One-dimensional Path Convolution
Authors: Xuanshu Luo
Martin Werner
Venue: ICML 2025 poster (2025)
Keywords: 1D convolution
Hilbert curve
Z-order curve
convolutional neural network
deep learning
computer vision
Primary area: deep_learning
TL;DR: This paper presents path convolution (PathConv), a novel CNN design exclusively with 1D operations, achieving ResNet-level accuracy using only 1/3 parameters.

Abstract: Two-dimensional (2D) convolutional kernels have dominated convolutional neural networks (CNNs) in image processing. While linearly scaling 1D convolution provides parameter efficiency, its naive integration into CNNs disrupts image locality, thereby degrading performance. This paper presents path convolution (PathConv), a novel CNN design exclusively with 1D operations, achieving ResNet-level accuracy using only 1/3 parameters. To obtain locality-preserving image traversal paths, we analyze Hilbert/Z-order paths and expose a fundamental trade-off: improved proximity for most pixels comes at the cost of excessive distances for other sacrificed ones to their neighbors. We resolve this issue by proposing path shifting, a succinct method to reposition sacrificed pixels. Using the randomized rounding algorithm, we show that three shifted paths are sufficient to offer better locality preservation than trivial raster scanning. To mitigate potential convergence issues caused by multiple paths, we design a lightweight path-aware channel attention mechanism to capture local intra-path and global inter-path dependencies. Experimental results further validate the efficacy of our method, establishing the proposed 1D PathConv as a viable backbone for efficient vision models.

One-dimensional Path Convolution
Xuanshu Luo"
Martin Werner
Abstract
son, ID convolution operating sequentially along a single dimension is inadequate for image processing due to its in- ability to preserve the spatial continuity of adjacent pixels in both directions. Nonetheless, vision models still incorporate 1D convolution through factorized convolution (Rigamonti et al., 2013; Szegedy et al., 2016; 2017; Peng al., 2017; Guo et al., 2022) or oriented kernels (Freeman et al., 1991; Weiler et al., 2018; Li et al., 2021: Kirchmeyer & Deng, 2023) to reduce computational complexity. The primary rationale arises from the architectural simplicity of ID con- volution scaling linearly with the kernel size rather than quadratically like 2D convolution.
Two-dimensional (2D) convolutional kernels have dominated convolutional neural networks (CNNs) in image processing. While linearly scaling ID convolution provides parameter efficiency, its naive integration into CNNs disrupts image locality, thereby degrading performance. This paper presents path convolution (PathConv). a novel CNN design exclusively with 1D operations, achieving ResNet-level accuracy using only 1/3 parameters. To obtain
image traversal paths, we analyze Hilbert/Z-order paths and expose fundamental trade-off: improved proximity for most pixels comes at the cost of excessive distances for other sacrificed ones to their neighbors. We resolve this issue by propos- ing path shifting, a succinct method to reposition sacrificed pixels. Using the randomized round- ing algorithm, we show that three shifted paths are sufficient to offer better locality preservation than trivial raster scanning. To mitigate poten- tial convergence issues caused by multiple paths, we design a lightweight path-aware channel at- tention mechanism to capture local intra-path and global inter-path dependencies. Experimental re- sults further validate the efficacy of our method, establishing the proposed 1D PathConv vi- able backbone for efficient vision models.
These insights drive this paper's pursuit of ostensibly paradoxical objective: constructing a vision model that ex- clusively utilizes ID convolution to achieve superior parame- ter efficiency while simultaneously overcoming its intrinsic limitations to match the capabilities of conventional 21) CNNs. The sequential architecture comprising solely 1D operators facilitates pipelining implementation on hardware (Mukherjee & Mukhopadhyay, 2018; Cheng & Parhi, 2020; Narayanan et al., 2021), thereby substantially reducing end- to-end latency. Consequently, using only ID convolution offers apparent advantages in computational efficiency. The primary challenge lies in enabling ID CNNs to effectively interpret visual information.
This work employs only 1D operators to construct CNNs, necessitating 2D images to be flattened. Trivial image traver- sal by raster scanning disrupts the spatial locality of images, which is essential for detecting primitive visual features, such as edges, textures, and shapes. Unlike raster scan- ning, the Hilbert (Hilbert, 1935) and Z-order (Morton, 1966) curves are two alternative traversal paths with recognized spatial proximity preservation capability (Jagadish, 1990; Dai & Su, 2003; Wang al., 2022; Wu et al., 2024), indicat- ing their great potential to be the key ingredient to ID CNNs to achieve comparable performance to 2D CNNs. However, this paper argues that the locality preservation properties of these two paths have a critical trade-off: compared to raster scanning, while the majority of pixels maintain closer proximity to their adjacent pixels in the resulting ID pixel streams, the remaining pixels are sacrificed due to exces- sively long distances neighbors, even sometimes leading to greater total distances than raster scanning. This find- ing persists for multiple image resolutions, demonstrating
1. Introduction
Convolutional Neural Networks (CNNs) have emerged as the predominant paradigm for computer vision tasks across diverse domains (LeCun et al., 1989; Fukushima, 1988;
Krizhevsky et al., 2017; Girshick, 2015; Chen et al., 2018: Ravanbakhsh et al., 2016), with two-dimensional (2D) con- volution serving as the fundamental operation. In compari-
'Professorship of Big Geospatial Data Management, Tech- nical University of Munich, Germany. Conrespondence to: Xuanshu Luo cxuanshu.luo@tum.de Martin Werner <mar- wemerr tum.de>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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One-dimensional Path Convolution
99
2. Related Work
ResNet-18/99
S
68
2.1. ID convolution in CNNs
Imageluc:4 Top-1 (6
97-
The linear scaling cost of ID convolution with the kernel size presents compelling opportunities for efficient 2D CNN architectures. We separately discuss them based on their distinet theoretical foundations.
96
66
CIFAR 10/5VHN
65
95
Factorized convolution. The core idea of factorized convo- lution builds on a fundamental principle from linear algebra that a 2D convolution kernel can be approximated or exactly decomposed into sequential ID convolutions. For instance, rank-1 separable kernel is equivalent to the product of consecutive horizontal and vertical ID kernels (Rigamonti et al., 2013; Jaderberg et al., 2014; Jin et al., 2014), enabling k x k kernel to be substituted by two ID kernels with 2k parameters in total. Such parameter reduction becomes increasingly significant as increases while maintaining identical receptive field sizes, so CNN architectures involv- ing large kernel sizes widely adopt this idea (Szegedy et al., 2016; 2017: Peng et al., 2017: Guo et al., 2022; Huang et al., 2023). However, 2D kernels in CNNs do not inherently possess low-rank properties. Consequently, many methods have emerged to better approximate 2D kernels by learning low-rank kernels and applying singular value decomposi- tion, aiming to minimize potential performance degradation (Denton et al., 2014; Lin et al., 2018; Yang et al., 2020).
94-
63
93
66
CIFAR-10
SVHN
ImageNect.G.
Figure 1. The performance comparison between PathConv and ResNet models on three datasets, with circle radii denoting the number of model parameters. PathConv models achieve ResNet- level performance using only 1/3 of the parameters.
that direct usage of both paths cannot globally guarantee better locality than raster scanning, inducing suboptimal performance of ID CNNs relative to 2D architectures.
This paper first presents that the positions of these sacrific- ing pixels can be shifted by sampling from their spatially expanded variants, inspiring us integrate a set of carefully selected shifted paths to ensure pixels at all positions are closer to their neighbors than raster scanning for at least one path. Meanwhile, limiting the number of paths is essential to avoid significant computational overhead, as input im- ages must be traversed once for each path. Determining the minimal set of paths to ensure better locality preservation than raster scanning for all pixels is equivalent the NP set cover problem (Vazirani, 2003). Using the randomized rounding algorithm (Bertsimas & Vohra, 1998), we show that merely three paths suffice for Hilbert and Z-order paths. A customized CUDA kernel further accelerates the image traversal procedure up to 73x, making its costs negligible.
Oriented convolution. Unlike factorized convolution, ori- ented kernels place parameters along multiple angles beyond solely horizontal and vertical orientations (Li et al., 2021; Kirchmeyer & Deng, 2023). The underlying motivation could be illustrated by a trivial example: a full-rank 2D kernel denoted by a diagonal matrix is hard to decompose into horizontal and vertical ID kernels but can be directly represented by a single oriented ID kernel. However, these methods generally require dedicated software and hardware implementation for efficiency and optimization, as popular deap-lea libraries do not natively support them. A conceptually related technique is steerable filters, which enhance their expressiveness by learning rotation- equivalence/iivariince features, which are proven effec- tive in capturing directional characteristics like textures and edges (Freeman & Adelson, 1991; Cohen & Welling, 2016; Worrall et al., 2017; Weiler et al., 2018). These methods are usually more complex than oriented kernels and do not necessarily use ID convolution.
Building upon this efficient image traversal layer, we pro- pose path convolution models adapted from the macro de- sign of depthwise separable convolution (Chollet, 2017) and inverted bottleneck (Sandler et al., 2018) but using ex- clusively ID operators. Notably, multiple traversal paths require ID kernels to process distinct regions of images con- currently, which presents challenges for model convergence. Hence, we propose a path-aware channel attention mecha- nism to capture both local intra-path and global inter-path dependencies to achieve dynamic focus images, thereby reasonably enabling rapid receptive field growth. Experi- mental results demonstrate that our models achieve compa- rable performance to ResNet (He et al., 2016) while utilizing only 1/3 parameters, indicating that the proposed path con- volution effectively maximizes the parameter efficiency of ID kernels with minimal performance degradation.
In comparison, the proposed path convolution model com- prises only 1D operations to facilitate pipelining and max- imize the parameter efficiency of ID kernels, processing pixel streams flattened from input images, which differen- tiates this paper from all methods above incorporating ID convolution while maintaining 2D input/output signatures. However, images inherently have a spatial structure where
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One-dimensional Path Convolution
proximate pixels have strong correlations and form mean- ingful patterns. Flattening images into 1D pixel streams disrupts these crucial locality features. To address this issue, this paper uses traversal paths based on Hilbert and Z-order curves for better locality preservation than raster scans.
Hilbert curves
2.2. Hilbert and Z-order curves
Z-order curves
Hilbert and Z-order curves exemplify space-fillingg curves (Peano, 1890) passing through all unit squares (pixels) ex- actly once in two- or higher-ding nigher-dimensional discrete spaces with topologiaal self-organizing properties, i.e., invariant ge- ometric structure across multiple scales. Concretely, one can observe a pattern topologically equivalent to the complete curve any segment for arbitrary magnification levels, dif- fering only in scale and/or rotational orientation, as shown in Figure 2. This characteristic originates from their recursive definitions, creating a scale-invariant 1D-to-2D mapping inherently constraining the spatial distances of points in the output space relative to their input space proximity (Lempel & Ziv, 1986; Dai & Su, 2003). Hence, the self-organizing nature of Hilbert and Z-order curves enables their locality preservation properties for multiple resolutions, empower- ing applications in image compression (Ansari & Fineberg, 1992; Wang et al., 2022), databases (Kamel & Faloutsos, 1994; Castro et al., 2005), parallel computing (Yoo et al.,
21Z 2 214 Z7Z
Figure 2. Hilbert and 7-order curves for multiple sizes. Hilbert curves repeat a U-shape pattern with rotations, whereas Z- curves replicate Z shapap pattern without rotations.
(a) raster sean path
(b) Hilbert path
(c) Z-urder puth
Figure 3. Pixel-wise comparison between (a) raster scan and (b) Hilbert/ / (c) Z-order paths. Pixels with shorter cumulative distances to all their neighbors than (a) are in apricot, otherwise purple.
2003; Böhm et al., 2018), and point cloud processing (Chen
et al., 2022: 2023; Wu et al., 2024).
However, this paper traverses images along paths defined by Hilbert and Z-order curves, mapping from 2D to 1D rather than the reverse, thus hardly ensuring equivalent strength in locality preservation. The fundamental limitation arises because while two points may be proximate in 2D space, ID paths might traverse a long distance to connect them, particularly for pixels at (sub)quadrant boundaries, result- ing in some spatially adjacent points being mapped farther apart in the 1D sequence than raster scanning, as depicted in Figure 3 (see Appendix A for more examples in mul- tiple resolutions). Consequently, the direct application of Hilbert or Z-order paths for image traversal produces subop- timal results in image recognition tasks, as evidenced by our experiments (Table 4). In this paper, we mitigate this lim- itation by integrating multiple shifted paths sampled from their spatially expanded variants as elaborated in Section 3.2, thereby ensuring better locality preservation for every pixel than raster scanning in at least one path.
locality preservation. To address this deficiency, we pro- pose concise path-shifting method to relocate pixels with long distances to their neighbors. While combining multiple shifted paths suffices the locality constraint, minimizing the number of paths remains essential for optimizing traversal efficiency. Our analysis indicates that determining the mini- mal set of paths fulfilling the locality constraint constitutes an NP problem. We use the randomized rounding algorithm to solve the problem and find that three paths suffice the locality constraint for commonly used image resolutions.
3.1. Locality measurements
A straightforward and convincing approach to assessing the locality preservation capability of traversal path is to calculate the cumulative distances between each pixel and all its adjacent neighbors in resulting 1D pixel streams. Comparing to the raster scan path served as the baseline, except for comparing the total distances of all pixels in a path, pixel-wise comparisons are also taken into account by calculating the proportion of pixels with shorter distances to their neighbors at the same positions, denoted by Psa-
3. Locality-preserving Paths
This section aims to find set of paths satisfying the locality constraint, wherein each pixel maintains shorter cumulative distances to its neighbors than its corresponding pixel in the raster scan path. Our locality measurement reveals that conventional Hilbert and Z-order paths possess imperfect
Table I reports the locality measurements of Hilbert and Z- paths versus raster sean paths for multiple resolu- tions. Although Hilbert paths guarantee that P.d increases as the resolution grows, the total distances are contradic-
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One-dimensional Path Convolution
(a) the original path
(b) sampling from larger paths
(c) the resulting shifted path (d) locality preservation of pixels
Figure 4. The path shifting procedure. Given (a) His and {6, [1, 5). 0}. expanded (h) H22 (16 6) is firstly generated. Then,
16 x 16 sampling window, with its bottom left coordinate at [1, 5]. is applied to (b) to obtain (c) His The coloring schema of Figure 3 applied to (a) and (c). Jointly considering (a) and (c). (d) shows the locality preservation status of all pixels compared with Rıs. where (a) initially enables apricot pixels to have better locality preservation. (c) further enables light apricot pixels, demonstrating the effectiveness of the proposed path-shifting method for relocating sacrificed pixels. Purple pixels still have longer distances their neighbors.
illustrates the detailed procedure using Hilbert path as an example. Given images of size & x 8, shifted paths are sampled from their spatially extended variants, with sizes equal to the sum of s and the padding size p. Subsequently, the sampling window of dimension cording to its bottom left coordinates [ where i, integers in range [0, p). Besides, shifted paths could be ro- tated r degrees, where r E {0, 90, 180, 270}, to achieve more diverse repositioning of sacrificed pixels. A configura- tion e {p. [r,j].r) r) thus specifies a path-shifting operation. Accordingly, we denote shifted Hilbert and Z-order paths of size s with configuration e as HC and Z respectively, The original Hilbert and Z-order paths conform to this tation with omitted, as all parameters equal 0, yielding H. and 2,, respectively. Similarly, we have R., to denote the raster scan path. The proposed shifting approach ef- fectively relocates sacrificed pixels previously located at (sub)quadrant boundaries, as shown in Figure 4, revealing the feasibility of meeting the locality constraint. When set of c generates shifted paths that fulfill the locality constraint, denote it as C. In other words, set of paths H. or Z can guarantee that there are no sacrificed pixels.
Table 1. Locality measurements of the raster scan, Hilbert, and Z-order paths multiple resolutions.
Path
Resolution
Total Distance
Pad
raster scan
1.88 x 10a
s positioned ac-
Hilbert
32 32
2.15 x 10a
79.10%
Z-order
1.80 x 10
74.61%
raster scan Hilbert
1,54 x 106
64
1.80 10"
80.96%
Z-order
1.49 10
81.35%
raster
1.24 10?
Hilbert
128 x 128
1.48 107
86.51
Z-order
1.22 x 107
84.50%
raster scan
1.00 10
Hilbert
256 x 256
1.20 10
89.04%
Z-order
0.99 x 10
89.17%
torily longer. This is because most pixels at the border of quadrant boundaries are sacrificed to keep the recursive, self- similar structures of Hilbert paths, necessitating non-local multi-step connections between spatially adjacent pixels located in different quadrants of recursive subdivisions, as previously shown Figure While Z-order paths maintain shorter total distances than the raster scan path, they fail to consistently provide higher P.d for the same reason Hilbert paths, not to mention the locality constraint implies P.a 100%. Consequently, directly applying Hilbert and Z-order paths yields suboptimal spatial locality preservation, thereby constraining the model performance.
We only consider square images here for simplicity. Ap- pendix B further indicates that this shifting method also works for non-square resolutions and substantially broad- ens the applicability of Z, by removing its limitation of [2n I NJ, where N is the of natural numbers.
3.3. The minimal C satisfying the locality constraint
The cardinality of C. denoted [C), represents the num- ber of shifted paths. While large |C] trivially meets the locality constraint, simultancously introduces substantial overhead by traversing images for [C] times. Therefore, the ideal solution is determining C with the smallest possible
3.2. Path shifting
This paper introduces a simple method for repositioning sacrificed pixels in Hilbert and Z-order paths, eliminating their concentration (sub)quadrant boundaries. Figure

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Onc-dimensional Path Convolution
4. Path Convolution Model
Table The configurations denoted as C satisfying the locality constraint when [C] 3 for Hilbert and Z-order paths.
This section introduces the path convolution (PathConv) model, a CNN architecture with only ID operators pro- cessing flattened ID images. Figure 5 shows its overall architecture similar canonical 2D CNNs (He et al., 2016; Liu et al., 2022). Input images are first transformed into ID pixel streams by the path traversal layer. then fed into the stem layer, followed by PathConv blocks organized in four stages with downsampling layers in between. Besides, we propose a path-aware channel attention mechanism as an imperative component in our model to effectively process pixel segments from different regions of images. Detailed designs are elaborated in the following subsections.
Path
8
Configurations (C*)
D, 0, 01,
32
(24, [14, 11, 180 [18, 7, 01, 270]
Hilbert
{ 0, 0, 01, 0)
64
{ 7, 77, 01, 180 [40, [13,
1], 0
[32, [16, 261, 180)
32
32, 114, 161, 0
Z-order
(32, [12, 201, 0f
[64, [36, 321, , 270
64
64, 136, 341, , 90
{64, [28, 321, 0}
Stage 1
Stage 2
Stage
Stage
cardinality satisfying the locality constraint. In Appendix C. we demonstrate that this problem is polynomial-time reducible to the NP cover problem (Vazirani, 2003). Hence, existing algorithms and theoretical bounds of the set cover problem could be applied to our problem to find and evaluate satisfactory C. We employ the randomized rounding algorithm (Raghavan & Tompson, 1987) trying to find the minimal C satisfying the locality constraint. First, the universe C must be specified given image size s². |C| should be sufficiently large to ensure adequate diversity in the distribution of sacrificed pixels while remaining con- strained to maintain computational feasibility given the NP complexity. In each configuration c {p, [i.j],,), r,, where r E {0, 90, 180, 270} and [ is determined by p with J E [0, pl. only p requires detailed consideration.
path traversal layers
stem layers
O PathConv blocks
downsampling layers
classification head
Figure 5. The overall architecture of PathConv models
Figure 6. Given distinct paths (left: Hie right: Hig. where [6. [0. 5), 0}), the model simultaneously observes (nearly) aligned regions in different orders (e.g., Region a: first 16 pixels), or disparate regions (c.g., Region b: last 16 pixels).
Both H, and Z, are defined recursively and self-replicated when is doubled. When p 2 8, the same [i, derives similar shifted paths where the positions of sacrificed pixels exhibit no significant differences between H. / 2 and HS/ / 2 (see Appendix D). Consequently, [1, 8) is a reasonable parameter space for p. In practice, while H, can directly apply this setting, Z, only accept s E T. where T -22 n N}. Hence, we set p t s for Z,, where i is the smallest element in T larger than s. In this way, C could be determined given for both kinds of paths. Solving by the randomized rounding algorithm, we find that [C] = 3 sufficiently satisfics the locality constraint. The corresponding are given in Table 2. Considering the theoretical approximation ratio of In 22 x 6.93 and 8.32 for s 32 and 64, respectively (Bertsimas & Vohra, 1998: Feige, 1998), [C] = 3 highly likely optimal, as one path cannot meet the locality constraint. We designate these C as C* to represent the default configurations for model construction, which theoretically yield optimal path sets for efficient locality-preserving preserving image traversal. More details of parameter selections, algorithm specifications, calculation time, etc., are provided in Appendix D.
4.1. Path traversal layer
The path traversal layer flattens input images into ID pixel streams following traversal orders specified by a set of paths consisting of sequential pixel coordinates. Note that this transformation is read-only, presenting significant potential for efficiency optimization. For example, we implement CUDA kernel to perform fast image traversal by leveraging NVIDIA GPU parallelism, demonstrating up to 73-fold ac- celeration compared to a single-thread thread CPU implementation. We provide more details in Appendix E. We emphasize that the path traversal layer remains hardwarr-agnostic. Its over- head can become negligible through common optimization strategies, thus hardly downgrading the model efficiency.
The path traversal layer samples pixels along distinct paths. Hence, PathConv models simultaneously process pixel seg- ments (a) from spatially aligned regions in varying orders or (b) from different spatial regions, as shown in Figure 6.
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Onc-dimensional Path Convolution
Both patterns are worth discussion as they fundamentally differ from conventional CNNs, generally processing the same spatial areas in the same wiiddo order.
(b) PathConv block
PACA
(a) stem layer
Case (a) implies that ID kernels handle pixels within (nearly) overlapping regions for multiple orders, which is similar to applying depthwise ID convolution grouped by paths. This pattern also conceptually resembles oriented ID convolution discussed in Section 2.1, where varying orienta- tions (paths) enhance kernel expressiveness. Whereas case (b) necessitates the model to observe different regions of original images, effectively expanding the receptive fields. However, the corresponding kernel sizes remain the same, which may provide insufficient model capacity, potentially impeding model convergence.
dy, K= dim
pig, k
pi exp. dim
dw, k-11
7. GELU
pww dim
LN
O-pos, enc.
Figure 7. The stem layer and PathConv block designs.
culated through a multi-layer perception:
f [WW[GGGGUUWWWNGG + b1) + b2)
where W1 RR bj E R2P, W22 E R2PxP, by E IR are parameters of two dense layers, LN denotes layer nor- malization (Ba et al., 2016), introducing O(P2) parame- ters in total. GELU serves as the activation function here (Hendrycks & Gimpel, 2016). The final attention weights w combine both intra- and inter-path attentions by w = 2 O f, where @ denotes element-wise multúplication with broad- casting. We can apply to scale X across channels with global context relationships produce the final output.
4.2. Path-aware channel attention
To address this issue, we introduce a path-aware channel at- tention (PACA) module that captures both path-specific and cross-path dependencies, enabling models to focus dynami- cally discriminative features. The parameter efficiency is maintained by using only channel-wise and path-wise attention, thereby avoiding excessive additional parameters. Given an input tensor X E x where B is the batch size, H is the number of channels, and L is the input length, PACA processes features within and across paths.
With appropriate feature scaling by PACA, we mitigate the convergence issue presented in case (b). Meanwhile, con- sidering the number of paths P is constant (c.g.. |C| = 3), the entire PACA only introduces Ollog G) additional pa- rameters, growing logarithmically with model width, thus maintaining parameter efficiency. PACA modules are inte- grated into PathConv blocks introduced later.
We first organize X into path-specific groups to calculate intra-path path attention, building upon efficient channel atten- tion (Wang et al., 2020). For P paths and G channels per path, X is reshaped into X E BBxxxXXL We compute dual-scale attention weights for each path p using adaptive kernel sizes h1 = + (1 t mod 2) and l2 = 2k1 - 1, where t = [log, + 8/7], y and B are hyperparame- ters controlling attention scales. We set 7 = 2 and B in experiments. Then for each p, the corresponding X, is averaged over L to obtain Y/ BxIXG Subsequently, the intra-patt attention weights ep are computed by
4.3. Other building blocks
Stem design. Figure 7(a) illustrates the design of stem lay- ers. A pointwise ConviD first increases the channel dimen- sion, followed by depthwise Conv1D with a kernel size of 11. Compared to depthwise separable convolution (Chol- let, 2017), reordering depthwise and pointwise Conv1D operations reduces the number of parameters. Addition- ally, learnable positional encoding (Dosovitskiy et al., 2021) enables the model retain awareness of critical spatial in- formation of pixels, with its necessity demonstrated by the ablation study (Table 5).
p ConviD(yp:ki)
6 = ConviD(yp, 1z)
2
ep (Woll)) + bo)
where ConvlD are 1D convolutional layers for ef and eg, Wo E R1x2 and by E IR are learnable real-number pa- rameters processing concatenated e and eg, and a is the sigmoid activation function. The intra-path attention cap- tures channel-wise dependencies at two scales, requiring merely ((P log G) parameters to obtain more robust chan- nel scaling through dynamic balance between scales.
Coth Blocks.
The PathConv model utilizes PathConv
blocks as the main building block, incorporating PACA modules before inverted bottlenecks (Sandler et al., 2018) with an expansion ratio of 4, as depicted in Figure 7(b). In the ablation study (Table 5), we validate that PACA is indispensable for PathConv models.
Others. The downsampling layers contain depthwise Conv1D with kernel sizes of 9 and strides of 2, followed by LN layers. The classification head is simply a dense layer transforming the dimensionality into the number of classes.
To model inter-path relationships, the concatenated path- specific ep are firstly averaged across G to obtain & E R BxP. Then, the inter-path attention weights f are cal-
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Onc-dimensional Path Convolution
5. Experiments
Table 3. Performance comparison for three datasets. Numbers in parentheses show accuracy differences between PathConv variants and their ResNet counterparts with similar FLOPs. Green indicates improvement, red shows 2 2% degradation, and amber denotes others. For WRN (WRN-n-w), a indicates the number of layers, and w denotes the width multiplier.
We present experimental settings and results to evaluate the proposed PathConv model. Analysis of path selection demonstrates that C brings optimal performance. Ablation studies further validate the PathConv design.
5.1. Settings
model
#param.
FL.OPs
top-1 acc.
Datasets. We evaluate PathConv models using three datasets: CIFAR-10 (Krizhevsky al., 2009), SVHN (Net- zer et al., 2011), and ImageNet-64 (Chrabaszcz et al., 2017), a downsampled version of ImageNet- IK (Russakovsky et al., 2015) with the same image count and dataset splits with a resolution of 64 x 64 across 1000 classes. We use low-resolution datasets as they amplify locality preservation issues (as shown in Table 1, Pad grows for both paths as image resolution increases), allowing clear attribution of performance gains to our path-shifting method and design of components in PathConv models.
CIFAR-10
(s
32)
ResNet-18
11.68M
1.11G
93.17%
PathConvS- "
3.62M
1.14G
92.45%
(-1.02%
PathConvS 22
3.62M
1.14G
92.66% (-0.51%)
PathConvS-PR R,
3.61M
1,14G
89.56% (-3.61%)
ResNet-50
25,55M
2,60G
93.95%
PathConv MahhConBB-7"" "
7.82M
2.50G
93.97% (+0.02%)
PathConvB
7.82M
2.50G
93.56% (-0.39%)
PathConvB-R.
7.82M
2.49G
90.02% (-3.93%
SVHI (s
32)
ResNet-18
11.68M
1.1IG
97.66%
PathConvS-P(
3.62M
1.14G
97.12% (-0.54%)
Models. PathCony models are constructed in two variants, small and base, denoted as PathConvS/B, with compara- ble floating-point operations (FLOPs) as ResNet-118555 (He al., 2016), respectively. The number of blocks (#bl.) in each stage and the corresponding number of channels (#ch.) are listed below:
PathConvS-Z
3.62M
1.14G
96.57% (-1.09%)
PathConvS-RR
3.61M
1.14G
91.60% (-6.06%)
ResNct-50
25.55M
2.60G
97.86%
PathConVB-P4
7.82M
2.50G
97.81% (-0.05%)
PathConVB-Z2
7.82M
2.50G
97.58% (-0.28%)
PathConvB-R.
7.82M
2.49G
94.36% (-3.50%)
PathConvS: #b 2, 2), #ch.=(48, 96, 192, 384) PathConvB: #bl.=(2, 2, 3)3,, #6h.=(60, 120, 240, 480)
ImageNet 64 (s 64)
WRN-40-2
9.33M
3.83G
61.23%
ResNet- -8
11.68M
4.44G
63.43%
We remove pooling layers from ResNet and change their large kernels into x 3 to better adapt dataset resolu- tions. For ImageNet-64, the wide residual network (WRN) (Zagoruyko, 2016) is also considered, which is used in the paper introducing ImageNet-64 (Chrabaszez et al., 2017).
PathConvS-74 -
3.76M
4.58G
62.26%
(-1.17%)
PathConvS-22 2
3.76M
4.58G
62.73% (-0.70%)
PathConvS-RR.
3.76M
4.58G
58.04% (-5.39%)
WRN-40-5
57.24M
23.78G
70.32%
ResNet-50
25.55M
10.39G
68.43%
Training. PathCony models employs the Adam opti- mizer (Loshchilov & Hutter, 2019), while ResNet models utilize stochastic gradient descent with Nesterov momen- tum (Sutskever et al., 2013). All models are trained for 300 epochs with 10-epoch linear warmup. The learning rate follows the cosine annealing schedule (Loshchilov & Hutter, 2017). We adopt RandAugment (Cubuk et al., 2020) for data augmentation. For ImageNet-64, we additionally apply Mixup (Zhang et al., 2018) and CutMix (Yun et al., 2019). Stochastic depth (Huang et al., 2016) and label smoothing (Szegedy et al., 2016) are also used for regularization. More experimental settings are provided in Appendix F.
PathConvB-PP -
8.01M
9.98G
68 68.46% (+0.03%)
PathConvB-22
8.01M
9,98G
68.83% (+0.40%)
PathCouvB
8.00M
9,97G
60.40% (-8.03%)
We can observe that PathConv models with H. and 2 consistently achieve comparable performance ResNet while requiring < 1/3 #param. and similar FLOPs. Notably, on ImageNet-64, PathConvS-7( / /2, outperforms WRN- 40-2 despite the latter's 's higher parameter count. PathConvB- H.a /2." also perform better than ResNet-50. These re- sults demonstrate the parameter efficiency of the proposed pure ID PathConv architectures. Conversely, PathConv models utilizing R, have significantly lower performance than ResNet, indicating that prreetiig paths gen- erated by our path-shifting method (C*) are more effective for image traversal in ID PathConv models than raster scan paths, corroborating the analysis presented in Section 3 and also illustrating that PathConv models effectively leverage the
5.2. Results
Table shows the comparison results for three datasets regarding parameter counts (#param.), FLOPs and top-1 ac- curacy. PathConvSBB models incorporate suffixes denoting paths for the path sampling layers. Except for H." and 2 (see Table 2), R. serves as a bascline for PathConv models.
properties of HC and 2

---

One-dimensional Path Convolution
Table Performance comparison of PathConvB models using varying image traversal paths. Numbers in parentheses indicate accuracy differences of " from H. /2,"" and 2.,
Table 5. Results for PathConv BC, three datasets following the ablation of positional encoding and PACA components, with the coloring scheme in Table 3.
The same color schema Table is applied here.
pos. enc.
PACA
top-1 acc.
model
#param.
FLOPs
top-1 acc.
CIFAR-10 (s = 32)
CIFAR- 10 (s 32)
93.97%
PathConvB-743
7.82M
2.50G
93.97%
X
90.13% (-3.84%)
92.34% (-1.63%)
PathConvB-PP "
7.82M
2.50G
91.01%
(-2.96%)
X
86.23% (-7.74%)
MathConvB-44
7.82M
2,49G
94.12% (+0.15%)
PathConvB-22 -
7.82M
2.50G
93.56%
SVHN (s 32)
97.12%
PathConvB B-2.
7.82M
2.50G
90.98% (-2.58%)
X
94.20% (-2.92%)
PathConv B-2,
7.82M
2.49G
93.23% (-0.33%)
95.91% (-1.11%)
SVHN
(8
32)
X
89.77% (-7.35%)
PathConvB-P
7.82M
2.50G
97.81%
ImageNet-644 (s 64)
PathConv B-
7,82M
2.50G
96.57% (-1.24%)
68.45 08.46%
x
60.93% (-7.33%)
PathConvB-PP
7.82M
2.49G
97.23% (-0.58%)
X
/
63.85% (-4.61%)
PathConvB-22 7.82M
2.50G
97.58%
X
52.91% (-15,55%)
PathConvB-22
7.82M
2.50G
97.04% (-0.44%)
PathConvB-2,, 7.82M
2.49G
97.31% (-0.27%)
ImageNct-64
(s = 64)
ever, P= performs significantly worse than C* and C+ for PathConvB, as it fails to meet the locality constraint, thus providing insufficient model capacity for large datasets (ImageNet-64). In conclusion, |C*| = 3 perfectly balances model efficiency and performance. These insights empiri- cally suggest C* as the optimal set for PathConv models.
PathConvB -
8.01M
9.98G
68.46%
PathConv B-H.
8.01M
9.98G
62.23% (-6.23%)
MathConvB-P7
8.00M
9.97G
60.53% (-7.93%)
MuthConvB-2""
8.01M
9.98G
68.83%
PathConvB-22 B
8.01M
9.98G
63.07% (-5.76%)
PathConvB-22
8.00M
9.97G
60.01% % (-8.82%)
5.4. Ablation study
5.3. The impact of path selection
We validate that both positional encoding and PACA are essential for PathConv models by ablation. Table presents the corresponding results of removing either both com- ponents from "ithConvB-74," These results indicate that removing either module substantially degrades model per- formance, with PACA's 's removal having more pronounced effect. The absence of PACA significantly impairs the abil- ity of PathConv models to process cases (b). Given the small #param. requirements for both modules, the architecture of PathConv models has proven effective and efficient.
We now examine the influence of path selection on PathConv model performance. The results in Table 3 demonstrate negligible performance variations between 7C5 and _ This invariance to path type indicates the robustness of our path-shifting method, as both paths effectively satisfy the locality constraint, enabling PathConv models to achieve performance comparable to ResNet models.
Beyond path types, we also investigate whether |C*|= = 3 is optimal for PathConv models. To this end, C+ is defined of presented in Table 10 (Appendix F) with doubled cardinality (i.e.. P 6) satisfying the locality constraint. Meanwhile, the original H., and Z, are also path candidates for p= 1. We apply these paths to PathConvB models for three datasets, with results shown in Table 4.
6. Conclusion
This paper presents a novel approach to constructing CNNs using exclusively 1D operations, achieving parameter effi- ciency while maintaining performance comparable to 2D CNNs. We introduce a path-shifting method to effectively preserve the locality of flattened 2D images, requiring only three paths to outperform trivial raster scan paths. Building upon this foundation, our PathConv architecture, featuring path-aware channel attention, matches the performance of ResNet while using only 1/3 of the parameters across multi- ple datasets. This work reveals new paradigm for efficient CNN design based purely on 1D operations.
For small datasets, P brings limited adverse effects, occasionally providing better performance (CIFAR-10) than C* while P results in more severe performance drops. These findings indicate that PathConv models achieve better convergence with fewer paths on small datasets, enabling better results even without satisfying the locality constraint. Meanwhile, despite PACA, P = exacerbates the prob- lem of the case (b), impeding model convergence. How-
8

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One-dimensional Path Convolution
Acknowledgements
Chollet, F. Xception: ble convolutions. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp. 1251- 1258, 2017.
Deep learning with depthwise separa-
This work was supported by the Federal Ministry Educa- tion and Research of Germany (BMBF) under grant number 16DHBKI020.
Chrabaszez, P., Loshchilov, I., and Hutter, F. downsam- pled variant of imagenet an alternative to the cifar datasets. arXiv preprint
Impact Statement
2017.
This paper presents work that aims to advance the field of computer vision. There are many potential societal con- sequences of our work, none of which we feel must be specifically highlighted here.
Cohen, T. and Welling, M. Group equivariant convolu- tional networks. In International Conference on Machine Learning, pp. 2990-2999, PMLR, 2016.
Cubuk, E. D., Zoph, B., Shlens, J., and Le, Q. V. Ran- daugment: Practical automated data augmentation with a reduced search space. In Proceedings of the IEEF/CVF Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, June 2020.
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One-dimensional Path Convolution
A. Pixel-wise comparison of locality preservation for multiple resolutions
extend pixel-wise comparison locality preservation with the same comparison method and coloring schema as Figure to multiple resolutions. Figure 8 illlustrates pixel-wise locality preservation status for Hilbert/Z-order paths compared raster scan paths of the same sizes. We can observe that the shape of purple pixels also roughly grows with topologiaa self-organizing properties, aligning with the recursive definition of both paths.
Hilbert paths
Z-order paths
2/22
2/2
72
2022
2202
72
2222
2322
2222
2022 22
2222
22122
777 777240777
2222
2222
22 22
272 22
22 7202
2022
24242222
22
Figure Applying the same comparison method and coloring schema utilized in Figure with 82, 162 323. 64 resolutions. Pixels with shorter cumulative distances to all their neighbors than raster scan paths are in apricof, otherwise purple.
B. Supports for rectangular images
Except for R, inherently accommodating various resolutions, H. also supports rectangular images through minor modifica- tions, such as generalized Hilbert curves (gilbert) (Červený, 2024). Figure (a) shows an example of 18 21 generalized
12

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One-dimensional Path Convolution
Hilbert path using gilbert. In comparison, original 2g only accepts squares with {2" n E NJ, where N is the of natural numbers. The proposed path-shifting method manages to fill this gap, shown in Figure 9 (b). We obtain this 18 21 Z-order path by sampling from Z32 with i 0, 5, 0. Hence, our shifting method also unintentionally extends the application scope of 2, for arbitrary image sizes. Consequently, our method could be applied to any image dataset without compatibility constraints.
2/7
(a) 18x21 generalzeed Hilbert path
(b) 18x21 Z-order path by path shifting
Figure 18 paths as examples of paths for rectangular sizes.
C. Reduction to the set cover problem
We first formally describe our problem determining C with the smallest possible cardinality to meet the locality constraint and the set cover problem, then show the reduction procedure.
PROBLEM (P1) Minimal C with the locality constraint. Let C = foI, ..., Cn be the universe of all possible configurations given each configuration Cg is associated with
binary matrix M2 E (0, 1]*x* representing pixel-wise locality preservation status as shown in Figure 4(d). If pixel at [j, has shorter distances its neighbors, M, [j+ otherwise M, k] 0.
a distance measurement dy E N recording the total distance of this path (the same as Table 1).
Determine subset C C C minimizing
(1)
subject to the locality constraint
[MG 2c 1, (0,
(2)
We formulate this problem optimization procedure without loss of generality as minimizing (1) inherently prevents involving unnecessary paths.
PROBLEM 2 (P2) Weighted Set Cover Problem. Given a universe U, family of sets F = (FI,.., Fr), where each set F u, and cost function w : F + N. Find subset F C F minimizing
Σ E( (w(FF)
FEF) FFF)
(3)
subject to
U(FI U[FIFEF) =u
(4)
Reduction. Consider the following reduction 7 mapping any instance of P1 to instance of P2 follows:
Let A[ Vj.
[0, })), where each element pixel position in P1.
13

---

One-dimensional Path Convolution
For each configuration Cg C, create F.g {(x,y) M2 v) = 1]}. Then, F (Fg i [1,]]].
For every Fg corresponding to cz. let w/ (Fa) = dg.
Following this mapping, the reduction 1 respectively suffices to these three properties:
is polynomial-tine time computable, as only (ns²² operations are required to construct U and F from an instance of Pl, where n is the number of configurations and s² the size of images.
preserves solution mapping. This is because for any C of an instance of PI, the set (F, cq CJ is also solution to the corresponding instance of P2, as any pixel (j. k) with Myli, k] corresponds to (j, k) covered by Eg. and vice versa.
! maintains solution optimality by construction. For any C of an instance of Pl and its corresponding solution I,, E(d Cq E C} = (F) e F}.
Therefore, PI is polynomial-time reducible to P2. Existing algorithms and theoretical bounds of the set cover problem could be used to find and evaluate satisfactory C in our problem.
D. More details of obtaining C
In the previous section, have presented detailed steps map our problem of finding C* an instance of the weighted set cover problem. This formulation enables to employ the randomized rounding algorithm (Raghavan & Tompson, 1987), widely adopted approach for solving set cover problems.
Let 2g be binary variable indicating whether configuration cy is selected. We reuse definitions in Appendix C. Then, finding C* can be formulated as the following integer linear program (ILP):
minimize
Σ (dgpgg cg EC}
subject the locality constraint
Σ (M nnyl Co EC} 2 K (0, (0,
where x,, E 10. The randomized rounding algorithm begins by relaxing the constraint 2g t, S1 E IR thereby transforming this ILP into a linear program (LP) solvable in polynomial time. After obtaining the optimal fractional solution á, the algorithm executes &-round iterations. During each iteration, selects G, with probability min(a 1), where a denotes a scaling factor. When the selected configuration set satisfies the locality constraint (2), the algorithm updates the optimal solution if shorter total distance is achieved. Our implementation utilizes parameters t 30 and 1.2. While employed this study, the randomized rounding algorithm is not our contribution. Detailed algorithmic analysis is available in (Raghavan & Tompson, 1987).
Table 6. Comparison of the randomized rounding and greedy algorithms on finding C², showing average computational time and the cardinality of the worst-case solution (C*) Greedy algorithm cannot always guarantee |C*|
Randomized rounding algorithm
Greedy algorithm
8
Path
[C]
avg. time worst case |C*)
avg. time
worst case |C*
Hilbert Z-order
39684
67. Is
8.3s
3
32
3969
2.0s
0.1s
3
Hilbert
333316
7134.6s
3
634.6s
4
64
Z-order
16129
321.7s
3
51.9s
4
While the randomized rounding algorithm provides a theoretical approximation ratio of O s2) with high probability (Feige, 1998), its computational cost significantly exceeds that of the trivial greedy algorithm (Table 6). This motivates an investigation into whether the greedy algorithm can achieve comparable solution quality with reduced computational
14

---

One-dimensional Path Convolution
Table Runtime comparison hetween CUDA kernel for image traversal and its CPU counterpart for varying batch sizes, s, and the number of paths P. Each measurement represents the average execution time over 1,000 iterations, recorded in milliseconds (ms).
P 3
P = 5
batch size
CPU
Ours
CPU
Ours
32
0.755ms
0.083ms 9.1
0.830ms
0.076ms (10.909x)
128
64
1.648ms
0.075ms (22.063x)
2.400ms
0.094ms (25.489x)
224
25.920ms
0.369ms (70.250x)
1
0.555ms (61.026x)
32
1.506ms
0.077ms (19.521x)
2.141ms ms
0.093ms (22.933x)
512
64
8.074ms
0.158ms (51.181x)
10.070ms
0.212ms (47.393x)
224
80.864ms
1.319ms (61.287x)
123.716ma
2.09 (59.166x)
32
4.083ms
0.105ms (38.860x)
5,297ms
0.133ms (39.962x)
1024
64
15.877ms
0.253ms (62.823x)
21.508ms
0.366ms (58,690x)
224
190.583ms
2.587ms (73.659x)
254.165ms
4.160ms (61.101x)
Table Training settings of PathConv models for three datasets.
Setting
CIFAR-10
SVHN
ImageNet-64
initial learning rate
4e-3
4e-3
le-2
weight decay optimizer
2e-4
2e-4
2e-4
Adam W
Adam W
Adam W
optimizer momentum batch size scheduler warmup epoch warmup schedule
B1, 32 = 0.9,0.999 0.999
B1, B2 = 0.9, 0.999
$1, 32 = 0.9, 0.999
512
512
384
cosine annealing
cosine annealing 10
cosine annealing 10
10
linear
linear 7
linear
RandAugment
7
9
Mixup
0.2
CutMix
0.6
label smoothing
0.05
0.05
0.05
overhead. We implement the greedy solver by iteratively picking e that covers the most pixels with longer distances to their neighbors than raster scanning. Table 6 provides the comparison results of using randomized rounding and greedy algorithms to find C The average computational time calculated across three executions of each algorithm. Despite the significantly shorter computational time of the greedy solver, does not consistently guarantee |C*| Consequently, the randomized rounding algorithm remains the preferred choice, particularly given that C* computation is required only once.
E. CUDA implementation of the path sampling layer and efficiency comparison
We implemented a specialized CUDA kernel to accelerate the image traversal procedure and integrated it into the path sampling layer for maximum efficiency. The kernel assigns one thread per pixel position to compute corresponding index in the resulting ID pixel stream. We specify 256 threads grouped together for every CUDA block, enabling shared GPU memory access, cooperation, and synchronization among threads within each block. Table 7 compares runtimes between CUDA kernel implementation and CPU version for varying batch sizes, and the number of paths P on NVIDIA A100 GPU. Our CUDA kernel achieves up to 73.659-fold acceleration compared to the CPU implementation, demonstrating the efficiency of the path sampling layer.
We emphasize that the image traversal procedure is read-only, enabling optimization through various parallel processing approaches. Our CUDA implementation here is one example, selected here due to its direct benefits for model training.
15

---

One-dimensional Path Convolution
Table 9. Training settings of ResNet/WRN models for three datasets.
Setting
SVHN CIFAR-10
ImageNet-64
initial learning rate
2e-2
2e-2
2e-2
weight decay
2e-4
2e-4
2e-4
optimizer
SGD
SGD
SGD
optimizer momentum
0.9
0.9
0.9
batch size
512
512
384
scheduler warmup epoch
cosine annealing
cosine annealing
cosine annealing
10
10
10
warmup schedule
linear
linear
linear
Rugment
7
7
9
Mixup
0.2
CutMix
0.6
label smoothing
0.05
0.05
0.05
Table 10. The configurations denoted C+ satisfying the locality constraint when |C+| = 6 for Hilbert and Z-order paths. We specify C+ for Section 5.3 to investigate the impact path selection.
Path
s
Configurations (C+)
0, 0, 0].
0). { 5, [ 0, 4]
0
Hilbert
32/64
0, 4, 0].
0). { 0, [ 4, 4]
0}.
{ 0, 2, 2], 0).{ 0, [ 2, 4],
0}
0, 0, 0],
03,(32, [16, 26], 0}.
32
{32, [14, 16],
180),(32) [12, 20], 180).
{32, 8, 4],
(32, [ 4, 81, 0}
Z-order
0, 0, 0],
0),(64, [36, 32]
270].
64
{64, 136, 34),
90). (64, [28, 321,
0).
{64, 8],
0}, (64,
[ 8,, 4], 0}
F. Experimental settings
We provide detailed training settings for PathConv models and 2D CNNs in Table 8 and Table 9, respectively. We do not apply weight decay on all normalization layers (LN). To ensure fair comparison, all models are trained from scratch. This eliminates confounding factors from transfer learning and allows direct attribution performance differences to architectural choices.
To investigate the impact of path selection, we also specify C+ with |C+| 6 for Section 5.3 as given in Table 10.
16


Reviews:
[{"review_id": "7GxLIrYGgR", "reviewer": "Reviewer_z3TX", "summary": "This paper presents an enhancement to 1D convolutional networks with space-filling curves (traversal path) for the image classification task.\nSpace-filling curves are paths that pass through every point within a higher-dimensional discrete space. Recently, 1D convolution has successfully been applied to several domains in computer vision using space-filling curves. This paper aims at showing that the most used space-filling in this domain (i.e. Hilbert curves and Z-order curves) lacks a locality preservation property. This work proposes a method to preserve locality in Hilbert curves and Z-order curves by sampling these last from an expanded dimension and relocating the most distant pixels in the path via a path-shifting method that relies on an attention mechanism. Determining the set of minimal paths that fulfills the locality property constitutes an NP problem, therefore, the paper uses an approximation based on heuristics, which belongs to the randomized rounding algorithms family. The cardinality of the set of minimal paths was found to be 3 via this heuristic for images of size 32x32 and 64x64.\nThe paper then tests and compares the approach with simple raster scans, Hilbert curves, and Z-order curves for the single-label image classification task on CIFAR-10, SVHN, and a downsampled version of ImageNet1k: ImageNet-64.\n\n## update after rebuttal\nThe rebuttal addressed some of my concerns about scalability to higher resolutions, but questions remain about the significance of the accuracy results for the same budget of FLOPs and/or parameters. I, therefore, maintain my previous rating.", "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 sincerely appreciate your effort for your review and insightful comments. The key concerns raised will be addressed in the following sections.\n\n## Scalability to higher resolutions\n\n> As this problem is NP-hard, there is no evidence that finding such paths for practical resolutions used in ImageNet1k (224x224, or 256x256) is possible.\n\nSection 3.3 and Appendix C demonstrate that finding $\\mathbb{C}$ with minimal cardinality satisfying the locality constraint is NP. Given a large $s$, the size of universe $|\\mathcal{C}|$ becomes too large for NP problems. For instance, given $s=224$, we have $|\\mathcal{C}| = 14785796$ for Hilbert paths $\\mathcal{H}_s$, making it impractical to calculate $d$ (following Table 1) for every configuration, not to mention as the input of an NP problem. Therefore,\n\n- For $\\mathcal{H}_{224}$, we decrease the parameter space of $p$ into $[1, 16]$ to obtain a much smaller $|\\mathcal{C}| = 4484$.\n- For $\\mathcal{Z}_{224}$, we only consider $p \\in \\set{0, 32}$ (explained in Section 3.3), resulting in $|\\mathcal{C}| = 3969$.\n\nWe find that the above settings of $p$ generate sufficiently large $|\\mathcal{C}|$ to find a $\\mathbb{C}$ with $|\\mathbb{C}|=3$ when $s=224$, aligning with the results of low-resolution cases in this paper. We also denote such $\\mathbb{C}$ as $\\mathbb{C}^{\\ast}$. Following the structure of Table 2, we have\n\n|Path|$s$|$\\mathbb{C}^{\\ast}$|\n|-|-|-|\n|||{13, [10,  5], 270}|\n|Hilbert|224| {10, [ 5,  0], 270}|\n|||{ 7, [ 7,  0],  90}|\n|||{ 0, [ 0,  0],   0}|\n|Z-order|224| {32, [11, 14], 180}|\n|||{32, [16, 14],   0}|\n\nSimilarly, we extend Table 6 for $s=224$\n\n|s|Path|\\|$\\mathcal{C}$\\||Randomized rounding algorithm||Greedy algorithm||\n|-|-|-|-|-|-|-|\n||||avg. time|worst case \\|C*\\||avg. time|worst case \\|C*\\||\n|224|Hilbert|4484|2.3s|3|0.1s|4|\n||Z-order|3969|2.0s|3|0.1s|4|\n\nThese results indicate that\n\n- $|\\mathbb{C}^{\\ast}|=3$ consistently holds for higher resolution ($224^2$). Hence, to find $\\mathbb{C}^{\\ast}$ is possible using the proposed method for practical resolutions.\n- The greedy algorithm is unable to consistently derive $|\\mathbb{C}^{\\ast}|=3$.\n- In the original paper, $p\\in [1, s)$ may be unnecessarily large for finding $\\mathbb{C}^{\\ast}$, where $|\\mathbb{C}^{\\ast}|=3$.\n\n## Evaluation\n\n> ... This is not conclusive. A better approach would have been to compare the difference in accuracy for the same number of parameters.\n\nIf #param. are set to the same, the corresponding FLOPs will be higher than ResNet counterparts. In this case, even though PathConv models outperform ResNet, it is hard to conclude anything, as PathConv is with the same #param. but require higher FLOPs compared to ResNet. Hence, we make a harsher case that PathConv is with the same FLOPs but with much fewer parameters than ResNet. Additionally, the current setting can also represent the parameter efficiency of 1D kernels, which is the key motivation of this work.\n\n> ...to make a graph accuracy/parameters and a graph accuracy/FLOPs!\n\nThis is an excellent suggestion for determining the optimal #param. (FLOPs)/acc trade-off. We will apply your suggestion in subsequent studies. We express our sincere gratitude.\n\n## Reference to discuss\n\n> The usage of Hilbert and Z-order curves for image classification was presented in anterior work like Wang, Hanyu, et al. (2022) \"Neural space-filling curves.\" and for point clouds in computer vision in Wu, Xiaoyang, et al. (2024) \"Point transformer v3: Simpler faster stronger.\".\n\nNeural space-filling curves are designed to learn a spatially coherent path primarily for image compression rather than classification. Point transformer v3 employs SFCs for the essential linearization of point clouds as input for transformers. Consequently, these two references are not particularly relevant to this article, as they do not share similar motivations and functions with this paper.\n\n\n## Regarding your questions\n\n> Since the locality preserving curves are independent of the images, wouldn't it be more convenient to pre-calculate/find all of them up to a certain dimension beforehand and save the corresponding path traversal for ulterior use?\n\nYou are absolutely correct. Actually, that's exactly how we implement `PathConv`. The paths will be only calculated once during instantiation. We will articulate this more explicitly in the main text of the manuscript.\n\n> From a biological perspective, how does locality in space filling curves relate to human or mammal eye-tracking (gaze path)?\n\nThis is a very insightful question. Mammals often employ a coarse-to-fine visual search mechanism, starting with broad scanning before focusing on areas of interest [(Hochstein et al., 2002)][1]. This is similar to how space-filling curves recursively divide space and cover broad areas before filling subregions (Figure 2).\n\n[1]: Hochstein, S., & Ahissar, M. (2002). View from the top: Hierarchies and reverse hierarchies in the visual system. Neuron, 36(5), 791-804.", "claims_and_evidence": "In general, the claims made by the paper are clear and supported; however, it seems that the proposed method, although preserving locality, should rely further on an attention mechanism as the depthwise 1d convolution mechanism used on the space filling curve sometimes necessitates the model to observe different regions of original images (see Figure 6). This questions the relevance of the 1D convolution on this representation of data as a space-filling curve.\n\nMoreover, the paper finds traversal paths that comply with the locality constraints for resolutions up to 64x64. As this problem is NP-hard, there is no evidence that finding such paths for practical resolutions used in ImageNet1k (224x224, or 256x256) is possible. Not to mention downstream tasks such as segmentation, which are missing from this work and which sometimes require 512x512 or full resolution.", "methods_and_evaluation_criteria": "The proposed metric that calculates the proportion of pixels with shorter distances to their neighbors at the same positions (Psd) makes sense for evaluating the locality constraint. The datasets also make sense, however, their size is limited.\n\nThe improvement described in the paper only encompasses a reduction in the total number of parameters with more or less the same accuracy. In many cases, this does not allow us to decide whether the approach is beneficial or not for the task. For example, in Table 3, the authors report an accuracy of 93.17 for ResNet-18 (11.68M params) on CIFAR10 and an accuracy of 92.66 for PathConvS-Z (3.62M params) - a reduction of a factor of 3.5 in the number of params for a 0.5 reduction in accuracy. \nThis is not conclusive. A better approach would have been to compare the difference in accuracy for the same number of parameters. Or better still, to make a graph accuracy/parameters and a graph accuracy/FLOPs!\n\nIn terms of FLOPs, the proposed methods have approximately the same cost, with marginally better or worse accuracy.\n\nIn addition, the chosen baselines are not state-of-the-art. More recent models achieve better accuracy for less compute budget. There is no mention of seed variability or the sensitivity of the results with respect to randomness.\n\nIn Table 4, however,  the locality property put forward by the authors seems confirmed as having a significant impact on performance.", "theoretical_claims": "I was not able to fully check the theoretical claim about the minimal C satisfying the locality constraint and the associated appendices C and D due to lack of time. Another reviewer might have checked it. Other theoretical claims seem to be correct.", "experimental_designs_or_analyses": "The paper uses depthwise separable convolution and an inverted bottleneck design in the blocks. There is also a notion of stem layer. The paper builds those techniques on top of a ResNet architecture. Perhaps it would have been much easier to start from a modernized architecture that already integrates all of those techniques like ConvNeXt (Liu, Zhuang, et al. \"A convnet for the 2020s.\"), for example, or any SOTA convolutional model, and compare the performance of the approach on top of it.", "relation_to_broader_scientific_literature": "The article follows the line of research concerning 1D convolutional neural networks with an application to image classification.", "essential_references_not_discussed": "The usage of Hilbert and Z-order curves for image classification was presented in anterior work like Wang, Hanyu, et al. (2022) \"Neural space-filling curves.\" and for point clouds in computer vision in Wu, Xiaoyang, et al. (2024) \"Point transformer v3: Simpler faster stronger.\". The current work cites those two papers and others, specifying that its contribution is an enhancement of this idea:\n\n> This paper first presents that the positions of these sacrific-\ning pixels can be shifted by sampling from their spatially\nexpanded variants, inspiring us to integrate a set of carefully\nselected shifted paths to ensure pixels at all positions are\ncloser to their neighbors than raster scanning for at least one\npath.", "other_strengths_and_weaknesses": "In general, the idea has a good motivation and is supported theoretically. Although not completely novel, the originality of the work lies in addressing the problem of locality in space-filling curves for 1D CNN.\n\nThe major concern, however, is the significance of the results with respect to the baseline and the lack of evidence for the usability of the results in higher resolutions.", "other_comments_or_suggestions": "The style of the text is a little repetitive and sometimes incoherent, as if it had been written by a machine. For example, it is very hard for the reader to understand sentences like:\n\n>  we analyze Hilbert/Z-order paths\nand expose a fundamental trade-off: improved\nproximity for most pixels comes at the cost of\nexcessive distances for other sacrificed ones to\ntheir neighbors.", "questions_for_authors": "- Since the locality preserving curves are independent of the images, wouldn't it be more convenient to pre-calculate/find all of them up to a certain dimension beforehand and save the corresponding path traversal for ulterior use?\n\n- From a biological perspective, how does locality in space filling curves relate to human or mammal eye-tracking (gaze path)?", "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": "DQsA5dYMyS", "reviewer": "Reviewer_SGYX", "summary": "This paper proposes a path convolution (PathConv) architecture based on one-dimensional convolution, aiming to solve the problem that traditional one-dimensional convolution destroys the spatial locality of the image. By introducing the Hilbert/Z-order curve as the image traversal path and combining the path shifting technology to adjust the position of the sacrificed pixel, only three shifting paths are needed to globally retain the locality that is better than traditional raster scanning. In addition, a lightweight path-aware channel attention (PACA) mechanism is designed to alleviate the convergence problem caused by multiple paths. Experiments show that PathConv achieves ResNet-level performance on datasets such as CIFAR-10, SVHN, and ImageNet-64 with only 1/3 of the parameters.\n\n*----------------------------------------------------------------------------------------------------------------------------------------------------------*\n\n## Response to the Authors' rebuttal:\n\nThanks for your response. Although some parts have not addressed in a good shape, e.g., 1) no experimental results for ImageNet-256 or non-square images (such as COCO) are provided. Only a hierarchical path hypothesis is proposed, which lacks actual data support; 2) The basis for selecting translation parameters (e.g., fill size) is still unclear, and no parameter sensitivity analysis is provided, which affects the reproducibility of the method; 3) The computational complexity of path selection for ultra-large images such as 1024×1024 has not been analyzed, and the scalability of the NP problem is questionable, the core contribution of the paper (achieving locality preservation and parameter efficiency of one-dimensional CNN through path translation and set covering theory) has good theoretical and engineering value, and the experimental design fully verifies the basic assumptions on CIFAR/SVHN/ImageNet-64. Hence, I tend to keep my original score unchanged.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful that the motivation and proposed method of this paper have received your endorsement. We will try to address your concerns in the following sections.\n\n## Performance on high-resolution/non-square images\n\n> The actual performance of high-resolution (such as 256×256) or non-square images was not tested...\n\n> The experiment is extended to higher resolution (such as ImageNet-256) and non-square images to verify the generalization of the method.\n\nWe fully understand the concern regarding the absence of PathConv performance for high-resolution images. We posit that hierarchical paths across different resolution levels can be an effective approach for higher-resolution images. The locality features of high-resolution images may involve much more pixels than low-resolution ones, which probably prove challenging to capture using the current method. In this context, a ViT-like architecture may be a better solution. This is also what we will focus on following this work.\n\nThe inclusion of analysis for non-square images would indeed enhance the comprehensiveness of this paper. Performance testing for downstream tasks will be conducted in future work. Images in common object detection datasets are usually non-square (e.g., MSCOCO), thus providing a valid way to address the stated concern.\n\n\n## Other backbones\n\n> Only ResNet and WRN are compared, lacking horizontal comparison with efficient models, such as MobileNet and EfficientNet.\n\nWe agree that comparisons with efficient models would strengthen the manuscript. The core contribution of this work is demonstrating the viability of using only 1D kernels to build CNNs to achieve superior parameter efficiency. ResNets are widely adopted and chosen as a foundation for many CNN architectures (e.g., ConvNeXt). The performance of mentioned backbones is not notably superior to ResNet. Hence, existing results sufficiently validate the effectiveness of the proposed methods. Furthermore, especially for EfficientNet using NAS, direct comparison incurs risks conflating the benefits of our method with orthogonal optimizations.\n\n## Path shifting parameters\n\n> The selection basis of specific shift parameters (such as padding size) in Table 2 is not detailed...\n\nThe parameter spaces are provided in Section 3.3. Following these settings, the universe $\\mathcal{C}$ can be determined. We then apply the randomized rounding algorithm to obtain Table 2 following the steps described in Appendix D.\n\n## Paths for high-resolution cases\n\n> The NP nature of set coverage may cause a sharp increase in the computational cost of path selection for larger-scale images...\n\nWe conducted an additional experiment applying the path-shifting method to high-resolution images, where $s=224$. We find that a small range of $p$ is enough to find $\\mathbb{C}$ with cardinality of 3. Following the structure of Table 2, we have\n\n|Path|$s$|$\\mathbb{C}^{\\ast}$|\n|-|-|-|\n|||{13, [10, 5], 270}|\n|Hilbert|224|{10, [5, 0], 270}|\n|||{7, [7, 0], 90}|\n|||{0, [0, 0], 0}|\n|Z-order|224|{32, [11, 14], 180}|\n|||{32, [16, 14], 0}|\n\nDue to space limitations, please refer to the first section in the response to reviewer z3TX for details.\n\n## Questions\n\n1. Can the choice of path shift parameters (e.g., $p$) be optimized through automated search (e.g., NAS)?\n\n    We believe this is possible. We are also considering adding more flexibility by learning a space-filling curve dependent on input images. Both methods are interesting to investigate in future works. We appreciate your inspiring comments.\n\n2. How to handle spatiotemporal consistency of paths in dynamic inputs (e.g., videos)?\n\n    A trivial way is to obtain embedding by frame and employ a temporal architecture to model sequential features. Moreover, both Hilbert and Z-order paths could be extended to higher dimensions (e.g., 3D Hilbert curve).\n\n3. What is the parallel computing potential of PathConv? Is multi-path processing limited by GPU memory bandwidth?\n\n    Path sampling operations are read-only, thereby amenable to optimization by various parallel computing strategies. This paper provides a CUDA-based solution. If GPU memory is the bottleneck, sampling operations can be offloaded to CPU while maintaining efficiency. Additionally, we try to find $\\mathbb{C}$ with minimal cardinality is to minimize GPU memory requirements. Hence, we see a large potential in parallelism.\n\n4. Can it be combined with Transformer (e.g., ViT) to further enhance global modeling capabilities?\n\n    We believe this is possible as a feasible solution, especially for high-resolution cases, as explained in the first section.\n\n5. Will the robustness of PathConv drop significantly in extremely low-parameter scenarios (e.g., edge devices)?\n\n    This is a question worth exploring. A feasible solution is to follow the suggestion from reviewer z3TX to make acc./#param. and acc./FLOPs graphs and check when there is a catastrophic performance degradation to answer this question quantitatively.", "claims_and_evidence": "(1) Claim: Path shifting can effectively improve locality preservation.\n\n(1) Evidence: Table 1 shows that the ratio of sacrificed pixels ($P_sd$) of the Hilbert/Z-order path increases with the resolution, but the total distance may still be inferior to raster scanning; Figure 4 proves that after path shifting, the sacrificed pixels are repositioned and the locality coverage is more comprehensive.\n\n(2) Claim: Three shift paths are sufficient to satisfy the locality constraint.\n\n(2) Evidence: The problem is modeled as a set cover (NP problem) via a randomized rounding algorithm. Table 2 shows that the three-path configuration ($]\\mathbb{C}$) covers all pixels in the experiment (Table 4).\n\n(3) Claim: PathConv outperforms traditional 2D CNN in parameter efficiency.\n\n(3) Evidence: Table 3 shows that PathConv-S/B has the same accuracy as ResNet-18/50 with 2/3 fewer parameters, and even slightly better in some scenarios.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "(1) Locality preservation theory: The recursive self-similar structure of Hilbert/Z-order curves can preserve spatial proximity at multiple scales, but the existence of sacrificed pixels needs to be redistributed through path shifting.\n\n(2) Set cover modeling: The minimum path number problem can be polynomially reduced to a set cover problem, and a near-optimal solution is obtained through a random rounding algorithm (Appendix C).\n\n(3) Attention mechanism design: PACA balances the information fusion of multi-path inputs through dynamic weighting of channel and path dependencies (formula and Figure 7).", "experimental_designs_or_analyses": "Yes, the experimental setup appears well-motivated.", "relation_to_broader_scientific_literature": "This paper combines the theoretical advantages of space-filling curves and the dynamic modeling capabilities of attention mechanisms in a one-dimensional convolutional framework through path shifting and PACA mechanisms, and solves the path selection problem with the help of combinatorial optimization theory. This work fills the gap between locality preservation and parameter efficiency of full one-dimensional CNNs, and provides a new direction for the design of hardware-friendly and efficient visual models.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n\n(1) The path selection problem is transformed into a set covering problem, and mathematical reduction and approximate solutions are given, with sufficient theoretical support.\n\n(2) Covering multiple data sets (low/medium resolution), ablation experiments (Table 5), path configuration comparison (Table 4), verifying the necessity of each component.\n\n(3) CUDA acceleration is used to implement the path traversal layer, with a speed increase of 73 times (Table 7), ensuring the practicality of the method.\n\nWeaknesses:\n\n(1) The actual performance of high-resolution (such as 256×256) or non-square images was not tested, and only briefly mentioned in Appendix B.\n\n(2) Only ResNet and WRN are compared, lacking horizontal comparison with efficient models, such as MobileNet and EfficientNet.\n\n(3) The selection basis of specific shift parameters (such as padding size) in Table 2 is not detailed, and may depend on parameter adjustment.\n\n(4) The NP nature of set coverage may cause a sharp increase in the computational cost of path selection for larger-scale images (such as 1024×1024), which is not discussed.", "other_comments_or_suggestions": "(1) The experiment is extended to higher resolution (such as ImageNet-256) and non-square images to verify the generalization of the method.\n\n(2) Compared with other efficient models (such as MobileNetV3), showing the parameter efficiency of the proposed PathConv.", "questions_for_authors": "(1) Can the choice of path shift parameters (e.g., $p$) be optimized through automated search (e.g., NAS)?\n\n(2) How to handle spatiotemporal consistency of paths in dynamic inputs (e.g., videos)?\n\n(3) What is the parallel computing potential of PathConv? Is multi-path processing limited by GPU memory bandwidth?\n\n(4) Can it be combined with Transformer (e.g., ViT) to further enhance global modeling capabilities?\n\n(5) Will the robustness of PathConv drop significantly in extremely low-parameter scenarios (e.g., edge devices)?", "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": {"ethical_review_flag": "Flag this paper for an ethics review.", "ethics_expertise_needed": ["Other expertise"], "ethical_review_concerns": "N/A"}}, {"review_id": "ZgXFZ9XJgX", "reviewer": "Reviewer_5Xg4", "summary": "The use of 1D CNN kernel can greatly reduce the model size comparing aginst the traditional 2D. However, the lack of locality information hidders the use of 1D kernel. This paper proposes a new traversal mechanism which can be used for Hilbert and Z-order scans for 1D CNN model. The result shows the proposed 1D CNN design can achieve competitive result to the traditional 2D model with fewer parameters.\n\n## update after rebuttal:\nAccording to the guideline, my review after rebuttal is attached in the rebuttal comment below, \"Rebuttal Comment by Reviewer 5Xg4\".", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the reviewer's constructive feedback. We have summarized your concerns and will respond to each accordingly, followed by addressing your questions.\n\n## Related works & experimental settings\n\n> Moreover, convolution can be used in other vision tasks, validating the method on other tasks, e.g., detection or segmentation could be more convincing, like oriented-1D did [Kirchmeyer & Deng, 2023].\n\n> However, the comparison between this study (scanpath) and their methods are not clear, \"popular deep-learning libraries do not natively support them\" is not convincing to me, especially for theory discussion.\n\n> ...I wonder why the proposed method cannot be compared against those oriented-1D or steerable solutions?\n\nWe thank the reviewer for these insightful suggestions. We emphasize that this paper aims to build a vision model only using 1D kernels to maximize parameter efficiency and achieve locality preservation by space-filling path shifting and the PathConv model design. Whereas oriented-1D utilizes a different solution based on directional 1D kernels requiring non-trivial implementations. In contrast, the proposed method is conceptually distinct from oriented-1D. It is logical to compare oriented-1D (with different principles) after demonstrating the viability of our method in this paper. Hence, we agree on this point and will present a comprehensive comparison in future studies.\n\nFurthermore, we fully acknowledge the absence of PathConv performance for high-resolution images. We hypothesize that hierarchical paths across different resolution levels can be an effective approach for higher-resolution images. The locality features of high-resolution images may involve much more pixels than low-resolution ones, which probably prove challenging to capture using the current method. When solving this problem, we will evaluate the performance of downstream tasks, whereupon the comparison with oriented-1D will be more comprehensive. Nevertheless, we emphasize that the conclusions of this paper hold substantial value for related research. The comparison with oriented-1D is essential and will constitute our main focus in future work.\n\n## Choice of paths\n\n> One critical issue is the choice of P in this design. From section 5.3, it seems the selection of an appropriate cardinality is important, but this also makes the whole design more complicated. And this choice could be dataset specific, otherwise it will \"impede model convergence\", which makes the whole design more complicated, as discussed in App.D. And choosing the right 3 is more complicated.\n\nThe impact of path selection is investigated in Section 5.3, with corresponding results presented in Table 4. While $\\mathcal{H}_s/\\mathcal{Z}_s$ occasionally perform better on small datasets, $\\mathbb{C}^{\\ast}$ can stably deliver robust performance for all datasets, especially for ImageNet-64. Therefore, we can conclude that $\\mathbb{C}^{\\ast}$ is the best choice to balance model efficiency and performance, independent of the dataset. Consequently, $\\mathbb{C}^{\\ast}$ can be universally applied to PathConv models in practice, where detailed configurations are provided in Table 2.\n\nDetermining the appropriate set of paths is indeed challenging. However, this issue has already been addressed in Section 3.3 and Appendices C and D. As a result, we have $\\mathbb{C}^{\\ast}$ in Table 2 for different $s$. Moreover, $\\mathbb{C}^{\\ast}$ is dataset independent, thereby only needing to be calculated once. Hence, although determining the right three paths is complicated, we contend that this challenge has been substantially resolved.\n\n## Question\n\n> Page 4, the \"sacrificed\" information lost in hilbert and Z-order are the purple points in Fig. 3, (please correct me if I am wrong). If so, it might be better to highlight this main drawback in a rectangle in Fig. 3...\n\nYes, you are correct. The sacrificed pixels are in purple in Figure 3. And yes, highlighting sacrificed pixels would be more intuitive for readers. We may consider to apply this idea. Thank you for your advice!\n\n> In Table 4, C+ represents the doubled cardinality P=6, but the FLOP numbers are all the same, ... Higher cardinality represents more (configurations) input data, am I rigtht?\n\nYes, you are correct. Thank you for pointing this out. This is of vital importance for us. Some fields have rounding problems. We provide correct figures below.\n\nPathConvS (s=32):\n\n|P|#param.|FLOPs|\n|-|-|-|\n|1|3,614,912|1,144,381,120|\n|3|3,615,676|1,144,966,848|\n|6|3,617,248|1,145,844,096|\n\nPathConvS (s=64):\n\n|P|#param.|FLOPs|\n|-|-|-|\n|1|3,762,368|4,575,092,416|\n|3|3,763,132|4,577,447,616|\n|6|3,764,704|4,580,979,072|\n\nPathConvS (s=32):\n\n|P|#param.|FLOPs|\n|-|-|-|\n|1|7,820,316|2,494,394,480|\n|3|7,821,274|2,495,123,760|\n|6|7,823,230|2,496,214,560|\n\nPathConvS (s=64):\n\n|P|#param.|FLOPs|\n|-|-|-|\n|1|8,004,636|9,974,468,720|\n|3|8,005,594|9,977,409,840|\n|6|8,007,550|9,981,818,400|", "claims_and_evidence": "1. It can be easily proven that the use of 1D filter can reduce the model size of 2D baseline, the number of parameters, this claim is shown in Tab. 1.\n\n2. The main issue is the lost locality information. This study shows how Hilbert and Z-order curve can somehow solve this problem, the scanpath can still capture the local feature, and this claim is clearly showed in Fig. 3. \n\n3. However, the use of the scanpath still sacrifice the information near quadrant boundaries, thus shifted + rotated scanpath are added as supplementary to fix this missing information. This main proposal is shown in Fig. 4 and Section 3.2.\n\nThe claims are easy to follow, the only question is if the result is convincing, (discussed below).", "methods_and_evaluation_criteria": "The method is proposed to improve the 1D convolution process, which can be used for a wide range of vision tasks. The current draft is evaluated based on image classification, accuracy is reported to compared against the traditional raster scan. \n\nAccuracy on classification can somehow show the effectiveness of the method, but it is better to compare the method on \"standard\" resolution (256*256) or larger. Moreover, convolution can be used in other vision tasks, validating the method on other tasks, e.g., detection or segmentation could be more convincing, like oriented-1D did[Kirchmeyer & Deng, 2023].\n\nOne main drawback in the evaluation is the baseline setup, please see the weaknesses section.", "theoretical_claims": "The theoretical claims in this paper are easy to understand, including:\n1. 1D filter requires less parameters than 2D, Table 3.\n2. Flattening the input image for 1D filter will destroy the locality information.\n3. Hilbert and Z-order scan can focus more on locality, Figs. 2.\n4. There still exists information lost (wrong neighbour distance) in step 3, Fig. 3.\n5. Shifting and rotating the traversal path could capture the lost information in step 4, Fig. 4.", "experimental_designs_or_analyses": "The proposed method is validated based on image classification, thus it is reasonable to compare the proposed method with traditional model using the widely used measure, accuracy. The experiment was conduceted on CIFAR, SVHN(32*32) and ImageNet-64(64*64). The result shows the proposed method can achieve competitive results against the traditional 2D model, ResNet. This result seems \"partially\" convincing to me to validate the proposed method.\n\nHowever, I also have several questions regarding the experiments shared in the secion below.", "relation_to_broader_scientific_literature": "This paper studies an essential problem that how to organize and process the input information. This question is essential to almost any convolutional systems, which can be used in various vision applications.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "I generally like the writing of this draft, the problem is introduced clearly and solution seems easy to understand. However, a main problem in the draft is about the related work. The references could be adequate, but the difference is not discussed in depth.\n\nI can find those related literatures in the draft for oriented filter, or steerable filters. However, the comparison between this study (scanpath) and their methods are not clear, \"popular deep-learning libraries do not natively support them\" is not convincing to me, especially for theory discussion. \n\nFuther, the only baseline used for the experiment is the standard raster scan with ResNet, this baseline is not convincing, I wonder why the proposed method cannot be compared against those oriented-1D or steerable solutions?\n\nOne critical issue is the choice of P in this design. From section 5.3, it seems the selection of an appropriate cardinality is important, but this also makes the whole design more complicated. And this choice could be dataset specific, otherwise it will \"impede model convergence\", which makes the whole design more complicated, as discussed in App.D. And choosing the right 3 is more complicated.", "other_comments_or_suggestions": "Please see below.", "questions_for_authors": "Page 4, the \"sacrificed\" information lost in hilbert and Z-order are the purple points in Fig. 3, (please correct me if I am wrong). If so, it might be better to highlight this main drawback in a rectangle in Fig. 3, which is solved by the proposed shifting method.\n\nIn Table 4, C+ represents the doubled cardinality P=6, but the FLOP numbers are all the same, I am not sure if this is a typo, or I might miss sth in the context. Higher cardinality represents more (configurations) input data, am I rigtht?", "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": "NE7G39G65g", "reviewer": "Reviewer_kfqd", "summary": "The authors propose an alternative for 2D convolutional networks that is lighter. They use a topological 1D path across all the pixels based on Hilbert or Z paths. They use more than one such path at each layer to make up for locality. Other than the convolution layer they have several other building blocks such as channel attention inter and intra paths, activations and more. They show results comparable to resnets with more parameters. They have implemented their layers in CUDA.\n\n\nThroughout the fields below, strength, weaknesses and questions will be marked by **(S), (W), (Q)** . \n\n\n## update after rebuttal\nI thank the authors for their response. The authors acknowledged some of the concerns I raised. For one concern they provided some scaling evidence, applying their method on ImageNet resolution. While not fully convinced that generally scaling is practical, I'm glad they provided this example and at the very least showed that there are ways to adjust. I maintain my opinion that this paper is novel and has significant scientific contribution, but not a method ready for actual practical use (which is fine). I think this paper should be accepted and I maintain my original favorable score.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are profoundly grateful for your exceptionally insightful and supportive evaluation of our work. Such constructive affirmation not only underscores the significance of this research within the current paradigm but also inspires our dedication to advancing this direction toward practical applications.\n\nIn addition, we would like to thank you for clearly marking strengths, weaknesses, and questions to facilitate our communication. We sincerely appreciate your thoughtful recognition of the novelty of this paper. To further strengthen the discussion and address your valuable concerns, we now elaborate on the limitations raised.\n\n## Weakness\n\n> **(W1)** There is no evidence of availability to scale up...\n\nWe conducted an additional experiment applying the path-shifting method to high-resolution images, where $s=224$ follows the common setting of ImageNet. We find that a small range of $p$ is enough to find $\\mathbb{C}$ with cardinality of 3. Following the structure of Table 2, we have\n\n|Path|$s$|$\\mathbb{C}^{\\ast}$|\n|-|-|-|\n|||{13, [10, 5], 270}|\n|Hilbert|224|{10, [5, 0], 270}|\n|||{7, [7, 0], 90}|\n|||{0, [0, 0], 0}|\n|Z-order|224|{32, [11, 14], 180}|\n|||{32, [16, 14], 0}|\n\nDue to space limitations, please refer to the first section in the response to reviewer z3TX for details. In terms of experiments on normal-resolution datasets and performance for downstream tasks, we indeed do not have sufficient results at the current stage and will address this limitation in our future work.\n\n> **(W2)** ...An experiment that would have been good is taking an existing CNN and replacing its layers with the 1D layers... Another thing ...see the effect of some of the other introduced blocks in baseline CNNs.\n\nThis is a very inspirational suggestion. This paper seeks to validate the feasibility of building a CNN using only 1D kernels using space-filling shifted paths. Hence, the current experiments, including ablation studies, focus on the PathConv model design and path selection. We admit that the current experimental design is a bit rough. To replace layers in 2D backbones with 1D layers is a feasible way to identify critical block design as suggested. We intend to draw on this approach in our next research. We express sincere appreciation for this suggestion.\n\n> **(W3)** ...The two more important elements are wall clock time and the memory size of the activations...it is hard to judge whether some other design choices help more than the main story and maybe such that could be combined to regular CNNs.\n\nWe totally agree that memory consumption represents a key consideration with respect to efficiency and edge computing. In fact, we plan to validate the efficiency by implementing and deploying the PathConv model on hardware platforms like FPGAs. As path sampling is read-only, substantial optimization opportunities exist at the hardware level. Subsequently, memory consumption comparisons can be conducted directly on the board, closely approximating real-world deployment conditions.\n\nRegarding the next comment on whether alternative design choices may help, further exploration is indeed necessary. As your previous comments (W2) suggest, a viable approach might be to substitute layers and find the decisive block(s). We also wonder whether some different design principles would be more appropriate than adapting principles from existing 2D backbones. We will investigate this in future work. We sincerely appreciate your insight in distinguishing between theoretical significance and implementation maturity.\n\n## Regarding the `other comments or suggestions` section\n\nWe are deeply honored by your recognition of this work's novelty and potential. Your characterization of the contribution as *out of the mainstream, daring, and mathematically creative* resonates with our original aspiration to challenge conventional assumptions. Meanwhile, we also concur that framing this paper as a *practically useful method* in the current state is sub-optimal. I will take this advice to heart and adjust the writing style.\n\n## Questions\n\n> I'm wondering about striding freedom- It seems that having this fractal structure comes with a ratio by which the fractal repeats. This suggests that one cannot just choose whatever stride to take in the 1d convolutions that would still locally make sense. So for example stride 3 seems like it couldn't work. Thanks!\n\nYour observation about stride constraints is insightful.\nWe tend to believe the stride size may not have a big influence. $\\mathbb{C}$ has guaranteed that regardless of which pixel to start, we will always obtain better locality preservation than raster scanning in at least one path. Meanwhile, there are multiple different paths. Even if a spatially discontinuous sequence of pixels is obtained on one path, potentially, it is possible to avoid this phenomenon on other paths. Still, we think it's worth exploring. We will investigate this issue in future research. Thank you very much!", "claims_and_evidence": "**(S1)** As far as I could see, the claims made are backed by existing theory, proofs or empirical evidence.", "methods_and_evaluation_criteria": "**(W1)** There is no evidence of availability to scale up. I am not convinced by the statement in lines 344-345: \"We use low-resolution datasets as they amplify locality preservation issues...\". In practice there are many great ideas that can be demonstrated on lighter data. It is ok if some idea is not ripe enough for bigger scales. But it should be laid out that currently we have not shown to be of practical use.\n\n**(S2)** The results of the experiments are well specified, fairly testing various setups. \n\n**(S3)** There is an ablation study that sheds light on the need of the positional encoding and the PACA.\n\n**(W2)** However, it feels like the ablation is not testing the core elements. An experiment that would have been good is taking an existing CNN and replacing its layers with the 1D layers. It could be that it doesn;t make sense without additional building blocks, in which case these ones need to be considered as part of the basic layer. Another thing that would have been enlightening is, if possible, see the effect of some of the other introduced blocks in baseline CNNs.", "theoretical_claims": "Theoretical claims are backed.", "experimental_designs_or_analyses": "See Methods And Evaluation Criteria.", "relation_to_broader_scientific_literature": "**(S4)** Details, examples and proofs are well provided in the appendix.", "essential_references_not_discussed": "Couldn't find any.", "other_strengths_and_weaknesses": "**(S5)** This is a refreshing original idea, bringing mathematical allegedly unrelated elements into improving deep learning. It is novel and original, creative and interesting. I find that it has significant scientific value.\n\n **(W3)** The gain of having less parameters in the final model is not the most important thing in efficiency. The two more important elements are wall clock time and the memory size of the activations. The former seems to be very similar to baselines and the latter is not mentioned. It is important to know that there is no significant activations memory cost because this is the true practical obstacle for using models in different environments. The parameter count is important usually because of its correlation with the other two elements and because of theoretical value about expressiveness. I think that reduction to 1/3 parameters is not a super-impressive result. It is less than an order of magnitude, and it is hard to judge whether some other design choices help more than the main story and maybe such that could be combined to regular CNNs.", "other_comments_or_suggestions": "This paper has a novel and significant contribution. This is a type of works we should see more- out of the main stream, daring and using math for creatively solving problems. I think this paper should be accepted and fits the venue. I believe it has potential of opening a door to a new line of research. I do have to say that I think that the authors made a mistake, branding their work as a working useful method. There is not enough evidence to back this up. This is more of first-step kind of work, a POC perhaps. A respectable, elegant POC but not yet something we can genuinely offer practitioners as a method to use.", "questions_for_authors": "I'm wondering about striding freedom- It seems that having this fractal structure comes with a ratio by which the fractal repeats. This suggests that one cannot just choose whatever stride to take in the 1d convolutions that would still locally make sense. So for example stride 3 seems like it couldn't work. Thanks!", "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: The paper received all accept recommendations. After considering the author rebuttal and reviewer discussion/comments, the area chair agrees with this recommendation. The authors are encouraged to incorporate the rebuttal and reviewer suggestions to further improve the final version of the paper.
Decision: Accept (poster)