On the Fundamental Limits of Integrated Sensing and Communications Under Logarithmic Loss

Jun Chen; Lei Yu; Wen Tong; Wuxian Shi; Yiqun Ge; Yonglong Li

REVIEW 2 major objections 6 minor 2 cited by

On the Fundamental Limits of Integrated Sensing and Communications Under Logarithmic Loss

T0 review · 2 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper derives a closed-form capacity-distortion-power function for Gaussian integrated sensing and communications under squared-error distortion, covering the previously open non-degraded regime.

desk verdict Exact binary and Gaussian ISAC limits with a clever state-splitting trick; Theorem 5's single-letter starting point is asserted, but the paper deserves a serious referee. read the letter →

arxiv 2502.08502 v1 pith:U3WHGFBU submitted 2025-02-12 cs.IT math.IT

Jun Chen , Lei Yu , Yonglong Li , Wuxian Shi , Yiqun Ge , Wen Tong This is my paper · ORCID

classification cs.ITmath.IT MSC 94A1594A1794A24

keywords integratedsensingandcommunicationscapacity-distortionfunctionlogarithmiclossbroadcastchannelsuperpositioncodingextremalinequalitystateestimationGaussian

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper derives the fundamental tradeoff between how fast a transmitter can send digital messages and how accurately a separate sensing receiver can estimate an environment state, when the sensing receiver is required to output a soft (posterior-like) estimate scored by logarithmic loss. The central object is the capacity-distortion function $C(D)$, and the paper gives matching lower and upper bounds whenever the channel to the communication receiver is degraded relative to the sensing receiver, or vice versa. For two concrete channel models the bounds are evaluated exactly: a binary-symmetric pair with additive Bernoulli state, and a Gaussian pair under squared-error distortion. The Gaussian result is a closed-form capacity-distortion-power function that covers the previously open non-degraded regime in which neither receiver's channel is degraded with respect to the other. If correct, these formulas settle the information-theoretic cost of sensing in those settings and show that partial decoding at the sensing receiver is an emergent feature of the optimum.

What carries the argument

The load-bearing machinery is superposition coding with an auxiliary random variable $U$: the transmitter splits the message into a common part decodable by both receivers and a private part decodable only by the communication receiver, which yields the single-letter rates $I(X;Y_1|U)+I(U;Y_2)$ and the distortion $H(S|U,Y_2)$. Matching upper bounds use additional auxiliary variables and standard converse steps, and when one channel is degraded the bounds collapse to simple maximizations over $p_X$ or $p_{UX}$. Exact evaluation rests on two extremal inequalities: a binary-entropy inequality (112) that controls $I(X;Y_1|U)+I(U;Y_2)-\lambda H(S|U,Y_2)$, and a Gaussian entropy-power inequality variant (125) used for the Gaussian converse. The Gaussian squared-error result adds a state-splitting trick that splits $S$ into two independent Gaussians and rewrites the non-degraded parameter range $N_1\in(N_2,N_2+N_S)$ as an equivalent degraded system with effective sensing-noise variance $N_1$, so the previously solved case applies.

What would settle it

Fix binary parameters, say $\beta_1=0.24$, $\beta_2=0.2$, $\beta_S=0.1$, and search over all $p_{UX}$ with $|\mathcal{U}|\leq|\mathcal{X}|+1$ and $\lambda\geq 0$ to evaluate both sides of (112); a single distribution violating the claimed upper bound would refute Theorem 3. For the Gaussian formula, the same test can be run by numerically maximizing the single-letter expression in (86)–(88) for $N_1\in(N_2,N_2+N_S)$, e.g. $N_1=2.5$, $N_2=2$, $N_S=1$, $P=1$, and comparing with (93) at several $D$ values.

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Extended reading notes

Core claim

On its own terms, the paper's central claim is that the capacity-distortion function for ISAC under logarithmic loss is characterized by a superposition-coding expression $C(D)=\max_{p_{UX}}\min\{I(X;Y_1),I(X;Y_1|U)+I(U;Y_2)\}$ subject to $H(S|U,Y_2)\leq D$, and that this expression is tight in the two degraded orders. The decisive new results are exact evaluations. For the Gaussian channel $Y_1=X+Z_1$, $Y_2=X+Z_2+S$ with independent noises $Z_1\sim\mathcal{N}(0,N_1)$, $Z_2\sim\mathcal{N}(0,N_2)$, state $S\sim\mathcal{N}(0,N_S)$, power constraint $P$, and squared-error distortion, Theorem 5 states that $C'_G(D,P)=\frac{1}{2}\log\left(\frac{P+N_2+N_S}{N_1 N_S^2}\left((N_1-N_2)N_S+(N_2+N_S-N_1)D\right)\right)$ for $N_1<N_2+N_S$, and $C'_G(D,P)=\frac{1}{2}\log\left(\frac{P+N_1}{N_1}\right)$ for $N_1\geq N_2+N_S$, on the full distortion interval $D\in\left[\frac{N_2 N_S}{N_2+N_S},\frac{(P+N_2)N_S}{P+N_2+N_S}\right]$. The binary-symmetric counterpart, Theorem 3, gives the same kind of complete description for all degradation orders, with the intermediate regime requiring a linear timesharing segment. The paper also establishes that when the communication channel is degraded relative to the sensing channel, rate and distortion decouple in the signaling strategy, extending the monostatic decoupling principle.

Load-bearing premise

The exact binary characterization stands on the extremal inequality (112), which asserts that a particular linear combination of mutual informations and conditional entropy is maximized by a Bernoulli input with a single bias parameter; if that inequality fails for any channel parameters, the binary capacity-distortion formula collapses.

Editorial extensions

If this is right

In Gaussian ISAC with squared-error sensing, the exact rate-distortion-power curve is now known, so any proposed waveform can be checked against a fundamental limit instead of a bound.
When the communication channel is degraded with respect to the sensing channel, rate and sensing distortion decouple: for a fixed input distribution any rate up to $I(X;Y_1)$ is achievable, and sensing performance depends only on the signaling distribution.
When the sensing channel is degraded with respect to the communication channel, the optimal strategy makes the sensing receiver partially decode the message, and the rate cost of reducing distortion is exactly quantified by the formula.
In the binary BSC-Bernoulli model, the capacity-distortion curve has three explicit regimes: a constant rate for light sensing requirements, a straight timesharing segment in the intermediate regime, and a nonlinear curve in the low-distortion regime.
The Gaussian formula makes the same capacity-distortion-power function valid for both sequence-wise and symbol-wise logarithmic loss in the degraded and low-noise regimes, and for squared error it becomes exact even in the intermediate non-degraded regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The state-splitting trick is likely portable: other Gaussian additive-noise ISAC models whose non-degraded parameter gap can be absorbed into a split state may also admit closed-form solutions, even where the logarithmic-loss version remains only bounded.
The closed form implies that in the non-degraded regime the marginal rate gain from transmit power is $1/(2\ln 2\,(P+N_2+N_S))$, independent of the target distortion $D$; power and sensing accuracy therefore shift the rate-distortion curve additively rather than reshaping it.
A cheap numerical grid search over binary parameters could verify the extremal inequality (112) for finite alphabets, providing strong evidence for the binary theorem before the full analytic proof is trusted.
The paper's open question of whether sequence-wise and symbol-wise logarithmic loss always coincide could be tested in the binary intermediate regime, where the paper currently has only a lower bound for sequence-wise loss.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid information-theory paper with two genuinely new exact results—the binary-symmetric capacity-distortion function under log loss (Theorem 3) and the Gaussian squared-error capacity-distortion-power function (Theorem 5) via a state-splitting trick. The bounds in Theorems 1–2 are extensions of [12] to log loss, but the matching conditions in Corollaries 1–2 are clean, and the binary evaluation is new.

The state-splitting in Theorem 5 is the cleverest part. Splitting S into S' + S'' so that the effective sensing noise has variance N1 turns the previously open non-degraded case N1 in (N2, N2+N_S) into the already-solved subcase. I checked the algebra around equations (98)–(100) and it works. If the single-letter characterization in (83)–(88) is granted, the proof of Theorem 5 is complete.

The soft spots are real but not fatal. First, Theorem 1's achievability is only sketched, as an "adaptation" of [12, Theorem 1]. For an ISAC audience that is probably fine, but a referee will want the differences spelled out. Second, and more important: the single-letter characterization for the squared-error case, equations (83)–(88), is asserted without proof. The paper calls it a 'simple extension' of Corollaries 1 and 2, but those corollaries are proved for logarithmic loss over finite alphabets. For squared error you need a variance lower bound instead of the conditional-entropy lower bound, plus the Gaussian discretization/weak-convergence argument. The footnote cites standard techniques, but the converse for the squared-error constraint is not actually written down. This is the main load-bearing step for Theorem 5, and it is a gap in the written proof—not a demonstrated error. The achievability direction follows from the same superposition coding, and a standard broadcast-channel converse with U as the time-sharing variable should work, so I expect it is true. The extremal inequality (112) in Appendix B is dense, but I followed the second-derivative analysis and it appears correct; it is the engine for the binary result.

Overall this is a genuine contribution with honest treatment of its open case (Gaussian log-loss for N1 in (N2, N2+N_S)). My recommendation: send to peer review. In revision, require the authors to prove the squared-error single-letter characterization (83)–(88) fully, or to state it as a lemma with a complete converse and explicit Gaussian extension. Without that, Theorem 5 lacks a foundation; with it, the paper is a clean accept. I'd bring it to reading group.

Referee Report

2 major / 6 minor

Summary. The paper studies the capacity-distortion tradeoff for an integrated sensing and communication (ISAC) system with a communication receiver and a sensing receiver, where the sensing distortion is measured by logarithmic loss (both sequence-wise and symbol-wise). The main results are: (i) lower and upper bounds on the capacity-distortion function C(D), which coincide when the two receiver channels are degraded in either order; (ii) an explicit characterization of C(D) for a binary symmetric channel pair with additive Bernoulli state (Theorem 3); and (iii) Gaussian counterparts, including a closed-form capacity-distortion-power function under squared error distortion obtained via a state-splitting argument (Theorem 5). The paper also develops original extremal inequalities in the appendices.

Significance. If correct, the results constitute substantial progress on the information-theoretic limits of ISAC. The paper provides the first exact characterization of the log-loss capacity-distortion function in degraded broadcast scenarios, resolves a previously open non-degraded binary case, and offers a closed-form expression for the Gaussian squared-error case that was not previously known. The extremal inequalities in Appendices B and C are technically nontrivial and appear to be proven carefully. The connection between log-loss soft estimation and conventional hard estimation is also a useful conceptual contribution. However, the Gaussian squared-error result depends on a single-letter characterization that is asserted without proof, and this is a load-bearing gap that must be addressed before the claims can be fully accepted.

major comments (2)

[Section IV-B, Eqs. (83)-(88)] The single-letter characterization of C'_G(D,P) under squared error distortion is asserted without proof. The text states that it 'closely resembles' the log-loss counterpart and footnote 2 refers to a 'simple extension' of Corollaries 1 and 2, but those corollaries are proved only for logarithmic loss over finite alphabets. For squared error, the converse requires a lower bound on the conditional variance rather than the conditional entropy, and the Gaussian case requires discretization and weak-convergence steps that are not given. Since Theorem 5's closed-form formula (93) and the state-splitting argument rest entirely on (86)-(88), this is a load-bearing gap. The authors should either provide a full proof of (83)-(88) (including the continuous-alphabet converse) or give a precise citation of a prior result that establishes it.
[Section IV-B, Proof of Theorem 5] The state-splitting equivalence used for the subcase N1 ∈ (N2, N2+NS) is stated in a single sentence: 'In light of (99), the original ISAC system with distortion constraint D is equivalent to the new ISAC system with distortion constraint D''.' Although (99) gives the linear relationship between the average conditional variances of S and S', the proof does not explicitly show that the two capacity-distortion-power functions are the same under the mapping (100), i.e., that achievability in one system implies achievability in the other at the corresponding distortion level. This equivalence is essential for the proof of Theorem 5 and should be elaborated in detail.

minor comments (6)

[Section II, notation paragraph] There are multiple typos in the notation paragraph: 'sufﬁcienty' should be 'sufficiently', 'discret' should be 'discrete', and 'contiunous' should be 'continuous'.
[Section III-A, Theorem 1 proof] The phrase 'adaption of [12, Theorem 1]' should be 'adaptation of [12, Theorem 1]', and the word 'achievability' is misspelled as 'achievability' in several places.
[Section III-B, after Theorem 2] The sentence 'the timesharing variable can be obsorbed into the auxiliary random vaiable U' contains two typos: 'obsorbed' should be 'absorbed' and 'vaiable' should be 'variable'.
[Section IV-A, Fig. 2 caption] The caption misspells 'symbol-wise' as 'sybmol-wise'.
[Section IV-B, Eq. (62) and following text] The phrase 'average power contraint' should be 'average power constraint', and 'incorprating' should be 'incorporating'.
[References] Reference [5] has an erroneous leading '5.' in the author list, and reference [3] is missing a closing period after 'detection'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central ISAC results are proved by original extremal inequalities and reductions; the only self-citation is an independent DSBS lemma, and the squared-error single-letter gap is an omitted proof, not a circular step.

full rationale

The claimed derivation chain is not circular. Theorems 1 and 2 are proved directly via superposition coding and a standard converse (Appendix A), and Corollaries 1 and 2 follow algebraically from degradedness. Theorem 3's binary formula is obtained from the lower bound R_B(D), the extremal inequality (112) proved in Appendix B, and a second-derivative analysis of J(alpha). The only self-citation, [19, Lemma 5], is used in Appendix B to convert the sign pattern of J''(alpha) into concavity or convexity of R_B(D); because [19] is a parameter-free statement about the minimum-relative-entropy region of the DSBS, not a restatement of the ISAC result, it is independent support and does not create circularity. Theorem 4's Gaussian log-loss result is proved in Appendix C through the entropy-power-inequality-based extremal inequality (125), again without assuming the answer. Theorem 5's squared-error formula (93) is derived by sandwiching the explicit lower bound R'_G(D,P) in (97) with the log-loss upper bound C_G(1/2 log(2pi e D),P), and then reducing the remaining subcase N1 in (N2,N2+N_S) to the already solved subcase via the state-splitting transformation (98)-(100); neither step presumes formula (93). The honest weakness is that the squared-error single-letter characterization (83)-(88) is asserted as 'closely resembl[ing]' the log-loss counterpart rather than proved, which is an omitted-proof or correctness gap rather than a circular reduction; that is the appropriate target for skepticism.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard information-theoretic machinery (channel coding theorem, support lemma, EPI) and on the memoryless i.i.d.-state channel model. The paper introduces no fitted constants: all channel parameters (beta1, beta2, betaS, N1, N2, NS, P) are given model inputs. The only non-standard technical engines are extremal inequalities proven in Appendices B and C, which the paper supplies rather than importing. No new physical entities are postulated.

assumptions (7)

standard math Channel coding theorem, converse, and Fano's inequality (standard single-letterization tools).
Used in the converse proof of Theorem 2 (Appendix A) and in the achievability sketch of Theorem 1 to justify rate constraints I(X;Y1), I(X;Y1|U)+I(U;Y2), etc.
standard math Support lemma (Csiszar and Korner) for cardinality bounds on auxiliary random variables U and V.
Invoked in Remarks 2 and 3 to justify the 'max' in (15), (22), (24) and the cardinality bounds |U| <= |X|+1 or +2.
standard math Entropy power inequality (EPI) for Gaussian random variables.
Used in Appendix C, equation (130), to prove the extremal inequality (125) needed for Theorem 4 case 3.
domain assumption Memoryless broadcast channel model with i.i.d. state S independent of input X, and finite alphabets (or Gaussian model in Section IV-B).
The entire capacity-distortion function is defined for this model in Section II; without memorylessness and independence, the single-letter bounds do not follow.
standard math The capacity-distortion function depends only on p_{Y1|X} and p_{Y2S|X}.
Stated in Remark 1 as easy to verify; it justifies coupling Y1 and Y2 arbitrarily in the Gaussian degradedness proof (Appendix C) and the binary proof.
standard math Posterior distribution achieves minimal expected logarithmic loss (equations (4) and (7)).
Used to convert log-loss distortion constraints into conditional entropy constraints H(S|U,Y2) in the single-letter formulas.
standard math Lemma 5 from [19] (Lei Yu, arXiv:2106.03654) on convexity and concavity of envelopes of the minimum-relative-entropy region for the DSBS.
Self-cited in Appendix B to infer that the unique maximizer of J(alpha) implies R_B(D) is concave (or convex) in D, which is essential for Theorem 3.

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Pith. "Pith review of On the Fundamental Limits of Integrated Sensing and Communications Under Logarithmic Loss." pith.science (2026). https://pith.science/paper/U3WHGFBU

@misc{pith2026250208502,
  author       = {Pith},
  title        = {Pith review of: On the Fundamental Limits of Integrated Sensing and Communications Under Logarithmic Loss},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U3WHGFBU}},
  note         = {Machine review of arXiv:2502.08502}
}

read the original abstract

We study a unified information-theoretic framework for integrated sensing and communications (ISAC), applicable to both monostatic and bistatic sensing scenarios. Special attention is given to the case where the sensing receiver (Rx) is required to produce a "soft" estimate of the state sequence, with logarithmic loss serving as the performance metric. We derive lower and upper bounds on the capacity-distortion function, which delineates the fundamental tradeoff between communication rate and sensing distortion. These bounds coincide when the channel between the ISAC transmitter (Tx) and the communication Rx is degraded with respect to the channel between the ISAC Tx and the sensing Rx, or vice versa. Furthermore, we provide a complete characterization of the capacity-distortion function for an ISAC system that simultaneously transmits information over a binary-symmetric channel and senses additive Bernoulli states through another binary-symmetric channel. The Gaussian counterpart of this problem is also explored, which, together with a state-splitting trick, fully determines the capacity-distortion-power function under the squared error distortion measure.

Figures

Figures reproduced from arXiv: 2502.08502 by the authors.

**Figure 2.** Plots of CB(D) under sybmol-wise logarithmic loss with β2 = 0.2 and βS = 0.1 are shown for β1 = 0.3, β1 = 0.24, and β1 = 0.18, corresponding to cases 1), 2) and 3) in Theorem 3, respectively. The plots for β1 = 0.3 and β1 = 0.18 also apply to sequence-wise logarithmic loss, while the plot for β1 = 0.24 serves as a lower bound on CB(D) under sequence-wise logarithmic loss. replacing probability mass functions with pr… view at source ↗

**Figure 3.** Plots of CG(D, P) with P = 1, N2 = 2, and NS = 1 are shown for N1 = 3.5, N1 = 2.5, ad N1 = 1.5, corresponding to cases 1), 2) and 3) in Theorem 4, respectively. They apply to both sequence-wise and symbolwise logarithmic loss. The plots for N1 = 3.5 and N1 = 1.5 depict the exact values of CG(D, P), while the plot for N1 = 2.5 serves as a lower bound on CG(D, P). As discussed in Section II, there exist intimate conn… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Plots of C′ G(D, P) under the squared error distortion measure with P = 1, N2 = 2, and NS = 1 are shown for N1 = 3.5, N1 = 2.5, and N1 = 1.5. Here, N1 = 3.5 corresponds to case 1), while both N1 = 2.5 and N1 = 1.5 correspond to case 2) in Theorem 5. Different from its …

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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