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HingePlace: Harnessing the neural thresholding behavior to optimize Transcranial Electrical Stimulation

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Replacing zero-forcing penalties with a symmetrized hinge loss that ignores sub-threshold electric fields yields more focal neural responses in transcranial stimulation simulations, with off-target activation reduced by up to about 60…

desk verdict Solid unification theorem and a genuinely new loss idea, but the headline gains are inflated by tuning tolerances on the evaluation metric; still worth refereeing. read the letter →

arxiv 2502.01382 v1 pith:KYGTSLET submitted 2025-02-03 eess.SY cs.SY

Chaitanya Goswami , Pulkit Grover This is my paper · ORCID
classification eess.SYcs.SY
keywords transcranialelectricalstimulationelectrodeplacementalgorithmshingelossneuralthresholdingconvexoptimizationfocalityLCMV-Eseaofneuronsmodel
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

All major electrode-placement algorithms for transcranial electrical stimulation share a common restriction, the paper argues: they drive the off-target electric field toward zero, even though neural response is thresholded and non-zero fields below the threshold produce little or no direct stimulation. Extending a known unification result, the paper proves that the four main algorithm families—weighted least squares, reciprocity-based, constrained directional maximization, and LCMV-E—are essentially equivalent, all minimizing off-target field magnitude. On that basis it introduces HingePlace, a convex algorithm whose symmetrized hinge loss penalizes only off-target field components above user-set tolerances. Across an MRI head model and a biophysically realistic 'sea of neurons' model, HingePlace montages consistently reduced off-target activation relative to LCMV-E, by as much as about 60 percent and up to 75 percent in one case. If the thresholding premise holds in real tissue, the result reframes montage design from creating focal electric fields to creating focal neural responses.

What carries the argument

The load-bearing object is the symmetrized hinge loss, L_hinge(I) = ‖Γ_C L_HP(I)‖_p, where L_HP(I) = max(0, T_c I − E⁺_tol) + max(0, −T_c I − E⁻_tol) for the off-target forward matrix T_c, voxel-volume weights Γ_C, and user-set upper and lower tolerance vectors. It is what lets the algorithm leave sub-threshold off-target fields unpenalized while still punishing supra-threshold leakage. The argument also rests on Theorem 1, which proves that the CDM and LCMV-E optimization problems are essentially equivalent (for α < α_max they are dual with identical unique solutions; for α ≥ α_max, LCMV-E picks the CDM solution with least off-target energy), so the four major traditional algorithm families are all special cases of HingePlace at zero tolerances.

What would settle it

In an animal or ex-vivo preparation, compare HingePlace and LCMV-E montages delivering matched target activation while recording off-target spiking or calcium activity across the stimulated region: if LCMV-E produces equal or fewer off-target responses, the thresholding-based focality gains claimed here do not transfer to real tissue.

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Extended reading notes

Core claim

The paper's central discovery is that the traditional objective in electrode placement—minimizing the off-target electric field magnitude—is stricter than the neural response requires, because neurons respond nonlinearly and only fire or change state when the local field crosses a threshold. Theorem 1 establishes that the CDM and LCMV-E formulations are essentially equivalent (they are each other's dual for α < α_max, and for larger α, LCMV-E selects the most relevant CDM solution), thereby completing the unification of the four major algorithm families and showing that all of them force off-target fields to be close to zero. HingePlace replaces the quadratic off-target term with a symmetrized hinge loss, L_hinge(I) = ‖Γ_C L_HP(I)‖_p, whose per-voxel components max(0, T_c I − E⁺_tol) + max(0, −T_c I − E⁻_tol) are zero whenever the field stays within user-set tolerances; the problem stays convex and reduces to LCMV-E when p=2 and the tolerances are zero. In two simulation platforms, HingePlace montages generated more focal neural responses than LCMV-E montages, with off-target activation volume or spike counts reduced by up to about 60 percent (MRI head model) and up to 75 percent (sea-of-neurons model), and the paper shows that the L1L1-norm method is a special case of HingePlace with equal tolerances in all directions.

Load-bearing premise

The load-bearing premise is that sub-threshold off-target electric fields are behaviorally irrelevant—fields below a neuron's activation threshold have no meaningful off-target effect—so if they still modulate plasticity, network activity, or subthreshold membrane dynamics, the simulated focality gains could shrink or vanish in real tissue.

Editorial extensions

If this is right

  • All four major traditional algorithm families (WLS, reciprocity-based, CDM, LCMV-E) are subsumed by HingePlace at zero tolerances, so the measured focality gains apply to the whole classical family, not just the LCMV-E baseline used in the simulations.
  • Direction-specific tolerances matter: encoding the neuronal preference for certain field orientations (instead of one scalar tolerance) substantially improves focality, which means physiological orientation-sensitivity should be built into the loss.
  • Since the hinge-loss problem is convex, the thresholding idea can be added to existing electrode-placement pipelines with modest extra computation, the only added hyperparameters being the three tolerance vectors.
  • The same loss applies to other multi-electrode stimulation modalities—ECoG, deep brain stimulation, TTFields, temporal interference—wherever below-threshold off-target fields are known to be inert.
  • The correct design target is the focal neural response, not the focal electric field, opening the door to incorporating further neural response properties (e.g., orientation selectivity, adaptation) into montage optimization.

Reading between the lines

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

  • If sub-threshold off-target fields in real tissue still modulate plasticity, network synchrony, or subthreshold membrane dynamics, the simulated focality gains could shrink or vanish; a direct plasticity assay comparing HingePlace and LCMV-E montages at matched target activation would settle this.
  • The tolerance vectors could be made patient-specific by deriving them from individual input-output curves or neuron models, which the paper tunes only by grid or random search in simulation; this may increase the gains beyond the reported values.
  • Because the paper proves equivalence of the classical families, attaching the same hinge loss to reciprocity-based or weighted-least-squares formulations should yield comparable focality gains without re-deriving the algorithm.
  • A testable prediction of the thresholding hypothesis: the relative focality gain should be largest when the desired target field is just above the neural threshold, and should shrink when off-target fields are far supra-threshold.
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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

3 major / 4 minor

Summary. The paper proposes HingePlace, a convex optimization algorithm for multi-electrode transcranial electrical stimulation montages that penalizes off-target electric field components only above user-specified tolerance levels, thereby exploiting the thresholding behavior of neural response. It extends the unification result of Fernández-Corazza et al. by proving (Theorem 1, Appendix A) that the CDM and LCMV-E formulations are essentially equivalent, and shows (Appendix C) that the L1L1-norm optimization of Prieto et al. is a special case of HingePlace with p=1 and direction-independent tolerances. The algorithm is evaluated in two simulation platforms: an MRI head model (ROAST) using the VTh metric, and a sea-of-neurons model combining spherical head models with biophysically realistic neurons using the Nact metric. The authors report consistent focality improvements over LCMV-E in both platforms, with reductions in off-target activation as high as 60-75% in some conditions.

Significance. If the empirical claims hold, the paper makes two contributions: a clean theoretical unification of major electrode placement algorithms, and a novel, convex design objective that permits sub-threshold off-target fields to improve focality. The formal equivalence proof in Appendix A is detailed and accompanied by numerical illustrations, and the sea-of-neurons evaluation is a strong independent check that goes beyond common field-based surrogates. The code is publicly available, which supports reproducibility. However, the headline performance numbers currently rest on simulation studies whose tolerance hyperparameters are selected to minimize the very metric used for evaluation, and the sub-threshold irrelevance assumption is embedded in both the loss and the MRI-head-model metric. The significance is therefore conditional on a more rigorous validation protocol.

major comments (3)
  1. [Appendix D.3 / Sec. 5] The tolerance hyperparameters are selected on the evaluation metric. For the MRI studies, Etol is chosen by grid search and the paper states 'We always present the result with the least VTh'; for the sea-of-neurons studies, [Extol,Eytol,Eztol] is chosen by a 30-point random search minimizing Nact, and if HingePlace underperforms LCMV-E at some (Isafe, Imultot) the tolerances are re-tuned at that operating point. Because the same 800 sampled neuron locations are used both for tuning and scoring, and because no cross-validation, independent tuning set, or bootstrap confidence intervals are reported, the reported 60-75% reductions in off-target activation are not protected against selection bias and are likely optimistic. Please add an independent tuning/validation split (or fix tolerances a priori from biophysical thresholds), and report means and confidence intervals over repeated neuron samples.
  2. [Sec. 5.2.1 / Appendix C] The comparison against 'L1L1-norm optimization' may not implement Prieto et al.'s method correctly. Appendix C shows that the l1 regularizer in (C.1) is equivalent to reducing the total-current constraint to Itot=∥I∗∥1/2 on the HingePlace side, and that the target-fit term becomes a hard constraint. The text in Sec. 5.2.1 says only that L1L1 was analyzed 'by implementing the HingePlace algorithm with p=1 and the same tolerance across all directions,' with a footnote replacing the one-sided l∞ constraint. If the reduced Itot and/or the l1 regularizer are not included, the baseline is not the [28] algorithm, and the conclusion that direction-specific tolerances are essential may rest on an unfair comparison. Please state the exact optimization solved and verify the equivalence numerically for the reported operating points.
  3. [Sec. 1, Eq. (9), Appendix D.1] The hinge-loss objective and the MRI-head-model metric VTh both presuppose that sub-threshold off-target fields are harmless: the loss does so through E+tol and E−tol, and VTh through its 80%-of-Edes activation threshold. Consequently, the MRI results cannot independently validate the thresholding hypothesis, and the sea-of-neurons results test only direct spiking, not possible subthreshold modulation of plasticity or network activity (a limitation acknowledged in Sec. 6 but not examined). Since the paper's central claim is that HingePlace produces more focal neural responses in real tissue, please either provide a sensitivity analysis over tolerance values (for example, showing the focality gain as a function of the assumed threshold) or explicitly bound the conditions under which subthreshold fields can be ignored.
minor comments (4)
  1. [Abstract / Sec. 1 / Sec. 5.2] The maximum reported focality improvement is stated inconsistently: the abstract says 60%, Sec. 1 says '~30% (MRI) and ~60% (sea of neurons)', Sec. 5.2.1 reports up to 75%, and Sec. 5.2.3 reports up to 65%. Please harmonize the headline numbers with the exact values in the results.
  2. [Fig. 7 caption] The sentence 'TP refers to target point in l and k' is unclear because k and l are subplot labels; please specify which panels are meant and what TP denotes.
  3. [Sec. 5.1.4] The text says 'We run the same study as Sec. 5.1.4' but should refer to a preceding study (e.g., Sec. 5.1.3 or Sec. 5.1.1); please correct the cross-reference.
  4. [Appendix D.3] The grid for p=2,3 is listed as {0.01Vm−1, 0.65Vm−1, 0.55Vm−1, 0.35Vm−1}; please explain the ordering and the inclusion of 0.01, and state whether the same grid was used for both p=2 and p=3.

Circularity Check

2 steps flagged · score 6.0 of 10

Reported focality gains are in-sample optima: HingePlace's tolerances are tuned on the exact VTh/Nact metrics used to measure improvement.

  1. fitted input called prediction [Appendix D.3 ('Implementing LCMV-E and HingePlace') and Sec. 5.1 (MRI head model results)]
    "We used a grid-search to determine the value of Etol. For p = 1 and p = 2, 3, we searched across the values {0.1Vm−1, 0.5Vm−1, 0.6Vm−1, 0.7Vm−1} and {0.01Vm−1, 0.65Vm−1, 0.55Vm−1, 0.35Vm−1}, respectively. We always present the result with the least VTh."

    The MRI-head-model comparison in Sec. 5.1 reports the relative reduction in VTh as HingePlace's measured focality gain. But the HingePlace tolerance Etol is selected by grid search to minimize exactly this VTh metric, and the paper always presents the least-VTh result. LCMV-E has no analogous tuned parameter. Hence the reported improvement is not a prediction from a fixed algorithm; it is the minimum of a hyperparameter search over the evaluation metric itself, so the numerical gain is partly forced by the selection rule.

  2. fitted input called prediction [Appendix D.3 and Sec. 5.2 (sea of neurons studies)]
    "We determined the value of [ Ex tol, Ey tol, Ez tol] using a random search across 30 points for each sea of neurons model study. We chose [ Ex tol, Ey tol, Ez tol] with the least Nact value... If these chosen values of [ Ex tol, Ey tol, Ez tol] were severely sub-optimal for some values of Itot and Isaf e leading to HingePlace performing worse than LCMV-E, then we re-tuned [ Ex tol, Ey tol, Ez tol] at those values of Isaf e and Itot through the same random-search."

    Nact is the exact metric used to report the up-to-75% reductions in off-target stimulation in Sec. 5.2.1. The tolerance vector is chosen by random search to minimize Nact on the same 800 neuron locations used to score the final montage, with no separate tuning set or cross-validation described. Re-tuning whenever HingePlace underperforms LCMV-E makes the comparison additionally favorable. The headline neural-activation gains are therefore an in-sample optimum of the evaluation label rather than an independent estimate of HingePlace's advantage.

full rationale

The theoretical contributions are not circular: Theorem 1 (CDM/LCMV-E equivalence) is proved in Appendix A via KKT conditions with stated assumptions, and the unification with WLS/reciprocity-based methods is imported from Fernández-Corazza et al. as external prior work. The hinge loss in Eq. (9) is not definitionally equal to the VTh or Nact metrics, and the sea-of-neurons spiking metric is an independently biophysical outcome, so there is no self-definitional collapse. However, the paper's central empirical claim—that HingePlace montages reduce off-target neural response by up to 60-75%—is partially self-confirming: in every simulation study the tolerance hyperparameters are selected by grid or random search to minimize the exact metric used for comparison, the same 800 sampled neuron locations are used for both tuning and scoring, and re-tuning is performed when HingePlace underperforms. No held-out validation, independent tuning set, or error bars are reported, so the magnitude of the headline gains is statistically forced by the selection procedure rather than demonstrated by an out-of-sample comparison. This warrants a score of 6 rather than a lower score: the algorithmic idea and theoretical unification have independent content, but the quantitative focal-neural-response claim reduces in part to fitting the tolerance to the evaluation metric.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced; HingePlace is only a new loss function and optimization program. The central contribution takes the thresholding behavior of neurons as an input rather than deriving it, and the main numerical gains depend on tolerance hyperparameters that are fitted to the evaluation metrics. The proof itself relies on standard convex analysis plus a full-rank assumption for Ac.

free parameters (5)
  • E_tol (MRI head model) = p=1 grid {0.1,0.5,0.6,0.7} V/m; p=2,3 grid {0.01,0.35,0.55,0.65} V/m, chosen by least VTh
    Appendix D.3: HingePlace tolerance for the MRI platform is selected by grid search and the paper always presents the result with the least VTh, so this number is fitted to the evaluation metric.
  • [E^x_tol, E^y_tol, E^z_tol] (sea of neurons) = 30-point random search between 0.1 and 0.7 Edes, selected by least Nact, then retuned if HingePlace underperformed
    Appendix D.3: direction-wise tolerances are tuned using the same Nact metric used to report improvement, so the comparison is not parameter-free.
  • p (HingePlace loss exponent) = 1, 2, 3 after preliminary studies over p=1..5 using Nact
    Appendix D.8: p is selected on the basis of the outcome metric, and some studies drop p=3 after observing worse performance, which is model selection on the evaluation endpoint.
  • Edes (desired target field) = 1 V/m for MRI; 84.93, 84.94, 70.27 V/m for sea-of-neurons studies
    Appendix D.3: Edes is selected from baseline montages or visual input-output curves; it is shared by HingePlace and LCMV-E, so it does not explain relative gains but is a hand-chosen input.
  • Etol for L1L1-norm comparison = grid 0.1 to 0.9 Edes, least Nact per operating point
    Appendix D.3: the baseline L1L1 method gets its own tuned threshold; this affects the comparison in Sec. 5.2.1 but is not needed for the primary HingePlace claim.
assumptions (7)
  • domain assumption The forward model TI=E is linear under quasi-static Maxwell assumptions and provided by ROAST or the 4-sphere analytical model.
    Sec. 2.1, Eq. (1)-(2); all optimizations and simulations rely on this linearity and on the head-model conductivities.
  • domain assumption Neural response under tES is threshold-like, so off-target fields below a non-zero threshold have no significant effect.
    Sec. 1 and the hinge loss (9); this premise justifies not penalizing below-threshold off-target fields.
  • domain assumption Cortical pyramidal neurons are activated primarily by the radial-in component of the electric field.
    Sec. 5.1 and Appendix D.1; the VTh metric counts only radial-in field above 0.8 Edes.
  • domain assumption The 80 percent of Edes threshold from Forssell et al. [43] approximates the true activation boundary.
    Appendix D.1; used to compute VTh in the MRI platform.
  • standard math Ac is full rank, and Slater's conditions hold for the convex problems.
    Appendix A.3 P2/P4; needed for strict convexity of LCMV-E and the KKT-based proof of Theorem 1.
  • domain assumption The biophysical neuron models from Aberra et al. [37] and Blue Brain [55] predict human cortical activation thresholds.
    Appendix D.2; the sea-of-neurons Nact metric counts spikes from these models.
  • domain assumption All major traditional algorithms are representable by CDM, LCMV-E, or the magnitude-maximization-D trick.
    Sec. 3, Appendix B, and [13]; used to claim that the LCMV-E comparison covers WLS, reciprocity, and CDM.

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Cite this review

Pith. "Pith review of HingePlace: Harnessing the neural thresholding behavior to optimize Transcranial Electrical Stimulation." pith.science (2026). https://pith.science/paper/KYGTSLET

@misc{pith2026250201382,
  author       = {Pith},
  title        = {Pith review of: HingePlace: Harnessing the neural thresholding behavior to optimize Transcranial Electrical Stimulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYGTSLET}},
  note         = {Machine review of arXiv:2502.01382}
}
read the original abstract

Transcranial Electrical Stimulation (tES) is a neuromodulation technique that utilizes electrodes on the scalp to stimulate target brain regions. tES has shown promise in treating many neurological conditions, such as stroke rehabilitation and chronic pain. Several electrode placement algorithms have been proposed to optimize tES-based therapies by designing multi-electrode montages that create focal neural responses. We first extend a well-known unification result by Fernandez-Corazza et al. to unify all major traditional electrode placement algorithms. We utilize this unification result to identify a common restriction among traditional electrode placement algorithms: they do not harness the thresholding behavior of neural response. Consequently, these algorithms only partially harness the properties of neural response to optimize tES, particularly increasing the focality of neural response. We propose a new electrode placement algorithm, HingePlace, that utilizes a symmetrized hinge loss to harness the thresholding behavior of neural response. We extensively compare the HingePlace algorithm with traditional electrode placement algorithms in two simulation platforms. Across both platforms, we find that HingePlace-designed montages consistently generate more focal neural responses -- by as much as 60% -- than the electrode montages designed by traditional electrode placement algorithms.

Figures

Figures reproduced from arXiv: 2502.01382 by the authors.

Figure 1
Figure 1. A representation of the HingePlace algorithm utilizing the hinge loss (top-left) to allow sub-threshold electric field amplitudes in off-target regions (shown in the top￾middle) to reduce off-target stimulation volume (represented by the smaller stimulated region depicted in top-right). In contrast, traditional electrode placement algorithms uti￾lize the l2 loss (bottom-left) that minimizes the electric field amplit… view at source ↗
Figure 2
Figure 2. A visualization of the l2 (square) loss and the HingePlace loss with different values of p as a function of electric field amplitude at a single voxel in the off-target region. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. a and b present a sagittal and horizontal view of the realistic MRI head model discussed in Sec. 5.1. The electrodes are shown in blue, and the brain is represented using the salmon color. c shows the five different target locations used in the simulation studies described in Sec. 5.1. MC denotes a target location in the motor cortex, and the MNI coordinates of target locations 1, 2, 3, and 4 are provided in Appendi… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: A representative plot of the electric field along the radial-in direction and the cor [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: a-c plot the relative decrease in the stimulated volume between the HingePlace and LCMV-E montages for the simulation study described in Sec. 5.1.1. a, b, and c correspond to different values of I mul tot , namely, 2, 4, and 8, respectively. Similarly, d, e, and f plot…
Figure 6
Figure 6. Figure 6: A representative plot of the electrode montage, the current density magnitude [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: a-c, d-f, and g-i plot the relative decrease in the off-target stimulation between the HingePlace and LCMV-E montages for the simulation studies described in Sec. 5.2.1, Sec. 5.2.2 and Sec. 5.2.3, respectively. The groups of {a, d, g}, {b, e, h}, and {c, f, i} correspo…

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    Note that for Edes < Af I∗ M Ethe corresponding α = ∥AcI∗ 2∥2 2 < αM AX thereby ensuring that λ(1)∗ ̸= 0. Furthermore, Observe that I∗ 1 = I∗ 2, λ(1)∗ = −1/β(2), µ(1)∗ = −µ(1)∗ /β(2)∗, κ(1)∗ = −κ(1)∗ /β(2)∗,ν(1)∗ = −ν(1)∗ /β(2)∗, and δ(1)∗ = −δ(1)∗ /β(2)∗ solves the K.K.T. sys...

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    (C.27) Now note that due to (C.17), we know that ∂L(3)(I∗

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    and conse- quently, must contain the element 0N . Hence, L(3)(I∗

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    ∋ 0N . (C.28) 59 Similarly, 1T N I∗ 1| {z } =0 due to (C.18) = 0, (C.29) δ(1)∗(∥I∗ 1∥1 − 2 ˜Itot)| {z } =0 due to˜Itot=∥I∗ 1 ∥1/2 = 0, (C.30) ν(1)∗ j (eT j I∗ 1 − Isaf e) | {z } =0 due to (C.20)] = 0 ∀ j ∈ {1, . . . , N}. (C.31) Hence, the solution set {I∗ 1, µ(1)∗, δ(1)∗ +αζν...

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    Furthermore, by the particular choice of gEdes = Tf I∗ 3, we know that I∗ 3 lies in the constraint set of (C.33)

    Now note that (C.33) is a con- vex optimization problem with only affine inequalities. Furthermore, by the particular choice of gEdes = Tf I∗ 3, we know that I∗ 3 lies in the constraint set of (C.33). Consequently, (C.33) satisfies the relaxed Slater’s condition and strong dua...

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    , N}, (C.38) Tf I∗ 2 − gEdes = 0|F |

    = ∂Lhinge(I∗ 2)+ TT f β∗ +δ(2)∗∥∂I∗ 2∥1 + µ(2)∗1N + NX j=1 ν(2)∗ j ej ∋ 0N , (C.35) 1T N I∗ 2 = 0, (C.36) δ(2)∗(∥I∗ 2∥1 − 2 ˜Itot) = 0, (C.37) ν(2)∗ j (eT j I∗ 2 − Isaf e) = 0 ∀ j ∈ {1, . . . , N}, (C.38) Tf I∗ 2 − gEdes = 0|F |. (C.39) 61 From lemma 4, we also can simplify (C...

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    (C.40) Let T ∈ ∂∥Tf I∗ 3 − Edes∥, b ∈ ∂∥I∗ 3∥1, and c ∈ ∂Lhinge(I∗

    + TT f (ν∂ ∥Tf I∗ 3 − Edes∥1) + δ(3)∗∂∥I∗ 3∥1+ NX j=1 ν(3)∗ j ej + µ(3)∗1N ∋ 0N . (C.40) Let T ∈ ∂∥Tf I∗ 3 − Edes∥, b ∈ ∂∥I∗ 3∥1, and c ∈ ∂Lhinge(I∗

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    (C.41) Note that by (C.23), we are guaranteed the existence of at least one such T, b, and c

    such that: c+TT f (νT)+ δ(3)∗b+ NX j=1 ν(3)∗ j ej +µ(3)∗1N = 0N . (C.41) Note that by (C.23), we are guaranteed the existence of at least one such T, b, and c. Then, we make the following substitutions in K.K.T. conditions of (C.33), i.e., in (C.35)- (C.35): I∗ 2 = I∗ 3, (C.42...

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    (C.47) We know that T ∈ ∂Lhinge(I∗

    =∂Lhinge(I∗ 3)+ TT f (νb)+ δ(3)∗∂∥I∗ 3∥1 +µ(3)∗1N + NX j=1 ν(3)∗ j ej. (C.47) We know that T ∈ ∂Lhinge(I∗

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    Hence, we can conclude that: ∂L(2)(I∗

    and c ∈ ∂∥I∗ 3∥1 by their definition in (C.41). Hence, we can conclude that: ∂L(2)(I∗

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    (C.48) Combining (C.48) and (C.41), we can conclude that: ∂L(2)(I∗

    ∋ c+TT f (νT)+ δ(3)∗b+ NX j=1 ν(3)∗ j ej +µ(3)∗1N . (C.48) Combining (C.48) and (C.41), we can conclude that: ∂L(2)(I∗

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    ∋ 0N . (C.49) 62 Similarly, 1T N I∗ 3| {z } =0 due to (C.24) = 0, (C.50) δ(3)∗(∥I∗ 3∥1 − 2 ˜Itot)| {z } =0 due to (C.25) = 0, (C.51) ν(3)∗ j (eT j I∗ 3 − Isaf e) | {z } =0 due to (C.26) = 0 ∀ j ∈ {1, . . . , N}, (C.52) Tf I∗ 3 − gEdes | {z } =0|F |due to gEdes=Tf I∗ 3 = 0|F |....

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    We first solve the L1L1-norm optimization problem and obtain the solution I∗ L1L1

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    Using I∗ L1L1, we specify the values of the values of Edes = Af I∗ L1L1, Itot = ∥I∗ L1L1∥1/2, and E+ tol = E− tol = νϵ1|C| in the modified HingePlace optimization problem (stated in (C.3)), and obtain the corresponding solution I∗ HP

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    64 We quantified the difference between I∗ L1L1 and I∗ HP for five different target locations

    We calculate the difference between I∗ L1L1 and I∗ HP empirically by, ∥I∗ L1L1 − I∗ HP ∥1 ∥I∗ HP ∥1 × 100 . 64 We quantified the difference between I∗ L1L1 and I∗ HP for five different target locations. The corresponding spherical coordinates for the five targets are as follow...

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    (C.64) Proof. From (C.12), we know that: L(I) = Lhinge(I) + ν∥TT f I − Edes∥1. (C.65) 68 Taking sub-differential on both sides, we obtain: ∂L(I) = ∂Lhinge(I) + ∂ν ∥TT f I − Edes∥1. (C.66) where the above inequality comes from property (R2). Using property (R1) and (R3) in the ...

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    sea of neurons

    Randomly choose 25% of the rest of the voxels and add them to C. Construct the matrix Tc by only considering the voxels that lie on C. The above sub-sampling procedure utilizes the smoothness of transcranial fields to significantly reduce the size of Tc, thereby allowing faste...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.