DISK: Differentiable Sparse Kernel Complex for Efficient Spatially-Variant Convolution
Authors: Zhizhen Wu
Cao Zhe
Yuchi Huo
Venue: ICLR 2026 Poster (2026)
Keywords: Kernel Approximation
Differentiable Filtering
Spatially-Varying Convolution
Efficient Image Processing
Primary area: optimization
TL;DR: This work introduces a differentiable framework for decomposing complex kernels into optimized sparse layers, enabling high-performance, spatially-varying filtering via a filter-space interpolation scheme.

Abstract: Image convolution with complex kernels is common in photography, scientific imaging, and animation, but dense convolution is too expensive for resource-limited devices. Existing approximations, such as simulated annealing and low-rank decompositions, are either slow or struggle with non-convex kernels.
We present a differentiable kernel decomposition framework that represents a spatially variant dense kernel with a small set of sparse samples, assuming the target dense kernel is known for both optimization and filtering. Our method provides (i) end-to-end differentiable sparse-kernel optimization, (ii) shape-aware initialization for non-convex kernels, and (iii) kernel-space interpolation for efficient, multi-dimensional spatially varying filtering without retraining or added runtime cost.
Across Gaussian and non-convex kernels, our method achieves higher fidelity than simulated annealing and lower cost than low-rank decomposition. It is practical for mobile imaging and real-time rendering, and integrates cleanly into learning pipelines.

DISK: DIFFERENTIABLE SPARSE KERNEL COMPLEX FOR EFFICIENT SPATIALLY- VARIANT CONVOLUTION
Zhizhen Wu Zhe Cao Yuchi Huo
State Key Lab of CAD&CG, Zhejiang University, China zh www.weeeeweeeeee edu en, caozhe022@qq.co com, huo. yuchi sc&gma com
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b) Differentiable Optimization F(x,y)
c) Spatially Varying Filtering
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An overview of our method. We represent a dense filter as a Sparse Kernel Complex, a sequence of sparse layers whose parameters θ are learned via differentiable optimization. We apply our filter Fe to an impulse to yield a synthesized kernel Ksyn, and minimize a loss against the target Kegt to learn arbitrary shapes. These optimized kernels form basis for high-performance spatially varying filtering, achieving quality close to ground truth with up to 20x speedup.
Figure I:
ABSTRACT
Image convolution with complex kernels common in photography, scientific imaging, and animation, but dense convolution is too expensive for resource- limited devices. Existing approximations, such simulated annealing and low- rank decompositions, are either slow or struggle with non-convex kernels. We present differentiable kernel decomposition framework that represents spa- tially variant dense kernel with small set of sparse samples, assuming the target dense kernel known for both optimization and filtering. Our method provides (i) end-to-end differentiable sparse-kernel optimization, (ii) shape-aware initial- ization for non-convex kernels, and (iii) kernel-space interpolation for efficient, multi-dimensional spatially varying filtering without retraining or added runtime cost. Across Gaussian and non-convex kernels, our method achieves higher fi- delity than simulated annealing and lower cost than low-rank decomposition. It is practical for mobile imaging and real-time rendering, and integrates cleanly into learning pipelines.
INTRODUCTION
From rendering realistic depth-of-field effects (Sakurikar Wu et al. 2022) in computational photog- raphy to modeling the intricate point spread functions (Liu al. [002 Shajkofei Liebling 2020) of optical systems, the ability to apply large, complex convolution kernels is fundamental building block modern vision and graphics computing systems. This creates a fundamental tension: while larger, more intricate kernels enable higher-fidelity results, their quadratic computational cost ren- ders direct implementation impractical for interactive applications on devices ranging from mobile phones to high-end GPUs.
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To bridge this gap, many works have focused approximation strategies. For specific cases like Gaussian blur, elegant constant-tine solutions (Zing 2010 Kovesi 2010) exploit the filter's 's analytic structure. However, such specialized methods are not applicable to the arbitrary, often non-convex kernels needed for advanced effects. More general approaches, such as low-rank matrix decomposi- tion (McGraww) 2013), support arbitrary kernels but typically reduce the computation to sequence of smaller dense convolutions, which limits sparsity and caps efficiency gains.
A more direct and efficient approach (Schuster et al. 2020) is approximate dense kernel with a truly sparse one, drastically reducing the number of required computations. It relies on heuristic search via parallel simulated annealing to identify sparse sampling patterns for arbitrary kernels. Despite its generality, it requires many iterations and still misses high-fidelity solutions due to the non-convex optimization landscape. This motivates a more principled and efficient way to obtain high-quality sparse kernel representations.
In this work, we introduce differentiable kernel decomposition framework to address this chal- lenge. Like prior approximation methods, our formulation assumes access to the dense target kernel throughout optimization and filtering. Unlike traditional low-rank methods, we optimize a sequence of natively sparse kernels, yielding an efficient representation for runtime inference. We cast the decomposition as an end-to-end optimization problem and apply gradient- heuristic search such simulated annealing, enabling more reliable convergence to high- high-fidelity solutions in far fewer iterations. To improve stability for non-convex target kernels, we further pro- pose a two-part part initialization strategy: structure-aware sampling capture fine shape details and deterministic radial initialization to stabilize and accelerate convergence.
methods instead of
Beyond single-kernel approximation, our framework supports efficient, multidimensional spatially varying (SV) filtering. The main challenge is the cost of generating a unique kernel per pixel, which often creates a significant performance bottleneck. We address this with filter-space interpolation: we precompute an optimized basis of sparse filters spanning the target effect range, then synthesize per-pixel kernels at runtime by interpolating this compact basis. This strategy reduces per-pixel synthesis to a minimal set of multiply-ade operations, decoupling senee-generation cost from image resolution and enabling sophisticated SV effects with negligible overhead.
Our contributions are as follows:
A differentiable framework for decomposing dense, arbitrary kernel into sequence of optimized sparse layers, enabling efficient, high-fidelity approximation.
An initialization scheme, combining general radial strategy for stable convergence with sparse sampling method for capturing non-convex kernels.
A filter-space interpolation method for segg-performance, spatially-varying filtering that decouples kernel synthesis cost from image resolution.
RELATED WORK
2.1 PIGH-PPRFORMANCE KERNEL
Given that Gaussian blur is computationally expensive, numerous methods have been proposed to accelerate it. Common examples include O(1) approximations such as the Extended Binomial Fil- ter (Zing 2010) and Summed-Are Table-based methods (Kovesil 2010). However, their reliance on precomputation or inherently sequential steps makes them ill-suited the massively parallel design of modern GPUs.
better match for real-time rendering is Kawase blur (awasse 2003), a multi-pan malti-pass filter that uses only four texture samples per pass. As an extension, Dual Filtering (Martin et al.) 2015) intro- duces downsampling followed by upsampling, reducing bandwidth and computation by operating on lower-resolution textures. However, such pipelines trade regular downsampling for extra recon- struction work (often additional convolution) and may lose fine detail, whereas we stay at a single resolution and reduce cost via flexible sparse sampling while matching the target kernel response. A further practical limitation is the lack of a systematic mapping from a target Gaussian strength (c.g., given a) the corresponding Kawase or Dual Filtering parameters; our work addresses this gap.
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2.2 SPATIALLY- VARIANT FILTERING
A substantial body of work learns spatially varying, per-pixel convolution kernels, with applica- tions to video prediction, frame interpolation, denoising, and deblurring (Jia et al. 2016 Niklaus et al.] [017]] Mildenhall et al. 2018 Zhou et al., 2019 (021). Unlike approaches that directly
predict dense per-pixel kernel map, our method decouples filter generation from spatial resolu- tion. Specifically, we learn compact lookup table (LUT) that parameterizes a continuous family of filters, enabling flexible and efficient spatially-varians filtering. Spatiotemporal
Grided Filtering (Schied al. 2017) forms per-pixel filter mixtures guided by estimated spatial and tem- poral variance. Differently, we condition filter generation on an input per-pixel blur-intensity map. This allows to synthesize filters that match any target spatially varying blur, without intermediate variance estimation or other content statistics.
2.3 KERNEL APPROXIMATION AND DECOMPOSITION
Image Filters (Schuster et al. 2020) uses parallel tempering to optimize sparse sampling pattems, but its optimization stability can be limited by sensitivity to many hyperparameters compared with our gradient- brasee formulation. Ker- nel decomposition is also used to reduce cost in other settings: 2D kernels into pairs of 1D kernels for video interpolation (Niklaus et al., 2017); ; 3D spatiotemporal kernels into spatial and temporal atoms (STDCF) (Schied et al. 2017); ; standard convolution into depthwise and 1 pointwise op- erations (Howard et al. 2017 Chollet 2017; Ramadhani et al. 2024); dynamic weights into static bases plus residuals (Li al. 2021 2024); and related decompositions for traph--ransformen atten- tion (KDLGT) (Wu et al. 2023) and large depthwise kernels (LKD-Net) (Luo et al. 2023).
Inspired by Kawase blur Kawase 2003).
Despite their success, these methods are largely limited to structured factorizations or heuristic op- timization. Our framework addresses the missing capability: a differentiable, general solution for sparse approximation of arbitrary (including non-convex) kernels, together with an efficient mecha- nism for spatially varying kernel synthesis.
PRELIMINARY
3.1 KERNEL-BASED FILTERING
Kernel-based filtering is fundamental to many image processing tasks. This process takes an input image In and computes each pixel's value for the output image Lout as a weighted average of its local neighbors within Lnn- Formally, this operation is expressed a 2D convolution, defined as:
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Tout [x, vl = (Jan * K)E v)
Σ
Σ
In la + +1 + 1) K[.,),
-=
where the matrix K is the M M convolution with kernel size M E RR+,, whose elements K Kii, J] are weights that determine the contribution of each neighboring pixel to the final filtered value.
3.2 FILTER REPRESENTATION
The dense matrix representation for the kernel K in Eq. (1) is straightforward. However, its O(M²) computational cost presents significant bottleneck. This is especially true for filters with large spatial support, such as Gaussian blur with a large a, where the cost becomes prohibitively expen- sive for real-time applications that demand high frame rates.
Our key insight is to approximate this expensive operation by structuring the filter sequence of lightweight convolutional layers, where the output of one layer serves as the input for the subse- quent one. Each layer applies highly efficient sparse kernel, K Ksparse. which we define by small collection of N samples with fffset-weight pairs:
K Ksparse {(o,, w:)
(2)
where Oş E IR2 is the spatial offset and w; is its corresponding weight.
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The complete operation, consisting of L such layers with kernels (K1, K2 ... KL), can be expressed as a nested convolution:
Lout = (...((IIn K1) K2) * ... K1).
(3)
This multi-layer filter reduces the cost to O Ni) per pixel. Since this sum is far smaller than the number of weights the target dense kernel C Ni < M the approach offers a dramatic speedup.
METHODOLOGY
4.1 DIFFERENTIABLE MULTI-LAYER KERNEL COMPLEX
Overview Sparse filters offer a computationally efficient alternative dense kernels; however, they often fail to capture the intricate structure of large, complex filters. 'he core challenge lies determining the optimal parameters- the spatial offsets and weights-for a sequence of sparse kernels to accurately reconstruct target. Manually designing these parameters using traditional, non-differentiable methods is a formidable task.
To overcome this, our key contribution is to frame the decomposition as differentiable optimization problem. This enables the simultaneous end-to-end learning of all sparse kernel parameters across all layers. We define the complete set of these learnable parameters as θ = {(o1,, w1,)))=)), L.,,
which includes the offsets and weights for N2 samples in each the layers.
Our goal is find the optimal parameters @ by minimizing loss function that measures the discrepancy between our approximation and the target kernel:
θa arg min L(Ktarget F Fopprew (Q)),
(4)
Fapprea (0) Ks,1 Ks * ... * K
where Krarget is the desired dense filter and Papproz (0) is the composite kernel formed by the convolution of the learned sparse kernels.
Learnable Parameter Our optimization strategy treats the offsets and weights of each sample as independent, learnable parameters. Specifically, for each layer / and for each of the Ni sampling points within it, we simultaneously optimize both the 2D offset vector O,, and its corresponding scalar weight Wh,s-
The complete set learnable parameters for the entire model, denoted by θ, is therefore the collec- tion of all such weight pairs:
θ Uf wi )
(5)
Initialization A well-designed parameter initialization is crucial for stable optimization conver- gence. Heuristic methods, such as Kawase (Kawase 2003) and Dual Filtering (Martin et al. (015). have fixed schemes tailored to specific filter types; however, a general approach is required for arbi- trary target kernels of different sizes.
To address this, we propose radial initialization strategy. The core idea is to initialize the sampling points in each layer to be uniformly distributed on the circumference of this circle increasing linearly with the layer index. This progressive expansion ensures that the effective receptive field of the composite kernel grows with each subsequent layer, making the initial configuration capable of spanning a large-area target kernel from the outset. The radius for layer denoted rı, is governed by a step size A, derived from the target kernel's 's spatial extent and the total number of layers L (see Appendix for derivation). The corresponding weights in each layer initialized uniformly.
circle, with the radius
This initialization is formally defined as:
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4.2 SPARSE SAMPLING OF ARBITRARY KERNEL
A common way to initialize filter offsets is by sampling random positions within a local neighbor- hood. While this approach is general, it often traps the optimization in poor local optima, especially for kernels with complex or non-convex shapes.
Our method decomposes dense kernel into series of sparse ones. The first of these, K,,1 s (Eq. 4 has the greatest influence on the final filtered output, so its initialization is critical. A simple im- provement over purely random sampling is confine samples the minimal bounding box of the kernel's 's non-zero pixels. This ensures most samples fall near the target shape, but is still inefficient for non-convex kernels, whose bounding boxes can contain large empty regions.
To overcome this limitation, we propose a more sophisticated initialization strategy leveraging re- jection sampling. Instead of drawing samples from the kernel's bounding box, method samples directly from the support of the kemel, i.c., its non-zero locations. We first quantify the effective sampling area, denoted by S, as the count of these non-zero pixels. A sampling radius is subse- quently derived based on the desired number of samples, N.:
N The detailed procedure is provided in the appendix. This initialization sets the first sparse kernel's offsets to closely match the target shape. By restricting samples to relevant regions, it avoids van- ishing gradients and reduces the risk of converging to poor local optima.
4.3 SPATIALLY VARYING FILTERING
Next, we propose decomposition method for spatially varying filtering.
Spatially varying filtering generalizes convolution by applying a unique filter at each pixel (z, y). The filter's propertie--such
its blur radius, orientation, shape-are are determined by corre- sponding value P(x, y) from a parameter map. The core challenge lies in efficiently synthesizing and applying these unique per-pixel kernels.
Conventional spatially varying filtering is often impractical. Generating dense kernels on the
fly (Wang et al., 2023) too slow, while precomputing them (Kovesil 2010) incurs prohibitive memory costs; both are poorly suited to modern parallel hardware. Faster alternatives (Leimkühler al. 2018 restrict filters to simple analytic forms (e.g., Gaussians). but this sacrifices the expres- siveness needed for complex, non-convex point spread functions (PSFs).
We observe that, in prior work, the cost of generating or storing spatially varying kernels grows linearly with image resolution. To remove this bottleneck while retaining expressive, sparsely op- timized kernels, we introduce Filter-Space Interpolation, which decouples kernel-generation com- plexity from image size.
Our spatially varying filtering is built on an ordered set of M basis sparse filters, which discretely sample continuous, one-dimensional space F of filters. Each basis filter, St, corresponds to scalar parameter pk (with P1 < P2 < ... < PM) and consists and weights. This design allows basis to represent wide range of filter behaviors across the parameter space, from applying arbitrary linear transformations to a kernel to simply varying the standard deviation (a) of a Gaussian. We define the basis
unique set of N sampling offsets
(8) We divide the approach into offline pre rre-computation stage and runtime inference stage. In the offline stage, we optimize each basis filter fı individually to represent the ideal filter effect at its parameter value pk.
F = (fk(p)) k= 1, MY, where fa = ((Oki, was)
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At runtime, synthesize unique sparse filter for each pixel (z. y), which is guided by per- pixel parameter map. P. From the parameter value at each coordinate, P(z, y), we determine a corresponding vector of M interpolation weights, axz y) (a1,..... ..y). ). These weights specify how to blend a compact set of basis filters, [ to reconstruct the final filter instance.
The final sparse filter for given pixel, (x; y), is synthesized as direct convex combination of the basis filters:
f(x Σ ag y) Jk:
subject to the constraint that Σ (x( v) and (1; v) 2 0.
By directly interpolating basis-filter offsets and weights, we sidestep the costly on-the-fly gener- ation of kernels from analytical functions. This reduces the computational overhead of spatially varying kernel synthesis to minimal set of parallelizable multiply-add operations. Furthermore, the interpolatable nature our basis filters makes the entire set highly compressible, allowing us to significantly reduce the memory footprint required to achieve wide range of expressive effects while offering flexible control over the quality--erformen trade-off.
4.4 IMPLEMENTATION DETAILS
To ensure our learned filter parameters are generalized and not overfit to spe- eific dataset, we adopt image-agnostic optimization strategy. We leverage core principle of Linear Shift-Invariant (LSI) systems (Goodman 2005): a filter is fully characterized by its impulse response.
Training Process
First, we synthesize the effective kernel of our multi-pass filter, Fs. by applying it to a discrete Dirac delta function, 8. The resulting output is the synthesized impulse response, K syn- The impulse ō is an image with a single non-zero pixel at its center coordinate c:
(1 if
Ksyn (8). where
8[n]
(10)
otherwise.
Here, 0 represents the learnable parameters of our filter and n denotes the discrete pixel coordinates.
Loss Design Second, we define our loss function, L, as the Charbonnier LI loss (Charbonnier al. 1994 between the synthesized kernel Ksyn and target kernel Kygi
L = C( Kynn Keg).
(11)
This reppuss-response-eee based supervision allows us to "collapse" the entire multi-layer filtering se- quence into a single, equivalent kernel for direct and precise approximation of the target.
5 EXPERIMENTS
In this section, we conduct series experiments to evaluate our differentiable kernel decom- position framework thoroughly. We first describe the experiment details and evaluation protocol Section 5.1 Next, Section 5.2] we assess our method's ability to approximate single, complex kernels, comparing it against state-of-the-art techniques. We extend this analysis to the more chal- lenging task of spatially varying filtering in Section 5.3 To validate our specific design choices, we present series of ablation studies in Section 5.4
5.1 SETUP
Baselines. We compare method against several baselines. For both single kernel and spa- tially varying filtering, we include low-rank decomposition (LowRank) (MeGraw) 2013) and the optimization-base based method of Parallel Tempering (PST) (Schuster et al. 2020).

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Figure 2: Comparison of Gaussian kernel approximation with varying We compare method against PST using two sparse configurations (8 layers x samples and 12 layers x sam- ples). LPIPS scores appear in the top-right corner (lower is better). The top-left inset visualizes the emror map, with positive errors shown in red and negative errors in green.
Datasets and Kernels. To evaluate the versatility of our method, we use a diverse set of target kernels and images. This set includes standard analytical shapes, such as Gaussian kernels (with values from 5 to 11). To assess performance on more complex targets, we additionally use a suite of arbitrary kernels comprising simple geometric primitives (disks, rings). regular polygons (4-sided and 6-sided), non-convex shapes (a heart, four-pointed star, and an ampersand symbol), more complex shapes (animal silhouettes), and optical PSPs (coma and spherical aberration). For the spatially varying filtering experiments, we use five high-resolution photographs selected to represent realistic scenarios with complex textures and both 1D and 2D spatial variations.
Implementation and Evaluation Metric. We implement our methods in PyTorch and perform all optimization on a single GPU with 24 GB of memory, offering computational power comparable to NVIDIA RTX 4090. For all configurations of kernels and layers, we use the same Adam optimizer with learning rate linearly decayed from 1 x 10 3 to 1 10-4, We use 1000 optimization steps per kernel for method. For comparison, with 10 parallel candidates, for a total of 100,000 optimization steps. For the LowRank method, we utilize decompositions with ranks 1,2 and 3, chosen maintain comparable number of samplings.
the PST algorithm for 10,000 iterations
For runtime analysis, we benchmark our approach representative mobile device equipped with Qualcomm Snapdragon Gen 3 SoC, and report la- tency in milliseconds (ms). We evaluate both merical fidelity and perceptual similarity using Peak Signal-to-No -o-iiee Ratio (PSNR), Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al. (018)), and FLIP-LDR (Andersson et al. al.]| 2020). Higher val- ues indicate better performance for PSNR, and lower values are better for LPIPS and FLIP-LDR.
Rank (M)
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5.2 SINGLE KERNEL.
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Fig. 3 shows that our method consistently achieves a superior balance between reconstruction quality and inference speed compared to all other approaches. For our method and PST. the 'S', 'M'. and 'L' correspond to total sample counts of 48 (12x4), 96 (24x4), and 128 (32x4), respectively. The LowRank's 'M' and 'L' use 98 (49x2) and 196 44 parameters.
Figure 3: Speed, accuracy, and samples com- parison. The figure plots quality against latency (lower is better for both). The size each bubble represents the total sample count.
Next, present comparison of Gaussian kernel approximation with varying standard deviations a in Fig. 2 6-layer, 8-sample (8x6) configuration, our method achieves high-fidelity results with low perceptual error, whereas PST exhibits visible noise and artifacts. This performance gap widens sparser 12x4 setup. As o increases, PST's approximation degrades severely, while our result remains visually coherent and maintains substantially lower I.PIS error. These results show

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Figure 4: Comparison of Single kernel approximation. Compared to baselines, SVD-based de- composition (LowR.) (MeGraw) 2015) and Parallel Simulated Tempering (PST) (Schuster et al.] 2020), our approach (blue) better preserves sharp features on non-convex targets, resulting in lower LPIPS scores (lower is better). The top-left inset visualizes the error map, with positive errors shown in red and negative errors in green.
that our gradient-based optimization yields more accurate approximations than PST, consistently producing stable solutions even in challenging sparse configurations.
Our method's accuracy extends beyond Gaussian kernels to the more general case of arbitrary single- kernel filters, as shown Fig. 4 Our method preserves structure across both simple and complex shapes, while LowRank introduces blocky artifacts and PST yields noisy results at low sample counts. Quantitatively, our method achieves the lowest LPIPS across all tests, often by a large margin. It is also much more efficient, requiring only 1/100 the iterations of PST.
5.3 SPATIALLY VARYING KERNEL
We present three spatially varying filtering examples in Fig- 5 The first is ID spatially varying blur that uses a pseudo-depth map to simulate tilt-shift camera effect. The other two are 2D anisotropic effects: a rotational bokeh blur and radial motion blur, both controlled by two parameters blur intensity and local blur angle.
Our results visually indistinguishable from the ground truth. As highlighted in the red and green insets, our method reproduces the complex structure of the GT kerelss. In contrast, PST introduces noise and LowRank oversmooths, and neither recovers the correct kernel shape, while directly applying GT kernels is prohibitively slow. Quantitatively, our method achieves the highest PSNR among all methods while maintaining real-time performance.
This performance difference stems from how well each method's base kernels handle filter-space interpolation. While approaches use interpolation to generate the varying filter parameters, our optimization-based kernels are better conditioned for this process and appear to vary more linearly. Consequently, they interpolate smoothly to form sharp, complex patterns. PST's kernels, however, suffer from poor optimization quality, and interpolating between them simply produces more noise. Similarly, interpolating the basis kernels from Low Rank's 's decomposition causes them to average into indistinct blurs rather than preserving the target structure.
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Figure 5: Visual comparison of diverse spatially varying (SV) effects. We evaluate three SV con- figurations: ID tilt-shift blur (top), 2D rotational blur (middle), and 2D radial motion blur (bottom). We compare our method against Parallel Simulated Tempering (PST) and Low-Rank Decomposi- tion (LowRank).
5.4 ABLATIONS
We conduct ablation studies to validate our main design choices, focusing on both initialization strategies and different layer configurations.
We first evaluate different initialization schemes across multiple kernels, as shown Fig. 6 Both our method and Parallel Simulated Tempering (PST) benefit from the proposed Sparse Sampling (SS) initialization, which consistently outperforms the Increasing Radial (IR) initialization, while the Random (Rand) initialization performs worst. Although SS accelerates convergence for both our method and PST. PST still requires more than 30x the number iterations to converge compared with ours, and our final reconstruction quality is significantly higher.
We further study the influence of different configurations, varying the number of layers and the num- ber of samples, shown in Fig. 7 The convergence curves show that all configurations converge stably, and configurations with more samples and layers tend to achieve higher quality. Compared with PST, our method delivers consistent behavior and better quality across all tested configu- rations.
For additional results, please refer the Appendix, which includes ablations Gaussian kernels with fewer samples and quantitative evaluations of initialization and regularization strategies on arbitrary kernels.
DISCUSSION AND CONCLUSION
We introduced differentiable framework that recasts the challenging problem of approximating large, complex convolution kernels as end-to-end optimization task. Our approach supports wide range of kernels from simple Gaussians to complex, non-convex forms and converges to high-fidelity solutions far more efficiently than prior methods. We extend this with filter-space in-

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Figure 6: Ablation of initialization strategies the Flower and Dove kernel. We evaluate both our method and Parallel Simulated Annealing (PST) combined with three initialization schemes: Random (Rand), Increasing Radial (IR), and Sparse Sampling (SS).
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terpolation, enabling multi-dimensional spatially varying effects with minimal per-pixel overhead. A current constraint of formulation is that it requires access to the target dense kernel during optimization and filtering. This work opens several promising avenues for future research, includ- ing multi-dimensional parameter maps for simultaneous control over kernel attributes and the use filter decompositions. Overall, our
of neural architecture search to discover hardware-optings method provides a practical, high performaef solution for advanced image filtering real-tima applications such as computational photography, while remaining fully differentiable and thus us- able as a trainable layer within modern deep learning pipelines.
ACKNOWLEDGMENTS
This work was partially supported by National Key R&D Program of China (No. 2024YFB2809104), NSFC (No. 52532013). and Key R&D Program Zhejiang (No: 2025C01064). We also thank the anonymous reviewers for their constructive comments.
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Reviews:
[{"review_id": "WKvEWTHpWj", "reviewer": "Reviewer_cyNF", "summary": "This work addresses a significant limitation in practical applications for spatially varying blur effects in computation photography using  image convolution with kernels. Usually Gaussian kernel is considered a standard kernel. However, Gaussian kernels are slow on computationally low-end devices (due to their dense M^2 complexity). To encounter, this paper proposes layer-wise filtering where a kernel is represented by L layers, where each layer has a kernel K with only N samples. Each kernel K is applied to an image resulting in much smllaer number of weights compared to M^2 weights in an ordinary dense kernel of pixel footprint M^2.", "questions": "- How do you obtain a blur intensity map that is conditioned for the filter generation process? \n- How the whole process of structuring a filter as layers different from a convolution followed by e.g, apply max pooling to an image and then applying the same filter to the image? It will automatically increases the receptive field of the kernel without any additional cost to computing kernel weights\n- What is the similarity of the approach to the A-trous filters where the number of learnable parameters remain the same despite the receptive field?\n- How the kernels adapt the anisotropy of the signal? What if you want nice anisotropic bokeh effect for artistic purposes?\n- What are “filter objects” in L291? \n- It is not clear what objects are interpolated? Is it the interpolation between the trained parameters \\alpha_k?\n- I would like more explanation on the implementation details. How the optimization works? You convolve a kernel with a Dirac function, this gives you an image with a single non-zero pixel. This image is then matched to a target image with a target kernel applied to a Dirac function. Is my understanding correct?\n- Also, how this idea of layered kernels different from what is used in CNNs or U-Nets?\n\nFigure 1 should appear early in the paper (on the second page), or better as a teaser on the first page.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Propose a robust and fast approach to perform image convolutions\n- Decouples filter generation from the  spatial resolution by introducing filter space interpolation\n- The idea is simple and effective\n- Results are very convincing.", "weaknesses": "- The paper needs to do a bit more job in explaining the relation wrt A-trous filters or max-pooling like approaches used in U-Nets for increasing the receptive field of kernels.\n\n- Please always mention how to read different metric values (higher is better or lower is better). \n\n- There are no theoretical guarantees provided that explains why layered kernels would be robust in general. \n\nPlease see my questions below that also directly highlight my concerns.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7CMV9Pshcs", "reviewer": "Reviewer_jzA7", "summary": "This paper formulates the efficient convolution problem as an optimization problem involving a few \"layers\" of sparse discrete convolution operators that are applied in succession. This differentiable formulation enables gradient-based optimizers to find sparse filters that approximate target convolutions by matching their impulse response to the target. The resulting method is quick to optimize, quick to run, and uses little runtime memory. The authors further propose a robust initialization scheme which they verify improves the optimization quality via ablations. Additionally, they propose an time-and memory-efficient way to allow for spatially varying convolutions -- albeit only allowing for a 1D parameter value -- by linearly interpolating between pre-computed \"basis\" sparse filters where the interpolation weights are determined by the per-pixel parameter value. Overall, the paper presents an effective, simple-to-implement way of performing efficient discrete image convolutions.", "questions": "In addition to latency/FPS numbers, can you also provide peak VRAM or CPU RAM usage numbers? This seems important to me because being more memory efficient is one of the method's advantages. \n\nHave you considered second-order-ish optimizers? In many cases the number of parameters sounds small enough that it's probably tractable. But then the optimization is also probably not-so-convex.\n\nMinor typos/suggestions:\nI believe most of the citations in the paper should be of the form \\citep instead of \\citet. \nAt L283, it's not clear how the actual interpolation weights are decided from P(x, y)\nL341 says \"single GPU with computational power equivalent to an NVIDIA RTX 4090.\" I'm guessing that it's just an L40S, RTX 6000 Ada, or a RTX 4090D? I think it's okay to omit the details, but it'd be helpful to include the VRAM capacity.\nL347 is probably talking about a \"Snapdragon\" 8 Gen 3? The \"Snapdragon\" is missing from the text\nIn Fig. 4 \"ring\", any idea how \"Ours 24×4\" looks better than \"Ours 36×4?\" Is it just due to random initialization?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- Good-looking results! The supplementary video shows the proposed sparse convolutions producing results visually indistinguishable from the ground-truth.\n- Speed and memory requirements are both greatly improved from competing efficient convolution approaches.\n- Simple to implement on desktop GPUs or mobile platforms (though pseudocode or code would be appreciated)", "weaknesses": "Since (I believe) the optimization problem in eq. (11) is non-convex in the (offset, weight) parameters, partly due to the dependence of one layer's input on the previous layer's outputs, it's unclear to me how robust the optimization is with respect to vastly different number of samples, and number of layers, or how well it works with even more intricate point-spread functions. If the authors included more robustness tests and described how they tuned (if at all) the optimization hyperparameters such as the optimizer and learning rates, I would find the results much more convincing. \n\n(Minor weakness) Unclear how well linear interpolation will work with multi-dimensional parameter values: naively, the number of basis filters will need to grow exponentially in D, the dimensionality. On the other hand, only having one spatially-varying parameter value feels limiting to me.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JtfDp09Njy", "reviewer": "Reviewer_BSUK", "summary": "The paper proposes a gradient-based optimization strategy to learn a sequence of sparse convolutional kernels whose sequential application approximates a large, dense convolutional kernel using far less computation than the standard dense convolution. The main innovation in implementing this idea is an initialization strategy for the positions of the sparse (nonzero) components in each of the sequential sparse filters, which is fairly heuristic (balancing fit with the target kernel and leveraging of kernel support expansion via repeated convolution) yet empirically effective. This idea is also extended to spatially varying convolutional kernels, e.g. to model a realistic point spread function as it varies across the field of view. This is done by pre-training a set of basis filters (each of which uses sequential sparse convolution) and then combining these basis filters with spatially varying weights.", "questions": "As the focus is mainly on improving the speed at which a convolutional kernel can be applied to an image, I wonder what is the computational overhead of pre-optimizing a sparse sequence of filters?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Figure 1a is a compelling illustration of the main idea of the method, approximating a dense (e.g. Gaussian) kernel with fairly large spatial extent as a sequence of sparse kernel convolutions whose net computational cost is far less than the cost of a standard convolution with the dense kernel.\n\nI appreciate that the experiments seem to be fair (or more than fair) towards the baselines, in the sense that equal or greater parameters and computation time were afforded to the baselines as to the proposed method. However, I am not an expert in the kernel approximation literature so I take the appropriateness of the chosen baselines at face value.\n\nFigure 5’s illustration of efficient spatially varying filtering is particularly compelling.", "weaknesses": "There are aspects of the presentation that could be improved:\n- The abstract describes “convolution with complex kernels”--at this point in the paper, it is unclear whether “complex” means “real plus imaginary” or “complicated”. More generally, the abstract does not clearly explain the task being solved, specifically whether the dense, spatially-varying kernel to be approximated is known in advance or unknown and to be inferred from measurements (after reading the rest of the paper, it seems the former is the goal). This should be made clear early on.\n- Section 2.3 summarizes a bunch of related papers, but does not explain their limitations or their relationship with the proposed method. \n- The Charbonnier L1 loss in equation 11 is not defined.\n- The font sizes in figures 3 and 4 are too small. It might be preferable to choose one Gaussian scale for the main paper so that the figures could be larger and arranged with PST in one row and “ours” in the other, making the figure easier to interpret (the other Gaussian scale could be shown in the supplement if there is not enough space in the main text). Both sets of figures also include a green and red visualization in the top left of each subfigure (perhaps an error map?) that is not described. Figure 4 also has a number (perhaps a measure of computation time?) in the bottom right of each subfigure, that is not described. Figure 4 caption says “our approach (blue)” but I do not see any blue in the figure.\n\nIn terms of the method itself, the main limitation is that it requires knowledge of the target dense filter, for example by calibration. For settings where the kernel is not known or is difficult to calibrate, it would be preferable to be able to jointly solve for the (spatially varying) PSF and apply it efficiently. The primary obstacle to using the proposed method for that goal is that the initialization strategy for the sparse kernels requires knowledge of the dense kernel. I’d at least like to see this limitation mentioned/discussed, even if addressing it is beyond the scope of this paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rqK7ybST1B", "reviewer": "Reviewer_1x3B", "summary": "Image convolution with large and complex kernels is computationally heavy. This paper follows a line of research seeking to reduce computational costs through kernel decomposition, using an approximation that achieves both expressibility and efficiency. The paper proposes a method for initializing a differentiable set of sparse kernels and a kernel-space interpolation method for spatially varying kernels. In both single-kernel and spatially-varying-kernel applications, the proposed method shows improvements in speed, quality, and sample efficiency.", "questions": "- Could the authors provide more insight from an optimization perspective on why the proposed sparse kernel set initialization outperforms existing methods? For example, how it alleviates issues in previous approaches and leads to better convergence or performance, potentially supported by optimization curves?\n- For Fig. 7 in the appendix, why do the curves for 'Ours 32 x 4' and 'PST 32 x 4' fluctuate instead of showing a monotonic downward trend?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Consistent and strong performance improvement over baselines across two different sets of experiments (single kernel and spatially varying kernel).\n- Well-written paper, easy to follow and understandable with proper figures.", "weaknesses": "- **Role of initialization in optimization**: The relationship between initialization and optimization, which appears central to the paper, could be emphasized more clearly. Considering the components affecting both single- and spatially-varying kernel cases, the initialization of the sparse kernel set seems to be a core contribution. Other concepts, such as sparse kernel sets and Dirac delta function-based optimization, have been explored in prior work. Therefore, it is plausible that the improved results primarily stem from the proposed initialization. The paper would be strengthened by an analysis from an optimization perspective, explaining why existing methods struggle, how the proposed initialization addresses these issues, and how it leads to better performance—potentially supported by optimization curves.\n- **Limited experiments**: Adding a comparison for single-kernel filtering (i.e., applying the kernels directly to images, not just approximating them) would offer a more complete evaluation of the method’s effectiveness.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)