HPENets: Efficient 3D Point Cloud Processing

• HPENets are a family of MLP networks that implement a two-stage ABS-REF architecture with high-dimensional positional encoding to capture intrinsic local 3D geometry.
• They replace costly local MLP operations with efficient non-local MLPs by explicitly injecting geometric information via additive HPE, improving feature refinement.
• Empirical evaluations show HPENets outperform PointNeXt with reduced FLOPs and enhanced mIoU across multiple point cloud benchmarks.

Searching arXiv for the specified HPENets paper and closely related context. arXiv Search Query: id:(Zou et al., 4 Mar 2026) HPENets are a suite of MLP networks for point cloud processing introduced in “Efficient Point Cloud Processing with High-Dimensional Positional Encoding and Non-Local MLPs” (Zou et al., 4 Mar 2026). They are organized around a two-stage abstraction and refinement (ABS-REF) view, and their central technical contribution is a High-dimensional Positional Encoding (HPE) module that explicitly utilizes intrinsic positional information in local 3D neighborhoods. Within this formulation, HPENets replace time-consuming local MLP operations with non-local MLPs for efficient non-local information updates, while using HPE to preserve effective local information representation. The resulting design is presented as readily deployable in MLP-based architectures and compatible with transformer-based methods.

1. ABS-REF formulation and network family

The point of departure for HPENets is an architectural reinterpretation of point cloud processing as a sequence of abstraction and refinement stages. In this view, early models focused on ABS stages, whereas more recent techniques devised sophisticated REF stages to attain performance advantages (Zou et al., 4 Mar 2026). The framework is defined over point positions and features at layer $l$, denoted $\mathcal{P}^{l}$ and $\mathcal{F}^{l}$.

In the ABS stage, point resolution is reduced by farthest point sampling,

$\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$

followed by neighborhood construction,

$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$

and intra-set aggregation,

$$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$

The REF stage keeps the point set resolution fixed and performs same-resolution grouping,

$$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$

followed by inter-set refinement,

$$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$

HPENets instantiate this ABS-REF paradigm as a family rather than a single network. The paper distinguishes HPENet and HPENet V2, and also reports scaled variants such as HPENet V2-S and HPENet V2-XL. A plausible implication is that “HPENets” is best understood as the architectural family formed by combining the ABS-REF decomposition, HPE-based geometric encoding, and scalable refinement stages, rather than as a single fixed backbone.

2. High-dimensional Positional Encoding

The HPE module is defined on point clouds

$$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$

with

$$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$

For local neighborhoods centered at $\mathcal{P}^{l}$0, the geometrically meaningful quantity is the relative coordinate

$\mathcal{P}^{l}$1

The method emphasizes relative coordinates so that the encoding is translation invariant.

The paper first defines two lower-dimensional positional encodings used as baselines. The sinusoidal version allocates channels in groups of six: $\mathcal{P}^{l}$2

$\mathcal{P}^{l}$3

$\mathcal{P}^{l}$4

$\mathcal{P}^{l}$5

$\mathcal{P}^{l}$6

$\mathcal{P}^{l}$7

The learnable low-dimensional baseline is

$\mathcal{P}^{l}$8

HPE introduces a two-step construction: high-dimensional projection followed by alignment to the backbone feature space. Its sinusoidal form is

$\mathcal{P}^{l}$9

and its learnable form is

$\mathcal{F}^{l}$0

In the original HPENet, $\mathcal{F}^{l}$1. In HPENet V2, $\mathcal{F}^{l}$2.

Although these equations are written for a generic point coordinate $\mathcal{F}^{l}$3, the practical operator is applied to relative coordinates: $\mathcal{F}^{l}$4 A local feature update can therefore be written as

$\mathcal{F}^{l}$5

Under a global translation $\mathcal{F}^{l}$6,

$\mathcal{F}^{l}$7

so

$\mathcal{F}^{l}$8

which yields translation invariance.

The paper presents this as an extension of positional encoding from Transformer literature to a 3D, local, relative, high-dimensional geometric encoding tailored to point clouds. This suggests that HPE is not merely a replacement for raw coordinate concatenation, but a feature-space geometric operator for irregular 3D neighborhoods.

3. Local aggregation, non-local MLPs, and HPE-mediated geometry injection

HPENets use HPE to separate explicit geometric encoding from feature propagation. The grouped feature tensor after neighborhood construction is augmented by HPE: $\mathcal{F}^{l}$9 where $\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$0. This additive fusion aligns positional information with the same $\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$1-channel space as grouped features.

The paper analyzes three aggregation operators. Traditional local processing is written as

$\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$2

while non-local alternatives are

$\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$3

for PreConv, and

$\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$4

for ProConv. Here $\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$5 denotes grouping, $\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$6 a set operation or MLP, and $\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$7 a reduction such as max pooling.

The central architectural claim is that local MLPs can be reduced or replaced by non-local MLPs if local geometric information is injected explicitly. PreConv is efficient but weak on local geometry by itself; HPE supplies the missing local 3D structure. The paper therefore adopts a hybrid strategy: in the first ABS stage it keeps a stronger local operator to preserve fine detail, while later ABS and REF stages prefer efficient non-local MLPs, supported by HPE. In REF, because the stage does not change point geometry, the paper further states that HPE is shared within each REF stage to reduce computational costs (Zou et al., 4 Mar 2026).

This division of labor is foundational to HPENets. Non-local MLPs handle efficient feature updates, while HPE carries the local geometry that would otherwise be encoded by repeated local MLP operations.

4. Architectural composition and segmentation pipeline

HPENets are encoder-centric point cloud models built from ABS and REF blocks. A compact architecture description is

$\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$8

For HPENet V2-XL, the paper gives

$\mathcal{P}^{l+1} = \mathrm{FPS}(\mathcal{P}^{l}),$9

The article’s details distinguish HPENet and HPENet V2 at the block level. In the ABS-stage pseudocode, HPENet applies a local MLP after HPE-enhanced grouping,

$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$0

whereas HPENet V2 uses

$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$1

This suggests that HPENet V2 pushes the efficiency agenda further by relying more directly on HPE-enhanced grouped features and less on expensive local MLP stacks.

For segmentation, the encoder is complemented by a Backward Fusion Module (BFM). BFM takes high- and low-resolution features, extracts global statistics from $$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$2 using max and mean pooling, refines them via inverted-residual MLPs, and uses the resulting attention to update both streams. The overall segmentation pipeline can therefore be summarized as

$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$3

The paper also states that HPE is compatible with transformer-based methods. This matters because the HPE module is presented not as an MLP-only trick, but as a general positional encoding block for local geometric representation in point cloud backbones.

5. Empirical performance and ablation evidence

The paper reports experiments on seven public datasets across four different tasks and summarizes the most direct comparison against PointNeXt in the abstract (Zou et al., 4 Mar 2026).

Dataset	Reported advantage over PointNeXt	FLOPs relative to PointNeXt
ScanObjectNN	$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$4 mAcc	$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$5
S3DIS	$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$6 mIoU	$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$7
ScanNet	$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$8 mIoU	$$\mathcal{D}_{ABS}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l},\mathcal{F}^{l}),$$9
ShapeNetPart	$$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$0 Cls. mIoU	$$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$1

The benchmark sections provide the corresponding concrete model figures. On ScanObjectNN, PointNeXt-S reports $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$2 mAcc, $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$3 OA, and $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$4, whereas HPENet V2-S reports $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$5 mAcc, $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$6 OA, and $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$7; HPENet V2-S* reaches $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$8 mAcc and $$f_m^{l+1} = h_{ABS}\!\left( \mathcal{D}_{ABS}^{l+1}(p_m^{l+1}), \mathcal{K}_{ABS}^{l+1}(p_m^{l+1}) \right).$$9 OA. On S3DIS 6-fold, PointNeXt-XL reports $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$0 mIoU and $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$1, while HPENet V2-XL reports $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$2 mIoU and $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$3. On S3DIS Area-5, PointNeXt-XL reports $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$4 mIoU and $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$5 mAcc, while HPENet V2-XL reports $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$6 mIoU and $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$7 mAcc, with a best single run at $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$8 mIoU. On ScanNet, PointNeXt-XL reports $$\mathcal{D}_{REF}^{l+1}(p_m^{l+1}),\ \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) = \mathrm{KNN}(p_m^{l+1},\mathcal{P}^{l+1},\mathcal{F}_{ABS}^{l+1}),$$9 mIoU and $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$0, while HPENet V2-XL reports $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$1 mIoU and $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$2. On ShapeNetPart, PointNeXt-S(c=160) reports $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$3 Cls. mIoU, $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$4 Ins. mIoU, and $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$5, while HPENet V2-S(c=160) reports $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$6 Cls. mIoU, $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$7 Ins. mIoU, and $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$8, with a best single run of $$f_m^{l+1} = h_{REF}\!\left( \mathcal{D}_{REF}^{l+1}(p_m^{l+1}), \mathcal{K}_{REF}^{l+1}(p_m^{l+1}) \right).$$9 Cls. mIoU and $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$0 Ins. mIoU.

Ablation results establish the role of HPE itself. On S3DIS Area-5 for HPENet V2-XL, using no positional encoding gives $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$1 mIoU, $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$2, and $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$3. Using PE in ABS+REF gives $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$4 mIoU, $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$5, and $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$6. Using HPE in ABS+REF gives $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$7 mIoU, $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$8, and $$\mathcal{P}=\left\{p_m \in \mathbb{R}^{1\times 3}\right\}_{m=1}^{N}, \qquad \mathcal{F}=\left\{f_m \in \mathbb{R}^{1\times C}\right\}_{m=1}^{N},$$9. With BFM, the result is $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$0 mIoU, $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$1, and $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$2. This indicates that HPE introduces moderate overhead relative to PE, while still improving mIoU.

The ablations also clarify several design choices. Replacing relative-coordinate HPE with absolute-coordinate HPE (“HPE (abs)”) drops S3DIS Area-5 mIoU from $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$3 to $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$4. Multiplicative fusion (“HPE (mul)”) gives $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$5 mIoU, below additive fusion. The dimensionality study over

$$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$6

reports $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$7, $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$8, $$p_m = [p_m^x, p_m^y, p_m^z], \qquad f_m \in \mathbb{R}^{C}.$$9, $\mathcal{P}^{l}$00, and $\mathcal{P}^{l}$01 mIoU, respectively, making $\mathcal{P}^{l}$02 the best tradeoff in HPENet V2.

The paper also evaluates HPE beyond MLP backbones. On S3DIS Area-5, Point Transformer improves from $\mathcal{P}^{l}$03 mIoU at $\mathcal{P}^{l}$04 to $\mathcal{P}^{l}$05 mIoU at $\mathcal{P}^{l}$06 after adding non-local MLPs, HPE, and BFM. Stratified Transformer improves from $\mathcal{P}^{l}$07 mIoU at $\mathcal{P}^{l}$08 to $\mathcal{P}^{l}$09 mIoU at $\mathcal{P}^{l}$10. This reinforces the paper’s claim that HPE is compatible with transformer-based methods.

6. Scope, interpretation, and limitations

HPENets are best characterized as point cloud networks in which local 3D geometry is made explicit through a high-dimensional, translation-invariant, relative-coordinate positional embedding,

$\mathcal{P}^{l}$11

Their distinctive contribution is therefore not merely the use of positional encoding, but the use of positional encoding as a first-class geometric representation aligned with backbone feature channels.

This framing also resolves a common misconception. HPENets do not simply concatenate raw coordinates to point features. Instead, they project relative coordinates into a higher-dimensional latent geometric space and then align that representation with the backbone feature dimension. The paper explicitly criticizes low-dimensional alternatives $\mathcal{P}^{l}$12 and $\mathcal{P}^{l}$13 as remaining effectively low-dimensional relative to the complexity of local geometric relationships in irregular 3D point sets (Zou et al., 4 Mar 2026).

A second misconception is that HPENets are transformer-specific because HPE extends the “positional encoding” concept from Transformer literature. The paper states the opposite architectural scope: HPE can be readily deployed in MLP-based architectures and is compatible with transformer-based methods. The method is therefore local and geometric in its operational role, even though its conceptual lineage comes from Transformer positional encoding.

The main trade-off made explicit in the ablations is computational rather than conceptual. Relative to standard PE, HPE increases S3DIS Area-5 compute from $\mathcal{P}^{l}$14 to $\mathcal{P}^{l}$15 and parameters from $\mathcal{P}^{l}$16 to $\mathcal{P}^{l}$17, while improving mIoU from $\mathcal{P}^{l}$18 to $\mathcal{P}^{l}$19. This suggests that HPE is not costless. The paper’s broader claim is instead that this cost is justified because HPE enables a larger architectural simplification: replacing time-consuming local MLP operations with non-local MLPs while preserving effective local information representation.

In that sense, HPENets occupy a specific design position in point cloud learning. They do not eliminate local geometry; they relocate it into a dedicated high-dimensional positional representation, and then use that representation to support efficient ABS-REF feature extraction.