Is Haar Enough? Exploring Symlets and Coiflets for Wavelet Convolution Layers

Md Rifat Ur Rahman

cs.CVAug 18, 2026

Is Haar Enough? Exploring Symlets and Coiflets for Wavelet Convolution Layers

Authors: Md Rifat Ur Rahman

Organizations: Bangladesh Univeristy of Engineering and Technology

Abstract

Wavelet convolution layers have recently emerged as an efficient mechanism for enlarging receptive fields through multiresolution analysis, but prior work has fixed the wavelet basis to Haar or Daubechies at a chosen decomposition depth, leaving open whether a different basis can shift the underlying efficiency frontier. We identify and characterize a previously unexplored trade-off in this setting: bases with stronger approximation properties (longer filters) can reduce the decomposition depth required for competitive accuracy, yielding a net reduction in parameters and FLOPs despite higher perlevel transform cost. We formalize this as an F-vs.-L tradeoff (filter length vs. decomposition levels) and study it systematically across Haar, Daubechies, Symlets, and Coiflets under controlled architectures and budgets. On image classification (CIFAR-10, ImageNet-1K) and semantic segmentation (Cityscapes), Coiflet-based wavelet convolutions match Haar at deeper levels with approximately 32% fewer additional parameters and 33% fewer additional FLOPs, providing a concrete and actionable design choice for practitioners building wavelet-based architectures.

Figures & tables

Figure 1: Two-level 2D-DWT decomposition used in our wavelet convolution layer. The first level decomposes the feature map into $(LL_{1},LH_{1},HL_{1},HH_{1})$ , and the second level further decomposes only $LL_{1}$ into $(LL_{2},LH_{2},HL_{2},HH_{2})$ . We evaluate alternative wavelet bases by replacing the fixed analysis/synthesis filter banks.

Family	Filter	Coefficients
Haar (db1)	$h_{0}$	$[\tfrac{1}{\sqrt{2}},\;\tfrac{1}{\sqrt{2}}]$
	$h_{1}$	$[-\tfrac{1}{\sqrt{2}},\;\tfrac{1}{\sqrt{2}}]$
Daubechies (db2)	$h_{0}$	$[-0.12940952,\;0.22414387,\;0.83651630,\;0.48296291]$
	$h_{1}$	$[-0.48296291,\;0.83651630,\;-0.22414387,\;-0.12940952]$
Daubechies (db3)	$h_{0}$	$[0.03522629,\;-0.08544127,\;-0.13501102,\;0.45987750,\;0.80689151,\;0.33267055]$
	$h_{1}$	$[-0.33267055,\;0.80689151,\;-0.45987750,\;-0.13501102,\;0.08544127,\;0.03522629]$

Table 1: 1D analysis filter taps used in our 2D-DWT. The 2D sub-band filters are formed via outer products in Eq. ( 6 ). Coefficients are taken from a standard wavelet library for reproducibility [ 18 ] .

Panel of Figure 2 — Figure 2: Two-level WTConv layer ( $L=2$ ). Top row (analysis): the input $X$ is decomposed by DWT into four sub-bands; only the approximation band $LL_{1}$ is recursively decomposed at level 2. The deepest level’s packed sub-bands are processed by a learnable depthwise convolution (groups $=4C$ ). Bottom row (synthesis): reconstruction proceeds level-by-level via IDWT from $i=L$ down to $i=1$ , with detail bands $\{LH_{1},HL_{1},HH_{1}\}$ bypassing the deeper level via a skip path. The choice of wavelet basis affects only the fixed DWT/IDWT filters: longer-support bases (db $N$ , sym $N$ , coif $N$ ) raise per-level transform cost but, with stronger approximation, can reduce the required depth $L$ — the $F$ -vs.- $L$ trade-off central to this work.

Model	Top-1 (%) $\uparrow$	Params (M) $\downarrow$	FLOPs (G) $\downarrow$
ConvNeXt-T [ 15 ]	81.0	28.6	4.5
ConvNeXt-T + GFNet [ 19 ]	81.2	32.9	4.7
ConvNeXt-T + FNO [ 13 ]	81.3	31.8	4.9
ConvNeXt-T + AFF [ 9 ]	81.4	31.2	4.8
ConvNeXt-T + WTConv (coif1)	81.7	30.6	4.7

Table 2: CIFAR-10 classification comparison (120 epochs).

Model	Wavelet / Levels	Top-1 (%) $\uparrow$	Params (M) $\downarrow$	FLOPs (G) $\downarrow$
ConvNeXt-T [ 15 ]	—	82.1	28.6	4.5
ConvNeXt-T + WTConv	Haar, $[5,4,3,2]$	82.8	30.6	5.0
ConvNeXt-T + WTConv	db2, $[5,4,3,2]$	82.7	30.6	5.0
ConvNeXt-T + WTConv (ours)	coif1, $[4,3,2,1]$	82.8	30.0	4.8

Table 3: ImageNet-1K classification (ConvNeXt-T backbone, 300 epochs,

224{\times}224

Levels	Kernel	Wavelet	D-W Param. (M) $\downarrow$	D-W FLOPs (G) $\downarrow$	Top-1 (%) $\uparrow$
$[4,3,2,1]$	$3\!\times\!3$	Haar (db1)	0.50	0.12	81.24
$[4,3,2,1]$	$5\!\times\!5$	Haar (db1)	1.38	0.33	81.32
$[4,3,2,1]$	$7\!\times\!7$	Haar (db1)	2.70	0.65	81.56
$[5,4,3,2]$	$3\!\times\!3$	Haar (db1)	0.73	0.17	81.49
$[5,4,3,2]$	$5\!\times\!5$	Haar (db1)	2.04	0.49	81.75
$[5,4,3,2]$	$7\!\times\!7$	Haar (db1)	3.99	0.95	81.69

Table 4: CIFAR-10 ablation study on ConvNeXt-T + WTConv (120 epochs). D-W Param./FLOPs denote WTConv additional cost.

Model	mIoU (%) $\uparrow$	Params (M) $\downarrow$	FLOPs (G) $\downarrow$
ConvNeXt-T [ 15 ]	80.0	28.6	4.5
ConvNeXt-T + GFNet [ 19 ]	80.3	32.9	4.7
ConvNeXt-T + FNO [ 13 ]	80.4	31.8	4.9
ConvNeXt-T + AFF [ 9 ]	80.5	31.2	4.8
ConvNeXt-T + WTConv (coif1)	80.8	30.6	4.7

Table 5: Cityscapes semantic segmentation comparison.

Levels	Kernel	Wavelet	D-W Param. (M) $\downarrow$	D-W FLOPs (G) $\downarrow$	mIoU (%) $\uparrow$
$[4,3,2,1]$	$3\!\times\!3$	Haar (db1)	0.50	0.12	80.29
$[4,3,2,1]$	$5\!\times\!5$	Haar (db1)	1.38	0.33	80.37
$[4,3,2,1]$	$7\!\times\!7$	Haar (db1)	2.70	0.65	80.61
$[5,4,3,2]$	$3\!\times\!3$	Haar (db1)	0.73	0.17	80.54
$[5,4,3,2]$	$5\!\times\!5$	Haar (db1)	2.04	0.49	80.80
$[5,4,3,2]$	$7\!\times\!7$	Haar (db1)	3.99	0.95	80.74

Table 6: Cityscapes ablation study on ConvNeXt-T + WTConv. D-W Param./FLOPs denote WTConv additional cost.

Explore similar work

Fast and Memory-Efficient Wavelet Convolutions via I/O-Aware Reformulation
Aug 11, 2026Amit Aflalo, Shahaf E. Finder, Roy Amoyal +2GPU Kernel Optimization Wavelet Methods
The Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Israel
COLORA: Efficient Fine-Tuning for Convolutional Models with a Study Case on Optical Coherence Tomography Image Classification
May 23, 2025Mariano Rivera, Angello HoyosLow-Rank Adaptation Convolutional Neural Networks
Centro de Investigación en Matemáticas, A.C. (CIMAT) Guanajuato, Guanajuato 36023, México
WaRA: Wavelet Low-Rank Adaptation for Medical Image Classification
Jun 25, 2025Moein Heidari, Yijin Huang, Yasamin Medghalchi +3Vision Foundation Model Adaptation Low-Rank Adaptation
School of Biomedical Engineering, The University of British Columbia, Vancouver, BC, Canada · Department of Electronic and Electrical Engineering, Southern University of Science and Technology, Shenzhen, China · Department of Electrical and Computer Engineering, The University of British Columbia, Vancouver, BC, Canada +2

Aug 11, 2026J/K move · Enter open · S save

Fast and Memory-Efficient Wavelet Convolutions via I/O-Aware Reformulation

Amit Aflalo, Shahaf E. Finder, Roy Amoyal, Eran Treister+1

The Stein Faculty of Computer and Information Science, Ben-Gurion University of the Negev, Israel

GPU Kernel Optimization Wavelet Methods Convolutional Neural Networks

Open paper AskPDF arXiv

Wavelet convolution (WTConv) has emerged as an increasingly popular drop-in replacement for standard convolutions, expanding a network's receptive field exponentially with the number of decomposition levels while keeping the parameter count linear. However, its reference implementation is severely memory-bound due to excessive data movement through high-bandwidth memory (HBM). We develop an I/O model of WTConv to characterize this bottleneck and use it to guide three algebraic reformulations: (1) recomputing the inexpensive Haar analysis butterfly on chip, (2) collapsing the multi-level synthesis cascade into a single closed-form pass indexed by output-coordinate bits, and (3) folding learned per-channel scales into the convolution weights. Together, these reformulations enable an I/O-aware fused implementation that substantially reduces HBM traffic. We evaluate the WTConvNeXt configuration across decomposition levels and a broad range of tensor shapes. Despite performing comparable arithmetic, the reference WTConv is substantially slower than the depthwise convolution it replaces. Our reformulation reduces modeled HBM traffic by approximately

2.55\times

, yielding up to a

4.35\times

training speedup over the reference while roughly halving peak memory usage. Thus, our reformulation preserves the benefits of WTConv while substantially reducing its execution time and memory footprint, removing the systems overhead that previously limited its practical efficiency.

Abstract

Explore similar work

Fast and Memory-Efficient Wavelet Convolutions via I/O-Aware Reformulation

COLORA: Efficient Fine-Tuning for Convolutional Models with a Study Case on Optical Coherence Tomography Image Classification

WaRA: Wavelet Low-Rank Adaptation for Medical Image Classification