spatial coordinates by maximizing the likelihood under the assumption that contact frequencies follow a Poisson distribution conditioned on 3D po- sitions.
A.3 DEEP LEARNING BASED METHOOSS
Unlike traditional distance and probability-based approaches, deep learning offers a fundamentally different paradigm for reconstructing 3D chromosome structures from Hi-C data. HiC-GNN (Hov- enga et al. 2023) pioneered this line of work by applying graph convolutional networks to capture neighborhood structures Hi-C interaction graphs, directly inferring 3D conformations from large- scale data. Recently, HiCEGNN (Wang & Cheng] (025) extended this idea by employing an SO(3)- equivariant graph neural network (EGNN) to account for translational and rotational symmetries, enabling accurate prediction of the 3D coordinates of genomic loci. These graph- based deep learn- ing approaches demonstrate not only the feasibility but also the advantages of deep learning for 3D genome modeling: they are faster and more straightforward than traditional optimization methods, and they can leverage large Hi-C datasets to capture common structural patterns often missed by conventional approaches. Building on this line of work, we propose new deep learning frame- work. In contrast to existing graph-bases methods, our approach demonstrates that a Transformer architecture can provide superior modeling capacity for 3DD chromosome structure prediction.
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B DERIVATION OF 3D GEOMETRIC POSITION EMBEDDING
We aim to extend the rotary position embedding (RoPE) mechanism to 3D Euclidean space for chromosome structure modeling. The goal is to design geometric position embedding such that the relative rotation between two positions satisfies:
R2 32 Ruy 22,31
(17)
To achieve this, first recall that any imaginary number as:
bi can be written in polar form
r cos 0 + ir sin 0 re*
(18)
where r = |a| and 0 = arg(z). This can further be expressed in matrix exponential form using the identity:
rexp(0.) = r exp(())
(19)
0
with /
which satisfies n -1, -J, = Thus,
0
r rexp(0J) = (0J)
0²
0
= ++ 0J
J + +...)
(20)
n!
2!
3!
n=0
Grouping terms by and odd powers:
1)m
1-1mm22m-1
rexp(((J) r
(2n) 1
Σ
(21)
(2n + 1)!
cos 0 I + sin J.J
(22)
cos 0
sin 0 cos
(23)
sin
Hence, the 2D rotation matrix can written as Rm exp(m)), and it follows that:
R Rr exp(-m.)) exp(n.J) exp((n mm)J) Rn-m-
(24)
Now, generalize this to 3D. We define 3D rotation operator Ra,,y, the exponential of skew-symmetric matrix:
0
0
0
x
0
0
0
0
0 0
R = exp
(25)
0
8
0
which decomposes into three independent 2D rotations along the (xr, v). (u, 2), and (z,ar) planes.
Expanding this matrix exponential yields:
cos
sin z0
sin r0
cos x0
0
0
0
0
0
0
cos y
sin y
0
R
(26)
0
0
sin ye
y
0
0
0
0
cOs 28
sin 20
0
0
0
sin 20
cos 20
When applied to a 6-dimensional token embedding q
[qu, 41 93. 94, 9s
we obtain:
cos x0
41
sin x0,0
Q1
cos zθ
Q0
sin 26,
R
Q2
yog
+
Q3
sin you
(27)
Q3
cos yo
sin you
Q4
20g
45
sin 20n
cos z0,
sin z0,
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where 8g is the base frequency (e.g., 10000 2/0), and denotes element wise multiplication. This construction ensures that the inner product satisfies:
(Bei.i.. q) (Brz. k) q Rz -1.vi 02.21
(28)
required. Therefore, our 6D geometric embedding naturally captures relative spatial relationships through rotational invariance.
C DETAILED LOSS FUNCTIONS
STRUCTURAL-LEARNIGG Loss
Let the spatial distance representation be
= (D1, D., Dat:
D; [D .i. D
(29)
and the 3D coordinate representation be
$ = (S), SS..,, Sn), S = (wi, Mi, zi).
(30)
Neighborhood Probability Distributions. probabilities are:
Based on heterogencity between genomic bins, the
exp( si 15
PIR Enji - 4iji Exer exp( 1ap(-11s: 5k (12)
(31)
where ||st - syll2 1 is the squared Euclidean distance between bins and j.
Bidirectional KL Divergence. Let P [P,...,PP Pr} the distance distribution and Q {Q1,... Qn the 3D-space distribution. We define
KL(PIIQ) = Σ
Pile log
Pj
(32)
Ozli
i
KL ΣΣ iti
Tole
log
Ozlk
(33)
Piji
Balancing false positives and misses via parameter A, the final diructural-lcarnnng loss is:
Lstruct X (I((I)Q) (1 A) KI(QIIP)
= Σ Σ
Pije log
Pile
+ (1- A) Σ
Til log
Oile (34)
4zls
j4
C.2 VALUE- WEIGHTED MSE LOSS
To emphasize smaller distances (high-intensity contacts), we weight the MSE by ranks. For a batch with n distinct true distances, the weight for each value is computed as:
rank(i)
wy nn + 1)/2'
(35)
where rank(i) is the ascending rank order value (smallest value has rank and receives the highest weight).
The final weighted MSE loss is: Lweighted-mse
Σ(u p N
p:)a
Σ
W/
(36)
iEvalues
where N; the frequency of true value in the current batch.
By combining Equation (34) and Equation (36), model simultaneously learns accurate local distances and a consistent global structure, enabling more accurate and robust 3D chromosome reconstruction.
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D DERIVATIONS FOR INERTIAL ALIGNMENT STABILITY
We summarize here the derivation and the empirical checks that complement the main text.
Notation. E RN denotes the contact matrix, D = f(C) its corresponding distance matrix ob- tained through the monotone transform f(-), H = I 11 the centering matrix, B - - HD
the Gram matrix, and X E Nx3 the centered 3D coordinates whose sample covariance is Σx- N X
First-order perturbation. Let AC' be small perturbation of C. By Taylor expansion of we have to first order
AD f'(C) AC.
Consequently, the Gram perturbation is
AB H (D AD) H.
Taking operator norms and using || yields the bound
|| X || (C) ||||||
Thus the size of AB scales linearly with the contact perturbation and with the current dis- tances.
Effect principal components. Let un be the leading eigenvector of B and iy that of B AB. By the Davis-Kahan theorem (Equation (16) in the main text) we have
|| 8B
sin L(u1, w1)
where dg is the spectral gap of B (difference between its first and second eigenvalues). Hence, very small perturbations of C' can produce large rotations of the leading direction whenever d is small; conversely, a large gap yields robust orientation.
Empirical validation (Chromosome 3 at 320 kb). We found through numerical calculations that many of the chromosome 3D coordinates used as input in grams are coplanar or almost coplanar, which directly indicates the problem of using grams as input. Of course, we still validated the difference between using two types of methods as inputs through our approach. We have selected chromosome 3 for validation here because it is a gram as the input that there is no coplanar sample. Table reports the quantitative stability metrics of the Gram-based embedding and the 3DMax reconstruction for the same chromosome. We observe that the Gram top eigenvalues are very close (spectral gap å 0), making its orientation unstable, 3DMax, on the other hand, shows much larger spectrum spread (6 18.6) and correspondingly stable orientation.
Table 4: Stability metrics for Chromosome 3 (320 kb).
Top-3 spectrum
Rotation stability
Gram 3DMax
[0.00547, 0.00546, 0.00502), 5 = 0.0000 [70.62, 52.04, 41.30], 5 18.58
2.83
4.49c-5
Noise experiment. Figure shows the median angle of the first principal component (PC1) function of injected noise scale. The Gram embedding (blue) displays rapid growth of the PC1 angle once noise is added, whereas the 3DMax embedding (orange) remains essentially unaffected. This is consistent with the theoretical bound above: vanishing spectral gap makes the orientation of the Gram embedding highly sensitive to perturbations. These observations jointly validate our perturbation analysis: even when distances and neighborhood structure are perfectly preserved, near-degenerate spectrum leads to unstable principal directions, whereas a large spectral gap confers rotational stability.
IG-Gram experiment. Table illustrates the performance Gram reconstruction as input into InertialGenome.
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PC1 angle vs noise
Gram PC1 angle
80
3DMax PC1 angle
60
40
Modian
20
0-
10-3
10
10aa
10
Noise
Figure 5: Median PCI angle versus noise scale for the Gram and 3DMax.
Table 5: dSCC and dRMSE of IG-Gram on different resolutions.
320KB
160KB
80КB
40KB
Method
dSCC1
dRMSEJ
dSCC1
dRMSEL
dSCC
dRMSEL
dSCC1
dRMSEJ
IG-Gram
0.4770
0.3728
0.1079
0.4656
0.0524
0.4811
0.0083
0.5383
E SUPPLEMENTARY FOR ABLATION STUDY
E. LOSS COMPONENTS
We analyze the contribution of the structural stability loss (Catruet) versus the coordinate regression loss (Lweighted.mse)) by varying their weighting ratio a/ in the total objective Liotal a.Cutruct + &L where 3 = a. As shown Table [ removing structural supervision entirely (a 0) yields reasonable performance at coarse resolutions but leads to significant degradation in dRMSE at fine scales (e.g., 40kb), indicating poor geometric consistency. improves dRMSE at coarse resolutions but harms distance correlation (dSCC). A certain proportion of structural regularization (a 0.1, 0.5) consistently achieves the best trade-off across all resolutions, validating its role in enforcing biologically plausible 3D genome topology without sacrificing coordinate fidelity.
contrast, using
only structural loss
Table 6: Ablation study about IG-3DMAX on the loss weighting between structural stability and coordinate regression. Reported metries: distance-based Spearman correlation (dSCC t) and root mean square error (dRMSE 4) across four genomic resolutions. Best results per resolution are bolded.
320KB
160KB
80KB
40KB
Ratio (a/3)
dSCCt
dRMSEJ,
dSCC1
dRMSE,
dSCCt
dRMSEJ
dSCCt
dRMSEJ
0.0 1.0
0.9030
0.1728
0.8627
0,1935
0.7532
0.2152
0.7132
0.2410
0.1 0.9
0.9029
0.1696
0.8595
0.1848
0.7663
0.2197
0.7158
0.2407
0.5 0.5
0.9002
0.1671
0.8580
0.1879
0.7741
0.2266
0.7203
0.2445
1.0
0.8815
0.1453
0.8484
0.1775
0.7677
0.2297
0.7192
0.2788
E.2 COMPONENT ABLATION
Results using IG-3DMAX in Table show: (1) Without inertial alignment, dRMSE increases across all resolutions (e.g., from 0.1547 to 0.1611 at 320 kb), indicating its role in stabilizing global struc- ture; (2) Removing RoPE consistently degrades both dSCC and dRMSE, confirming that relative positional encoding essential for structural fidelity; (3) Disabling the Nyström branch leads the
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largest performance drop fine scales (e.g., dRMSE rises by 0.0114 at 40kb). demonstrating its critical contribution to modeling long-range pairwise distances. The full model achieves the best trade-off between structural consistency and coordinate accuracy, outperforming all ablated variants at every resolution.
Table 7: Ablation study of key components in our model on single-cell Hi-C data (Frontal cortex test set). Reported metrics: distance-based Spearman correlation (dSCC f) and root mean square emror (dRMSE J) across four genomic resolutions. Best results per resolution are bolded.
320KB
160KB
80KB
40KB
Model
dScC1
dRMSEJ
dScC1
dRMSE,
dSCC1
dRMSEL
dSCC1
dRMSEJ
Full (Ours)
0.9030
0.1547
0.8621
0.1809
0.7757
0.2035
0.7297
0.2382
w/o Inertial
0.9008
0.1641
0,8598
0,1869
0.7737
0.2185
0.7226
0.2385
w/o RoPE
0.8976
0.1613
0.8566
0.1894
0.7709
0.2229
0.7213
0.2454
w/o Nyström
0.9002
0.1659
0.8607
0.1998
0.7746
0.2218
0.7214
0,2496
F EXPERIMENTAL VALIDATION
F.1 VALIDATION VIA TAD DOMAIN CONSISTENCY
Topologically Associating Domains (TADs) are contiguous genomic regions within which loci tend to interact more frequently with each other than with loci outside the domain (Dixon et al.) 2012). Validating the biological plausibility of predicted 3D chromosome structures can therefore be per- formed by examining whether spatial distances among loci within the same TAD significantly shorter than those between loci from different TADs.
TAD Boundary Acquisition TAD regions can be obtained from public annotations (e.g.. ENCODE project or 3D Genome Browser) or identified directly from Hi-C contact matrices using established
TAD callers such as: TopDom (Shin al. 2016), Armatus (Filippova et al. 2014), and HiCExplorer Wolff et al. 2020). In our workflow, TADs were identified using Tom following the procedure
Serra al. (2017).
Mapping 3D coordinates to TADs: Each bin in the 3D structure corresponds to a genomic interval based on Hi-C resolution (e.g., 250 kb). For bin indexed by i, its genomic span is (i 250 kb, (i+ 1) x 250 kb). We assign each 3D point to a TAD by checking whether its genomic start coordinate falls within the start and end of any TAD interval.
Evaluation Metric (Mean intra-/inter- TeADiitttr--ADDDDD distance): For each TAD, we compute the average pairwise Euclidean distance Dintra among all loci inside the same TAD. We also compute the average distance Dinter between loci across different TADs. A well-structured 3D prediction should satisfy:
mmna) < mean(Diner),
(37)
indicating spatial clustering of loci within the same TAD, consistent with known biological organi- zation.
Statistical Testing We assessed whether intra-TAD distances are significantly smaller than inter- TAD distances using the Mann-Whitney U test. Table summarizes the mean intra- and inter-TAD distances, intra/inter ratios, and corresponding p-values for IG-3DMAX and IG-LorDG reconstruc- tions across chromosomes 4, 14, 18, and 20.
These metrics together provide quantitative assessment of whether the reconstructed 3D structures preserve known domain-level chromatin organization. In particular, while IG-3DMAX consistently shows significantly lower intra-TAD distances with low intra/inter ratios across all chromosomes, HiCEGNN exhibits weaker separation and a non-significant result for chromosome 20, indicating comparatively less biologically plausible domain organization in that case.
F.2 VALIDATION VIA A/B COMPARTMENTALZATIIN
To assess the biological plausibility of our predicted 3D chromosome structures, we validate them based A/B compartment organization. In eukaryotic nuclei, chromosomes spatially segregate
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Table 8: TAD-based validation statistics for IG-3DMAX and HiCEGNN reconstructions. Mean intra- and inter- TAD distances, intra/inter ratio, and Mann-Whinng U test p-values are reported for selected chromosomes.
Model
Chromosome
Intra-TAD
Inter-TAD
Ratio (Intra/Inter)
test p-value
IG-3DMAX
Chr4
0.852
1.122
0.760
IG-3DMAX
Chrl4
0.957
1.203
0.796
8.95e-93
IG-3DMAX
Chr18
0.882
1.083
0.814
1.55c-66
IG-3DMAX
Chr20
0,941
1.177
0.800
2.83e-47
HiCEGNN
Chr4
0.408
0.442
0.925
2.61e-34
HiCEGNN
Chrl4
0.405
0.425
0.952
1.09e-4
HiCEGNN
Chrl8
0.380
0.416
0.914
3.16e-10
HiCEGNN
Chr20
0.451
0.454
0,993
0.159
into two major compartments- A (active) and (inactive)- -with loci from the same compartment (A-A or B BB) tending be spatially closer than loci from different compartments (A-B).
We follow the process of Lieberman-Aider et al. (2009) to assign compartment labels. For each chromosome, we compute the Pearson correlation matrix of the normalized Hi-C contact matrix, then extract the first principal component (PCI). Loci with positive PCI values are assigned to compartment A, and those with negative PC1 values to compartment B.
Distance computation and grouping. From the predicted 3D coordinates we compute pairwise Euclidean distances and categorize them into three groups:
intra-A: distances between loci within compartment A;
intra-B: distances between loci within compartment B;
inter-AB: distances between loci from compartments A and B.
Statistical analysis and visualization. We compare the distributions of these three groups using box plots and permutation test. A valid 3D structure should exhibit significantly shorter distances for intra-compartment pairs (A-A and B-B) than for inter-compartment pairs (A-B), indicating that the predicted structures preserve the compartmental spatial organization.
IG-3DMAX pairwise distances)
HICEGNN (all pairwise distances)
2.0
Euclidean distance
Euclidean distance
0.6
1.0
0.4
0.2
0.0
A-inter-8
A-Inter-B
Distance type
Distance Syyp
Figure A/B compartment validation. Box plots show intra- and inter-compartmeet distances for IG-3DMAX (left) and HICEGNN (right).
F.3 FISH-BASED VALIDATION OF 3D STRUCTURES
FISH experiments provide direct measurements of the spatial distances between genomic loci the nucleus. To validate our 3D chromatin structure predictions, compared the predicted distances between key regions- L1, L2, and with the experimentally observed FISH distances reported by Rao (Rao al. 2014). These regions were identified loop anchors using HiCCUPS
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analysis, with L1 and L2 forming strong interaction peak, while L3 served as internteree control.
We evaluated model outputs at 250 kb resolution, where sufficient structural detail is preserved to accurately localize these regions. For each chromosome, we computed the Euclidean distances between L1-L2 and 1.2-L3 in the predicted 3D structures, along with their corresponding contact probabilities from the original Hi-C data.
As shown in Table the predicted L1-L2 distances consistently shorter than the L2-L3 distances across all chromosomes, consistent with the experimental observations. Moreover, the contact prob- abilities exhibit an inverse trend: higher values for L.--I2 compared to 1.2-1.3, reflecting stronger physical proximity. This agreement between predicted distances and Hi-C contact frequencies sup- ports the biological plausibility of our modeled structures.
Table 9: FISH validation results on GM12878 chromosomes 11, 14, and 17 250 resolution. The table shows the predicted L1-L2 and L2-LX distances, and the corresponding KR- e--nommaized contact probabilities from Hi-C data.
Chromosome
L1-L2 Distance
L1-L2 Probability L2-L3 Distance
L2-L3 Probability
11
0.8
2.74 x 10*
3.3
3.92 x 10
14
2.2
9.10 103
13.1
2.07 10
17
2.3
2.05 10a
3.7
1.32 x 10a
The results demonstrate that our model captures the expected spatial organization: looped regions (L11L12) are closer in space and exhibit higher contact frequencies than non-looped regions (I.2-L3).
This consistency with both FISH measurements and Hi-C data confirms that our method produces biologically realistic chromatin structures.
21


Reviews:
[{"review_id": "HmtcvJfjte", "reviewer": "Reviewer_A2GD", "summary": "This paper proposed InertialGenome, a novel transformer-based framework for robust and resolution-agnostic chromosome reconstruction.  With this new design, it greatly improved the accuracy compraed to previous methods.", "questions": "1. Problem setting for real chromatin tracing data.\n2. Ablation study to understand the key contribution of the designs.\n3. The biological validation is not so meaningful, since you should expect to have similar TAD/compartment observations if you can make distance map close to the original distance/contact map, but that does not mean", "limitations": "", "rating": 4, "confidence": 5, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "1. The background and problem setting is clearly addressed.\n2. The network architecture is well designed and improved over previous graph-neural-network based approaches.\n3. The performance improvement is impressive.", "weaknesses": "1. I think the problem setting has a siginificant limitation: there is no other ways to validate the predicted coordinates. In your setting, you compared the projected contact map from your 3D coordinates and the original contact map. However, single-cell Hi-C is very sparse and it can not fully capture the real 3D coordinates. Therefore, the reconstructed structure can be wrong even it has high agreement with the contact map.  Instead, I suggested to include chromatin tracing data to validate your methods, which you can refer to Higashi, which adopted some popular dataset.\n\n2. The ablation study of the design is very limited, which I think need substantial improvement.\n2.1 How to balance the structural stability loss and MSE loss? Are both losses needed?\n2.2 What is the performances without inertial frame canonicalization?\n2.3 What is ROPE's contribution to the performances?\n2.4 What is the contribution of Nystr¨ om Approximation for Structure Tokenization?\n\n3. The cross-resolution transfer benchmark is not so useful. We should expect we have different models for different resolutions, since they have very different biological focuses.\n\nINERTIAL FRAME CANONICALIZATION", "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": "9JPySjIz21", "reviewer": "Reviewer_YNMs", "summary": "This paper introduces InertialGenome, a novel Transformer-based framework for 3D chromosome reconstruction that is designed to be robust to different resolutions. The author proposes an inertial frame canonicalization and geometry-aware positional encoding scheme that combines 3D RoPE and Nyström approximation. The method demonstrates strong empirical results, outperforming baselines.", "questions": "1. On Line 219, the phrase \"relationships between anchor points\" is used twice in the same sentence. Is this a typographical error?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. The use of inertial frame canonicalization to achieve pose invariance is a reasonable and well-justified preprocessing step.\n\n2. The model consistently outperforms existing baselines in reconstruction metrics, TAD consistency and cross-resolution generalization.", "weaknesses": "1. In Sec 3.2, the textual description (L180-183) appears to contradict the mathematical formulation in Eq 5, where 'Selective' and 'Separate' modes seem functionally identical.\n\n2. In Eq 2, the query/key vectors are intended to be 6D vector. Can we understand six as a pair, just as we considered two as a pair in RoPE? This lack of clear explanation hinders understanding.\n\n3. It is not explicitly stated, but is the 'vocab_size' simply the total number of bins for a given chromosome (248,947 at 1Kb resolution)? This confirmation would be helpful for understanding.\n\n4. The paper claims RoPE-3D has limitations in 'capturing pairwise distance relationships' (L214), but this claim is not sufficiently substantiated or demonstrated. The paper should provide an ablation study to demonstrate the contribution of this Nyström approximation .\n\n5. In Sec 3.2, $u_i$, $A$, and $O$ are used in the main text but are only clearly defined several lines later, which hinder understanding.\n\n6. The paper relies exclusively on distance-based metrics (dSCC, dRMSE). For a task focused on 3D structure reconstruction, TM-score or RMSD are essential for evaluating the global structural similarity of the predicted conformations.\n\n7. The paper claims the method is 'resolution-agnostic' and attributes this to the inertial-frame alignment and RoPE. This explanation is high-level and lacks a deep analysis. A more thorough explanation is needed to precisely understand why this specific combination of components grants robustness to resolution changes, which is a key claim 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": "sn1Szmir20", "reviewer": "Reviewer_FLe5", "summary": "This model tackles the problem of inferring genomic 3D coordinates from a distance matrix derived from Hi-C data.  The approach involves training a transformer model to infer coordinates that, essentially, maximize the agreement between the observed distance matrix and the inferred one.  The novel aspects include using something called the Nystrom method to infer candidate sets of points from an RBF kernel matrix, and using RoPE embeddings in 3D.", "questions": "Why did you only use two single-cell Hi-C datasets?  There are many more available.\n\nHow did you decide which methods to compare your method against?  Did you run them yourself, or just compare to published results?\n\nI don't actually understand the sentences (lines 81-83) that describe how an \"inertial frame\" is used.  Some definitions of terms would be helpful.  What do you mean by \"inertial frame\" here, and \"position canonicalization\"?", "limitations": "", "rating": 2, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The overall approach contains several novel ideas, as outlined above.\n\nThe paper contains fairly extensive empirical results, largely following the evaluation protocols proposed in previous work in this area.", "weaknesses": "Overall, I found the empirical validation of the methods pretty unsatisfying.  In general, it's difficult to benchmark 3D genome reconstruction methods, because it is hard to find an orthogonal source of gold standard structures.   One common approach is to aggregate the single-cell reconstructions and compare the result to a bulk Hi-C dataset from the same type of cells.  Another approach is it to take similar approach using imaging data (e.g., chromatin tracing).\n\nThe list of methods to infer 3D chromatin contacts (line 44) is radically incomplete.  You should focus on methods that infer genome-wide contacts, which would leave off 3C, 4C, and 5C, but would include GAM, SPRITE, ChIA-PET, and many others.\n\nSimilarly, the list of methods for 3D reconstruction (lines 72-77) is incomplete.  The list later on the same page is better.  You should re-organize the text so that all the related work is discussed at once.\n\nThe critique of related work is pretty vague.  You just say that all of these methods \"rely on simplistic modeling of contact matrices as the sole input, lacking deeper structural interpretation, and their model expression ability may be constrained and limited.\"  It's not clear what you mean by \"lacking deeper structural interpretation\" nor what the \"model expression\" refers to.  I'd rather see specific critiques of particular methods, or at the very least a more precise critique.\n\nline 110: Not every method uses these two steps.  You should make clear that the problem does not have to be solved in this fashion.  And even if it is broken down into two steps, there is not agreement in the field about the proper transfer function to translate from counts to distances.\n\nMinor points:\n\nThe abstract should make it clear earlier (around line 15)  you are talking about single-cell Hi-C data.  You should also be careful to describe it as such in the intro to Section 4.  If you just say \"Hi-C,\" that typically means bulk Hi-C.\n\nline 107: The number of bins in the genome depends on what reference genome you use, not what cell line you are looking at.\n\nline 116: What does \"canonicalized\" mean here?\n\nline 216: Give the Williams & Seeger citation when you first mention the Nystrom method.\n\nline 218: I thought the phrase \"and the effectiveness of this approach has been confirmed by research\" was oddly vague.  What kind of research?\n\nA brief description of the Nystrom method (as in lines 218-221, but shorter) should appear in the Introduction.", "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": "woQTAlO3Kg", "reviewer": "Reviewer_M6rw", "summary": "This paper introduces InertialGenome, a novel Transformer-based framework for reconstructing 3D chromosome structures from Hi-C data. The core contributions are a two-stage process: first, it canonicalizes the initial 3D coordinates using an inertial frame to achieve pose-invariance, removing translational and rotational ambiguity. Second, it employs a geometry-aware Transformer, enhanced with 3D rotary position embeddings (RoPE) and Nyström-based structural features, to refine these coordinates. The proposed method demonstrates state-of-the-art performance on two single-cell datasets across four resolutions, showing particular strength in generalization and cross-resolution transfer learning.", "questions": "1. Comparison with CHROMFORMER: Could the authors please comment on the CHROMFORMER (NIPS'2022) model? Given its use of a Transformer for the same task, it appears to be a crucial baseline. Can you provide a comparison, or a compelling argument for its exclusion, to better situate your performance results?\n\n2. Novelty of 3D-RoPE: The recently proposed STRING (ICML'2025) introduced a similar 3D rotary position encoding. Could you please clarify the novelty of your 3D-RoPE implementation in relation to this prior work?\n\n3. Ablation Study for Loss Parameter α: Your hybrid loss function in Equation (12) is controlled by the parameter α. Could you please provide an ablation study showing how performance (e.g., dSCC and dRMSE) varies with different values of α? This is important for understanding the interplay between the two loss components and for reproducibility.\n\n4. In Section 3.4, you motivate the Value-Weighted MSE by stating that Hi-C data has higher reliability for smaller distances. However, another common approach for handling data with varying reliability or potential outliers is to use the Mean Absolute Error (MAE), which is inherently less sensitive to large errors than MSE. Could you please elaborate on the reasoning for choosing a weighted MSE scheme over a simpler, more robust loss function like MAE?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The primary contribution is the elegant and effective use of inertial frame canonicalization as a preprocessing step for 3D chromosome structure refinement. This method directly addresses the fundamental problem of pose variance in geometric deep learning, providing a principled way to standardize inputs. This standardization appears to be the key driver behind the model's impressive resolution-agnostic capabilities and strong generalization performance.\n\nThe paper's novelty lies not in a single component, but in the intelligent synthesis of several ideas. While inertial frames and Transformers are not new in themselves, their application and combination in this domain are novel and well-motivated. The theoretical analysis of the inertial frame's stability (Section 3.5) using the Davis-Kahan theorem is a significant strength.", "weaknesses": "The most critical weakness is the omission of highly relevant and recent related work in both the discussion and the experimental comparison. This significantly impacts the claims of novelty and the comprehensiveness of the evaluation.\n\n1. Missing Baseline: A key missing baseline is CHROMFORMER (NIPS'2022), which also uses a Transformer-based architecture for this exact problem. Without a direct comparison, it is difficult to ascertain whether the performance gains of InertialGenome stem from the novel inertial frame canonicalization or simply from the power of the Transformer backbone, which CHROMFORMER already established.\n\n2. Unaddressed Novelty: The paper's geometry-aware positional encoding (3D-RoPE) is presented as a novel adaptation. However, the recent work \"Learning the RoPEs: Better 2D and 3D Position Encodings with STRING (ICML'2025)\" appears to propose a very similar 3D extension of RoPE. The lack of citation and discussion of this work obscures the precise novelty of the authors' formulation.", "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)