Synergistic Localization and Sensing in MIMO-OFDM Systems via Mixed-Integer Bilevel Learning

Kai Yang; Rui Zhang; Zelin Zhu

REVIEW 5 major objections 7 minor 55 references

Synergistic Localization and Sensing in MIMO-OFDM Systems via Mixed-Integer Bilevel Learning

T0 review · 5 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Pairing localization and sensing in one bilevel training loop with a shared binary subcarrier selector improves both tasks, and the stochastic proximal algorithm provably converges at the standard stochastic-gradient rate.

desk verdict New formulation, broken convergence proof; deserves a referee but only with major revisions. read the letter →

arxiv 2507.07118 v1 pith:QFD42J3O submitted 2025-07-07 cs.NI cs.LG

Zelin Zhu , Kai Yang , Rui Zhang This is my paper · ORCID

classification cs.NIcs.LG MSC 90C2690C1190C15

keywords mixed-integerbileveloptimizationwirelesslocalizationandsensingMIMO-OFDMsubcarrierselectionstochasticproximalgradientintegratedcommunicationWi-FiCSIconvergenceanalysis

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

Localization and sensing in a MIMO-OFDM system receive the same channel-state information, and a subset of subcarriers carries most of the target-dependent signal. This paper claims the two tasks are best learned together, with localization treated as the coarse-grained version of sensing, and formulates joint training as a mixed-integer bilevel optimization: an outer sensing model uses the subcarrier weights that are optimal for an inner localization model, with each weight constrained to be binary. Because exact mixed-integer bilevel problems are combinatorial and non-convex, the paper relaxes the integer weights to the unit interval, approximates the inner optimality condition with cutting planes, and solves the relaxed problem with a stochastic proximal gradient algorithm it calls SPG-MIBO. On three Wi-Fi CSI datasets, the joint SPG-MIBO training improves both localization and sensing over separate training, and the paper proves an $O(1/\sqrt{T})$ convergence rate for the relaxed problem. If the result holds, it gives a principled way to exploit task hierarchy in integrated sensing and communication systems rather than treating each task in isolation.

What carries the argument

The load-bearing object is the SPG-MIBO update scheme: Lagrangian relaxation of the single-level reformulation, a closed-form proximal operator for the distance-to-binary regularization $G(w)=\|w - \hat{x}\|^2$, and a cutting-plane polytope $P_t$ that is updated as the lower-level solution is re-estimated each iteration. The proximal operator maps fractional weights toward $\{0,1\}$ in two linear branches depending on whether each coordinate is below or above $1/2$, which is what keeps relaxed solutions close to the integer constraint. The cutting-plane update is what justifies replacing the exact lower-level optimality condition by a convex feasible set. The convergence theorems follow a standard $L$-smooth stochastic-gradient template applied separately to three variable groups: relaxed subcarrier weights, continuous parameters, and dual variables.

What would settle it

Compute the directional derivative of the Lagrangian in Eq. (9) at a point where the selected subcarrier count equals $N_{\min}$ or $N_{\max}$; the subgradient of the count term is multivalued there, so no finite $L$ can satisfy the $L$-smooth inequality in Assumption 1. A concrete empirical check is to run SPG-MIBO on a problem whose optimum lies exactly on the cardinality boundary and measure whether the squared iterate gap or gradient norm actually decays like $1/\sqrt{T}$.

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Extended reading notes

Core claim

The central discovery is that a shared binary subcarrier-selection vector, optimized so that it solves an inner localization problem while an outer sensing loss is minimized, produces better localization and sensing models than training either alone. The paper models the sensing loss as the upper-level objective and the localization loss as the lower-level objective, enforcing $w_i \in \{0,1\}^{N_{\text{subs}}}$ with a cardinality constraint $N_{\min} \le \|w_i\|_1 \le N_{\max}$. SPG-MIBO relaxes $w$ to $[0,1]^d$, adds a distance penalty $G(w)=\|w - \hat{x}\|^2$ where $\hat{x}$ is the nearest binary vector, replaces lower-level optimality by a polytope built from cutting planes, and updates via stochastic proximal gradient. The paper proves Theorems 1–3 showing an $O(1/\sqrt{T})$ convergence rate for the relaxed variables, the continuous parameters, and the dual variables. Empirically, joint training beats separate training on the Wi-mans dataset, a simulated dataset, and a Wi-Respiration dataset, and the proximal method achieves a smaller integer feasibility gap than a penalty-based approach.

Load-bearing premise

The convergence theorems assume the objective changes smoothly at every point, but the objective contains terms that measure the total count of selected subcarriers via a sum of absolute weights, and such sums are not differentiable exactly at the lower and upper count boundaries; if smoothness fails there, the stated $O(1/\sqrt{T})$ guarantees do not apply to the algorithm as written.

Editorial extensions

If this is right

On all three datasets, joint SPG-MIBO training improves both localization and sensing losses relative to separate training under identical model architectures.
The proximal relaxation reaches a smaller integer feasibility gap than the penalty method, so the relaxed binary selectors are closer to true on/off decisions.
The algorithm is stochastic and mini-batch based, so it can be applied to high-dimensional MIMO-OFDM CSI without storing full-batch gradients.
The proof of an $O(1/\sqrt{T})$ convergence rate places SPG-MIBO in the same complexity class as plain SGD for nonconvex problems, so the integer constraint handling does not asymptotically slow training.

Reading between the lines

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

The paper proves convergence for the relaxed continuous problem; going from a relaxed solution to a provably good integer solution would require a separate bound on the relaxation gap, which the paper does not provide.
If localization as a coarse-grained sensing prior is what drives the gain, the same bilevel template could be tested on other task hierarchies in wireless sensing, such as presence detection as the lower level and activity recognition as the upper level.
The integer feasibility gap measures closeness to the nearest binary vector, but not whether the chosen subcarriers are the physically informative ones; a natural extension is to correlate the learned selector with per-subcarrier dynamic-path power in the channel model.

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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

The paper has a genuinely new idea: casting joint localization and sensing in MIMO-OFDM as a mixed-integer bilevel learning problem with a shared subcarrier selection module. That framing is not in the prior literature, and the algorithmic recipe — stochastic proximal gradient with cutting planes — is a reasonable way to attack it. The experimental trend across three datasets, including a public one, is consistent. So the paper is worth engaging with.

The problem is the theory. The convergence analysis in Section V and Appendix B does not hold up. The clearest issue is in the proof of Theorem 1: in Eqs. (42)–(45) the term E[∇wL(ξ)^T (w_{k+1} − w_k)] is cancelled using unbiasedness, but w_{k+1} depends on the same ξ_k. The expectation of the product does not factor. That breaks the descent estimate from which the O(1/√T) rate is derived. The same type of reasoning appears in the other theorem proofs.

The stress-test note adds a deeper problem that I think is correct: the dual updates for λ and μ in Eq. (11) are unconstrained gradient ascent with no projection onto the nonnegative orthant. The stationarity measure in Theorem 3, (1/T)ΣE||∇v L||^2, measures stationarity of the unconstrained Lagrangian, not KKT of the constrained relaxed problem (8). Negative multipliers are allowed, so the claimed convergence to a solution of the surrogate problem is not established. Even if the algebra in Appendix B.C were correct, the theorem would not say what the authors claim.

One thing the reader flagged is not a real flaw: the l1 terms in the Lagrangian are not non-differentiable after relaxation, because w̄ is confined to [0,1]^d, making those terms linear. So that particular concern should be dropped.

The empirical section is thin — no error bars, no code or data release — and the gains are modest. But the experiments are not the decisive issue. The decisive issue is that the central theoretical contribution, the convergence guarantee, does not support the claims as written.

My take: this is a solid problem with a novel formulation and a plausible heuristic. The theory needs a major fix or a substantially weakened claim before it is publishable as presented. It deserves a serious referee because the question matters and the formulation is new. I would not cite it as-is, but I would want to see a revision.

Referee Report

5 major / 7 minor

Summary. The paper proposes a mixed-integer bilevel optimization (MIBO) formulation for jointly training localization and sensing models in MIMO-OFDM systems with a shared subcarrier-selection vector. The binary selection constraints are relaxed to [0,1], the lower-level optimality constraint is approximated by a cutting-plane polytope, and the resulting single-level constrained problem is solved by a stochastic proximal gradient method with gradient ascent on Lagrangian multipliers (SPG-MIBO). The central theoretical claim is an O(1/√T) convergence rate (Theorems 1–3, Section V). Experiments on three datasets report that joint training outperforms separate training for both tasks.

Significance. If the theoretical claims were correct, the paper would offer a scalable stochastic-gradient method for mixed-integer bilevel learning with convergence guarantees, which is a useful direction for wireless ISAC applications. The problem formulation is sensible, the proximal closed-form expression in Eq. (13) is correct, and the empirical comparison against separate training is a reasonable first evaluation. The main weakness is that the convergence proofs contain several load-bearing errors, so the advertised O(1/√T) guarantee is not established. The experiments are suggestive but lack error bars or significance tests.

major comments (5)

[Appendix B.A, Eqs. (42)–(45)] The proof of Theorem 1 cancels the stochastic cross term E[∇_wL(u_k,v_k,w_k;ξ_k)^T (w_{k+1}−w_k)] against ∇_wL(u_k,v_k,w_k)^T(w_{k+1}−w_k) using the unbiasedness assumption (44). This cancellation is invalid because w_{k+1} is generated by the proximal update (36) and therefore depends on the same mini-batch ξ_k. Unbiasedness of the gradient estimator does not imply that the product expectation factorizes; the two terms do not cancel. Consequently, the descent inequality (45) and the bound in Theorem 1 are not established.
[Section V, Theorem 1] The stationarity measure for w, (1/T)Σ E[||w_{k+1}−w_k||²], is not a meaningful measure of convergence to a stationary point. Even at a point where the true gradient ∇_wL is zero, the stochastic update gives w_{k+1}−w_k ≈ −η_k∇_wL(ξ_k) after the proximal step, so E||w_{k+1}−w_k||² is at least of order η_k²σ_w². With η_k = η/√k, the average is O(log T/T), which tends to zero regardless of whether the iterates approach a stationary point. Thus the O(1/√T) rate is an artifact of the step-size schedule rather than evidence of convergence.
[Section IV-C and Appendix B.C] The dual variables λ and μ are updated in Eq. (11) by unconstrained gradient ascent, with no projection onto ℝ_+. For the inequality-constrained problem (8), dual feasibility requires nonnegative multipliers. The stationarity measure used in Theorem 3, (1/T)ΣE||∇_vL||², can vanish at points with negative multipliers that violate dual feasibility and complementary slackness, so the theorem does not certify convergence to a KKT point of (8). Additionally, the derivation in Eqs. (76)–(78) has a sign error: the coefficient (Lη²logT/2 − 2η√T) is negative for large T, and replacing E||∇_uL||² by its upper bound G_u² reverses the inequality; the claimed upper bound in Eq. (78) does not follow from Eq. (76).
[Section V vs Algorithm 1] The convergence analysis in Theorems 1–3 is carried out for a single fixed Lagrangian L with a fixed polytope P_t. However, Algorithm 1 updates the polytope via Eqs. (5)–(6) whenever lower-level constraints are violated, which changes the Lagrangian and its gradient from one iteration to the next. The analysis contains no term accounting for this time-varying modification, so the theorems do not apply to the algorithm as stated.
[Section IV-A, Eqs. (2)–(6)] The relaxation of the lower-level optimality constraint in Eq. (2) and its cutting-plane approximation in Eqs. (4)–(6) are not shown to preserve the solution set of the original MIBO problem (1). For a nonconvex lower-level problem, requiring the loss to be within ε of L_l(ŵ,θ̂_1) is neither necessary nor sufficient for near-optimality in the decision variables, and nothing guarantees that the polytope P_t contains the true lower-level solution or that solving (8) yields a solution close to (1). The paper claims to solve the mixed-integer bilevel problem, but the link between (8) and (1) remains heuristic.

minor comments (7)

[Throughout] There are numerous typos and grammatical errors, including 'precent', 'Interger', 'Wirelesss', 'leaning', 'the the', and a missing citation marker 'asynchronous manner [?]' in Section IV-A. These should be corrected.
[Eq. (5)] The quantity ν_i^l used in the drop condition is never defined, making the cutting-plane removal criterion incomplete.
[Assumption 3, Eq. (18)] The variance bounds are written as inequalities between vectors and scalars; they should be stated as bounds on the squared norm, e.g., E[||∇_wL(ξ)−∇_wL||²] ≤ σ_w².
[Table III] The performance comparison reports no error bars, confidence intervals, or significance tests. The improvements are modest (e.g., 0.24→0.21 MSE on Wi-respiration localization; 0.701→0.736 accuracy on Wi-mans), so the claim of consistent gains is not yet statistically supported.
[Eqs. (10)–(11)] Eq. (10) introduces two multiplier sets μ_{1,i} and μ_{2,i} for the two sides of the cardinality constraint, while Eq. (11) writes a single multiplier vector μ; the grouping should be made explicit.
[Algorithm 1] The line 'Drop cutting planes' in the pseudocode is incomplete; the drop criterion is only indirectly defined in Eq. (5) and should be stated in the algorithm.
[Assumption 1] The concern that Assumption 1 is invalid because L contains l1-norm terms does not land for the relaxed domain [0,1]^d, where ||ar w_i||_1 is linear in ar w; the nonsmooth term G(ar w) is outside L and is handled by the proximal step. This point is not a reason to reject.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity in the central derivation; one supporting integer-feasibility evaluation is self-definitional because the proximal step minimizes the same distance metric used to measure success.

self definitional [Section VI-B3 (Integer-Constrained Variable Analysis), with Eqs. (7), (12), and (25)]
"G( ¯w) = ∥ ¯w − ˆx∥2 s.t. ˆx = arg min x ∥ ¯w − ¯x∥2, x ∈ {0, 1}d, ¯w ∈ [0, 1]d. (7) ... Integer Feasibility Gap = |xrelaxed − xint| / |xint| (25) ... In Fig. 8 the relaxed variables optimized with SPGD and our proximal operation design are much closer to the original {0, 1}d space, indicating a tighter relaxation for the {0, 1} integer constraint."

The proximal update (12) is defined to minimize G, and G (Eq. 7) is exactly the squared Euclidean distance from ¯w to the nearest vector in {0,1}^d. The Integer Feasibility Gap (Eq. 25) measures essentially the same distance. Hence the observation that SPG-MIBO yields a smaller gap (0.041 vs 0.116) is guaranteed by construction of the proximal operator, not an independent confirmation that the relaxation is tighter. The comparison is still meaningful as a head-to-head between proximal and penalty surrogates, and this tautology does not affect the central convergence theorem or the joint-vs-separate training comparisons.

full rationale

The paper's central derivation chain is not circular. The O(1/√T) convergence claims (Theorems 1-3) are derived, in Appendix B, from the stated L-smoothness, unbiased-gradient, boundedness, and variance assumptions applied to the algorithm's own update equations; they do not presuppose the conclusion. The relaxed problem, cutting-plane polytope, proximal operator, and Lagrangian are independently motivated, and the numerical comparisons are against separately trained baselines and an external benchmark (Wi-Mans), not against the paper's own fitted outputs. Self-citations to the authors' earlier bilevel and cutting-plane work ([24], [49], [50]) are methodological and are not used as an unverified uniqueness oracle; the proof is self-contained. The one genuine circularity is minor and localized: the integer-feasibility result in Section VI-B3 is self-definitional because the proximal operation minimizes the same distance-to-{0,1} function G that the Integer Feasibility Gap metric measures. This does not propagate to the main claims, so the overall circularity score is low.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on smoothness and unbiasedness assumptions that are partly violated by the paper's own objective, plus an unquantified cutting-plane approximation of the lower-level problem. These are the main sources of theoretical uncertainty; no new physical entities are introduced.

free parameters (2)

Nmin, Nmax = one-third and one-half of Nsubs
Hyperparameters bounding the number of selected subcarriers; set based on observations in [55] rather than derived or cross-validated within the paper (Section VI-A).
epsilon = not specified
Relaxation bound for the lower-level optimality constraint (Eq. 2); no value or tuning procedure is given, yet it controls the looseness of the cutting-plane polytope.

assumptions (5)

ad hoc to paper Lagrangian L is L-smooth (Assumption 1, Eq. 14)
Stated to enable the proof, but the actual Lagrangian contains l1-norm terms and is non-differentiable.
standard math Stochastic gradients are unbiased (Assumption 2)
Standard assumption for mini-batch training.
standard math Gradients are bounded and F has a lower bound (Assumption 3)
Standard boundedness and coercivity assumptions.
domain assumption The polytope Pt with cutting planes from linearized J accurately approximates the lower-level feasible region
The relaxation (4)-(6) is assumed to track the lower-level optimum, but no error bound is provided, and in practice only K gradient steps are used to form it.
domain assumption K-step gradient descent yields a good estimate of the lower-level solution
Algorithm 1 step 3 uses K inner steps; the theory treats the lower-level optimum as known.

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Cite this review

Pith. "Pith review of Synergistic Localization and Sensing in MIMO-OFDM Systems via Mixed-Integer Bilevel Learning." pith.science (2026). https://pith.science/paper/QFD42J3O

@misc{pith2026250707118,
  author       = {Pith},
  title        = {Pith review of: Synergistic Localization and Sensing in MIMO-OFDM Systems via Mixed-Integer Bilevel Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QFD42J3O}},
  note         = {Machine review of arXiv:2507.07118}
}

read the original abstract

Wireless localization and sensing technologies are essential in modern wireless networks, supporting applications in smart cities, the Internet of Things (IoT), and autonomous systems. High-performance localization and sensing systems are critical for both network efficiency and emerging intelligent applications. Integrating channel state information (CSI) with deep learning has recently emerged as a promising solution. Recent works have leveraged the spatial diversity of multiple input multiple output (MIMO) systems and the frequency granularity of orthogonal frequency division multiplexing (OFDM) waveforms to improve spatial resolution. Nevertheless, the joint modeling of localization and sensing under the high-dimensional CSI characteristics of MIMO-OFDM systems remains insufficiently investigated. This work aims to jointly model and optimize localization and sensing tasks to harness their potential synergy. We first formulate localization and sensing as a mixed-integer bilevel deep learning problem and then propose a novel stochastic proximal gradient-based mixed-integer bilevel optimization (SPG-MIBO) algorithm. SPG-MIBO is well-suited for high-dimensional and large-scale datasets, leveraging mini-batch training at each step for computational and memory efficiency. The algorithm is also supported by theoretical convergence guarantees. Extensive experiments on multiple datasets validate its effectiveness and highlight the performance gains from joint localization and sensing optimization.

Figures

Figures reproduced from arXiv: 2507.07118 by the authors.

**Figure 2.** Shared Subcarrier Selection for Task Models Learning [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Experimental Setup for Wi-Respiration Dataset Collection [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Respiration Pattern Acquisition Using a PPG Biophysical Signal [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 7.** Figure 7: Loss Curves for SGD-Penalty and SPGD with Mixed-Integer Machine [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 6.** Figure 6: Training Loss Comparison Between Synergistic Fusion Training and [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 8.** Figure 8: Optimized Relaxed Variables from Integer Constraint with SPGD and [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Simulation Scenario Diagram APPENDIX B CONVERGENCE ANALYSIS Assume L(u, v, w) is L-smooth where u ∈ R n1 , v ∈ R n2 ,w ∈ [0, 1]d represents continuous variables, dual variables, and relaxed variables, respectively. The objective function with regularization term G(w)…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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