Test-time Adaptation for Image Compression with Distribution Regularization
Authors: Kecheng Chen
Pingping Zhang
Tiexin Qin
Shiqi Wang
Hong Yan
Haoliang Li
Venue: ICLR 2025 Poster (2025)
Keywords: test-time adaptation
image compression
entropy coding
Primary area: transfer learning, meta learning, and lifelong learning

Abstract: Current test- or compression-time adaptation image compression (TTA-IC) approaches, which leverage both latent and decoder refinements as a two-step adaptation scheme, have potentially enhanced the rate-distortion (R-D) performance of learned image compression models on cross-domain compression tasks, \textit{e.g.,} from natural to screen content images.  However, compared with the emergence of various decoder refinement variants, the latent refinement, as an inseparable ingredient, is barely
 tailored to cross-domain scenarios. To this end, we are interested in developing an advanced latent refinement method by extending the effective hybrid latent refinement (HLR) method, which is designed for \textit{in-domain} inference improvement but shows noticeable degradation of the rate cost in \textit{cross-domain} tasks. Specifically, we first provide theoretical analyses, in a cue of marginalization approximation from in- to cross-domain scenarios,  to uncover that the vanilla HLR suffers from an underlying mismatch between refined Gaussian conditional and hyperprior distributions, leading to deteriorated joint probability approximation of marginal distribution with increased rate consumption. To remedy this issue, we introduce a simple Bayesian approximation-endowed \textit{distribution regularization} to encourage learning a better joint probability approximation in a plug-and-play manner. Extensive experiments on six in- and cross-domain datasets demonstrate that our proposed method not only improves the R-D performance compared with other latent refinement counterparts, but also can be flexibly integrated into existing TTA-IC methods with incremental benefits.

Published as a conference paper at ICLR 2025
TEST-TIME ADAPTATION FOR IMAGE COMPRESSION WITH DISTRIBUTION REGULARIZATION
Kecheng Chen, Pingping Zhang, Tiexin Qin, Shiqi Wang, Hong Yan, Haoliang Li" City University of Hong Kong. Hong Kong SAR. "Corresponding author. cs ckc9kk 66 gmail. com;
11 11 110my- cit cityu, edu.hk
ABSTRACT
Current test- or compression-time adaptation image compression (TTA-IC) ap- proaches, which leverage both latent and decoder refinements as a two-step adapta- tion scheme, have potentially enhanced the rate-distottion (R-D) performance of learned image compression models cross-domain compression tasks, e.g. from natural to screen content images. However, compared with the emergence of vari- ous decoder refinement variants, the latent refinement, an inseparable ingredient, is barely tailored to cross- scenarios. To this end, we aim to develop an ad- vanced latent refinement method by extending the effective hybrid latent refinement (HLR) method, which is designed for in-domain inference improvement but shows noticeable degradation of the rate cost cross- coos-ddmain tasks. Specifically, we first provide theoretical analyses, in a cue of marginalization approximation from in- to cross-domain scenarios, to uncover that the vanilla HLR suffers from an underlying mismatch between refined Gaussian conditional and hyperprior distributions, lead- ing to deteriorated joint probability approximation marginal distribution with inereased rate consumption. To remedy this issue, we introduce simple Bayesian approximation-endovee
distribution regularization to encourage learning a better joint probability approximation in plug-and-plas manner. Extensive experiments on six in- and cross-domain datasets demonstrate that our proposed method not only improves the R-D performance compared with other latent refinement counterparts, but also can be flexibly integrated into existing TTA-IC methods with incremental benefits. Our code is available at http::/tonyckc. www.://...giiuub.boTTTT.......
INTRODUCTION
With rapid developments in data streaming techniques, fruitful high-resolution images need to be transmitted online between edge devices. It therefore imperative to develop more efficient, effective, and versatile image compression approaches for better storage and transmission. To this end, we have witnessed learning- based image compression (LIC) methods (Ballé al., 2016; 2018; Cheng et al., 2020; Kim et al., 2024) significantly outperform conventional codecs, such as VVC (Bross et al., 2021) and JPEG (Wallace, 1991) for rate-distortion (R-D) performance. Such gains mainly derive from the unprecedented non-linear transform capacity of deep neural networks (DNN) and accurate probability representations for entropy coding in an end-to-end R-D cost-guided learning framework.
Nevertheless, these DNN-based LIC approaches inevitably inherit the identically and independently distributed (i.i.d)) assumption between the source (training data) and target (testing data) domains, which may not always hold in versatile image compression scenarios, e.g., there is a significant distribution gap between natural and screen content images. Compared with in-domain compression (i.e., i..dd. assumption holds), such domain shifts would deteriorate the R-D performance of DNN- based codecs in cross-domain compression. For example, most advanced LIC models leverage a byperprior-based entropy framework (Ballé et al., 201 2018), where the hyperprior model extracts the side information Z of the latent variable y to capture Gaussian conditional probability p(y|z) for coding, and the side information coding by the hyperprior probability p(2) learned from entropy bottleneck. In cross-domain scenarios, the discrepancy in statistical property between source and target domains will cause inaccuracy or ineffectiveness of learned probability models (Ulhaq Bajić, 2024), leading to suboptimal entropy coding with additional bit consumption. Moreover, it is difficult for a source-trained decoder to render high-fidelity target images due to domain shifts.

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To this end, it is necessary to conduct test- or compression- time adaptation for image compression (TTA-IC), One ap- proach (Zou et al., 2021) is update the decoder of the entropy model, which involves the transmission of adapted parameters to the decoder side. Although recent parameter- efficient transfer learning-based (Hu et al., 2021) exten- sions (Shen et al., 2023) reduce the huge bitrate overhead of parameter transmission to an acceptable level, the entropy optimization of updated parameters and extra bit consumption are still troublesome. Another promising approach (Djelouah Schroers, 2019) is to directly refine the latent variables without altering any model parameters. Compared with the first approach, such a branch is still effective for many special- ized neural decoders, where model parameters are hard-coded and non-modified (Dass et al., 2023).
Aöpp (Bits per pisel)]
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Figure 1: Comparison of various la- tent refinement methods under cross- domain tasks (SIQAD screen content dataset). The Cheng20 (Cheng et al., 2020) (quality= pre-trained on nat- ural images) model is used.
Here, we focus the second approach, i.e., latent refinement, due to its simple optimization, friendly bitrate, and natural immunity to catastrophic forgetting. Some state-of-the-an (SOTA) TTA-IC methods (Campos al., 2019; Shen et al., 2023) actually have introduced basic latent refinement (BLR) (Djelouah & Schroers, 2019) into their two-step adaptation framework including latent and decoder refinements, e.g., updating the latent variable y by R-D objectíve. Yet, the Gaussian conditional probability p[y|z) and hyperprior p(z) are still inaccurate without refining the side information 2, leading suboptimal cross-domain R-D performance. A potential solution may be hybrid latent refinement (HLR) ( (Yang et al., 2020) that conducts joint update of latent variable and side information 2. Although the vanilla HLR, tailored to in-domain inference improvement, exhibits better R-D performance compared with BLR in in-domain TTA-IC tasks ( (Yang et al., 2020), it suffers from a significant downside in rate cost while enhancing the reconstruction quality in cross-domain scenarios, as depicted in Figure 1. This degradation in rate cost is, however, ignored by existing two-step TTA-IC approaches (Lv et al., 2023; Tsubota et al., 2023; Shen et al., 2023) that directly impose the vanilla HLR their inseparable ingredient.
Motivated by the abovementioned analyses, this study aims to develop an advanced latent refinement method that can be adaptive to cross doos-n TTA-IC with consistent R-D gains. Such a scheme can not only render better latent representations with zero model update and transmission but also enhance the R-D performance of existing SOTA TTA-IC approaches latent refinement. With these goals in mind, we propose to tailor the vanilla HIR method designed for in-domain inference improvement to cross-domain cases. To achieve this, theoretical analysis. in cue of marginalization approximation from in- to cross-domain scenarios. is provided to reveal the degradation reasons of the vanilla HLR in the cross-domain scenario. In a nutshell, we uncover that the underlying mismatch between refined Gaussian conditional and hyperprior distributions may trigger the deteriorated joint probability approximation marginal distribution, leading to increased rate consumption. To remedy this issue, we introduce a novel distribution regularization to the existing R-D objective, which encourages learning a better joint probability approximation from theoretical perspective. Moreover, impose a Bayesian approximation of the proposed distribution regularization to circumvent any model modification in a plug-an and--aay manner. Experiments on in- and cross-domain tasks demonstrate that our proposed method surpasses other latent refinements approaches and contributes to SOTA TTA-IC models. Our contributions are summarized as follows.
effective alternative to existing
To the best of our knowledge, we are the first to reveal the degradation reasons of advanced LIC and corresponding latent refinement methods in marginalization approximation perspective. We propose Bayesian approximation-ende
distribution regularization to encourage learning a better joint probability approximation in soug-and-play manner.
The experimental results demonstrate our proposed method not only surpasses existing latent refinement counterparts but also can be integrated into SOTA TTA-IC methods.
RELATED WORKS
Various TTA approaches have been proposed to tackle distribution shifts at test time. For instance, Wang et al. (2020) introduced TENT that updates normalization statistics and optimizes channel-wir affine transformations. Niu al. (2022) adapted models based on active sample selection and Fisher
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regularize. Darestani et al. (2022) combined self-vuuervision during training with test-time training- Recent works extend TTA to handle more challenging scenarios in image restoration tasks, such as image supervision (Deng et al., 2023) and blind image quality assessment (Roy et al., 2023). Notably, Xu et al. (2023) is the first attempt to optimize the downscaled representation instead of the model.
To address the domain gap in TTA-IC tasks, various studies have explored strategies like fine-tuning the encoder during inference to refine latent variables without extra bit transmission. For example, Djelouah & Schroers (2019) introduced BLR, which uses gradient descent on latent representations guided by R-D cost, while Guo et al. (2020) extended BLR with a two-step approach optimizing both latent variables and side information. Beyond BLR, Yang et al. (2020) proposed HLR, which jointly refines latent variables and side information while eliminating discretization gaps. While HIR improves R-D performance for in-domain tasks, its effectiveness in cross-doman scenarios remains unclear. Decoder refinement is another popular approach, where decoder parameters are updated and transmitted during test-time (Lam et al., 2020; Zou et al., 2021), though often incurs higher bit costs (Kim et al., 2024). Recent SOTA TTA-IC methods inspired by parameter-efficient transfer learning (PETL) introduced decoder adaptors to improve reconstruction with lower bit consumption (Shen et al., 2023; Lv et al., 2023; Tsubota et al., 2023), as well as combining either BLR or HLR as the first step of their two-step framework. In a nutshell, compared with the emergence of various PETL-bbased decoder refinement variants, the latent refinement barely tailored to cross- domain scenarios, thereby exhibiting underexplored space.
METHODOLOGY
We first explain existing learned image compression from a perspective of marginalization approx- imation in the context of in-domain R-D cost-guided training. Then, analyze the practical marginalization approximation when applying the learned image codec to cross-domain image com- pression. We further render a simple solution achieve better R-D performance for cross-domain image compression based on the analysis of practical marginalization approximation.
Preliminary. Existing advanced LIC approaches mainly adopt hyperprior- based entropy frame- work (Ballé et al., 2018). As illustrated in Figure 2(a), given an image x, an analysis transform Ga can compute a latent representation y= (x)), which is then quantized and transmitted to the receiver in the context of an entropy model p(y). A synthesis transform Je can finally render the reconstruction. To enhance the bitrate, hyperprior is typically introduced to approximate the entropy model p(y) (marginal distribution) as a joint probability. which involves a hyper latent variable = ha (v) (a.k.a side information) calculated by a hyper analysis transform has and hyper synthesis he usually output the mean and scale of Gaussian distribution as the entropy model p(v)=) =) for the entropy coding of latent representation y. One line of LIC works aims to achieve better joint probability approximation of marginal distribution by various model elaborations (e.g. self-attention (Zou et al., 2022) and autoregressive (Minnen& & Singh, 2020)). Our work mainly analyzes the marginalization approximation from in- to cross-domain and proposes solution for better cross-domain performance.
3.1 IN-DOMAIN MARGINALIZATION APPROXIMATION
First, we introduce the marginalization approximation used by existing entropy models (Ballé et al., 2018; Cheng et al., 2020; Zou et al., 2022). Specifically, in the context of hyperprior-baxee entropy models, the following result holds,
Lemma 1 (Ballé et al. (2018)). Let plulz) and p(z) be accessed, a joint probability over y and z can be constructed to appreximate the true marginal probability over
p(u) f plu, 2)dz, plu.2) =p(l2)) p(2), s.t., ga(x), h ha(v),
(1)
Lemma implies that the compression of the raw image involves the compression of using the learned prior distribution p(z) and further compression of y using the learned conditional distribution p(yl=). Practically, we implement p(z) by a non- parammeeee,i, fully-factorized density model (a.k.a.. entropy bottleneck) (Ballé et al., 6). and implement p(y]z) by mean-scale Gaussian model, i.e.
p(z) = p(=lp)(((())))))))) p(y|=) p(y|2 On.)) = (N a), μ, = O,.),
(2)
where denotes the learnable parameters of the density model characterized by the univariate distribution, and h.).) denotes synthesis transform function of the hyperprior model parameterized
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by learnable . The learning of these parameterized probability models derives from training data, which means that the i..d assumption between training and testing data is implicitly needed. We therefore call such marginalization approximation as the in nomain one.
The widely used R-D objective can also be formulated by maximizing the probability of joint distribution p(y, 2) and the posterior distribution p(alyy 9g), ). i.e., Ledd
Pe-ppe)) (. logíp(ylz) p(=)] + X log p(=b))) E, (lgop(ulz)) - log)(z) + A|[= (v)ll
Rate
Distortion
Minimizing Eq. (3) can contribute to an optimal joint probability ply, z), which approximates the marginal distribution p(y). i.e., Lemma holds. Formally, the optimal probability representation is defined as follows.
Definition 1. (Optimal Probability Representation). By minimizing Eq. 3, there exist learnable parameters 0% and s" that achieve minimum rate cost for ply.z) and p(z). According to Shannon' entropy theorem, the learned encoding distributions p(y|z) and p(z) thus converge optimal probability representations p(y" (==)) and p(z") that exactly match the true underlying distributions:
p(y = ply 2, = p 1=²)
p(z) = p(= (4' p(=").
However, such marginalization approximation requires more information to be encoded (Townsend al., 2019). Theoretically, we can quantify extra encoding information of marginalization approxi- mation using the entropy (-) of distribution as the rate consumption, i.e.,
Proposition 1. Let and z be the latent and hyper latent variables, and these variables with the asterisk be their optimal representations. In the context of in-domain image compression, if an optimal joint probability approximation of true marginal distribution be achieved by minimizing Eq. (3), the extra rate consumption of marginalization approximation is
AH H Hy, 2) H(y") (E...-p(..))) log p(=lw)),
Proof. On the one hand, with joint probability and the Bayesian rule ply") = ply" p(w"l=")p(s")
we have
p(=1
H (y.2) z) = Epl- p(vl)) logp(=)), ( Ep[- Log logp(=") log (og"(""lu")) (5)
Then, we have
AH = Ep
- (- log [>)) + Bp1- log p(=) (- log p(= })] + - log "v""). (6)
On the other hand, as there exist optimal probability representations for p(vl2) and p(z) for in-domain image compression by minimizing Eq. (3), we have
|pp1- log p(ul=) (- logp(y" (==(v"==)))) [=)))]
0, stt p(ulz) = p(y (2") and p(="),
(7)
Ep[ logp(=) - (- logg(==)))))) = 0, s.t p(z) = p(=")
(8)
Thus,
AH =lg===>p log p(y [2*)] = Ey, log p(zy))
(9)
For in-domain image compression, Eq. (7), Eq. (8), and Eq. (9) hold, p(ylz), p(2), and p(=ly)
close to optimal probability representations p(y" p(²*), and p(=*ly") due to the assumption of the optimal joint probability approximation of true marginal distribution.
Lemma 1 and Prop. are mainly adaptive to in-domain image compression. By this cue, will discuss practical marginalization approximation and its impact on extra rate consumption in the context of cross-domain image compression.
3.2 GENERALIZE TO
MARGINALIZATION APPROXIMATION
When we apply the learned image codec to cross-domain scenarios, e.g. assuming the source-domain image x, and the target-domain image Ft, the i.i.d. assumption between source and target domains violates. We can derive that Lemma may not hold due to the following insight:
The practical joint probability p(y; 24) on cross-domain images will deteriorate to make it not equivalent to an optimal joint probubility plyi= =t) (that exactly match the true underlying distributions, in other words, p(yī ( and p(2?) are also optimal as Definition 1) that also corresponds to an optimal marginal approximation distribution p(yi), ) leading to deteriorated marginalization approximation, i.e.,
Proposition 2.
plui xt) # p(y,, zt).p()) # plue, 2x), P(y,, =x) p(va 2e, 0%.) -((ell")).
(10)
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Proof. It is straightforward induce the proof of this proposition based commonly used con- clusions and existing observations. Specifically, the distribution shifts, f.e. p(zz) * p p(zx), would result in the entropy bottleneck p(z+(p") parameterized by 8 (which replaces up in Eq. (2) to em- phasize its correlation with source-domain images x:s) is quite poor at specializing and encoding 2, for cross-domain images (Ulhaq & Bajić, 2024). Also, p(ye 2t, 0f ),, is inaccurate encode y due to source image-correlated 0% and unreliable z (Camposs et al., 2019). As the deterioration of independent distributions results the deterioration of the joint distribution, the joint probability p(ye, 2t) significantly deteriorates compared with the optimal one p(vi : 24) and further causes an unfavorable marginalization approximation p(y") &6 p(y:. 2t).
In light of Prop. 2, a cross-domain extension of Prop. 1 can be derived as follows.
Corollary 1. The extra rate consumption of cross-domate marginalization approximation AH will be larger than that of in-domain marginalization approximation, i.e., AH > AH", due to deteriorated joint probability.
Proof. Due to distribution shifts and suboptimal source-domain parameters (6 40), Eqs. (7) and (8) in Prop. can be further represented based on Prop. 2 as follows,
E-(- 11-logp(yulz 0%.)) (-
O, s.t. p(u,2,, 0%..) ) # plui =i) and ((ll))))))) # p(xf),
(11)
E (1-logp(==/<>)) (- po;)( > O, s.l. p(=ll))) # p(=i),
(12)
where p(ue /2, 0g Fp(( [={) holds due to source ioage-correlatdd 0% and unreliable 2t. p(2)((=")
# p(={) holds due to poor specialization of entropy bottleneck 40" Let p(uelz )= 2,) and pl p(=(1)))==p(zz) for brevity. We have AH-
pol-logp(uel:: (4) - (- (-logp(0f1=))) + Ep[- ((x) - (- log pog 112) > AH (13)
Since optimal probabilities always match the true underlying distributions from Prop. 2, p( 1 is equivalent to
in Eq. 9 for optimal posterior distributions. Thus, 13 holds and implies more rate consumption is potentially incurred in cross-domain scenarios.
Besides, distortion error is increased as the posterior probability p(z, Wy: 07 correlated with source- domain images v, is suboptimal for cross-domain images Tz. To enhance the R-D performance on cross-domain images Xt, latent refinement proposed to optimize latent variables while making model parameters unchanged. We summarize existing latent refinement methods below.
Assume that the initial latent representations y/ obtained by analysis transform as yf and the initial latent representations 2, obtained by hyperprior transform as =Q. The BLR scheme (refers to Figure 2) only updates the latent representation of with M > 1 steps, for each step m:
dLur Letr Re Be. log log [=,, (R.) +X(+ logpaall" (0,)),; (14) Tr..
st '/"
where the R-D cost excludes - log p(2) in Eq. (3) as the latent representation z is unchanged. For BLR, the gain of R-D performance is limited, as it is difffcultt to obtain updated yt" that can simultaneously lead to minimal rate cost and minimal distortion error in Eq. (14) due to fixed z/ Instead, the HLR in (Yang et al., 2020) not only conducts a joint amortization gap minimization but also eliminates the discretization gap for latent representations using the following optimization step,
vith DLhir
aChir
V
Dy. 21
21
Oz
Caie Bee-pe *. |- logooogpul ply"" 12"" 0%_) log p(z,"" Il0)) " + A(- log bud"
(15)
By simultaneously optimizing Ye and 21, Yang et al. (2020) reported significant R-D gains compared with BLR for in-domain adaptive compression. However, when we impose the HIR scheme to cross- domain adaptive compression, such gains are marginal as shown in Figure 1, where the reconstruction quality of the HLR achieves significant gains compared with BLR, but the HLR consumes more bits.
Discussion - we provide the analyses of degradation reasons of the vanilla HIR in the cross-domain scenario using Cor I and Prop. 2. First, compared with in-domain compression, the biggest trouble in the cross-domain scenario is that the static entropy bottleneck correlated with the distribution property of source-domain images cannot well model the true distribution of cross-domain latent variable as in Prop. 2. Second, although updating = to appropriate 2!"h that can render minimal negative log- log-likeldoood of p(="") (sp)) is possible, ensuring z!" appropriate Gaussian condition
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of yi" is questionable for minimal negative log-likelihood of p(y]" Porp. 2 and Cor. 1, more bit consumption by the HLR implies that the finally updated joint probability p(yl". 2, may be further away from the optimal (unknown) joint probability ply 27), and even the initial joint probability p(vl.=²) This can be directly reflected by more rate consumption of marginal approximation by extending the result of Cor. to specific update steps.
Finally, by recalling
AH AH > AH"
(16)
(s*)) To conclude, the underlying mismatch between p(=)""(>"" and p(yl"l | 02) (in the first two points) triggers deteriorated joint probability approximation of true marginal distribution (in the third point).
3.3 DISTRIBUTION REGULARIZATION VIA BAYESIAN APPROXIMATION
Thus, we are interested in an advanced latent refinement method, which not only can significantly improve the reconstruction quality but also can enjoy mild rate cost after latent refinement. Motivated by cross-domain joint probability approximation in Prop. and corresponding impact for rate consumption in Cor. we propose to introduce a simple yet efficient distribution regularization into the objective of vanilla HLR as follows, LDR =
Es Po [- log |=""" , .h..) logp(=""l<") + 18 log p(z" )(") + X(- log p(aalp )
(17)
where the third term is the distribution regularization, which serves as an empirical approximation of optimal posterior distribution p(z) ly²) that can be optimized directly using this objective. The proposed distribution regularization has two advantages as follows.
(i) If the balance coefficient 3 minimizing the entropy of probability estimates (i.e., the first two terms in Eq. (17)) is equivalent to minimizing the distribution gap between estimated probability p(ylz) or p(2) and (unknown) optimal probability p(y*|2*) or p(=") (MacKay, 2003). This coincides with the objectives of the first two terms in Eq. (13) of Cor. 1. Thus, LDR can be derived as follows,
LDR x AH + A(- logg((x)(lv)))
(18)
By recalling Eq. (13) in Cor. and Eq. (16), Eq. (18) implies that the minimiza- tion of extra rate consumption of marginalizatton approximation corresponds encouraging the deteriorated joint probability approximation to approach the initial and even optimal ones. In other words, AH
Iill Tees Toai
the lower bound of the first term of Eq. (18), i.e., the better joint probability ap- proximation, the closer to the lower bound. LD can potentially remedy the addi- tional rate of consumption of vanilla HLR.
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PEPD)
h
APO
APO)
UIQ
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0,39
9
9.
Ap(Hly)
Ap(+)y)
(ii)- For the optimization process in Eq. (15), there no explicit constraint to en- sure that =/" can well match its posterior distribution un- der the condition of y," leading to deteriorated joint probability. As practical implementation of the ideal posterior distribution, the proposed distribution regularization can eliminate such an issue.
(a) BLR
(b) HLR
(c) Ours
Figure 2: Architectures of different latent refinement TTA-IC methods.
U represents the quantization and entropy coding.
To implement the LDR, it is necessary to model the estimated posterior distribution p(a (("") (v()) of z!" given y/"", However, for existing hyperpiror models, the practical aT conditioned by yi i.., 2mh hy O,,,), is deterministic output without distribution property. Although it is feasible to approximate the 27" as the mean j.m' of a Gaussian distribution (as the assumption of posterior distribution) while constructing a variance branch from scratch (Vahdat & Kautz, 2020), the optimization of the variance model may be difficult when model parameters are fixed during latent refinement phase. To address these issues, utilize the dropout variational inference (Gal & Ghahramani, 2016) as a Bayesian approximation to existing hyperprior models for modeling the posterior distribution. Specifically, without introducing any network, the deterministic

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pre-trained model Oh. can be formulated probabilistic one p(Oh.), by treating the weights as distributions. Thus, the estimated posterior estimation can be represented as follows,
p( y(rh ~ p(z4" ly(") """"""(0"""
(19)
where the posterior distribution estimated and assumed as a fully factorized Gaussian distribution. The estimated mean fi("" and variance (dr"))² can be represented as follows,
P4
Σ ha ha ; 0h.))
Σ [ha (hh) T
1
1 Σ ha 0h 11" 0h, ~ (o(m)))
(20)
For the practical dropout variational inference (DVI) (Gal Ghahramani, 2016), the dropout strat- egy (Srivastava et al., 2014) is conducted to render approximate samples from the posterior dis- tribution, which equals to use a Bernoulli variational distribution qo(On, parameterized by to approximate the true model weight posterior p(O,.). ). By conducting T times of Monte Carlo (MC) sampling from qo (f,,,)) ). we can estimate the mean and the variance of the posterior distribution p(z]" (, The dropout probability related to the Bernoulli variational distribution is set to 0.5. The computation consumption of the DVT is mild, as a single instance can parallelly sample 7 masked weights by one inference in batch of repeated instances. Moreover, it is flexible to degenerate to vanilla deterministic networks when the dropout probability is 1. Finally, we compute the negative log-likelihoods of the estimated posterior distribution given the current latent variable z!"
Discussion - connection with Bit-back coding (BBC). Both BBC (Townsend et al., 2019; Ruan et al., 2021; Ho et al., 2019) and our method derive from the joint probability approximation of marginal distribution. However, BBC usually specializes in in-domain image compression to narrow the marginalization gap, i.e., transforming learned optimal joint probability to true marginal probability compression time by minimizing the following objective, LBBC
Ex, Pe |- log |z[", .(h.)) - p(2""((""'>)>>>>>>''"))""")
+ A(- pogp((>))""")) []]] (21)
For cross-domain scenarios, such optimal joint probability does not hold due to mismatched encoding distribution as discussed in Eqs. (13) and (16). Instead, we focus on refining deteriorated joint probability to potentially optimal (even initial) one by minimizing the extra rate consumption of marginal approximation, as discussed in Eq. (18). More differences refer to Appendix.
4 EXPERIMENTS
Datasets. By following previous literature (Lv et al., 2023; Tsubota et al., 2023; Shen et al., 2023), we collect six different datasets with four types of image styles to comprchensively evaluate the R-D performance of different approaches on cross-domain TTA-IC tasks, including natural image (Kodak), screen content image (SIQAD Yang et al. (2015), SCID (Ni et al., 2017), CCT (Min al., 201 7)), pixel-style gaming image (Lv et al. (2023)' self-collected), and painting image (DomainNet (Peng al., 2019)) datasets. The details of the used dataset can be found in the Appendix. Specifically, we consider the natural image dataset as in-domain evaluations, and others as cross-domain evaluations.
Implementation Details. We use CompressAl (Bégaint et al., 2020) to implement our proposed and baseline methods. Two widely-used backbone models are adopted, including the base hyperprior- based entropy model proposed by Ballé et al. (2018), namely Hyperprior, and the autoregressive context-based entropy model adopted by Cheng et al. (2020), namely AR-CM. Both are trained on natural images with various settings. We use the pre-trained models provided by CompressAL. For TTA-IC, we use the same value of hyperparameter À = |0.0018. 0.0035. 0.0067, 0.013, 0.025. 0.048] for latent refinement. The Adam optimizer is utilized to update the latent variables in a learning rate of 1 1x10-3 with 2000 iterations. T is empirically set to 20 for MC sampling. We discuss different implementations and hyperparameter settings (e.g., 3) of dropout variational inference in sec. 4.3.
Baselines. For latent refinement methods, we adopt two common baselines: (i) BLR (Campos et al., 2019), as formulated in Eq. (14). (ii) HLR (Yang et al., 2020), as formulated in Eq, (15). For HLR, we follow Yang et al. (2020)
a temperature annealing schedule with defaulted hyperparameters, where Tu = 0.3, co- = 0.001, and tn = 700.
Plug-and-play Validation. We also replace different latent refinement methods of existing SOTA TTA-IC approaches (Tsubota et al., 2023; Shen et al., 2023) with our proposed latent refinement counterpart in a plug-and-play manner. We follow their defaulted hyperparameter settings.
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07 64 0.
Figure 3: R-D curves on six datasets using different latent refinement methods. Two different base model architectures including AR-CM and Hyperprior are used.
Table I:
Comparison of our method with existing latent refinement approaches. The evaluation is measured in terms of average BD-rate savings (% 4) using the respective base models (without test-time adaptation). Smaller values indicate superior performance.
Method
In-domain dataset Kodak
Cress-ommii darasets
SIQAD
SCID
CCT
Pixel
DomainNer
Averagel
CM (Cheng
BLR
-6.55
-1875
-18.31
-25.33
-26.2
-18.78
-21 41
HLR
-12.73
-21.88
-15.11
-3.088
-3.76
-15.61
-21
Ours
.16.11
-2.63
-28.66
-3589
.27.00
Hypertior
BLR
.6.99
13.16
-16.71
-1.99
-15.12
-22.9
-15.4
HLR
-11.32
.13.9
-1.599
-13.22
-1889
-16.68
-1.87
-1482
-20.55
23355
-23.58
-22.2
-23.32
-20.78
Table 2: Integration of our method with the SOTA TTA-IC approaches. The evaluation is measured in terms of average BD-rate savings (% 1) using the respective base models (without test-time adaptation). Smaller values indicate superior performance. Matrix decomposition- based adaptor. Entropy efficient adapter. Stages and 2 denote the latent refinement and decoder adaptation phases.
In-dumaiin Kodkk
Croso-dommin
Method
Type
Averagei
SIQAD
SCTD
OCT
Pisrl
DomoinNer
HHLR
-3.50
-13.20
-12.34
-17.15
-11.08
-2.94
-1002
Stage
et al, (2023)
Oues
(b)
-14.32
-1462
-1898
-1301
-846
-1201
Aileptor!
-2.98
20.73
-34.0
-14.9
Stage
(b) A
(d)
-5.48
-12.70
-22.5
-36.0
-19.99
4.51
-1695
Stage
BLR
10.95
-3305
4.74
Shen et al, (2071)
Ouzs
(0
.5.45
-14.32
-1463
-18.95
-13.01
-5.6
-1201
Stage
(6) Mitpporr
(D) Adaptez2
(b)
.6.0
-20.3
21.03
-2423
14.99
-5.99
-15.26
4.1 COMPARISON WITH LATENT REFINEMENT METHODS
R-D Performance. We compare our proposed method with BLR and HLR. Note that the model without adaptation is also involved, namely No Adaptation. The peak signal-to-noise ratio (PSNR) and the data rate in bits per pixel (bpp) on different quality levels (refers to different À levels) are calculated to evaluate the R-D performance of different methods. Then, the R-D curves be plotted. The results can be found in Figure 3. We can observe that all latent refinement methods achieve comparable performance, which may be reasonable the Kodak dataset can be regarded as the
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045
0.12
2000
2004
2-40
Dub: sarag E-
044
ELA Crappeepp
3.325
042
0.79
6/1
150
(a)
(C)
(d)
Figure 4: Entropy curves of different probabilities with the iteration m, including (a) - p(y(")
(b)- log p(z)) (c) -logp(2;" ( (d) validation loss as calculated in Eq. (3).
PSNW/Ep
PSNDapp
Baseline Ground-Truti
Asia smog At scene
Standing ovation for
31/37)/373
Asia smog Al scene turnt land
Standing Duation robot musrciimm 25.00 296
30.700
Figure 7: R-D performance on medical images, where pathological breast cancer images used for cross-domain compression.
Medical Image
+ BLR
Asia smog Al scene
Standing Dialton for rbbot
Table 3: Ablation study of proposed distribution regularization D in terms of BD-rate (% 4) on SIQAD. DL denotes the number of the dropout layer. f; assume o = without Bayesian ap- proximation. 1; minimizing Eq. (21).
Quaerii eoeee axdioon robo music carr
33.8437
+ HLR
Asin smog At acene burst land
Standing ovalion for Standing ovatiin robot municians
Baseline (HLR)
0: As an anchor
Ours
DL
0.1
10
of bumnt
proposed CDR
-10.01
-10.02
-7.07
-10.03
-9.62
-5.5
3
-10.06
9961
4.95
Image
Deterministic Cig
-3.54
BBC (Bang et al., 2020)
0.15
Figure 6: Qualitative results
in-domain data without domain shifts. The gains of HLR and our proposed method mainly derive from the minimization of the discretization gap compared with BLR. For out-of-domain images such as SIQAD, SCID, CCT, Pixel, and DomainNet, our proposed method outperforms other approaches with clear margin regardless of different backbones.
Bjontegaard Delta bit-rate (BD-rate). We compute the BD-rate (Bjontegaard, 2001) for perfor- mance evaluation. A lower BD rate demonstrates better performance. The baseline model without adaptation is the anchor model for BD-rate calculation. The results are shown in Table 1. First, the HLR and our proposed method outperform the BLR with a clear margin for the in-domain Kodak dataset, which coincides with Yang et al. (2020)' obervations. However, on the cross-domain tasks, the HLR suffers from significant degradations compared with BLR, where HLR performs below the HLR on three out of five tasks, e.g., SCID and CCT. This is reasonable, as the static entropy bottleneck correlated with the distribution property of natural images cannot model the true distribution of latent variable well. Finally, our method can achieve obvious gains on all cross-domain tasks.
4.2 PLUGGING INTO SOTA TTA-IC
Moreover, we replace the existing latent refinement methods used by SOTA TTA-IC approaches with our proposed method. The full R-D curves are in the Appendix. We present the BD-rate Table 2. is obvious that proposed method can improve the performance of these TTA-IC methods all tasks. Moreover, such gains are roughly maintained at a similar level when adopted by the decoder adaptation. Due to the benefit of hybrid latent refinement, these gains are significantly enlarged for Shen et al. (2023). Thus, our proposed method enjoys plug-and-pp and-play property, benefiting existing TTA-IC methods.

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4.3 IN-DEPTH ANALYSES OF OUR PROPOSED METHOD
Why does the proposed distribution regulariza- tion improve the R-D performance? (i) From the- oretical analyses in section 3.2, the domain shifts and the underlying mismatch between p(=;" (0a) and p(y"") 24"h, )%. may trigger the deteriorated joint probability approximation ply:", 2²1 ) of true marginal distribution p(y?) for vanilla HLR. By em- pirical experiments in Figure 4, we observe that it is indeed difficult to jointly optimize latent variable
Table 4: Correlation between adaptation per- formance and adaptation time (using a single NVIDIA GeForee 3090 GPU) on SIQAD.
500
1000
2001
Runtime (Ang. Sec//Img.
BLR
HLR
0.07
0.07
14,86s
56.25
57.58
Oars
0.06
15,45
20.254
59.70s
BLR
-1001
19.23
20.57
21,04
21.27
BDDrrate(S
HLK
-0.92
-12.75
-14.73
-19.32
-23.68
Oum
-0.95
-24.3
-29.88
-3.15
-34.00
ye and side information 2; for vanilla HLR as shown in Figs. 4(a) and 4(b), where - log p(+"") converges first and degrades with the iterations while - log p(v,") converges well. In contrast, due to the lack of optimization for slide information Σt. there is limited convergence of - log (((") for the BLR while po²" is unchanged. Thus, as shown in Figure 4(c), there is an obvious degradation for vanilla HLR, i.e. A < AHM, reflecting more rate consumption of marginal approximation. This implies that the finally updated joint probability ,( 22M) ) is further away from the optimal joint probability p(v?, =f) ) and even the initial joint probability p(vl' =)). (ii) By the proposed distribution regularization (refers to Figure 4(c)), the degradation of joint probability approximation is alleviated well (blue curve) compared with vanilla HLR. Even, our method can achieve good convergence, i.e.,
AH A which is reasonable as we directly minimize AHl as shown in Eq. (18).
In short, the distribution regularization encourages the deteriorated joint probability approximation to approach the initial and even unknown optimal ones with a good convergence of rate consumption.
How about the effectiveness of Bayesian approximation for distribution regularization? First, we ablate the number of DVI layers. Intuitively, more dropout layers facilitate more accurate estimate of the posterior distribution. As shown in Table 3, there are marginal gains with improving dropout layers. Moreover, over-regularivation, Le., , = 10, will result in obviously negative effects. Thus, to avoid performance degradation, it is better to construct more dropout layers with more accurate probability representation and set a lower regularization coefficient (such as 0.1) to alleviate the werrregularized downside. Second, try a deterministic version of our distribution regularization by removing the Bayesian approximation, i.e., assuming o = 1 without the dropout layer. As we can see, deterministic LD performs significantly below that of the Bayesian approximation--asee based one, which is reasonable as the over- or below-estimated variance of the posterior distribution is inaccurate compared with the practical DVI in MC sampling. Finally, we observe negligible gains of the BBC on cross-domain image compression, which coincides with our discussions in section 3.3.
How about the adaptation efficiency compared with baseline methods? As illustrated in Table our proposed method has obvious performance gains in different adaptation steps. In contrast, the additional adaptation time taken by ours is mild in the same adaptation step which is reasonable as an instance can parallelly sample T masked weights by one inference in a batch of repeated instances.
What is the ability to generalize for larger distribution discrepancy? We explore whether our proposed method can be scalable to more challenging medical images, where the image distribution of medical images is quite different from natural images. The qualitative and quantitative results in Figures. 6 and show our method can improve R-D performance with better texture preservation.
5 LIMITATION AND CONCLUSION
Limitation. Due to the introduction of the DVI, our proposed method has a minor increase in adaptation cost compared with baseline methods, as illustrated in Table 4. developing more efficient solution and using more powerful GPU on the server can reduce the adaptation time in the future.
We approach an advanced latent refinement method by tailoring the vanilla HLR designed for in- domain inference improvement to cross-domain cases. Specifically, we provide a theoretical analysis to uncover that the underlying mismatch between refined Gaussian conditional and hyperprior distributions may trigger the deteriorated joint probability approximation of marginal distribution. Then, we introduce a Bayesian paaroximationneeddwed distribution regularization to encourage learning better joint probability approximation in a plug-and dlug-and-play and-play manner. Extensive experiments demonstrate that our proposed method can achieve promising performance.
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ACKNOWLEDGMENTS
This work was supported in part by the Hong Kong Innovation and Technology Commission (TTC) (InnoHK Project CIMDA), in part by the Institute of Digital Medicine of City University of Hong Kong (Project 9229503), in part by the Hong Kong Research Grants Council under Projects 21200522, 11200323 and 1200220,, in part by Chow Sang Sang Donation and Matching Fund (Project 9229 161), and in part by the Hong Kong Innovation and Technology Commission (Project GHP/044/21SZ).
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A FULL DERIVATIONS
Proposition 1. Let and a be the latent and hyper latent variables, and these variables with the asterisk be their optimal representations. optimal joint probability approximation of true marginal distribution can be achieved by minimizing Eq. (3), the extra rate consumption of marginalization approximation
the context of in-domain image compression, if an
AH - H(y") =.. p(g
(22)
Proof. On the one hand, with joint probability and the Bayesian rule p(y") = p(y )p(=')))p(=")
we have
p(=
Hl,, 2) = bol log p(u|=)] + E - 1og
(23)
H H - -ogg log + Ex-=p(=)) logp)>l"logp>"
pyr.:-=lorr.-))--- logg(=""ll (24)
Then, we have
AH
= R y, sy. -ot.:laapppelloooppoo By x xppp 1e (ptte log]l((''o"""""""""""
+ Ex-p(z) (ogpp=)-loop(=)) - E. E=-- p(g)(=)))]]
+ Ey = log p(= |v*)]
(25)
On the other hand, as there exist optimal probability representations for p(y|z) and p(2) for in-domain image compression by minimizing Eq. (3), we have
F py.a-ptuls)paa) -lg
log p(yl=)] Eny -p (xte)) log [=))]] (( (y" = 0,
s.t. p(ylz) = 12*) and p(z) p(="), ((=-lo(((=))))
(26)
E Expp(=) log [())p())]
s.l. p(=) = =p(=))
(27)
Thus,
AH pp ply [2*)] = -E Ey. log p(2lv)]
(28)
For in-domain image compression, Eq. (26), Eq. (27), and Eq. (28) hold, as p(vl / 2), p(2), and p(=lv) are close to optimal probability representations p(y" [y"1 2=1 1=*) p(2"), and p(=" ((="))") due to the assumption of the optimal joint probability approximation of true marginal distribution.
Corollary The extra rate consumption of cross-domait marginalization approximation AH will be larger than that of in-domain marginalization approximation, i.e., AH > AH", due to deteriorated joint probubility.
Proof. Due to distribution shifts and suboptimal source-domain parameters (0% (4") Eqs. (26) and (27) in Prop. can be further represented based on Prop. 2 as follows,
E po )r(=)) log p(vel ze, 0%.) =o0 (o puy [z]))] 0,
s.t. p(yal 24, 0%_ # p(u (2) and p(zipp)) # p(2;);
(29)
E Bec log p(=)((")) R log (ex)()))] O, s.t. p(a (ee) * p(z)), (30)
where p(pclze. 0% #p(( (2²) holds due source image-correlane 0% and unreliable 21 ((2))p)) # p(zt) holds due to poor specialization of entropy bottleneck sp" Let p(yc|ze, = and pl p(zel=")=p(( for brevity. We have
AH
E py log p(v)22))) log
Ey "pp"""" (og.
+ (o-p(s)-loggrll - E2 [ [(- [[- logn(=")))
Ey lop p( |u)) > AH*
(31)
Since optimal probabilities always match the true underlying distributions from Prop. 2, p(27 lu?) is equivalent to p(="l=") in Eq. 28 for optimal posterior distributions. Thus, Eq.31 holds and implies more rate consumption is potentially incurred in cross-domain scenarios.
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Here, we further explain why the Eqs. (29) and (30) hold as follows.
According to Shannon's entropy theorem, the optimal probability distribution p(vil2?) achieves the theoretical minimum coding length, which equals the entropy of the true data distribution. Any non-optimal coding distribution (c.g., p(ya|2, 24, _) must have higher expected code length than the theoretical minimum. The entropy consumption of coding distribution reflect this to make the following inequality strictly hold:
E pyssmptoulls)ptaal Lorpolloogppuue p(uel, 0%, 0%, 2 y p (()(())) log puz 12]),
(32)
where the equality holds only if p = p(yn zr; 0 and p(= (pf) = p(=t).
Since we have assumed that p(vll= F.. 0 ) * ply (zt) holds due to source image-correlated 0% and unreliable p(zelp" kx*) * p(=() on cross-domain scenarios. Thus,
By ((.emp(w((c))((a)) log p(u (2, (%, Bg p(v]
(33)
holds. Thus, the following inequality can be naturally derived,
yy.emp(onss)))aa)) log p(u1 0% Eny appl beoo)r))) log. p(bf(2)) 0.
(34)
To summarize, it is clear Eq. (29) holds. The same applies to Eq. (30).
B DETAILS OF ADOPTED DATASETS
By following previous literature (Lv et al., 2023; Tsubota et al., 2023; Shen et al., 2023), we collect six different datasets with four types of image styles to comprehensively evaluate the R-D performance of different approaches on cross-domain TTA-IC tasks, including natural image (Kodak)) )) screen content image (SIQAD Yang al. (2015), SCID (Ni al., 2017), CCT (Min al., 2017)), pixel-style gaming image (Ly et al. (2023)' self-collected), and painting image (DomainNet (Peng et al., 2019)) datasets. The details of the used dataset be found in the Table Specifically, we consider the natural image dataset as in-domain evaluations, and others as cross cross- domain evaluations.
C COMPARISON WITH SOTA TTA-IC METHODS
We provide the full R-D curves in Figure when we compared our proposed method with SOTA TTA-IC methods. Note that our proposed method based on (Tsubota et al.. 2023) with our proposed distribution regularization. We can observe that all latent refinement methods achieve comparable performance, which may reasonable as the Kodak dataset can be regarded as the in-domain data without domain shifts. The gains of HLR and our proposed method mainly derive from the minimization of the discretization gap compared with BLR. For out-of-domain images such as proposed method outperforms other approaches
SIQAD, SCID, CCT, Pixel, and DomainNet, with clear margin regardless of different backbones. Especially, a better R-D performance on the low-bit conditions can be observed for AR-CM-bas CCsed realizations the Pixel dataset.
D CONNECTION WITH BIT-BACK CODING
1) In-domain Cross-domain: Both BBC (Townsend et al., 2019; Ruan et al., 2021; Ho et al., 2019) and our proposed method derive from the joint probability approximation of marginal distribution. However, BBC usually specializes in in-domain image compression to narrow the marginalization gap, i.e., transforming learned optimal joint probability to true marginal probability at compression time. Instead, in the context of cross-domain image compression, such optimal joint probability does
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SCDD
Out
Out
Out
32
1300
30
34
0.1
24
03
04
2.5
(b)
CCT
Foel
Demain
Cura
0ui
Dull
0.9
92
05
04
31
3.
85
1.
38
(d)
(f)
Figure 8: R-D curves of test-time adaptation of different latent refinement approaches on six datasets using different latent refinement methods. Two different base model architectures including AR-CM and Hyperprior are used.
Table 5: The datasets for evaluation. The symbols and denote in- and cross-domain datasets, respectively.
Dataset
Description
Num.
Avg. Resolution
Kodak
Natural
576x
SIQADÉ
Screen Content
24
685x739
SCID
Screen Content
720x 1080
CCTI
Screen Content
24
915x 1627
PixcIt
Gaming
25
746x850
DomainNet
Painting
25
492x640
hold due mismatched encoding distribution (e.g., entropy bottleneck), discussed in Eqs. (13) and (16).
2) Minimizing posterior probability v.s. Maximizing posterior probability: BBC minimizes the second term of the first line of Eq. (22) as the true entropy of the marginal distribution, i.e.,
LBBC log p(y""' 24 0% log p(="""(²"") (- log p(2;"' ))r X(- log p( lv.")) (35)
where the third term corresponds to minimizing the posterior probability based on the in-domain optimal joint probability assumption of BBC. In contrast, proposed method focuses refining deteriorated joint probability to optimal one by minimizing the extra rate consumption of marginal approximation, discussed in Eq. (18).
3) Performance and implementation differences: Our experiments Table 3 observe negligible gains of BBC cross-domain image compression. More importantly, in order acquire distribution prop- erty, our proposed distribution regularization relies on more flexible variational Bayesian inference rather than introducing additional networks like BBC (Yang al., 2020).
E MORE PLUG-AND-PLAY VALIDATIONS
To further the effectiveness of this plug-and ments on more baseline methods. The results can be observed in Table As we can see, our proposed method can improve the performance of these TTA-IC methods in plug-and-play manner, which is
aad--add-pay -pay advantage, have conducted additional experi-
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Table Integration of our method with the SOTA TTA-IC approaches. The evaluation is measured in terms of average BD-rate savings (% 4) using the respective base models (without test-time adaptation). Smaller values indicate superior performance. & Matrix decomposition- based adap- tor. Entropy efficient adapter. dynamic low-rank adaptor. H: Only latent representation y is Optimized.Stages and 2 denote the latent refinement and decoder adaptation phases.
Mathud
Type
In-dimain Kodee
SIQAD
SCID
Crus-drmmin datinsts
CCT
Poudl
DarasiNNe
Averagel
WACNN (Zon stall, 2007)
0
0
Tscbota or al, (3001)
Stage
Ours
(
(A)
-5.45
+13.20
-14.22
-12.4
-1462
+17.15
+1895
-13.1
-5.66
-10.02
-12.01
Stage
(8) Adaptoo)
(b) +Adapoor!!
(
-5.48
-32.70
30.94
-22.6
-26.0
-19.37
4.51
-16.55
WACNN
Stage
BLR
(6)
183
-10.8
-9.6
-13.05
-3.8
4.83
4.76
Siten al. (2023)
-Ore
-.45
-14.2
-1462
-1898
-1301
-8.6
-12.11
Stage
(G) Alleptoe2
-1..1
-18.4
-22.17
-10.44
03 Adaioor"
4.01
-20.3
-1102
-24.3
-34.9
-5.99
-15.66
2023)
Lt (2033)
Stage
+II gil
0
-2:35
-11.00
-10.56
-18.5
-14.00
-3.87
-1.54
-9.6
Stage
(3 Allgpoora Adhpior3
(
-345
-1678
-2.05
22.59
18.0
3.25
-1357
-5.9
-18.99
-19.99
-25.88
-21122
4.21
-16.66
Table 7: Comparison of method with existing latent refinement approaches. The evaluation is measured in terms of average BD-rate savings (% 4) using the base models (without test-time adaptation). Smaller values indicate superior performance.
In-domain datasets
Method
SetS
Sct14
BSD100
Urban100
Average|
AR-CM (Cheng et al., 2020)
BLR
-13.27
-15.32
-5.92
-16.25
-12.69
HILR
-15.01
-1.29
-6.61
-21.42
-15.33
Ours
-21.47
-24.88
-11.23
-26.68
-21.06
reasonable as our proposed method can alleviate deteriorated joint probability approximation, leading to better bitrate budget in a two-stage TTA-IC framework.
F MORE VALIDATIONS ON POPULAR DATASETS
We utilize Set5, Set14, BSD100, and Urban benchmark datasets for further evaluations, where Set 5 and Set14 include 5 and 14 images, respectively, both BSD100 and Urban 100 have 100 high- resolution images. As illustrated in Table 7, our proposed method can surpass baseline methods in terms of the BD-rate. Such performance gains show a good generalization ability of our proposed method on diverse datasets, exhibiting its usability in the real world.
G ADAPTATION COST
We compute the adaptation cost of different methods in terms of GPU memory usage and GFL.Ps. As illustrated in Table 8, our proposed method has minor increase in adaptation cost, which reasonable as the dropout variational inference is introduced during optimization for better R-D performance.
Notably, although our proposed method increases the adaptation cost, such latent refinement stage is usually conducted on the encoding side of the server, which means that there is no effect on the real-time decoding edge devices, making proposed method applicable real-world scenarios. Moreover, our proposed method has no model modification to obtain probability representation based on dropout variational inference. This simplification also enhances its usability.
17

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Published as a conference paper at ICLR 2025
Table 8: Average adaptation of different methods Kodak and CCT datasets.
BLR 0.72/3.67 98.98/525.8
HLR
Ours 0.79/3.98 110.58/587.47
GPU Memory (GB) GFLOPs
0.74/3.92 101.23/563.24
H MORE DISCUSSION ABOUT LEARNING MATCHED POSTERIOR DISTRIBUTION
Here, we discuss potential method, namely direct inferring, which conducts directly inferring through a hyper analysis transform address the mismatched problem. Although it is feasible to conduct a direct inferring through a hyper analysis transform in out-of-distribution (OOD) scenarios, we observe that the result is significantly behind the proposed distribution regularization. For example, by using the ablation study in Table 3, the result achieved by direct inferring is -4.84, which seems to be similar to the setting with DL=3 and 3 = 10 (-4.95) and the deterministic distribution regularization (-3.54).
We conjecture that the direct inferring may be a deterministic over-regularization method. Since the optimization of latent representation directly affected by the reconstruction loss and rate cost of latent representation, the usage of direct inferring result still cannot be ensured to lead to minimal rate cost of side information and appropriate conditional distribution p(y]=). Moreover, the optimization of side information uses the direct inferring result may result in potential optimization difficulty, due continually variational z obtained by hyper analysis transform.
Therefore, seems that an appropriate regularization like the proposed distribution regularization is necessary to alleviate the optimization difficulty between latent representation and side information, leading to more matched joint probability.
18


Reviews:
[{"review_id": "12CtWUSUev", "reviewer": "Reviewer_5FRR", "summary": "This paper addresses a latent refinement techqniue in test-time adaptation methods for image compression.\nEspecially, the paper focuses on methods that use a hyper-prior model, where the image is encoded into a latent variable $y$ and a hyper-latent variable $z$ in a hierarchical way.\nThe paper analyzes the issue of recent latent refinement methods not generalizing well in cross-domain settings, arguing the cause to a mismatch between the hyper-prior $p(z)$ and the gaussian conditional $p(y|z)$ used in refinement methods.\nBased on this conjecture, the paper proposes a distribution regularization term that uses dropout variational inference, claiming it can improve probability approximation. \nExperimental results show that the proposed method continuously improves latent refinement and demonstrates effectiveness even when integrated into state-of-the-art methods that jointly utilize latent and decoder refinement.", "questions": "* Please address the weakness above.\n* The writing is a bit difficult to understand, especially in the discussion parts of both the methodology and experiment sections", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The proposed method can be applied without altering any model parameters, making optimization easy and immune to catastrophic forgetting.\n2. The paper tries providing detailed discussions on the proposed method, both in a principled and empirical way.\n3. The experimental results consistently demonstrate the effectiveness of the proposed method in cross-domain image compression across different settings: (1) with only latent refinement, (2) integrated with state-of-the-art TTA-IC methods, which further include decoder adaptation, and (3) on medical images.", "weaknesses": "Most importantly, as i understand, the theoretical analysis and subsequent derivation of the proposed method, which are claimed as major contribution in this paper, contain many unclear and misleading aspects. \n1. (L173, 193) The definition of \"optimal probability representation\" is not clearly defined, throughout the paper.\n2. The entropy function, i.e., $H(X)=-\\mathbb{E}_{x\\sim p(x)}[\\log p(x)]$, is interpreted in the form of log probability, and it seems that the expectation term is not considered in the analysis and derivation.\n3. In the Proposition 1, assuming an optimal joint probability approximation of the true marginal distribution $p(y^*)=\\int p(y,z)dz$, the proposition simply interprets $H(Y,Z) - H(Y^*) = H(Z|Y) + H(Y) - H(Y^*) = H(Z|Y)$ when $H(Y) = H(Y^*)$. However, the assumption $p(y^*)=\\int p(y,z)dz$ may not induce equations (7,8), which can be misleading.\n4. In the proof of Proposition 2, the authors argue that the entropy bottleneck $p(z_t)$ and the posterior $p(y_t|z_t)$ are learned from the source dataset (L212, source image correlated) and may not work effectively. While this makes sense, it is not a theoretical analysis, in my point of view.\n5. For Corollary 1, it is unclear whether $z^*$ and $y^*$ in equation 13 correspond to in-domain or cross-domain cases. In the cross-domain case with $y^*_t$ and $z_t^*$ as in Proposition 2, equation 13 is as follows: $\\triangle H = H(Y, Z) - H(Y^*_t) > H(Z^*|Y^*)$, which holds according to equations (11, 12) only if $H(Y^*_t)=H(Y^*)$ and $H(Z^*_t|Y^*_t)=H(Z^*|Y^*)$. However, there is no guarantee for the condition.\n6. Importantly, in Section 3.3, the paper proposes distribution regularization using equation 13, interpreting the first three terms in equation 17 as corresponding to $H(Y|Z), H(Z), H(Z^*|Y^*)$ (the three terms at the front) in equation 13. However, it is unclear how $H(Z^*|Y^*)=\\mathbb{E}_{y^*,z^*\\sim p(y^*,z^*)}[p(z^*|y^*)]$ leads to $-\\log p(z^m_t|y^m_t)$ in the optimization objective. Note that $H(Z^*|Y^*)$ includes expectation from $p(y^*,z^*)$.\n\nIn conclusion, the theoretical analysis in this paper seems to have many over claimed or misclaimed elements.", "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": "PhWsnOKKsX", "reviewer": "Reviewer_GYDX", "summary": "The authors propose a new latent refinement method to enhance the rate-distortion (R-D) performance for test-time adaptation image compression. They theoretically analyze that existing methods require higher rate consumption to estimate the marginal probability of the latent variables in cross-domain scenarios. To address this, the authors incorporate a new distribution regularization term into the R-D objective, promoting a better approximation of the latent representation. Experimental results across six cross-domain datasets demonstrate the effectiveness of the proposed approach, showing its superiority over existing methods. Please see my detailed comments below.", "questions": "1. In Table 3, the results indicate that increasing the number of dropout layers can negatively impact image compression performance. Please add further discussion on how to effectively approximate the distribution regularization to avoid such performance degradation.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The authors offer a theoretical analysis highlighting limitations of existing methods, which informs their introduction of a distribution regularization technique. This technique effectively adapts the compression model to test data with domain shifts.\n2. They provide detailed analysis results on the entropy curves of different probabilities about different methods. The results in Figure 4 demonstrate that the proposed method can lead to a good convergence of rate consumption and better R-D performance.\n3. Experimental results demonstrate that the proposed method can be easily integrated with existing methods to enhance R-D performance across cross-domain image datasets.", "weaknesses": "1. The test datasets used in the experiments contain a limited number of images, which may affect the robustness of the results. To strengthen the findings, please consider providing additional results on larger-scale datasets.\n2. It would be helpful to include a preliminary section explaining the image compression process before the theoretical analysis of existing methods. This section could clarify the steps involved in obtaining the latent variable y and hyper-latent variable z (side information), enhancing readers' understanding of the compression model and the subsequent theoretical analysis.\n3. In addition to the adaptation cost of different methods, it is better to provide the compression cost of the proposed methods, which would offer a more comprehensive view of its performance. \n4. The authors do not provide a limitation analysis of the proposed methods. For instance, the long adaptation time may limit the proposed methods to be used in real-time applications.\n5. More discussion on recent test-time adaptation works would be beneficial to place this study in context. Relevant works to consider include, but are not limited to, [A-E].\n\n[A] Test-Time Training Can Close the Natural Distribution Shift Performance Gap in Deep Learning Based Compressed Sensing, ICML 2022.\n\n[B] Efficient Test-Time Model Adaptation without Forgetting, ICML 2022.\n\n[C] Tent: Fully Test-Time Adaptation by Entropy Minimization, ICLR 2021.\n\n[D] Efficient Test-Time Adaptation for Super-Resolution with Second-Order Degradation and Reconstruction, NeurIPS 2023.\n\n[E] Test Time Adaptation for Blind Image Quality Assessment, ICCV 2023.", "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": "7XSerAgzrM", "reviewer": "Reviewer_7uht", "summary": "Current Test-Time Adaptation Image Compression (TTA-IC) methods improve rate-distortion (R-D) performance on cross-domain tasks by refining both latent variables and decoders in a two-step process. However, latent refinement has not been specifically optimized for cross-domain scenarios. The authors conduct theoretical analyses to identify a mismatch between refined Gaussian conditionals and hyperprior distributions in the vanilla Hybrid Latent Refinement (HLR), leading to poorer joint probability approximations and higher rate consumption. To address this issue, they introduce a Bayesian approximation-based distribution regularization technique that enhances joint probability modeling in a plug-and-play manner. Extensive experiments across six in-domain and cross-domain datasets demonstrate that the proposed method outperforms existing latent refinement approaches in R-D performance.", "questions": "Will the authors make their code public to assist the community upon acceptance?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper effectively identifies the limitations of existing Hybrid Latent Refinement (HLR) methods in cross-domain image compression, particularly highlighting the mismatch between refined Gaussian conditionals and hyperprior distributions.\n\n2. It provides a thorough theoretical analysis using marginalization approximation, establishing a solid foundation for the proposed improvements.\n\n3. The introduction of a Bayesian approximation-based distribution regularization technique successfully addresses the identified mismatch, enhancing joint probability approximation. \n\n4. The method shows improvements in rate-distortion (R-D) performance compared to other latent refinement approaches, validating its effectiveness.", "weaknesses": "1. The paper does not include a comparison of GPU memory usage and GFLOPs, which are crucial metrics for evaluating the complexity of various methods. Including these metrics would illustrate the trade-off between performance gains and computational complexity more clearly. It would be beneficial to present memory usage across different datasets, especially since DVI is introduced during optimization, making it reasonable to assess the additional complexity incurred.\n\n2. In [1], the posterior distribution of $\\hat{z}$ is assumed to be a uniform distribution. It is unclear whether directly inferring $\\hat{z}$ through a hyper analysis transform in out-of-distribution (OOD) scenarios is the better choice? Because this approach not only avoids incurring a larger $\\Delta H$ but also naturally matches its posterior distribution during pre-training. \n\n3. The paper introduces a regularization term based on HLR in TTA-IC. While this contribution is valuable, its impact on the community appears moderate.\n\n[1] Ballé, J., Minnen, D., Singh, S., Hwang, S. J., & Johnston, N. (2018). Variational image compression with a scale hyperprior. arXiv preprint arXiv:1802.01436.", "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": "g14omqJIlO", "reviewer": "Reviewer_ShHq", "summary": "This article investigates test-time adaption image compression method on cross-domain compression tasks from the perspective of latent refinement, by extending hybrid latent refinement and introducing a simple Bayesian approximation-endowed distribution regularization. The authors provide theoretical analysis and experimental results to demonstrate the performance of this method. My detailed comments are as follows.", "questions": "The compared methods are too old. More popular image reconstruction/compression datasets should be considered for comparisons.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1.\tThe authors propose their method from the perspective of latent refinement, which hasn’t been fully investigated. Compared with HLR, the method proposed by the author greatly improves the performance cross-domain compression tasks.\n2.\tThe authors provide sufficient theoretical analysis on the method proposed in the article.", "weaknesses": "1.\tThe experimental results of this plug-and-play method are relatively weak, which cannot illustrate the generalization ability of this method on other TTA-IC approaches. It would be better to apply this method to more baseline approaches.\n2.\tIt would be better to report the image compression results on top of popular image reconstruction benchmarks, such as Set5, Set14, BSD100, Urban100, etc.\n3.\tThis paper is hard to follow and the writing could be further improved. It would be better to explicitly highlight the key contributions of this paper.\n4.\tThe compared methods BLR and HLR are too old since both are published before 2021. It would be better to compare with more recent work, such as [A].\n[A] Downscaled representation matters: Improving image rescaling with collaborative downscaled images, ICCV 2023.", "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": "ZW1bKqOSJ7", "reviewer": "Reviewer_5d5v", "summary": "The author propose a novel test-time adaptation framework for image compression through directly refine the latent variables without altering any model parameters.", "questions": "None", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This research topic of this work seems a bit a minority (image compression + test-time adaptation) and contains a lot of background knowledge. After hard attempts, I think I still can't give an accurate evaluation. It is recommended that AC find another highly relevant reviewer in order to make a more comprehensive judgment.", "weaknesses": "None", "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)