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Capacity of the Binary Energy Harvesting Channel

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read For the noiseless binary energy-harvesting channel, the paper proves that capacity can be computed to any desired precision by solving two convex optimization problems whose gap vanishes as $1/N$.

desk verdict This paper gives the first computable capacity bounds for the binary energy harvesting channel, and the convergence proof holds up under scrutiny; send it to review. read the letter →

arxiv 2502.07566 v1 pith:AYLV7X4Y submitted 2025-02-11 cs.IT math.IT

Eli Shemuel , Oron Sabag , Haim H. Permuter This is my paper · ORCID
classification cs.ITmath.IT MSC 94A1594A2494A4090C25
keywords binaryenergyharvestingchannelcapacityfinite-stateconvexoptimizationQ-graphsfeedbackdirectedinformation
topics P versus NP
open problems P versus NP
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

The paper claims that the capacity of the binary energy-harvesting channel—a unit-sized battery, binary inputs, random energy arrivals, noiseless output—can be pinned down to any specified precision, something no earlier capacity expression achieved in computable form. It constructs two sequences of single-letter bounds, a lower bound and an upper bound, each obtained by optimizing over an $N$-node auxiliary directed graph called a Q-graph. The two sequences squeeze the true capacity from below and above, and the gap between them is at most $\max_p H(p)/(Np+1)$, which goes to zero as $N$ grows. This gives a practical recipe: pick $N$ to meet a target accuracy, run the convex programs, and read off the capacity. The paper demonstrates the recipe for every $\eta \in \{0.1,\dots,0.9\}$, reporting $10^{-6}$ precision and tightening the best previously published bounds for all of them.

What carries the argument

The load-bearing object is the Q-graph: a directed graph with $N+1$ nodes, node $q$ recording 'q consecutive zeros have been sent since the last 1'. Each node has outgoing edges labeled by the transmitted symbol $X=0$ or $X=1$, and the graph's transitions define the auxiliary variable $Q$ in the single-letter bounds. The lower bound uses the graph where the $X=0$ edge from node $N$ returns to node $0$; the upper bound uses the graph where that edge self-loops at node $N$, which makes the capacity of a modified 'boosted' channel that dominates the original. The auxiliary random variable $U(Q=q)=\{0,\dots,q\}$ encodes the time since the encoder last attempted a $1$, and the strategy $f(U^+,S)=S\cdot\mathbf{1}\{U^+=0\}$ sends a $1$ only when the battery is full and the policy chooses $U^+=0$. The machinery turns the capacity problem into two finite-dimensional convex programs with $O(N^2)$ linear constraints, whose objectives are conditional mutual informations $I(U^+,U;X|Q)$.

What would settle it

Run the convex programs (26) and (28) for $\eta=0.05$ at $N=10{,}000$ and check whether the gap exceeds $\max_p H(p)/(Np+1) \approx 10^{-3}$; a violation, or a direct violation of the monotonicity inequality (62) for any $N$, would refute the claimed $O(1/N)$ convergence.

Watch

Extended reading notes

Core claim

The central discovery is that the notoriously hard noiseless binary energy-harvesting channel admits a computable capacity expression in the form of a limit of convex programs. The encoder's battery state is a finite-state channel state known causally to the encoder; the difficulty is that the state process has unbounded memory because a run of zeros can be arbitrarily long. The paper's key move is to truncate that memory to the last $N$ inputs, encoded as a Q-graph with $N+1$ nodes whose transitions track the number of consecutive zeros since the most recent transmitted one. For the lower bound, the last node sends a zero back to node $0$; for the upper bound, that edge becomes a self-loop, which corresponds to a 'boosted' channel where $N$ consecutive zeros guarantee a charged battery and therefore a channel whose capacity is at least the true one. Both bounds are optimized over a finite auxiliary variable $U$ that records how many steps ago the encoder last attempted to send a $1$, and both optimizations are convex. The upper and lower sequences converge to the same limit because the only discrepancy between them is concentrated at the last node, and the stationary probability of that node is at most $1/(Np+1)$ under any optimal policy. Therefore the capacity equals the common limit, and the gap is bounded by $\max_p H(p)/(Np+1) = O(1/N)$.

Load-bearing premise

The proof assumes that in the optimal upper-bound coding distribution, the chance of being in any intermediate node of the tracking graph is never higher than in the matching lower-bound distribution; this monotonicity, justified by a short self-loop argument at the last node, is what makes the gap shrink as $1/N$.

Editorial extensions

If this is right

  • For any target accuracy $\epsilon$ and any harvesting probability $\eta$, choosing $N$ large enough (e.g. $N=10{,}000$ guarantees about $10^{-3}$ for all $\eta$) yields computable lower and upper bounds enclosing $C_{\mathrm{BEHC}}$ within $\epsilon$, without needing a closed-form capacity expression.
  • The computed values for $\eta \in \{0.1,\dots,0.9\}$ at $10^{-6}$ precision improve on the best previously reported bounds, with the largest gains at small $\eta$.
  • The same Q-graph plus MDP machinery gives numerical achievable rates for noisy BEHCs with feedback; for $\eta=1$ the achievable rate for a BSC($p$) equals the ordinary BSC capacity $1-H(p)$.
  • The convex formulations are explicit optimization problems (Eqs. 26 and 28) with $O(N^2)$ constraints, so the method is directly reproducible from the paper and the released code.
  • Because the gap bound is independent of $\eta$, the method provides a uniform precision guarantee across all harvesting probabilities.

Reading between the lines

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

  • The same truncate-and-squeeze idea could extend to larger finite batteries and non-binary harvest amounts: the state is still a finite automaton whose memory is a run of zeros, so an $N$-node Q-graph with appropriately adjusted edges may give analogous convex bounds, though the cardinality bound on $U$ would need reworking.
  • A testable practical consequence is that the achieved precision for small $\eta$ is limited mainly by the $O(N^2)$ constraint-construction cost, so a sparse or iterative solver could push the method to $N \gg 10^4$ and sub-$10^{-6}$ precision cheaply.
  • The upper bound's 'boosted channel' construction—forcing a charge after $N$ consecutive zeros—is a general-purpose upper-bounding trick for any channel with a battery that is guaranteed to fill after enough idle slots; identifying such monotonicity in other state-dependent channels could yield computable upper bounds where none exist.
  • The equivalence between the auxiliary variable and 'time since last attempted 1' suggests a natural coding interpretation: near-optimal codes on the BEHC are renewal-like, attempting a 1 after a geometrically distributed number of zeros.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the noiseless binary energy harvesting channel (BEHC) with a unit-sized battery and i.i.d. Bernoulli(η) energy arrivals. Its main contribution is a computable characterization of the capacity: a lower bound obtained by specializing the Q-graph lower bound of [26] to an N+1-node Q-graph, and an upper bound obtained from a modified 'boosted' channel that charges the battery after N consecutive zeros. Both bounds are formulated as convex optimization problems (Eqs. (26) and (28)), and the paper proves that the two optimal values converge to the same limit, with a gap bounded by max_p H(p)/(Np+1). Numerical capacities for η ∈ {0.1,...,0.9} are reported to six decimal places in Table I, improving the bounds of [14]. The paper also uses an MDP/value-iteration approach to compute achievable rates for the noisy BEHC with feedback over a binary symmetric channel.

Significance. This is a substantial contribution: it converts the previously uncomputable multi-letter capacity expression for the BEHC into a genuinely computable convex optimization problem, and it supplies the first Q-graph upper bound for this channel, via the boosted-channel construction. The paper is transparent about building on the general Q-graph lower bound of [26], and it provides public MATLAB code for the numerical results. I specifically checked the step flagged in the stress-test note, inequality (62) in Theorem 6: writing the stationary distributions of the two Q-chains in closed form confirms that ̄P*_Q(q) ≤ ̃P_Q(q) for q<N, so the convergence proof's load-bearing inequality is sound. The main weaknesses are an incorrect asymptotic-order statement in the abstract/Remark 2 and a few local proof gaps that are readily fixable; they do not undermine the central claim.

major comments (2)
  1. [Abstract; Section III (p. 4); Remark 2, Eq. (64)] The paper states that the gap between the bounds vanishes as O(N), and Remark 2 writes ψ(N)=O(N). This is inconsistent with Eq. (61), which gives b_N - a_N ≤ max_p H(p)/(Np+1) =: ψ(N). Since ψ(N) tends to 0 — in fact the current bound gives ψ(N)=O(log N / N), not O(N) — the advertised central claim of a vanishing gap is misstated in all three locations. Please correct the asymptotic notation uniformly.
  2. [Lemma 4, Section V-A] The induction proving π(S=0|u,q) = ̅η^{u+1} does not explicitly cover the case q=N with u>0. This case is needed because the convex formulation (26) and the policy constraints (24) require π(·|u,N) for every u∈[0:N]. The missing case follows by the same predecessor argument, since (u,N) can only be reached from (u-1,N-1) with X=0, but as written the proof of Theorem 2 is incomplete at this point. Please add the missing induction step.
minor comments (5)
  1. [Theorem 6, first paragraph] "For the sequence of solutions to the convex optimization problems (28) for the lower bound" should refer to problem (26), not (28), for the lower bound.
  2. [Theorem 5, Step 3, Section V-B] The passage "we set |̃U(Q=N)|=0" followed immediately by "we set ̃U(Q=N)=[0:N]" is internally inconsistent. The final auxiliary sets (38) are clear, but the proof should explain that the variable ̃U_{i-1} is retained at Q=N only as a dummy for stationarity.
  3. [Appendix A, proof of Lemma 7] The Markov-chain step in Eq. (66)(b) is delegated to "an identical derivation of [30, Lemma 21]" without reproducing the argument. Since this chain is used in the upper-bound proof, please expand the derivation or state the referenced lemma explicitly.
  4. [Section V-B, before Lemma 5] The claim C_BEHC < ̅C_BEHC(N) should be relaxed to C_BEHC ≤ ̅C_BEHC(N), since equality can occur in degenerate cases such as η=1; the upper-bound argument only needs the non-strict inequality.
  5. [Eq. (61), Step (e)] The notation ̄I_{̄P*} is used before being defined; please define it explicitly as the conditional mutual information evaluated under the distribution ̄P*.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lower bound rests on a prior general Q-graph theorem, the upper bound is newly derived from a converse inequality, and the convergence proof compares explicit stationary distributions rather than reusing the target result.

full rationale

The derivation is self-contained apart from genuinely independent prior results. The lower bound invokes Theorem 1, quoted as "[26, Th. 5]" and described as a single-letter Q-graph lower bound on feedback capacity for FSCs with feedback and causal state information. Although [26] shares authors with the present paper, it is a previously published general theorem with its own proof; its assumptions are about arbitrary strongly connected FSCs and do not include the BEHC capacity as an input, so it is independent evidence rather than a definitional loop. The upper bound is new: Theorem 3 and Theorem 4 derive it from the general converse inequality (40), taken from [26, Eq. (42)], and from a constructed "boosted" BEHC(N) whose capacity dominates CBEHC. The O(1/N) convergence claim in Theorem 6 is established by explicitly constructing a lower-bound distribution from the optimal upper-bound distribution and then bounding the gap in (61). The only terse step is inequality (62), which asserts a stationary-distribution ordering between the upper-bound and lower-bound Q-chains. That inequality is not circular: the two chains share all transition probabilities on nodes q<N and differ only at node N, where the upper-bound chain has a self-loop and the lower-bound chain does not; the stationary distributions can be compared directly from the stated transition kernels, and the claimed ordering follows. No parameter is fitted to a subset of capacity data and then renamed a prediction: the numerical values in Table I are outputs of the convex programs (26) and (28), and they are benchmarked against the independent bounds of [14]. The self-citations to [26] and [29] are load-bearing but are established general results that do not reduce to the present claim. Accordingly, no circular step is present, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities or fitted parameters. The only auxiliary constructions are the Q-graphs and the boosted BEHC(N), which are mathematical devices for bounding, not new physical postulates. The central claim rests on prior information-theoretic results and the assumed channel model.

assumptions (4)
  • standard math General Q-graph lower bound on feedback capacity of FSCs with SI at the encoder (Theorem 1, from [26, Th. 5])
    The lower bound for the BEHC in Theorem 2 directly invokes this prior theorem; it is an accepted published result, but it is assumed without reproof here.
  • domain assumption The BEHC is a strongly connected FSC with channel law PS+,Y|X,S as in (3)-(4)
    The finite-state model and Bernoulli energy arrival process define the channel; this is the modeling assumption that the capacity analysis is built on.
  • standard math Feedback does not increase the capacity of a noiseless channel
    Used to justify that the noiseless BEHC capacity equals the capacity of the corresponding FSC with feedback, enabling the use of [26] results.
  • domain assumption CVX or an equivalent convex solver returns the global optimum of the formulated convex problems
    The precision claims (e.g., 1e-6) depend on solving the convex optimization problems (26) and (28) to global optimality; numerical tolerances of the solver are not analyzed in the paper.

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Pith. "Pith review of Capacity of the Binary Energy Harvesting Channel." pith.science (2026). https://pith.science/paper/AYLV7X4Y

@misc{pith2026250207566,
  author       = {Pith},
  title        = {Pith review of: Capacity of the Binary Energy Harvesting Channel},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYLV7X4Y}},
  note         = {Machine review of arXiv:2502.07566}
}
abstract

The capacity of a channel with an energy-harvesting (EH) encoder and a finite battery remains an open problem, even in the noiseless case. A key instance of this scenario is the binary EH channel (BEHC), where the encoder has a unit-sized battery and binary inputs. Existing capacity expressions for the BEHC are not computable, motivating this work, which determines the capacity to any desired precision via convex optimization. By modeling the system as a finite-state channel with state information known causally at the encoder, we derive single-letter lower and upper bounds using auxiliary directed graphs, termed $Q$-graphs. These $Q$-graphs exhibit a special structure with a finite number of nodes, $N$, enabling the formulation of the bounds as convex optimization problems. As $N$ increases, the bounds tighten and converge to the capacity with a vanishing gap of $O(N)$. For any EH probability parameter $\eta\in \{0.1,0.2, \dots, 0.9\}$, we compute the capacity with a precision of ${1e-6}$, outperforming the best-known bounds in the literature. Finally, we extend this framework to noisy EH channels with feedback, and present numerical achievable rates for the binary symmetric channel using a Markov decision process.

Figures

Figures reproduced from arXiv: 2502.07566 by the authors.

Figure 1
Figure 1. The energy-harvesting model with a finite battery. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FSC with feedback and SI available causally to the [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. An instance of a 2-node Q-graph with ternary output. T (s) and input distribution of the form {P(xi |si−1)} T(s) i=1 that may depend on s, such that PT(s) i=1 P(Si = s|S0 = s ′ ) > 0. B. Preliminary on Quantized-graphs The Q-graph is a method for deriving single-letter lower and upper bounds on the capacity of a setting with feedback from a multi-letter expression. This technique is established on Quantized-graphs (… view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: A Q-graph for upper bounding CBEHC in Theorem 3. In particular, this policy structure, combined with (12), always attempts to transmit X = 1 at node Q = N, with success determined by whether the battery is charged. While the BCJR constraint of the policy in its general…
Figure 5
Figure 5. Figure 5: Notably, unlike the general Q-graph lower bound in Theorem 1, no general Q-graph upper bound on Cfb-csi has been established in [26], primarily due to the lack of a cardinality bound on U. Theorem 4 (Step 2). The capacity of the boosted BEHC(N) is upper bounded by C¯ B…

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