Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
Authors: Maria Despoina Siampou
Jialiang Li
John Krumm
Cyrus Shahabi
Hua Lu
Venue: ICML 2025 poster (2025)
Keywords: geometry encoding
geospatial applications
GeoAI
Primary area: general_machine_learning->everything_else
TL;DR: A new unified encoding approach for encoding point, polyline and polygon geometries.

Abstract: Encoding geospatial objects is fundamental for geospatial artificial intelligence (GeoAI) applications, which leverage machine learning (ML) models to analyze spatial information. Common approaches transform each object into known formats, like image and text, for compatibility with ML models. However, this process often discards crucial spatial information, such as the object's position relative to the entire space, reducing downstream task effectiveness. Alternative encoding methods that preserve some spatial properties are often devised for specific data objects (e.g., point encoders), making them unsuitable for tasks that involve different data types (i.e., points, polylines, and polygons). To this end, we propose Poly2Vec, a polymorphic Fourier-based  encoding approach that unifies the representation of geospatial objects, while preserving the essential spatial properties. Poly2Vec incorporates a learned fusion module that adaptively integrates the magnitude and phase of the Fourier transform for different tasks and geometries.
We evaluate Poly2Vec on five diverse tasks, organized into two categories. The first empirically demonstrates that Poly2Vec consistently outperforms object-specific baselines in preserving three key spatial relationships: topology, direction, and distance. The second shows that integrating Poly2Vec into a state-of-the-art GeoAI workflow improves the performance in two popular tasks: population prediction and land use inference.

Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
Maria Despoina Siampou"
Jialiang Li
John Krumm
Cyrus Shahabi Hua Lu
Abstract
1. Introduction
The increasing availability of geospatial data from sources such as satellites, ground- based sensors, and crowdsourced platforms like OpenStreetMap (OSM)' (Lee & Kang, 2015; Jokar Arsanjani et al., 2015; Basiri et al., 2019), com- bined with the recent advancements in machine learning (ML) (Vaswani, 2017: Bommasani et al., 2021), has fu- eled significant progress in geospatial artificial intelligence (GeoAl) (Smith, 1984; Couclelis, 1986; Janowicz al., 2020; Gao et al., 2023). GeoAI leverages ML models to analyze geospatial objects, such as points of interest (POIs), building footprints, and vehicle trajectories, thereby extracting valuable insights that enable a variety of decision- making applications, including transportation network op- timization (I.i et al., 2018b; Mirowski et al., 2018), urban planning (Zhang et al., 2021; Wu al., 2022), energy man- agement (Sun al., 2020), and improved emergency re- sponse strategies (Kyrkou et al., 2022), to name few.
Encoding geospatial objects fundamental for geospatial artificial intelligence (GeoAI) appli- cations, which leverage machine learning (ML) models to analyze spatial information. Common approaches transform each object into known for- mats, like image and text, for compatibility with ML models. However, this process often discards crucial spatial information, such as the object's po- sition relative the entire space, reducing down- stream task effectiveness. Alternative encoding methods that preserve some spatial properties are often devised for specific data objects (e.g., point encoders), making them unsuitable for tasks that involve different data types (i.e., points, poly- lines, and polygons). To this end, we propose POLY2VEC, polymorphic Fourier-based encod- ing approach that unifies the representation of geospatial objects, while preserving the essen- tial spatial properties. POLY2VEC incorporates a learned fusion module that adaptively integrates the magnitude and phase the Fourier transform for different tasks and geometries. We evaluate PoLy2VEC on five diverse tasks, organized into two categories. The first empirically demonstrates that POLY2VEC consistently outperforms object- specific baselines in preserving three key spatial relationships: topology, direction, and distance. The second shows that integrating POLY2VEC into state-of-ther of-the-r-- GeoAl workflow improves the performance two popular tasks: population prediction and land use inference.
A fundamental step in GeoAI pipelines is the transforma- tion of geospatial data into latent representations that can be casily processed by ML models, a step formally known as encoding. common approach to encoding converts coordinate-best based geospatial data into formats compatible with established feature extraction models. Although effec- tive for specific tasks, this conversion often discards crucial spatial information, significantly limiting the generalizabil- ity of these models. For example, building footprints are frequently rasterized into images and processed with vision- based models for urban prediction tasks (Li et al., 2023; Balsebre et al., 2024). While this approach captures object shapes, it neglects important spatial relationships, such as the relative positioning and alignment of objects within the area. Similarly, POIs that are represented as text, by using attributes like category as input to language-basee based models, capture semantic relationships but omit precise spatial loca- tions (Huang et al., 2022). As a result, these approaches are application-spen -specfff and struggle to generalize across tasks that require a deeper understanding of spatial relationships.
"Equal contribution. Jialiang Li's work was conducted while the author was a visiting student at USC's InfoLab. 'Department of Computer Science, University of Southern California, Los An- geles. USA 'Department of People and Technology. Roskilde University, Denmark Department of Computer Science, Aalborg University (Copenhagen campus), Denmark. Correspondence to: Maria Despoina Siampou <siampou@ edc.edu>.
To address the aforementioned limitations, spatially explicit encoding techniques have been proposed. These methods
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
"hups://opensttretmap..gg

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Poly2 Polymorphic Fourier- Barier--Beeddd- Encoding of Geospatial Objects for GcoAI Applications
We formally define four key properties, shape preservation, direction preservation, distance preservation, and task flexi- bility, as essential criteria for evaluating the effectiveness of geometry encoding. These properties ensure the produced embeddings accurately capture the essential geometry char- acteristics while remaining versatile across different tasks, To demonstrate that POLY2VEC preserves these proper- ties, we conduct a two-part evaluation. First, we evaluate POLY2 on spatial reasoning tasks, showing that it out- performs the state-of-the-art specialized baselines by up to 17% for topological classification, 26% for directional classification, and 75% for distance estimation. Second, we show that integrating POLY2VEC into a state-of-the-arr GeoAl workflow reduces prediction error by 14% and 5% in population prediction and land use inference.
(a)
(
(c)
Figure| I: Visualization of the Fourier transform magnitude and phase of (a) road segment, (b) building, and (c) POI.
In summary, our contributions are:
preserve crucial spatial properties, while remaining com-
We introduce POLY2VEC, the first encoding framework that unifies the representation of various 2D geometries.
patible with downstream ML models. For instance, THE-
ORY (Mai et al., 2020) encodes the absolute positions of
We propose a 2D continuous Fourier transform-based encoding approach to capture crucial spatial properties, in- cluding shape, distance, and direction.
POIs using sinusoidal functions with varying frequencies. Xu et al. (2018) directly encodes the coordinates of trajec- tories using multi-layer perceptons and feeds to a GRU, capturing their sequential nature. For polygons, NUFT- SPEC (Mai et al., 2023) maps geometries into the spectral domain, effectively preserving key polygon properties such as topology awareness. However, the design of these meth- ods inherently limits their applicability, as they only capture the properties of the specific geospatial object they are de- vised for. This restricts their generalizability tasks involv- ing mixed geospatial object types, such as land use classifi- cation, where integrating points (e.g., POIs) and polygons (e.g., buildings) requires simultaneously preserving their spatial properties as well as relationships between them.
We design learned fusion strategy to adaptively combine Fourier magnitude and phase for diverse objects and tasks. Our experiments show that POLY2VEC preserves crucial geometry encoding properties, demonstrating its versatility in handling diverse geospatial objects, and task-flexibility when integrated into state-of-the-ee GeoAl pipelines.
2. Preliminaries
2.1. Problem Formulation
Definition 1 (Geospatial Object). A geospatial object g 2² is represented by aray P E Nx2, where each row is point (z. v). and N is the total number of points. The type of geometry (e.g., point, polyline, or polygon) is determined by the organization and relationships among these points.
In this work, we introduce POLY2VEC", a polymorphic encoding framework that unifies the representation of 2D geospatial objects, including points, polylines, and polygons. At its core, POLY2 leverages the Fourier transform to encode essential spatial properties, transforming the input geometries' into the frequency domain. Given that this transformation results in complex-valued features, the mag- nitude and phase components are extracted. As shown in Figure 1. these components complement each other: the magnitude reflects spatial extent, being uniform for points with no shape and varying for polygons and polylines, while the phase highlights directionality, such as the alignment of polyline. To combine these components into single repre- sentation, POLY2VEC incorporates a learned fusion module that adaptively balances their contributions based on the task and geometry type, producing a real-valued geometry embedding that ensures compatibility with ML models.
Polymorphic Encoding of Geospatial Objects. Given a dataset of geospatial objects = {0} E RN the goal is to define an encoding function es(g) : Nx2 + Rd. parameterized by 0, that maps each geometry 9 to d- dimensional vector V. termed as geometry embedding. The embedding dimension d remains constant across different geometry types, making es polymorphic. The encoding is intended to capture the following key properties.
Property (Shape Preservation). For any geometry g E G, its embedding should capture its structural characteristics: shape and boundary for polygons, length for polylines, and the lack of spatial extent for points.
Property 2 (Direction Preservation). For any geometries Ii. 9j E G, es should ensure their embeddings v., V, reflect their relative orientation.
*Code available at USC- USC-InfelI wninfllla//pooo yyo
https: htt///gttttutt com/
Wee refer to geometries and geospatial objects interchangeably.
Property 3 (Distance Preservation). For any geometries
2

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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAT Applications
Poly2Vec
Input
2D Continuous Fourler transform
Point f2(x,))
Learned Fusion
GeoAI Applications
Fourier transform
Polyline
fg,yy)
computation
Fo
v
f((x,y)
1 decomposition
Polygon
(x,33
(F, v))
Ø
Affine transform from the
rect60
(ii) Affine transform from the triangle A with vertices qrs to the canonical trangie with vertices
Fog (x.
transformation
affine
artitrary line segment to the canonical line segment 4,-
McCr, A),
(b) Affine transform arbitrary geometry to its corre- sponding canonical geometry.
(a) The workflow of POLY2VEC.
Figure 2: Overview of POLY2VEC.
Pi E G. the similarity of their embeddings v,, V, should monotonically decrease as their spatial distance [6: ااll increases, and vice versa.
Hermitian Symmetry. For real-valued functions f(z v). F(u, v) satisfies F(u, v) = F-u,,, (--u,-p -v), where F (u, v) denotes the complex conjugate.
Property 4 (Task Flexibility). The encoder eg should facil- itate multiple tasks without requiring modifications.
Magnitude and Phase. The Fourier Transform F(u, v) is complex-valued function composed of real part, Re( F(u, vi). and an imaginary part, Im (f((u, u)). The mag- nitude z(u, v) and phase 6(u, v) are defined as:
Properties 1-3 ensure that v captures all essential spatial information, while Property 4 guarantees flexibility for use across GeoAl models. Section empirically demonstrates that our proposed Eg satisfies these properties.
2(u, v)
/Re(F(u, x))² Im(F(u, (4)
(u,, u, v) atan2((m(F((uu)) Re(F(), (5)
2.2. 2D Continuous Fourier Transform Properties
key component of our encoding approach is the computa- tion of the 2D continuous Fourier transform (CFT) 4. For given 2D function fxx y). its Fourier transform is denoted 7 (s(x, v)) F(u, u)$ and is formally defined
3. Methodology
Figure 2 illustrates our proposed POLY2VEC, which uni- formly encodes arbitrary geospatial objects for GeoAI ap- plications. We first describe how the Fourier transform is derived for each geometry type, and then outline the learned fusion module for deriving the final geometry embeddings.
F(u, u) 1- / /- f(x, y) (eee-r(((utot))) dz dy
(1)
where j j= -1 and u, e are the frequency samples.
3.1. 2D Continious Fourier Transform of Geometries
We now summarize Fourier transform properties relevant to our approach following (Gaskill, 1978).
1.1.1. FOURIER TRANSFORM OF A POINT
A point p = (Tp: Up) E IR. is modeled as a 2D Dirac delta function, which represents the point as a distribution concentrated entirely at (Zp. Up), and can be expressed as:
Linearity. The Fourier transform of sum of functions de- noted as f(x y). is the sum of their corresponding Fourier transforms (1, w):
Sp(2, v) 5(x Tpp 1/ Wp)
(6)
a.f((zz v)
Σ (,, a e
(2)
To that extent, the Fourier transform of fp y) is given by: = ( (12((t,)));))))
Affine Transformation. For an
function [(Ax +T), where x = [x. v) its Fourier transform is:
where (u, v) are the frequency components.
F[[(Ax+T)) |det(A)]
1
22n A TuF(AAAT u) (3)
The Fourier transform magnitude for any point is constant, (p v) 1, while the phase Φ,,u,, v) encodes its location.
where [u, A e R2x2 is the affine matrix, and
As shown in Figure 2, deriving the Fourier transform for polylines and polygons involves additional steps. Polylines are divided into line segments, and polygons are triangulated into oveeovellg triangles. The Fourier transform
T E R is the translation vector.
"We use Fourier Transform and CFT interchangeably. FFor compactness, we use F(u, vi) to describe the CFT.
3

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Poly2Vec: Polymorphic fourier-Based Encoding of Geospatial Objects for GeoAI Applications
computed for each component by affine transforming them to their canonical geometry, and the linearity property of Eq. (2) is used to compute the Fourier transform of the
1.3.3. FOURIER TRANSFORM OF A POLYGON
To compute the Fourier transform of a polygon we decom- pose it into set of non-overlapping triangles using standard triangulation techniques'. We thus begin with the Fourier transform of a canonical isosceles right triangle and then generalize to its computation for arbitrary triangles.
original geometry". Details for polylines and polygons are specified below, with derivation details in Appendix A.2.
3.1.2. FOURIER TRANSFORM OF A POLYLINE
Consider the canonical isosceles right triangle 4: with ver- tices a=(0, 0), b=(1, 0). and c= 1). represented as:
We begin by deriving the Fourier transform of a canoni- cal line segment and then generalize to any arbitrary line segments. Consider the canonical line segment le, which extends from (- 2 0) to b (1) 0) in R2, as shown in Figure 2b. Then, le can be represented as:
1, if S r S 1 and 0 S y S 2, (12)
(z,
otherwise.
(z. y) = rec)()))(y)
(8)
The Fourier transform of fa. (2. y) is then given by
where (yy) represents Dirac delta function ridge along the ar-xiis and rect(x)) restricts the ridge to the interval |x]| S 3-
F4 (u, v) S/
//a () dy dr
The Fourier transform of (a, y) is given by:
4T²uv(u v) ((u + cos((ru))
1
Fi, (u, u) = sinc(u)
(9)
ucoo((2T (u + t)) -v- - -j((u + v) (2Tu)
Now consider an arbitrary line segment I with endpoints q = (Iigs (g)) and (r), To compute the Fourier transform of I, map it to the canonical line segment le using the affine transformation property. To compute this, we first introduce an auxiliary point e = (2. 1) to the structure of 4 so that it is not colinear with ab. This point maps to another auxiliary point s introduced in the structure of the arbitrary line segment 1. The auxiliary point s is defined as s r +++ n, where n = (bg - Gr. - xzy] representing 90 clockwise rotation of the vector r q- Note that the line segments qr and rs have the same length.
-usin(22(u + +)]]]
(13)
Next, we compute the Fourier Transform of an arbitrary triangle A, with vertices = (Fig. Bg). r r= - yr). and = (v), by mapping to the canonical triangle using the affine transformation property (Figure 2b). The affine transformation matrix is defined A [a bellqrs]]-
By substituting the vertices of A into A and applying Eq. (3), the Fourier Transform of the triangle (u, v) can be calculated. In this computation, the determinant of A, (et((A) = 1 where a is the area of the triangle A.
Given the points q.r. r, and b, we then construct the affine transformation matrix A = [a b cllqqr r s]-1. By applying Eq. (3), the Fourier transform of an arbitrary line segment 1, with endpoints q. r, is expressed as:
2a
Finally, the Fourier transform of an arbitrary polygon pg. given the linearity property of Eq. (2), can be computed as:
v) = de)(
1
12 Tu T (A u)
Tpu (u, = > FA (u.
(14)
= ||q 12x
att "sine T (r q)) (10)
At (u, v) 0), the Fourier transform is (1
where FA, (u, v) is the Fourier transform of the i-th triangle, and Tpg is the total number of extracted triangles.
||q 1//a, the squared length of the line segment.
Finally, following Eq. (2), the Fourier transform of an arbi- trary polyline pl is computed as:
Building on the Fourier transform computation described now extract the frequency representation of given geometry g, expressed spatial function fg (z. y) over I22 as, F. = [F1. Fz..... Fw]T E CW, where W is the number of frequency components, and F, = F(u,, vs) represents the value of the Fourier transform evaluated at the specific frequency coordinates (u,, v:)..
earlier,
Tr
(u v) = Σ (u(,v))
(II)
where (u, v) is the Fourier transform of the i-th line segment and Ti is the total number of line segments.
"We adopt Constraint Delauney triangulation in this paper. SSpecial cases where u. v, and u-v v approach zero are handled separately, and presented in Appendix A.2.3.
'The same methodology can be adopted compute the CFT of polyyo..
4

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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAT Applications
To sample the frequency components, we employ a geo- metric series sampling strategy (Mai al., 2023), which balances low and high-frequency components to capture both global and local details. We also experimented with learned frequency sampling but found that the two strategies produced nearly identical results. Detailed comparisons are presented in Appendix A.3.2.
4.1. Spatial Reasoning Tasks
This section addresses RQ1 and RQ2, empirically evaluat- ing POLY2 ability to preserve the properties of Sec- tion 2.1, against specialized basclines. We frame these eval- uations as spatial reasoning tasks, which are fundamental to broader applications like geospatial question answering (GeoQA), and require accurate spatial understanding (Pun- jani et al., 2018; Papamichalopoulos et al., 2024).
3.2. Learned Fusion of Fourier Transform Features
Datasets. We evaluate our approach on two OpenStreetMap (OSM) datasets from Singapore and New York. Each dataset includes three types of geospatial objects: POIs (represented as points), main roads (as polylines), and buildings (as poly- gons). All input geometry coordinates are normalized to the range [-1, x [-1, 1]. Additional dataset statistics and preprocessing details are provided in Appendix A.4.
Given that F, consists of complex values, we decompose in two real-valued vectors of the magnitude z and the phase 6, computed as in Eqs. (4) and (5), respectively. This transformation ensures the representation is compatible with downstream ML models, which typically operate on real- valued inputs. Furthermore, the magnitude captures the intensity of contributions at different frequencies, reflect- ing the geometry's 's size and overall shape, while the phase & encodes positional and orientational information of the geometry's 's features (Zahn& & Roskies, 1972).
Baselines. We include three categories of baselines: point encoders: (i) DIRECT, directly utilizing coordinates (Chu et al., 2019), (ii) TILE, a discretization method (Berg et al., 2014), (iii) WRAP, a coordinate wrapping mecha- nism (Mac Aodha et al., 2019), (iv) GRID, inspired by posi- tion encoding (Mai et al., 2020), and (v) THEORY, a multi- scale encoder (Mai et al., 2020). All point encoders are extended to other geometries handling them as sequences of points, following Rao et al. (2020); Xu al. (2018). poly- line encoder: (i) T2VEC classic trajectory encoder (Li et al., 2018a). polygon encoders: (i) RESNETID ID (Mai et al., 2023) and (ii) NUFTSPEC (Mai et al., 2023). More details on baselines can be found in Appendix A.5.
While the final geometry embedding can be created by simply concatenating z and Φ. their relative importance should vary with the geometry type and the downstream task. For instance, the magnitude of points is always 1, whereas it encodes the shape and size of polygons. There- fore, when encoding points, the phase should contribute more than the magnitude in the representation. To this end, POLY2VEC adaptively learns the importance of mag- nitude and phase through two separate transformations (z) and (h)) where h,: RW + W and h RW + W are separate MLPs for z and d respectively.
4.1.1. TOPOLOGICAL RELATIONSHIP CLASSIFICATION
Finally, the transformed vectors z E RW and o' E RW are concatenated and passed through a final MLP h: BRW. + Rd to produce the final geometry embedding == d")= E IRd, which can be inputted any ma- chine learning model M, such that (v) y, where y represents task-specific outputs. We will empirically verify that v preserves the key properties in Section 4.
This task classifies topological relationships defined by the DE-9IM model (Clementini et al., 1993) for geospatial ob- ject pairs. We present the supported relationships in Table 3.
Settings. The geometry embeddings of each pair are con- catenated, passed through a 2-layer MLP with Nc output units,, corresponding to the number of relationships. We adopt cross-entropy loss for optimization. Performance is measured by accuracy, precision, recall, and FI-score. Ac- curacy results are presented in Table 1, while the remaining metrics are presented in Appendix A.10.
4. Experiments
In this section, we conduct experiments to evaluate the effec- tiveness of POLY2VEC across four key research questions: [RQ1) Does POLY2VEC effectively preserve the critical geometric properties of shape, direction, and distance?
Results. From Table 1, we observe that POLY2VEC con- sistently outperforms all baselines across all experiments. Unlike specialized encoders that excel only for specific pairs, POLY2VEC's performance is consistent across all geometries, highlighting its versatility and generalization capabilities. The second-best performing models vary by geometry type, with T2VEC for polylines and NUFTSPEC for polygons. This shows that simply extending point en- coders to handle all geospatial objects is not adequate, as it fails to preserve characteristics like the object's 's shape and position, leading to decreased performance. Finally, all mod-
[RQ2] How does POLY2VEC perform in comparison to baseline encoding methods tailored for specific object types? [RQ3] Can integrating POLY2VRC into existing workflows lead to improvements in their performance? [RQ4] Does learned fusion boost POLY2VEC performance?
5

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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAT Applications
Table I: Model accuracy on topological relationship classification. Best and second best are highlighted.
Singapore
New York
Methods
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon- polygon
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon-
polygon
RESNETID NUFTSPEC
0 4,477
0.452mms
602
0.5855mm
T2VEC
0.728
0.807o.121
DIRECT
0.8230ors
0.8430o0s
0.733
0.368o.11
0.357ams
0.8466011
0.909
0.745o..o
0.495
0.446ums
TILE
0.790me
0.700sow
0.5055aos
0.459k.0r3
0.41 1ow9
0.659nuss
0.783
0.502n.00
0.494kons
0.405umx
WRAP
0. 0,886m00
0.880s.os
0.716
0,4762010
0.349
0.886m.06
0,8 0,8800017
0.7330.00
0.550
0.38 I
GRID
0.8 0.8460.04
0.844o,04
0.697
0.458o.004
0.335
0.822aus9
0.89 1oms
0.7390.00
0.51 6ooos
0.38 nus
THEORY
0.892
0.90 0,900s.ors
0.7 11uus
0,450
0.46 1oo4
0.897ance
0. 9.909
0.7340.00s
0.59 1.00
0.455tu41
POLY2VEC
0.955mmm
0.949aam
0.812mmm
0.509
0.702amm
0.953umm
0.980
0.8.00..044
0.641amm2
0.684mmm
Table 2: Model accuracy on directional relationship classification. Best and second best are highlighted.
Singapore
New York
Methods
point- point
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon- polygon
point- point
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon-
polyyon
RESNETID
1.819
0.747amo
NUFTSPEC
0.807a
0.698 ov
T2VEC
0.268xe15
0. 0.249
DIRECT
0.880uos
0.841nom
0.844mox
0.820kcee
0.830scoo
0.752>mr
0.877
0.766ums
0. 06836
0.653
0.78-4
0.6944aeer
TILE
0,253mxm
0.268wazz
0.273waos
0.326xcoa
0.454a.e:
0.394ame
0,245
0,258
0.3
0.217
0.466ua
0.349emz
WRAP
0,861oms
0.804eam
0. 0.8030034
0.78 Toose
0.831oce
0.778
0.809
0. 0.669am
0. 0.749kms
0.5960019
0.772n.0
0.602x
GRID
0.882omm
0.728ma
0.77 1003
0 0.699%..
0.64Ia.0
0.534.
0,868
0.590mes
0. 0,64600a
0.4380o04
0.752
0.
THEORY
0.912aars
0.867mam
0 8.858
0.834ac
0.860
0.735am
0.892ami
0. 0.760mar
0. 0.826mm
0. 6..844mm
0.7 0.775uu00
0.555am1
POLY2VEC
0.932aas
0.935
0.925ma22
0.906xs*
0.907axer
0.833axee
0.909
0.89
0.8 0.883
0.863
0.876
0.785axe3
Table 3: Topological relationships of geometry pairs.
Settings. We follow the same setting as Section 1.1.1, with N₂ 16, and report the same metries. Accuracy results are in Table 2, with the rest in Appendix A.11.
Geometry Pair
Topological Relationships (a relationship b)
point-pe polyyiel
disjoint, intersects
point-polygon polyline-polyline polyline polygon
disjoint. contains disjoint, intersects disjoint, touches, intersects, within
Results. From Table 2, we observe that POLY2 consis- tently outperforms all baselines across all experiments. This demonstrates its ability to effectively preserve the direction (Property 2) among diverse geometry types. While polygon encoders outperform the extended point encoders also in this task, T2 VEC underperforms. This is due to T2 V2C'''s strat- egy of assigning coordinates to grid cells during encoding, which is effective for trajectory related tasks, but introduces discretization artifacts that affect angular relationships. A similar limitation is observed in the performance of TiLe, which also relies on discretizing points into grid cells. In contrast, POLY2VEC encodes geometries holistically, pre- serving their relative orientation and avoiding these pitfalls.
polygon-polygon
disjoint, touches, intersects, contains, within, equals
els perform better when detecting relationship is a binary classification (e.g., point-polyline in Table 3), compared to multi-classification (e.g., polygon-polygon in Table 3). This is expected, as the latter requires capturing fine-grained spatial nuances, posing greater difficulty. In summary, these results emphasize the importance of preserving shape (Prop- erty 1), and distance (Property 3) in geometry embeddings. POLY2VEC's ability to do so, along with its unified frame- work, enables it to consistently outperform baselines.
4.1.3. DISTANCE ESTIMATION
This task evaluates whether geometry embeddings preserve pairwise distances (Property 3).
4. 1.1. DIRECTIONAL RELATIONSHIP CLASSIFICATION
This task classifies the directional relationships defined by the 16-compass direction model of two geospatial objects.
Settings. The original distance is estimated by the Euclidean distance of the geometry embeddings. The mean squared
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Poly 2Vec: Polymorphic Baurier-Based Encoding of Geospatial Objects for GcoAT Applications
10
onggnnl
100
oeginal
1,0
eniiinl prddited
10
orggana prodicual
Lo
ociginal prodictad
1.0
Daturaoss
peaiical
Distances
Distapocs
Distancess
85
0.5
0.5
000
0.0
0.0
0.5
on
0.
0.
1.
Original distances
Originel distances
Original distances (e) Wrap
Original distancees
Original distances
Original distances
(4) Direct
(b) Tile
(d) Grid
(c) Theory
(1) Poly2Voc
Figure 3: Distance scatter plots of point-polygon pairs on Singapore dataset for different encoders.
error (MSE) is adopted as loss function. We compare the differences between the predicted and original distances in Figure 3 and report the mean absolute error (MAE) in Appendix A.
the variants following the original work. For land use in- ference, we report L L1-distance, KL-divergence, and cosine similarity metrics. For population prediction, report MAE, root mean squared error (RMSE), and coefficient of determination (1²²).
Results. Figure 3 depicts that the predicted distances gener- ated by POLy2VEC are closely aligned with the original dis- tances, whereas the predicted distances from other point en- coders appear more scattered. This highlights POLY2VEC's superior ability to preserve spatial distance relationships across various geometry types. Methods like DIRECT are overly simplistic, while approaches such as TILE, GRID, and WRAP introduce location distortions through discretization or periodic transformations, affecting the distance preser- vation. By leveraging the Fourier transform, POLY2V effectively captures both the positions and relative spatial relationships of the geometry pairs, enabling it to implicitly encode distance as a core property into its embeddings.
The results for both tasks are presented in Table 4. Removing the distance-bias term from REGIONDCL, leads to a noticeable drop in performance, emphasizing the im- portance of encoding the spatial location and alignment of objects for accurate land use and population predictions. When POLY2VEC is added, the performance improves sig- nificantly. This shows that POLY2VEC can adequately cap- ture the shape and orientation of objects, similar to the initial image-based features, while also benefiting from the inclusion of object's 's location. Overall, POLY2VEC encodes spatial information directly into its embeddings, removing the need for additional mechanisms like the distance- term. This improves performance while simplifying the pipeline, showcasing the task flexibility of POLY2VEC and its potential for effective integration into GeoAl workflows.
Results.
4.2. Integration In an End-to-End GeoAI Pipeline
The section addresses RQ3, demonstrating the benefits of integrating POLY2VEC into existing GeoAl workflow.
4.3. Ablation Study
Dataset. We utilize the same dataset as in Section 4.1. The regions for both cities are extracted using the administrative boundaries of Singapore Subzones and NYC Census Tracts
This section addresses RQ4, highlighting the benefits of the proposed learned fusion module.
Settings. We include three variants: (1) w/mag uses only the Fourier transform magnitude, (2) w/phase uses only the phase, and (3) w/concat combines both via concatenation.
Baseline. We adopt REGIONDCL (Li et al., 2023), an unsupervised urban region representation learning frame- work that uses buildings and POIs from OSM for land use inference (predicting urban functional distributions) and population prediction (estimating region population). REGIONDCL encodes building footprints by converting their coordinates into images and extracting features using ResNet18 while using categorical features for POIs. To address the loss of location information, REGIONDCL em- ploys iistancee-baseed transformer, which introduces bias in the self-attention mechanism to prioritize closer objects. We provide more details in Appendix A.9.
w/ Mag
w/
Phase
concat
Poly2Vec
Accuracy
3.39
Accuray
389
3.6
264
0.40
Singapore New York
0.40
Singapore
New York
Singapore
New York
(o) Topolopical relationship
(b) Directional relationship
(c) Distance estimation
classification
classification
Figure 4: Ablation study for the point-polygon dataset.
Settings. We evaluate three variants: (1) REGIONDCL, the original framework, (2) REGIONDCL w/o distance-bias removes the distance-biased term, and (3) REGIONDCL. w/ Vol removes the distance-biased term and replaces the encoding part of the pipeline with POLY2VEC. The training and evaluation strategies remain unchanged across
Results. As shown in Figure 4, among the variants, w/ mag performs the worst across all tasks, particularly in direc- tional relationship classification, as the Fourier transform magnitude primarily captures shape, which is insufficient on its own to address these tasks. In contrast, w/ phase, which

---

Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAT Applications
Table 4: Comparison of methods for Land Use Classification and Population Prediction. Best values are highlighted.
Land Use Classification
Methods
Singapore
New York
L1 +
kL.
Cosine f
Ll 4
KL
Cosine t
RegionDCL.
0.498 0.088
0.294 0.047
0.879 + 0021
0,418 = 0.022
0.229 = 2003
0.912 0,006
RegionDCL w/o distance-bias
0.558 0.043
0.369
= 0.067
0.844 + 0023
0.439
= 0.012
0.244 = 0012
0.904 = 0.005
RegionDCL w/ Poly2Vec
0.484 0,921
0.278 0.025
0.881 + mn22
0.397 0.010
0.212 0.011
0.923 0,007
Population Prediction
Methods
Singapore
New York
MAE |
RMSE 4
R² f
MAE L
RMSE 4
R² 1
RegionDCL
5807.54 52222
7942.74
77944
0.427 1 0.100
5020,20
A 21663
6960.51
1 22235
0,575
0,009
RegionDCL w/o distance-bias
6018.94 98.94 1:67711
8214.58
95111
0.385
1 0.87
5293.04
1 27131
7348.86
1 314022
0,532
AL 0.000
RegionDCL w/ Poly2Vec
4957.58 506.82
6874.47
551.7
0.561 + 0.117
4602.75
179.66
6393.38
279.70
0.621 0,037
encodes location information, performs better since relative location, here, is more crucial. Combining both through w/ concat shows improvements, highlighting the importance of integrating both shape and location information. In contrast, VEC outperforms all variants by employing learned fusion strategy that adaptively balances the contribution of magnitude and phase based on the task and geometry type. Particularly, this strategy benefits POLY2 more in tasks such point-related distance estimation, where points lack spatial extent, and thus magnitude should contribute signifi- cantly less than the phase containing location information. More ablation studies are presented in Appendix A.13.
resented as ordered sequences of points. Most approaches rely discretization. For instance, Li et al. (2018a) uses grid-based encoding, training an RNN on degraded data to infer missing information and embedding grid cells to capture relative spatial positions. Other approaches directly use coordinates, leveraging sequential models (i.e. RNNs) to process the encodings (Feng et al., 2018; Xue et al., 2021; Rao et al., 2020; Xu al., 2018), but require strict sequential ordering and may overlook geometric relationships.
Polygon encoding has gained significant attention. Veer et al. (2018) 8) employ elliptic Fourier descriptors to approximate polygon outlines and utilize bidirectional L.STM and 1D CNNs to encode vertex sequences. Mai et al. (2023) used a ID ResNet architecture with circular padding for loop origin invariance. Other approaches use the non-uniform Fourier transform (NUFT) to map polygons the spec- tral domain, converting them back into images via inverse Fourier transforms (IDFT), though this suffers from the limitations of grid-based approaches (Jiang et al., 2019a;b), Mai et al. (2023) refine this approach by omitting the IDFT. POLY GONGNN (Yu et al., 2024) encodes multipolygons, modeling their shape details and inter-polygonal relation- ships through heterogeneous visibility graphs
5. Related Work
Existing geometry encoding approaches often focus on one shape type, with point encoders receiving the most atten- tion. Direct point encoding methods simply feed raw co- ordinates into neural networks but fail to capture details of location distributions (Xu et al., 2018; Chu et al., 2019). Dis- cretization methods assign points to predefined grid cells, as seen in approaches leveraging location context for im- age classification (Tang et al., 2015; Berg et al., 2014), but struggle with fixed resolution and imprecise representations. Sinusoidal methods encode normalized coordinates using sinusoidal functions, such as WRAP, which captures cyclic patterns (Mac Aodha et al., 2019). Extensions like multi- scale encoder (Zhong et al., 2020) introduce multiple si- nusoidal scales. THEORY improves this by computing the dot product of coordinates with unit vectors separated by 120% (Mai et al., 2020). There are also point encoders that jointly model location and neighborhood features (Qi et al., 2017; Yin et al., 2019; Zhou & Tuzel, 2018).
While effective for specific geometry types, existing ap- proaches are devised for specific geospatial objects. En coding heterogeneous coordinate-based data remains a chal- lenge, as current methods, in such cases, either use separate encoders for different object types, thereby adding complex- ity, or convert geometries into known formats (i.e., image, text), leading to a loss of spatial precision. This limitation is particularly critical for GeoAI models that aim to incor- porate coordinate-bant based geospatial data as an additional modality (Zhang et al., 2024; Mai et al., 2024). POLY2VEC addresses this gap by uniformly encoding points, polylines, and polygons within the same framework, offering a level
Unlike points, there are no dedicated approaches for encod- ing polylines in their generic form. The closest relevant work trajectory encoding, where trajectories are often rep-
8

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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
of versatility not demonstrated by prior methods.
Berg, T., Liu, J., Woo Lee, S., Alexander, M. L.,, Jacobs, D. W., and Belhumeur, P. N. Birdsnap: Large-scale fine- grained visual categorization of birds. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 201 1-2018, 2014.
6. Conclusion and Future Work
We proposed POLY2VEC, a unified encoding framework for geospatial objects that preserves essential spatial properties, including topology, directionality, and distance. By out- performing object-specific baselines and improving down- stream tasks like population prediction and land use infer- ence, POLY2 demonstrates its versatility and effective- ness in GeoAl pipelines. Future work will explore extending POLY2V EC to highee-dimensional geometries, including 3D shapes, and its integration into Geo-Foundation models as a unified representation for coordinate data modalities.
Bommasani, R. Hudson, D. A., Adeli, E., Altman, R., Arora, S., von Arx, S., Bernstein, M. S., Bohg. J., Bosse- lut, A., Brunskill, E., et al. On the opportunities and risks of foundation models. arXiv preprint 10:::0208.02555 2021.
Chu, G., Potetz, B., Wang, W., Howard, A., Song, Y., Brucher, F., Leung, T., and Adam, H. Geo-aware net- works for fine-grained recognition. In Proceedings of the IEEE/CVE International Conference on Computer Vision Workshops, pp. 0-0, 2019.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here. Our improved representation of 2D geometry for deep models could lead to more accu- rate, versatile GeoAl applications. leading to better under- standing the Earth and improvements for the environment, transportation efficiency, and access equity.
Clementini, E., Di Felice, P., and Van Oosterom, P. A small set of formal topological relationships suitable for end- user interaction. In International symposium on spatial databases, pp. 277-295, Springer, 1993.
Couclelis, H. Artificial intelligence in geography: Conjec- tures on the shape of things to come. The professional geographer, 38(1):1-11, 1986.
Feng, J., Li, Y., Zhang, C., Sun, E. Meng, F., Guo, A.,
Acknowledgments
and Jin, D. Deepmove: Predicting human mobility with attentional recurrent networks. In Proceedings of the 2018 world wide web conference, pp. 1459-1468, 2018.
This research has been funded in part by the NIH award R01LM014026 and NSF award DMS-2428039, Any opin- ions, findings, and conclusions, or recommendations ex- pressed in this material are those of the authors and do not necessarily reflect the views of the sponsors such as the NIH and NSF. J. Li and H. Lu's work was supported by Indepen- dent Research Fund Denmark (No. (032-00481B) Part of H. Lu's work was done when the author was employed at Roskilde University.
Gao, S., Hu, Y., and Li, W. Handbook of geospatial artificial intelligence, 2023.
Gaskill, J. D. Linear systems, Fourier transforms, and optics.
John Wiley & Sons, 1978.
Huang, J., Wang, H., Sun, Y., Shi, Y., Huang, Z., Zhuo,
A., and Feng, S. Ernie-geol: A geography-and-langun pre-trained model and its applications in baidu maps. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 3029-3039, 2022.
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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
A. Appendix
A.I. Geospatial Objects Definitions
Definition 2 (Point), point is a cero-dimensional geometric entity in R², defined by single coordinate (æ, y), where
y E R. point represents specific location in the plane but has no extent, size, nor dimension.
Definition (Line Segment), A line segment a one-dimensional geometric object R?, defined as a straight line segment between distinet endpoints p1 (æ1, v)) and p2 (22, 1/2).
Definition (Polyline). A polyline one-dimensional object in RR represented by array P E RNx22 where each row point ps (I,, 34). It consists of connected line segments formed by consecutive points ps and P++1 for S < N, with P1 + PN- Definition (Polygon). A polygon is a two-dimensional geometric object in R²², represented as closed sequence of points forming its boundary. It is defined by an array P E IR,x2, where each row corresponds to a point (Ii,W)) E IR² and (I1, v1) (x).
A.2. Analytical Calculations of Fourier Transform
A.2.1. FOURIER TRANSFORM OF A POINT
By representing a point (Ip, yp) E RR2 as 2D Dirac delta function [2(2, y) = 8(x Tp. Up) the Fourier transform of f1( y) can be derived as follows:
u)(u, = (1f((r,))
fpl2, y)e j2m( dr dy
8(x Jpp 11 yr)e 12 (2x dx dy
(2
where (u, 2) are the frequency components.
A.2.2. FOURIER TRANSFORM OF A POLYLINE
Canonical line segment. We express the canonical line segment le extending from (-2, to bo (2. b as
fi, (I. 3) = rec)(z)8(y). where rect(z) restricts the ridge to |a| < 2. and o(y) represents a Dirac delta function along the --axis. The Fourier transform of ft. (x. 9) is:
Fu v) $ (2,
fi. (z, y)e (2 do dy
rect(z)o(y)e (2 dx dy
Using the sifting property of the Dirac delta function, the integral over evaluates to the value of the integrand at y = 0:
Ft. (u, v) =
rect(a:) f2zux e (f2m((()) da
rect(a:) e ffzuu da
sinc(u)
where (u, u) are the frequency components and v = 0.
Arbitrary line segment. We consider an arbitrary line segment with endpoints q = (xg. My) and r (x,, to compute the Fu v), we map it to the canonical line segment 1, using affine transformation. For this purpose, we introduce an
12

---

Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
auxiliary point (5: 1) the structure of le that it is colinear with ab. This point maps another auxiliary point introduced the arbitrary line segment 1. The auxiliary point is defined representing 90" clockwise rotation of the vector r q. Note that the vectors qr and rs are the same length.
n, where (yg Ur, 3ir Tig)
Given all the above, the affine transformation matrix A is defined as:
@1 bi
A =
ba
Then the values of are computed as follows:
[qrs| = b
A aa b [
2
Zqq
Ir
Er + 14
yr
0
Yq
Ur
Dr
+ Fp
Zig
1
1
1
(u) 1 22 -
-Eg + Kr
V/
+
3/
D
3g
+ Z2
2 + x Fgyr
0
D
where
D det(A)
(xgg
xz,² 2 + (g4g yr )2
the determinant of A.
Following the affine Fourier transform property from Eq. (3), the Fourier transform of an arbitrary line segment / with endpoints (1igs 94) and (Ir, yr) is:
Fi(u, = 2 (2,
1
-2me A
det det(A))
which can be rewritten as:
Fi(u,))
(20
byz - 022
byu + a1
(15)
D
D
D
where b (b102 - byc1) and 1 (Ugc1 (2)))
By substituting the specific values into Eq. (15), Fi (u, v) can be simplified to:
j2m fetter a votur w) sinc((a,, xz)U + (yr, 8y)u
Fi(uv)) (Igg - x2, (yg yr
A.2.3. FOURIER TRANSFORM OF A POLYGON
Isosceles canonical right triangle. The canonical isosceles right triangle 4 with vertices a = (0,0), b (1, and (1, 1), is represented by the function f4, (x,y) y) which equals inside the triangle and 0 otherwise.
13

---

Poly2Vec: Polymorphic Tourier-Based Encoding of Geospatial Objects for GeoAI Applications
The Fourier transform of L4 (2, y) is computed as:
F2 (u, v) & (5a (2,
-f
La (2, v) (2r((rto))))) dy dr
!'" () dy dr
E f2x((tt) E edauur da
-j2TU
1
Vo +22x(uto) dx (( e- dx
j2TU
[uu+ ule
f2au -12a ue (ee-((()))))
(16)
4 2 + b)
Using Euler's formula (eJ0 cos 0 + sin 0), we can expand Eq. (16) to:
F4 (u, uu + v) ((u + v) cos(2mu)= cos(2 o)) v) -j((u sin((( usinn (i (u + v)))
This equation is undefined for some values of (u, v). We present the Fourier transform for each special case:
(0,
(0,
(j2TO + cos(( - ssinnnntn -1)
2
(u,
1
(cos(2TT)) 2Tusin((TTt - -))(sssn(22r)) 2T cos cos (2T))
27222
Fa(-u,u) = 2 j2mo + cos(2 + (iin(2 -1)
Arbitrary triangle. We calculate the Fourier transform of an arbitrary triangle A, with vertices q, r, s by using the affine transformation property. To that extent the affine transformation matrix A defined as:
Q1
Ci
A
02
b2
C2
0
Then the values of A are computed as follows:
A sq = a c
A b c| qqrs--
0
Fqq
Kr
+ Vq yr
ys Yr
Fr -
(n - xx) + F. - -Ba)
0
0
Yy
yr
+
Ey
= D
1/4 yr
Igg
IzYr Bq
1
0
D
where
|D|
3eg 2y(Ur Uo) x, Ug) + x. (Vg Vr
the determinant of A.
If the area of A is then D
2n
Finally the Fourier transform FA v) can be calculated by substituting the affine transform parameters into Eq. (3).
For the case of (0, 0) we get that: Fa (0 0) 1 F4 (0,0) = 2/ a, which is the area of A.
14

---

Poly2Vec: Polymorphic Courier-Based Encoding of Geospatial Objects for GcoAT Applications
A.3. Frequency Sampling Strategy
A.3.1. DETAILS ON THE GEOMETRIC FREQUENCY SAMPLING
We sample frequencies as a geometric series balance the contribution of low and high-frequency frequency components. Formally,
ft = fuin Ps. i= 000,,,.... W -1
where ft is the i-th frequency, fmin. fmax correspond the minimum and maximum frequencies and W is the number of sampled frequencies in each dimension. Pj the step ratio and is defined as ps (Jmsx)
Using this sequence, we construct 2D meshgrid of frequencies, denoted as (U.V), centered around zero. Due to the Hermitian symmetry property of the Fourier transform, we only compute frequencies for half of the plane.
While uniform sampling is an alternative, previous studies suggest geometric sampling is better suited for tasks like ours, as naturally balances the significance of low- and high-frequency components (Mai al., 2023).
A.3.2. ADDITIONAL STRATEGIES
learned sampling
geometric frequeney mapping
1.0
1.0
0.05
Accuracy
Accuracy
0.8
0.6
0.6
MAE
0.03
0.4
0.4
0.02
0.2
0.2
0.01
0.0
Singapore New York
0.0
Singapore New York
0.00
Singapore New York
(a) Topological relationship (b) Directional relationship
(c) Distance estimation
classification
classification
Figure 5: The effect of frequency sampling strategy on point-polygon pairs.
To investigate whether learning the frequency values would improve performance, we conducted an experiment where the frequencies were treated as learnable parameters and optimized alongside the model. Our results are reported in Figure We observe that learning the frequencies does yield significant improvements over fixing the frequencies any of the tasks. This suggests that the geometric sampling approach is sufficiently effective for balancing low- and high-frequency contributions, and learning the frequencies does not provide additional benefits for the tasks considered.
A.4. Dataset Details
We utilized publicly available OpenStreetMap (OSM) datasets for Singapore and New York, obtained from Geofabrik in osm. pbf format. Geospatial objects, including POIs, roads, and buildings, were extracted using OSM-specific tags (amenity, shop, tourism, leisure for POIs, motorway, trunk, primary, secondary for roads, and building for buildings). Region partitions were derived from Singapore Subzones 10 and NYC Census 1racts Dataset statistics are presented in Table
City
# POIs
# roads
# buildings
# regions
Singapore
4,347
45,634
109,877
304
New York
14,943
139,512
1,153,088
2,324
Table 5: Statistics of the Singapore and New York datasets.
page
15

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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
Labels for the land classification task were sourced from the Singapore Master Plan 201912 and NYC MapPLUTO Following previous approaches (Li et al., 2023), we merge the fine-grained land use classes into five major categories, including Residential, Industrial, Commercial, Open Space, and Others. Population estimation labels were obtained from WorldPop" 1 for both cities.
For the remaining tasks, the labels are generated manually. Specifically, for the topological classification task, the number of relationships depends on the types of objects being compared. Point/polyline, point/polygon, and polyline/polyline pairs can belong to one of two classes: disjoint or not disjoint. Polyline/polygon pairs, however, have four distinct relationship classes, while polygon/polygon pairs include six classes, following the DE9IM model. To eliminate redundancy, we remove equivalent relationships such within and contains, keeping only one representative relationship from each pair of equivalents. To create balanced dataset across all relationship classes, we generate geometry pairs by slightly adjusting the positions of the original geospatial objects and randomly selecting 5,000 pairs for each class within a group.
For the directional relationship classification task, we classify the spatial relationships between two geometries into one of 16 compass directions based on their angular relationship. These 16 classes are derived from the cardinal and intercardinal directions: north, northeast, east, southeast, south, southwest, west, northwest, and their boundary counterparts (e.g.. north-northeust, east-northeast) Labels are computed based on the relative orientation of the geometries' centroids. Similar to the topological classification task, we randomly select 5,000 pairs for each directional class to ensure balanced dataset.
For the distance estimation task, labels are computed using the actual spatial distance between the centroids of the two geometries. The spatial distance is calculated using Euclidean distance for planar geometries, for topological and directional relationship classification. We randomly select 10,000 geometry pairs for this task.
A.5. Baselines
We now describe the baseline methods used to evaluate POLY2VEC.
Point encoders
DIRECT: Feeds directly the geometry's s input coordinates the downstream model, without any encoding mechanism.
TILE: Partitions the study area into uniform grid with cells of size Each grid cell is assigned an embedding, which serves as the encoding for the points assigned to that cell (Berg et al., 2014; Adams et al., 2015; Tang et al., 2015).
WRAP: Uses a wrapping mechanism [sin(T)); cos(T)] to encode point (Mac Aodha et al., 2019).
GRID: Follows the Transformer's position encoding model (Vaswani, 2017), representing spatial positions through multi-scale sine and cosine transformations. At each scale the encoding is given by PE! (p)
sin these multi-scale representations, capturing spatial structures across different resolutions (Mai al., 2020).
008
where = Amaa controls the frequency range. The final encoding concatenates Amin
gg
TheoRy: Encodes spatial positions using dot products with unit vectors separated by 120". At each scale s, the encoding
is given by PE (t) (p) cos
(p.oz) sin Amaia A 3
(p.a.)
Vj E (1,2,3), where [1, 0J" a2 = 1 "T
Aminng 9 I
and 03 = | 2 I are unit vectors spaced at 120%. The final encoding concatenates these multi-scale representations across all vectors (Mai al., 2020).
2. Polyline encoders
T2VEC: First uniformly partitions the whole space into grid cells, and map each trajectory point into the grid cell. Through this tokenization, each trajectory is converted to a sequence of grid cell IDs. Then adopts a GRU encoder to encode the sequence and an end-to-end training paradigm that amis to reconstruct the original trajectories from the distorted/doownsamplee ones (Li et al., 2018a).
3. Polygon encoders
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16

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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
RESNETI D: Adapts the ID variant of the Residual Network (ResNet) architecture, incorporating circular padding effectively encode the exterior vertices of polygons (Mai al., 2023).
NUFTSPEC: Transforms polygons into the spectral domain using the Non-Uniform Fourier Transformation (NUFT) and j j-simplex meshes and then learns polygon embeddings from these spectral features using MLPs (Mai et al., 2023).
A.6. Hyperparameter Configuration
The coordinates of the input geometries are normalized to lie within the range [-, x [-1,1], based on the bounding box of the corresponding area of interest. We set the minimum frequency fmin = 0.1. the maximum frequency fmax 1.0 and W = 10, resulting in 210 frequencies. We set the final size of the geometry embedding v to d = 32. All the MLPs consist of two layers with ReLU activation functions.
1. HYPERPARAMETERS OF SPATIAL REASONING TASKS.
For training on the spatial reasoning tasks, we utilize the AdamW optimizer and set the learning rate (r 10-4 and weight decay wd 10 8 The batch size is set to 128, and the downstream models were trained for 20 epochs. The training, validation, and testing ratios for the datasets corresponding to these tasks is 60:20:20. All experiments were run 5 times and we report average performances and standard deviation.
A.6.2. HYPERPARAMETERS OF GROAI TASKS.
We follow the same hyperparameters as presented by Li et al. (2023), to keep our comparison consistent.
A.6.3. HYPERPARAMETERS OF OTHER BASELINES.
The implementation of baselines follows the corresponding papers, along which each method's specific hyperparameters. The rest of hyperparameters related to downstream tasks are kept consistent with approach.
A.7. Experimental Environment
Our experiments performed on a cluster node equipped with 18-core Intel 19-9980XE CPU, 125 GB of memory, and two 11 GB NVIDIA GeFFrree RTX 2080 Ti GPUs. Furthermore, all neural network models are implemented based on PyTorch version 2.3.0 with CUDA 11.8 using Python version 3.9.19.
A.8. Training Details of Evaluation Tasks
We use cross entropy loss to train the downstream model on the topological and directional relationship classification tasks. The loss is defined as:
LCF(O)
Vs.c log (og(úr.c),
where N is the number of samples, C is the number of classes (C = 2 for binary classification), Vi.c E {0, 1} is the one-hot encoded ground-truth label for class c, and Ūr.c E [0, 1] is the predicted probability for class c.
For the distance preservation task, the model is evaluated using the mean squared error (MSE) loss, defined as:
N (ya 0:)
(MSE =
N
where ys is the ground-truth distance for the i-th sample, and 0, is the predicted distance.
We note that for the population prediction and land use classification tasks, POly2VEC used input to the pretrained urban region representation model REGIONDCL (Li et al., 2023), and thus follow the same training and evaluation procedure was originally presented by the authors.
17

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Poly2Vec: Polymorphic Fourier-Bascd Encoding of Geospatial Objects for GeoAT Applications
A.9. Further Details Poly2Vec Integration into End-to-End GeoAl Pipeline
In this section, we describe in more detail how Poly2Vec is integrated into RegionDCL.
We utilize RegionDCL end-to-end GeoAl pipeline to demonstrate the utility of Poly2Vec. Poly2Vec is used the input encoding in RegionDCI, replacing its original input representation. RegionDCI. originally rasterizes OSM building footprints, converting coordinate data into image inputs so it can leverage convolutional encoders like ResNet-18. This rasterization leads to the loss of important spatial information, such as the absolute location of each building. To mitigate this, RegionDCL introduces Distance-biased transformer encoder, where the bias term consists of pairwise distances between buildings and POIs reintroduce spatial context.
In our experiments, (1) replaced the inputs with Poly2 encodings, and (2) replaced the distance-biased transformer encoder with a standard transformer encoder, because our inputs from (1) capture the necessary spatial information. The fact that Poly2Vec improves performance even without the distance bias demonstrates its ability to inherently retain spatial and positional information.
A.10. Supplementary Results on Topological Relationships Classification
Table 6: Overall model Performance on topological relationship classification. Best and second best are highlighted.
Singapore
New York
Metric
Methods
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon- polygon
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon- polygon
RESNETID
0.3 0.398ams
0.42 0.42loom Inos
NUFTSPEC
0.588ama
0.562axs2
T2VEC
0.768mea1
0.74Somz
DIRECT
0.859
0.83lamr
0.637wesz
0.415amz
0.328 uos
0.835uee2
0.933uaw
0.661omz
0.498a
0.439maae
Precision
TILE
0.735
0.705
0.505
0.490auss
0.4399ms
0.6644nss
0.789usee
0.502amx
0.494uma
0.418moce
WRAP
0.874xa11
0.865um15
0,6450ece
0.453am25
0.4055o00
0.879
0,915
0.655ama
0.586ame
0.4050000
GRID
0.799
0.84 Iomo
0.626mesr
0.405aoss
0.288
0. m6
0.904nais
0.658
0.513umm
0.355mor7
THEORY
0. 0,903
0.874mm04
0,651ocse
0.432am1s
0.478amc
8866
0.8930a17
0. 0.602a
0.431
POLY2VEC
0.91
0.924um
0.779
0.506
0.694um
1.92
0.979kua
0.745
0.63
0.698usec
RESNET D
0.455am11
0.452ness
NUFTSPEC
0.572mxe
0.592
T2VEC
0.732nx24
0.718
DIRECT
0.792xa11
0.838
0.997
0.414amm 14mm
0.450kma
0.838ams
0.888
0.987azz
0.497aom
0.43
Recall
TILE
0.894ooss
0.695am04
1.0x00
0.463000
0.413a.04
0. 0.659k.0
0.769ko1
1.00tvo0
0. 0.4999.c0
0.405e004
WRAP
0,903sass
0.901ams
0. 0.992weor
0 0.477oa12
0.380m.os
0.8 0,894m.c
0, 0.842aas
0. 0.9863
0.55 15.51o.0s
0.3800.06
GRID
0.92 Ioots
0.8 m4
0.980 0.980ne
0.465a
0.339umm
0. m.333mm
0.88 Inoos
0.99 0.995
0.51 0.514am 44
0.382nom
THEORY
0.986
0.9 0.933umc
0.972we12
0.451oma
0.4677u0
0.923mxu4
0.91 20017
0.782
0.61
0.41
POLY2VEC
1.0
0.974um
1.0
0.498
7.697
1.0
0.989
1.0km
0.638kmm
0.697kee7
RESNET ID
0. 0.399m.017
0.399
NUFTSPEC
0.574mms
I
T2VEC
0.732o00
0. 0.741am0
DIRECT
0.824xx0
0.834mm
0.777ness
0.402amm
0.3 m4
0.836
0.910mmm
0.792>um
0.4633
0.403
TILE
0.805ea13
0.6944.m7
16671os04
0.412aum
0,3840x0
0.66 1.661o.o
0.779us04
0.6688300
0.453u.oi
0.369axe
WRAP
0.888
0.882 om
0.78 Incos
0.450amo
0.339k
1..86
0.876m
0.787ome
0.51 7a
0.339
GRID
0.855mao
0.844m om
0.764n
0.41 loms
0.2677oos
0. 0.842nms
0. 0.892now
0.792oms
0.463mne
0.322nos
THEORY
0.938
0.903ams
0. 788
0.438omz
0.425muss
0.883mux
0,891
0.726
0. 0.549muw
0.419 9osoo
POLY2VEC
0.955ea11
0.948k.m
0.831
0.483
0.682ama
0.959k.om
0.984aa12
0.854amm
0.588
0.679kme
18

---

Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAT Applications
A.11. Supplementary Results on Directional Relationship Classification
Table 7: Overall model Performance on directional relationship classification. Best and second best are highlighted.
Singupore
New York
Metric
Methods
point- point
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon- polygon
point- point
point- polyline
point- polygon
polyline- polyline
polyline- polygon
polygon- polygon
NUFTRESNET
0.828koe
0.783mm
NUFTSPEC
0.832om
12VEC
0.227
0.232am
DIRECT
0.882uxe
0.84Guue
0.847uxx
0.825wee
3.813
0. 0.765un
0.880
0.767
0. 0.843uee
0.687wes
0.794mes
0.774uxe
Precision
TIL.
0.259kmm
0.260ses
0.286ios
0.370uess
0.466xm0
0.415
0.29
0.279kmm
0.322mes
0.280uess
0.496imm
0.376mms
WRAP
0.8633uo
0.810
0.6064
0.790uoe
0.835uxr
0,789kus
0.809
0.684ooxc
0.759uus
0.610uuz
0.78
0.667ame
GRID
0.884ume
0.733wer
0.775mue
0, 0.708
0.6 0.653umm
0.545uu
0.872wxz
0.605uee
0. 0.670uu
0.44Iues
0.766wes
4.51
THEORY
0.908
0.872
3.06s
0.8 5
0. 7.9
0,88 loon
0. 0.774mm
0. 1.809
0.692m
0.789kes
0.538
POLY2VEC
0.928
0.942am22
0.9 0.918
0.898
1.92
0.889
0. 0.875ue
0.889
0.853
0. 0.792axe
NUFTRESNET
0. 7.74mm
NUFTsPEC
0.792kx0
0.685
12VEC
0.21 Guoes
0,253uoz
DIRECT
0.879ux
0.84 luxe
0.845ums
0.8 0.820ume
0.830uuse
0, 0.752umr
0.877uxx
0. 0,766uee
0.836
0.653mer
0,784mes
0. 0.094mxs
Recall
TILE
0.253
0.269kue
0.273wocs
0.324mo0
0.454omz
0.395
0.248
0.257
0.3166ess
0.217
0.466xm
0.348
WRAP
0.861uxo
0.80Maxe
0.803300
0.782uxa
0.83
0.779kux
0.810
0.669kue
0.759kus
0,598ums
0.772uxx
0. 0.6022uee
GRID
0.882uxe
0.729kue
0.722
0. 1.99
0.64 1onns
0,533kus
0.868ux:
0. 0.590une
0. 0.647ues
0.437uez
0,752mes
0.483uus
THEORY
0.883ums
0.867
0.855
3um
0.502
0.897
0. 0.783
lonee
0.823
7.709ee
0.80
0.567mm
POLY2VEC
0.946um7
0.947aa
0.933mm
0. 3uu0s
0.838
0.826xue
0,923
0. 0.89-
0. 0.886
0.878um
0.875
0.793ko1
NUFTRESNET
0.82 loms
0. 7566mm
NUFTSPEC
0.8022oas
0.667wo
T2VEC
0.219kwe
0 0,252unss
DIRECT
0.880
0.84 looes
0.845uees
0.82 looe
0.840ues
0, 0.754mm
0.876
0. 769
0.838
0.6566ue
0.71 0,784mes
TILE
0.21.5mm
0.226xos
0.247oou
0.309
0.447oom
0.388ro01
0.236wo
0.288
0.1933ue
0.439
0.339kus
WRAP
0.861axo
0.80Muue
0.8033mp
0.782uxx
0.83 loune
0.780uom
0.805
0.6688oc
0.752
0.590uez
0.769uux
0.613
GRID
0.882ame
0.728kx
0,772uxee
0. 1.698um
0.640um
0,530kus
0.868ue
0.588uxe
0. 0.649uum
0.409hes
0,749uees
0. 0.460um
THEORY
0.903um
0.852ume
0.855ure
0. 0.842unz
0.845uue
0. 0,741uxx
0.884um
0.752umr
1881sses
0. 6668uee
0.756ucs
0.537am
VOLY2VBC
0.928
0.927
18918
0.90 Hours
0.899
0.827
0.892uma
0.883um
0. 0.9003um
0.877uas
0.832ues
0.769um
A.12. Supplementary Results on Distance Estimation
Table 8: Overall model performance on distance estimation. Best and second best are highlighted.
point- point
point- polyline
point- polygon
Dataset
Model
DIRECT
0.088 e004.
0,093
0.084+0023
TILE
0.252
0. 2.1
0. 0.157
WRAP
0.085
0. 0.1062a.02
0. 0000
Singapore
GRID
0.087 1
0.107 1.104 100ms
0. 10.08 1to0z
THEORY
0.065
0.083+4017
0.079=0021
VEC
0.016=
0.043
0.029=0000
DIRECT
0.075
0. 26
0.1 1 12003
TILE
0 20005
0. nos
0. 89
WRAP
0. 10000
0.148 10mm
0. 10000
New York
GRID
0.073+a00:
0. 1.1244..mm
0.1 1.11 18=0011
THEORY
0.068+8008
0.089
0. 100-+0008
POLY2VEC
0.030
0.049 jame
0.042.n001
mni predittd
original pedicted
original predicted
orgginal predicted
original
1.0 ongiral
Distanees
Distances
Nistaness
Distances
Distances
peodicted
Distanees
predicted
0.5
R. 0.D
0LD
0.5
1.0
60
255
1.0
06
0.5
Lo
0.5
1.0
0.
01.
1.0
60
0.5 Original distances
1.0
Original distances Direct
Original distanoss
Original distances
Original distanoes
Original distanees
(b) Tile
(e) Wrap
(d) Grid
Theory
(f) Poly2Vee
Figure 6: Distance seatters of point-polygon pairs New York dataset for different encoders.
19

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Poly 2Vec: Polymorphic Baarier-Based Encoding of Geospatial Objects for GcoAT Applications
10
onganal
1.0
oeginal
1,0
ongion
orgeanl 10
10
ociginal
1.0
Disuraoss
Distances
Disturr noee
Distanoos 8.
Distancess 6.5
Distuncess
0.5
0.5 Original distances
1.0
Original (4) Direct
Original distances
Original distarsces
Original distances
Original distances
(b) Tile
(e) Wtap
(d) Grid
(c) Theory
(1) Poly2Voc
Figure 7: Distance scatters of point-polyline pairs on Singapore dataset for different encoders.
onggrnl preficeed
original prdiieed
otginal predited
original prodicted
ongiial predicted,
Distnness
Distances
Distances
Distanness
S Distances
prodicted
Dictanoss
0.0
0.0
60 00
0.0
0.0 0.0
0.0 00
00
1.
0.5
1.0
0.5
0.5 Original distances
10
Original distances
Original distanecs
Original distances
Original distauces
Original distanecs
(a) Direct
(b) Tile
(c) Wrap
(d) Grid
(e) Theory
(f) Poly2Vec
Figure 8: Distance scatters of point-polyline pairs on New York dataset for different encoders.
1.0
original predicted
1.0
neiginal
original predicted
1.0
original predicted
sniginal prodicted
1.0
oiigiaal
Dis ansie
Distances
Mistances
Disiancees
Distances
Distances
predicted
0.5
000
U.S
01
0.5
00
1.0
Original distances Direct
Original distances
Original distances
Original distancess
Original distances
Original distances
(b) Tile
(c) Wrap
(d) Grid
(e) Theory
(i) Poly2Vec
Figure 9: Distance scatters of point-point pairs on Singapore dataset for different encoders.
1.0
orgginl prodicuo
Distances
proliooa
Distunces
grsdicted
Mosunces
prolsted
Distances
Distunces
prodictod,
Distonees
predicted
0.5
0.5
R5
2.5
0.0
0.0
60
0.0
0.0
60
00
10
0.5
1.0
0.0
L0
000
0.5
1.0
00
Original distances (a) Direct
Otiginal distances
Original distances
Original distaucess
Original distances
Original distances
(b) Tile
(c) Wrap
(d) Grid
(e) Theory
(f) Poly2Vec
Figure 10: Distance scatters of point-point pairs on New York dataset for different encoders.
A.13. Supplementary Ablation Experiments
1..1 EFFECT OF LEARNED FUSION MODULE
We' ve shown the effect of learned fusion on point- saint-polygor gon tasks in Section 4.3. We demonstrate its effect on the rest of spatial reasoning tasks in Figures 12a, 12b, 11a, and lb.. We again observe similar trends as reported the main evaluation.
w/ Mag
w/ Phase
w/ concat
Poly2Vee
Mag
w/ Phase
w/ concat
Poly2Voe
0.7
1.0
0.7
1.0
Accuracy
0.6
Accuracy
Accuracy
Accuracy
0.5
0.6
0.5
0.6
0.4
0.4
0.4
Q4
0.3
0.2
0.3
0.2
Singapore York
Singapore New York
0.2
Singapore New York
Singapore York
(a) Topological relationship
(b) Directional relationship
(a) Topological relationship
(b) Directional relationship
classification
classification
classification
classification
(a) Dolyline-polygon pairs
(b) Polygon-polygon pairs
Figure 11: Effect of learned fusion on polyline-polyse polygon and polygon-polygon pairs
20

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Poly2Vect Polymorphic Fourier-Based Encoding of Geospatial Objects for GeoAI Applications
w/ Mag
Phase
w/
Poly2Vec
w/ Mag
w/ Phase
w/ concat Ni
Poly2Vee
1.0
1.00
1.00
Accuracy
0.44
Accurcey
0.88
Accurcy
0.88
0.4
0.8
0.76
0.76
MAE
0.3
0.4
0.22
0.2
0.1
0.0
Singapore New York
0.00
Singapore New York
Singapore New York
Singapore New York
Singapore New York
Directional relationship
(b) Distance estimation
(a) Topological relationship
(b) Directional relationship classification
(c) Distance estimation
classification
classification
(a) Point-point pairs
(b) Point-polyline pairs
Figure 12: Effect of learned fusion on sont-polygon and point-polyline pairs
A.13.2. EFFECT OF EMBEDDING SIZE d
Topological Relationship Classification Singapore
Topological Relationship Classification New York
1.0
0.8
0.6
Accuracy
0.6
0.4
0.2
0.2
16
128
256
Embedding Size
Embedding Size
Figure 13: Effect of embedding size on topological relationship classification
Directional Relationship Classification Singapore
Directional Relationship Classification New York
1.0
0.8
0.8
Accurity
Accuracy
0.6
0.4
0.4
0.0
128
256
0.0
64
128
256
Emhadding Size
Embodding Size
Figure 14: Effect of embedding size on directional relationship classification
Distance Estimation Singapore
Distance Estimation New York
0.10
MAC
0.
MAE
0.06
0.04
0.04
0.02
0.02
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Figure 15; Effect of embedding size in distance estimation
21

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Poly2Vec: Polymorphic Fourier-Based Encoding of Geospatial Objects for GcoAI Applications
A.13.3. EFFECT OF NUMBER FREQUENCY COMPONENTS W
Topological Relationship Classification Singapore
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Figure Effect freq. components on topological relationship classification
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Figure 17: Effect of # of freq. components directional relationship classification
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Figure 18: Effect of # of freq. components in distance estimation
22


Reviews:
[{"review_id": "tFwXqp0vRw", "reviewer": "Reviewer_wSi2", "summary": "This work considers the problem of encoding points, polylines, and polygons for the purposes of prediction tasks that require understanding of spatial relationships such as topology, direction, and distance. Similar to previous point and polygon encoding approaches (Space2vec, NUFTspec), Fourier transformation is used to transform vector space data to fixed length presentations. To achieve more detailed representation of alignment both magnitude and phase formation are encoded. To achieve encoding of mixed item types (point, polyline, polygon) three versions of Fourier transformations are used together.\n\nFor basic spatial relationship tasks (topology, direction, distance), experiment shows improvement over previous approaches such as Space2vec, ResNet1D, T2Vec and NUFTspec.\n\nFor mixed item type tasks (Land use classification, population prediction), an unsupervised  approach RegionDCL is used as the baseline. Improvement of RegionDCL performance is achieved by replacing its distance based attention bias with the proposed Poly2Vec approach. However, not much detail is given in the paper about how exactly is Poly2Vec integrated to RegionDCL. It would also be useful to provide visualization of a few examples to help the reader understand how Poly2Vec helps RegionDCL.\n\nOverall the proposed approach is reasonable, but there is significant detail missing which makes it hard for me to understand the result.", "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 thank the reviewer for the insightful comments.\n___\n\n***Reviewer:*** Similar to previous point and polygon encoding approaches (Space2vec, NUFTspec), Fourier transformation is used to transform vector space data to fixed length presentations.\n\n***Authors:*** We note that our Poly2Vec differs from Space2Vec in that (1) Poly2Vec has the full complement of frequency components available to it, as opposed to Space2Vec’s more tightly specified sine and cosine frequencies on a grid of scales and (2) Poly2Vec works uniformly for points, polylines and polygons, while Space2Vec is limited to points. Compared to NUFTspec, Poly2Vec presents a unified, consistent Fourier transform encoding for points and polylines, as well as polygons. Our approach for computing the Fourier Transform for polylines and polygons (through decomposition and affine transformation) is also a novel component of our approach. \n___\n***Reviewer:*** Improvement of RegionDCL performance is achieved by replacing its distance based attention bias with the proposed Poly2Vec approach. However, not much detail is given in the paper about how exactly is Poly2Vec integrated to RegionDCL.\n\n***Authors:*** We agree that clearly describing Poly2Vec’s integration into RegionDCL would improve the manuscript’s clarity. Due to space constraints and since RegionDCL itself was not our primary contribution, we focused instead on detailing our novel components.\n\nTo clarify, Poly2Vec is used as the input encoding in RegionDCL, replacing its original input representation. RegionDCL originally rasterizes OSM building footprints, converting coordinate data into image inputs so it can leverage convolutional encoders like ResNet-18. This rasterization leads to the loss of important spatial information, such as the absolute location of each building. To mitigate this, RegionDCL introduces a distance-biased transformer encoder, where the bias term consists of pairwise distances between buildings and POIs to reintroduce spatial context. In our experiments, we (1) replaced the inputs with Poly2Vec encodings, and (2) replaced the distance-biased transformer encoder with a standard transformer encoder,  because our new inputs from (1) capture the necessary spatial information. The fact that Poly2Vec improves performance even without the distance bias demonstrates its ability to inherently retain spatial and positional information.\n\nWe will add these additional details in the appendix to clearly describe the integration and highlight the differences from the original setup of RegionDCL.\n___\n\n***Reviewer:*** It would also be useful to provide visualization of a few examples to help the reader understand how Poly2Vec helps RegionDCL.\n\n***Authors:*** We provide some visualization examples in [Link1](https://anonymous.4open.science/r/r-0752/), showing building footprints as polygons and their spatial relationships — information that, discarded by the original design of RegionDCL but captured by Poly2Vec, contributes to improved classification performance in RegionDCL. We will add such visualizations of selected regions in the new manuscript.", "claims_and_evidence": "see summary", "methods_and_evaluation_criteria": "see summary", "theoretical_claims": "see summary", "experimental_designs_or_analyses": "see summary", "relation_to_broader_scientific_literature": "No", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "see summary", "other_comments_or_suggestions": "see summary", "questions_for_authors": "see summary", "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": "WjMfmXQ28W", "reviewer": "Reviewer_Bw6M", "summary": "The paper presents a Fourier based encoding strategy for geospatical principles, i.e., points, lines, and polygons, into a deep-learning compatible vector format. The methodology is well-explained and founded in signal processing theory. It well-explains Fourier analysis and inherent properties (Affine Transformation, Symmetry) that are used in a effective way to first (affine) transform geospatial primitives and then encode them into frequency domain to a vector representations for amplitude and phase. Then two MLPs fuse these representations to map the features vector to a target variable, for instance, for point-in-polygon testing.\n\nOverall, it is well-written. The principles are well-explained and the results well structured into underlying research questions and numerically convincing, as the proposed Poly2vec representation outperforms existing point-based and line-based methods. In my opinion, a strong accept.", "questions": "", "limitations": "", "rating": 5, "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": "Thank you for the thorough review and encouraging comments.\n\n___\n\n***Reviewer:*** The experiments are well-structured according to 4 research questions. However, all results seem to be based on the classification of point-to-polyline etc relationships, which is reasonable for this experimental setup, but a more realistic use case, like predicting, for instance, the building use (residential, factory, shopping mall) from the polygon geometry may be interesting as well.\n\n***Authors:*** While we do not perform single-building classification, our RegionDCL experiments address a more general version of that task: predicting region-level attributes from groups of buildings. This can be seen as a superset of building classification, as it involves not just the shape of a single polygon, but also the locations and inter-relationships of multiple buildings, which is particularly beneficial for our downstream tasks where spatial relationships between objects play a critical role.\n\n___\n\n***Reviewer:*** some more realistic datasets could be tested. E.g., classification of building type, or land use from the geometry shape\n\n***Authors:*** Same as our first response above.\n\n___\n\n***Reviewer:*** Can the Fourier transformation be inverted to map back from embedding space to polygon/line/points?\n\n***Authors:*** We thank the reviewer for the thought-provoking question. We believe this can be achieved by simultaneously training a decoder to map embeddings back to geometries (i.e., shape decoder). So far, we have experimented with applying the inverse Fourier transform to reconstruct approximations of the original shapes from the raw Fourier samples. This is an exciting direction and we will consider it in future research toward generative models that can reconstruct realistic spatial geometries from learned embeddings.\n\n___\n\n***Reviewer:*** What use cases do the authors see for Poly2Vec in a more complex setting, where the fourier features are integrated in a deeper neural network rather than individual MLPs for point-in-polygon tests.\n\n***Authors:*** Our experiments with RegionDCL show one example of integrating Poly2Vec into a transformer-based geospatial pipeline, by replacing its input representation with Poly2Vec to classify regions and estimate population from geometric map features. Looking forward, we believe Poly2Vec can serve as a valuable component in multimodal geo-foundation models, when one of the input modalities is vector-based spatial data (i.e., coordinates).", "claims_and_evidence": "The paper claims to unify representation of geospatial objects, namely points, lines, and polygons. This claim is supported, as the results show better representation accuracy over point- and line-specific models.", "methods_and_evaluation_criteria": "Methods:\nThe underlying method is well-explained first on simple numerical examples that are then abstracted to the generic case. Overall, very intuitive and understandable\n\nEvaluation\nThe experiments are well-structured in 4 accurate reserach questions that are systematically investigated in the results.", "theoretical_claims": "There are no proofs in the paper, but properties of Fourier transform (Linearity, Affine Trasnformation, Hermitian Symmetry, Magnitude and Phase) are well-explained. All used theory is supported by intuition and concrete experiments on simple cases.", "experimental_designs_or_analyses": "The experiments are well-structured according to 4 research questions. However, all results seem to be based on the classification of point-to-polyline etc relationships, which is reasonable for this experimental setup, but a more realistic usecase, like predicting, for instance, the building use (residential, factory, shopping mall) from the polygion geometry may be interesting as well. For classic geospatial operations like point-in-polygon prediction highly optimized algorithms exist that achieve 100% accuracy (<- but this is still a good problem to compare different embeddings)", "relation_to_broader_scientific_literature": "This paper presents a crucial step forward embedding geospatial primitives, which is highly helpful in learning from geospatial data in Geospatial Foundation Models. The unifying framework that includes points, lines, polygons and outperforms methodologies designed for single primitives like points, will be important for the geospatial machine learning community and subsequent research fields like Geo-Information Science.", "essential_references_not_discussed": "I think the references are all justified. No additions requested.", "other_strengths_and_weaknesses": "Strengths\n* Convincing results against baselines focusing on a singly type of geometric primitive\n* well-explained methodology and well-structured results\n\nWeakness:\n* some more realistic datasets could be tested. E.g., classification of building type, or land use from the geometry shape", "other_comments_or_suggestions": "no typos found.", "questions_for_authors": "* Can the Fourier transformation be inverted to map back from embedding space to polygon/line/points?\n* What usecases do the authors see for Poly2Vec in a more complex setting, where the fourier features are integrated in a deeper neural network rather than individual MLPs for point-in-polygon tests.", "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": "WyFVzbqMo9", "reviewer": "Reviewer_qMT2", "summary": "The authors introduce a method of encoding representations of geospatial objects which they call Poly2Vec. This method is capable of representing points (e.g. points of interest), polylines (e.g. roads), and polygons (e.g. buildings) while competing methods struggle to represent all of these different formats. Poly2Vec is a fourier transform based approach which seems to preserve some important attributes of the encoded objects such as location, shape and direction. This method is compared to a wide range of other approaches on OpenStreetMap based datasets for tasks such as determining the distance between objects or their orientation, as well as for use as input encodings for a modern machine learning approach to land use classification and population prediction. Poly2Vec performs best on all of these tasks.\n\n## update after rebuttal\n\nThe author response clarified my one concern and I believe the paper should be accepted.", "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": "Thank you for your insightful comments.\n\n___\n\n***Reviewer:*** (Note we grouped these comments, because our answer is essentially the same for all of them.) The method involves learning how to encode objects in a way that seems to allow it to be fine tuned to specific tasks which might not be helpful for a downstream user who does not wish to train a model in order to encode their geospatial objects, and might limit the transferability of these representations. … I also wonder if including a similar neural network for some of the baseline approaches and training these to produce encodings that are useful for specific tasks would also improve performance of these baselines and make Poly2Vec comparatively less high performing. … The method involves learning how to encode objects in a way that seems to allow it to be fine tuned to specific tasks which might not be helpful for a downstream user who does not wish to train a model in order to encode their geospatial objects, and might limit the transferability of these representations. Do the authors know if including a similar neural network for some of the baseline approaches and training these to produce encodings that are useful for specific tasks would also improve performance of these baselines and make Poly2Vec comparatively less high performing? This will allow me to better judge the performance of the proposed method.\n\n***Authors:*** We understand this question and appreciate the trigger to think about this more deeply. You are right that we train our MLPs specifically for the task. Please note that we do the same for the baselines, so our performance comparisons are fair. We will clarify this in the next version of the paper. As we explain in the paper (Section 3.2), different parts of the Fourier transform are important for different tasks. E.g. the Fourier transform’s magnitude generally encodes shape, and its phase generally encodes location. Thus the targeted training tends to extract the right information for the specialized task.  Learning general-purpose embeddings is also an interesting direction that we have considered. Such representations could be learned using unsupervised or self-supervised techniques, such as encoder-decoder frameworks and contrastive learning. \n\n___\n\n***Reviewer:*** Well put together paper and not many mistakes that I can see, but on line 875 we have a couple of “Figure ??”s.\n\n***Authors:*** Thank you for finding these omissions. We will fix them all in the next version.", "claims_and_evidence": "Yes. The authors seem to compare to a wide variety of competing approaches and show favourable performance across a range of tasks and appropriate metrics.", "methods_and_evaluation_criteria": "Yes. The method seems sensible. The OpenStreetMap based datasets seem sensible for the task and similar datasets are used in the literature (“Urban Region Representation Learning with OpenStreetMap Building Footprints” (KDD '23), “Urban2Vec: Incorporating Street View Imagery and POIs for Multi-Modal Urban Neighborhood Embedding” (AAAI '20)). The evaluation metrics seem appropriate. Ground truth data for tasks such as land use classification has been gathered similarly to previous works in the area.", "theoretical_claims": "I couldn’t see any problems with the formulation of the fourier transform based method.", "experimental_designs_or_analyses": "All experiments seem to be well designed.", "relation_to_broader_scientific_literature": "Some existing works involve creating representations of buildings or places of interest but these make use of semantic information such as text or photos of the location. “Urban Region Representation Learning with OpenStreetMap Building Footprints” (KDD '23)\n\nMany works involve generating an embedding of points and some of these are compared against in this work (e.g. WRAP method in \"Presence-Only Geographical Priors for Fine-Grained Image Classification\" (ICCV '19)). These can be extended to represent lines as sequences of points. (e.g. \"LSTM-TrajGAN: A Deep Learning Approach to Trajectory Privacy Protection\" (GIScience 2021))\n\n“Towards General-Purpose Representation Learning of Polygonal Geometries” (GeoInformatica 27(4):1-52) produces encodings of polygons that are suitable for machine learning approaches.\n \n“Graph Convolutional AutoEncoder models” can also produce encodings of polygons or lines.\n\nSpectral domain polygon encoders do exist such as “NUFTSPEC”. The authors do cite this work. (“Towards General-Purpose Representation Learning of Polygonal Geometries” (GeoInformatica 27(4):1-52)).\n\nPoly2Vec does seem to uniquely create a unified embedding space for the different geometries that are discussed in the work (polygons, polylines, points)\n\nDeterministic spatial reasoners such as postGIS can do many of the tasks that are discussed in this paper such as accurately measuring distances between polygons, lines and points, and determining directions. However these do not produce an encoding that is suitable for many machine learning approaches.", "essential_references_not_discussed": "Essential references seem to be included.", "other_strengths_and_weaknesses": "The work seems to be quite original in furthering spectral based methods of encoding polygons to also include points and lines.\n\nThe paper is clear and the work done is well described. Baseline approaches are covered quite well.\n\nThe tasks seem well constructed and a wide range of approaches are compared against.\n\nSeveral different evaluation metrics are used and the proposed method is consistently well performing.\n\nThe method involves learning how to encode objects in a way that seems to allow it to be fine tuned to specific tasks which might not be helpful for a downstream user who does not wish to train a model in order to encode their geospatial objects, and might limit the transferability of these representations. I also wonder if including a similar neural network for some of the baseline approaches and training these to produce encodings that are useful for specific tasks would also improve performance of these baselines and make Poly2Vec comparatively less high performing.", "other_comments_or_suggestions": "Well put together paper and not many mistakes that I can see, but on line 875 we have a couple of “Figure ??”s.", "questions_for_authors": "1. The method involves learning how to encode objects in a way that seems to allow it to be fine tuned to specific tasks which might not be helpful for a downstream user who does not wish to train a model in order to encode their geospatial objects, and might limit the transferability of these representations. Do the authors know if including a similar neural network for some of the baseline approaches and training these to produce encodings that are useful for specific tasks would also improve performance of these baselines and make Poly2Vec comparatively less high performing? This will allow me to better judge the performance of the proposed method.", "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": "PAfZGIDLue", "reviewer": "Reviewer_kc8b", "summary": "This paper proposes Poly2Vec, a Fourier-transform approach to encoding shapes (points, lines, polygons) for geospatial tasks. The authors find that Poly2Vec outperforms baselines in preserving shape topology, direction, and distance over OSM datasets for two cities, New York and Singapore.\n\n## Update after rebuttal\nI think this paper is good and recommend it for acceptance. I would have liked to see experiments integrating Poly2Vec with a larger geospatial workflow as I think it can benefit geospatial tasks such as super-resolution, conditional generation, and may even benefit unsupervised learning methods for geospatial data. While evidence for these points would have made the paper stronger, in my opinion the contribution as presented by the authors is sufficient to clear the bar for acceptance.", "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": "Thank you for these detailed, thought-provoking comments.\n___\n***Reviewer (paraphrased)***: Existing baselines use OSM data, including LLMs[1]. ResNet-18 is weak given [2,3,4].\n\n***Authors:*** Poly2Vec is designed to handle arbitrary geometric data described by coordinates (vector spatial data), rather than pixel-based images such as satellite imagery. Thus, methods explicitly requiring imagery are not directly applicable to our scenario.\n\nRegionDCL converts vector data of buildings into images, using ResNet-18 for image encoding and distance-biased transformer for restoring location information of buildings and POIs. We hypothesize that RegionDCL adopted this approach because no effective unified encoding existed for vector data before Poly2Vec. This vector-to-image conversion (rasterization) step inherently introduces information loss. We demonstrate that directly encoding vector data with Poly2Vec is more effective. While using a more sophisticated image-based model might yield incremental improvements, the fundamental loss of spatial detail due to rasterization would persist.\n\nOn the suggested references: [1] primarily evaluates large language models, while [3] and [4] focus explicitly on satellite images, and therefore Poly2Vec does not directly apply. Nevertheless, the broader trend in foundation and large language models emphasizes multimodal data integration. [2] exemplifies this integration, combining both images and geographical coordinates; however, its focus appears more oriented toward geographic positioning rather than the locations, topology, or spatial relationships that Poly2Vec exploits. Poly2Vec could be integrated into [2] to replace/complement its location encoder, to support more sophisticated downstream tasks. Similarly, [1], [3], and [4] can benefit from Poly2Vec if they can be extended to incorporate supplementary vector data inputs.\n___\n\n***Reviewer (paraphrased)***: Benchmark datasets too narrow in scope. … broader claim of relevance to an end-to-end Geospatial workflow is not sufficiently tested over OSM data with two cities.  Need experiments on larger, more recent workflows.\n\n***Authors:*** The two cities, New York City and Singapore, have significant differences in building structures, spatial layout, and data density (as visualized in [Link1](https://anonymous.4open.science/r/r-0752/)). New York City has a large, diverse urban environment with varied functional zones and more regular building shapes, while Singapore features a much higher population density, more data-sparse regions, and more irregularly shaped footprints [1]. Poly2Vec focuses on encoding vector data, which serves as the foundation for many geospatial applications. We believe that improving encoding vector data will facilitate a broader range of geospatial tasks.\n\n[1] Urban Region Representation Learning with OpenStreetMap Building Footprints, ACM SIGKDD 2023\n\n___\n***Reviewer (paraphrased):*** Linear probing embeddings from satellite image foundation models for distance or topological details of shapes in the image can yield the same accuracies? … demonstrate the value of Poly2Vec’s embeddings in improving the embeddings of geospatial image foundation models.\n\n***Authors:*** Similar to the first point. On the one hand, satellite image foundation models cannot apply to our settings since we focus on vector data. On the other hand, if a geospatial image foundation model can take vector data as input, so as to inject the spatial information into the embedding, we believe Poly2Vec can be a good choice to encode the coordinates.\n___\n***Reviewer (paraphrased):*** More ablations required.\n\n***Authors:*** We tuned the main hyperparameters before submission. Please find the hyperparameter study in [Link1](https://anonymous.4open.science/r/r-0752/), including an additional sampling technique, namely uniform sampling.\n____\n\n***Reviewer (paraphrased):*** Can Poly2Vec  improve generative image foundation models?\n\n***Authors:*** See 3rd rebuttal response to Reviewer Bw6M.\n____\n***Reviewer (paraphrased):*** Using Fourier-based encodings is not new\n\n***Authors:*** Agreed. Our novel contributions: (1) Leverage Fourier transform to create a unified, polymorphic encoding across different geometry types, with easy extensions to any 2D shape. (2) Fourier transform of line segment is new. Novel polygon approach. (3) Flexibly mixing different geometry types for geometric inferences. (4) Consistently outperforming baselines, including previous frequency-based.\n____\n\n***Reviewer (paraphrased):*** Supplementary material input/output of each task can be clearer. Link A.4.1 in main text for clarity.\n\n***Authors:*** For the sake of reproducibility, we provide the details of all our experimental settings. But given the space limitation, we can only present them in the appendix. To improve clarity and accessibility, we will link to each appendix subsection in the main paper and also add backward references in the appendix.\n____", "claims_and_evidence": "A few key claims and my discussion of the evidence used to support them:\n\n**Poly2Vec outperforms baselines in preserving shape topology, direction, distance:**  \nI believe evidence is shown to support this claim, as Poly2Vec seems to clearly outperform baselines like Direct, Tile, Wrap, Grid, Theory, and T2Vec. The datasets are limited to two cities, and broader geographic diversity would be welcome though, as I’m guessing the shapes in NYC vs Singapore aren’t too different.\n\n**Integrating Poly2Vec in an end-to-end GeoAI workflow improves performance in population prediction and land-use inference**  \nThis is the more pertinent/interesting claim to me and I think the claim is not strongly supported here. The authors pick a specific baseline, RegionDCL, and show that replacing its distance-biased encodings with Poly2Vec improves the pipeline. This is too narrow of a result to support the broader claim. There are important baselines that use OSM data including through large language models (eg: [1]) that can be used for population prediction. Moreover, a ResNet-18 is too weak of a baseline given that newer, larger foundational models [2,3,4] are used for creating embeddings of locations or images. Does integrating Poly2Vec into these workflows improve the quality of embeddings? I think that would be very valuable to test.\n\n---\nReferences:\n\n[1] Large language models are geographically biased, _ICML 2024_.  \n[2] Satclip: Global, general-purpose location embeddings with satellite imagery, _AAAI 2025_.  \n[3] SatMAE: Pre-training Transformers for Temporal and Multi-Spectral Satellite Imagery, _NeurIPS 2022_.  \n[4] Scale-MAE: A Scale-Aware Masked Autoencoder for Multiscale Geospatial Representation Learning, _ICCV 2023_.", "methods_and_evaluation_criteria": "The benchmark datasets are a bit too narrow in scope. They do the job of demonstrating that Poly2Vec might outperform other shape-encoding baselines, but the broader claim of relevance to an end-to-end Geospatial workflow is not sufficiently tested over OSM data with two cities.\n\nMoreover, I’m curious to know if linear probing embeddings from satellite image foundation models for distance or topological details of shapes in the image can yield the same accuracies that Poly2Vec displays. This to me is also an important comparison- the authors can use this experiment to demonstrate the value of Poly2Vec’s embeddings in improving the embeddings of geospatial image foundation models.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "I think more ablations are required, e.g., over the number of samples in the frequency domain W, the embedding size, more sampling strategies beyond learned vs geometric etc.", "relation_to_broader_scientific_literature": "The authors proposed method is promising for geospatial applications, but the key evidence required to demonstrate its utility is through integrations in much larger, more recent geospatial workflows. This is currently lacking in the paper, which instead discusses improvements over other shape encoding baselines (necessary but not sufficient). The authors should spend time surveying more recent geospatial use-cases where poly2vec could demonstrate value, and then share the results of experiments where poly2vec is used to provide embeddings.", "essential_references_not_discussed": "See above.", "other_strengths_and_weaknesses": "Some other strengths:\n* Paper is clearly written, easy to follow.\n* Improvements over other shape-encoding baselines specifically for topological/directional metrics is clear.\n\nSome other weaknesses:\n* Lack of thorough experimental validation (over multiple cities/geographies)\n* Insufficient experimental results on more recent geospatial workflows\n* Scope of claims is too narrow\n* Using Fourier based encodings is not new\n\nI am still slightly leaning towards acceptance because I do think the paper is promising, provided the authors demonstrate empirical evidence over Poly2Vec’s broader utility.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See above.\n\nI'm also curious about the ability of Poly2Vec to improve generative image foundation models for geospatial applications (e.g. diffusion), although this is not required to test.", "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: After the discussion phase, all reviewers recommended acceptance (Strong Accept, Accept, 2x Weak Accept) noting that the paper is well written, clearly motivated, has novelty, and is sufficiently evaluated. The rebuttal addressed many of the reviewer concerns, such as clarifying details of the baselines. As a result, the AC decided to accept the paper. Please take the reviewer feedback into account when preparing the camera-ready version.
Decision: Accept (poster)