Optimal Neural Compressors for the Rate-Distortion-Perception Tradeoff
Authors: Eric Lei
Hamed Hassani
Shirin Saeedi Bidokhti
Venue: NeurIPS 2025 spotlight ()
Keywords: neural compression
rate-distortion-perception
lattice quantization
dithering
Primary area: deep_learning

Abstract: Recent efforts in neural compression have focused on the rate-distortion-perception (RDP) tradeoff, where the perception constraint ensures the source and reconstruction distributions are close in terms of a statistical divergence. Theoretical work on RDP describes properties of RDP-optimal compressors without providing constructive and low complexity solutions. While classical rate-distortion theory shows that optimal compressors should efficiently pack space, RDP theory additionally shows that infinite randomness shared between the encoder and decoder may be necessary for RDP optimality. In this paper, we propose neural compressors that are low complexity and benefit from high packing efficiency through lattice coding and shared randomness through shared dithering over the lattice cells. For two important settings, namely infinite shared and zero shared randomness, we analyze the RDP tradeoff achieved by our proposed neural compressors and show optimality in both cases. Experimentally, we investigate the roles that these two components of our design, lattice coding and randomness, play in the performance of neural compressors on synthetic and real-world data. We observe that performance improves with more shared randomness and better lattice packing.

Optimal Neural Compressors for the Rate-Distortion-Perrettooo Tradeoff
Eric Leit", Hamed Hassani', Shirin Saeedi Bidokhti
UPMMrrgaChhase Global Technology Applied Research, 'University of Pennsylvania eric. Fricclei@jpmchase com, (hassani, saeedi /asaas.uennnn www.n..
Abstract
Recent efforts in neural compression have focused on the rate-distortion o (RDP) tradeoff, where the perception constraint ensures the source and reconstruc- tion distributions are close terms of statistical divergence. Theoretical work on RDP describes properties of optimml compressors without providing construc- tive and low complexity solutions. While classical rate- distorisoo theory shows that optimal compressors should efficiently pack space, RDP theory additionally shows that infinite randomness shared between the encoder and decoder may necessary for RDP optimality. In this paper, we propose neural compressors that are low complexity and benefit from high packing efficiency through lattice coding and shared randomness through shared dithering over the lattice cells. For two impor- tant settings, namely infinite shared and zero shared randomness, we analyze the RDP tradeoff achieved by proposed neural compressors and show optimality in both cases. Experimentally, we investigate the roles that these two components our design, lattice coding and randomness, play in the performance of neural com- pressors on synthetic and real-world data. We observe that performance improves with more shared randomness and better lattice packing.
Introduction
Neural compressors learned from large-scale datasets have achieved state-of-the-art performance in terms of the rate-distortion tradeoff (Ballé et al., 2020; Yang et al., 2023), especially when trained to produce reconstructions that align well with human perception (Mentzer et al., 2020; Tschannen al., 2018: 8; Agustsson et al., 2019; Muckley et al., 2023). To achieve this, an additional perception loss term is used, typically defined a statistical divergence between the reconstruction and source distributions. As such, recent focus has shifted to the rate-distortion, uut-ddstorrionn-peeeptio (RDP) framework. where compressors explore triple tradeoff between rate, distortion and perception (Blau and Michaeli, 2019). The RDP function source X ~ Px. defined as
R(D,P) P) min PRix
I(X;X)
(1)
Exx A S Px)
where A is distortion function, has emerged describe this fundamental tradeoff (Matsumoto, 2018; Blau and Michaeli, 2019; Li et al., 2011). Several RDP coding theorems have recently been proven (Theis and Wagner, 2021; Wagner, 2022; Chen et al., 2022), providing operational meaning to (1) as fundamental limit of lossy compression for the RDP tradeoff
In this paper, we investigate how neural compressors may achieve RDP optimality, and what compo- nents are necessary for good RDP performance. The RDP coding theorems, while non-constructive,
Prepared prior to employment JPMorganChase.
SSpecifically, (1) is achievable by sequence source codes, and no source code can do better than (1).
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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3
d Unitli/AAll
Ua
Qx
9x
9a
QA
9a
Q^
9x
9a
Q
9
Pg
Peiu
(a) Lattice transform cod- (b) Private-dither LTC (c) Shared-dither LTC (SD- (d) Quantized-shaardd-ittee ing (LTC); deterministic. (PD-LTC); R. = 0. LTC); R. 00.
LTC (QSD-LTC) finite R..
Lattice transform coding (LTC) with R. bits of (shared) randomness using dithering; u ~ Uni and u ~ (V((g) are continuous, and ú ~ (n/ is discrete, where A, Ay f are nested lattices. LTC/PD-LTC entropy-code Qy(y) with likelihoods Pý- SD-LTC and QSD-LTC entropy-code (x uy-yyu u) and Qa(y - ú) with likelihoods Peju and Pelé. respectively.
Figure 1:
shed light on properties of opP-opt compressors. In contrast to the classical rate-distortion func- tion Ri D) = R(D, oo), which is asymptotically achievable by fully deterministic codes, achieving the RDP function may require not only stochastic encoding/decoding but also infinite randomness shared between the encoder and decoder. Devising neural compressors that may achieve RDP optimality at low complexity is an important step in advancing the theory and practice of neural compression. Moreover, infinite shared randomness may not always be available. In settings where shared randomness is limited, or anavailable, we are still interested in the best possible schemes.
Infinite shared randomness has previously proved successful in RDP-oriented compressors such as Theis al. (2022) that are based on reverse channel coding (RCC) (Theis and Ahmed, 2022: Li and El Gamal, 2018; Cuff, 2013). RCC enables communication of a sample from prescribed distribution (e.g., one that good for RDP) under a limited rate constraint. The performance of RCC provably near-optimal, but this comes at the cost of high complexity. Moreover, RCC heavily relies on infinite amount of randomness and does not allow for limited or zero randomness. We seek to develop methods with much lower complexity and allow for zero, limited, or infinite amount of randomness.
In the classical rate-distortion framework, recent work has investigated whether neural compressors are optimal, where vector quantization (VQ) (Gersho and Gray, 2012) is known to be optimal, but suffers from high complexity. In Lei al. (2025), it was shown that lattice transform coding (LTC), which uses lattice quantization (LQ) in the latent space, can achieve performance close to VQ at significantly lower complexity: see also Zhang and Wu (2023); Kudo et al. (2023). VQ and LTC provide near-optimal RD performance due to high space-packing efficiency. However, it is not clear first glance whether VQ-like coding good for the RDP setting, where randomized reconstructions are required to satisfy the perception constraint (Tschannen et al., 2018 8). It is further unclear how randomness should be incorporated with quantization a way that RDP-optimal.
Dithering is a common method of randomizing a quantizer. Classically, a shared random dither can help quantization noise admit desired statistical properties, and has applications such as universal quantization (Ziv, 1985), and practical training methods for neural compression (Ballé et al., 2020). For the RDP setting, using dithering to introduce randomness has been investigated recently; Theis and Agustsson (2021) show on a simple source how dithered scalar quantization (SQ) benefits RDP.
In this paper, introduce randomness, shared and private, into neural compressors via architectures that build on LTC (Fig. 1), and investigate the roles that randomness and quantization play in how neural compressors perform for the RDP tradeoff, while managing complexity. We study LTC where randomness takes the form of a random lattice dither vector. When the randomness is private (i.e., none shared), the dither is only added at the decoder (PD-LTC: Fig. lb). When the randomness is shared, the dither is subtracted at the encoder, and added back at the decoder (SD-LTC and QSD-LTC; Figs. 1c, 1 respectively). This work unifies transform coding, lattice coding, and randomized coding, identifying the roles each play when used together for RDP. Our contributions are the following".
1. We propose LTC with infinite private dither (PD-LTC) respectively, and describe the benefits of the former in Sec. 3. We then propose a discrete dithering scheme defined via nested lattices that allows for fmite randomness to be shared between the encoder and decoder. This scheme, QSD-LTC, interpolates between PD- and SD-LTC, enabling control over the rate of shared randomness.
shared randomness, using shared dither (SD-LTC) or
2. We theoretically analyze and show optimality of SD-LTC and PD-LTC on the Gaussian source with squared 2-Wasserstein perception and asymptotic blocklength in Sec. 4. For the former
"ood be found at https://github htt.u//.o.ommmeee////eee/e

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(where infinite shared randommess is available), we use the sphere-like behavior of lattice cells and the AWGN-like behavior of dithered LQ to show that SD-LTC achieves the RDP function R(D, P). For the latter (where no shared randomness is available), we use lattice Gaussian PD-LTC achieves R(D/2, x)), which coincides with the fundamental RDP limit under no shared randomness.
coding methods to show that under a perception constraint of P
3. We empirically study PD-LTC, SD-LTC, and QSD-LTC performance on synthetic and real- world sources in Sec. 5. We verify our theory and show that RDP performance improves with increased shared randomness and better lattice packing efficiency.
2 Background and Related Work
Neural Compression for RDP. Most neural compressor designs that account for perception are derived from the nonlinear transform coding (NTC) setup (Ballé et al., 2020). These models are parameterized by analysis transform 9a, synthesis transform 9s, and entropy model Pe; see Fig. la. To compress a source x, the encoder computes the latent y ga(z), which gets scalar quantized via rounding. The codeword or quantized latent ÿ = Qx (y) is then entropy coded using an entropy model Pg- The decoder provides the reconstruction á Is (t). The model is trained end-to-end via
min E[ - (og + À E| (x, })] + )20(Pa: Pa),
(2)
and X1 A2 2 0 control the RDP tradeoff. methods (Tschannen et al., 2018; Agustsson et al., 2019; Mentzer et al., 2020; Muckley et al., 2023; He et al.. 2022; Zhang et al., 2021) optimize (2), primarily differing in the choice of A and the way d is estimated in practice, which is typically done using an adversarial loss involving discriminator neural network. The use of randomness (shared or not) in these methods has not always been consistent, nor fully explored in its relation to RDP optimality. While Blau and Michaeli (2019) add uniform noise the quantized latent, Tschannen et al. (2018); Agustsson et al. (2023) concatenate noise to the quantized latent, and Mentzer et al. (2020); Muckley et al. (2023) do not use randomness at all. Zhang et al. (2021) use (shared) dithered SQ with NTC, but do not explore why or how it is good for RDP. In contrast, we study LQ with dithering under varying levels of shared randomness, and show that one needs both the improved packing efficiency of lattices along with shared lattice dithering to achieve best performance. While there exist few RDP-on Dori-nieeted neural compressors that do not fit the NTC framework (Theis et al., 2022: Yang et al., 2024; Yang and Mandt, 2024) and instead leverage diffusion models, our work focuses on NTC-style neural compressors for RDP, as they remain the most pervasive type of neural compressor in use, and do not suffer from the higher complexity of RCC or diffusion models.
where 0 denotes the parameters of the codec (ga: Is. Po), Ps.Po).
Many
Analysis of RDP. The RDP function in (1) was formally introduced and widely adopted following Blau and Michaeli (2019); see also carlier related work by Matsumoto (2018); Li et al. (201 1); Saldi al. (2015). (1) a purely informational quantity, i.e., function of the source Px, and thus does not have a meaning as fundamental limit of compression without a corresponding coding theorem describing it as such. The first RDP coding theorem was provided by Saldi et al. (2015), who show that for perfect perception (P- = 0), (1) is achievable under infinite shared randomness, and conversely that no compressor can outperform (I). For general P, Theis and Wagner (2021) establish optimality (i.e., achievability and converse) of (1) when infinite shared randomness is available. Saldi et al. (2015) further characterize the fundamental RDP limit at P 0 when only R. bits per sample of shared randomness is allowed, which only coincides with (1) when R. = oo; a smaller R, results in a strictly worse fundamental limit (see also Wagner (2022)). This establishes the necessity of infinite shared randomness to achieve (I). Li et al. (2011) use dithered LQ to show achievability of (1) for P = 0: in contrast, we show this to be true for general P and also analyze the R. O case, which requires new set of proof techniques based on lattice Gaussian coding (Ling and Belfiore, 2014). Under private randomness (R.- = 0). Yan et al. (2021) establish R(D/2, 20) the fundamental limit for P = 0: Hamdi et al. (2024a) show that randomized encoders do not help. Chen et al. (2022) study the RDP tradeoff when the perception constraint is strong- weak-sense". Under weak-sense, it was shown that R(D, P) is achievable without shared randomness, whereas under strong-sense, shared randomness is necessary, agreeing with Saldi al. (2015); Wagner (2022). In our work, we focus on strong-sense, since that is typically how the
iiformation-Thhoreee
On vectors x, & E IR", strong-sense denotes 5(1', Pa) < P: weak-sense denotes 8 Pa.) < P. Vi.
3

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perception is measured in practice (i.e., in (2)). In particular, we focus on strong- sense Wasserstein, since that aligns with 8's and evaluation metries chosen in practice such as Fréchet inception distance.
Coding theorems use schemes that are typically not constructive (e.g., random coding) and/or not practical (e.g., high complexity RCC schemes). They do, however, provide insights on structures that may be useful or even necessary for optimality. In addition to the necessity of infinite shared randomness to achieve (I), we show in Sec. A how the RCC scheme of Theis and Wagner (2021) implies that a good RDP compressor should behave like a randomized VQ: it should have VQ-like packing efficiency (i.e., good for distortion) combined with random codewords that follow the right distribution (i.e, good for perception). In our work, the former is handled with LQ (at low complexity), while the latter handled with dithering; our schemes are further shown optimal on Gaussians.
The benefits of (potentially shared) randomness have also been discussed outside the context of coding theorems. Tschannen al. (2018) show how randomized decoders are necessary achieve perfect perceptual quality. Theis and Agustsson (2021) illustrate how quantizers benefit from shared randomness on a toy circle source. Similarly, Zhou and Tian (2024) demonstrate how staggered SQ can use limited shared randomness to improve performance on the circle. In contrast, our work presents a more general approach for infinite, limited, and no shared randomness with LQ that empirically shows its benefits and is provably optimal on Gaussians.
Lattice Quantization. Lattice quantization (LQ) involves lattice A, which consists of countably infinite set of codebook vectors in n-dimensional space (Conway and Sloane, 1999; Zamir et al., 2014). We denote Qa(x)) arg minxe || x 2 as the LQ of a vector æ E TR² The fundamental cell, or Voronoi region, of the lattice is given by (A) == {ar E R Qx(x) = 0), i.e., the set of all vectors quantized to O. We denote the lattice volume V(A) == SVo(A) dax, the lattice (²())= == 1 (u--Uair(V)(A))) (1x=]])) and the normalized second moment (NSM)
second moment as
(A) G(A) = (V(A))a>= The lattice's packing efficiency can be measured by how small its NSM is; it is
known that there exists sequences of lattices (*()) ]=z that achieve the sphere lower bound, i.e., lim,-+0o (()())) Ieee where the lattice cells become sphere-like (Zamir et al., 2014, Ch. 7). The closest vector problem (CVP), which finds (2). is NP-hard in general, but many lattices with low NSM (e.g., Es. Barnes-Wall, Leech) have efficient CVP solvers. Recently proposed polar lattices (Liu et al., 2021), which have polynomial time CVP, were shown to be pphere-bound-achivvin et al. 2024). Recently, LQ explored in neural compression (Zhang and Wu, 2023; Kudo et al., 2023) as a low-complyxity method that improves the poor packing efficiency of SQ, equivalent to the integer lattice Zn Lei et al. (2025) showed that NTC transforms are insufficient to overcome the poor packing efficiency of a suboptimal lattice, leading to the lattice transform coding (LTC) framework; performance improves with increased lattice packing efficiency.
(Liu
3 Lattice Transform Coding for RDP
We seek to design compressors that are RpP-optimal given constraints on the amount of shared randomness available and are also low complexity. As mentioned in Sec. 2, show in Sec. A that a good RDP scheme should implement randomized VQ. Dithering is a method that can enable a quantizer to be randomized, In the context of neural compression, NTC transforms generate VQ-like regions in the source space due to the limited packing efficiency of latent space SQ (Lei et al., 2025). The LTC framework, which uses latent LQ, can provide the benefits of VQ-like regions in the source space. LQ naturally supports dithering that is uniform over the lattice cell. Thus, dithered LQ emerges as a promising scheme that is both randomized and VQ-like, while maintaing low complexity. In the following, we describe how LQ with dithering can be integrated into the LTC framework and trained end-to-end, We present three architectures that handle the cases of infinite-, no-, and shired randomness via shared-, private- and quantized shared-dither respectively.
unable to
3.1 LTC with Infinite Shared Randomness
We first define the shared-dither LTC, which assumes infinite shared randomness is available between the encoder and decoder. We denote the fundamental cell (A(A) as Vo for ease of notation.
Definition 3.1 (Shared-Dither Lattice Transform Code (SD-LTT); Fig. 1c). A SD-LITC is a triple (go, 9c, A), with mappings Zo. 9e and lattice A. A random dither (ni)()o(A)). uniform over the

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lattice cell, is shared between the encoder and decoder. The SD-LTC computes the latent y = (a)
entropy codes c = Qx u) at the encoder, and the decoder outputs ã g. + u).
Since u is available at the encoder and decoder (which utilizes the availability of infinite shared randomness), the operational rate of SD-LTC is given by
(clw)) - Eyyul log Petu (c/u)) - E - log
Py-w/ul (wll (w)2))
(3)
Vo-c
= Ly,ul- log Ew Iyy (Q u) + u u'lll.
(4)
where v' ~ Un). As an alternative, we can make use of the additive channel equivalence (Q - u) + d - y + uea, where Usa ~ Unii(Vo(A)) (Zamir et al., 2014, Thm. 5.2.1), yielding H(clu) = 1 u) = h(y + u) -()) = By log Py (y + u)] - log V (A)
(5)
= Ey.. - llog
Py (yww) = Ey log Eur Ia (3 + u + u III.
(6)
(o(())+++
where h(-) denotes differential entropy, V(A) is the lattice volume, and (6) holds since Py+u
2) = V(Aj (Vo(A)+y+u (y(w)duu The objective is trained with ming H (clu) + A E| (x. +)] A20(Pr. Pa). with learned parameters 0. Either (4) or (6) can be used for Hclu): (4) requires the straight-through estimator due to non-differentiabiliyy of the quantizer. In the following, we comment on several connections between (4), (6), and other work in the neural compression literature.
Integrating the learned py The inner integral in (4) and (6) can be computed exactly under SQ (equivalently, A = z") using the CDF of Py. following Ballé et al. (2018 8). For general lattices, the inner integral can be estimated using Monte-Carlo following Lei et al. (2025), by sampling vectors uniform over the lattice cell (Conway and Sloane, 1984).
Noisy proxies and operational rates. The equivalence of (4) and (6) was shown in Ballé al. (2020) for A = Zn Their equivalence for general lattices, as shown above, follows from Zamir et al. (2014). Ballé et al. (2020) uses this equivalence support (6) training objective, which is known the noisy proxy to quantization in the literature. This is useful since (6) is differentiable with respect to y. whereas (4) is not. Here, we emphasize that both (4) and (6) represent the operational rate for SD-LTC. For deterministic NTC/LTC trained with the noisy proxy, a deterministic dither (perhaps 0) chosen test time, resulting in a different operational rate that does not average over u. Agustsson and Theis (2020) proposed dithered SQ as universal quantization; however, they were motivated by reducing train/test rate mismatch rather than the RDP tradeoff, which the focus of our work.
3.2 LTC with No Shared Randomness
We now consider when no shared randomness is available. As mentioned, decoder randomness is needed for the perception constraint, and this manifests itself as private dither at the decoder.
Definition 3.2 (Private- Dither Lattice Transform Code (PD-LTC); Fig. 1b). A PD-LTC is a triple (ga, 9s. (ga.9a, A), with transforms Ja and ys and lattice A. A random dither u Uno (A)), uniform over the lattice cell, at the decoder only. The PD-LTC entropy-codes ý = Q (ga (ar)), and the decoder outputs à = ( + su), where > 0 is parameter that controls the dither magnitude.
For PD-LTC (shown in Fig. 1b), the operational rate is the same as in deterministic LTC, given by H = Egy -- log P. (g))) as there is no shared dither. Therefore, the training objective remains the same (2). The following proposition provides some intuition why shared- dither quantization (Fig. Ic) is superior to private-dither quantization (Fig. 1b) in terms of distortion.
Proposition 3.3. Define æSD Qx u) + u, 2DD = (z) + For any and 1,
E E[|x 2pp s² [[ 2sD + E[[ Q (=) E[1
æsD 2 [1
(7)
The proof is provided in Sec. D; note that here is an arbitrary vector (not necessarily the source). Prop. 3.3 implies the PD error is the sum of the SD) error and the error under deterministic quantization; Fig. 2. While dşD is random over the blue cell, incurring error equal to the lattice second moment, ēPD is random over the gray cell, incurring additional quantization error. Regarding & > 1. lower PD error is possible with < but the support of ý + su would have gaps within each lattice cell, which may make the perception constraint difficult to satisfy. In Sec. 4, we show that s > 1 is necessary for PD-LTC achieve optimality on the Gaussian source.
5

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While Prop. 3.3 provides an idea of how SD-LTC and PD-LTC may perform distortion wise, the rate and perception are more complicated. For perception, the induced reconstruction distributions are different, since that of æsp is convolution between the source and a zero-mean dither, whereas that of ēPD is mixture of dithers centered on lattice vectors. For the rate, compared to H (ÿ). H (c/u) has additional randomness due to the averaging over u, and Figure 2: Reconstr. of thus we may expect the rate of SD-LTC to be larger than that of PD-LTC if x under PD (gray) or they share the same transforms. Ideally, this potential increase in rate can help SD (blue); s = 1.
a X
achieve overall superior RDP tradeoff for SD-LTC. In Sec. 4, we show that this is true on the Gaussian source, and verify it empirically on real-world sources as well in Sec. 5.
3.3 LTC with Finite Shared Randomness
While the availability of truly infinite shared randomness may be difficult to obtain in practice, the availability of finite shared randomness may be more feasible. Let R. denote the rate of shared randomness in bits per dimension. RDP theory informs us that performance should improve as Rt increases (See. 2). To make use of finite shared randomness, we propose scheme (Fig. Id) that interpolates between SD-LTC and PD-LTC by using nested lattices (Zamir et al., 2014, Ch. 8). Lattices A, y are nested if A is sub-lattice of the fine lattice Ay. We restrict our attention to self-similar nested lattices A = a where > O is an integer. We denote A/A, = {A E A, X E (A(A) as the fine lattice vectors contained in the fundamental cell of A.
Definition 3.4 (Quantized Shared-Dither Lattice Transform Code (QSD-LTC); Fig. Id). A QSD-LTC is given by (gasgs: Gs. A, Af), with mappings Ia. Is and nested lattices A, A, A random discrete dither á ~ Unif(A/Ay) is shared between the encoder and decoder, and a continuous dither (ni((Vo(AA)) uniform over Ay's cell is at the decoder only. The QSD-LTC computes uy = gu (2), entropy codes c = Qx y - û). and the decoder outputs 2 = ge(c + û + + The rate of shared randomness is Ri = L log |A/Ayl bits, i.c., there are 2d possible shared dither vectors.
Remark 3.5. QSD-LTC recovers PD-LTC and SD-LTC when R. 0 and R. oo, respectively:
When R. = 0, we have that A, = A, û. = 0, and uj ~ (n)f())(AA). and therefore
Q^ (y ú) + ú + suy Q1(y) + suy..
(8)
When R. = 0o, we have that A, = R", â ~ Unif (Vo (A)). U = 0, and therefore
Q1 ú) + ú + suy Q (y ù) + ú.
The operational rate, H( follows (4), except replacing u with The training objective is ming (c(() + X (ax, .)] + 220( Pa. P=). Unlike SD-LTC, the additive channel equivalence does not apply, since the support of y ū QA û) is random and does not always equal A/A,-
We note that the R. values that QSD-LTC may achieve are limited to log I, where I E Zt, a positive integer, is the nesting ratio of A, Ay (Zamir et al., 2014, Ch. 8). This is due to the structure of nested lattices. To achieve R. values between O and 1, a non-uniform distribution for the shared dither vector would need be employed; we leave this future work.
Remark 3.6. One may ask whether infinite shared randomness can be obtained by sending a pseudorandom seed and drawing continuous dither vectors (A) from a random number generator (RNG) based on the seed. If one compresses a source realization e to a bitstream b, a pseudorandom seed of A bits would imply that only 2* possible dither vectors could be used the decoder to decode b; this is noted by Hamdi et al. (2024b). Therefore, sending a pseudorandom seed insufficient simulate infinite shared randomness; rather, implements a scheme with finite shared randomness. Furthermore, finite shared randomness with a constant number of bits per dimension necessary to achieve the fundamental limits (Wagner, 2022); this impossible to satisfy with a random seed of fixed number of k bits for high-dimensional sources. In addition to having the capability of imposing a R, bits per dimension of shared randomness, QSD-LTC has the additional advantage of ensuring the dither vectors are drawn uniformly from the fine lattice vectors in the lattice cell (A). These are spread out uniformly throughout (A( due to the structure of nested lattices. In comparison, dither vectors drawn from a random seed have no guarantee on where they may land in (A),,, and would depend on the RNG and seed used, As an example, for poorly chosen seed and RNG, the dither vectors generated could all be concentrated near the center of (o(A), which would effectively yield no shared randomness. Therefore, in settings where infinite shared randomness impractical, QSD-LTC enables structured way of using finite shared randomness.

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On complexity. SD., PD., and QSD-LTC rely on LQ. For fixed lattice up to dimension 24 (Sec. 2). the closest codebook vector search (also used to generate dithers) can be performed any rate in fixed number of operations, and is significantly faster than that of VQ, which is exponential in the rate. For higher dimensions, polar lattices have complexity polynomial in the dimension.
4 Achieving the Fundamental Limits
We now theoretically analyze the performance of SD-LTC and PD-LTC on the Gaussian source, and describe the RDP tradeoff asymptotically achievable by SD-LTC and PD-LTC. While the operational rate and distortion are given by the per-dimension versions of those in Sec. 3, the operational perception used per-dimension squared 2- 2-Wasserstein distance. We leave full proofs to Sec. D.
4.1 Infinite Shared Randomness
Proposition 4.1 (RDP function for Gaussian source (Zhang et al., 2021)). Let Px = NN(0,
A (x, x)) = (x 2), and 8(1, v) = Wz(,,,) )) be squared 2- 2-Wasserstein distance. Then
log
for /P <P<o- -22- D),
P)
o² ppp
yp D)a for 2220-
(10)
max
log
o² D 0
Remark 4.2. When vP < -2
D], the optimal X in (1) is jointly Gaussian with marginal X (O, (a - vP)²). and covariance max ( + (a vP)² D).0}. When vP 2 / 02 - Di, X is jointly Gaussian with marginal X ~ ~ D).
The following theorem shows that R(D, P) is achievable with SD-LTCs, and is an extension of Li et al. (201 1), whose authors addressed the P = 0 case.
Theorem 4.3 (Optimality of SD-LTC for Gaussian source). Let X1, X2,.. iid. NO o²). ) For any (n) (m)
P E [0, o2), D E [0, ) there exists a sequence of SD-LTCs ((9)) .gx ) such that
1 H((QAm (g. (r) (X") - u)(u)= = R)D,P),
lim n n-+000
(11)
lim _ _ 81113] SDD
(12)
lim W2 (Px,, S
S.
(13)
n = n
where X = g0 (Q (0) (n) (X") - -) u) and (")
Remark 4.4. The proof of Thm. 4.3 relies on a sphere-bound-achiewee scalar transforms Ja. As dimension grows, the dither u becomes Gaussian-like, and the latent dithered LQ acts like an additive Gaussian channel (Zamir and Feder, 1996), imposing joint Gaussian relationship between X and X as desired by the RDP solution; see Remark 4.2 and Fig.
sequence of lattices with
4.2
No Shared Randomness
We now consider the case when the encoder and decoder not have access to any shared randomness. For simplicity and ease of presentation, we consider the regime of near-perfect and perfect perception, corresponding to P < e for any e > 0 and P O, respectively. This is the only regime of perception where lower bounds on the RDP achievable under no shared randomness are known (Saldi et al., 2015; Wagner, 2022; Chen et al., 2022: Yan et al., 2021), which corresponds to R/2222 oo), and evaluates to 2 log 5 on the i.i.d. Gaussian source. The following theorem shows that R(P/2, oc) is achievable with PD-LTCs under near-perfect perception, which can be easily extended to perfect perception (see Remark D.8), As discussed in Sec. 2, R(D,0) 0) < R(P/2, oo); see Fig. 4 for visualization.
7

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QA
yLG
Figure 3: In the latent space, SI>-LTC (left) becomes AWGN-like PD-LTC (right) models the sum of lattice Gaussian and Gaussian; must be large enough for the sum to he Gaussian.
Theorem 4.5 (Optimality of PD-LTC for Gaussian source). Let X1, X2,... Ltd. NNO, o²) For any
D satisfying 0 < D SD2 there exists sequence of PD-LTTCs {(ga .90
(x) , ( such that
1 HH(QA (x")) - (0)/
lim n n
(14)
lim - - - X11 - D,
(15)
4=00 nt
1 lim _ W2((xx Px.) 00 0,
(16)
n-000 n
(n) (X")) + su). u ~ (V((()()(())) and s Vo2- (n) (Q (9a where X = gs
Remark 4.6. PD-LTC does not use a shared dither; we cannot make use of classical results (Zamir and Feder, 1996) to analyze the statistical behavior of latent LQ as an additive channel (as opposed to Thm. 4.3). Instead, Thm. 4.5 relies on lattice Gaussian coding methods (Ling and Belfiore, 2014); see Sec. D.3 for details. Using scalar transforms, QA (z)) behaves like lattice Gaussian. The additive private dither su (which becomes Gaussian -ouus makes the reconstruction Gaussian-like (Fig. 3). It then suffices for 9s to scale X to impose the desired variance o². In classical rate-distortion, dithered LQ and lattice Gaussian coding can both achieve optimality, so their differences are primarily practical (e.g.. restrictions on lattice families, availability of a shared dither). Thms. 4.3, 4.5 show that with a perception constraint, the two proof techniques lead to different fundamental limits as well.
Remark 4.7. We note that the choice of = 0/ D/2 > 1 depends on the regime of the rate- distortion tradeoff; it is approximately for small D, and becomes large for D - 22 At low rates, this means the effective dither su "leaks" outside the lattice cell rather significantly. This is required to ensure the perception term vanishes (16). Specifically, su needs to be large enough relative for Qxte> (n) (X")) + su to approximate Gaussian of covariance σ² Ln arbitrarily accurately. The nature of this approximation is based on the flatness factor (Ling et al., 2014, Def. 5) used in lattice Gaussian coding, which is elaborated on in Remark D.7.
to the quantization steps of Q(al")(X)))
(g"")(""))
5 Experimental Results
Experimental setup. We denote the source dimen- sion and ny the latent space dimension. We denote LTC as NTC when the lattice is chosen be the integer lattice, i.e., A = Zmy- We train the PD-, SD- and QSD- LTC models using their RDP objectives discussed in Sec. 3. For the rates reported at test time, we use the (4) version of H (clu) for SD-LTC, l-I-log log Pa ( for PD-LTC, and (c))ū) for QSD-LTC; all require hard quantization. These rates are cross-entropy upper bounds on the true entropy, due to the learned Py den-
2.5
QSD NSC Z,, 1.55
SD-NTC R 30
PDALTC EG, R--0
1.5
8, 1151
Race
SD-LTC & & =
1.0
RED 0)
0.5
R(D)22 mc)
0.0
0.1
0.2
0.3
0.4 0.5
1.0
2.0
sity. We use MSE distortion A(x, .) _ || 212 For perception, to obtain reliable estimates higher di- mensions, we use squared sliced Wasserstein distance
Distortion
Figure 4: Effect of lattice choice and shared
randomness on Gaussians at P P= 0.
(Bonneel et al., 2015) of order 2, 6(Pa, Pa) SSW Pa). During training, we use the straight- through estimator with hard quantization. See Sec. for further details on architecture/traning..
5.1 Synthetic Sources
We first evaluate the i.i.d. Gaussian source of dimension n = =8. In Fig. 4, we plot the equi-perception curves with P = 0 to compare the methods under the perfect realism setting, As shown, for fixed
8

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MNIST
Physics
Speech
20
NIC
NTC
Rate
PD-NTC SDATC
Rate
0.4
PD-NTC SD-NTC
PD-NTC
J0
804
SD-NTC
0,050
0.075
1.100
0.125
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Distortion
Distortion
Disortion
20
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Rate
PD-ITC SD-1CC
Rate
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0.5
PD-LIC
SD-LTC
0.3
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0.00
0.01
0.02
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0.04
0.005
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0.015
0.020
Pereeption
Pereeption
P Perception
Figure 5: LTC, PD-LTC, and SD-LTC on real-world sources; R-D (top) and R-P (bottom).
MNIST
Physics
Speech
0.50
0.6
Rate
PD-LIC QSD-LTC-2
gate
PD-LTC
205
PD-LTC QSD- LTC-2
QSD 2IC
0.25
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Distortion
Distuction
Distortion
20
QSD-LTC-3
0.50
QSD-1TC-3 SQALIC
0.5
QSD-1TG-3
Rate
5D-1TC
SIALTC
0.25
0.3
0.00
0.05
0.10
0.15
0.002
0.004
0.006
0.006
0.008
0.010
0.012
Perception
Perception
Perception
Figure 6: QSD-LTC-I' (I' the nesting ratio) on real-world sources; R-D (top) and R-P (bottom).
amount of shared randomness, a more efficient lattice improves performance. Analogously, for a fixed lattice more shared randomness improves performance. We additionally evaluate QSD-LTC with finite shared randomness. The fine lattice A in Def. 3.4 is set to be self-similar with A, with a nesting ratio of 3. This results in R. log 3 bits per dimension. As shown, the QSD-LTC performance is nearly able to achieve that of SD-LTC, despite not having infinite shared randomness; this verifies that QSD-LTC allows one to interpolate between SD-LTC and PD-LTC. At low rates, the LTC with Es has some suboptimality; this is due to the Monte-Carlo estimation when computing the integral for latent likelihoods (Lei al., 2025).
When compared to the fundamental limits described in Sec. 4, we see that the PD models are lower bounded by R (D22, oo), and the SD models are are lower bounded by the RDP function R(D, R(D,0) but outperform R(D/2 P/2, oo) when the rate is not too small. Although the fundamental RDP limits should be interpreted operationally with perception measured Wasserstein and not sliced Wasserstein, empirically verify that on Gaussians, sliced Wasserstein is faithful to Wasserstein in Appendix. E. The performance of
RCC (i.e., each dimension of x: is compressed with RCC separately) and n = 8 RCC is shown in Fig. 7a. Performance improves with increasing dimension. However, the 8-dimensional RCC is outperformed by SD-LTC, and additionally has complexity exponential in dimension and rate; SD-LTC does not suffer the same high complexity. We show the RDP achieved by deterministic NTC in Fig. 7b: at low rates the lack of randomness prevents from enforcing the perception constraint. At larger rates, its performance coincides with SD-NTC.
5.2 Real-Worke Sources
We use MNIST (Lecun et al., 1998), Physics and Speech datasets (Yang and Mandt, 2022). These contain grayscale images of dimension 28 x 28, physics measurements of dimension 16, and audio signals of dimension 33 respectively. We use a latent dimension of nt = 8 for the first two and n 16 for Speech. We use the integer lattice for NTC models; for MNIST/Physics. we use the Eg lattice and for Speech, we use the A16 lattice (Barnes and Wall, 1959). The corresponding RDP tradeoff is shown in Fig. Due to lack of randomness, deterministic NTC and LTC are unable to enforce the perception constraint at lower rates, no matter how large X2 in (2) is set. Similar to the Gaussian case, performance improves with better lattices and increased shared randomness, demonstrating the benefits of lattices and shared randomness described in the prior sections translate to real-world sources that require nonlinear transforms. For QSD-LTC, we use self-similar nested lattices with a nesting ratios of 2 and 3, corresponding to R. log 2 1 and R. log 3 ~ 1.58

---

respectively. Performance increases with R. (Fig. 6). A full comparison across all models/datasets shown in Figs. and 10. Overall, SD-LTC achieves the RDP tradeoffs. For fixed lattice, QSD-LTC, with R. = log 3, can nearly achieve the performance of SD-LTC, which uses R. 0o.
5.3 Ablation Study
We perform an ablation study on lattice choice and training of PD-LTC with STE or the noisy proxy. We use the D lattice for SD-LTC in Fig. 11 on Speech. Its performance lies between integer and Barnes-Wall lattices, which aligns with the fact that the Dn packing efficiency lies between those two lattices. This supports our result in Thm. 4.3 that performance is optimal when the lattices pack space more efficiently. For the noisy proxy, we use it to train PD-LTC, but this may result a train/test mismatch, since unlike SD-LTC, the noisy proxy does not equal the rate under hard quantization. A comparison of the two in Fig. 12 shows there is not much difference in the resulting performance.
6 Conclusion and Limitations
We investigate low-complexity. high-performing compressors for the RDP tradeoff. We propose combining dithered LQ with neural compression, which supports different amounts of shared random- ness. Under infinite and no shared randomness, we show SD-LTC and PD-LTC achieve optimality on the Gaussian source. We empirically verify that performance improves with increased shared randomness and improved lattice efficiency across synthetic and real-world data. Future work may address: (i) expanding the range of R. in QSD-LTC, (ii) theoretical analysis of QSD-LTC, which may require new tools developed for dither vectors defined over nested lattices, and (iii) extension to SOTA image compression architectures, which would require careful integration of random dithering with hyperprior models that implicitly use a deterministic dither.
This work was supported by The Institute for Learning-enabled Optimization at Scale (TILOS), under award number CSSFCCCF-211265.
Acknowledgements.
Disclaimer. This paper was prepared by Eric Lei prior to his employment at JPMorgan Chase & Co.. Therefore, this paper is not a product of the Research Department of JPMorgan Chase & Co. or its affiliates. Neither JPMorgan Chase & Co. nor any of its affiliates makes any explicit or implied representation or warranty and none of them accept any liability in connection with this paper, including, without limitation, with respect the completeness, accuracy, reliability of the information contained herein and the potential legal, compliance, tax, or accounting effects thereof. This document is not intended as investment research or investment advice, or as a recommendation. offer, or solicitation for the purchase or sale of any security, financial instrument, financial product or service, to be used in any way for evaluating the merits of participating in any transaction.
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A Reverse Channel Coding
A.1 RCC Preliminaries
Reverse channel coding (RCC), also known as channel simulation, devises schemes for sending sample from a prescribed distribution with a rate constraint (Cuff, 2013; Li, 2024), and typically assumes shared randomness. Since RDP requires randomized reconstructions, RCC can be applied to RDP, and in fact is a technique used to prove RDP coding theorems; Theis and Wagner (2021) use the one-shot RCC scheme of Li and El Gamal (2018), and Saldi et al. (2015); Wagner (2022) use channel synthesis results of Cuff (2013). We focus our discussion here on the particular RCC scheme of Li and EI Gamal 2), defined below.
Definition A.I (One-shot reverse channel coding (RCC) via the Poisson functional representation (Li and EI Gamal, 2018)), Let X ~ Px be the source, and xxx be a channel we wish to simulate.
Define Q to be the X-marginal of the joint distribution Px P xxxx Suppose that the same sequence of T1 72, (xp and X1,X1.... Ltd. Qx are generated at both the encoder and decoder (requiring infinite shared randomness U). Let W = E==1 Ty. Given source realization X = the encoder computes
dQx (Xa),
K = arg min W;
(17)
dP xxxxx (-1z)
and entropy-codes it. The decoder simply outputs Ax. By Li and El Gamal (2018), it holds that Xx|{X- = x} ~ Pxxx(+a)) Additionally, the rate satisfies
H(K(U) X(X:)) + (og( + +
(18)
Remark A.2. The one-shot RCC technique above enables immediate achievability results for informational quantities, such as the rate-distortion-prrcepe the channel x that achieves the infimum in (1). Due to the fact that the reconstruction Xx has distribution equal to the channel x any constraint in the informational quantity, such as the expected distortion constraint, or the perception constraint, is automatically satisfied by RCC. The I(X; X) term in (18) then becomes equal to the informational quantity R(D, P). Applying this scheme to i.i.d. blocks yields the asymptotic result. This is the approach used to prove achievability in Theis and Wagner (2021).
function R(D,, P), by simulating
A.2 RCC Randomized VQ
The scheme in Def. A.1 essentially chooses random codebook at both the encoder and decoder. The encoder chooses codeword X according to the criterion in (17), which may appear abstract first glance. However, the following two propositions show that when the channel Px used to simulate chosen to be RD- or RDP-achieving, it becomes clear that (17) uses a minimum dinimum-distance codebook search similar VQ.
Proposition A.3 (RCC on the rate- riie-distortion-acheee channel; Prop. of Lei et al. (2022)). Let Px|x be the anteddsstrrion-acchiving channel, i.e., the channel achieving the infimum of R(D, oo).
Then the density ratio satisfies
d. x1x (-1x) (i)
c B ((..)
(19)
dQ, x
E Ex |e -ALA,"1'
and (17) is equivalent to
K = arg ((x, X.) + In In
(20)
where 3 > 0 is the unique Lagrange multiplier determining I(;;8) = R(D, oc) at D =
Epx Xxix [AX, X)).
14

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Proposition A.4 (RCC on the RDP RDP-achieving channel with f f-divergence perception), Assume that the perception is measured by 8(Px, (x(Px11Px), a f f-divergence. Let Px|x be the RDP-achieving channel that achieves (1). Serra et al. (2023) show that the density ratio satisfies
dP (1xx((12))
e A(r,8) 9
(21)
dQx
E ExIQ
Bi Aan 1x.
where g(Px. (x) f dPx (8) xQxx
dPx (3) and B1, 82 0 are the unique dPx (z)0f x1xx xix
Lagrange multipliers determining Dy(Px|1Px), Therefore, (17) is equivalent
= R D, P). at D = Epx xxx (XX, and P =
32
K arg A (z, X;) +
g(x, Qx X.) Bi
In W..
(22)
3
Proof. The proof follows Lei et al. (2022, Prop. 1), using (21) instead of (19).
Remark A.5. The above two propositions imply that RCC with the RD- or RDP-achiewee channels result the encoder searching for the random codeword in (X,, Xg....> that is closest to the source realization x in terms of the distortion metric A, regularized by In W,, which enforces the rate constraint: for the RDP case, it is additionally regularized by g(Px, +x+ X²) which enforces the perception constraint.
Thus, when taking distortion into account, RCC can be seen as sort of randomized VQ that finds the minimum-distance codeword (which are random). Despite its optimality for RDP, RCC is of high complexity (exponential in rate and dimension), and requires infinite shared randomness.
Remark A.6. A closed-form for the density ratio of the RDP-achieving channel Pxxx for when the perception is measured by squared 2- Wasserstein (which is the focus of our paper) is currently not known. However, conjecture that similar (19) and (21), it will consist of e- A(a,2) term in the numerator, which will result in (17) having A(x, Xx) in the objective. Another slight discrepancy to our setup is that the n-letter operational perception in Serra et al. (2023) for the f-divergence perception RDP function is measured in the weak-sense (see Sec. 2). We focused on the strong-sense setting as it more faithfully describes practical usage.
A.3 RCC Implementation Details
To simulate RCC, one can implement the scheme in Def. A.1 and use (20) or (22) for finding the index to entropy-code. For the Gaussian source, the RDP-achieving channel and output marginal is given in Remark 4.2; we can directly implement (17) in closed-form since these distributions Gaussian. This is what is done for the results in Fig. 7a. Since it is not possible to generate an infinite number of samples X, we generate codebook N 10, 000 samples instead. Following Li and El Gamal (2018), the index K is entropy-coded using a Zipf distribution with parameter = +1+1 1/ X + e - log e + 1). A full algorithm describing the encoding and decoding process can be found in Theis and Ahmed (2022) well as Lei et al. (2022).
B Additional Experimental Details
For the synthetic (i.i.d. Gaussian source), we set ni 8, Ia and g, to be linear functions, as the constructions in Thm. 4.3 and Thm. 4.5 suggest this to be sufficient for optimality, and use the Za and Es lattices. To cover the RDP tradeoff, we sweep a variety of X1 A2 values. For the Speech and Physics datasets, we use MLPs for Ja and gs of depth 3, hidden dimension 100, and softplus nonlinearities. For MNIST, we follow the same exact experimental setup of Blau and Michaeli (2019), including model architecture, and using the test Wasserstein distance via the discriminator neural network. For NTC models, we use the factorized py of Ballé et al. (2018), and for LTC models, we use the RealNVP normalizing flow (Dinh et al., 2017). All models are trained for 100 epochs on the
15

---

training data split, and reported metrics are averaged over the test split. Training is performed on NVIDIA RTX5000 GPU.
For the real-world sources, we swept A (the distortion weight) and kept 22 (the perception weight) fixed positive value. This allows compare rate-distortion and rate-perception tradeoff's across methods; note that the two plots should be examined jointly together, and that the individual plots do show the rale-perception (or rate-distortion) performance for fixed distortion (or perception).
C Additional Empirical Results
Figures pertaining to the experimental evaluation in Sec. 5, such as ablation studies, are shown here.
1,00
2.0
RR,D,O)
2.0
PD-NTC Z. R.
PD-NTC Zr. R
1.5
24.
0.50
Rate
1.0
1.0
0.25 A
SD LTC E, -o
0.5
RCC's 000000
PD-ETC 8s R==0
RCC
0.00
0.2
0.3
1.0
2.0
0.1
0.2
0.3
0.4
0.5
1.0
2.0
(a) Comparison with reverse channel coding (RCC).
(b) Comparison with deterministic NTC.
Figure 7: Gaussian RDP results, comparing PD-LTC and SD-LTC with RCC and deterministic NTC.
NIC
PD-NIC
QSD-NIC2 QSD-LTC-2
QSD-NIC-
SD-NTC SD-LIC
LTC
PD-LIC
QSD-LIC-
20
20
15
10
Rac
10
Distoction
Perception
Figure 8: RDP tradeoff of all models on MNIST. QSD-NTC/LTC-T corresponds to QSD-NTC/LTC with nesting ratio of T.
16

---

NIC
PD-NTC
QSDANC-22
QSD-NTC3 QSD-LTC-3
SD-NTC SD-LTC
QSD-LTC-2
0.5
0.5
0.4
0.4
Rate
0.3
0.3
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.002
0.004
0.006
0,008
0.010
0.012
Distortion
Peroeption
Figure 9: RDP tradeoff of all models on Physics. QSD-NTC/LTC-- corresponds to QSD-NTC/LTC with nesting ratio of r.
NIC
PD-NIC
QSD-NTC-2
QSD-NTC-3 QSD-LTC
SD-NTC
LTC
PD-LTC
QSD-LTC-2
SD-LTC
0.60
0.60
0.35
0.55
0.50
0.50
Rate
0.45
0.45
0.40
0.40
0.35
0.35
0.30
0.30
0.6
Distortion
Perception
Figure 10: RDP tradeoff of all models on Speech. QSD-NTC/LTC-- corresponds to QSD-NTC/LLCC with a nesting ratio of r.
0.6
SD-NTC Z. SD-LT SD-LTC D
Rate
0.5
Rate
0.4
PD-NTC Noisy
PD. STE.
0.3
0.4
0.5 Distertion
0.6
0.4
0.5
0.6
Distortion
0.6
PD-NTC Noisy
PD-NTC
0.5
Rate
SD-NTC 7.0
0.5
SD-LTC Ags
SD-LTC D2
0.3
0.006
0.008
0.010
0.012
0.014
0.006
0.007
0.008
0.009
0.010
Perception
Perception
Figure 12: Comparing straight-through esti- mator (STE) with hard quantization vs. noisy proxy for PD-NTC, on Speech.
Figure 11:
Comparing different lattice choices for SD-LTC, Speech.
17

---

D Proofs
D.I Proof of Proposition 3.3
Proof.
By the crypto lemma (Zamir et al., 2014), we have that æso = Q1(æ u) + u
x + u.
Then
E || 2SD
B|||u|11 (2(),
(23)
n
the second moment of the lattice. Additionally,
1
E [[112 ēpp 12] = B[[1 Qx)x) sull
(24)
1 B[1x - [xx))]
_ E u||i]
(25)
n
n
sO)) + 1 n 2 307(A))
-11==-Qaaa Qx))
(26)
(27)
2 (A),
(28)
as desired.
D.2 Proof of Theorem 4.3
Theorem D.1 (Optimality of SD-LTC for Gaussian sources (Thm. 4.3 in main text)). Let X2,..... Lid. ~ o(0, For any P and D satisfying 0 s P S 0² and O < D S 20², there (n) (n)
exists sequence of SD-LTCs {(g6") ,9 ) such that the achieved rate, distortion, and perception satisfy
lim H(Q (g. (x") - u)(w) - R)D,P),
(29)
n-+00 1
lim 1on-x1133 X11 DD
(30)
1+000 n
lim W2 X²) < P,
(31)
n+000
where 8 g. (QAio (ga (1) (X") u) + u). and (AA(
Proof. We first focus on the case when /P <0- V D). Define the sequence of DLTCs ((gb") (n) (n)
99s A )m as follows. Choose (x) to be a sequence of sphere-bound-achievimn lattices, ie.. 1im (() 2xe where G(-) is the normalized second moment, such that the second 20
moment (A((m)) = =1² = 0²
Set go (m) (0) = v to identity mapping, H(-HH--PP--D)
and set 96 (n) (v) --v/P Vaa
We first verify the perception constraint satisfied. Fix > From Zamir et al. (2014, Thm. 7.3.3),
we have that
2
DKL (ze
(32)
7
20
18

---

for n sufficiently large, where un)) ~ (1AA is uniform over the fundamental cell of A A(n),
and z ~ NN(0 Let Ppn = -o (n). Then, for n sufficiently large,
1 (P
- DKL
T-VP VP (x" + ||
0 -- (x" + +2)
(33)
7
n2
V + np
00² + 1p
1 DKL(X" + +u'"" +
(34)
11 1 (KL(u(")12))
(35)
n
e²
(36)
202
where (33) holds by the crypto lemma (Zamir et al., 2014, Ch. 4.1), (34) holds since KL-diierreecce is invariant to affine transformations, and (35) is by data-pr -ata-proesssing inequality. Thus, for sufficiently large,
1 Wx()
1
S W2 (Px,
W2 (2(Pxn:Ppr Pr.)
(37)
VTi
Vn
1
1
(((Pxx,Py)) +
v 2222. DKL (P (Pxx))Pp))
(38)
vn
1 W2(Px", Py.) +
(39)
1
Σ
W3
+
(40)
Vnl
WW 02) ((00((0-V)))) te
(41)
= =VP+c
(42)
where (37) holds since Wasserstein distance satisfies triangle inequality, (38) is by Talagrand (1996), and (40) holds by properties of 2-Wasserstein distance on product measures (Panaretos and Zemel, 2019). By continuity of + 2², we have lim [ X²) < P.
The rate achieved will satisfy
lim H(QAsw (X" - (m))) +
lim 1 I(X"; X" + ())
(43)
n0000 n
=1(X;- X + Z)
(44)
1 log 1
σ²)
(45)
2
112
1
log
1+
(46)
2
102 (0P
(VP-VP
log
2: (47)
2
0² - VP2 1 02 + (o - -VPp- D
where (m)) is uniform over (m), and Z ~ \o 12) Here, (43) holds by Zamir et al. (2014,
Thm. 5.2.1), and (44) is due to Zamir and Feder (1996, Thm. 3).
19

---

The distortion satisfies
lim E x
1-1
lim E
X"
-- -P
(X + u (1))
(48)
Va² + +1p
0 0 vP 0² +1
lim 2-000
E
X
(X + 2
(49)
where 7n (0, 1) Here, (48) holds by crypto lemma, and (49) holds since X and u (n) independent, so the squared norm becomes a sum of second moments of X and u ("), and B (7) = |||| n 2]. Since X" and z" are now iid.,
lim 1 E x 8"
0 VP
E
X
(X + x+
(50)
0
VP
(0 lo- vPJ2
0²
1?²
(51)
002 + 172
0 + 1イ
20 0 0 --VP
0² + (o vp)2
(52)
= = = 4 vP (0² + vp D)
(53)
= D
(54)
For the case when vP 2 o vo0- we use a sequence of DLTCs with (v) v.
gs (n)
(v)
-==00 o² v. and a sequence of sphere-bound-acee -ccheviih lattices ( with second moment 1/5-1/02- For the perception constraint, by following the proof of the perception constraint
(A
-D),) and .V (0, (10-1122 In)) have
in the previous case, except with py... that
W2(Pxx, Px=) S
+e
(55)
= 1022 D1 +
(56)
SVP+G
(57)
for any >> O and sufficiently large, where the last step is by the assumption that /P
Vi02 DD. The result follows again by continuity of + z², For the rate and distortion constraints
of
1 (Q (x" u (11)) (m) = max log a² 0
lim n-+00 n
D D.O
(58)
2
and
lim 1 E || X x-u1[]] D,
(59)
n-oo n
the proof follows that of Zamir al. (2014, Thm. 5.6.1).
D.3 Proof of Theorem 4.5
We now consider the case when the encoder and decoder do not have access to infinite shared
randomness. Unlike Thm. 4.3, Thm. 4.5 cannot make use of the additive channel equivalence, and we instead rely on results from lattice Gaussian coding (Ling and Belfiore, 2014). We first introduce several important concepts of lattice Gaussians, then establish several lemmas that will be used to prove Thm. 4.5.
20

---

Definition D.2 (Lattice Gaussian Distribution). A lattice Gaussian random variable y ~ NN(cc, supported on (shifted) lattice A + F has PMF
Pa(A)
9y (x Pa c)
XEA+c,
(60)
where (B) = z llvll and (A) ExEAPO
For more in-depth introduction, see Crephens-Davidowiy (2017). The proof of Thm. 4.5 relies on known results regarding the lattice Gaussian second moment and entropy; see Ling and Belfiore (2014); Regev (2009); Banaszczyk (1993) for details.
Two tools which will be used throughout the proof are the (i) flatness factor of a lattice, and (ii) a vanishing error probability of maximum posteriori (MAP) decoding of lattice Gaussian signal sent over an AWGN channel. For (i), the flatness factor (Ling et al., 2014) of lattice is defined as
e (T) = max (V()))))())) 1/ =EV((A)
(61)
1m-a12 24
where Py,A (x) (2m9-)"/2 ΣxE
For (ii), suppose we wish send y Na o² v) over an AWGN channel with noise z ~ (O, vln). Ling and Belfiore (2014) show that given the received signal ã y z, the MAP decoder of y given ä is given by Qx-e(ax)) where
The high-level approach the proof of Thm. 4.5 is to analyze the rate, distortion, and perception when compressing ã y z, following Liu et al. (2021; 2024). This is because Ps approximates
(O In), with the approximation error given by the flatness factor eΛ
[o2-w)e (Ling and
Belfiore, 2014; Regev, 2009). With vanishing MAP error probability, QA (ax) R y. and the overall system be statistically analyzed using the properties of y, which is lattice Gaussian. We refer the reader to Ling et al. (2014); Ling and Belfiore (2014); Liu et al. (2021) for further details of lattice Gaussian coding.
The next result describes the scaling of the lattice covering radius Teor Teov(A) ((0,r) r) is covering of R"), where A + ((0,r) is the set composed of spheres of radius cen- tered at all lattice vectors of A (Zamir et al., 2014). This allows one to bound the e² error between vector and its lattice-quantize version by O(n\/2)
min A
Lemma D.3. Let A be a n--dimensional lattice with volume 0T/22 Then its covering radius satisfies
Teov S C2//
((1
1/2
[
2
1/n
Onn
(62)
π
for positive constant Cz-
Proof. Using results in Zamir et al. (2014, Ch. 3),
Tenv (A) Peow (A) Toff (ot((A)
(63)
V(A)
1/n
< C
V
Vn
(64)
1/v.
C
πE
C
(65)
(n/2jJ
1/2
C1
CV
((n/2)
(66)
(m')2)
(67)
where C' is a constant, Peov is the covering efficiency, reff is the effective lattice radius, and V, is the volume of a n-dimennsionll unit ball, following Zamir et al. (2014, Ch. 3).
The next lemma bounds the second moment of Q(aæ)), which is used to bound the error terms.
21

---

Lemma D.4. Let c y NN u) and z ~ NO, vla). Let بيبة If [A J==1 a sequence of lattices that is AWGN-good with vanishing error probability of MAP decoding of y given ã. then for any 6 >> 0,
= +e,
(68)
for sufficiently large
Proof. The second moment of Q(ax) satisfies
B[I[Q(aa)]13 = + +
(69)
= |1111111 + +[[[((0a]]1((1]]
(70)
S || +
(71)
[|lv|]] + [[[[((()))
(72)
S [1]]]]]] + [ P(E),
(73)
where in (71) we use Hölder's inequality, and the last step is by Cauchy-Schwarz. The second term involving the error event vanishes as follows. We have that
[[((a))]** v (()((a))) az/|* + 8|
(74)
S 8| y - oz + 8 |
(75)
8|| - a) - az 8llayy + az
(76)
where (74) by triangle inequality and Cauchy-Schward and (75) holds since Q finds the closest
vector to ozC We have that the expectations satisfy
[[I((1 - a)y - a=
(77)
s (1 a²3²
x + 2(1 - a) [
(78)
< 3(1 (0² - +(n+
[1 || B[[[=|1]]
(79)
< 3(1 a)²(0² - -) - - 2) v)
(80)
O
(81)
and
E Elllay + = =1
2) 20 E
||11|111211
(82)
<00 -tlooo-nn-tttnn(-)))) B | ||
(83)
30" -
3o 2 v)un?
(84)
= O(n)),
(85)
for sufficiently large. This implies that
E [[I[((a0)]1 < 8 a)y-
+ +8E[ (10y-02211 1=]44]
+ az|l = O(1²),
(86)
and therefore
/EIlQ(a)))11PC <e
(87)
for n sufficiently large since lim, P(E) = 0.
L
Before proving Thm. 4.5, we first prove a lemma which describes an achievable RDP region of pre/post-scaled lattice quantization with private dithering on the Gaussian source that makes the relationship between the flatness factors and the scaling s explicit. Ensuring that all flatness factors vanish would imply a single-letter characterization of the achievable RDP region for the near-perfect perception regime, which is what Thm. 4.5 provides.
Lemma D.5. Let X1, X2,... iid. N((,02). Leive U E (0,0²), a d' and > 0, 9 > be constants satisfying
-Qv v)v
0²
(02 -v + +
(88)
02
B2
22

---

Let y y NA(c, v) and = ~ that AWGN- with vanishing error probability of MAP decoding of y given æ, and is also guantization-godd then for any 0.
vI, If {A Jn=I is sequence of lattices
0² H H H(QA))(aX))) - -e. log
+ Ch +e).
(89)
B 81111 s tc= (1- 3)2 -v)
-v² + 2 102 v)w
+ t0)
(90)
Be: (2 0202 -2 + 2z²
lo²
-
v)u
W2(Pxx, Pgn) S V 880 +8//6
(21 4ez
2
te,
(91)
0²
for n sufficiently large, where X = B ((Qam)) + su), u (n))( In the above, ch 1og(1-e1 + x(1- +1)et and m1(-(1)
V -N -V
(V(o--100 v)v
(V-zz 1 v) -
EA(m)
62 = (((o)
eu == (xin)
0² 52
(92)
are flatness factors of lattices A Am)) with 0 +<t<< 1/e.
Proof. Since the lattice sequence is AWGN-good, choose the fundamental volume to satisfy ((A())) (2mee (o²- In the following, denote the quantizers QAGi Q with the dependence on the lattice implicit.
x² (1) $ where c2 "= 0, following that of Ling and Belfiore (2014).
Define & (Q(aæ) o y as the "error" event of MAP decoding of y given ø (Ling and Belfiore, 2014, Lemma 11). By assumption, lim P(E) 0. Finally, since the lattices are sphere- achievinhe we have that the lattice NSM satisfies lim,-+x (() + T
1. The vanishing
error probability and sphhre-bound-achieee properties of the lattices will be used to establish the convergence.
We first address the perception constraint. By triangle inequality,
1 W2 Px.)) Vn
1 Vn
1
W2(Pxe Po(Q(o))+suu) /n X,, Wa Po(((oa))su))-
(93)
We first bound the second term on the right. By Regev (2009, Claim 3.9) and Ling and Belfiore (2014, Lemma 9), [dP dPx (x)) S (x)) Yo. By the change of variable formula, this implies IdPaz (x) - dP (x)xx(x)l S 4e,dPax (z). Additionally, we have that for any A E A,
dPas S /
Pr(Q(aæ) A) = No+A
(1 + 4e: = (1 + de,) Pr(Q(aX)) = A), (94)
Vota
and Pr(((att) = A) 2 (1 - (c,) 4cz) Pr(Q(ax)))) = A) follows similarly. Thus Pr(Q(aā) X) Pr((Q(xX)) A)1 S 4e, Pr(Q(ax") A).
(95)
Finally, we have
dP tPQla (w) Σ Pr(Q((at) = A) A)
(96)
xEA
<
Σ (1+ + Pr(Q(ax") = A)dPou(W A)
(97)
= (1 + 4c, )d PPQlaax*)+suutl
(98)
where the other direction follows similarly. Thus
( (w)) d(Q(ax))+su (cddPa(ax))-au(u))
(99)
23

---

for any 20. This implies
1
1
B Wo (Pxpx= P(()(oa)+su))
W2( W2(Po(ox))+to Por
(100)
vn
2 f [101 d dPQ(axx)+su (w))))
$ vn
(w)
(101)
$ 8c, [lll((aX") + sull?),
(102)
where (101) is by Villani (2016, Thm. 6.15). To bound the second norm, first get
[] + su [|] - [II n |1|lsu1??
(103)
"
2 )1I 28 (02 uiv
(104)
x²
The second moment of (ax")) satisties
[11(((ax))133 = | =1)
(105)
Σ (A(2(P)(Q(at)) = A) -Pr(Q(ax") A)) + Σ |||1 Pr Q(až) = A)
(106)
S 4r, BlIl((x"))) =(llQ(())))))))))((
(107)
where the last step is by (95). Thus for n sufficiently large,
|] 4e, B[[lQ(aa)]
(108)
4e, (0² (o² v),
(109)
where the last step is by Lemma D.4 and Banaszczyk (1993). Combining with (102) and (104),
W2 Ps(Q(ae)) S 81/ 88g 2 4r,
v + 252
2222 (02
v)u
(110)
o²
For the first term the right of (93), we again apply triangle inequality:
1 Vn
W2(Px*, su)) S W2(PxM, Po(v+ou)) + W2 (P(PsiQraa) Pplerts))
Vn
(111)
For term let ã v((0, (02- (o2-ur1 In) and N((0, so.n)) Then
Vn W2(Pr. Pytoz) 5 Vn /
2
|| - dPytosldu ldw
(112)
8en E 1|
(113)
8e x²
(114)
g2
24

---

where (112) is by Villani (2016. Thm. 6.15), and (113) holds since s, satisfy (88) and by applying Ling and Belfiore (2014, Lemma 9). Therefore
1 Byn
W2(Px=, Po(ytsw)) = W2(Pr, Pu-su)
(115)
Vn
1
(2 Pa p=-s + 1 vn
W2( ytoE Pytou)
(116)
vn
8c
+
8
W2( (u, Pz)
(117)
B²
vn
0²
(0² (o²-v)j
v/v
8tn
+
8
2
DKL (ul())
(118)
B2
Vn
0²
0²
BEN
+
(119)
B2
28
where (117) holds by Ling and Belfiore (2014, Lemma 9) for n sufficiently large, and data processing
inequality for W2 (Santambrogio. 2015, Lemma 5.2). (118) holds by Talagrand (1996), and (119) holds because
DKL (ull))= = n
En [ logge(()) H(u)
(120)
7
72
1
Eul Eull|z|]
+ n
log
2n (0² Lo2-vju- v)v
}
log ((A(m)
(121)
22 2
2
o²
n
o²
(((())) 2
+
62
1
log
1
(122)
2ln
2
c(1 + (2)
1
(G(A())) 2Te(1 + 62) -1)
log(1 + E2)
(123)
2n2
2
m0 0,
(124)
where we use the fact that Bll B|||u||]] G(A(m)) 2Te (o-=1))0 (1 + 62) and the lattice sequence is Thus
A Vn
W2(Px"; Po(v+t)) S V802en 2
(125)
For term B, let us first divide by 3 and analyze the squared 2-Wasserstein; this gives us ( (o(o))+su Pytsu) S W3 (Q((a)) Py) by Santambrogio (2015, Lemma 5.2). Let æ be the coupling between Po(aa), Py induced by the joint Pa.yi i... ÿ. y i means that ý Q(a(( + z)) with x ~ NO0,vI,,, as defined above. Then,
1 (3(( Pa)= n
1 min
Eg By [l|g]
(126)
n x (lIE(()
1
1
Eg y [l|g] -11
1yy [llg-- ]]]
(127)
n
n1
1
Ey
||ا 1/13 [0] P(E9)
+
1
Fy
F
[1] [1)]
(128)
n
n
Ey
[lli - vl 118]
(129)
1
<
v
E و y 11()
(130)
n
where E = (Q(aæ) # y) = ((a - 1)y + az & Vo(A)) the "error" event that quantizing aT does not equal the lattice Gaussian y, and the last step is by Cauchy- Schhar. For (129), we have
25

---

that Ey | y/| [0] -y. ||| y y) 0. Note that
1
y/1 = - az + az
yl1 I / aze + 1 cvēl
min y'EAM)
1/ - axt
+ 11 a.
(131)
y
+ 11 - az6l = 2 11 -
(132)
This implies that
B y||*
(133)
16 166B[1ly o:30 [=]
(134)
E [[[(1 ax y - [=
(135)
16(1 - aa |||l|/|| + 16a -|||=||*| -1a) x) + 32(1 - a)²a² Bllly] ||
(136)
where we use the fact that 2 are independent. Note that E1lby| v 3(0² 2² for sufficiently
large (Zhao and Qian, 2024; Micciancio and Regev, 2004). ||||=||*| = v2n(n+ and E (y z)²1 <
E 1 [| I B |22] [|| x by Cauchy-Schware and independence. Additionally, E|1by
n(o² v) by Banaszczyk (1993). Therefore
E l1 - yl|* v 48(1 - a)* (o² - =v)nn + 16a v²n(n 2) + 96a - a)² B[
B =|
(137)
$ 48(1 - a)* (o² v + 16a 22n + + 96a 2 - nn²n - v)v
(138)
(
(139)
By the choice of a, (ax) computes the MAP estimate of y (Ling and Belfiore, 2014, Prop. 3). Therefore,
E1 P()) I
(140)
432
for sufficiently large, since the error of the MAP estimate satisfies (Pohn+so((()) 0. Hence
B
W Wa(Ps(Q(aa)+su) PB(gg)))
W2 (PQ(aa)) Pa)
(141)
Vn
Vn
(129)
8
E Ey.
11l b1121(8)]
(142)
(140)
8
(143)
23
In conclusion, we have shown
02 -v v
(02-1)0 v)v
Vn W2(Px", Pxa) 2o + B
Se:
2 21 4r,
+ 2s²
te.
(144)
Next, we address the distortion term. We have that
1 Exn,a [|X" -((((ax"))) - suu]ll] En
1 I | --(((0x)) sud)pp (x) dd2(x)))dk
1
Ei
||| -B((((0()) s2)
(145)
n
82
26

---

By Regev (2009, Claim 3.9) and Ling and Belfiore (2014, Lemma 9), (d))(x) - dP)x((x) 1 4e,dPx x- (x), Vx. The first term S₁ can be written as
- S S 4c; 1 n / || (ol((=))>s')l(PPpppp = Acc Ex -1011=1 =(Q((xx)) - su')1l?
(146)
for any u' and therefore
Ex a [11" 1 - su)l 1 1 4ez S2.
(147)
n
We focus on the S₂ term. As before, let & == {Q(aæ)) + y) be the "error" event. Then
1
_
E
|
B(Q(aæ)) su)l|
E [ ((Q(aa)) -u))l=
(148)
71
n
< -
1
E
|
B(Q( (1 +
1
/ / B(Q(az)) su]ll{]((E))
(149)
where we use Cauchy-Schward Schwarz. Note that the term with the 4-th moment satisfies
[| ((Q( - s1
(150)
s 8(1 30) |" + 8 [IQ(až) az + 83
(151)
The second term is (n²) by following (136), and the third term satisfies
8, B/ su ||]]
8,35* (A Onn²)
(152)
by Lemma D.3. For the first term,
B|| x|| []-
B[Ig + = S BB|ly||11 + ||||=||] (11l)1 6 ||||=| ||3]
(153)
(154)
S 3(0² - 2) + v2n(n + 2) + 6(o² - v)un?
(155)
= O(n²)
(156)
for n sufficiently large, by the independence of y and z, and using 4th moment results on lattice
Gaussians from Zhao and Qian (2024); Micciancio and Regev (2004). Since lim P(E)= = 0, we
have that 1 / E E ((Q(az)= su) [+] ((E) < 2 for n sufficiently large. Thus
S n S² E || - B(Q(az)) - su) (2))|11(88)
+ 2
(157)
| - B(y - su) su) |3 + e
(158)
2
| - B(y su)
11 1
0
11o0 + 2
(159)
1
E
Bly I||@
su) 12| + 2/2
(160)
1
E
[I[[1
B)y+ + Bsul| || ?|
+ e
2/2
(161)
n
= (1 3)2_ E[lb| 21 1 |||| + 2 _ Elll ul $1 + c
(162)
71
2
(02 -v² v)v
< (1 8) v) + 5²32 (()(m)) 2Te
(1 62) +
(163)
o²
(1 (1- 3)2 v) + 32 (02 v)v
+c,
(164)
for sufficiently large, where (159) is by Hölder's inequality, and (163) holds by Banaszezyk (1993), and since the lattice second moment satisfies J [ = G(A())) ((A(m))2/n and the lattice sequence is quantization-gont (i.e., baahere-boun--chieving The result follows by combining (147)
and (164).
27

---

Finally, we address the rate term. We have
1 H(Q(ax))) = Ex |- log Pe (((x)))
(165)
n n
1 Ex [- log d ((aa)
log (og(dPQqcom)) (dPxx(xr) (dPP) - dPa()))) +
n
R₂
(166)
The first term RS will vanish as + co, due to the following. By Regev (2009, Claim 3.9) and Ling and Belfiore (2014, Lemma 9), we have that [dP=(x=) - dPx (ar) & 4x.dPx= (ær), Va, by the choice of the lattice sequence A Am),, for n sufficiently large. Therefore,
1 /
R1 -
log d (ddx (2)) dP&
(167)
1
/
S 4e, n
log (d ((2))()))))pxx
(168)
4e, n
H(Q(ax)))
(169)
which implies that
H(Q(ax")) $ 1 4c,
1
R2,
(170)
n
for n sufficiently large.
For R2, this is the per-dimension entropy of ((az). Let p(A) = Pr(Q(ax) = A) be the PMF of
Q(ax)) supported on À A, and let qy (A) be the lattice Gaussian PMF of y (0, v). Then
H(Q(aT)) = Σ p(A) log
(171)
AEA
Σ p(A) log (A)
(172)
AEA
Σ p(A)
2( -1j |||12 + log log (A)
(173)
AEA
1
E B[1Q(a0))11] + log (A).
(174)
2( v)
Combining, we have that
1
R₂ -H(((aæ))
(175)
11
1 [[((((az)) [| + log (A)
(176)
2(02
D)
+ +e(eo² - w) E |byl2
+ log (ogP(oz_
(177)
2(02 D)
1
1
1 n
E [[
log ( te
(178)
20 v) n
1 Σ
My (A) log 9, (A) +e
(179)
dEA
1
(()) +
(180)
28

---

for n sufficiently large, where (177) is by Lemma D.4. Therefore, for n sufficiently large, we have
1 H(Q(ax))) = 1 - 4e;
1
H(y)
+
(181)
n
1
[
o²
$
log
+ + +e
(182)
1 4e
2
v(1 + E2)
1
[
0²
log
+ EN +E
(183)
1 4
2
where (181) is due to (170) and (180). of volume, for sufficiently large.
(182) holds by Ling and Belfiore (2014, Lemma 6). the choice
We now prove Thm. 4.5, which is restated below. Theorem D.6 (Optimality of PD-LTC for Gaussian sources (Thm. 4.5 in main text)). Let
X1, X iid. o(0,o²). For any D satisfying O < D S 20", there exists a sequence of PD-LTCs
((90)) Is , ) such that
(n) (n)
lim - [H(QAmm (x²x))))) S R(P/2, 11000 n
(184)
lim 1 = 8011 s D,
(185)
a n
1
lim WP (Px,, Px=) = 0,
(186)
n n2
where 8 = go (Q (ga" ( (X")) su). u ~ (V(((( and s Va D/2
Proof. Set 8
9 1, and v/ = D o Vo²-n"
which satisfies (88). Choose the lattices A (n) as polar 2
lattices (Liu et al., 2021: 2024). By the choice of s, Gu E == Eg(n) (V (a2 2/v
and therefore both flatness factors will vanish exponentially fast as n + oo by Liu et al. (2021, Prop. 1). Additionally, by using the fact that E S (Λts)
(V (Voe2 \V=))) and since Es vanishes, taking t - 0 results in vanishing E and therefore vanishing ENS see Ling and Belfiore (2014, Sec. III). By Liu et al. (2019), polar lattices are AWGN-good and have exponentially decaying MAP error probability for the Ø y AWGN channel with lattice Gaussian input, any signal-to-noise ratio (and therefore any v E (0, o²)). By Liu et al. (2024), the polar lattices are also quantization-good Therefore, by Lemma D.5 we have that the rate, distortion, and perception satisfy
1 20
lim n-+0c
H(QAio) (x"))
log
(187)
D
- 1 -=[1X"X""11" D.
lim
(188)
n-000 n
1
lim
W2(Pxx,, Pxm) 0,
(189)
200 n
when the transforms are chosen to be g!! (v) ar and g (1) (v) Bv.
Remark D.7. In the proof of Lemma D.5 (and therefore Thm. 4.5), the condition in (88) ensures that the distribution of X" is approximately Gaussian with covariance x² I,, and the closeness of this approximation is quantified by cu. This essentially controls how well the perception constraint can be enforced, which is why s shows up in the flatness factor. One can see that 8 needs to be sufficiently large to ensure c. 6z which guarantees e, vanishes by Liu et al. (2021, Prop. 1). If is too small, en c, and it is not guaranteed it will vanish. We see that the choice of Vo- 2/2
indicates
a dither su that "bleeds" outside the lattice cell at low rates sufficient to ensure a vanishing 6, and therefore vanishing perception. This seems to imply it is not possible to choose smaller than
29

---

the one chosen and still make e, vanish; if this were possible, it would imply that a near- pear-perfect perception rate-distottion tradeoff below R(2/2, oo) is achievable, which would contradict results in the RDP literature that RDD/2,00 00) is the lower bound (Yan et al., 2021; Wagner, 2022; Chen et al., 2022).
On the other hand, is used to approximate Px" Re Pa. and En. is used to approximate the entropy rate y; shown in the proof, vanishing e, implies vanishing e/-
Remark D.8. The sequence of PD-LTCs in Thm. 4.5 that satisfy the near-perfect perception constraint in (16) can be upgraded to a sequence of codes satisfying a perfect perception constraint of Px" Px= with the same asymptotic rate and distortion, by following the "coupling argument" of Saldi al. (2015). That is, if "x"Xx" the coupling that satisfies
n W (Pxx. Px.)) ) e, then can use X" X"" as the reconstruction instead of X". This would ensure Px- Px* and the
distortion - [[[[| - X -=1I < 72 _ - [²] Since this change only affects the decoder, rate remains the same as before.
Remark D.9. The choice of polar lattices in the proof of Thm. 4.5 may not be necessary, and other lattice families may also work. The essential requirements are that both e, and 6, vanish, AoGN-goodness, crantization-goodnee
and vanishing MAP error probability for the ã y AWGN channel. While the latter three can be simultaneously satisfied by other lattice families, such as the mod-y lattices (Loeliger, 1997), to the best of the authors' knowledge, the only currently known lattice family with known results on vanishing and is the polar lattice. This does not preclude the existence of other lattice families that may satisfy all the aforementioned requirements.
E Empirical Evaluation of Sliced Wasserstein
estimate the squared 2- Wasserstein Distance (P, with the sliced Wasserstein distance (P, Q) (Bonneel al., 2015). Let P (1, I,), and (O,, 2I,)) Then _ (P.QQQQ) = 2. Shown in Table 2, 1, sliced Wasserstein provides fairly accurate estimate of the true Wasserstein for Gaussian samples, where N is the number of samples. This supports the use of sliced Wasserstein as a proxy for the Wasserstein distance in our experiment surrounding the Gaussian source (as we would expect the reconstruction distribution to be near- Gaussian). Therefore, the theoretical bounds are a meaningful comparison, as they align with the operational quantities the corresponding coding theorem.
Here, assess how accurate
N
Estimate
Std. Error
100
2.129
0.282
12
1000
1.999
0.187
5000
2.004
0.158
10000
1.994
0.164
100
2.118
0.235
1000
2.017
0.192
= 24
5000
2.005
0.188
10000
2.008
0.184
Table 1: Estimating W3 using weede-WWasersttinn with 50 projections.
N
Estimate
Std. Error
100
2.142
0.329
1000
1.987
0.299
5000
1.999
0.274
10000
2.009
0.276
Table 2: Estimating W3 using Bliced-Wasseree with 20 projections.
30

---

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Reviews:
[{"review_id": "BIbk66pqOC", "reviewer": "Reviewer_RfEa", "summary": "This paper investigates how to effectively use randomness in neural compression to balance the trade-off between rate, distortion, and perception (RDP). The core method is based on Lattice Quantization, introducing randomness via \"dithering\". The authors clearly present three architectures for scenarios with infinite shared randomness (SD-LTC), no shared randomness (PD-LTC), and a novel scheme for finite shared randomness (QSD-LTC). The paper not only proves the optimality of its methods on an ideal Gaussian source but also experimentally validates that more shared randomness and better lattice structures improve performance. While I am not an expert in vector quantization, I believe this work clearly connects information-theoretic concepts with practical compressor design.", "questions": "none", "limitations": "yes", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1.A major strength is the paper's solid theoretical foundation. The authors prove that their proposed methods achieve the theoretical RDP limits under the standard Gaussian source assumption, which provides strong justification for their model design.\n2.The paper does an excellent job of explaining the roles of different types of randomness (shared, private, finite) in compression and provides corresponding solutions. The model architectures are intuitive and easy to understand.\n3.The experiments effectively support the paper's theory. The results demonstrate that using better lattices (i.e., with better space-packing efficiency) and more shared randomness indeed leads to better RDP performance.\n\nWeaknesses:\nThe experiments are primarily conducted on relatively small-scale datasets (e.g., MNIST). State-of-the-art image compression techniques are typically applied to high-resolution images with much more complex models. It is therefore unclear how well the proposed methods would perform in these more challenging, real-world scenarios (e.g., Kodak dataset), which limits the assessment of their practical impact.", "was_revised": true, "final_justification": "The authors' rebuttal has adequately addressed my concerns, and I am inclined to accept this paper.", "rebuttal": "We thank the reviewer for their helpful comments and feedback, and for recognizing the contributions of our work. \n\nExtending the proposed framework to state-of-the-art high-resolution image compressors is an important direction of our current research. The approach described in the current work could be extended to NTC architectures that handle image data. State-of-the-art architectures for RDP would include HiFiC [1] and follow-up works such as ILLM [2], which are architecturally derived from [3]. Therefore, following [4], we can apply lattices product-wise across the channel dimension, without incurring any notable increase in computational complexity due to the use of dithering or lattice quantization. PD-LTC would involve adding a random dither vector at the decoder in the latent space. The advantage it provides would be an ability to enforce tighter perception constraints compared to deterministic codecs, which we showed for the Speech and Physics datasets. We have preliminary results that show an improvement of FID scores with PD-LTC using the integer lattice with the ILLM model for images: at 0.06 bpp, FID reduces from 5.2 to 4.7 while PSNR remains around 29.1, and at 0.12 bpp, FID reduces from 4.1 to to 3.8 while PSNR improves 30.5 to 30.7. The gain is expected to improve for more efficient lattices. Incorporating SD-LTC in state of the art image compression methods such as ILLM requires some adjustments and variations of the vanilla setting discussed in the present paper. This is because state-of-the-art methods such as [2, 3] use hyperprior entropy models where the latent $y$ is first mean-shifted before quantization and entropy-coding: $y$ is quantized as $Q(y-\\mu)+\\mu$ where $\\mu$ is the mean output from the hyperprior model. We believe that understanding the interplay between $\\mu$ and the random dither vector is important in analyzing the role of dithering in coordinating the encoder and decoder and investigating the benefit when random dithering is incorporated in such entropy models. As a result, we believe that a full set of results on image sources requires a more careful investigation of how random dithering should be integrated with the hyperprior models, and this deserves its own separate work.\n\nNonetheless, our current set of experiments on audio and scientific measurement data, which demonstrate performance benefits for moderate-dimensional data, already illustrate the practical benefits of our design for RDP-oriented compression of scientific and audio sources. \n\n**References**\n\n[1] Mentzer, F., Toderici, G. D., Tschannen, M., and Agustsson, E. High-fidelity generative image compression. NeurIPS, 2020.\n\n[2] Muckley, M. J., El-Nouby, A., Ullrich, K., Jégou, H., and Verbeek, J. Improving statistical fidelity for neural image compression with implicit local likelihood models. ICML 2023.\n\n[3] Ballé, J., Minnen, D., Singh, S., Hwang, S. J., and Johnston, N. Variational image compression with a scale hyperprior. ICLR 2018. \n\n[4] E. Lei, H. Hassani, and S. Saeedi Bidokhti. Approaching rate-distortion limits in neural compression with lattice transform coding. 2024."}, {"review_id": "FyHriLypYR", "reviewer": "Reviewer_GpJK", "summary": "Although the rate–distortion–perception literature is theoretically well developed, it still lacks a practical scheme with rigorous guarantees. This work fills that gap by introducing lattice transform coding (nonlinear transform coding with a lattice-coded bottleneck) evaluated under three shared-randomness regimes: none, infinite, and finite (quantized). The authors prove the scheme is rate–distortion optimal for Gaussian sources and support their theory with experiments on more complex data sets.", "questions": "- As noted in lines 15–16 and 81–82, the RDP performance improves with increasing amounts of shared randomness. Isn’t this relationship somewhat expected? Or is there a nontrivial aspect I’m overlooking? If it’s not trivial, it would be helpful to clarify why highlighting this point as part of the contribution is justified.\n\n- Doesn’t the rate–perception curve for MNIST (Figure 5, bottom left) exhibit somewhat unexpected behavior? It appears that increasing the rate does not consistently lead to improved perception, which seems counterintuitive. Could the authors comment on this?\n\n- Could similar optimal theoretical results be established for QSD-LTC of Gaussian sources, analogous to Theorems 4.3 and 4.5? Since QSD is not mentioned in Section 4, it might be helpful for the reader to include a remark discussing why extending such results to QSD-LTC could be challenging—if that is indeed the case.", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The literature review and problem statement are beautifully written—clear, concise, and comprehensive.\n- The theoretical results for Gaussian sources are particularly valuable and of clear interest to the community.\n\nWeakness:\n- Unless I’m missing something, the experimental results do not seem to offer significant new insights into the benefits of using LTC over NTC—this has already been demonstrated by Lei et al. (2025). Similarly, the observed improvements with increased shared randomness are expected. Overall, the experimental section doesn’t appear to add much beyond the valuable theoretical contributions presented in Section 4.\n\n- Although line 992 mentions that the code is provided in the supplementary materials, I wasn’t able to find an anonymized link to the code or any supplementary files. I’m not sure if this is an issue with the OpenReview system or if I simply missed it.", "was_revised": true, "final_justification": "I have read the authors’ responses and found their explanations convincing on all points, so I have revised my rating accordingly. The paper already presents interesting and relevant results for the community, and the minor concerns I raised have been satisfactorily addressed in the response.", "rebuttal": "We thank the reviewer for recognizing the quality of the writing and theoretical contributions of our work.\n\nWe would like to first clarify several points of our work related to weakness 1 and question 1. On shared randomness, while it may be expected that more shared randomness should not hurt the performance, it is not immediately clear how shared randomness, infinite or finite, could be used to boost the performance in an efficient/optimal manner as promised by results from information theory. The extent of this performance gain is also unclear in practical settings (finite dimension $n$, real-world sources beyond Gaussian, etc). In the classical rate-distortion (RD) setting, no randomness is needed to achieve optimality. As a result, most neural codecs, whether oriented for RD or rate-distortion-perception (RDP), do not necessarily use shared randomness or randomness at all as a resource. Recent information-theoretic work (see Sec. 2) has shown that not only randomized reconstructions are needed to satisfy the perception constraint, but also _infinite_ shared randomness is needed to achieve the RDP function. Any less shared randomness results in worse performance. However, these works do not provide constructive schemes that one may implement in practice. In contrast, our work provides a practical scheme that provably achieves the optimal tradeoffs for infinite and no shared randomness as derived by the information theory literature. The experimental results show that our proposed construction utilizes the shared randomness in an efficient manner as predicted by prior information theory work and the theory we established. These trends extend to compression of real-world sources in addition to Gaussians. Furthermore, the theory we provide is asymptotic (e.g., in the limit of large dimensions), but the experimental results show that the predicted behaviors are still exhibited by moderate-length lattice codes (such as $E_8$), which is enough to significantly bridge the gap to $R(D, P)$ and $R(D/2, \\infty)$. Finally, while a theoretical proof of QSD-LTC (the finite randomness case) is left for future work (see below), the experimental results show that it can nearly achieve the performance of SD-LTC despite not having infinite shared randomness. \n\nRegarding the benefits of LTC over NTC, Lei et al. (2025) consider the RD setting but do not address the RDP setting. As mentioned before, randomized reconstructions are necessary for RDP. It is therefore not immediately clear whether VQ-like coding (which is optimal for RD) is good for RDP; it is further unclear how VQ should be integrated with randomized codecs as VQ is deterministic. Our work reveals that VQ-like coding is indeed good for RDP (Appendix A), and then showed in Section 3 how lattices, which provide VQ-like regions, could be incorporated with shared or private randomness for neural compression via shared/private dithering. Thus, our experimental results confirm that under the RDP setting, SD/PD-LTC is indeed superior to SD/PD-NTC, is able to approach optimality (as predicted by Thms. 4.3, 4.5), and extends to general sources.\n\nQ2. Regarding the MNIST figure, this is not unexpected. The behavior is the result of the RDP tradeoff being a triple tradeoff, the regimes where the perception constraint is active or not, and the way the compressors were trained. To generate the curves for the general sources, we swept $\\lambda_1$ (the distortion weight) and kept $\\lambda_2$ (the perception weight) fixed to a positive value. At large rates, the distortion being small makes a small perception easy to achieve. At near-zero rates, the objective is essentially just rate and perception, so the model would prioritize making the perception small. At moderate rates, there is more tension between the perception and distortion constraints, and the tradeoff regime depends on how $\\lambda_1$ and $\\lambda_2$ relate. If a larger $\\lambda_2$ value was chosen for the moderate rate points (e.g., for PD-LTC/NTC), then their perception values would be lowered, at the cost of worse distortion. However, the current figure 5 still illustrates our main arguments: (i) deterministic NTC/LTC cannot enforce the perception constraint at low rates compared to PD-NTC/LTC and SD-NTC/LTC, and (ii) SD-LTC achieves a superior rate-distortion tradeoff compared to PD-LTC while also achieving a lower perception. If PD-LTC’s perception were lowered, it would only worsen its rate-distortion performance, but SD-LTC is already outperforming PD-LTC. If $\\lambda_1$, $\\lambda_2$ were further chosen more particularly, a nicer-looking convex plot would be seen. However, this type of training and figure presentation that we do is consistent with prior RDP works such as [1, 2], and as such the two plots should be examined jointly. \n\nQ3. Regarding QSD-LTC, this is indeed more difficult. Similar to the proof of PD-LTC, QSD-LTC cannot make use of the additive channel equivalence of dithered lattice quantization. Therefore proving a fundamental limit for QSD-LTC would likely require a similar approach to PD-LTC based on lattice Gaussian analysis. However, unlike PD-LTC, it would require further understanding of the behavior of a discrete dither vector that is distributed over the nested lattices. To the best of our knowledge, this type of dithering has not been studied before in the context of lattice Gaussian coding. Therefore the QSD-LTC analysis deserves its own treatment based on new tools developed for dither vectors defined over nested lattices. We will update the paper with a remark reflecting this. \n\nW2. The code was added to an anonymous link, however, it appears that an error occurred during the submission and this was not included in the supplementary material. While it could be added here, the new rebuttal policies disallow sharing of anonymous links, and we apologize for this inconvenience.\n\n**References**\n\n[1] Y. Blau and T. Michaeli. Rethinking lossy compression: The rate-distortion-perception tradeoff. ICML 2019.\n\n[2] Muckley, M. J., El-Nouby, A., Ullrich, K., Jégou, H., and Verbeek, J. Improving statistical fidelity for neural image compression with implicit local likelihood models. ICML 2023."}, {"review_id": "ILKH9OY2bo", "reviewer": "Reviewer_jGLB", "summary": "- Authors propose LTC with infinite or no shared randomness, using a shared dither (SD-LTC) or\nprivate dither (PD-LTC) respectively, and describe the benefits of the former. \n- The authors propose a discrete dithering scheme, defined via nested lattices, which enables finite randomness to be shared between the encoder and decoder (QSD-LTC). This scheme provides an interpolation between PD- and SD-LTC, thereby enabling control over the rate of shared randomness.\n- The authors analyze SD-LTC theoretically and demonstrate its optimality, while studying the performance of PD-LTC, SD-LTC and QSD-LTC on synthetic and real-world sources empirically.", "questions": "- Does it have any advantages over other lattice quantization methods? For example, paper [a] discusses image compression using lattice vector quantization. In terms of RD curve efficiency and computational efficiency, what effect will the proposed method have compared to this?\n\n[a] Xi Zhang, et al. “Learning Optimal Lattice Vector Quantizers for End-to-end Neural Image Compression”, NeurIPS 2024", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength\n- Fundamental improvements in quantization in neural compression are expected.\n- It is interesting to note that LTC using shared randomness improves compression coding efficiency, both theoretically and experimentally (synthetic and real-world sources across multiple domains).\n- By treating finite shared randomness (QSD-LTC), the work may bridges the gap between purely theoretical extremes and practical systems.\n\nWeakness\n- As this paper covers fundamental content, it is reasonable that the experiments are performed on relatively low-dimensional data. However, in the modern field of deep image compression, datasets such as Kodak and CLIC are often used for evaluation rather than small-scale data such as MNIST.", "was_revised": true, "final_justification": "After reading the authors' responses, I have decided to maintain my positive evaluation.", "rebuttal": "We thank the reviewer for taking the time to provide valuable feedback and comments, and recognizing our work. \n\nExtending the proposed framework to state-of-the-art high-resolution image compressors is an important direction of our current research. The approach described in the current work could be extended to NTC architectures that handle image data. State-of-the-art architectures for RDP would include HiFiC [1] and follow-up works such as ILLM [2], which are architecturally derived from [3]. Therefore, following [4], we can apply lattices product-wise across the channel dimension, without incurring any notable increase in computational complexity due to the use of dithering or lattice quantization. PD-LTC would involve adding a random dither vector at the decoder in the latent space. The advantage it provides would be an ability to enforce tighter perception constraints compared to deterministic codecs, which we showed for the Speech and Physics datasets. We have preliminary results that show an improvement of FID scores with PD-LTC using the integer lattice with the ILLM model for images: at 0.06 bpp, FID reduces from 5.2 to 4.7 while PSNR remains around 29.1, and at 0.12 bpp, FID reduces from 4.1 to to 3.8 while PSNR improves 30.5 to 30.7. The gain is expected to improve for more efficient lattices. Incorporating SD-LTC in state of the art image compression methods such as ILLM requires some adjustments and variations of the vanilla setting discussed in the present paper. This is because state-of-the-art methods such as [2, 3] use hyperprior entropy models where the latent $y$ is first mean-shifted before quantization and entropy-coding: $y$ is quantized as $Q(y-\\mu)+\\mu$ where $\\mu$ is the mean output from the hyperprior model. We believe that understanding the interplay between $\\mu$ and the random dither vector is important in analyzing the role of dithering in coordinating the encoder and decoder and investigating the benefit when random dithering is incorporated in such entropy models. As a result, we believe that a full set of results on image sources requires a more careful investigation of how random dithering should be integrated with the hyperprior models, and this deserves its own separate work.\n\nNonetheless, our current set of experiments on audio and scientific measurement data, which demonstrate performance benefits for moderate-dimensional data, already illustrate the practical benefits of our design for RDP-oriented compression of scientific and audio sources. \n\nRegarding the types of lattice quantization methods, our theoretical results in Section 4 suggest that one needs a sphere-bound-achieving sequence of lattices to be optimal on the Gaussian source. In other words, on the iid Gaussian source, performance should be best when the normalized second moment (NSM) of the lattice is minimized, and therefore other lattice quantizers would be suboptimal (unless they coincide the NSM-optimal quantizer in each dimension). The lattices chosen, such as the moderate dimension ones used in the experiments, or the high-dimensional ones such as the polar lattices (mentioned in Sec 4), all have low complexity CVP and low NSM. For real-world sources, while our experiments in Section 5 suggest that lower NSM lattices provide better overall RDP performance, it is possible that even better RDP performance could be achieved using the methods in paper [a] which optimize the lattice directly while using the Babai rounding estimate to perform approximate CVP, which avoids the exponential lattice vector search, as there are currently no theoretical guarantees on more general sources. Combining the ideas in [a] with our proposed methods could be an interesting area for future research.\n\n\n\n**References**\n\n[1] Mentzer, F., Toderici, G. D., Tschannen, M., and Agustsson, E. High-fidelity generative image compression. NeurIPS, 2020.\n\n[2] Muckley, M. J., El-Nouby, A., Ullrich, K., Jégou, H., and Verbeek, J. Improving statistical fidelity for neural image compression with implicit local likelihood models. ICML 2023.\n\n[3] Ballé, J., Minnen, D., Singh, S., Hwang, S. J., and Johnston, N. Variational image compression with a scale hyperprior. ICLR 2018. \n\n[4] E. Lei, H. Hassani, and S. Saeedi Bidokhti. Approaching rate-distortion limits in neural compression with lattice transform coding. 2024."}, {"review_id": "hBlT2jqEg2", "reviewer": "Reviewer_MW7C", "summary": "This paper introduces a neural compression models that aim to the theoretical optimum under the rate-distortion-perception (RDP) tradeoff. Building on recent non-constructive RDP coding theorems, the authors propose constructive implementations leveraging lattice-based transform coding with varying degrees of shared randomness. Their framework includes deterministic LTC, private and shared dither variants (PD-LTC and SD-LTC), and quantized shared dither (QSD-LTC), which are theoretically motivated and experimentally evaluated. Via experiments, this work demonstrates how shared randomness on lattice quantization affect RDP performance.", "questions": "1)\tIn the QSD setting, was the shared randomness rate (Rc = 1.0 or 1.58) included in the total rate?\n\n2)\tI understand that the proposed method uses a much more structured form of randomness compared to approaches based on pseudo-random seeds. In addition, the authors claim that the seed-sharing approach using k bits is not suitable for high-dimensional data due to the limitation of having only 2^k distinct seeds. If multiple groups of data dimensions share the same pseudo-random seed, would the proposed method still show a clear advantage? For example, in the case of 2^(k+2)-dimensional data, it seems possible to divide the data into 2^2 groups and assign a shared pseudo-random seed to each group based on a predefined rule, thereby constructing group-wise shared dithers within the lattice space. I am curious whether the proposed method demonstrates a clear benefit compared to such a simple implementation.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "[Strength]\n\nOverall, the paper is well-structured and contributes both theoretically grounded and empirically validated insights into the design of RDP-efficient compressors. Particularly, this work effectively bridges the gap between theory and practice by demonstrating how lattice packing and shared randomness affect RDP performance on both synthetic and real-world datasets.\n\n[Weakness]\n\n1)\tHowever, one notable limitation is that, although the authors claim the main advantage of their method over existing approaches, such as RCC, lies in its practicality, the experimental validation is conducted only on the limited, low-dimensional real-world datasets. While the authors mention this as future work, the paper would have been significantly stronger if such validation had been included.\n\n2)\tWhile the proposed architecture is theoretically motivated by the RDP optimality framework, the empirical results suggest that even with infinite shared randomness (i.e., SD-LTC), there remains a non-negligible gap from the theoretical lower bound R(D, 0), indicating practical limitations in achieving RDP-optimality. In particular, as shown in Fig. 4, even under high-distortion constraints, the rate achieved by the proposed method appears to exhibit a form of lower-bound behavior. This observation could be a critical limitation, especially considering that recent perceptual codec techniques are primarily focused on improving subjective quality in the low-rate regime.\n\n3)\tFor synthetic sources, it is reasonable to compare distortion performance under the constraint P = 0. However, for real-world sources, a trade-off between perception and distortion is observed in some data types (particularly speech). A discussion on this phenomenon is needed.\n\n5)\tFor synthetic sources, the magnitude \"s\" of the PD scheme is deterministically derived. In contrast, for real-world sources, the authors should clarify how the magnitude \"s\" was selected.\n\n6)\t(minor) In Section 3.2, the input to lattice quantization is y, the output of g_a. Therefore, it would be more appropriate to use the variable y instead of x in the notation. Alternatively, if this reviewer has misunderstood the proposed method, clarification would be appreciated.", "was_revised": true, "final_justification": "Thank you for your thorough response. Many of the reviewer’s concerns have been addressed through the rebuttal, and therefore, I am updating my rating to 5 accordingly.", "rebuttal": "We thank the reviewer for their helpful feedback and comments, and recognizing the work as bridging the gap between theory and practice. We address the comments in the following.\n\n**Responses to weaknesses:**\n\nW1. Extending the proposed framework to state-of-the-art high-resolution image compressors is an important direction of our current research. The approach described in the current work could be extended to NTC architectures that handle image data. State-of-the-art architectures for RDP would include HiFiC [1] and follow-up works such as ILLM [2], which are architecturally derived from [3]. Therefore, following [4], we can apply lattices product-wise across the channel dimension, without incurring any notable increase in computational complexity due to the use of dithering or lattice quantization. PD-LTC would involve adding a random dither vector at the decoder in the latent space. The advantage it provides would be an ability to enforce tighter perception constraints compared to deterministic codecs, which we showed for the Speech and Physics datasets. We have preliminary results that show an improvement of FID scores with PD-LTC using the integer lattice with the ILLM model for images: at 0.06 bpp, FID reduces from 5.2 to 4.7 while PSNR remains around 29.1, and at 0.12 bpp, FID reduces from 4.1 to to 3.8 while PSNR improves 30.5 to 30.7. The gain is expected to improve for more efficient lattices. Incorporating SD-LTC in state of the art image compression methods such as ILLM requires some adjustments and variations of the vanilla setting discussed in the present paper. This is because state-of-the-art methods such as [2, 3] use hyperprior entropy models where the latent $y$ is first mean-shifted before quantization and entropy-coding: $y$ is quantized as $Q(y-\\mu)+\\mu$ where $\\mu$ is the mean output from the hyperprior model. We believe that understanding the interplay between $\\mu$ and the random dither vector is important in analyzing the role of dithering in coordinating the encoder and decoder and investigating the benefit when random dithering is incorporated in such entropy models. As a result, we believe that a full set of results on image sources requires a more careful investigation of how random dithering should be integrated with the hyperprior models, and this deserves its own separate work.\n\nNonetheless, our current set of experiments on audio and scientific measurement data, which demonstrate performance benefits for moderate-dimensional data, already illustrate the practical benefits of our design for RDP-oriented compression of scientific and audio sources. Moreover, the theoretical results themselves bridge several sub-fields of neural compression and information theory, including dithered lattice quantization, lattice Gaussian coding, rate-distortion-perception theory, and nonlinear/lattice transform coding. We believe the proof techniques and theorems to be of potential independent interest to all these sub-fields, providing additional value to the paper’s contributions.\n\nW2. The theoretical results provided are asymptotic, i.e., to achieve the fundamental limits, a very large dimensional lattice would need to be used. For practical uses, a finite-dimensional lattice must be chosen. The experimental results show that the predicted behaviors are still exhibited by finite-length lattice codes (such as $E_8$), which is enough to significantly bridge the gap to $R(D, P)$ and $R(D/2, \\infty)$. Using higher-dimensional lattices would help bring the performance even closer to the fundamental limits. Regarding the low-rate regime in Fig. 4, as mentioned in lines 335-336, this is an artifact due to the limitations regarding the Monte-Carlo integration technique for LTC borrowed from Lei et al (2025); see the ablation studies of Lei et al (2025) for more details. Designing lattice-specific entropy models that can avoid this issue is important future research, and is out of the scope of the current work.\n\nW3. For the real-world sources, we swept $\\lambda_1$ (the distortion weight) and kept $\\lambda_2$ (the perception weight) fixed to a positive value. This allows us to compare rate-distortion and rate-perception tradeoffs across methods; note that the two plots should be examined jointly together, and that the individual plots do not show the rate-perception (or rate-distortion) performance for a fixed distortion (or perception). We can see that the pairs of figures show how better lattice packing and more shared randomness improve the rate-distortion tradeoff for the perception constraints shown. In addition, (i) deterministic NTC/LTC cannot enforce the perception constraint at low rates compared to PD-NTC/LTC and SD-NTC/LTC, and (ii) SD-LTC achieves a superior rate-distortion tradeoff compared to PD-LTC while also achieving a lower perception. While not directly shown, the perception-distortion tradeoff depends on the rate regimes, and the way the compressors were trained. At large rates, the distortion being small makes a small perception easy to achieve; on Speech (PD-LTC), this would be observed if larger rates were shown. At near-zero rates, the objective is essentially just rate and perception, so the model would prioritize making the perception small. At moderate rates, there is more tension between the perception and distortion constraints, and the tradeoff regime depends on how $\\lambda_1$ and $\\lambda_2$ relate. If a larger $\\lambda_2$ value was chosen for the moderate rate points (e.g., for PD-LTC/NTC), then their perception values would be lowered, at the cost of worse distortion. If $\\lambda_1$, $\\lambda_2$ were further chosen more particularly, a nicer-looking convex plot would be seen. However, this type of training and figure presentation that we do is consistent with prior RDP works such as [2], and as such the two plots should be examined jointly. We will include this discussion in a future revision of the paper.\n\nW4. For both synthetic and real-world sources, the $s$ parameter is set as a learnable scalar parameter, constrained to be greater than 0. \n\nW5. Regarding notation in Sec. 3.2, we agree it is a bit confusing, and will clarify the notation in a future revision of the paper. \n\n**Response to questions:**\n\nQ1. In QSD-LTC/NTC, $R_c$ is not included in the compression rate $R$. It is simply assumed that a back channel consisting of random bits at a rate of $R_c$ bits per latent dimension is available for use. This matches the setup and assumptions made by the theorems in Section 4, as well as prior work in information theory. \n\n\nQ2. The proposed group-wise dithering is an interesting idea that may have potential in developing alternative efficient dithering methods when finite shared randomness is available. It seems that the proposed group-wise dithering would impose a hierarchical structure on how different dimensions of the dither vector are sampled. Thus, we believe it would implement some sort of nested dithering, which is in line with our proposed QSD-LTC method that is constructed with self-similar nested lattices. However, as mentioned in the last part of Remark B.1 in the paper, QSD-LTC ensures that the finite set of dither vectors chosen are uniformly distributed throughout the coarse lattice cell and thus maximally cover the cell, which is guaranteed by the properties of self-similar nested lattices. It is unclear whether the group-wise approach would select an equally good set of dither vectors. In addition, the example provided for the group-wise approach would still only generate $2^k$ total possible dither vectors of dimension $k+2$ if the \"predefined rule\" assigning seeds to groups is deterministic. Specifically, in the example above, let $s$ be a realization of the $2^k$ possible seeds (available at encoder/decoder), and $i \\in \\{0,1,2,3\\}$ be the group index. If $f(s, i)$ is the rule that assigns a seed to group $i$, there are still only $2^k$ possible combined assignments $[f(s, 0), f(s, 1), f(s, 2), f(s, 3)]$ that generate dithers for the whole vector of dimension $k+2$, by enumerating $s$. If $f$ was random, then more than $2^k$ possibilities could be generated for the whole vector, but then the decoder would be unable to decode unless it had access to $f$’s randomness, which would necessitate additional shared randomness beyond the $k$ already shared. Thus it seems unlikely the group-wise approach would alleviate the challenges of a fixed $k$-bit random seed in high-dimensional cases. Overall, the group-wise approach could be promising, but it is unclear whether it provides nice properties on the dithers that QSD-LTC provides due to the self-similar nested lattices.\n\n\n**References**\n\n[1] Mentzer, F., Toderici, G. D., Tschannen, M., and Agustsson, E. High-fidelity generative image compression. NeurIPS, 2020.\n\n[2] Muckley, M. J., El-Nouby, A., Ullrich, K., Jégou, H., and Verbeek, J. Improving statistical fidelity for neural image compression with implicit local likelihood models. ICML 2023.\n\n[3] Ballé, J., Minnen, D., Singh, S., Hwang, S. J., and Johnston, N. Variational image compression with a scale hyperprior. ICLR 2018. \n\n[4] E. Lei, H. Hassani, and S. Saeedi Bidokhti. Approaching rate-distortion limits in neural compression with lattice transform coding. 2024."}]

Author rebuttal:

Meta-review: This paper proposed a neural compression models to the theoretical optimum under the rate-distortion-perception tradeoff. It is well-written, well-structured, and provides clear theoretical contributions. All reviewers give clear positive reviews (all clear accept each with a score of 5) and in their final justification there is no clear further issues remained. Then, all four reviewers acknowledge the clear theoretical contribution for the compression, i.e., "solid theoretical foundation" (reviewer RfEa), "theoretical results... particularly valuable and of clear interest" (reviewer GpJK), "fundamental improvements" (reviewer jGLB), "effectively bridges the gap between theory and practice" (reviewer MW7C). Thus, the AC decides to accept this paper.
Decision: Accept (spotlight)