Adaptive Resolution Residual Networks — Generalizing Across Resolutions Easily and Efficiently
Authors: Léa Demeule
Mahtab Sandhu
Glen Berseth
Venue: Accepted by TMLR ()

Abstract: The majority of signal data captured in the real world uses numerous sensors with different resolutions. In practice, most deep learning architectures are fixed-resolution; they consider a single resolution at training and inference time. This is convenient to implement but fails to fully take advantage of the diverse signal data that exists. In contrast, other deep learning architectures are adaptive-resolution; they directly allow various resolutions to be processed at training and inference time. This provides computational adaptivity but either sacrifices robustness or compatibility with mainstream layers, which hinders their use. In this work, we introduce Adaptive Resolution Residual Networks (ARRNs) to surpass this tradeoff. We construct ARRNs from Laplacian residuals, which serve as generic adaptive-resolution adapters for fixed-resolution layers. We use smoothing filters within Laplacian residuals to linearly separate input signals over a series of resolution steps. We can thereby skip Laplacian residuals to cast high-resolution ARRNs into low-resolution ARRNs that are computationally cheaper yet numerically identical over low-resolution signals. We guarantee this result when Laplacian residuals are implemented with perfect smoothing kernels. We complement this novel component with Laplacian dropout, which randomly omits Laplacian residuals during training. This regularizes for robustness to a distribution of lower resolutions. This also regularizes for numerical errors that may occur when Laplacian residuals are implemented with approximate smoothing kernels. We provide a solid grounding for the advantageous properties of ARRNs through a theoretical analysis based on neural operators, and empirically show that ARRNs embrace the challenge posed by diverse resolutions with computational adaptivity, robustness, and compatibility with mainstream layers.

Published in Transactions on Machine Learning Research (06/2025)
Adaptive Resolution Residual Networks Across Resolutions Easily and Efficiently
Generalizing
Léa Demeule
len. demente@mila, cemeete@mila..ucheee
Mila Quebec Al Institute, Université de Montréal
Mahtab Sandhu
mahtab. mahlab.sandduude candbu@mille quebee
Mila Quebec Al Institute, Université de Montréal
Glen Berseth
glen. bersethomiila quebec
Mila Quebec Al Institute, Université de Montréal, and CIFAR
Reviewed on OpenReview: https: // openreviee.. net/ forum? id- kd.5tFFFd
Abstract
The majority of signal data captured in the real world uses numerous sensors with different resolutions. In practice, most deep learning architect ures are fised-resobulion; they consider a single resolution at training and inference time. This is convenient to implement but fails to fully take advantage the diverse signal data that exists. In contrast, other deep learning architectures are adaptive- resolution; they directly allow various resolutions be processed at training and inference time. This provides computational adaptivity but either sacrifices robustness compatibility with mainstream layers, which hinders their use. In this work, we introduce Adoplive Resolulion Residual Nelworks (ARRNs) to surpass this tradeoff. We construct ARRNs from Laplacian residuals, which serve as generic adaptive- resolution adapters for fized-resolution layers. We use smoothing filters within Laplacian residuals to linearly separate input signals over a series of resolution steps. We can thereby skip Laplacian residuals to cast high-resolution ARRNs into low-resolution ARRNs that are computationally cheaper yet numerically identical over low-resolution signals. We guarantee this result when Laplacian residuals are implemented with perfect smoothing kernels. We complement this novel component with Laplacian dropout, which randomly omits Laplacian residuals during training. This regularizes for robustness to a distribution of lower resolu- tions. This also regularizes for numerical errors that may occur when Laplacian residuals are implemented with approzimate smoothing kernels. We provide a solid grounding for the advantagoous properties of ARRNs through a theoretical analysis based on neural operators, and empirically show that ARRNs embrace the challenge posed by diverse resolutions with computational adoptivity, robustness, and compatibility with mainstream layers.
Efficient problem-solving strategies typically allocate effort according to difficulty. Efficient deep learning architectures may therefore be created by incorporating form of computational adaptivity that is con- scientious of the difficulty of individual data points. The case of tasks involving image data, audio data, volumetric data, any other form of signal data is uniquely positioned to benefit from such an approach. Signals have no universal resolution; there is, instead, a diversity of resolutions that are contingent on the sensors used at the time of capture; there is, therefore, a concrete opportunity for impact given the varying difficulty of individual data points across common data modalitios. Signals also conform a mathematical structure that is well understood, which enables the implementation of inductive biases that can aid in finely quantifying and decomposing this notion of difficulty within architecture.

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To this end, we propose Adaptive Resolution Residual Networks (ARRNs), novel architecture for tasks involving signal data that addresses gap in the capabilities of prior methods, which either lack compulational adaptivity, lack robustness, or lack compatibility with mainstream layers. We begin by outlining a typology of existing methods. We identify three categories: fized-resolun resobutio;;; adaptive-resolution through variable sampling window; adaplive-resokulion through variable sampling densily.
rixeutoesolution. In the case of transformer architectures and certain other architectures, resolution a property that must remain fixed for evaluation to be possible, although signals of different resolutions can be interpolated to the resolution of the architecture to allow evaluation. This solution is sufficient for certain applications. This solution is however incapable of delivering computational adaptivily in the sense that it cannot reduce its computational cost at lower resolutions. We thus find this approach unsatisfactory for our purpose.
Adaptive-resolution through variable sampling window. In the case of fully convolutional architec- tures and a range of other architectures, resolution is a property that can be adapted at evaluation. This solution achieves computational adaptivity through translation equivariant layers or permutation equivariant layers that can conform to arbitrary resolutions. This solution more formally adapts its resolution by varying its sampling window rather than its sampling density, which is critically important consideration for robust- ness. These two means of varying resolution are not equivalent: varying the sampling window is analogous to resizing the frame delineating painting to offer larger or smaller field of view into the scene; varying the sampling density is analogous to changing the quantity and size of brush strokes that make a painting to convey the scene in a finer or coarser degree of detail. We illustrate this distinction in Figure 10 and Figure 11. In the context of natural images, the sampling window tends to be consistent, as camera systems have similar optics; the sampling density is however subject to vary significantly, as camera systems have sensors that range widely in resolution. This hints at a mismatch between the nature of resolution changes within mainstream architectures and within natural image datasets. In certain applications, it is possible for architectures with an adaptive sampling window to learn some degree of invariance to sampling densily, which can attemuate this mismatch. In Figure I however, we demonstrate the adaptive sampling window of mainstream architectures can be inappropriate for handling even mild distribution shifts between training resolution and inference resolution in the case of natural images. The training resolution of each architecture is set to be the maximal resolution of each dataset. The inference resolution of each architecture is then swept across a range of lower resolutions, evaluating directly at that resolution. Further experimental details are given in |section 4 We observe regression to near-random accuracy by the point inference resolution is halved relative to training resolution, highlighting the lack of robustness of this approach.
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Figure 1: Accuracy of mainstream architectures at various resolutions after training at the full dataset resolution. Evaluation is performed directly al the inference resolution.
In Figure 2, we show these architectures display much greater robustness across various resolutions when evaluating after an interpolation step that ensures the inference resolution matches the training resolution. This effectively holds constant the sampling window while accounting for the change in resolution by varying the sampling densily, which addresses exactly the discrepancy we have described in the case of natural images. This has the shortcoming of negating any computational benefit that would be gained by performing inference at lower resolution, as amounts to treating the architectures as fized-resolution. We cannot find a satisfying solution to our problem through this approach, as trades away computational adoptivity for robustness.
2

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CIARRO
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Accuracy of mainstream architectures at various resolutions after training at the full datasct resolution, Evaluation is performed after interpolation to the training resolution.
Figure 2:
Adaptive through variable sampling density. We wish to obtain both computational adaptivity and robustness, therefore we must find a way to vary the sampling densily of an architecture directly. We effectively dedicate our contribution to solving this problem elegantly. We first overview prior methods that achieve both computaationl adaptivity and robustness, but which all lack compatibility with mainstream layers Isection 1). We provide an overview of the fundamental notions of signal processing that allow our work to be formulated and thoroughly define our notation (subsection A. We introduce Lapla- cian pyramids as a stepping stone towards our contribution (section 2p. We introduce Laplacian residuals and show that high-resolution ARRNs can be turned into low-rcsolution ARRNs that are computationally cheaper by simply skipping Laplacian residuals (subsectton 3.1p. We prove that skipping Laplacian residuals yields numerically údentical results compared to using all Laplacian residuals when the input signal has a low-resolution and when Laplacian residuals are implemented using perfect smoothing kernels. We formulate Laplacian dropout as training augmentation that randomly omits Laplacian residuals (subsection 3.2p. We theoretically motivate its usefulness as regularizer for low-resolution robustness and as error correction mechanism for Laplacian residuals implemented using approzimate smoothing kernels. We perform set of experiments showing (subseetion 4.1] that our method yields stronger robust ness at lower resolutions com- pared to mainstream methods; (subsection 4.2) that our method enables significant computational savings through adaptation; (subscction 4.3) that our method is capable of generalizing across layer types a way that far surpasses prior adaptive-resoluliin architectures with variable sampling densily; (subsection 4.4) that our theoretical guarantee for adaptation using perfect smoothing kernels holds empirically; (subsec- tion 4.5b that our theoretical interpretation of the dual regularizing effect of Laplacian dropout also holds empirically.
1 Related Works
We review related works that allow the formulation of adaptive-resolution architectures with varying sampling densily. We also survey other related works that bear some similarity to Laplacian residuals in section A.2.
Adaptive-rasolution through variable sampling density with neural operators. We begin search for means of varying the sampling density of whole architectures by first. considering single con- volutional layer. We can see that varying the sampling density of the layer implies changing the sampling density of the kernel itself while preserving the role it plays as a feature extractor. A useful tool to this end is functional analysis; we can think of the kernel as a discrete function k| In Xn, + Rfuisfi defined over a discrete coordinate space X,, C Rd; we can imagine that that holds finite number of samples X of an underlying continuous function k(-): X - Rf-ixfe defined over a continuous coordinate space then make some assumptions about the space of functions are working with to allow interpolation into an equivalent discrete function k[ lu X, + Rf+ii that more finely (Xxll > x),)) or coarsely (Ix,l < (X,l) covers the same continuous coordinate space; we can thus cast the layer from its original sampling density |X,l/area(X) (X) to a new sampling density XX.l/area(X)) A next logical step is to conceptualize every layer that composes an architecture as an operator that expresses a map between inputs and outputs that are functions; we gain the ability to adapt the sampling density of the entire architecture if we ensure every layer can be converted between a discrete operator form and a continuous operator form (Bartolucci al.] 2023); we dedicate part of our appendix to more formal definition of
X C Rd where X, X;
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this constraint subsection A.1, Equation 36). This is the approach favored by neural operator methods Li et al. (2021): Kovachki et al. (2023); Fanaskov & Oseledets (2023). This approach provides the ability to vary sampling density and sampling window independently. This approach comes with an important drawback however: compatibility with moinstream layers is lost as the conversion between operator forms cannot be achieved without substantial alterations to typical layers (Bartolucci et al. 2023), which presents a significant barrier to more widespread adoption. In contrast, our approach follows the general paradigm of neural operators, but it entirely absorbs the conversion constraints within the fixed structure of Laplacian residuals, which provides compatibility with mainstreom layers; our approach also skips layers as it reduces its sampling density, which enhances compulational adaptivity-
Adaptive-resolution through variable sampling density with implicit neural representations. We can derive architectures with variable sampling density by leveraging an alternate representation of functions. While we can explicitly represent the input or output of layers as functions s(
through a set of samples {(x;, sxx;))[x; E X that is tied to a discrele coordinale space Xn. C X, we can instead implicitly represent functions s(: through parameter 0 that is attached to neural representation
n(.00) RJ s( which approxinates the function over the entire continuous coordinate space X. This later
approach is at the core of implicit neural representation methods Park et al.] [2019 Mescheder et al. (0019)
Sitzmann et al. (2020 Mildenhall et al.] (2021); Chen et al.] (2021) ; Yang et al. (2021) ; Lee & Jin (0022
Xu et al. 2022 Some of these methods directly leverage this scheme to reconstruct and render partially
obscrved volumetric data, image data and light field data with great success (Park et al., 2019 Meschoder
et al. 2019 Sitzmann et al. 2020 Mildenhall et al., 2021). Some of these methods expand on this scheme by splitting the parameter 9 into fixed part 1 that is shared across all data, points, and latent part tthtt is specific to individual data points, which enables forming more complex and reusable representations (Chen Yang et al.] 2021). This approach faces limited usefulness in the context of classification tasks, segmentation tasks, and diffusion tasks. This shortfall comes from a lack of compatibillty with mainstream layers, and more broadly from a set of challenges that arisc when mapping between implicit representations n(: ,,,,)) =1) + n0, 2411 directly through the latent space 2 + 1,+1, as such maps cannot easily preserve the symmetries of signals. This even renders difficult the implementation of convolutional layers (Xu et al. 2022 since there is priori simple relationship between the latent embedding 2 of a signal no( ,0, =)) and the latent embedding R1+1 of the same signal convolved against a kernel Ay there is within + #1+1 a nonlinear constraint n( .,0, =111 nl 0, =)) ky that must be satisfied everywhere over the continuous coordinate space X. In contrast, our method uses the most ubiquitous form of signal reprosentation, which straightforwardly enables compatibility with mainstream layers.
ct al. 2021 Lee & Jin 2022
2 Background
In our overview of background material, we introduce Laplacian pyramids as a stepping stone for the formu- lation of Laplacian residuals. In addition, we provide a discussion of signals in subsection A. A. that introduces the notation and fundamental concepts behind this work in way that aims for accessibility.
2.1 Laplacian pyramids
Laplacian pyramids (Burt & Adelson 1987) closely relate to Laplacian residuals, and are often used in vision techniques to decompose signals across range of resolutions.
Laplacian pyramids take some signal s, and produce series of lower and lower bandwidth signals plaw ......,ppyy,, low by convolving against a sequence of smoothing kornels then generate difference signals
Laplacian pyramids ddiff that isolate the part of the signal that was smoothed away at each step, which intuitively correspond to a certain level of detail of the original signal. The operations that compose a Laplacian pyramid can be captured by a base definition and two simple recursive definitions:
Jow * Φ1
S
(1)
low Jow = Pn-1 1 * *Øn
ES, = (s|s * On = s}
(2)
Pn
diff poo P low
E Dn = (d|d On dx Φn+1 d} C Sn
(3)
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In Figure 3
summarize the recursive formulation of Laplacian pyramids in three-block pyramid; this is intended to allow easy comparison with the Laplacian residuals we later illustrate in F Figure 4
Figure 3: Visualization of three Laplacian pyramid blocks.
Laplacian pyramids are convenient to implement as each level reduces computational cost relative to the preceding one. Each level fits within a stricter and stricter bandwidth constraint Si .... Sm. Each level therefore can be represented with lower and lower sampling density [X1//are(XX) > ... X //ml/areaXX (X)) We illustrate this in Figure 3 We represent signals in this way both in Laplacian pyramids and Laplacian residuals.
Laplacian pyramids enable reconstructing the original signal up to an arbitrary bandwidth Sn using only the last lower bandwidth signal Jow and a variable number of difference signals p ...... diff We illustrate this in the lower part of of Figure We express this as linear decomposition more formally below:
p
Pn
Jow
Pn
diff
+
diff P m
low Pm++
(4)
V
S
E Dr
Dm
31 E Sm+1
3! E Sm
31 E S.
Laplacian pyramids offer resolution with variable sampling window, since they rely entirely on convolutions that are equivariant to translation.
Laplacian pyramids also offer adaptive-resolutie ping. We can intuit this from Figure 3) if were to start with an 8 x 8 image of the bird, it would appear reasonable to skip the 16 x 16 level of the Laplacian pyramid. We rect from [Equatin 4 We observe that any discrete signal s[ lu that is correctly sampled must respect the bandwidth constraint S, associated with its sampling pattern X,,. We can therefore locate the band- width constraint of the signal relative to the bandwidth constraint of each level of the Laplacian pyramid 5, 3... S, S,, 5 ... Sm thus revealing the level of the Laplacian pyramid poo that fully captures the signal. We know that all prior difference signals pl pi....... ..... diff ,.. diff must be zero in this case. We can ignore
with variable sampling density through computation skip-
deduce this is formally cor-
the computation of earlier levels of the Taplacian pyramid and start the computation immediately at ddiff
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thus enabling computational adaptivity. We explicitly write the chain of zero terms that unravels in this scenario to later aid in understanding the case of Laplacian residuals, which is more complex:
P2 P1
low Q2
Pn
low Pn-1
Gn
low P1 Φ1
(5)
p Pl
P2
Pn Pn
Pm
Laplacian pyramids are typically formulated using Gaussian smoothing kernels, which violate Equation 35 and introduce errors in the sampling process. We require perfect Whittaker-Shannon smoothing kernels for the computation skipping property we show in Laplacian pyramids above and in Laplacian residuals later in Isussection 3.1 to hold exactly. We note that the form of decomposition yielded by Laplacian pyramids built using Shannon- Whitttkerr smoothing kernels is equivalent to the decompositions yielded by Shannon wavelets.
3 Method
In this section, build towards Laplacian residuals (subsection 3.1,, which are designed to allow the construction of adaptiverrsooluion architectures from fired-resolution layers, and Laplacian dropout (sub- section 3.2), which both serves a regularizer for robustnness at lower resolutions, and regularizer for error-corrccion when imperfect smoothing kernels are used.
3.1 Laplacian residuals for adaptive-resolution deep learning
Laplacian residuals are alike to Laplacian pyramids in the way they decompose a signal into lower and lower bandwidth signals using smoothing filters, and in the way they are able to operate at lower resolution by simply skipping computations. However, Laplacian residuals crucially differ in their ability to incorporate neural architectural blocks that enable deep learning.
In Figure 4] we summarize the formulation of Laplacian residuals in a diagram that illustrates chain of three Laplacian residuals, which allows easy comparison with the Laplacian pyramid shown in Figure
Laplacian residuals are formulated adeoliie layers with variable sampling density (X R( E S2 + (X + R(=+1) E Sn+1 that incorporate neural architectural blocks bn (X,, - R - (X,, R(a) that may be resouuteso layers or adaptive-resolute layers with a variable sampling window. the later case, Laplacian residuals also inherit adaptive- resouution with a variahle sampling window.
Laplacian residuals are compatible with any neural architectural block bn so long as it produces constant everywhere when its input is zero everywhere [Equation 6), which is trivially the case for linear layers, activation layers, convolutional layers, batch normalization layers, some transformer layers, and for any composition of layers that individually meet this condition. This wide compatibility with mainstream layers is unseen in prior adapläve- architectures with variable sampling densily.
b, [0} = where a E R
(6)
Each Laplacian residual block In incorporates single Laplacian pyramid block that is applied on the result of the preceding Laplacian residual rn-1:
1low (rn-1} Tn-1 On diff ph (r--1} = low (rn-1) low {ra-1}
Sn
(7)
E D, S,
The value of the zeroth Laplacian residual is taken to be linear projection Ao xo that raises the feature dimensionality from fo E N to fi E N+ before the first neural architectural block:
To Aos
(9)

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18.14
Gendled
Figure 4: Visualization of a chain of three Laplacian residual blocks.
The value of each subsequent Laplacian residual is obtained by applying a neural architectural block 6, on the difference signal pdift (ra-1}. J. by summing the lower bandwidth signal plr fra-1h. and by applying some some additional processing, which we define below and motivate more concretely next:
Tn(7n-13= An (bni (nn (n--1)) d Pnt1 + plo plow pho
Sn+1
(10)
We include constant rejection kernel de in Equation 10 to replicate the same computation skipping behaviour seen in Laplacian pyramids. This effectively subtracts the mean and ensures the neural architectural block b contributes zero to the residual signal r, the difference signal yn [r,-1} is zero, as shown in Equation We obtain this result thanks to the constraint set on the neural architectural block b, in Equation
bn {0) 1b
(11)
We include smoothing kernel Patt Equation 10 to ensure the output bandwidth of rn coincides with the input bandwidth of Tn+1, which is necessary for Laplacian residuals to follow the same general structure Laplacian pyramids.
We also incorporate a projection matrix A, R/++1/ in Equation 10 to allow raising the feature dimen- sionality from f. E N+ to fat1 E N+ at the end of each Laplacian residual, so that more capacity can be allocated to later Laplacian residuals.
We note that the neural architectural block bn { in Equation 10 written shorthand, and stands for the more terse expression J, (b. (6, }}}. This formal trick that enables our analysis by casting the neural architectural block from operator
discrele signals (X,, - R/2) + (X, R(a) to operator continuous signals (X + BC) E Sn + (X R(*) Sn that is equivalent in the sense of Equation 36 This does not directly reflect the implementation of the method, as all lincar operators are analytically composed then cast to their discrete form to maximize computational efficiency.
We add that we can alter the formulation Equation 10 to let the neural architectural block perform a parameterizod downsampling operation (X,, - R (X,+1 - R()) by changing the interpolation operator discussed above from D, (X, - R5a) + (x + R S₁ Jnt1 (Xn+1 + R (X RR))) Sa+1 and by dropping the smoothing kernel Φx+1 from Equation 10

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Adaptation to lower resolution signals with perfect smoothing kernels. We guarantee that evalu- ating ARRNs while skipping Laplacian residuals is exactly identicnl to evaluating ARRNs while performing the interpolation step typically used to provide adaptive-resolation capability to fized-resolution architec- tures. We spocifically require the use of perfect smoothing kernels for this to hold. This guarantee provides strong icel backing to the validity of our method, and also supported by empirical evidence in subsection
We prove this property following the same argument leveraged to show computational adaptivity in the case of Laplacian pyramids. We are given a discrete signal sf lu that respects the bandwidth constraint S, implied by its sampling pattern X,. We know that this bandwidth constraint can be located relative to the bandwidth constraints of the Laplacian residuals S 3 .... Sn 2 S, 5 .... S.. however we cannot immediately jump to claiming pf ,.... pdiff must be zero because of the nonlinear effect of the inner We must instead follow the terser process hinted at in Eqqation 5) with
architectural blocks b Laplacian pyramids and explicitly unroll the chain. We can see that the membership of s to the bandwidth constraints S, S, c... Si tells us is left unchanged by all of their smoothing filters. We can thus unroll the chain of Laplacian residuals while leveraging that s Ou On b, (0) th = 0 to arrive at our desired result:
s Ø1 and that
To Ags
(12)
P (ro} ro Φ1
(13)
Ags
(14)
P2 (ro) = ro tt2
(15)
Ags
(16)
PI
difl
(ro) = Jow {ro}
p2
low {ro}
(17)
(18)
ritro) = A1 61 (1 [ro}} * + 02 + p2 low {ro}
(19)
= Aps
(20)
Pn (nn-2) -nn22 * Un
low
(21)
=An-2" ... Aos
(22)
Pn-1 ( = =Pn-1 1 - Pn (ra-2}
diff
(23)
=0
(24)
Tn-1(rn-2) An-1
diff L
2/ * on + (oww(rn-)) (nn-2)
(25)
-A Aps
(26)
We can therefore evaluate chain of Laplacian residuals at lower sampling density by skipping higher sam- pling density Laplacian residuals :......'. Tn-1 and instead starting the computation at rn while carrying over the linear projection A,-1 ... Ag- This provides adaptive-resohuiin with variable sampling density through compulational adaplivily without compromise in robusiness and without compromise in compalibility with mainstream layers.
We can use this result to precisely state the cquivalence between ovaluation using all Laplacian residuals after interpolation (Equation 27) and evaluation using the strictly necessary Laplacian residuals (Equation 28
Emfrmf ... ro(Go(Pulsl- Jal]} ... }}
(27)
= =mm(mm ... rn vn(An-I... AnG 1aH) ... ".
(28)
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Adaptation to lower resolution signals with approximate smoothing kernels. We show that using opprosimale smoothing kernels causes some numerical perturbation when skipping the computation of higher sampling density Laplacian residuals. This observation motivates the use of Laplacian dropout, a training augmentation we later introduce in subsection 3.2 that addressos this limitation while also improving robustness.
When using approzimate smoothing kernels iPn On the guarantee we provide does not hold exactly, We consider the case case where On would leave signal unchanged, and note that yn would disturb the signal by small error signal En:
8 SE = S 0n =s = * Pn = = s (Pn pn)
(29)
We note the error above would cascade through every intermediate zero term that leads to Equation 26 therefore discarding unnecessary Laplacian residuals (Equation 28) would not be exactly equivalent to re- taining all Laplacian residuals (Equation 27). We observe this is not simply constrained to linear effect, as t will for instance affect b1, which has nonlinear behavior.
3.2 Laplacian dropout for effective generalization
In this section, we introduce Laplacian dropout, a training augmentation that is specially (ailored to improve the performance of our method at effectively no computational cost.
We formulate Laplacian dropout by following the intuition that Laplacian residuals can be randomly disabled during training to improve generalization. We only allow disabling consecutive Laplacian residuals (using the logical or operator) to ensure that Laplacian dropout does not cut intermediate information flow:
glndep B(1 duate)
(30)
Jchain d judee @ dehnin
(31)
diff Pn [n--1] = Johain (p/ low (rn-1) Pati (nn-1))
low
(32)
Next, we provide a theoretical interpretation that identifies two distinct purposes that Laplacian dropout fulfills our method. We see this dual utility as a highly desirable feature of Taplacian dropout.
Regularization of robustness at lower resolution. Since Laplacian dropout truncates Laplacian resid- uals in the same way they are truncated when adapted to lower sampling densities, Laplacian dropout is identical to randomly lowering sampling density when using perfect smoothing kernels. This acts as train- ing augmentation that promotes robustness over distribution of lower resolutions. We perform a set of classification tasks in Isubsection 4.1 that show this regularizing effect sometimes doubling accuracy over certain lower resolutions without adversoly affecting accuracy at the highost resolutions.
Regularization of errors introduced by approximate smoothing kernels. Since Laplacian dropout truncates Laplacian residuals in the same way they are truncated when adapted to lower sampling densities, Laplacian dropout exactly replicates numerical errors produced by approzimate smoothing kernels in Equa- tion 29 This allows learning a form of error compensation that offsets the effect of approximate smoothing kernels. We demonstrate in subsection 4.5 that this allows the use of very coarsely approximated smoothing kernels that otherwise impart a significant performance penalty on our method.
4 Experiments
We present à set of experiments that demonsttate our method's robuslness across resolutions, its compula- tional adaptivity, and its compatibility with mainstream layers. We show (subsection 4.1) that our method is highly robust across diverse resolutions; (subsoction 4.2) that adaptation provides our method with a significant computational advantage; (subscction 4.3) that our method can generalize across layer types in way that exceeds the capabilities of prior adaptiive-reollution architectures with variable sampling densily;

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(subecction 4.4) that our theoretical guarantee for adaptation to lower resolutions with perfect smoothing kernels holds empirically; and Subsection 4.5) that our theoretical interpretation of the dual regulariza- tion effect of Laplacian dropout coincides with the behaviour we observe empirically when isolating the effect of approzimate smoothing kernels, which improves the robustness of our method through two distinct mechanisms.
Experiment design. We compare models in terms of their robustness across resolutions, their computa- tional scaling relative to resolution, and their ease of construction. We follow a typical use case where we train cach model at a single resolution and then evaluate over range of resolutions. We consider the fluctu- ation of accuracy and inference time over resolution as the metrics of interest for our discussion. We perform set of classification tasks that require models to effectively leverage the information of low-resolution to medium-resolution images; CIFAR10 (32 x 32) (Krizhevsky et al. 2009), CIFAR100 (32x 32) KKrizhvssy
ct al. 2009).
TinylmageNet (64 x 64) (Le & Yang 2015) and STL10 (96 x 96) (Coates et al.
2011
Model design and selection. For most of our experiments Subsectionn 4.1, subsection 4.2
subscction 4.4
and subsection 4.5), we construct ARRNs by using layers that take inspiration from MobileNet V2 (Sandler et al. (018)) and V2 (Tan & Le] 2021); detailed design choices are documented in subsection A.3,
For the experiment that investigates genoralization across layer types dsubsoction 4.3), we construct ARRNs by transplanting layers that are found across a range of mainstream architectures that support adaptive- resolution with variable sampling window but without variable sampling denssity: ResNet18, ResNet550, ResNet101 (11.1M-42.5M) (He et al., 2016),, NedeRes WideRessNet1112 (66.8M-124M) [Zagorykko & Komodakis 2016), MobileNet V3Small, et al.] 2019). We splice the sequence of layers that composes each mainstream architecture at points where resolution changes occur and nest cach resulting subsequence of layers in a Laplacian residual with matching resolution. We discard the first two Laplacian residuals for MobileNet as the resolution of the tailing Laplacian residuals otherwise becomes very small. For our choice of baseline methods, we consider main- stream architectures that again support adaptive-resolution with variable sampling window but without variable sampling densily. We show they compromise robustness across diverse resolutions, yet they have no substantial advantage in ease of implementation or compolibility with mainslream layers compared to our method. We include all mainstream architectures discussed previously in this comparison, along with EfficientNeet2S, EfficientNet EfficientNett222 (20.2M-117.2M) (Tan & Lc 2021). For the ex- periments that validate our theoretical analysis (subsection 4.4 and Subsectiio 4.5), we perform an ablation study over the quality of the smoothing kernel, the use of Laplacian dropout at training time, and the use of adaptation at inference time.
(Howard
Model training and evaluation. All models are trained for 100 epochs at the full dataset resolution with identical hyperparametens that are described in /subsection A.3] All models are then evaluated at the full dataset resolution and at a range of lower resolutions that are generated by interpolation. We have showed in our introduction that the mainstream architectures we cover display very weak robustness when evaluated directly (meaning with computational adaptivity: Figure 1), and that they perform much more reliably when evaluated after an interpolation slep that ensures the sumpling window is kepl. conslant (meaning without computational adaptivity; Figure 2). In the comparisons we make between baseline methods and our method (subsection 4.1 subsection 4.2) and suuseection 4.3), we display baseline methods evaluated with interpolation, as it provides the fairest chance to compete against our method. In our appendix (subsection A.3), we show a complimentary comparison that displays baseline methods evaluated directly, which most interesting in terms of inference time; our method displays more aggressive computational savings as resolution decreases. We underline that all baseline methods share the same set of training runs across these figures; their parame- ters are ezactly identical; only their mode of evaluation changes. In terms of evaluating our method, we skip unnecessary Laplacian residuals unless performing an ablation over adaptation. We note that in principle, we should always round up the number of required Laplacian residuals for in-between resolutions; however, we sometimes achieve greater robustness if we round down. This is the case with TinyImageNet and STL10 in ssubscction 4.1] and (subsoction 4.2] and with TinyImageNet in |subscction 4.3) This effect likely results from more consistent statistical properties encountered when only evaluating Laplacian residuals that have full access to the part of the signal they usually address.
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4.1 Robustness and the effectiveness of Laplacian dropout
We demonstrate that our method allows for greater low-resolution robustness than mainstream methods with- out compromise high-resolution performance. Figure 5) shows four ablations of our method corresponding to permutations of two sets: either with Laplacian dropout (red lines) or without Laplacian dropout (black lines); and either with adaptation (solid lines) without. adaptation (dashed lines). We see that with method outperforms every baseline method
Laplacian dropout and with adaptation (full red lines), across every resolution and every dalaset. We also find that, in contrast, without Laplacian dropout (black lines), our method shows much weaker generalization across resolutions, clearly demonstrating Laplacian dropout is effective as regularizer for robustness to diverse resolutions.
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Figure 5: Accuracy of all architectures at various resolutions after training at the full dataset resolution. Evaluation is performed after interpolation to the training resolution in the case of mainstream methods. Our method (red full line) displays the best accuracy at the highest resolution and robustly maintains its accuracy at lower resolutions.
4.2 Computational efficiency
We confirm the advantage granted by computational adaptivity by performing inference time measurements on the previous experiment. We use CUDA event timers and CUDA synchronization barriers around the forward pass of the network to eliminate other sources of overhead, such data loading, and these time increments over all batches of the full dataset. We repeat this process 10 times and pick the median to reduce the effect of outlicrs. Figure shows the inference time of ARRNs with adaptation (full red lines) and without adaptation (dashed red lines). Our method significantly reduces its computational cost (highlighted by the shaded area) by skipping the evaluation of Laplacian residuals at lower resolutions. Our method also has a reasonable inference time relative to well-engineered mainstream methods.
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Figure 6: Inference time of all architectures at various resolutions. The inference time for the entire dataset is considered. Evaluation is performed after interpolation to the training resolution in the case of mainstream methods. Our mothod (red full line) can adapt to lower resolutions by skipping Laplacian residuals, which results in significant computational savings (highlighted by the shaded area) compared to using all Laplacian residuals (red dashed line). Our method also displays a reasonable inference time relative to typical convolutional neural networks despite not having highly optimized implementation.
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4.3 Generalization across layer types
We demonstrate the ease of use of our method and its compatibility with mainstreom layers by constructing adaptive-resohutio architectures from range of mainstream architectures (ResNet18, 88, 5esNet50, ResNet100, WideRess WideRessNet10122 MobileNet V3Small and Mo- bileNet V3Lagge) Figure compares the accuracy of architectures in adaptive-resolutie form (red box plots) and in mainstream form (green box plots). The distribution of accuracies of the seven underlying ar- chitectures in the adaptive-resolution group and mainstream group visually conveyed by drawing a small box plot al every resolution. Our method consistently delivers better low-resolution performance, and simi- lar or better high-resolution performance. Our method achieves this while generalizing beyond the abilities of prior adaptive-reslution architectures with variable sampling density, which are incompatible with the layers used in this experiment.
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Figure 7: Accuracy of two groups of methods at various resolutions, where 7 adaptive-rosolution architec- tures (in red) are constructed by taking 7 mainstream architectures (in green) and wrapping their layers in Laplacian residuals. Evaluation is performed after interpolation to the training resolution in the case of mainstream architectures. Our method yields architectures that have stronger low-resolution performance, and similar or better high-resolution performance, which demonstrates ease of use and compatibility with mainslream layers.
4.4 Adaptation with perfect smoothing kernels
We perform an ablation study to verify our theoretical guarantee for numerically identical adaptation. Figure 8 displays a set experiments that use perfect quality Whittaker-Shaman smoothing kernels (in
the upper row of graphs in green) implemented through the Fast Fourier Transform (Cooley & Tukey 1965)
We showcase the usual set of ablations within this group of experiments; with Laplacian dropout (bright green lines) or without Laplacian dropout (dark green lines); with adaptation (full lines) or without adaptation (dashed lines). Our method evaluates practically identically whether unnecessary Laplacian residuals are discarded (with adaptation, full lines, |Equation 28), or whether all Laplacian residuals are preserved (without adaptation, dashed lines, Equation 27j. The discrepancies are either exactly zero, or are small enough to be attributed to the numerical limitations of floating point computation. The discrepancies are quantifiid thoroughly in the tables overlaid over each plot. Our method is thus able to skip computations without numerical compromise, as predicted by our thcoretical guarantee.
4.5 Adaptation with approximate smoothing kernels and the dual effect of Laplacian dropout
We extend the previous ablation study to verify our theoretical analysis of the dual effect of Laplacian dropout. Figure 8 introduces set of experiments that relies faár quality approximate Whittaker-Shanon smoothing kernels (in the middle row of graphs in red), and on poor quality truncated Gaussian smoothing kernels (in the bot row of graphs in blue). Figure 9 displays these same results in the form of a decision tree to help recognize the trends that are relevant to our discussion. This decision tree factors the impact of choosing a specific filter quality, choosing whether to use Laplacian dropout or not, and choosing whether to use adaptation or not. This analysis considers accuracy averaged over all resolutions and all datasets as the metric of interest. The numerical values displayed each node correspond to the average multiplicative
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Figure 8: Accuracy of ARRNs al various resolutions, where each ARRN is identical to the architect.ure used in the main experiments aside from the choice of smoothing kernel. The smoothing kernels each correspond a different row of graphs and a diffcrent huc. The shaded area spanning pairs of curves highlights the difference between the accuracy with adaptation and without adaptation. The overlaid tables display a statistical breakdown of this discrepancy with Laplacian dropout and without Laplacian dropout.
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Multiplicative factor analysis showing an ablation over key components of ARRNs. The perfor- mance metric considered for analysis the average accuracy over all resolutions and all datasets. The three levels of the decision tree show which filter is used, whether Laplacian dropout is used at training time, and whether adaptation is used at inference time. The numerical ratios indicate how average accuracy changes conditional to the decision associated with the node. The last two levels of nodes coloured black or white according to the absence or presence of error terms cn respectively; at training time, this is determined by use of Laplaciann dropout; al inference time, this is determined by use of adaplation. The symbols below the leaf nodes of the decision tree show the line style used in Figure to allow easily referencing the underlying experiments in detail.
Figure 9:
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change in the metric once decision is made. We claimed in subsection 3.2 that Laplacian dropout has two distinct purposes: it regularizes for robustness across a wider distribution of resolutions; it also mitigates mumerical discrepancies caused by approximate smoothing kernels when using adaptation (Equation 29). We have demonstrated the first effect in sisusscction and can also observe this effect clearly in the decision tree. We can only observe the second eflect. we consider the choice of smoothing kernel, which motivates this experimental setup. Our method will conform to our theoretical interpretation absence of error terms Gn at training time yields better performance in the absence of error terms en inference time, and vice versa; that is to say on the last two levels of the decision tree, on the black and white nodes, tracing path across two nodes of identical colour should yield multiplier greater than 1 at the last node, and conversely, tracing path across two nodes of opposing colour should yield multiplier smaller than 1 at the last node. Our method displays exactly this behaviour, with the discrepancy at the last level of the decision tree growing monotonically with decreases in filter quality. Our method therefore leverages Laplacian dropout not just to improve robustness across a wider distribution of resolutions, but to compensate numerical errors inducod by imperfect smoothing kernels, which enables the use of computationally greedy implementations.
5 Discussion
We have introduced ARRNs, class of adaptive- architectures that inherits the compatibility with mainstream layers of fized-resolution methods, and the computational adaptivity and robusiness of adaptive- resolution methods. ARRNs substitute standard residuals with Laplacian residuals which allow creating ressliin architectures using only fioed-resolution layers, and which allow skipping computations at lower resolutions without compromisc in numerical accuracy. ARRNs also implement Laplacian dropout, which allows training models that perform robustly at wide range of resolutions.
Future Work. We have provided evidence on classification tasks over low-resolution and medium- resolution image data; our method's ability to generalize is well supported by theoretical justification, but further experiments that include more challenging tasks and high-rosolution data are desirable. We have investigated form of Laplacian residual that decreases resolution, which addresses only limited variety of architectures. We have applied our method in two dimensions on image data, but it is theoretically valid with any number of dimensions; its application to audio data and volumetric data is of interest.
Impact Statement. This paper presents work whose goal is to advance fundamental research in the field of deep learning and machine learning. No specific real-world application is concerned, although this contribution may render certain forms of technology more accessible.
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A Appendix
A.1
Background
We survey fundamental concepts of signal processing and introduce our notation. We aim to provide mcan- ingful intuitions for readers who are not familiar with these principles, and to also rigorously ground our method and satisfy readers who are knowledgeable in this topic.
In Figure 10 we illustrate how the sampling density and bandwidth of signals intuitively relate to each other while also highlighting the notation and indexing scheme we use designate these characteristics our analysis. In Figure II we illustrate the effect of the sampling window over signal. In the paragraphs that follow, we break down these notions in greater detail.
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Diagram showing the same signal captured with three distinct sampling densities /areal/ < X,l//rea((X < [Xn-al/area(X) (laid out horizontally) and three distinct bandwidth constraints S.+1 C Sn C Sn-1 (laid out vertically). The sampling density corresponds to the number of samples per unit of space used to cast the continuous signal into a discrete signal. The bandwidth corresponds to the smoothness the underlying continuous signal. contimuous signal may only be captured into a discrete signal if sampling density sufficient for the bandwidth; the blacked out signals show cases where this condition not met.
Figure 10:
Samping Window
Figure 11: Diagram showing the same signal captured with three distinct sampling windows X narrower C
X C wider The sampling window effectively consists in the region of space that is captured in the signal.
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Signals in continuous form. We can represent signals as functions s( ): X + R that map continuous spatial domain X that is a nicely behaved subset of TRd to feature domain R.. This representation is useful because it allows us to leverage notions from functional analysis and calculus, and because it is fully independent from the way a signal is captured. We often refer to the bandwidth of signals in this work, which we can intuitively relate to the smoothness of continuous signals; low bandwidth signals are smooth, while high bandwidth signals are detailed.
Signals in discrete form. We can also represent signals as functions si I::::: + R/ that map from a discrete spatial domain X, C X to a feature domain Rf. This representation enables us to perform computations on signals as they are fully defined by a finite amount of information given by the |X,l individual samples x; E X,- We commonly point to the quantity of samples as the resolution of discrete signal. We associate indexing by n to distinct resolutions throughout this work, where resolution decreases increases, meaning [xm| |Xn+1l.
Casting continuous signals into discrete signals by sampling. We can easily take a continuous signal s(.) and create a discrete signal s{ ]n by sampling values at points x, X,, We notate this process as the sampling operator Gn (X+ ( (X., + RR):
6 )} x; + sxx,) Vx, E X.
(33)
Casting discrete signals into continuous signals by interpolation. We can reverse the process above and derive a continuous signal s( .) from discrete signal s( In by applying a convolution with a smoothing kernel on, more formally known as Whittaker-Shannot kernel (Whittaker 1915 1927). We note that sampling and interpolation are only inverses of each under certain important conditions we come back to later. We interpret the effect of the convolution against a smoothing kernel as filling in the gaps between the samples. We notate the process outlined here as the interpolation operator In : :X,, - TR/) - (X- RR)::
In = Σ X.
(x VXEX
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Restricting the bandwidth of signals. We need a slightly more formal way of designating signals that respect certain bandwidth constraints in order to better discuss the equivalence between discrete signals and continuous signals. We can use the smoothing kernels On we just introducod to define sets of continuous signals S, = {s|s*0n = F} that are already smooth enough to be left unchanged by the action of smoothing kernel. We underline that convolving smoothing kernel on against a signal s restricts its bandwidth such that it belongs to the corresponding set of signals Sn. We finally note that bandwidth constraints form an ordering S, 3 Su+1, meaning signal that respects a low bandwidth constraint S, also respects any arbitrarily high bandwidth constraint Sp-k v k 0.
Equivalence of continuous signals and discrete signals. We can use discrete signals continuous signals to designate the same underlying information when the Nyquist-Shannon sampling theorem is satisfied (Shannon 1949 Petersen & Middleton 1962). This theorem intuitively states that a contimuous signal with high bandwidth requires a discrete signal with correspondingly high sampling density for sampling to generally be feasible without error. This theorem more formally states that a discrete signal s[ In can uniquely represent any continuous signal s( ) that respects the bandwidth constraint encoded by membership S, = {s|s * on, = s), where the expression for the smoothing kernel on depends on the sampling density Xal/area (X) of the discrete spatial domain over the continuous spatial domain and on the assumptions the boundary conditions of the continuous spatial domain X (Whittaker] 1915 1927 Petersen & Middleton 1962). We summarize the Nyquist-Shannor sampling theorem by stating that the sampling operator and interpolation operator are only guaranteed to be inverses of each other when the bandwidth constraint. is satisfied:
E Sn = Jn Jn(6,, (3))
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Equivalence of operators acting upon continuous signals and discrete signals We can extend the notion of equivalence wetween continuous signals s(.)= X - R and discrele signals s| In X, - TR/ to encompass the actions that can be performed on the same signals using operators on continuous signals D : (X + R²) + (X + RR) and operators on discrete signals D, (X, + -R) + (X, + RR). We are especially interested in this notion as enables us to think of our neural architecture as a chain of operators that act on continuous signals that can be cast to act on discrete signals of any specific sampling density Xxnl/areaXX.. We often this property formally labeled as discretization invariance in the neural operator community and highlight this concept is key to other works which allow adaptation to different sampling densities (Li et al. 2021 Kovachki et al., 2023: Fanaskov & Oseledets, 2023 Bartolucci et al. 2023). We can formally express the equivalence between the continuous form D and discrete form D, of some operator as commulalivity over the sampling operator when the bandwidth constraint is satisfied:
SE SES, Sn
6,(D(s)) - (G,((()
(36)
A.2 Related works
We shortly cover architectures that implement forms of residual connections that are similar to Laplacian residuals.
Residual connections with filtering operations. Singh et al. (2024) incorporates filtering operations within residuals to separate the frequency content of convolutional networks, although it provides no adaptive- resolution mechanism. Lai et al. (2017) uses Laplacian pyramids to solve super-resolution tasks with adaptive oulpuu resolution, with residuals ordered by increasing resolution. This is unlike our method, which is well suited to tasks with adaptive input resolution, with residuals ordered by decreasing resolution.
Residual connections with dropout. Huang et al. (2016) implements a form of dropout where the layers nested within residual blocks may be bypassed randomly. This is somewhat similar to Laplacian dropout, however, this is not equivalent to form of bandwidth augmentation and does not result in the same improved robustness to various resolutions we show in subsection 4.1
A.3 Experiments
We include an alternate evaluation of the experiments we present in ssuseccton 4.1 and subsection 4.2 We also provide further details on our experimental setup in this section.
In Figure 3 and Figure 6 we show baseline methods evaluated after an interpolation step, which yields stronger robusiness but negales compulational adaplicily. Figure 12 and Figure 13) we instead show baselines evaluated directly, which yields weak robustness but provides some reduction in inference time lower resolution. We observe that our method provides vastly superior robustness while also reducing its inference time more steeply at lower resolutions.
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Figure 12: Accuracy of all architcctures at various resolutions after training at the full datasct resolution. Evaluation performed direclly in the case of mainstream methods. Our method (red full line) more markedly dominates mainstream methods in terms of robustness under this alternate mode of evaluation.
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Figure 13: Inference time of all architectures at various resolutions. The inference time for the entire datasct is considered. Evaluation is performed directly in the case of mainstream methods. Our method (red full line) skips the computations of layers at lower resolutions, which more steeply reduces inference time relative to mainstream methods, which must still evaluate all their layers.
Model design. We provide detailed illustrations for the full architecture designs we use with our method according to each dataset: Figure 14 for CIFAR1O (5.33M-8.0) M0MM); ;; Figure 15 for CIFAR100 (9.59M-14.5M); Figure 16 for Tiny Imageett (15 155.0M-19.8M) 8M); and Figure 17 for STL10 (13.8M-18.4M). We indicate not a single parameter count, but a range of paramcter counts for cach architecture design, as adaptation enables computation of the forward pass or backward pass using a variable subset of the underlying parameters. We follow a general design pattern inspired by MobileNetV2 (Sandler al.) 2018) and Efficient.Net (Tan & Le 2021) to derive these architecture designs. We nest inner architectural blocks (bn, in Equation 10) within series of Laplacian residual blocks of decreasing resolution and increasing feature count. We create these inner architectural blocks by composing depthwise 3 x 3 convolutions and pointwise 1 x convolutions in alternation. We set all depthwise convolutions to use edge replication padding to satisfy Equation 6 and ensure resolution remains fixed within each Laplacian residual block. We prepend this string of layers with a pointwise convolution that expands the feature channel count. We conversely terminate the sequence of layers with pointwise convolution that contracts the feature channel count inversely. We separate cach convolution with batch normalization şloffe & Szegedy 2015)) and SiLU activation function (Ellwing al., 2018), chosen for its tendency to produce fewer aliasing artifacts. We apply different Laplacian dropout rates [drate in Equation 30) depending on the dataset: 0.6 for CIFAR1O; 0.3 for CIFAR100; 0.3 for TinylmageNett 0.3 for STL10. We use a common classification head that consists of a single lincar layer with a dropout set to 0.2, which is applied after global average pooling. The designs were chosen by sweeping over different configurations for inner architectural blocks, and over different resolutions and number of features for the Laplacian residual blocks that contain them. The number of permutations per final sweep ranged between 18 216 for each dataset.
Model training hyperparameters. Wc provide the specific hyperparameters used during training.
For CIFAR10 and CIFAR100, across all methods, we use AdamW [Toshchilov & Hutter 2019) with a learning rate of 10 and (81,8)) = (0.9, 0.999), cosine annealing (Lohchhlov & Hutter 2022) to a minimum learning rate of 10-5 in 100 epochs, weight decay of 10-3, and a batch size of 128. We use a basic data augmentation consisting of normalization, random horizontal flipping with p = 0.5, and randomized cropping that applies zero-padding by along cach edge to raise the resolution, then crops back to the original resolution.
For TinyIImagel and STL10, across all methods, we use SGD with a learning rale of 10-2, cosine annealing [Loshchilov & Hutter 2022) to minimum learning rate of 0 in 100 epochs, weight decay of 10-3, and batch size of 128. We use PrivialAugment Wide (Müller & Hutter 2021 to augment training.
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Published in Transactions on Machine Learning Research (06/2025)
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Figure 14: Architecture design for our method on CIFAR1O.
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Figure 15: Architecture design for our method on CIFAR100.
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Published in Transactions on Machine Learning Research (06/2025)
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Figure 16: Architecture design for our method on TinylmageNet.
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Reviews:
[{"review_id": "tQXejD1rJ5", "reviewer": "Reviewer_N159", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### **Strengths**  \n- The problem this paper addresses is interesting. Resolution adaptability is a practical but often overlooked issue, especially in computer vision applications. This problem is a special case of transformation equivariance, and this paper can inspire follow-up research on this topic.\n\n- The Laplacian pyramid-inspired solution proposed in this paper is reasonable, leveraging conventional domain knowledge in convolutional neural networks. Additionally, the use of Whittaker-Shannon smoothing kernels is interesting and guarantees generalization across different resolutions.\n\n- Except for the small image resolution, the experimental items are sufficient.\n\n\n### **Weaknesses**  \n- The first major weakness of this paper is its claims regarding existing architectures:\n  - \"We find this approach to be unsatisfying, as it lacks computational adaptivity.\" However, this approach has proven very effective for current transformer-based architectures. For segmentation tasks, such as SAM, the input is resized to be compatible with the training resolution, yet the output achieves sufficiently good results after resizing back to the original resolution. Therefore, the authors should elaborate more on this claim and provide sufficient justification; otherwise, it contradicts established facts.\n  - \"In many practical applications, however, they offer weak generalization across resolutions...\" Although convolution operations themselves are only translation equivariant, the learned feature representations can be somewhat equivariant to more complex transformations including scaling (applying to different resolutions), moderate rotations, and perspective transformations, especially for encoder-decoder architectures. This capability is fundamental for applications such as feature descriptor matching.\n\n- The second major weakness is the organization and writing, especially Section 3:\n   - Figure 4 could be illustrated more clearly.\n   - The expanded equations 5-14 and 21-34 can be condensed; the current presentation is poorly structured.\n   - Many descriptions are dense without sufficient explanations. For example, \"This is a formal trick that enables our analysis by ....... Equation 19.\" The explanation appears to be in the appendix but would be better placed in the main paper.\n\n- The third major weakness is the method design. Specifically, to adapt to different resolutions, the authors only apply nonlinearity to the Laplacian residuals within each Laplacian residual block, while the low-pass components undergo only linear transformations. In the lowest resolution case, inference degrades to only using the last \"Inner Architectural Block.\" This effectively upgrades input RGB features directly to a much higher dimension (256/512) through linear transformation, contradicting best practices of progressively upgrading feature channels.\n\n- The final major weakness of this paper is that all experiments are conducted on inputs with extremely small resolutions, only up to 96x96 pixels. Conclusions drawn from experiments at such low resolutions do not generalize well to practical cases. A significant potential reason is padding. In this paper, the authors adopt replication padding, which, although better than default zero-padding for low-resolution images, still interferes with convolution operations. In more practical higher-resolution scenarios, this interference is less severe.\n\nIn summary, although this paper addresses an interesting problem, its significance under practical settings is limited unless proven otherwise. Additionally, both the method design and experimental setup have fundamental issues requiring considerable effort to resolve.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper works on the problem of how to adapt the network structure to inputs of different resolutions. To this end, the authors propose the ARRN architecture, which is inspired by the mechanism of the laplacian pyramid of images. To adapt the idea of the laplacian pyramid to neural networks, especially CNNs, the authors propose the laplacian residual blocks which only applies non-linearity to the laplacian residuals. The authors also further propose laplacian dropout to skip randomly skil laplacian residuals during training for better generalization. To demonstrate the effectiveness of the proposed method, the authors conduct experiments on various dataset with very small resolution (CIFAR-10&100, TinyImagenet, STL10) with various baseline architectures (Resnet, MobileNet and EfficientNet), which demonstrates better performance on varying, especially low resolution inputs.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please refer to the weakness and try to make the following changes:\n- Justify the claim about weaknesses of transformers and CNNs from a more practical perspective (wether the output really suffers from the claimed weaknesses) instead of only focusing on the model structure paradigm.\n- Reorganize the paper and make the method description in the main paper more self-contained and clear.\n- Discuss about the degradation case of the method, when the operation before the final inner block becomes linear.\n- Using practical image resolution for all experiments.", "broader_impact_concerns": "The broader impact has been discussed in the paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "zHzlLuuI7K", "reviewer": "Reviewer_ZLpD", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength: \n1. Dealing with various input resolutions is a practical challenge. Very interesting and important topic.\n2. The proposed method demonstrates better robustness against lower resolution inputs.\n3. Adaptation improves evaluation runtime-efficiency while Laplacian dropout improves accuracy and robustness.\n\nMy comments and questions:\n1. It would be interesting to explore how the proposed architecture generalizes to higher resolutions. Instead of training the model on the highest resolution (e.g., 96x96 in STL10), the authors could consider training on lower resolutions and evaluating the model's performance on the original resolution. In addition, experiments with real datasets would be more interesting.\n2. This paper demonstrates the effectiveness of the proposed methods primarily for classification tasks. How does the proposed method perform in regression tasks, such as image reconstruction? Although lower-resolution inputs contain less information, would classical neural networks still perform well given the expected lower-resolution reconstruction?\n3. It appears that while the baseline methods perform poorly at very low resolutions, they are relatively stable within the higher resolution range. In comparison to the solid black line in Figure 5, the proposed architecture without Dropout and adaptation, which is more sensitive to resolutions different from the training resolution, could you explain why this happens?\n4. How does the proposed method compare to other multi-scale architectures? While baseline models like ResNet and EfficientNet can handle different input sizes, they do not explicitly leverage multi-scale architectures. It is unclear whether it is the multi-scale architecture itself that promotes robustness to different resolutions.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents Adaptive Resolution Residual Networks (ARRNs), a novel deep learning architecture that effectively generalizes across varying image resolutions. Unlike traditional deep learning models that operate on fixed-resolution inputs, ARRNs overcome this limitation by incorporating Laplacian residual and Laplacian dropout modules. The paper demonstrates better robustness against lower resolutions for classification tasks.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. What are the key differences between the Laplacian Pyramid and other multi-scale image analysis methods, such as the Wavelet transform? I suggest adding a few sentences in the Related Works section to clarify these differences and provide a comparative analysis.\n2. How to choose the number of downsizing levels? How does the accuracy depends on n?\n3. See point 4 in my comments above, adding experiment comparisons to other multi-scale models.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "nGHCqQXOpT", "reviewer": "Reviewer_xWy1", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n1. The problem studied in this paper, i.e. making image processing neural networks easily adaptive to various resolutions, is important. Hence the technique developed by the authors and presented in this paper may have an impact on this area of research.\n2. The authors have conducted extensive experimental evaluation to demonstrate the characteristics of the proposed method. The authors also included a lot of details in the paper, which may benefit follow-up researches.\n3. The idea of adopting Laplacian pyramid for deep learning with variable resolutions is interesting.\n\n# Weaknesses\n1. Despite that the paper presented a lot of background and mathematical details for the proposed method, the motivation and key idea is still not clear to me. The paper mainly introduce the problem through an overview of the related work, but what is missing in the literature and where is the gap between the existing research and the goal is still unclear. From my point of view, the authors paid too much attention on the mathematical derivation of the method but neglected the introduction to the concept, idea, and philosophy, making the paper overall not so accessible to broader audience.\nSpecifically, the following are not so clear to me and are suggested to be revised:\n* p.2 The authors motivate the proposed method by saying that the existing form of representation has shortcomings when mapping functions to functions. This statement needs further elaboration. What does mapping functions to functions refer to? Why is this important?\n* p.6 The authors should define \"numerical perturbation\" in the context.\n\n2. As described in the experimental settings, for the compared methods handling a lower resolution, the input is first interpolated to a higher resolution. I wonder whether this is the most appropriate treatment. To my knowledge, if the original architecture is like some convolutions followed by an adaptive global pooling, then there is no need to first upsample the input and process. The architecture itself should naturally supports different resolutions. Hence I think the comparisons conducted in the study might not be fair. The authors should at least include a setting where the low resolution inputs are directly fed into the baseline networks.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a method that adapts common neural layers to various resolutions based on the Laplacian pyramid. The key idea is to build a Laplacian pyramid using low-pass filter, and make common neural layers operate on the residual signal. Experimental results show that the proposed method improves neural networks' generalization to downsampled inputs when trained on full resolutions.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The authors should revise the introduction of this paper by giving an overview of the problem that this paper is solving, especially stressing the gap between existing research and the goal that is achieved in this study.\n2. The authors should revise and improve the clarity of the paper, corresponding to the two points I mentioned in the **Weaknesses**.\n3. The authors are encouraged to include a comparison to the baseline methods where the low-res inputs are not upsampled before sent to the network.", "broader_impact_concerns": "There is no ethical concern.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper introduces the ARRN architecture, along with associated blocks and dropout, to enable networks to adapt to varying resolutions. Experimental results demonstrate enhanced performance. Post-discussion, two reviewers offer favorable feedback, while one presents a critical view. All reviewers find the proposed adaptive resolution approach intriguing and consider the Laplacian pyramid solution reasonable. However, the primary concern raised is that the empirical experiments predominantly rely on small datasets. The authors to are suggested incorporate additional experimental analyses and make more careful claims in the introduction.
Decision: Accept (poster)