Pith. sign in

REVIEW 4 major objections 5 minor 2 cited by

Thermodynamic Algorithms for Quadratic Programming

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Quadratic programs can be solved faster by letting a thermal physical system do the linear algebra inside an interior-point method; the paper claims a polynomial asymptotic speedup and predicts 10–30× speedups on support vector machines.

desk verdict A genuine algorithmic contribution — embedding a thermodynamic linear solver in an interior-point method with an O(n+m) diagonal-block update — but the asymptotic speedup claim is overstated and the inexact-IPM convergence is asserted rather than demonstrated. read the letter →

arxiv 2411.14224 v1 pith:RLNRJX7B submitted 2024-11-21 cs.ET cond-mat.stat-mechmath.OC

Patryk-Lipka Bartosik , Kaelan Donatella , Maxwell Aifer , Denis Melanson , Marti Perarnau-Llobet , Nicolas Brunner , Patrick J. Coles This is my paper · ORCID
classification cs.ETcond-mat.stat-mechmath.OC
keywords thermodynamiccomputingquadraticprogramminginterior-pointmethodslinearsystemsolvinganalogsupportvectormachinesportfoliooptimizationresistivenetworks
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

Quadratic programming is a convex optimization framework with applications in machine learning, finance, and circuit design, and its standard interior-point solvers repeatedly solve large linear systems. This paper claims that those linear-system solves can be offloaded to a thermodynamic computer—a physical system that relaxes to thermal equilibrium and whose time-averaged position solves a linear system—producing a hybrid digital-analog interior-point method. The claimed result is a polynomial asymptotic speedup over conventional digital solvers such as LU decomposition, with complexity $O(n^3+m^3+\sqrt{n+m}\log(1/\epsilon)[n^2+m^2+\kappa^2(n+m)\delta^{-2}])$ versus $O((n+m)^{7/2}\log(1/\epsilon))$. Simulations on a support vector machine task predict a 10–30× speedup at around a thousand dimensions with a small loss in training accuracy. If this holds, thermodynamic hardware becomes a practical accelerator for a broad class of optimization problems, not just a theoretical curiosity.

What carries the argument

The central object is the thermodynamic linear solver (TLS), which maps a linear system $Qx=c$ to a quadratic potential $V(x)=\frac12 x^\top Qx-c^\top x$ and reads off the solution as the time-averaged position $\langle x\rangle=\frac1\tau\int_{t_0}^{t_0+\tau}x(t)\,dt$ after the system thermalizes. The paper inserts this solver into the interior-point method by symmetrizing each KKT Jacobian $J_k$ into $\tilde{J}_k=J_k^\top J_k$, a positive-semidefinite matrix that the TLS requires; the crucial efficiency comes from the fact that only the diagonal blocks $X_k^2,Z_k^2,X_kZ_k$ change between iterations, so the GPU computes those $O(n+m)$ updates and transfers them to the analog chip while the expensive matrix-matrix products are done once at initialization.

What would settle it

One concrete test: run Algorithm 2 on a family of random QPs of increasing size $n$ with a fixed condition number $\kappa$ (e.g., by scaling $Q$), and count the number of interior-point iterations until the barrier parameter $\mu$ drops below a fixed threshold. If the iteration count grows faster than $O(\sqrt{n+m}\log(1/\epsilon))$, or if the algorithm fails to converge for moderately ill-conditioned problems with regularization $\lambda=0.1$, then the claimed polynomial speedup does not hold. A second check is to measure the actual residual $\|\tilde{J}_k\Delta r_k-\tilde{v}_k\|$ produced by the thermal time-average and compare it with the per-iteration error bounds in the inexact-IPM theorems the paper cites.

Watch

Extended reading notes

Core claim

The paper's central claim is that quadratic programs can be solved faster by replacing the linear-system solve inside an interior-point method with a thermodynamic linear solver. The key structural observation is that the symmetrized Newton system $\tilde{J}_k\Delta r_k=\tilde{v}_k$ with $\tilde{J}_k=J_k^\top J_k$ can be updated between iterations by changing only the diagonal blocks $X_k^2$, $Z_k^2$, and $X_kZ_k$, so the per-iteration cost of maintaining the matrix on analog hardware is $O(n+m)$ rather than $O(n^3+m^3)$. The paper states that this reduces the total runtime to $O(n^3+m^3+\sqrt{n+m}\log(1/\epsilon)[n^2+m^2+\kappa^2(n+m)\delta^{-2}])$, which is polynomially faster than the LU-based IPM's $O((n+m)^{7/2}\log(1/\epsilon))$, and it reports SVM simulations predicting 10–30× speedups at ~1000 dimensions with minimal accuracy loss.

Load-bearing premise

The load-bearing premise is that the interior-point iteration count $O(\sqrt{n+m}\log(1/\epsilon))$ remains valid when each Newton system is solved only approximately by the thermodynamic solver, whose output error is a zero-mean Gaussian whose magnitude depends on integration time and on an added $\lambda=0.1$ regularization; the paper cites two earlier convergence analyses for inexact interior-point methods but does not prove that this particular error model satisfies their residual or relative-error conditions.

Editorial extensions

If this is right

  • Quadratic programs of dimension roughly a thousand can be solved 10–30 times faster on hybrid GPU/thermodynamic hardware than with LU or conjugate-gradient solvers, at the cost of a small accuracy loss in SVM experiments.
  • The same diagonal-block update trick makes the interior-point linear solve a cheap $O(n+m)$ operation per iteration on the analog side, which is what turns the asymptotic gain into a practical one.
  • The method extends directly to equality-constrained QPs and to three showcased applications: support vector machines, portfolio optimization, and simulation of nonlinear resistive networks.
  • Because the complexity scales as $\kappa^2\,\delta^{-2}$ in the condition number and solver error, the speedup is largest when the symmetrized KKT matrix is well-conditioned and moderate accuracy suffices.
  • The paper states that the approach can in principle be carried by any hardware accelerator for linear algebra, not only resistive networks, so the speedup may transfer to other analog platforms.

Reading between the lines

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

  • The diagonal-only update structure may also appear in KKT symmetrizations for other convex programs, such as certain second-order cone or semidefinite programs, which would let the same thermodynamic speedup extend beyond QPs; the paper raises this possibility but does not analyze it in detail.
  • The zero-mean Gaussian error of the thermal average suggests a clean accuracy-versus-time trade-off: longer integration $\tau$ reduces $\delta$ and hence the $\delta^{-2}$ factor, so an optimal $\tau$ could be chosen per iteration to balance total runtime; the paper does not derive such a schedule.
  • The predicted 10–30× speedup assumes 16-bit digital-analog conversion at 100 Mb/s; if real applications require higher precision, the DAC/ADC transfer time will grow and could erode the advantage, making the speedup precision-dependent rather than absolute.
  • If the inexact-IPM convergence theorems are satisfied by the thermal error model, the same hybrid construction would also accelerate the sequential quadratic programming extension the paper sketches, effectively carrying the speedup over to general nonlinear optimization.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript presents a hybrid thermodynamic-digital algorithm for convex quadratic programming. It embeds a thermodynamic linear solver (TLS) from Ref. [20] into an interior-point method, solving the symmetrized normal equations J_k^T J_k Δr = J_k^T v_k. The paper claims a polynomial asymptotic speedup over conventional digital approaches, gives a complexity analysis in Table 1, and reports SVM simulations with predicted 10–30x speedups. It also sketches applications to portfolio optimization and nonlinear resistive networks.

Significance. If the central complexity and convergence claims were rigorously established, the paper would be a useful contribution to thermodynamic computing: it identifies a practical subroutine (IPM linear solves) that could be offloaded to analog hardware, and it provides a concrete hardware timing model and an open-source simulation package. The observation that only diagonal blocks of the symmetrized KKT matrix change between IPM iterations is a nice algorithmic insight. However, the missing error analysis for the inexact IPM and the use of fixed regularization mean that the asymptotic claims are not yet supported as stated.

major comments (4)
  1. [3.2.3, Table 1] The claimed asymptotic speedup over conjugate gradients is not present at the dominant order. Both the CG and thermodynamic rows contain O(n^3+m^3) for initialization and O(sqrt(n+m) log(1/eps)(n^2+m^2)) for the per-iteration work. The terms kappa^2(n+m)delta^{-2} and sqrt(kappa)(n+m)^2 are lower order than n^2+m^2 when n and m are of the same size and kappa and delta are treated as constants, so the two algorithms have identical leading-order complexity. The abstract's 'polynomial asymptotic speedup compared to conventional digital approaches' is therefore only valid relative to LU decomposition, not to CG; the paper should either restrict the speedup claim to LU or provide a complexity model that accounts for the different constant factors and hardware parallelism.
  2. [3.2.2, Eq. (16), Figure 2] The paper does not establish that the inexact-IPM iteration bound of Eq. (6) applies to the thermodynamic solver. Section 3.2.2 asserts that Refs. [33,34] provide update schemes, but it does not state the residual or relative-error condition required by those theorems, nor does it show that the zero-mean Gaussian error from Algorithm 1 satisfies it. Moreover, the implementation in Figure 2 solves (J_k^T J_k + 0.1 I) Delta r = J_k^T v_k rather than Eq. (16). The constant 0.1 regularization introduces a bias that is independent of the barrier parameter mu_k, so the iterates cannot approach the central path beyond a fixed tolerance; consequently, the O(sqrt(n+m) log(1/eps)) convergence to the QP solution is not established.
  3. [3.2.3, Table 1] The complexity table leaves delta, the accuracy of each thermodynamic linear solve, unspecified in relation to the QP accuracy epsilon. If the inexact IPM requires the Newton residual to decrease proportionally to mu_k (as is typical for path-following methods), then delta must decrease with epsilon (and possibly with conditioning), and the delta^{-2} factor would contribute an additional log(1/epsilon) or larger factor to the overall complexity. The paper should derive the required per-solve accuracy and include it in the complexity expression, rather than treating delta as an independent parameter.
  4. [4.1] The numerical SVM experiments report training accuracy but not the KKT residual or optimality gap. Given the fixed regularization lambda=0.1, the results may reflect a regularized proxy rather than the original quadratic program. To support the practical speedup claim for QP solving, the authors should report the achieved optimality gap or constraint violation for the thermodynamic algorithm and compare it with the digital baselines on the same metric.
minor comments (5)
  1. [3.1, Eqs. (10)-(12)] Equation (11) appears to have sign errors: from Eq. (10) the Lagrangian should read 1/2 x^T Q x + (A^T mu + c)^T x - mu^T b, and the subsequent stationarity condition in Eq. (12) should be Qx + A^T mu = -c (or the sign convention should be fixed consistently).
  2. [Algorithm 2] In the line 'rk+1 <- rk + (alpha_p Delta x, alpha_d Delta y, alpha_d Delta x)', the last component should presumably be alpha_d Delta z, not alpha_d Delta x.
  3. [4.2] The portfolio optimization problem in Eq. (22) includes an inequality constraint x^T r >= R, which is not in the standard form of Eq. (1); the conversion to an equality constraint via a slack variable should be stated.
  4. [2.3] The phrase 'identity vector' is nonstandard; it should be 'all-ones vector'.
  5. [References] Ref. [44] appears to duplicate Ref. [13] (both cite Kalinin et al. on the analog iterative machine); the two should be consolidated or distinguished.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the speedup claim is a complexity calculation from a published independent TLS result and standard IPM bounds, not a fit or a definitional reduction.

full rationale

I find no step in the paper's derivation chain that reduces, by the paper's own equations or by self-citation, to its own inputs. The central idea is to embed the thermodynamic linear solver (TLS) of Ref. [20] inside an interior-point method (IPM), symmetrizing the Newton system as J_k^T J_k and updating only diagonal blocks (Eqs. 15-17). The claimed asymptotic speedup in Table 1 is obtained by combining the standard IPM iteration bound N = O(sqrt(n+m) log(1/epsilon)) (Eq. 6, cited to Ref. [29]) with the published TLS runtime O(n kappa^2 delta^-2) from Ref. [20]. Although Ref. [20] shares authors with the present paper, it is a peer-reviewed, independently checkable complexity theorem with stated assumptions; it is not fitted here, nor is it redefined by the QP outcome, so the self-citation is real evidence rather than a circular premise. The convergence of the inexact IPM is asserted by citing external results [33,34] (Sec. 3.2.2); the paper does not prove that the zero-mean Gaussian TLS error satisfies the residual conditions of those theorems, but that is a completeness gap, not a circular reduction. Likewise, the SVM speedup prediction (Sec. 4.1, Appendix 2) rests on explicit hardware timing assumptions and a heuristically chosen integration time, which are empirical inputs and not outputs defined by the speedup claim. Consequently, no prediction in the paper is equivalent by construction to a fitted parameter or to a self-citation chain.

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

The central speedup claim leans on one externally supplied but heavily reused component: the thermodynamic linear solver of Ref [20], whose complexity and error scaling are taken as given. On top of that, the paper adds several hand-chosen numerical and hardware parameters (regularization, integration time, RC, precision, transfer speed) that set the predicted speedup. No new physical entities are postulated.

free parameters (5)
  • Regularization parameter lambda for SVM simulations = 0.1
    Added to stabilize the symmetrized KKT system J^T J in both CG and thermodynamic IPM; chosen by hand in Section 4.1 and affects both accuracy and runtime.
  • Analog integration time t per linear solve = not fixed; chosen heuristically by exploring convergence
    Appendix 2, step 3: 't was chosen heuristically by exploring convergence in the solutions of the problem of interest'. This parameter sets the runtime and accuracy of each thermodynamic linear solve and largely determines the predicted speedup.
  • RC timescale of the analog circuit = 1 microsecond, from R=1 kOhm, C=1 nF
    Appendix 2 timing assumptions. The runtime estimates use RC as the relaxation time, and the predicted speedup scales linearly with this assumed value.
  • DAC/ADC precision = 16 bits
    Appendix 2 assumption setting the conversion cost and accuracy of the thermodynamic solver.
  • Digital transfer speed = 100 Mb/s
    Appendix 2 assumption for uploading matrix blocks and downloading solutions; contributes to I/O time in the runtime model.
assumptions (5)
  • domain assumption Ergodicity of the thermodynamic linear solver: the time average of the physical state converges to the thermal equilibrium mean Q^{-1}c.
    Section 2.3 invokes the ergodic hypothesis [32] to justify Eq. (8); the convergence rate and burn-in time are taken from Ref [20] without independent verification.
  • ad hoc to paper Inexact interior-point convergence results (Refs [33],[34]) apply to the thermodynamic solver's Gaussian zero-mean error and to the regularized squared KKT system.
    Section 3.2.2 asserts this by citation; the residual conditions are not checked for the specific error distribution and lambda=0.1 regularization.
  • domain assumption The condition number kappa of J^T J is constant or mild across interior-point iterations.
    Section 3.2.3 says 'assuming it does not vary over iterations for simplicity'; as IPM approaches the boundary, X and Z approach zero, typically worsening conditioning of J^T J.
  • domain assumption The programmable resistor crossbar array realizes the equation of motion C dot V = -G V + R^{-1}V_in + I_n with Gaussian noise and negligible parasitic effects.
    Appendix 2, Eq. (31), based on prior crossbar hardware [19],[26]; the accuracy and bandwidth of programmable resistors are assumed.
  • standard math The QP data are real, Q is positive semidefinite, A has full row rank, and an interior feasible start exists.
    Problem statement in Section 2.1; standard assumptions for convex QP and IPM.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Thermodynamic Algorithms for Quadratic Programming." pith.science (2026). https://pith.science/paper/RLNRJX7B

@misc{pith2026241114224,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic Algorithms for Quadratic Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLNRJX7B}},
  note         = {Machine review of arXiv:2411.14224}
}
read the original abstract

Thermodynamic computing has emerged as a promising paradigm for accelerating computation by harnessing the thermalization properties of physical systems. This work introduces a novel approach to solving quadratic programming problems using thermodynamic hardware. By incorporating a thermodynamic subroutine for solving linear systems into the interior-point method, we present a hybrid digital-analog algorithm that outperforms traditional digital algorithms in terms of speed. Notably, we achieve a polynomial asymptotic speedup compared to conventional digital approaches. Additionally, we simulate the algorithm for a support vector machine and predict substantial practical speedups with only minimal degradation in solution quality. Finally, we detail how our method can be applied to portfolio optimization and the simulation of nonlinear resistive networks.

Figures

Figures reproduced from arXiv: 2411.14224 by the authors.

Figure 1
Figure 1. Overview of the hybrid digital-analog algorithm to solve quadratic programs. At initialization, J˜ 1 = J ⊤ 1 J1 and v1 are computed by a GPU (to benefit from the high parallelization of matrix-matrix and matrix-vector multiplications) and uploaded onto the SPU. After some dynamical evolution, the update ∆r1 is downloaded from the SPU, which enables the calculation of quantities X2, Z2, X2Z2 which are used to upload … view at source ↗
Figure 2
Figure 2. Comparison of digital and thermodynamic approaches to training SVMs. The left panel compares the training times when different subroutines for the IPM are used. The thermodynamic subroutine runtime is estimated with a timing model, detailed in Appendix 2. The right panel shows the associated training accuracy of the model, showing a minimal degradation in solution quality. Here regularization is added for both the C… view at source ↗
Figure 3
Figure 3. Quadratic programming approach to resisitve networks. The parameters of the resistive networks (i.e. voltage sources V, current sources I, diodes D, resistors R and the graph structure G of the network) are encoded in the parameters of a quadratic program (A, b and c). The thermodynamic interior point algorithm (Algorithm 2) is then used to solve the optimization problem and produces a close-to-optimal solution x ⋆ … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Circuit diagram for the thermodynamic device in the case of a single resistor array implementation for a 3-dimensional problem. The device is comprised of three voltage sources, each of which is connected to three resistors {Ri} . Each of these resistors is connected t…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Scalable Thermodynamic Second-order Optimization

    cs.ET 2025-02 conditional novelty 6.0 of 10

    Thermodynamic computers can accelerate K-FAC training by replacing matrix inversions with physical linear-system solves, yielding quadratic rather than cubic per-layer scaling.

  2. Theoretical Analysis of Thermodynamic Matrix Inversion: First-order Equivalence to Preconditioned Gradient Descent and Implications for Analog Computing

    math.NA 2026-08 conditional novelty 4.0 of 10

    Thermodynamic matrix inversion by an Ornstein-Uhlenbeck process is, to first order, exactly a gradient descent iteration on the residual, so the stochastic sampling is not needed in a digital implementation.

Reference graph

Works this paper leans on

54 extracted references · 38 canonical work pages · cited by 2 Pith papers

  1. [20]

    Thermodynamic natural gradient descent,

    K. Donatella, S. Du ffield, M. Aifer, D. Melanson, G. Crooks, and P. J. Coles, “Thermodynamic natural gradient descent,”arXiv preprint arXiv:2405.13817, 2024

    arXiv 2024

  2. [1]

    Ising machines as hardware solvers of combinatorial optimization problems,

    N. Mohseni, P. L. McMahon, and T. Byrnes, “Ising machines as hardware solvers of combinatorial optimization problems,” Nature Reviews Physics , vol. 4, no. 6, p. 363–379, May 2022. [Online]. Available: http://dx.doi.org/10.1038/s42254-022-00440-8

    doi:10.1038/s42254-022-00440-8 2022

  3. [2]

    E fficient optimization with higher-order ising machines,

    C. Bybee, D. Kleyko, D. E. Nikonov, A. Khosrowshahi, B. A. Olshausen, and F. T. Sommer, “E fficient optimization with higher-order ising machines,” Nature Communications , vol. 14, no. 1, Sep. 2023. [Online]. Available: http: //dx.doi.org/10.1038/ s41467-023-41214-9

    work page 2023

  4. [3]

    Application of ising machines and a software development for ising machines,

    K. Tanahashi, S. Takayanagi, T. Motohashi, and S. Tanaka, “Application of ising machines and a software development for ising machines,” Journal of the Physical Society of Japan , vol. 88, no. 6, p. 061010, Jun. 2019. [Online]. Available: http://dx.doi.org/10.7566/JPSJ.88.061010

    doi:10.7566/jpsj.88.061010 2019

  5. [4]

    Analog coupled oscillator based weighted ising machine,

    J. Chou, S. Bramhavar, S. Ghosh, and W. Herzog, “Analog coupled oscillator based weighted ising machine,” Scientific reports , vol. 9, no. 1, p. 14786, 2019

    work page 2019

  6. [5]

    Coherent ising machines—quantum optics and neural network perspectives,

    Y . Yamamoto, T. Leleu, S. Ganguli, and H. Mabuchi, “Coherent ising machines—quantum optics and neural network perspectives,” Applied Physics Letters, vol. 117, no. 16, p. 160501, 2020

    work page 2020

  7. [6]

    Oim: Oscillator-based ising ma- chines for solving combinatorial optimisation problems,

    T. Wang and J. Roychowdhury, “Oim: Oscillator-based ising ma- chines for solving combinatorial optimisation problems,” in Uncon- ventional Computation and Natural Computation: 18th International Conference, UCNC 2019, Tokyo, Japan, June 3–7, 2019, Proceedings

    work page 2019

  8. [7]

    Pass: An asynchronous probabilistic processor for next generation intelligence,

    S. Patel, P. Canoza, A. Datar, S. Lu, C. Garg, and S. Salahuddin, “Pass: An asynchronous probabilistic processor for next generation intelligence,” arXiv preprint arXiv:2409.10325 , 2024

    arXiv 2024

Show all 54 references
  1. [8]

    Perspectives of quantum annealing: methods and implementations,

    P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, “Perspectives of quantum annealing: methods and implementations,” Reports on Progress in Physics , vol. 83, no. 5, p. 054401, May 2020. [Online]. Available: http: //dx.doi.org/10.1088/ 1361-6633/ab85b8

    2020

  2. [9]

    p-bits for probabilistic spin logic,

    K. Y . Camsari, B. M. Sutton, and S. Datta, “p-bits for probabilistic spin logic,” Appl. Phys. Rev., vol. 6, no. 1, p. 011305, 2019

    2019

  3. [10]

    Life is probabilistic—why should all our computers be deterministic? computing with p-bits: Ising solvers and beyond,

    J. Kaiser, S. Datta, and B. Behin-Aein, “Life is probabilistic—why should all our computers be deterministic? computing with p-bits: Ising solvers and beyond,” in 2022 International Electron Devices Meeting (IEDM). IEEE, 2022, pp. 21–4

    2022

  4. [11]

    A full-stack view of probabilistic computing with p-bits: devices, architectures, and algorithms,

    S. Chowdhury, A. Grimaldi, N. A. Aadit, S. Niazi, M. Mohseni, S. Kanai, H. Ohno, S. Fukami, L. Theogarajan, G. Finocchio et al., “A full-stack view of probabilistic computing with p-bits: devices, architectures, and algorithms,” IEEE Journal on Exploratory Solid- State Computa...

    2023

  5. [12]

    All-to-all reconfigurability with sparse ising machines: the xorsat challenge with p-bits,

    N. A. Aadit, S. Nikhar, S. Kannan, S. Chowdhury, and K. Y . Camsari, “All-to-all reconfigurability with sparse ising machines: the xorsat challenge with p-bits,” arXiv preprint arXiv:2312.08748 , 2023

    2023 arXiv

  6. [14]

    Thermodynamic computing,

    T. Conte, E. DeBenedictis, N. Ganesh, T. Hylton, J. P. Strachan, R. S. Williams, A. Alemi, L. Altenberg, G. Crooks, J. Crutchfield et al. , “Thermodynamic computing,” arXiv preprint arXiv:1911.01968 , 2019

    1911 arXiv

  7. [15]

    ChatGPT’s Hunger for Energy Could Trigger a GPU Revolution — wired.com,

    W. Knight, “ChatGPT’s Hunger for Energy Could Trigger a GPU Revolution — wired.com,” https: //www.wired.com/story/ fast-forward-chatgpt-hunger-energy-gpu-revolution, 2024

    2024

  8. [16]

    Thermodynamic neural network,

    T. Hylton, “Thermodynamic neural network,” Entropy, vol. 22, no. 3, p. 256, 2020

    2020

  9. [17]

    Thermo- dynamic computing via autonomous quantum thermal machines,

    P. Lipka-Bartosik, M. Perarnau-Llobet, and N. Brunner, “Thermo- dynamic computing via autonomous quantum thermal machines,” Science Advances, vol. 10, no. 36, p. eadm8792, 2024

    2024

  10. [18]

    Springer, 2019, pp. 232–256

    2019

  11. [19]

    Thermodynamic state machine network,

    T. Hylton, “Thermodynamic state machine network,” Entropy, vol. 24, no. 6, p. 744, 2022

    2022

  12. [21]

    Thermodynamic linear algebra,

    M. Aifer, K. Donatella, M. H. Gordon, S. Du ffield, T. Ahle, D. Simp- son, G. Crooks, and P. J. Coles, “Thermodynamic linear algebra,” npj Unconventional Computing, vol. 1, no. 1, p. 13, 2024

    2024

  13. [22]

    Thermo- dynamic matrix exponentials and thermodynamic parallelism,

    S. Du ffield, M. Aifer, G. Crooks, T. Ahle, and P. J. Coles, “Thermo- dynamic matrix exponentials and thermodynamic parallelism,” arXiv preprint arXiv:2311.12759, 2023

    2023 arXiv

  14. [23]

    Thermodynamic AI and the fluctuation frontier,

    P. J. Coles, C. Szczepanski, D. Melanson, K. Donatella, A. J. Mar- tinez, and F. Sbahi, “Thermodynamic AI and the fluctuation frontier,” in 2023 IEEE International Conference on Rebooting Computing (ICRC). IEEE, 2023, pp. 1–10

    2023

  15. [24]

    Thermodynamic bayesian inference,

    M. Aifer, S. Du ffield, K. Donatella, D. Melanson, P. Klett, Z. Belat- eche, G. Crooks, A. J. Martinez, and P. J. Coles, “Thermodynamic bayesian inference,” arXiv preprint arXiv:2410.01793 , 2024

    2024 arXiv

  16. [25]

    Ther- modynamic computing system for AI applications,

    D. Melanson, M. A. Khater, M. Aifer, K. Donatella, M. H. Gordon, T. Ahle, G. Crooks, A. J. Martinez, F. Sbahi, and P. J. Coles, “Ther- modynamic computing system for AI applications,” arXiv preprint arXiv:2312.04836, 2023

    2023 arXiv

  17. [26]

    Error mitigation for thermodynamic computing,

    M. Aifer, D. Melanson, K. Donatella, G. Crooks, T. Ahle, and P. J. Coles, “Error mitigation for thermodynamic computing,” arXiv preprint arXiv:2401.16231, 2024

    2024 arXiv

  18. [27]

    Solving matrix equations in one step with cross-point resistive arrays,

    Z. Sun, G. Pedretti, E. Ambrosi, A. Bricalli, W. Wang, and D. Ielmini, “Solving matrix equations in one step with cross-point resistive arrays,” Proceedings of the National Academy of Sciences , vol. 116, no. 10, pp. 4123–4128, 2019

    2019

  19. [28]

    Linear programming: foundations and extensions,

    R. J. Vanderbei, “Linear programming: foundations and extensions,” Journal of the Operational Research Society , vol. 49, no. 1, pp. 94– 94, 1998

    1998

  20. [29]

    Foundations of operations research: From linear programming to data envelopment analysis,

    Z. Sinuany-Stern, “Foundations of operations research: From linear programming to data envelopment analysis,” European Journal of Operational Research , vol. 306, no. 3, pp. 1069–1080, 2023. [Online]. Available: https: //www.sciencedirect.com/science/article/ pii/S0377221722008578

    2023

  21. [30]

    Interior point methods 25 years later,

    J. Gondzio, “Interior point methods 25 years later,” European Journal of Operational Research , vol. 218, no. 3, pp. 587–601,

  22. [31]

    Hamming, Numerical methods for scientists and engineers

    R. Hamming, Numerical methods for scientists and engineers . Courier Corporation, 2012

    2012

  23. [32]

    S. P. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press, 2004

    2004

  24. [33]

    Convergence analysis of an inexact feasible interior point method for convex quadratic programming,

    J. Gondzio, “Convergence analysis of an inexact feasible interior point method for convex quadratic programming,” SIAM Journal on Optimization, vol. 23, no. 3, pp. 1510–1527, 2013

    2013

  25. [34]

    L. E. Boltzmann, Einige allgemeine S¨ atze ¨ uber W¨ armegleichgewicht. K. Akad. der Wissensch., 1871

  26. [35]

    Support-vector networks,

    C. Cortes and V . Vapnik, “Support-vector networks,” Machine learn- ing, vol. 20, pp. 273–297, 1995

    1995

  27. [36]

    Global and polynomial-time convergence of an infeasible-interior-point algorithm using inexact computation

    S. Mizuno and F. Jarre, “Global and polynomial-time convergence of an infeasible-interior-point algorithm using inexact computation.” Mathematical Programming, vol. 84, no. 1, 1999

    1999

  28. [37]

    Support vector method for novelty detection,

    B. Sch ¨olkopf, R. C. Williamson, A. Smola, J. Shawe-Taylor, and J. Platt, “Support vector method for novelty detection,” Advances in neural information processing systems , vol. 12, 1999

    1999

  29. [38]

    Training invariant support vector machines,

    D. DeCoste and B. Sch ¨olkopf, “Training invariant support vector machines,” Machine learning, vol. 46, pp. 161–190, 2002

    2002

  30. [39]

    Breast cancer wisconsin (diagnostic),

    O. M. William Wolberg, “Breast cancer wisconsin (diagnostic),”

  31. [40]

    Text categorization with support vector machines: Learning with many relevant features,

    T. Joachims, “Text categorization with support vector machines: Learning with many relevant features,” in European conference on machine learning. Springer, 1998, pp. 137–142

    1998

  32. [41]

    A universal approximation theorem for nonlinear resistive networks,

    B. Scellier and S. Mishra, “A universal approximation theorem for nonlinear resistive networks,”arXiv preprint arXiv:2312.15063, 2023

    2023 arXiv

  33. [42]

    Equilibrium propagation: Bridging the gap between energy-based models and backpropagation,

    B. Scellier and Y . Bengio, “Equilibrium propagation: Bridging the gap between energy-based models and backpropagation,” Frontiers in computational neuroscience , vol. 11, p. 24, 2017

    2017

  34. [43]

    thermox: Exact ou processes with JAX,

    S. Du ffield, K. Donatella, and D. Melanson, “thermox: Exact ou processes with JAX,” https: //github.com/normal-computing/thermox, 2024

    2024

  35. [44]

    Analog iterative machine (aim): using light to solve quadratic optimization problems with mixed variables,

    K. P. Kalinin, G. Mourgias-Alexandris, H. Ballani, N. G. Berlo ff, J. H. Clegg, D. Cletheroe, C. Gkantsidis, I. Haller, V . Lyutsarev, F. Parmigiani et al. , “Analog iterative machine (aim): using light to solve quadratic optimization problems with mixed variables,” arXiv prep...

    2023 arXiv

  36. [45]

    P. J. Van Laarhoven, E. H. Aarts, P. J. van Laarhoven, and E. H. Aarts, Simulated annealing. Springer, 1987

    1987

  37. [46]

    A fast algorithm to simulate nonlinear resistive net- works,

    B. Scellier, “A fast algorithm to simulate nonlinear resistive net- works,” arXiv preprint arXiv:2402.11674 , 2024

    2024 arXiv

  38. [49]

    A combinatorial, primal-dual approach to semidefinite programs,

    S. Arora and S. Kale, “A combinatorial, primal-dual approach to semidefinite programs,” J. ACM, vol. 63, no. 2, may 2016. [Online]. Available: https://doi.org/10.1145/2837020 Supplementary Materials

    2016 doi

  39. [50]

    This is achieved by first replacing the non-negativity constraints x≥ 0 with the an appropriate penalty function and forming the associated Lagrangian

    Interior Point Methods The starting point for any IPM algorithm is to transform a given optimization problem into a set of (potentially non- linear) equations. This is achieved by first replacing the non-negativity constraints x≥ 0 with the an appropriate penalty function and ...

  40. [51]

    The thermodynamic solver may be implemented on various physical platforms, but the most natural choice for very large-scale integration is a network of passive electronic elements

    Hardware implementation In this section we provide more details about the model we use to provide results about timing and accuracy on the SVM task. The thermodynamic solver may be implemented on various physical platforms, but the most natural choice for very large-scale inte...

  41. [52]

    Compute ˜vk, and use a Digital-to-analog converter (DAC) to set the ˜ vk vector to voltages Vin

  42. [53]

    For subsequent iterations, set the diagonals of the blocks to be updated (4 n elements with a given bit precision to set)

    For iteration 0: Set the configuration of the programmable resistors ((2 n + m)2 values with a given bit precision to set by calculating ˜J0 digitally. For subsequent iterations, set the diagonals of the blocks to be updated (4 n elements with a given bit precision to set)

  43. [54]

    Let the dynamics run for t (the analog dynamic time). Note that for experiments t was chosen heuristically by exploring convergence in the solutions of the problem of interest, and in general it will be proportional to the relaxation time of the physical system. Since the ⟨V⟩ ...

  44. [55]

    Analog-to-digital (ADC) conversion of the solution measured at nodes Vi to the digital device. The relaxation time of the system is: τ = RC αmin (33) where R is a resistance scale (which means that all resistances Ri j are a multiple of this), C is the capacitance (assuming al...

  45. [1993]

    Available: https: //archive.ics.uci.edu/dataset/17

    [Online]. Available: https: //archive.ics.uci.edu/dataset/17

  46. [2012]

    Available: https: //www.sciencedirect.com/science/ article/pii/S0377221711008204

    [Online]. Available: https: //www.sciencedirect.com/science/ article/pii/S0377221711008204

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.