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How Many RF Chains Does a Microwave Linear Analog Computer (MiLAC) Need to Match the Fully-Digital Cram\'er-Rao Bound?

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read A lossless reciprocal MiLAC matches the fully digital Cramér-Rao bound for direction-of-arrival estimation once its row space contains the 2K-dimensional steering-derivative subspace for K targets.

desk verdict MiLAC reaches the fully digital CRB for DOA estimation with two RF chains per target when its row space covers the steering-derivative subspace. read the letter →

arxiv 2606.23986 v1 pith:KOO5NSCR submitted 2026-06-22 cs.IT eess.SPmath.IT

Yuchen Zhang , Yu Ge , Bruno Clerckx , Tareq Y. Al-Naffouri This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords microwavelinearanalogcomputerMiLACdirection-of-arrivalestimationFisherinformationmatrixCramér-RaoboundRFchainscombinerlosslessreciprocalnetwork
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 examines direction-of-arrival estimation for K far-field targets when a tunable lossless reciprocal microwave network serves as the antenna-array front end. It demonstrates that the Fisher information matrix is governed solely by the orthogonal projector onto the row space of the combiner and cannot exceed the information available to a fully digital receiver. Equality with the digital bound is achieved exactly when that row space includes the joint steering and derivative subspace, which has dimension 2K. This yields a threshold of two RF chains per target. A dimension-counting argument also lower-bounds the number of tunable elements needed, and a stem-connected topology meets the bound with linear scaling in antennas and targets.

What carries the argument

The orthogonal projector onto the row space of the MiLAC combiner, which alone determines the Fisher information matrix.

What would settle it

Compute the empirical Fisher information for a concrete MiLAC whose row space misses one derivative vector of a target; the resulting matrix must be strictly smaller than the fully digital matrix if the claim holds.

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Extended reading notes

Core claim

For a tunable receive-side lossless reciprocal MiLAC combiner, the Fisher information matrix depends on the combiner only through the orthogonal projector onto its row space. This matrix never exceeds that of a fully digital receiver. Equality holds if and only if the row space contains the 2K-dimensional joint steering-derivative subspace. The stem-connected MiLAC attains the digital Cramér-Rao bound asymptotically, up to an antenna-count-independent additive overhead, while scaling linearly with the antenna and target counts. Unlike a phase-shifter front end with the same number of RF chains, the MiLAC can exactly attain the fully digital bound.

Load-bearing premise

The MiLAC is perfectly lossless and reciprocal, so its effect reduces exactly to an orthogonal projector onto its row space under the standard far-field point-target array model.

Editorial extensions

If this is right

  • Two RF chains per target are necessary and sufficient for zero performance gap with the fully digital receiver.
  • A lower bound on the number of tunable components follows from a dimension-counting argument on the required row space.
  • The stem-connected MiLAC topology achieves the bound with linear scaling in the number of antennas and targets, plus a fixed overhead.
  • Phase-shifter networks with the same number of RF chains cannot attain the digital bound, while MiLAC can.
  • The result applies to every target configuration once the row-space condition is met.

Reading between the lines

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

  • The same projector argument could be applied to other array processing tasks such as beamforming or parameter estimation beyond angles.
  • Hardware verification would need to confirm that real microwave networks satisfy the lossless reciprocal model closely enough for the bound to be reached.
  • The linear scaling suggests MiLAC architectures could remain practical even as antenna counts grow into the hundreds.
  • The 2K-dimensional subspace requirement may generalize to other manifold-based estimation problems where both position and slope information matter.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The manuscript analyzes direction-of-arrival estimation for K far-field targets using a tunable lossless reciprocal MiLAC combiner as an antenna-array front end. It shows that the Fisher information matrix depends on the combiner only through the orthogonal projector onto its row space and is upper-bounded by the fully digital receiver's FIM, with equality when the row space contains the 2K-dimensional joint steering-derivative subspace. This yields a zero-gap threshold of two RF chains per target, a dimension-counting lower bound on tunable components, and an asymptotic attainment result for the stem-connected MiLAC architecture (linear scaling in antennas and targets) that exactly matches the digital CRB, unlike phase-shifter networks.

Significance. If the central claims hold, the paper supplies a precise geometric characterization of when analog combiners can achieve fully digital sensing performance in multi-target DOA estimation, together with an explicit architecture that meets the derived bound. Strengths include the direct application of standard linear-algebra facts on the FIM under the semi-unitary projector induced by losslessness, the dimension-counting argument, and the explicit contrast with phase-shifter front ends. The result is falsifiable via the stated subspace condition and supplies a concrete scaling law for hardware design.

minor comments (2)
  1. [dimension-counting argument] The dimension-counting lower bound would benefit from an explicit statement of the manifold dimension and the precise count of free parameters in the stem-connected topology (e.g., in the section deriving the asymptotic attainment).
  2. [model and FIM derivation] Notation for the joint steering-derivative subspace should be introduced once with a clear definition before its repeated use in the equality condition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review, accurate summary of our contributions, and recommendation to accept. No major comments were raised that require point-by-point response.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained linear algebra

full rationale

The central result follows directly from the standard expression for the Fisher information matrix of a linear Gaussian observation model y = W H( heta) s + n, where the lossless reciprocal combiner W satisfies W^H W = I and thus acts exactly as the orthogonal projector P onto its row space. Substituting yields that the FIM is a non-increasing function of P and saturates at the fully-digital value precisely when the row space of W contains the 2K-dimensional joint steering-derivative subspace; both the saturation condition and the dimension-counting lower bound on tunable elements are immediate consequences of linear algebra and the rank of that manifold. No parameter is fitted to data and then re-used as a prediction, no uniqueness theorem is imported from the authors' prior work, and the MiLAC projector property is taken as an external modeling assumption rather than derived from the present FIM claim. The argument is therefore independent of its own outputs.

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

The paper relies on established mathematical tools from array signal processing and linear algebra without introducing new free parameters or invented entities in the abstract description.

assumptions (2)
  • standard math Standard array signal model for far-field targets with the Fisher information matrix formula involving steering vectors and their derivatives.
    This is the basis for analyzing how the combiner affects the FIM.
  • domain assumption The MiLAC is lossless and reciprocal so that its effect is captured by the orthogonal projector onto the row space.
    Explicitly stated as the type of MiLAC analyzed in the abstract.

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Cite this review

Pith. "Pith review of How Many RF Chains Does a Microwave Linear Analog Computer (MiLAC) Need to Match the Fully-Digital Cram\'er-Rao Bound?." pith.science (2026). https://pith.science/paper/KOO5NSCR

@misc{pith2026260623986,
  author       = {Pith},
  title        = {Pith review of: How Many RF Chains Does a Microwave Linear Analog Computer (MiLAC) Need to Match the Fully-Digital Cram\'er-Rao Bound?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KOO5NSCR}},
  note         = {Machine review of arXiv:2606.23986}
}
abstract

A microwave linear analog computer (MiLAC) is a tunable microwave network that performs linear operations directly on radio-frequency signals through wave propagation. Used as an antenna-array front end, it can map many antenna signals to a small number of active RF chains. While lossless reciprocal MiLACs have been shown to provide flexible or capacity-achieving beamforming for wireless communications, their sensing performance remains largely unexplored. We analyze direction-of-arrival estimation for $K$ far-field targets using a tunable receive-side lossless reciprocal MiLAC combiner. We show that the Fisher information matrix depends on the combiner only through the orthogonal projector onto its row space and never exceeds that of a fully digital receiver. Equality holds when the row space contains the $2K$-dimensional joint steering--derivative subspace, establishing a zero-gap threshold of two RF chains per target. A dimension-counting argument lower-bounds the number of tunable components required to achieve the digital Cram\'er--Rao bound for every target configuration. The stem-connected MiLAC attains this bound asymptotically, up to an antenna-count-independent additive overhead, while scaling linearly with the antenna and target counts. Unlike a phase-shifter front end with the same number of RF chains, MiLAC can exactly attain the fully digital bound. Numerical results validate the analysis.

Figures

Figures reproduced from arXiv: 2606.23986 by the authors.

Figure 1
Figure 1. Receiver architecture studied in this paper. The fully-digital baseline ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Reciprocal lossless MiLAC topologies for [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Average per-target CRB (lines) and MLE Monte Carlo MSE (markers) versus SNR when [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Per-target CRB versus NR when K = 2. 1 2 3 4 5 6 7 8 9 10 100 101 102 103 non-identifiable ident. + gap zero gap Number of RF chains LR CRB ⋆ MiLAC/CRBdig CRB = ∞ (non-identifiable) MiLAC optimal CRB ratio [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Per-target CRB ratio versus LR at K = 3, where red crosses mark the non-identifiable regime LR ≤ 4. penalty of Remark 1. At LR = 6 = 2K the gap closes exactly to unity and remains there for LR ≥ 6, matching Corollary 1. This three-regime structure is not special to K =…
Figure 6
Figure 6. Figure 6: Per-target CRB ratio versus the chain-budget offset [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: Average per-target CRB versus angular separation [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: CRB ratio r(∆ˆθ) versus steering mismatch |∆ˆθ| at θ = 20◦ . The thick gray dashed lines mark the zero-gap reference r = 1 and the 3-dB penalty r = 2. VI. CONCLUSION This paper recasts MiLAC-aided sensing as a Grassmannian geometry problem on Gr(LR, NR): the row space …

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Works this paper leans on

39 extracted references · 14 canonical work pages

  1. [1]

    Integrated sensing and communications: Toward dual-functional wireless networks for 6G and beyond,

    F. Liu, Y . Cui, C. Masouros, J. Xu, T. X. Han, Y . C. Eldar, and S. Buzzi, “Integrated sensing and communications: Toward dual-functional wireless networks for 6G and beyond,”IEEE Journal on Selected Areas in Communications, vol. 40, no. 6, pp. 1728–1767, 2022

    2022

  2. [2]

    On the fundamental tradeoff of integrated sensing and communications under Gaussian channels,

    Y . Xiong, F. Liu, Y . Cui, W. Yuan, T. X. Han, and G. Caire, “On the fundamental tradeoff of integrated sensing and communications under Gaussian channels,”IEEE Transactions on Information Theory, vol. 69, no. 9, pp. 5723–5751, 2023

    2023

  3. [3]

    Cram ´er–Rao bound optimization for joint radar-communication beamforming,

    F. Liu, Y .-F. Liu, A. Li, C. Masouros, and Y . C. Eldar, “Cram ´er–Rao bound optimization for joint radar-communication beamforming,”IEEE Transactions on Signal Processing, vol. 70, pp. 240–253, 2022

    2022

  4. [4]

    Beyond-diagonal RIS in multiuser MIMO: Graph theoretic modeling and optimal architectures with low complexity,

    Z. Wu and B. Clerckx, “Beyond-diagonal RIS in multiuser MIMO: Graph theoretic modeling and optimal architectures with low complexity,”IEEE Transactions on Information Theory, vol. 71, no. 11, pp. 8506–8523, 2025

    2025

  5. [5]

    Integrated sensing and communication for 6G: Ten key machine learning roles,

    U. Demirhan and A. Alkhateeb, “Integrated sensing and communication for 6G: Ten key machine learning roles,”IEEE Communications Magazine, vol. 61, no. 5, pp. 113–119, 2023

    2023

  6. [6]

    Enabling 6G performance in the upper mid-band by transitioning from massive to gigantic MIMO,

    E. Bj ¨ornson, F. Kara, N. Kolomvakis, A. Kosasih, P. Ramezani, and M. B. Salman, “Enabling 6G performance in the upper mid-band by transitioning from massive to gigantic MIMO,”IEEE Open Journal of the Communications Society, vol. 6, pp. 5450–5463, 2025

    2025

  7. [7]

    Massive MIMO for next generation wireless systems,

    E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,”IEEE Communications Magazine, vol. 52, no. 2, pp. 186–195, 2014

    2014

  8. [8]

    An overview of signal processing techniques for millimeter wave MIMO systems,

    R. W. Heath, N. Gonz ´alez-Prelcic, S. Rangan, W. Roh, and A. M. Sayeed, “An overview of signal processing techniques for millimeter wave MIMO systems,”IEEE Journal of Selected Topics in Signal Processing, vol. 10, no. 3, pp. 436–453, 2016

    2016

Show all 39 references
  1. [9]

    Spatially sparse precoding in millimeter wave MIMO systems,

    O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath, “Spatially sparse precoding in millimeter wave MIMO systems,”IEEE Transactions on Wireless Communications, vol. 13, no. 3, pp. 1499–1513, 2014

    2014

  2. [10]

    Dynamic metasurface antennas for 6g extreme massive MIMO communications,

    N. Shlezinger, G. C. Alexandropoulos, M. F. Imani, Y . C. Eldar, and D. R. Smith, “Dynamic metasurface antennas for 6g extreme massive MIMO communications,”IEEE Wireless Communications, vol. 28, no. 2, pp. 106–113, 2021

    2021

  3. [11]

    Sum-rate maximization for DMA-based wideband near-field systems with Lorentzian responses,

    R. Zhang, H. Zhang, Y . Zhang, H. Ruan, H. Chen, and Y . C. Eldar, “Sum-rate maximization for DMA-based wideband near-field systems with Lorentzian responses,” inICASSP 2026–2026 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2026, pp. 21 461–21 465

    2026

  4. [12]

    Holographic intelligence surface assisted integrated sensing and communication,

    Z. Liu, Y . Zhang, H. Zhang, F. Xu, and Y . C. Eldar, “Holographic intelligence surface assisted integrated sensing and communication,”IEEE Transactions on Signal Processing, vol. 74, pp. 1942–1957, 2026

    1942

  5. [13]

    Hybrid codebook design for localization using electromagnetically reconfigurable fluid antenna system,

    A. Fadakar, Y . Zhang, H. Chen, M. F. Keskin, H. Wymeersch, and A. F. Molisch, “Hybrid codebook design for localization using electromagnetically reconfigurable fluid antenna system,”IEEE Journal of Selected Topics in Signal Processing, vol. 20, no. 3, pp. 371–388, 2026

    2026

  6. [14]

    A survey on reconfigurable and movable antennas for wireless communications and sensing,

    W. Ma, L. Zhu, Y . Tan, B. Zheng, Y . Zhang, Y . Zhang, K. Ying, Z. Gao, H. Sun, X. Shao, Z. Xiao, D. Niyato, and R. Zhang, “A survey on reconfigurable and movable antennas for wireless communications and sensing,”IEEE Communications Surveys & Tutorials, vol. 28, pp. 4842–4882, 2026

    2026

  7. [15]

    Embracing reconfigurable antennas in the tri-hybrid MIMO architecture for 6g and beyond,

    M. R. Castellanos, S. Yang, C.-B. Chae, and R. W. Heath, “Embracing reconfigurable antennas in the tri-hybrid MIMO architecture for 6g and beyond,”IEEE Transactions on Communications, vol. 74, pp. 381–401, 2026

    2026

  8. [16]

    Tri-hybrid multi-user precoding using pattern-reconfigurable antennas: Fundamental models and practical algorithms,

    P. Zheng, Y . Zhang, T. Y . Al-Naffouri, M. J. Hossain, and A. Chaaban, “Tri-hybrid multi-user precoding using pattern-reconfigurable antennas: Fundamental models and practical algorithms,” arXiv:2505.08938, 2025, preprint. [Online]. Available: https://arxiv.org/abs/2505.08938 35

    2025

  9. [17]

    Tri-hybrid beamforming design for ISAC systems with reconfigurable antennas,

    J. Chen, X. Lei, Y . Zhang, K. Meng, and C. Masouros, “Tri-hybrid beamforming design for ISAC systems with reconfigurable antennas,” arXiv:2604.20676, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2604.20676

    2026 arXiv

  10. [18]

    Analog computing for signal processing and communications – part I: Computing with microwave networks,

    M. Nerini and B. Clerckx, “Analog computing for signal processing and communications – part I: Computing with microwave networks,”IEEE Transactions on Signal Processing, vol. 73, pp. 5183–5197, 2025

    2025

  11. [19]

    Analog computing for signal processing and communications – part II: Toward gigantic MIMO beamforming,

    ——, “Analog computing for signal processing and communications – part II: Toward gigantic MIMO beamforming,” IEEE Transactions on Signal Processing, vol. 73, pp. 5198–5212, 2025

    2025

  12. [20]

    Quantization-aware EE optimization and SE-EE tradeoff for MiLAC-aided MU-MISO beamforming,

    Y . Zhang, P. Zheng, and T. Y . Al-Naffouri, “Quantization-aware EE optimization and SE-EE tradeoff for MiLAC-aided MU-MISO beamforming,” arXiv:2604.24538, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2604.24538

    2026 arXiv

  13. [21]

    Microwave linear analog computer (MiLAC)-aided multiuser MISO: Fundamental limits and beamforming design,

    Z. Wu, M. Nerini, and B. Clerckx, “Microwave linear analog computer (MiLAC)-aided multiuser MISO: Fundamental limits and beamforming design,” arXiv:2601.10060, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2601.10060

    2026

  14. [22]

    MIMO systems aided by microwave linear analog computers: Capacity-achieving architectures with reduced circuit complexity,

    M. Nerini and B. Clerckx, “MIMO systems aided by microwave linear analog computers: Capacity-achieving architectures with reduced circuit complexity,”IEEE Transactions on Wireless Communications, vol. 25, pp. 14 597–14 610, 2026

    2026

  15. [23]

    Capacity of MIMO systems aided by microwave linear analog computers (MiLACs),

    ——, “Capacity of MIMO systems aided by microwave linear analog computers (MiLACs),” arXiv:2506.05983, 2025, preprint. [Online]. Available: https://arxiv.org/abs/2506.05983

    2025

  16. [24]

    Beamforming design for stem-connected microwave linear analog computer (MiLAC)-aided multiuser MISO downlinks,

    Y . Zhang, Z. Wu, B. Clerckx, and T. Y . Al-Naffouri, “Beamforming design for stem-connected microwave linear analog computer (MiLAC)-aided multiuser MISO downlinks,” arXiv:2606.14499, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2606.14499

    2026

  17. [25]

    On the performance of lossless reciprocal MiLAC architectures in multi-user networks,

    T. Fang, X. Zhou, and Y . Mao, “On the performance of lossless reciprocal MiLAC architectures in multi-user networks,” IEEE Wireless Communications Letters, vol. 15, pp. 2609–2613, 2026

    2026

  18. [26]

    Channel estimation in MIMO systems aided by microwave linear analog computers (MiLACs),

    Q. Zhang, M. Nerini, and B. Clerckx, “Channel estimation in MIMO systems aided by microwave linear analog computers (MiLACs),” arXiv:2601.11438, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2601.11438

    2026

  19. [27]

    Physics-compliant modeling and optimization of MIMO systems aided by microwave linear analog computers,

    M. Nerini and B. Clerckx, “Physics-compliant modeling and optimization of MIMO systems aided by microwave linear analog computers,” arXiv:2602.19379, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2602.19379

    2026

  20. [28]

    Hybrid digital and microwave linear analog computer (MiLAC)-aided beamforming for multiuser MIMO-OFDM systems,

    Y . Peng, Z. Wu, and B. Clerckx, “Hybrid digital and microwave linear analog computer (MiLAC)-aided beamforming for multiuser MIMO-OFDM systems,” arXiv:2604.26532, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2604.26532

    2026 arXiv

  21. [29]

    Two-layer microwave linear analog computer (MiLAC)-aided multi-user MISO networks,

    X. Zhou, T. Fang, Y . Mao, and B. Clerckx, “Two-layer microwave linear analog computer (MiLAC)-aided multi-user MISO networks,” arXiv:2604.24303, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2604.24303

    2026 arXiv

  22. [30]

    Microwave linear analog computer (MiLAC) for simultaneous active and passive beamforming,

    M. Nerini and B. Clerckx, “Microwave linear analog computer (MiLAC) for simultaneous active and passive beamforming,” arXiv:2605.31549, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2605.31549

    2026 arXiv

  23. [31]

    Lossy microwave linear analog computer (MiLAC) for future MIMO: Learning-based architecture designs for spectral and energy efficiency maximization,

    B. Zhou and B. Clerckx, “Lossy microwave linear analog computer (MiLAC) for future MIMO: Learning-based architecture designs for spectral and energy efficiency maximization,” arXiv:2606.02369, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2606.02369

    2026 arXiv

  24. [32]

    Analog computing with hybrid couplers and phase shifters,

    M. Nerini, X. Liu, and B. Clerckx, “Analog computing with hybrid couplers and phase shifters,” arXiv:2603.24604, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2603.24604

    2026

  25. [33]

    Microwave linear analog computer (MiLAC)-aided MIMO radar sensing: Transmit beamforming design and DoA estimation,

    Z. Liu, Z. Wu, and B. Clerckx, “Microwave linear analog computer (MiLAC)-aided MIMO radar sensing: Transmit beamforming design and DoA estimation,” arXiv:2605.21020, 2026, preprint. [Online]. Available: https://arxiv.org/abs/2605.21020

    2026 arXiv

  26. [34]

    MIMO radar with colocated antennas,

    J. Li and P. Stoica, “MIMO radar with colocated antennas,”IEEE Signal Processing Magazine, vol. 24, no. 5, pp. 106–114, 2007

    2007

  27. [35]

    Stoica and R

    P. Stoica and R. L. Moses,Spectral Analysis of Signals. Upper Saddle River, NJ, USA: Prentice-Hall, 2005. 36

    2005

  28. [36]

    Multiple emitter location and signal parameter estimation,

    R. O. Schmidt, “Multiple emitter location and signal parameter estimation,”IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276–280, 1986

    1986

  29. [37]

    R. A. Horn and C. R. Johnson,Matrix Analysis, 2nd ed. Cambridge, U.K.: Cambridge University Press, 2012

    2012

  30. [38]

    The geometry of algorithms with orthogonality constraints,

    A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,”SIAM Journal on Matrix Analysis and Applications, vol. 20, no. 2, pp. 303–353, 1998

    1998

  31. [39]

    The measure of the critical values of differentiable maps,

    A. Sard, “The measure of the critical values of differentiable maps,”Bulletin of the American Mathematical Society, vol. 48, no. 12, pp. 883–890, 1942

    1942

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