Gaussian Rate-Distortion-Perception Coding and Entropy-Constrained Scalar Quantization
Li Xie Liangyan Li Jun Chen Lei Yu Zhongshan Zhang








































































Gaussian Rate-Distortion-Perception Coding and Entropy-Constrained Scalar Quantization
    Li Xie, Liangyan Li,  Jun Chen, Lei Yu,
        and Zhongshan Zhang
    September 9, 2024
======================================================================================




§ ABSTRACT 

This paper investigates the best known bounds on the quadratic Gaussian  distortion-rate-perception function with limited common randomness for the Kullback-Leibler divergence-based perception measure, as well as  their counterparts for the squared Wasserstein-2 distance-based perception measure, recently established by Xie et al. These bounds are shown to be  nondegenerate in the sense that they cannot be deduced from each other via a refined version of Talagrand's transportation inequality. On the other hand, an improved lower bound is established when the perception measure is given by the squared Wasserstein-2 distance. In addition, 
it is revealed by exploiting the connection between rate-distortion-perception coding and entropy-constrained scalar quantization that all the aforementioned bounds are generally not tight in the weak perception constraint regime.











Entropy-constrained scalar quantizer,  Gaussian source, Kullback–Leibler divergence, optimal transport,  rate-distortion-perception coding, squared error, transportation inequality,  Wasserstein distance.





















§ INTRODUCTION


Rate-distortion-perception theory <cit.>, as a generalization of Shannon's rate-distortion theory, has received considerable attention in recent years. It provides a framework for investigating the performance limits of perception-aware image compression. This is partly accomplished by assessing compression results more comprehensively, using both distortion and perception measures. Unlike distortion measures, which compare each compressed image with its corresponding source image, perception measures focus on the ensemble-level relationship between pre- and post-compression images. It has been observed that at a given coding rate, there exists a tension between distortion loss and  perception loss <cit.>. Moreover,  the presence of a perception constraint often necessitates the use of stochastic algorithms <cit.>. In contrast, deterministic algorithms are known to be adequate for conventional lossy source coding.



Although significant progress has been made in characterizing the information-theoretic limits of rate-distortion-perception coding, existing results are almost exclusively restricted to  special scenarios with the availability of unlimited common randomness or with the perfect perception constraint (also referred to as perfect realism). 
To the best of our knowledge, the only exception is <cit.>, which makes an initial attempt to study the fundamental distortion-rate-perception tradeoff 
with limited common randomness by leveraging the research findings from  output-constrained lossy source coding <cit.>. In particular, lower and upper bounds on the quadratic Gaussian distortion-rate-perception function under a specified amount of common randomness
 are established in <cit.> for both  Kullback-Leibler divergence-based  and squared Wasserstein-2 distance-based perception measures.
 These bounds shed light on the utility of common randomness as a resource in  rate-distortion-perception coding, especially when the perceptual quality is not required to be perfect.  
 On the other hand, they in general do not match and  are  therefore inconclusive. Note that the aforementioned upper bounds are derived by restricting the reconstruction distribution to be Gaussian. A natural question thus arises whether this restriction incurs any penalty. A negative answer to this question is equivalent to the existence of some new Gaussian extremal inequalities, which are of independent interest. It is also worth noting that Kullback-Leibler divergence and squared Wasserstein-2 distance are related via Talagrand's transporation inequality <cit.> when the reference distribution is Gaussian. As such, there exists an intrinsic connection between the quadratic distortion-rate-perception functions associated with Kullback-Leibler divergence-based  and squared Wasserstein-2 distance-based perception measures.  
 This connection has not been explored in existing literature. 
 
 
 We shall show that the bounds on the quadratic Gaussian distortion-rate-perception function with limited common randomness for the  Kullback-Leibler divergence-based perception measure cannot be deduced from their counterparts for the  squared Wasserstein-2 distance-based perception measure  via a refined version of Talagrand's transportation inequality. In this sense, they are not degenerate. On the other hand, it turns out that the lower bound can be improved via an additional tunable parameter when the perception measure is given by the squared Wasserstein-2 distance. Furthermore,  all the aforementioned bounds are  generally not tight in the weak perception constraint regime. 
 
 
 We demonstrate this result by exploiting the connection between rate-distortion-perception coding and entropy-constrained scalar quantization. Our finding implies 
  that restricting the reconstruction distribuiton to be Gaussian may incur a penalty.  This is somewhat surprising in view of the fact that the quadratic Gaussian distortion-rate-percetion function with limited common randomness admits a single-letter characterization, which often implies the existence of a corresponding Gaussian extremal inequality <cit.>.


 
 The rest of this paper is organized as follows. Section <ref> contains the  definition of quadratic distortion-rate-perception function with limited common randomness and a review of some relevant results.  Our technical contributions are presented in Sections <ref>, <ref>, and <ref>. We conclude the paper in Section <ref>.
 
 
We adopt the standard notation for information measures, e.g.,  H(·) for entropy, h(·) for differential entropy,  I(·;·) for mutual information, and J(·) for Fisher information.
The cardinality of set 𝒮 is denoted by |𝒮|. For a given random variable X, its distribution, mean, and variance are written as p_X, μ_X, and σ^2_X, respectively. We use  Π(p_X,p_X̂) to represent  the set of all possible couplings of p_X and p_X̂.
For real numbers a and b, let a∧ b:=min{a,b}, a∨ b:=max{a,b}, and (a)_+:=max{a,0}.
 Throughout this paper, the logarithm function is
assumed to have base e.























§ PROBLEM DEFINITION AND EXISTING RESULTS


A length-n rate-distortion-perception coding system (see Fig. <ref>) consists of an encoder f^(n):ℝ^n×𝒦→𝒥, a  decoder g^(n):𝒥×𝒦→ℝ^n, and a random seed K. It takes an i.i.d. source sequence X^n as input and produces an i.i.d. reconstruction sequence X̂^n. Specifically, the encoder
 maps   X^n and  K to a codeword J in codebook 𝒥 according to some conditional distribution p_J|X^nK while the decoder  generates  X̂^n based on J and K according to some conditional distribution p_X̂^n|JK. Here, K  is assumed to be uniformly distributed over the alphabet 𝒦 and independent of X^n. The end-to-end distortion is quantified by 1/n∑_t=1^n𝔼[(X_t-X̂_t)^2] and the perceptual quality by 1/n∑_t=1^nϕ(p_X_t,p_X̂_t) with some divergence ϕ. It is clear that 1/n∑_t=1^nϕ(p_X_t,p_X̂_t)=ϕ(p_X,p_X̂), where p_X and p_X̂ are the marginal distributions of X^n and X̂^n, respectively. 
















	For an i.i.d. source {X_t}_t=1^∞, distortion level D is said to be achievable   subject to the compression rate constraint R, the common randomness rate constraint R_c, and the perception constraint P	
	if there exists a length-n rate-distortion-perception coding system such that 	
	
    1/nlog|𝒥|≤ R,
       1/nlog|𝒦|≤ R_c,
       1/n∑_t=1^n𝔼[(X_t-X̂_t)^2]≤ D,
       1/n∑_t=1^nϕ(p_X_t,p_X̂_t)≤ P,

	and the reconstruction sequence X̂^n is ensured to be i.i.d. The infimum of such achievable distortion levels D is denoted by  D(R,R_c,P|ϕ).


The following result <cit.>, which is built upon  <cit.> (see also <cit.>),  provides a single-letter characterization of D(R,R_c,P|ϕ).



	For p_X with 𝔼[X^2]<∞,
	
    D(R,R_c,P|ϕ)   =inf_p_UX̂|X𝔼[(X-X̂)^2]
          X↔ U↔X̂,
          I(X;U)≤ R,
          I(X̂;U)≤ R+R_c,
         ϕ(p_X,p_X̂)≤ P.






Explicit lower and upper bounds on D(R,R_c,P|ϕ) are established for p_X=𝒩(μ_X,σ^2_X) when ϕ(p_X,p_X̂)=ϕ_KL(p_X̂p_X) <cit.> or ϕ(p_X,p_X̂)=W^2_2(p_X,p_X̂) <cit.>,
where

    ϕ(p_X̂p_X):=𝔼[logp_X̂(X̂)/p_X(X̂)]

is the Kullback-Leibler divergence and 

    W^2_2(p_X,p_X̂):=inf_p_XX̂∈Π(p_X,p_X̂)𝔼[(X-X̂)^2]

is the squared  Wasserstein-2 distance. Let 

    ξ(R,R_c):=√((1-e^-2R)(1-e^-2(R+R_c))).

Moreover, let

    ψ(σ_X̂):=logσ_X/σ_X̂+σ^2_X̂-σ^2_X/2σ^2_X

and
σ(P) be the unique number  σ∈[0,σ_X] satisfying ψ(σ)=P.

















	For  p_X=𝒩(μ_X,σ^2_X),
	
    D(R,R_c,P|ϕ_KL)≤ D(R,R_c,P|ϕ_KL)≤D(R,R_c,P|ϕ_KL),

where

    D(R,R_c,P|ϕ_KL):=min_σ_X̂∈[σ(P),σ_X]σ^2_X+σ^2_X̂-2σ_Xσ_X̂√((1-e^-2R)(1-e^-2(R+R_c+P-ψ(σ_X̂))))

and

    D(R,R_c,P|ϕ_KL):=σ^2_X-σ^2_Xξ^2(R,R_c)+(σ(P)-σ_Xξ(R,R_c))^2_+.








For  p_X=𝒩(μ_X,σ^2_X),

    D(R,R_c,P|W^2_2)≤ D(R,R_c,P|W^2_2)≤D(R,R_c,P|W^2_2),

where

    D(R,R_c,P|W^2_2):=min_σ_X̂∈[(σ_X-√(P))_+,σ_X]σ^2_X+σ^2_X̂-2σ_X√((1-e^-2R)(σ^2_X̂-(σ_Xe^-(R+R_c)-√(P))^2_+))
	
and

    D(R,R_c,P|W^2_2):=σ^2_X-σ^2_Xξ^2(R,R_c)+(σ_X-√(P)-σ_Xξ(R,R_c))^2_+.





The next three sections are devoted to investigating the tightness of these bounds, which will shed light on 
rate-distortion-perception coding in general. 







§ KULLBACK-LEIBLER DIVERGENCE VS. SQUARED WASSERSTEIN-2 DISTANCE


For p_X=𝒩(μ_X,σ^2_X), Talagrand's transportation inequality <cit.> states that

    W^2_2(p_X,p_X̂)≤ 2σ^2_Xϕ_KL(p_X̂p_X),

which immediately implies

    D(R,R_c,2σ^2_XP|W^2_2)≤ D(R,R_c,P|ϕ_KL).

Note that Talagrand's transportation inequality does not impose any assumptions on p_X̂. However, when p_X=𝒩(μ_X,σ^2_X), it suffices to consider p_X̂ with μ_X̂=μ_X and σ_X̂≤σ_X as far as D(R,R_c,P|ϕ_KL) and D(R,R_c,P|W^2_2) are concerned <cit.>. With this restriction on p_X̂, we have the following refined version of Talagrand's transportation inequality, which leads to an improvement on (<ref>).

 
For p_X=𝒩(μ_X,σ^2_X) and p_X̂ with μ_X̂=μ_X and σ_X̂≤σ_X,

    W^2_2(p_X,p_X̂)≤ 2σ^2_X(1-e^-ϕ_KL(p_X̂p_X)).

As a consequence, 
	
    D(R,R_c,2σ^2_X(1-e^-P)|W^2_2)≤  D(R,R_c,P|ϕ_KL)

when p_X=𝒩(μ_X,σ^2_X).


	See Appendix <ref>.



















It is clear that (<ref>) and (<ref>) are  stronger than their counterparts in  (<ref>) and (<ref>) since 1+z≤ e^z for all z. Theorem <ref> implies that for p_X=𝒩(μ_X,σ^2_X), 	
	every lower bound on D(R,R_c,·|W^2_2) induces a lower bound on D(R,R_c,·|ϕ_KL) and every upper bound on D(R,R_c,·|ϕ_KL) induces an upper bound on D(R,R_c,·|W^2_2); in particular,  we have
	
    D(R,R_c,P|ϕ_KL)≥D(R,R_c,2σ^2_X(1-e^-P)|W^2_2)

	and
	
    D(R,R_c,P|W^2_2)≤D(R,R_c,ν(P)|ϕ_KL),

where

    ν(P)   :=log2σ^2_X/(2σ^2_X-P)_+.

It is thus of considerable interest to see how these induced bounds are compared to their counterparts in Theorems <ref> and <ref>, namely, 
	
    D(R,R_c,P|ϕ_KL)≥D(R,R_c,P|ϕ_KL)

	and
	
    D(R,R_c,P|W^2_2)≤D(R,R_c,P|W^2_2).

	The following result indicates that (<ref>) and (<ref>) are in general looser. In this sense, (<ref>) and (<ref>) are nondegenerate. 
	








	For p_X=𝒩(μ_X,σ^2_X),
	
    D(R,R_c,P|ϕ_KL)≥D(R,R_c,2σ^2_X(1-e^-P)|W^2_2)

		and
		
    D(R,R_c,P|W^2_2)≤D(R,R_c,ν(P)|ϕ_KL).



	See Appendix <ref>.		


	It be can seen from Fig. <ref> that D(R,R_c,2σ^2_X(1-e^-P)|W^2_2) is indeed a looser lower bound on D(R,R_c,P|ϕ_KL) as compared to 
	D(R,R_c,P|ϕ_KL) and the latter almost meets the upper bound D(R,R_c,P|ϕ_KL). 	
	Similarly, Fig. <ref> shows that  D(R,R_c,ν(P)|ϕ_KL) is indeed a  looser upper bound on D(R,R_c,P|W^2_2) 
	 as compared to D(R,R_c,P|W^2_2), especially in the low rate regime, where the latter has a diminishing gap from the lower bound D(R,R_c,P|W^2_2).
	 
	  
	
	







































§ AN IMPROVED LOWER BOUND


The main result of this section is the following improved lower bound on D(R,R_c,P|W^2_2).


	For p_X=𝒩(μ_X,σ^2_X),
	
    D(R,R_c,P|W^2_2)≥D'(R,R_c,P|W^2_2)≥D(R,R_c,P|W^2_2),

where	
	
    D'(R,R_c,P|W^2_2):=min_σ_X̂∈[(σ_X-√(P))_+,σ_X]sup_α>0σ^2_X+σ^2_X̂-2σ_X√((1-e^-2R)(σ^2_X̂-δ^2_+(σ_X̂,α)))

with
	
    δ_+(σ_X̂,α):=(σ_Xe^-(R+R_c)-√(σ^2_X-α(σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂))_+/α.

Moreover, the second inequality in (<ref>) is strict if and only if R∈(0,∞), R_c∈(0,∞), and P∈(0,σ^2_X(2-e^-2R-2√((1-e^-2R)(1-e^-2(R+R_c))))).


See Appendix <ref>.























The difference between D'(R,R_c,P|W^2_2) and D(R,R_c,P|W^2_2) against P is plotted in Fig. <ref> for the case  p_X=𝒩(0,1), R=0.1,  and R_c=0.1.
According to Theorem <ref>, for R∈(0,∞) and R_c∈(0,∞), we have D'(R,R_c,P|W^2_2)=D(R,R_c,P|W^2_2) when 

    P≥σ^2_X(2-e^-2R-2√((1-e^-2R)(1-e^-2(R+R_c)))).

Setting σ^2_X=1, R=0.1, and R_c=0.1 in (<ref>) gives P⪆ 0.692, which is consistent with the result shown in Fig. <ref>. Fig. <ref> plots the  difference between D'(R,R_c,P|W^2_2) and D(R,R_c,P|W^2_2) against R  for the case  p_X=𝒩(0,1), R_c=0.1, and P=0.1. As shown in Appendix <ref>, for R_c∈(0,∞) and P∈(0,∞], we can write
(<ref>)  alternatively as

    R≥
    	0	   P≥σ^2_X,
    
    	-1/2logζ_3/ζ_2   R_c=log2, P<σ^2_X,
    
    	-1/2logζ_2-√(ζ^2_2-4ζ_1ζ_3)/2ζ_1   ,

where

    ζ_1:=4e^-2R_c-1,
       ζ_2:=4e^-2R_c+2P/σ^2_X,
       ζ_3:=(4σ^2_X-P)P/σ^4_X.

Setting σ^2_X=1, R_c=0.1, and P=0.1 in (<ref>) gives R⪆1.052, which is consistent with the result shown in Fig. <ref>.


 









It is interesting to note that for p_X=𝒩(μ_X,σ^2_X),
	
    D'(R,0,P|W^2_2)   =D(R,0,P|W^2_2)
       =σ^2_Xe^-2R+(σ_Xe^-R-√(P))^2_+,

which coincides with the minimum achievable mean squared error  at rate R and squared Wasserstein-2 perception loss P when the reconstruction sequence is not required to be i.i.d. <cit.>. As shown in the next section, D(R,0,P|W^2_2) and D'(R,0,P|W^2_2) are actually strictly below D(R,0,P|W^2_2) for sufficiently large P. Therefore, a price has to be paid for enforcing the i.i.d. reconstruction constraint. 
This should be contrasted with the case R_c=∞ for which it is known that the rate-distortion-perception tradeoff remains the same regardless of whether the reconstruction sequence is required to be i.i.d. or not <cit.> <cit.>.






§ CONNECTION WITH ENTROPY-CONSTRAINED SCALAR QUANTIZATION


This section is devoted to  investigating the tightness of bounds in Theorems <ref>, <ref>, and <ref>. We shall focus on the weak perception constraint regime where P is sufficiently large.


To this end, it is necessary to first gain a better understanding of the properties of D(R,R_c,P|ϕ). Clearly,  the map (R,R_c,P)↦ D(R,R_c,P|ϕ) is monotonically decreasing in each of its variables. The following result provides further information regarding D(R,R_c,P|ϕ) under certain conditions.

	For p_X with bounded support, if p_X̂↦ϕ(p_X, p_X̂) is lower semicontinuous in the topology of weak convergence[It is known that 
			p_X̂↦ϕ_KL(p_X̂p_X) <cit.> and p_X̂↦ W^2_2(p_X,p_X̂) <cit.> are  lower semicontinuous in the topology of weak convergence.], then the infimum in (<ref>) can be attained and the map (R,R_c,P)↦ D(R,R_c,P|ϕ) is right-continuous in each of its variables.


	See Appendix <ref>.



Theorem <ref> is not applicable when p_X is a Gaussian distribution. However, it will be seen that assuming the attainability of the infimum in  (<ref>) greatly simplifies the reasoning and helps develop the intuition behind the rigorous proof of the main result in this section (see Theorem <ref>).

The next two results deal with the special cases ϕ(p_X,p_X̂)=ϕ_KL(p_X̂p_X) and ϕ(p_X,p_X̂)=W^2_2(p_X,p_X̂), respectively.

	For p_X=𝒩(μ_X,σ^2_X) and (R,R_c)∈[0,∞]^2, the map  P↦ D(R,R_c,P|ϕ_KL) is continuous[A map x↦ f(x) is said to be continuous at x=∞ if lim_x→∞f(x)=f(∞).] for P∈[0,∞].


	See Appendix <ref>.	






	
	








	
	























	For p_X with 𝔼[X^2]<∞ and (R,R_c)∈[0,∞]^2, the map P↦ D(R,R_c,P|W^2_2) is continuous  for P∈[0,∞].


See Appendix <ref>.


Now consider the extreme case  P=∞. 
In light of Theorem <ref>, for p_X with 𝔼[X^2]<∞,

    D(R,R_c,∞|ϕ)   =inf_p_UX̂|X𝔼[(X-X̂)^2]
          X↔ U↔X̂,
          I(X;U)≤ R,
          I(X̂;U)≤ R+R_c,

which does not depend on the choice of ϕ.
Moreover, it can be verified that for p_X=𝒩(μ_X,σ^2_X),

    D(R,R_c,∞|ϕ_KL)   =D(R,R_c,∞|W^2_2)
       =D'(R,R_c,∞|W^2_2)
       =σ^2_Xe^-2R
	
and

    D(R,R_c,∞|ϕ_KL)
    	   =D(R,R_c,∞|W^2_2)
       =σ^2_X-σ^2_Xξ^2(R,R_c).
 
Therefore, we shall simply denote D(R,R_c,∞|ϕ_KL) and D(R,R_c,∞|W^2_2) by D(R,R_c,∞), denote D(R,R_c,∞|ϕ_KL), D(R,R_c,∞|W^2_2), and D'(R,R_c,∞|W^2_2) by D(R,R_c,∞), and denote D(R,R_c,∞|ϕ_KL) and
D(R,R_c,∞|W^2_2) by D(R,R_c,∞).

It will be seen that neither D(R,R_c,∞) nor D(R,R_c,∞) is tight in general. This fact can be established by exploiting the connection between rate-distortion-perception coding and entropy-constrained scalar quantization. For p_X with 𝔼[X^2]<∞, let[The existence of a minimizer for the optimization problem in (<ref>) can be proved via an argument similar to that for Theorem <ref>. Here, it suffices to assume 𝔼[X^2]<∞ since the bounded support condition in Theorem <ref> is only needed to address the intricacy caused by the Markov chain constraint (<ref>).]
 
    D_e(R,R_c):=min_p_X̂|X: I(X;X̂)≤ R, H(X̂)≤ R+R_c𝔼[(X-X̂)^2],

which is the counterpart of D(R,R_c,∞) with the decoder restricted to be deterministic <cit.>. 
When R_c=0, the constraint I(X;X̂)≤ R is redundant; as a consequence, 

    D_e(R,0)=min_p_X̂|X:  H(X̂)≤ R𝔼[(X-X̂)^2],

which is simply the distortion-rate function for 
entropy-constrained scalar quantization. Note that  
 
    D_e(R,0)=min_p_X̂:  H(X̂)≤ RW^2_2(p_X,p_X̂)

as every coupling of p_X and p_X̂ induces a (possibly randomized) scalar quantizer. When p_X is absolutely continuous with respect to the Lebesgue measure, W^2_2(p_X,p_X̂) is attained by a coupling that transforms p_X to p_X̂ via a determinstic map <cit.>, 
so there is no loss of optimality in restricting the quantizer  to be deterministic. Moreover, if p_X has a piecewie  monotone and piecewise continuous density, then we can further restrict the deterministic quantizer to be regular <cit.>. On the other hand, when R_c=∞, the constraint H(X̂)≤ R+R_c is redundant; as a consequence, 

    D_e(R,∞)=min_p_X̂|X:  I(X;X̂)≤ R𝔼[(X-X̂)^2],

which is simply the classical distortion-rate function.
The following result reveals that D_e(R,R_c) is intimately related to D(R,R_c,∞).






 
 





 













	For p_X with 𝔼[X^2]<∞,
	
    D_e(R,R_c)≥ D(R,R_c,∞)≥ D_e(R,∞).

Moreover, if the infimum in (<ref>) can be attained[According to Theorem <ref>, this assumption holds for p_X with bounded support.], then

    D_e(R,R_c)> D_e(R,∞)  ⇔   D(R,R_c,∞)>D_e(R,∞).



	See Appendix <ref>.
	


The connection revealed in Theorem <ref> enables us to derive the following result, which indicates that
D(R,R_c,∞) and D(R,R_c,∞) are not tight in general.



	For p_X=𝒩(μ_X,σ^2_X),
	
    D(R,R_c,∞)>D(R,R_c,∞)

when R∈(0,∞) and R_c∈[0,∞), and 
	
    D(R,R_c,∞)<D(R,R_c,∞)

when R_c∈[0,∞) and R∈(0,χ(R_c)), where χ(R_c) is a positive threshold that depends on R_c.


	See Appendix <ref>.	



















































	



	









































	
































	
	

































For p_X=𝒩(μ_X,σ^2_X), we exhibit below an explicit improvement over D(R,0,∞) in the low rate regime. Consider the following binary quantizer:

    X̂=μ_X-σ_Xe^-θ^2/2/√(2π)Q(θ)   X-μ_X/σ_X<θ,
    μ_X+σ_Xe^-θ^2/2/√(2π)(1-Q(θ))   X-μ_X/σ_X≥θ,

where θ≥ 0 and Q(θ):=1/√(2π)∫_-∞^θe^-x^2/2dx. It can be verified 
that 

    𝔼[(X-X̂)^2]   =σ^2_X-σ^2_Xe^-θ^2/2π Q(θ)(1-Q(θ))
       =:D(θ)
	
and

    H(X̂)   =-Q(θ)log Q(θ)-(1-Q(θ))log(1-Q(θ))
       =:R(θ).

For R∈(0,log 2], define D_e(R,0)
via the parametric equations 
D_e(R,0)=D(θ) and R=R(θ).
Clearly, D_e(R,0) is an upper bound on D_e(R,0) and consequently is also an upper bound on D(R,0,∞) in light of Theorem <ref>.
It can be seen from Fig. <ref> that D_e(R,0)<D(R,0,∞) for R∈(0,log 2]. In particular, we have 

    D_e(log 2,0)=π-2/πσ^2_X≈ 0.3634σ^2_X

while

    D(log 2, 0,∞)=7/16σ^2_X=0.4375σ^2_X.

By contrast, although D(R,R_c,∞) is known to be loose for R∈(0,∞) and R_c∈[0,∞), no explicit impprovement has been found (even when R_c=0). So D(R,0,∞) could be situated anywhere between D_e(R,0) (inclusive) and D(R,0,∞) (exclusive except at R=0) in Fig. <ref>.




As shown by the following results, Theorem <ref> has implications to the weak perception constraint regime in general.


	For p_X=𝒩(μ_X,σ^2_X), 
	
    D(R,R_c,P|ϕ_KL)>D(R,R_c,P|ϕ_KL)

	when R∈(0,∞), R_c∈[0,∞), and P is sufficiently large; moreover,
	
    D(R,R_c,P|ϕ_KL)<D(R,R_c,P|ϕ_KL),

	when R_c∈[0,∞), R∈(0,χ(R_c)), and P is sufficiently large.


	See Appendix <ref>.





	For p_X=𝒩(μ_X,σ^2_X), 
	
    D(R,R_c,P|W^2_2)>D'(R,R_c,P|W^2_2)

when R∈(0,∞), R_c∈[0,∞), and P∈(γ'(R,R_c),∞], where γ'(R,R_c) is a positive threshold that depends on (R,R_c) and γ'(R,R_c)<P'(R,R_c):=min{P∈[0,∞]:D'(R,R_c,P|W^2_2)=D'(R,R_c,∞|W^2_2)}[It can be verified that P'(R,R_c)=σ^2_X(2-e^-2R-2√((1-e^-2R)(1-e^-2(R+R_c)))) for p_X=𝒩(μ_X,σ^2_X).]; moreover,

    D(R,R_c,P|W^2_2)<D(R,R_c,P|W^2_2)

when R_c∈[0,∞), R∈(0,χ(R_c)), and P∈(γ(R,R_c),∞], where γ(R,R_c) is a positive threshold that depends on (R,R_c) and γ(R,R_c)< P(R,R_c):=min{P∈[0,∞]:D(R,R_c,P|W^2_2)=D(R,R_c,∞|W^2_2)}[For p_X=𝒩(μ_X,σ^2_X), we have P(R,R_c)>0 when R∈[0,∞) and R_c∈[0,∞]
since D(R,R_c,0|W^2_2)=D(R,R_c,0|W^2_2)>D(R,R_c,∞|W^2_2)≥ D(R,R_c,∞|W^2_2).].


	See Appendix <ref>.























































According to <cit.>, the upper bounds D(R,R_c,P|ϕ_KL) and D(R,R_c,P|W^2_2) are tight when the reconstruction distribution p_X̂ is restricted to be Gaussian. In light of Corollaries <ref> and <ref>, this restriction incurs a penalty in the weak perception constraint regime. In fact, the connection with entropy-constrained scalar quantization suggests that discrete reconstruction distributions might be more preferable in this regime. This is somewhat surprising since D(R,R_c,P|ϕ) admits a single-letter characterization, which is typically associated with a Gaussian extremal inequality <cit.>, especially considering the fact that both ϕ_KL(p_X̂p_X) and W^2_2(p_X,p_X̂) favor Gaussian p_X̂ when p_X is a Gaussian distribution (see Lemma <ref>).



§ CONCLUSION


We have investigated and improved the existing bounds on the quadratic Gaussian distortion-rate-perception function with limited common randomness for the case where the perception measure is given by the Kullback-Leibler divergence or the squared Wasserstein-2 distance. Along the way, a refined version of Talagrand's transportation inequality is established and the connection between rate-distortion-perception coding and entropy-constrained scalar quantization is revealed. 


Note that the fundamental rate-distortion-perception tradeoff depends critically on how the perception constraint is formulated. Our work focuses on a particulr formulation where the reconstruction sequence is required to be i.i.d. Therefore, great caution should be executed when utilizing and interpreting the results in the present paper. It is of considerable interest to conduct a comprehensive comparison of different formulations regarding their impacts on the information-theoretic performance limit of rate-distortion-perception coding. 





















































§ PROOF OF THEOREM  <REF>


We need the following result <cit.> concerning the Gaussian extremal property of the Kullback-Leibler divergence and the squared Wasserstein-2 distance.  



	For p_X=𝒩(μ_X,σ^2_X) and p_X̂ with 𝔼[X̂^2]<∞, 
	
    ϕ_KL(p_X̂p_X)
    		   ≥ϕ_KL(p_X̂^Gp_X)
       =logσ_X/σ_X̂+(μ_X-μ_X̂)^2+σ^2_X̂-σ^2_X/2σ^2_X

	and
	
    W^2_2(p_X,p_X̂)   ≥ W^2_2(p_X,p_X̂^G)
       =(μ_X-μ_X̂)^2+(σ_X-σ_X̂)^2,

	where p_X̂^G:=𝒩(μ_X̂,σ^2_X̂).


Lemma <ref> indicates that when the reference distribution is Gaussian, replacing the other distribution with its Gaussian counterpart leads to reductions in both the Kullback-Leibler divergence and the squared Wasserstein-2 distance. 
These reductions turn out to be quantitatively related as shown by the next result.


	For p_X=𝒩(μ_X,σ^2_X) and  p_X̂ with 𝔼[X̂^2]<∞,
	
    W^2_2(p_X,p_X̂)-W^2_2(p_X,p_X̂^G)≤2σ_Xσ_X̂(1-e^-(ϕ_KL(p_X̂p_X)-ϕ_KL(p_X̂^Gp_X))).



	Note that
	
    W^2_2(p_X,p_X̂)	   =(μ_X-μ_X̂)^2+W^2_2(p_X-μ_X,p_X̂-μ_X̂)
       =(μ_X-μ_X̂)^2+σ^2_XW^2_2(p_σ^-1_X(X-μ_X), p_σ^-1_X(X̂-μ_X̂))
       (a)≤(μ_X-μ_X̂)^2+σ^2_X+σ^2_X̂-2σ^2_X√(1/2π ee^2h(σ^-1_XX̂))
       (b)=W^2_2(p_X,p_X̂^G)+2σ_Xσ_X̂-2σ^2_X√(1/2π ee^2h(σ^-1_XX̂)),

	where (a) is due to <cit.> and (b) is due to Lemma <ref>. Moreover,
	
    h(σ^-1_XX̂)	   =h(X̂)-logσ_X
       =1/2log2π eσ^2_X̂/σ^2_X-ϕ_KL(p_X̂p_X)+ϕ_KL(p_X̂^Gp_X).

	Substituting (<ref>) into (<ref>) proves Lemma <ref>.


Now we proceed to prove Theorem <ref>. In view of Lemmas <ref> and <ref>, 

    W^2_2(p_X,p_X̂)   ≤max_μ,ση(μ,σ)
         μ=μ_X,
         σ≤σ_X,
         (μ_X-μ)^2/2σ^2_X+ψ(σ)≤ϕ_KL(p_X̂p_X),

where

    η(μ,σ)   :=-2σ^2_Xe^(μ_X-μ)^2+σ^2-σ^2_X/2σ^2_Xe^-ϕ_KL(p_X̂p_X)+(μ_X-μ)^2+σ^2_X+σ^2

and ψ(·) is defined in (<ref>).
Since  ψ(σ)  decreases monotonically from ∞ to 0 as σ varies from 0 to σ_X and increases monotonically from 0 to ∞ as σ varies from σ_X to ∞,  there must exist σ≤σ_X and σ≥σ_X satisfying

    ψ(σ)=ψ(σ)=ϕ_KL(p_X̂p_X).

Note that (<ref>)–(<ref>) can be written compactly as

    W^2_2(p_X,p_X̂)≤max_σ∈[σ,σ_X]η(μ_X,σ).

For σ∈[σ,σ_X],

    ∂/∂ση(μ_X,σ)   =-2σ e^σ^2-σ^2_X/2σ^2_Xe^-ϕ_KL(p_X̂p_X)+2σ
       ≥ 0,

which implies the maximum in (<ref>) is attained at σ=σ_X. So we have

    W^2_2(p_X,p_X̂)   ≤η(μ_X,σ_X)
       =2σ^2_X(1-e^-ϕ_KL(p_X̂p_X)).

This  proves Theorem <ref>.






































Interestingly,  Talagrand's transportation inequality (<ref>)
corresponds to the relaxed version without the constraints (<ref>) and (<ref>), i.e., 

    W^2_2(p_X,p_X̂)   ≤max_μ,ση(μ,σ)
         (μ_X-μ)^2/2σ^2_X+ψ(σ)≤ϕ_KL(p_X̂p_X).

We now prove this. It can be verified that

    ∂/∂(μ_X-μ)^2η(μ,σ)=-e^(μ_X-μ)^2+σ^2-σ^2_X/2σ^2_Xe^-ϕ_KL(p_X̂p_X)+1.

Given σ<σ, there is no μ satisfying (<ref>).
Given σ∈[σ,σ_X], for μ satisfying (<ref>), we have

    ∂/∂(μ_X-μ)^2η(μ,σ)≥ 0,

which implies that the maximum value of η(μ,σ) over μ satisfying (<ref>) is attained when

    logσ_X/σ+(μ_X-μ)^2+σ^2-σ^2_X/2σ^2_X=ϕ_KL(p_X̂p_X).

Therefore, for σ∈[σ,σ_X],

    max_μ:(<ref>)η(μ,σ)=κ(σ),

where

    κ(σ):=2σ^2_X(ϕ_KL(p_X̂p_X)-logσ_X/σ+1)-2σ_Xσ.

Since the maximum value of κ(σ) over σ∈[σ,σ_X] is attained at σ=σ_X, it follows that

    max_σ∈[σ,σ_X]max_μ:(<ref>)η(μ,σ)=2σ^2_Xϕ_KL(p_X̂p_X).

Given σ∈(σ_X,√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X)),  for μ satisfying (<ref>), we have

    ∂/∂(μ_X-μ)^2η(μ,σ)≥ 0   (μ_X-μ)^2+σ^2-σ^2_X/2σ^2_X≤ϕ_KL(p_X̂p_X),
    
    		<0   (μ_X-μ)^2+σ^2-σ^2_X/2σ^2_X>ϕ_KL(p_X̂p_X),

which implies that the maximum value of η(μ,σ) over μ satisfying (<ref>) is attained when 

    (μ_X-μ)^2+σ^2-σ^2_X/2σ^2_X=ϕ_KL(p_X̂p_X).

Therefore, for σ∈(σ_X,√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X)),

    max_μ:(<ref>)η(μ,σ)=2σ^2_Xϕ_KL(p_X̂p_X).

As a consequence,

    max_σ∈(σ_X,√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X))max_μ:(<ref>)η(μ,σ)=2σ^2_Xϕ_KL(p_X̂p_X).

Given σ∈[√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X),σ],  for μ satisfying (<ref>), we have

    ∂/∂(μ_X-μ)^2η(μ,σ)≤ 0,

which implies that the maximum value of η(μ,σ) over μ satisfying (<ref>) is attained when 

    (μ_X-μ)^2=0,μ=μ_X.

Therefore, for σ∈[√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X),σ],

    max_μ:(<ref>)η(μ,σ)=κ'(σ),

where

    κ'(σ):=-2σ^2_Xe^σ^2-σ^2_X/2σ^2_Xe^-ϕ_KL(p_X̂p_X)+σ^2_X+σ^2.

Since the maximum value of κ'(σ) over σ∈[√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X),σ] is attained at σ=√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X), it follows that

    max_σ∈[√(2σ^2_Xϕ_KL(p_X̂p_X)+σ^2_X),σ]max_μ:(<ref>)η(μ,σ)=2σ^2_Xϕ_KL(p_X̂p_X).

Given σ>σ, there is no μ satisfying (<ref>).
Combining (<ref>), (<ref>), and (<ref>) proves (<ref>).





§ PROOF OF THEOREM <REF>


In view of the definition of D(R,R_c,P|ϕ_KL) and D(R,R_c,2σ^2_X(1-e^-P)|W^2_2), for the purpose of proving (<ref>), it suffices to show

    [σ(P),σ_X]⊆[(σ_X-√(2σ^2_X(1-e^-P)))_+,σ_X]

and

    σ^2_X̂-(σ_Xe^-(R+R_c)-√(2σ^2_X(1-e^-P)))_+^2≥σ^2_X̂-σ^2_X̂e^-2(R+R_c+P-ψ(σ_X̂))

for σ_X̂∈[σ(P),σ_X].
Invoking (<ref>) with p_X̂=𝒩(μ_X,σ(P)) (see also Lemma <ref> for the expressions of the Kullback-Leibler divergence and the squared Wasserstein-2 distance between two Gaussian distributions)

    (σ_X-σ(P))^2≤ 2σ^2_X(1-e^-P),

from which (<ref>) follows immediately.
Note that (<ref>) is trivially true when 	
e^-(R+R_c)≤√(2(1-e^-P)). When e^-(R+R_c)>√(2(1-e^-P)), it can be written equivalently as

    √(2(1-e^-P))≥ e^-(R+R_c)(1-e^-(P+σ^2_X-σ^2_X̂/2σ^2_X)).

Since e^-(R+R_c)≤ 1 and

    1-e^-(P+σ^2_X-σ^2_X̂/2σ^2_X)≤1-e^-(P+σ^2_X-σ^2(P)/2σ^2_X)

for σ_X̂∈[σ(P),σ_X], it suffices to show

    √(2(1-e^-P))≥1-e^-(P+σ^2_X-σ^2(P)/2σ^2_X).

According to the definition of σ(P),

    P=logσ_X/σ(P)+σ^2(P)-σ^2_X/2σ^2_X.

Substituting (<ref>) into (<ref>) gives

    √(2(1-e^logσ(P)/σ_X-σ^2(P)/2σ^2_X+1/2))≥ 1-σ(P)/σ_X.

We can rewrite (<ref>) as

    τ(β)≥ 0,

where 

    τ(β):=1-2β e^-β^2/2+1/2+2β-β^2

with β:=σ(P)/σ_X.
Note that	β∈[0,1]. We have

    dτ(β)/dβ   =-2e^-β^2/2+1/2+2β^2e^-β^2/2+1/2+2-2β
       ≤ -2(1-β^2)+2-2β
       =-2(1-β)β
       ≤ 0.

Since τ(1)=0, it follows that τ(β)≥ 0 for β∈[0,1], which verifies (<ref>) and consequently proves (<ref>).


Now we proceed to prove (<ref>), which is equivalent to 

    D(R,R_c,2σ^2_X(1-e^-P)|W^2_2)≤D(R,R_c,P|ϕ_KL).

Since D(R,R_c,P|ϕ_KL)=D(R,R_c,(σ_X-σ(P))^2|W^2_2), it suffices to show

    (σ_X-σ(P))^2≤ 2σ^2_X(1-e^-P),

i.e.,

    P≥log2σ^2_X/σ^2_X-σ^2(P)+2σ_Xσ(P).

Substituting (<ref>) into (<ref>) and rearranging the inequality yields

    logσ^2_X-σ^2(P)+2σ_Xσ(P)/2σ_Xσ(P)≥σ^2_X-σ^2(P)/2σ^2_X,

which is indeed true since 

    logσ^2_X-σ^2(P)+2σ_Xσ(P)/2σ_Xσ(P)   (a)≥ 1-2σ_Xσ(P)/σ^2_X-σ^2(P)+2σ_Xσ(P)
       =σ^2_X-σ^2(P)/σ^2_X-σ^2(P)+2σ_Xσ(P)
       ≥σ^2_X-σ^2(P)/2σ^2_X,

where (a) is due to 
log z≥ 1-1/z for z>0. 
This completes the proof of  (<ref>).



§ PROOF OF THEOREM <REF>



It is known <cit.> that

    D(R,R_c,P|W^2_2)≥   inf_p_X̂σ^2_X+σ^2_X̂-2σ_X√((1-e^-2R)(σ^2_X̂-D(R+R_c|p_X̂)))
         μ_X̂=μ_X,
         σ_X̂≤σ_X,
          W^2_2(p_X,p_X̂)≤ P,

where

    D(R+R_c|p_X̂):=inf_p_Ŷ|X̂:I(X̂;Ŷ)≤ R+R_c𝔼[(X̂-Ŷ)^2].

In light of Lemma <ref>, the constraints (<ref>)–(<ref>) imply σ_X̂∈[(σ_X-√(P))_+,σ_X]. The following result provides a lower bound on D(R+R_c|p_X̂) and proves

    D(R,R_c,P|W^2_2)≥inf_σ_X̂∈[(σ_X-√(P))_+,σ_X]sup_α>0σ^2_X+σ^2_X̂-2σ_X√((1-e^-2R)(σ^2_X̂-δ^2_+(σ_X̂,α))).






	For p_X=𝒩(μ_X,σ^2_X) and  p_X̂ with W^2_2(p_X,p_X̂)≤ P, 
	
    D(R+R_c|p_X̂)≥sup_α>0(σ_Xe^-(R+R_c)-G(α))^2_+/α^2,
	
	where
	
    G(α):=√(σ^2_X-α((μ_X-μ_X̂)^2+σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂).



	First let p_X and p_X̂ be coupled according to the joint distribution attaining W^2_2(p_X,p_X̂).
	Then add Ŷ into the probability space  such that X↔X̂↔Ŷ form a Markov chain and I(X̂;Ŷ)≤ R+R_c.  For any α>0,
	
    𝔼[((X-μ_X)-α(Ŷ-μ_Ŷ))^2]
       =𝔼[((X-μ_X)-α(X̂-μ_X̂))^2]+α^2𝔼[((X̂-μ_X̂)-(Ŷ-μ_Ŷ))^2]+2α𝔼[((X-μ_X)-(X̂-μ_X̂))((X̂-μ_X̂)-(Ŷ-μ_Ŷ))]
       ≤(√(𝔼[((X-μ_X)-α(X̂-μ_X̂))^2])+α√(𝔼[((X̂-μ_X̂)-(Ŷ-μ_Ŷ))^2]))^2
       ≤(√(𝔼[((X-μ_X)-α(X̂-μ_X̂))^2])+α√(𝔼[(X̂-Ŷ)^2]))^2
       =(√(σ^2_X-2αρσ_Xσ_X̂+α^2σ^2_X̂)+α√(𝔼[(X̂-Ŷ)^2]))^2,

where  ρ denotes the correlation coefficient of X and X̂.	On the other hand,
	
    𝔼[((X-μ_X)-α(Ŷ-μ_Ŷ))^2]   (a)≥σ^2_Xe^-2I(X-μ_X;α(Ŷ-μ_Ŷ))
       =σ^2_Xe^-2I(X;Ŷ)
       (b)≥σ^2_Xe^-2I(X̂;Ŷ)
       ≥σ^2_Xe^-2(R+R_c),

	where (a) and (b) are due to the Shannon lower bound <cit.> and the data processing inequality <cit.>, respectively. 
	Combining (<ref>) and (<ref>) yields
	
    𝔼[(X̂-Ŷ)^2]≥(σ_Xe^-(R+R_c)-√(σ^2_X-2αρσ_Xσ_X̂+α^2σ^2_X̂))^2_+/α^2.

	It can be verified that 
	
    P   ≥ W^2_2(p_X,p_X̂)
       =𝔼[(X-X̂)^2]
       =(μ_X-μ_X̂)^2+𝔼[((X-μ_X)-(X̂-μ_X̂))^2]
       =(μ_X-μ_X̂)^2+σ^2_X-2ρσ_Xσ_X̂+σ^2_X̂,

	which implies
	
    2ρσ_Xσ_X̂≥(μ_X-μ_X̂)^2+σ^2_X+σ^2_X̂-P.

	Substituting (<ref>) into (<ref>) proves Lemma <ref>.


To establish the first inequality in (<ref>), we shall demonstrate that “inf" in (<ref>) can be replaced by “min". It suffices to consider the case R∈(0,∞) and R_c∈[0,∞) since otherwise the infimum is clearly attainable. 
The problem boils down to showing that the map σ_X̂↦sup_α>0δ_+(σ_X̂,α) is continuous for σ_X̂∈[(σ_X-√(P))_+,P].


Obviously, sup_α>0δ_+(σ_X̂,α)=0 if and only if sup_α>0δ(σ_X̂,α)≤ 0, where

    δ(σ_X̂,α):=σ_Xe^-(R+R_c)-√(σ^2_X-α(σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂)/α.

Note that sup_α>0δ(σ_X̂,α)≤ 0 is equivalent to

    P≥sup_α>0σ^2_X+σ^2_X̂-σ^2_X(1-e^-2(R+R_c))/α-ασ^2_X̂.

Since

    sup_α>0σ^2_X+σ^2_X̂-σ^2_X(1-e^-2(R+R_c))/α-ασ^2_X̂=σ^2_X+σ^2_X̂-2σ_Xσ_X̂√(1-e^-2(R+R_c)),

one can rewrite (<ref>) as

    P≥σ^2_X+σ^2_X̂-2σ_Xσ_X̂√(1-e^-2(R+R_c)).



On the other hand, we have
sup_α>0δ_+(σ_X̂,α)=sup_α>0δ(σ_X̂,α)>0 when 

    P<σ^2_X+σ^2_X̂-2σ_Xσ_X̂√(1-e^-2(R+R_c)).

If σ_X̂=σ_X-√(P), then 

    δ(σ_X̂,α)=σ_Xe^-(R+R_c)-|σ_X-ασ_X̂|/α

and sup_α>0δ(σ_X̂,α) is attained at[It follows by σ_X̂=σ_X-√(P) and (<ref>) that σ_X̂>0.] 

    α=σ_X/σ_X̂.
 
If σ_X̂>σ_X-√(P), then

    σ^2_X-α(σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂>0

for α>0.
As shown below,  ∂/∂αδ(σ_X̂,α)=0 has a unique solution, denoted as α̂, for α>0. 
It can be verified that

    ∂/∂αδ(σ_X̂,α)=α(σ^2_X+σ^2_X̂-P-2ασ^2_X̂)/2α^2√(σ^2_X-α(σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂)-σ_Xe^-(R+R_c)-√(σ^2_X-α(σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂)/α^2.

Setting ∂/∂αδ(σ_X̂,α)=0 gives

    2σ^2_X-α(σ^2_X+σ^2_X̂-P)=2σ_Xe^-(R+R_c)√(σ^2_X-α(σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂).

Note that (<ref>) has a solution  in (0,2σ^2_X/σ^2_X+σ^2_X̂-P) since its left-hand side is greater than its right-hand side   when α=0 and is less than its right-hand side  when α=2σ^2_X/σ^2_X+σ^2_X̂-P.  By taking the square of both sides of  (<ref>) and simplifying the expression, we get

    α^2((σ^2_X+σ^2_X̂-P)^2-4σ^2_Xσ^2_X̂e^-2(R+R_c))-4ασ^2_X(σ^2_X+σ^2_X̂-P)(1-e^-2(R+R_c))+4σ^4_X(1-e^-2(R+R_c))=0.
 
If σ^2_X+σ^2_X̂-P=2σ_Xσ_X̂e^-(R+R_c), then (<ref>) has only one solution, given by 

    α̂:=σ^2_X/σ^2_X+σ^2_X̂-P,

which is also the unique solution to ∂/∂αδ(σ_X̂,α)=0.
If σ^2_X+σ^2_X̂-P<2σ_Xσ_X̂e^-(R+R_c), then (<ref>) has two solutions with different signs and only the positive one, given by

    α̂   :=2σ^2_X(σ^2_X+σ^2_X̂-P)(1-e^-2(R+R_c))-2σ^2_Xe^-(R+R_c)√((4σ^2_Xσ^2_X̂-(σ^2_X+σ^2_X̂-P)^2)(1-e^-2(R+R_c)))/(σ^2_X+σ^2_X̂-P)^2-4σ^2_Xσ^2_X̂e^-2(R+R_c),

is the solution to ∂/∂αδ(σ_X̂,α)=0 for α>0. If σ^2_X+σ^2_X̂-P>2σ_Xσ_X̂e^-(R+R_c), then (<ref>) has two positive solutions and only the small one, also given by (<ref>), is the solution to ∂/∂αδ(σ_X̂,α)=0.
Indeed, the large one is a solution to the following equation obtained by negating the left-hand side of (<ref>):

    α(σ^2_X+σ^2_X̂-P)-2σ^2_X=2σ_Xe^-(R+R_c)√(σ^2_X-α(σ^2_X+σ^2_X̂-P)+α^2σ^2_X̂).

This can be verified by noticing that (<ref>) has a solution in (2σ^2_X/σ^2_X+σ^2_X̂-P,∞) since its left-hand side is less than its right-hand side when α=2σ^2_X/σ^2_X+σ^2_X̂-P and is greater than its right-hand side when α is sufficiently large. For σ_X̂ satisfying σ^2_X+σ^2_X̂-P=2σ_Xσ_X̂e^-(R+R_c), we have

    lim_ϵ→02σ^2_X(σ^2_X+σ^2_X̂(ϵ)-P)(1-e^-2(R+R_c))-2σ^2_Xe^-(R+R_c)√((4σ^2_Xσ^2_X̂(ϵ)-(σ^2_X+σ^2_X̂(ϵ)-P)^2)(1-e^-2(R+R_c)))/(σ^2_X+σ^2_X̂(ϵ)-P)^2-4σ^2_Xσ^2_X̂(ϵ)e^-2(R+R_c)
       =σ^2_X/σ^2_X+σ^2_X̂-P,

where σ_X̂(ϵ)=σ_X̂+ϵ.
Moreover, setting σ_X̂=σ_X-√(P) in (<ref>) gives α̂=σ_X/σ_X̂.
Therefore,  (<ref>) and (<ref>) can be viewed as the degenerate versions of (<ref>). 
Since δ(σ_X̂,α)<0 when α is either close to zero from the positive side or sufficiently large,  α̂ must be the unique maximizer of both δ(σ_X̂,α) and δ_+(σ_X̂,α)
for α>0.
This implies the continuity of σ_X̂↦sup_α>0δ_+(σ_X̂,α) for σ_X̂ over the region defined  by (<ref>).


It remains to show that δ_+(σ_X̂,α̂)→ 0 as σ_X̂, confined to the region defined by (<ref>), converges to some σ satisfying P=σ^2_X+σ^2-2σ_Xσ√(1-e^-2(R+R_c)). First consider the scenario where σ=0, which implies P=σ^2_X. We have α̂→∞ as σ_X̂→ 0, and consequently

    lim_σ_X̂→0δ_+(σ_X̂,α̂)
    	   =lim_σ_X̂→0(σ_Xe^-(R+R_c)/α-√(σ^2_X/α^2-σ^2_X+σ^2_X̂-P/α+σ^2_X̂))_+
       =0.

Next consider the scenario where σ>0. We have α̂→σ^2_X+σ^2-P/2σ^2 as σ_X̂→σ, and consequently

    lim_σ_X̂→0δ_+(σ_X̂,α̂)   =δ_+(σ,σ^2_X+σ^2-P/2σ^2)
       =0.

This completes the proof of the first inequality in (<ref>).


















The second inequality in (<ref>) follows from the fact that 

    D(R,R_c,P|W^2_2)=min_σ_X̂∈[(σ_X-√(P))_+,σ_X]σ^2_X+σ^2_X̂-2σ_X√((1-e^-2R)(σ^2_X̂-δ^2_+(σ_X̂,1))).

Now we proceed to identify the sufficient and necessary condition under which this inequality is strict. It suffices to consider the case R∈(0,∞) and R_c∈[0,∞) since otherwise D'(R,R_c,P|W^2_2) clearly coincides with D(R,R_c,P|W^2_2). 
Note  that the minimum in (<ref>) is attained at and only at <cit.>

    σ_X̂   =σ̂:=σ_X√(1-e^-2R)   √(P)/σ_X≥(1-√(1-e^-2R))∨ e^-(R+R_c),
    σ_X-√(P)   √(P)/σ_X∈[e^-(R+R_c),1-√(1-e^-2R)),
    √(σ^2_X(1-e^-2R)+(σ_Xe^-(R+R_c)-√(P))^2)   √(P)/σ_X∈[ν(R,R_c),e^-(R+R_c)),
    σ_X-√(P)   √(P)/σ_X<ν(R,R_c)∧ e^-(R+R_c),

where

    ν(R,R_c):=e^-2R-e^-2(R+R_c)/2-2e^-(R+R_c).

We have the following observation:
D'(R,R_c,P|W^2_2)>D(R,R_c,P|W^2_2) if and only if

    sup_α>0σ^2_X+σ̂^2-2σ_X√((1-e^-2R)(σ̂^2-δ^2_+(σ̂,α)))>σ^2_X+σ̂^2-2σ_X√((1-e^-2R)(σ̂^2-δ^2_+(σ̂,1))).


“If" part: Assume the minimum in (<ref>) is attained at σ_X̂=σ̃. If σ̃=σ̂, we have

    D'(R,R_c,P|W^2_2)
    	   =sup_α>0σ^2_X+σ̂^2-2σ_X√((1-e^-2R)(σ̂^2-δ^2_+(σ̂,α)))
       >σ^2_X+σ̂^2-2σ_X√((1-e^-2R)(σ̂^2-δ^2_+(σ̂,1)))
       =D(R,R_c,P|W^2_2).

If σ̃≠σ̂, we have

    D'(R,R_c,P|W^2_2)
    	   =sup_α>0σ^2_X+σ̃^2-2σ_X√((1-e^-2R)(σ̃^2-δ^2_+(σ̃,α)))
       ≥σ^2_X+σ̃^2-2σ_X√((1-e^-2R)(σ̃^2-δ^2_+(σ̃,1)))
       (a)>σ^2_X+σ̂^2-2σ_X√((1-e^-2R)(σ̂^2-δ^2_+(σ̂,1)))
       =D(R,R_c,P|W^2_2),

where (a) is due to the fact that σ̂ is the unique minimizer of (<ref>). Thus, D'(R,R_c,P|W^2_2)>D(R,R_c,P|W^2_2) holds either way.


“Only if" part: This is because

    sup_α>0σ^2_X+σ̂^2-2σ_X√((1-e^-2R)(σ̂^2-δ^2_+(σ̂,α)))   ≥D'(R,R_c,P|W^2_2)
       >D(R,R_c,P|W^2_2)
       =σ^2_X+σ̂^2-2σ_X√((1-e^-2R)(σ̂^2-δ^2_+(σ̂,1))).

 






Equipped with the above observation, we shall treat the following two cases separately.

1) P≥σ^2_Xe^-2(R+R_c): In this case, δ_+(σ_X̂,1)=0. Therefore, (<ref>) holds if and only  if sup_α>0δ_+(σ̂,α)>0, which, in light of (<ref>), is equivalent to










    P<σ^2_X+σ̂^2-2σ_Xσ̂√(1-e^-2(R+R_c)).


For the subcase √(P)/σ_X≥(1-√(1-e^-2R))∨ e^-(R+R_c), we have σ̂=σ_X√(1-e^-2R), and consequently (<ref>) becomes

    P<σ^2_X(2-e^-2R-2√((1-e^-2R)(1-e^-2(R+R_c)))).

For the subcase √(P)/σ_X∈[e^-(R+R_c),1-√(1-e^-2R)), we have σ̂=σ_X-√(P),  and consequently (<ref>) becomes 

    0<(2σ^2_X-2σ_X√(P))(1-√(1-e^-2(R+R_c))),

which holds trivially. Combining the analyses for these two subcases shows that P/σ^2_X must fall into the following interval:

    [e^-2(R+R_c),2-e^-2R-2√((1-e^-2R)(1-e^-2(R+R_c)))).

Note that

    e^-2(R+R_c)   =.2-e^-2R-1-e^-2(R+R_c)/α-α(1-e^-2R)|_α=1
       ≤sup_α>02-e^-2R-1-e^-2(R+R_c)/α-α(1-e^-2R)
       =2-e^-2R-2√((1-e^-2R)(1-e^-2(R+R_c))),

where the supremum is attained at and only at α=√(1-e^-2(R+R_c)/1-e^-2R). 
Thus, the interval in (<ref>) is nonempty unless R_c=0.
	
2) P<σ^2_Xe^-2(R+R_c): In this case, δ_+(σ_X̂,1)>0. 




Clearly, δ_+(σ_X̂,α)=δ(σ_X̂,α) whenever δ_+(σ_X̂,α)>0. Since δ_+(σ_X̂,1)>0, we must have δ_+(σ_X̂,α)=δ(σ_X̂,α) in a neighbourhood of α=1.






Setting .∂/∂αδ(σ_X̂,α)|_α=1=0 gives

    σ_X̂=σ^*_X̂:=√(σ^2_X-2σ_Xe^-(R+R_c)√(P)+P).



For the subcase √(P)/σ_X∈[ν(R,R_c),e^-(R+R_c)), we have 
σ̂=√(σ^2_X(1-e^-2R+e^-2(R+R_c))-2σ_Xe^-(R+R_c)√(P)+P), which, in view of (<ref>), implies .∂/∂αδ(σ̂,α)|_α=1≠ 0 unless R_c=0. Note that in this subcase, P=0⇒ν(R,R_c)=0⇒ R_c=0. 
For the subcase √(P)/σ_X<ν(R,R_c)∧ e^-(R+R_c), we have
σ̂=√(σ^2_X-2σ_X√(P)+P), which, in view of (<ref>), implies .∂/∂αδ(σ̂,α)|_α=1≠ 0 unless P=0. Note that this subcase is void when R_c=0 since ν(R,0)=0. Combining the analyses for these two subcases shows that if R_c>0 and P>0, then .∂/∂αδ(σ̂,α)|_α=1≠ 0, which further implies (<ref>).

It remains to show .α̂|_σ_X̂=σ̂=1 when R_c=0 or P=0. This can be accomplished via a direct verification. The proof of Theorem <ref> is thus complete.







 
 
 
 
 
 
 
 
 






§ PROOF OF (<REF>)


Clearly, (<ref>) holds for all R≥ 0 when P≥σ^2_X. It remains to consider the case P∈(0,σ^2_X). We can write  (<ref>) equivalently as

    2-z-P/σ^2_X≤ 2√((1-z)(1-ze^-2R_c)),

where z:=e^-2R.
Note that 

    2-z-P/σ^2_X= 2√((1-z)(1-ze^-2R_c))

has a solution in (0,1) since its left-hand side is less than its right-hand side when z=0 and is greater than its right-hand side when z=1. By taking the square of both sides of (<ref>) and simplifying the expression, we get

    ζ_1z^2-ζ_2z+ζ_3=0.

It is easy to see that ζ_2>0 and ζ_3>0.
If ζ_1=0 (i.e., R_c=log 2), then  (<ref>) has only one solution, given by

    ẑ:=ζ_3/ζ_2,

which is also the unique solution to (<ref>). If ζ_1<0 (i.e., R_c>log 2), then  (<ref>) has two solutions with different signs and only the positive one, given by

    ẑ:=ζ_2-√(ζ^2_2-4ζ_1ζ_3)/2ζ_1,

is the solution to (<ref>) for z∈(0,1). If ζ_1>0 (i.e, R_c<log 2), then (<ref>) has two positive solutions and only the small one, also given by (<ref>), is the solution to  (<ref>). Indeed, the large one is a solution to the following equation obtained  by negating the left-hand side of  (<ref>):

    -2+z+P/σ^2_X= 2√((1-z)(1-ze^-2R_c)).

This can be verified by noticing that (<ref>) has a solution in (1,∞) since its left-hand side is less than its right-hand side when z=1 and is greater than its right-hand side when z is sufficiently large. Therefore, ẑ is the unique solution to (<ref>) for z∈(0,1), and consequently 
(<ref>) holds if and only if z∈[0,ẑ], i.e., R≥-1/2logẑ.



§ PROOF OF THEOREM <REF>



	For the optimization problem in  (<ref>), there is no loss of generality in assuming U=𝔼[X|U] almost surely and 𝔼[X̂^2]≤(1+√(2))^2𝔼[X^2].
	

	Note that
	
    D(R,R_c,P|ϕ)≤ 2𝔼[X^2]
 
	since we can trivially let X, U, and X̂ be mutually independent and p_X̂=p_X. Therefore, it suffices to consider p_UX̂|X with 𝔼[X̂^2]≤ (1+√(2))^2𝔼[X^2] because otherwise
	
    𝔼[(X-X̂)^2]   =𝔼[X^2]+𝔼[X̂^2]-2𝔼[XX̂]
       ≥𝔼[X^2]+𝔼[X̂^2]-2√(𝔼[X^2]𝔼[X̂^2])
       >𝔼[X^2]+(1+√(2))^2𝔼[X^2]-2(1+√(2))𝔼[X^2]
       =2𝔼[X^2].

	
	
	For any p_UX̂|X satisfying (<ref>)–(<ref>), let Û:=𝔼[X|U]. Construct p_U'X̂'|X such that 	
	X↔ U'↔X̂' form a Markov chain, p_U'|X=p_Û|X, and p_X̂'|U'=p_X̂|Û. Clearly, 
	
    I(X;U')=I(X;Û)(a)≤ I(X;U)≤ R,
       I(X̂';U')=I(X̂;Û)(b)≤ I(X̂;U)≤ R+R_c,

	where (a) and (b) are due to the data processing inequality <cit.>. Moreover, we have p_X̂'=p_X̂ and consequently
	
    ϕ(p_X,p_X̂')=ϕ(p_X,p_X̂)≤ P.

	It can also be verified that
	
    𝔼[(X-X̂')]   (c)=𝔼[(X-U')^2]+𝔼[(X̂'-U')^2]
       =𝔼[(X-Û)^2]+𝔼[(X̂-Û)^2]
       (d)=𝔼[(X-X̂)],

where (c) and (d) follow  respectively from the facts that  U'=𝔼[X|U',X̂'] and Û=𝔼[X|Û,X̂] almost surely.
Therefore, there is no loss of optimality in replacing p_UX̂|X with p_U'X̂'|X.




Now we proceed to prove Theorem <ref>. For any positive integer k, in light of Lemma <ref>, there exists p_U^(k)X̂^(k)|X satisfying

    I(X;U^(k))≤ R,
       I(X̂^(k);U^(k))≤ R+R_c,
       ϕ(p_X,p_X̂^(k))≤ P,
       U^(k)=𝔼[X|U^(k)],
       𝔼[(X̂^(k))^2]≤ (1+√(2))^2𝔼[X^2]

as well as the Markov chain constraint X↔ U^(k)↔X̂^(k)
such that

    𝔼[(X-X̂^(k))^2]≤ D(R,R_c,P|ϕ)+1/k.

The sequence {p_XU^(k)X̂^(k)}_k=1^∞ is tight <cit.> since given any ϵ>0,

    ℙ{X^2≤3/ϵ𝔼[X^2], (U^(k))^2≤3/ϵ𝔼[X^2], (X̂^(k))^2≤3(1+√(2))^2/ϵ𝔼[X^2]}
       ≥ 1-ℙ{X^2>3/ϵ𝔼[X^2]}-ℙ{(U^(k))^2>3/ϵ𝔼[X^2]}-ℙ{(X̂^(k))^2>3(1+√(2))^2/ϵ𝔼[X^2]}
       ≥ 1-ϵ/3-𝔼[(U^(k))^2]ϵ/3𝔼[X^2]-𝔼[(X̂^(k))^2]ϵ/3(1+√(2))^2𝔼[X^2]
       ≥1-ϵ

for all k. By Prokhorov's theorem <cit.>, there exists a subsequence {p_XU^(k_m)X̂^(k_m)}_m=1^∞ converging weakly to some distriution p_XU^*X̂^*. 
Since p_X has bounded support, it follows by <cit.> that

    𝔼[(X-𝔼[X|U^*,X̂^*])^2]≥lim sup_m→∞𝔼[(X-𝔼[X|U^(k_m),X̂^k_m])^2]=lim sup_m→∞𝔼[(X-U^(k_m))^2].

On the other hand,  as the map (x,u)↦(x-u)^2 is continuous and bounded from below, we have

    𝔼[(X-U^*)^2]≤lim inf_m→∞𝔼[(X-U^(k_m))^2],

which, together with (<ref>), implies

    U^*=𝔼[X|U^*,X̂^*].
  Moreover, by the lower semicontinuity of mutual information and p_X̂↦ϕ(p_X, p_X̂) in the topology of weak convergence,

    I(X;U^*)≤lim inf_m→∞I(X;U^(k_m)),
       I(X̂^*;U^*)≤lim inf_m→∞I(X̂^(k_m);U^(k_m)),
       ϕ(p_X,p_X̂^*)≤lim inf_m→∞ϕ(p_X,p_X̂^(k_m)).


Construct p_U'X̂'|X such that 	
X↔ U'↔X̂' form a Markov chain, p_U'|X=p_U^*|X, and p_X̂'|U'=p_X̂^*|U^*. In view of (<ref>)–(<ref>) and (<ref>)–(<ref>),

    I(X;U')=I(X;U^*)≤ R,
       I(X̂';U')=I(X̂^*;U^*)≤ R+R_c,
       ϕ(p_X,p_X̂')=ϕ(p_X,p_X̂^*)≤ P.

Similarly to (<ref>), we have

    𝔼[(X-X̂')^2]   =𝔼[(X-U')^2]+𝔼[(X̂'-U')]
       =𝔼[(X-U^*)^2]+𝔼[(X̂^*-U^*)]
       =𝔼[(X-X̂^*)^2].

Since the map (x,x̂)↦(x-x̂)^2 is continuous and bounded from below, it follows that

    𝔼[(X-X̂^*)^2]≤lim inf_m→∞𝔼[(X-X̂^(k_m))^2.
 
Combining (<ref>), (<ref>), and (<ref>) shows

    𝔼[(X-X̂')^2]≤ D(R,R_c,P|ϕ).
 
Therefore, the infimum in (<ref>) is attained
at p_U'X̂'|X. 
























The above argument can be easily leveraged to prove  the lower semicontinuity of (R,R_c,P)↦ D(R,R_c,P|ϕ), which implies the desired right-continuity property since the map (R,R_c,P)↦ D(R,R_c,P|ϕ) is monotonically decreasing in each of its variables.

The following subtlety in this proof is noteworthy. It is tempting to claim that the weak convergence limit  p_XU^*X̂^* automatically satisfies the Markov chain constraint X↔ U^*↔X̂^*. We are unable to confirm this claim. In fact, this claim is false if (<ref>) does not hold. For example, let U^(k):=1/kU and

    X=X̂^(k):=
    		1    U≥ 0,
    
    		-1    U<0,

where U is a standard Gaussian random variable. It is clear that X↔ U^(k)↔X̂^(k) form a Markov chain for any positive integer k. However, the Markov chain constraint  is violated by the weak convergence limit p_XU^*X̂^* since X and X̂^* are two identical symmetric Bernoulli random variables whereas U^* is a constant zero. Our key observation is that it suffices to have (<ref>), with which the Markov chain structure can be  restored without affecting the end-to-end distortion (see the construction of p_U'X̂'|X). Nevertheless, we only manage to establish (<ref>) when p_X has bounded support. Note that, according to the exampel above, the minimum mean square error is not necessarily preserved under weak convergence if  (<ref>) does not hold. Indeed, while 𝔼[(X-𝔼[X|U^(k)])^2]=0 for any positive integer k, we have 𝔼[(X-𝔼[X|U^*])^2]=1.
















§ PROOF OF THEOREM  <REF>


We need the following well-known result regarding the Ornstein-Uhlenbeck flow (see, e.g., <cit.>).


	For p_X=𝒩(μ_X,σ^2_X) and p_X̂ with μ_X̂=μ_X and 𝔼[X̂^2]<∞, 
	let X̂(λ):=μ_X+√(1-λ)(X̂-μ_X)+√(λ)(X̅-μ_X), where X̅ is independent of X̂ and has the same distribution as X.		
	The map λ↦ϕ_KL(p_X̂(λ)p_X) is continuous[ϕ_KL(p_X̂(λ)p_X) varies continuously from ϕ_KL(p_X̂p_X) to 0 as λ increases from 0 to 1. Note that ϕ_KL(p_X̂(λ)p_X)<∞ for λ∈(0,1] and
		 lim_λ→ 0ϕ_KL(p_X̂(λ)p_X)=ϕ_KL(p_X̂p_X) even if ϕ_KL(p_X̂p_X)=∞ (in this sense, the map λ↦ϕ_KL(p_X̂(λ)p_X) is continuous at λ=0).], decreasing, and convex for λ∈[0,1].

















	
	
	














	
	
	













Now we proceed to prove Theorem <ref>.  Given ϵ>0, there exists p_UX̂|X satisfying (<ref>)–(<ref>) with ϕ=ϕ_KL and 𝔼[(X-X̂)^2]≤ D(R,R_c,P|ϕ_KL)+ϵ. Without loss of generality, we assume μ_X̂=μ_X and σ_X̂≤σ_X <cit.>. For λ∈[0,1], let X̂(λ):=μ_X+√(1-λ)(X̂-μ_X)+√(λ)(X̅-μ_X), where X̅ is assumed to be independent of (X,U,X̂) and have the same distribution as X. Note that X↔ U↔X̂↔X̂(λ) form a Markov chain.
By the data processing inequality <cit.> and  (<ref>), 

    I(X̂(λ);U)≤ I(X̂;U)≤ R+R_c.


First consider the case P∈(0,∞). In light of Lemma <ref>, given P̃∈(0,P],  there exists λ̃∈[0,1] such that ϕ_KL(p_X̂(λ̃)p_X)=ϕ_KL(p_X̂p_X)∧P̃; moreover, we have[When ϕ_KL(p_X̂p_X)=0, we can set λ̃=0 and consequently λ̃≤P-P̃/P̃ still holds.]

    λ̃≤ 1-ϕ_KL(p_X̂p_X)∧P̃/ϕ_KL(p_X̂p_X)≤P-P̃/P̃.

It can be verified that 

    D(R,R_c,P̃|ϕ_KL)-D(R,R_c,P|ϕ_KL)
       ≤ D(R,R_c,P̃|ϕ_KL)-𝔼[(X-X̂)^2]+ϵ
       ≤𝔼[(X-X̂(λ̃))^2]-𝔼[(X-X̂)^2]+ϵ
       =2𝔼[(X-X̂)(X̂-X̂(λ̃))]+𝔼[(X̂-X̂(λ̃))^2]+ϵ
       ≤2√(𝔼[(X-X̂)^2]𝔼[(X̂-X̂(λ̃))^2])+𝔼[(X̂-X̂(λ̃))^2]+ϵ
       =2√(𝔼[(X-X̂)^2]((1-√(1-λ̃))^2σ^2_X̂+λ̃σ^2_X))+(1-√(1-λ̃))^2σ^2_X̂+λ̃σ^2_X+ϵ
       ≤(4√(2-2√(1-λ̃))+2-2√(1-λ̃))σ^2_X+ϵ
       ≤(4√(2-2√((2P̃-P)_+/P̃))+2-2√((2P̃-P)_+/P̃))σ^2_X+ϵ.

This proves

    D(R,R_c,P̃|ϕ_KL)-D(R,R_c,P|ϕ_KL) ≤(4√(2-2√((2P̃-P)_+/P̃))+2-2√((2P̃-P)_+/P̃))σ^2_X.


Next consider the case P=∞.  In light of Lemma <ref>, given P̃∈[0,∞), there exists λ̃∈(0,1] such that ϕ_KL(p_X̂(λ̃)p_X)≤P̃; moreover, we can require λ̃→ 0 as P̃→∞. It can be verified that  

    lim_P̃→∞D(R,R_c,P̃|ϕ_KL)   ≤lim_λ̃→ 0𝔼[(X-X̂(λ))^2]
       =𝔼[(X-X̂)^2]
       ≤ D(R,R_c,∞|ϕ_KL)+ϵ.

This proves

    lim_P̃→∞D(R,R_c,P̃|ϕ_KL)≤ D(R,R_c,∞|ϕ_KL).


Finally, consider the case P=0. Given P̃∈[0,∞] and ϵ>0, there exists p_UX̂|X 
satisfying (<ref>)–(<ref>), ϕ_KL(p_X̂p_X)≤P̃, and 𝔼[(X-X̂)^2]≤ D(R,R_c,P̃|ϕ_KL)+ϵ. Without loss of generality, we assume μ_X̂=μ_X and σ_X̂≤σ_X <cit.>. Let X̅' be jointly distributed with (X,U,X̂)
such that X↔ U↔X̂↔X̅' form a Markov chain, X̅'∼ p_X, and 𝔼[(X̅'-X̂)^2]=W^2_2(p_X,p_X̂). By the data processing inequality <cit.> and  (<ref>), 

    I(X̅';U)≤ I(X̂;U)≤ R+R_c.

It can be verified that

    D(R,R_c,0|ϕ_KL)-D(R,R_c,P̃|ϕ_KL)
       ≤ D(R,R_c,0|ϕ_KL)-𝔼[(X-X̂)^2]+ϵ
       ≤𝔼[(X-X̅')^2]-𝔼[(X-X̂)^2]+ϵ
       =2𝔼[(X-X̂)(X̂-X̅')]+𝔼[(X̂-X̅')^2]+ϵ
       ≤ 2√(𝔼[(X-X̂)^2]𝔼[(X̂-X̅')^2])+𝔼[(X̂-X̅')^2]+ϵ
       =2√(𝔼[(X-X̂)^2]W^2_2(p_X,p_X̂))+W^2_2(p_X,p_X̂)+ϵ
       (a)≤2√(2σ^2_X𝔼[(X-X̂)^2]ϕ_KL(p_X̂p_X))+2σ^2_Xϕ_KL(p_X̂p_X)+ϵ
       ≤2√(2σ^2_X𝔼[(X-X̂)^2]P̃)+2σ^2_XP̃+ϵ
       ≤(4√(2P̃)+2P̃)σ^2_X+ϵ,

where (a) is due to Talagrand's transportation  inequality <cit.>. 
This proves

    D(R,R_c,0|ϕ_KL)-D(R,R_c,P̃|ϕ_KL)≤ (4√(2P̃)+2P̃)σ^2_X.


In view of (<ref>), (<ref>), and (<ref>),  the desired continuity property follows by the fact that the map P↦ D(R,R_c,P|W^2_2) is monotonically decreasing.





























































§ PROOF OF THEOREM  <REF>



We need the following result <cit.> regarding the distortion-perception tradeoff in the quadratic 
Wasserstein space.


	For λ∈[0,1] and p_XX̂ with 𝔼[X^2]<∞ and 𝔼[X̂^2]<∞, let
X̂(λ):=(1-λ)X̃+λX̅, where X̃:=𝔼[X|X̂], and X̅ is
jointly distributed with (X,X̂) such that X↔X̂↔X̅ form a Markov chain, X̅∼ p_X, and 𝔼[(X̅-X̃)^2]=W^2_2(p_X,p_X̃). We have

    𝔼[(X-X̂(λ))^2]=𝔼[(X-X̃)^2]+(W_2(p_X,p_X̃)-W_2(p_X,p_X̂(λ)))^2

and

    W^2_2(p_X,p_X̂(λ))=(1-λ)^2W^2_2(p_X,p_X̃).

Moreover, for any X̂' jointly distributed with (X,X̂) such that X↔X̂↔X̂' form a Markov chain and 𝔼[(X̂')^2]<∞,

    𝔼[(X-X̂')^2]≥𝔼[(X-X̃)^2]+(W_2(p_X,p_X̃)-W_2(p_X,p_X̂'))^2_+.











Now we proceed to prove Theorem <ref>. Given ϵ>0, there exists  p_UX̂|X satisfying (<ref>)–(<ref>) with ϕ=W^2_2 and 𝔼[(X-X̂)^2]≤ D(R,R_c,P|W^2_2)+ϵ. Let X̅ be jointly distributed with (X,U,X̂) such that (X,U)↔X̂↔X̅ form a Markov chain, X̅∼ p_X, and 𝔼[(X̅-X̃)^2]=W^2_2(p_X,p_X̃), where
X̃:=𝔼[X|X̂]. Moreover, let 
X̂(λ):=(1-λ)X̃+λX̅ for λ∈[0,1]. Note that X↔ U↔X̂↔X̂(λ) form a Markov chain. By the data processing inequality <cit.> and (<ref>), 

    I(X̂(λ);U)≤ I(X̂;U)≤ R+R_c.


Given P̃∈[0,P], in light of (<ref>) in Lemma <ref>, there exists λ̃∈[0,1] such that W^2_2(p_X,p_X̂(λ̃))=W^2_2(p_X,p_X̃)∧P̃. We have

    D(R,R_c,P̃|W^2_2)-D(R,R_c,P|W^2_2)
       ≤ D(R,R_c,P̃|W^2_2)-𝔼[(X-X̂)^2]+ϵ
       ≤𝔼[(X-X̂(λ̃))^2]-𝔼[(X-X̂)^2]+ϵ
       (a)≤(W_2(p_X,p_X̃)-W_2(p_X,p_X̂(λ̃)))^2-(W_2(p_X,p_X̃)-W_2(p_X,p_X̂))^2_++ϵ
       ≤(W_2(p_X,p_X̃)-W_2(p_X,p_X̂(λ̃)))^2-(W_2(p_X,p_X̃)-(W_2(p_X,p_X̃)∧ P))^2+ϵ
       =(2W_2(p_X,p_X̃)-(W_2(p_X,p_X̃)∧ P)-W_2(p_X,p_X̂(λ̃)))((W_2(p_X,p_X̃)∧ P)-W_2(p_X,p_X̂(λ̃)))+ϵ
       ≤ 2W_2(p_X,p_X̃)((W_2(p_X,p_X̃)∧ P)-W_2(p_X,p_X̂(λ̃)))+ϵ
       ≤ 2σ_X((σ_X∧√(P))-(σ_X∧√(P̃)))+ϵ,

where (a) is due to (<ref>) and (<ref>) in Lemma <ref>. This proves

    D(R,R_c,P̃|W^2_2)-D(R,R_c,P|W^2_2)≤ 2σ_X((σ_X∧√(P))-(σ_X∧√(P̃))),

which, together with the fact that the map P↦ D(R,R_c,P|W^2_2) is monotonically decreasing, implies the desired continuity property.
































































§ PROOF OF THEOREM <REF>














Note that for any p_X̂|X such that I(X;X̂)≤ R and H(X̂)≤ R+R_c, the induced p_UX̂|X with U:=X̂ satisfies (<ref>)–(<ref>). This implies D_e(R,R_c)≥ D(R,R_c,∞). On the other hand, for any p_UX̂|X satisfying
(<ref>)–(<ref>), it follows by the data processing inequality <cit.> that I(X;X̂)≤ R. 	
Therefore, we must have D(R,R_c,∞)≥ D_e(R,∞). This completes the proof of (<ref>).

For the purpose of establishing the equivalence relationship (<ref>), it suffices to show that D_e(R,R_c)>D_e(R,∞) implies D(R,R_c,∞)>D_e(R,∞) since the converse is implied by (<ref>). 
To this end, we shall prove the contrapositive statement, namely, D(R,R_c,∞)≤ D_e(R,∞) implies D_e(R,R_c)≤ D_e(R,∞). 	
Assume that the infimum in (<ref>) is attained by some p_U^*X̂^*|X. Let Û^*:=𝔼[X|U^*]. 
We have

    D(R,R_c,∞)   =𝔼[(X-X̂^*)^2]
       (a)=𝔼[(X-Û^*)^2]+𝔼[(X̂^*-Û^*)^2],

where (a) holds because Û^*=𝔼[X|U^*,X̂^*] almost surely. Since	

    I(X;Û^*)≤ I(X;U^*)≤ R,

it follows that 𝔼[(X-Û^*)^2]≥ D_e(R,∞). Therefore, D(R,R_c,∞)≤ D_e(R,∞) implies 𝔼[(X-Û^*)^2]=D_e(R,∞) and 𝔼[(Û^*-X̂^*)^2]=0 (i.e., Û^*=X̂^* almost surely).
Note that

    I(X;Û^*)≤ I(X;U^*)   ≤ R

and

    H(Û^*)=I(X̂^*; Û^*)≤ I(X̂^*; U^*)≤ R+R_c.

As a consequence, we have 𝔼[(X-Û^*)^2]≥ D_e(R,R_c). This proves D_e(R,R_c)≤ D_e(R,∞).





§ PROOF OF THEOREM <REF>



	

	For p_X=𝒩(μ_X,σ^2_X),
	
    D_e(R,R_c)>D(R,R_c,∞)

	when R∈(0,∞) and R_c∈[0,∞), and 
	
    D_e(R,R_c)<D(R,R_c,∞)

	when R_c∈[0,∞) and R∈(0,χ(R_c)), where χ(R_c) is a positive threshold that depends on R_c.


	Let p_X̂^*|X be some conditional distribution that attains the minimum in (<ref>). Clearly, we must have μ_X̂^*=μ_X. Note that
	
    R   ≥ I(X;X̂^*)
       =h(X)-h(X|X̂^*)
       =h(X)-h(X-X̂^*|X̂^*)
       (a)≥ h(X)-h(X-X̂^*)
    
    		   (b)≥1/2logσ^2_X/D_e(R,R_c).

	The inequalities (a) and (b) become equalities if and only if X-X̂^* is independent of X^* and is distributed as 𝒩(0,D_e(R,R_c)), which, together with the fact p_X=𝒩(μ_X,σ^2_X), 
	implies p_X̂^*=𝒩(μ_X,σ^2_X-D_e(R,R_c)).
	This is impossible since H(X̂^*)≤ R+R_c<∞ whereas the entropy of a Gaussian distribution with positive variance[Since R>0, it follows that D_e(R,R_c)<σ^2_X.] is infinite. Therefore, at least one of the inequalities (a) and (b) is strict, yielding
	
    R>1/2logσ^2_X/D_e(R,R_c).

	Now one can readily prove (<ref>) by invoking (<ref>).
	
	
	
	According to <cit.>, 
	
    D_e(R,0)=σ^2_X(1-2R)+o(R),

where o(R) stands for a term that approaches zero more rapidly than R as R→ 0.
	On the other hand, it can be deduced from (<ref>) that
	
    D(R,R_c,∞)=σ^2_X(1-2R+2Re^-2R_c)+o(R).

	Combining (<ref>) and (<ref>) then invoking the fact D_e(R,R_c)≤ D_e(R,0) proves (<ref>).


Clearly, (<ref>) is a direct consequence of (<ref>) and (<ref>). Since 

    D(R,R_c,∞)=D_e(R,∞)

for p_X=𝒩(μ_X,σ^2_X), it is tempting to deduce (<ref>) from  (<ref>) and (<ref>). Unfortunately, (<ref>) relies on the assumption that the infimum in (<ref>) can be attained, which has not been verified for Gaussian p_X. Nevertheless, we show below that the key idea underlying the proof of (<ref>) and (<ref>), namely, violating (<ref>) necessarily forces Û^* to be Gaussian and coincide with X̂^*, can be salvaged without resorting to the aforementioned assumption 
 by treating (Û^*,X̂^*)  as a certain limit under weak convergence.

Assume that (<ref>) does not hold, i.e.,  

    D(R,R_c,∞)=σ^2_Xe^-2R.

For any positive integer k, there exists p_U^(k)X̂^(k)|X satisfying

    I(X;U^(k))≤ R,
       I(X̂^(k);U^(k))≤ R+R_c

as well as the Markov chain constraint X↔ U^(k)↔X̂^(k)
such that

    𝔼[(X-X̂^(k))^2]≤σ^2_Xe^-2R+1/k.

Let Û^(k):=𝔼[X|U^(k)] and V^(k):=X-Û^(k). Since X↔ U^(k)↔X̂^(k) form a Markov chain, it follows that

    𝔼[(X-X̂^(k))^2]=σ^2_V^(k)+𝔼[(Û^(k)-X̂^(k))^2],

which, together with (<ref>), implies

    σ^2_V^(k)≤σ^2_Xe^-2R+1/k.

Moreover, we have

    h(V^(k)|Û^(k))	   ≥ h(V^(k)|U^(k))
       =h(X|U^(k))
       =h(X)-I(X;U^(k))
       ≥ h(X)-R
       =1/2log(2π eσ^2_Xe^-2R),

and consequently

    h(V^(k))≥1/2log(2π eσ^2_Xe^-2R).

 Combining (<ref>) and (<ref>) gives

    ϕ_KL(p_V^(k)𝒩(0,σ^2_Xe^-2R))
    	   =-h(V^(k))+1/2log(2πσ^2_Xe^-2R)+σ^2_V^(k)/2σ^2_Xe^-2R
       ≤1/2kσ^2_Xe^2R.

Therefore, p_V^(k) converges to 𝒩(0,σ^2_Xe^-2R) in Kullback-Leibler divergence as k→∞. It can be shown that the sequence {p_XÛ^(k)V^(k)X̂^(k)}_k=1^∞ is tight (cf. the proof of Theorem <ref>). By Prokhorov's theorem <cit.>, there exists a subsequence {p_XÛ^(k_m)V^(k_m)X̂^(k_m)}_m=1^∞ converging weakly to some distribution p_XÛ^*V^*X̂^*. Clearly, we have p_V^*=𝒩(0,σ^2_Xe^-2R). Note that (<ref>) implies

    h(V^(k))≤1/2log(2π e(σ^2_Xe^-2R+1/k)),

which, together with (<ref>), yields

    I(Û^(k);V^(k))≤1/2log(1+1/kσ^2_Xe^2R).

By the lower semicontinuity of mutual informaiton in the topology of weak convergence,

    I(Û^*;V^*)≤lim inf_m→∞I(Û^(k_m);V^(k_m))=0.

Thus Û^* and V^* must be independent. Since p_Û^*+V^*=p_X=𝒩(μ_X,σ^2_X) and p_V^*=𝒩(0,σ^2_Xe^-2R), it follows that p_Û^*=𝒩(μ_X,σ^2_X(1-e^-2R)). 

It remains to show that Û^*=X̂^* almost surely. In view of (<ref>),
the Shannon lower bound gives

    σ^2_V^(k)≥σ^2_Xe^-2R,

which, together with (<ref>) and (<ref>), further implies

    𝔼[(Û^(k)-X̂^(k))^2]≤1/k.

As the map (û,x̂)↦ (û-x̂)^2 is continuous and bounded from below, 

    𝔼[(Û^*-X̂^*)^2]≤lim inf_m→∞𝔼[(Û^(k_m)-X̂^(k_m))^2]=0.

This leads to a contradiction with (<ref>) since

    lim inf_k→∞I(X̂^(k_m);U^(k_m))   (a)≥lim inf_k→∞I(X̂^(k_m);Û^(k_m))
       (b)≥ I(X̂^*;Û^*)
       =∞,

where (a) is due to the data processing inequality <cit.>, and (b) is due to the lower semicontinuity of mutual informaiton in the topology of weak convergence.

The above proof can be simplified by circumventing the steps regarding the convergence of p_V^(k) to 𝒩(0,σ^2_Xe^-2R) in Kullback-Leibler divergence. Indeed, by
Cramér's decomposition theorem, both  Û^* and V^* must be Gaussian if they are independent and their sum is Gaussian. Moreover, one can invoke the weak convergence argument to show that σ^2_Û^*≤σ^2_X(1-e^-2R) and 
σ^2_V^*≤σ^2_Xe^-2R. Since σ^2_Û^*+σ^2_V^*=σ^2_X, we must have p_Û^*=𝒩(μ_X,σ^2(1-e^-2R)) and p_V^*=𝒩(0,σ^2_Xe^-2R). However, the original proof provides more information as convergence in Kullback-Leibler divergence is stronger than weak convergence.

















§ PROOF OF COROLLARY <REF>





In light of Theorem <ref>,

    D(R,R_c,∞|ϕ_KL)>D(R,R_c,∞|ϕ_KL)

when R∈(0,∞) and R_c∈[0,∞). This implies that
(<ref>) holds for sufficiently large P 
since P↦ D(R,R_c,P|ϕ_KL) is monotonically decreasing while P↦D(R,R_c,P|ϕ_KL) is continuous at P=∞.

In light of Theorem <ref>,

    D(R,R_c,∞|ϕ_KL)<D(R,R_c,∞|ϕ_KL)

when R_c∈[0,∞) and R∈(0,χ(R_c)). 
This implies that (<ref>) holds for sufficiently large P since
P↦ D(R,R_c,P|ϕ_KL) is continuous at P=∞ by Theorem <ref> and  P↦D(R,R_c,P|ϕ_KL) is monotonically decreasing.



§ PROOF OF COROLLARY <REF>




In light of Theorem <ref>,

    D(R,R_c,∞|W^2_2)>D'(R,R_c,∞|W^2_2)

when R∈(0,∞) and R_c∈[0,∞). This implies
that (<ref>) holds 
for P above a postive threshold γ'(R,R_c) strictly less than P'(R,R_c) since P↦ D(R,R_c,P|W^2_2) is monotonically decreasing while
P↦D'(R,R_c,P|W^2_2) is continuous and remains constant over the interval [P'(R,R_c),∞].

In light of Theorem <ref>,

    D(R,R_c,∞|W^2_2)<D(R,R_c,∞|W^2_2)

when R_c∈[0,∞) and R∈(0,χ(R_c)). 
This implies that (<ref>) holds for P above a postive threshold γ(R,R_c) strictly less than P(R,R_c) since
P↦ D(R,R_c,P|W^2_2) is continuous by Theorem <ref> and remains constant over the interval [P(R,R_c),∞] 
while  P↦D(R,R_c,P|W^2_2) is monotonically decreasing.








































1
	
		Matsumoto18
	R. Matsumoto, “Introducing the perception-distortion tradeoff into the
	rate-distortion theory of general information sources," IEICE Comm.
		Express, vol. 7, no. 11, pp. 427–431, 2018.
	
	
	Matsumoto19
	R. Matsumoto, “Rate-distortion-perception tradeoff of variable-length source coding for general information sources,? IEICE Comm. Express, vol. 8,
	no. 2, pp. 38–42, 2019.
	
	BM19 Y. Blau and T. Michaeli, “Rethinking lossy compression:
	The rate-distortion-perception tradeoff," Proc. Int. Conf. Mach. Learn., vol. 97, pp. 675–685, Jun. 2019.

	TW21 L. Theis and A. B. Wagner, “A coding theorem for
	the rate-distortion-perception function," Proc. ICLR, pp. 1–5, 2021.
	
	YWYML21 Z. Yan, F. Wen, R. Ying, C. Ma, and P. Liu, “On
perceptual lossy compression: The cost of perceptual reconstruction
and an optimal training framework,"  Proc. Int. Conf. Mach. Learn., vol. 139, pp. 11682–11692, 2021.

ZQCK21 G. Zhang, J. Qian, J. Chen, and A. Khisti, "Universal
rate-distortion-perception representations for lossy compression,"
Proc. Adv. Neural Inf. Process. Syst., vol. 34, pp. 11517–11529, 2021.


QZCK22
J. Qian, G. Zhang, J. Chen, and A. Khisti, “A rate-distortion-perception
theory for binary sources," Proc. International Zurich Seminar on Information and Communication, pp. 34–38, 2022.	
	
LZCK22 H. Liu, G. Zhang, J. Chen, A. Khisti, “Lossy compression
with distribution shift as entropy constrained optimal transport,"
	in International Conference on Learning Representations,
	2022.
	
LZCK22_2	
H. Liu, G. Zhang, J. Chen, and A. Khisti, 	
	“Cross-domain lossy compression as entropy constrained optimal transport," IEEE J. Sel. Areas Inf. Theory, vol. 3, pp. 513–527, Sep. 2022,
	
CYWSGT22	
J. Chen, L. Yu, J. Wang, W. Shi, Y. Ge, and W. Tong, “On the rate-distortion-perception function," IEEE J. Sel. Areas Inf. Theory, vol. 3, no. 4, pp. 664–673, Dec. 2022.



SPCYK23
S. Salehkalaibar, T. B. Phan, J. Chen, W. Yu, and A. Khisti, 
“On the choice of perception loss function for learned video compression," Proc. Adv. Neural Inf. Process. Syst., vol. 36, 2023.

QSCKYSGT24
J. Qian, S. Salehkalaibar, J. Chen, A. Khisti, W. Yu, W. Shi, Y. Ge, and W. Tong, “Rate-distortion-perception tradeoff for  Gaussian vector sources," IEEE J. Sel. Areas Inf. Theory, under revision.

SCKY24
S. Salehkalaibar, J. Chen, A. Khisti, and W. Yu, “Rate-distortion-perception tradeoff based on the
conditional-distribution perception measure," 2024, arXiv:2401.12207. [Online] Available: https://arxiv.org/abs/2401.12207


BM18	
Y. Blau and T. Michaeli, “The perception-distortion tradeoff," in Proc.
	IEEE Conf. Comp. Vision and Pattern Recog. (CVPR), 2018, pp. 6288–6237.

FMM21
D. Freirich, T. Michaeli, and R. Meir, “A theory of the distortion-perception tradeoff in Wasserstein space," Proc. Adv. Neural Inf. Process. Syst., vol. 34, pp. 25661–25672, 2021.


FWM24
D. Freirich, N. Weinberger, and R. Meir,
“Characterization of the distortion-perception tradeoff for finite channels with arbitrary metrics,"
2024, arXiv:2402.02265. [Online] Available: 	https://arxiv.org/abs/2402.02265
	
	
	
	
		
	
	TA21 L. Theis and E. Agustsson, “On the advantages of
	stochastic encoders," Proc. ICLR, pp. 1–8, 2021.
	
	Wagner22
	A. B. Wagner, “The rate-distortion-perception tradeoff:
	The role of common randomness," 2022, arXiv:2202.04147. [Online] Available: 	https://arxiv.org/abs/2202.04147
	
	HWG24
	Y. Hamdi, A. B. Wagner, and D. Gündüz, “
	The rate-distortion-perception trade-off:
	The role of private randomness," 2024, arXiv:2404.01111. [Online] Available: https://arxiv.org/pdf/2404.01111
	
	
	XLCZ24
	L. Xie, L. Li, J. Chen, and Z. Zhang, “Output-constrained lossy source coding with application to rate-distortion-perception theory," 2024, 	arXiv:2403.14849. [Online] Available: 	https://arxiv.org/abs/2403.14849






LKK10 M. Li, J. Klejsa, and W. B. Kleijn, “Distribution
preserving quantization with dithering and transformation," IEEE Signal Process. Lett., vol. 17, no. 12, pp. 1014–1017, Dec.
2010.



LKK11 M. Li, J. Klejsa, and W. B. Kleijn. (2011). “On
distribution preserving quantization. [Online]. Available: http://arxiv.org/abs/1108.3728

KZLK13 J. Klejsa, G. Zhang, M. Li, and W. B. Kleijn, “Multiple
description distribution preserving quantization," IEEE Trans.
	Signal Process., vol. 61, no. 24, pp. 6410–6422, Dec. 2013.

SLY15J1 N. Saldi, T. Linder, and S. Yüksel, “Randomized
quantization and source coding with constrained output distribution,"
IEEE Trans. Inf. Theory, vol. 61, no. 1, pp. 91–106,
Jan. 2015.

SLY15J2 N. Saldi, T. Linder, and S. Yüksel, “Output constrained
lossy source coding with limited common randomness," IEEE
	Trans. Inf. Theory, vol. 61, no. 9, pp. 4984–4998, Sep. 2015.

Talagrand96
M. Talagrand, “Transportation cost for Gaussian and other product measures,"
Geometric Funct. Anal., vol. 6, no. 3, pp. 587–600, May 1996.

GN14
Y. Geng and C. Nair, “The capacity region of the two-receiver Gaussian vector broadcast channel with private and common messages," IEEE Trans. Inf. Theory, vol. 60, no. 4, pp. 2087–2104, Apr. 2014.

YWL22
Z. Yan, F. Wen, and P. Liu, “Optimally controllable perceptual lossy compression," Proc. Int. Conf. Mach. Learn., vol. 162, pp. 24911–24928, 2022.

QCYX24
X. Qu, J. Chen, L. Yu, and X. Xu, “Rate-distortion-perception theory for the quadratic Wasserstein space," IEEE Trans. Inf. Theory, to be submitted.

PW24
Y. Polyanskiy and Y. Wu, Information Theory: From Coding to Learning.  Cambridge, U.K.: Cambridge Univ. Press, 2024.

Villani08
C. Villani, Optimal Transport: Old and New. Berlin, Germany: Springer, 2008.


PZ20
V. M. Panaretos and Y. Zemel, An invitation to Statistics in Wasserstein Space.  Berlin, Germany: Springer, 2020.

GL02
A. György and T. Linder, “On the structure of optimal entropy-constrained scalar quantizers," IEEE Trans. Inf. Theory, vol. 48, no. 2, pp. 416–427, Feb. 2002.

BWO23
Y. Bai, X. Wu, and A. Özgür, “Information constrained optimal transport: From Talagrand, to Marton, to Cover," IEEE Trans. Inf. Theory, vol. 69, no. 4, pp. 2059–2073, Apr. 2023. 

CT91 T. M. Cover and J. A. Thomas, Elements of Information
	Theory. New York, NY, USA: Wiley, 1991.














WV12
Y. Wu and S. Verdú,  “Functional properties of minimum mean-square error and mutual information," IEEE Trans. Inf. Theory, vol. 58, no. 3, pp. 1289–1301, Mar. 2012.


WJ18
A. Wibisono and V. Jog, “Convexity of mutual information along the Ornstein-Uhlenbeck flow," 2018 International Symposium on Information Theory and Its Applications (ISITA), Singapore, 2018, pp. 55–59.

MN06
D. Marco and D. L. Neuhoff, “Low-resolution scalar quantization for Gaussian sources and squared error," IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1689–1697, Apr. 2006. 

































































































