Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations

Dante Kalise; Karl Kunisch; Sergey Dolgov

REVIEW 3 major objections 5 minor 66 references

Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that tensor-train methods can solve high-dimensional HJB equations for nonlinear feedback control with cost polynomial in the state dimension, demonstrated up to 121 dimensions.

desk verdict A real deterministic TT solver for stationary nonlinear HJB equations that reaches 100+ dimensions, with an honest LQ rank theorem and shipped code; the high-dimensional accuracy claims rest on empirically observed low-rank structure and LQR/uncontrolled comparisons, not yet on independent HJB verification. read the letter →

arxiv 1908.01533 v4 pith:UA2AGO6L submitted 2019-08-05 math.OC cs.NAmath.NA

Sergey Dolgov , Dante Kalise , Karl Kunisch This is my paper · ORCID

classification math.OCcs.NAmath.NA MSC 15A6915A2365F1065N2249J2049LXX49MXX

keywords tensortrainHamilton-Jacobi-BellmanequationoptimalfeedbackcontrolcurseofdimensionalitypolicyiterationAllen-CahnFokker-Planckhigh-dimensionalPDE

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 tries to show that Bellman's 'curse of dimensionality' can be partially circumvented for a broad class of nonlinear optimal feedback control problems by writing the value function of a Hamilton-Jacobi-Bellman (HJB) equation in tensor-train form. On top of this compressed representation it runs a Newton-like continuous policy iteration, solving each linearized equation with a shifted alternating least-squares method. The reported outcome is that the method's cost scales at most polynomially (observed order about 4) with the state dimension, allowing accurate feedback synthesis for nonlinear PDE-controlled dynamics with more than 100 variables. If this holds, deterministic PDE methods become a practical route to optimal feedback control in regimes where grid-based dynamic programming and linearized approximations such as LQR are either intractable or too crude.

What carries the argument

The central object is the tensor-train (TT) decomposition, which stores a d-dimensional array using about $dnr^2$ parameters by chaining small three-dimensional blocks with $r$ ranks; here the array holds the coefficients of a Legendre spectral expansion of the value function. Around this object the method wraps three mechanisms: continuous policy iteration (a Newton-type scheme that linearizes the HJB equation at each step), TT-cross interpolation to assemble the nonlinear drift and control terms directly in low-rank form, and a shifted AMEn (alternating minimal energy) iteration that solves the resulting nonsymmetric, degenerate linear systems while preserving the TT structure. The linear-quadratic rank bound comes from writing the Riccati solution as the inverse of a Kronecker-sum operator and approximating that inverse by a sum of matrix exponentials of bounded quasi-separable rank.

What would settle it

Fix a target accuracy and compute the maximal tensor-train rank of the value function for a family of nonlinear control problems (for instance, transport-dominated or nonlocal dynamics) as the state dimension $d$ increases; if the rank required to keep the error fixed grows faster than polynomial in $d$, the polynomial-scaling claim collapses even though each algorithmic step remains correct.

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Extended reading notes

Core claim

The central claim is that the value function of the stationary HJB equation associated with deterministic infinite-horizon optimal control can be represented and computed in the tensor-train format with low ranks, making the overall algorithm scale polynomially in the dimension instead of exponentially. The paper proves this rank structure rigorously for linear-quadratic problems: Theorem 3.1 bounds the TT ranks of the quadratic value function by the off-diagonal ranks of the linearized system matrix times a polylogarithmic factor in the desired accuracy. For nonlinear dynamics, the same low-rank behavior is established numerically: the Allen-Cahn stabilization problem is solved with state dimensions up to 121 and the Fokker-Planck problem with reduced dimensions up to about 20, with maximal TT ranks growing linearly (or stabilizing) in the dimension. The paper also shows that the resulting feedback laws stabilize unstable equilibria at substantially lower cost than the LQR feedback computed from the linearized system, and that control bounds can be enforced through penalty functions.

Load-bearing premise

The load-bearing premise is that the value functions of the nonlinear problems stay low-rank in tensor-train form as the dimension grows; for nonlinear examples this is observed numerically, not proved, and the conclusion explicitly leaves the identification of low-rank problem classes open.

Editorial extensions

If this is right

For dynamics whose value function admits small TT ranks, optimal feedback synthesis cost becomes polynomial in state dimension, so state-space dimension is no longer the decisive obstacle for deterministic HJB methods.
HJB-based feedback from the method can stabilize unstable nonlinear equilibria at much lower control cost than LQR linearization, as demonstrated for Allen-Cahn and Fokker-Planck models.
Control constraints can be incorporated through smooth penalty functions, at the price of larger but still manageable TT ranks.
The method extends from semi-discretized one-dimensional PDEs to two-dimensional spatial domains, producing 121-dimensional state systems that are solved in reasonable CPU times on a single core.
The linear-quadratic rank bound provides an a priori criterion: the off-diagonal ranks of the linearized system matrix (and actuator rank) indicate whether the TT approach is promising for a given control system.

Reading between the lines

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

The observed near-linear rank growth for Allen-Cahn suggests a testable conjecture: value functions for feedback stabilization of semilinear parabolic PDEs with localized actuators form a low-rank class, and a systematic study of rank growth across reaction-diffusion, convection-diffusion, and nonlocal systems would map the method's real boundary.
The Fokker-Planck treatment already uses balanced truncation before HJB synthesis; a natural extension is to feed the TT-HJB solver with data-driven or reduced-order coordinates for transport-dominated problems, where direct ranks are likely higher.
Because the rank bound ties feasibility to the off-diagonal ranks of the Jacobian, one could build a cheap pre-screening procedure: compute those ranks for a candidate dynamics before committing to the full HJB solve.
The shifted AMEn linear solver may transfer to time-dependent and stochastic (parabolic) HJB equations, where the same nonsymmetric-degenerate linearized systems appear; that connection is implicit in the paper but not developed.

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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

What you should know: this is the first deterministic, PDE-based tensor-train solver I know of for stationary fully nonlinear HJB equations that genuinely goes past 100 dimensions. The Allen-Cahn 121-dimensional example is the real headline, and the paper is honest about what it can and cannot prove.

The core engineering is sound. They combine continuous policy iteration with a Galerkin Legendre discretization, assemble the linearized HJB operator in the tensor-train format, and solve the nonsymmetric degenerate systems with a shifted AMEn iteration. That last piece matters: the shift makes the linear solves work despite the gradient operator's nullspace, and the geometric reduction of the shift as the policy iteration converges is a practical touch that clearly took experimentation to get right. The paper ships Matlab code, gives convergence studies in polynomial degree, TT threshold, and dimension, and reports CPU times on one core. That is reproducible work and should be credited.

The main theoretical contribution is Theorem 3.1: a parameter-free polylogarithmic TT-rank bound for linear-quadratic value functions, derived through Riccati/Lyapunov analysis and low-rank approximants to the matrix exponential. I checked the structure of the proof, not every constant, and it is plausible and genuinely useful. It is also clearly labeled as LQ-only, and Remark 3.3 correctly warns that TT ranks can grow with spatial degrees of freedom for 2D discretizations.

The soft spots are real but not fatal. The load-bearing assumption for the nonlinear results is that value functions of these controlled parabolic PDEs keep low TT rank as the dimension grows. That is observed numerically, with ranks growing roughly linearly in d, but not proven for nonlinear dynamics. The paper itself flags this as open in the conclusion, so the authors are not hiding it. Still, the polynomial-scaling claim is conditional on that empirical regularity. Second, the 121-dimensional Allen-Cahn result is validated by comparing running costs and controls against LQR and uncontrolled trajectories, not against an independent HJB solution. That shows the controller stabilizes and beats LQR, which is meaningful, but it does not quantify how close the computed value is to the true optimal value. Third, the Fokker-Planck pipeline inherits an unquantified bilinear balanced-truncation error; they estimate cost differences with respect to reduced dimension, but there is no a priori bound. These are addressable weaknesses, and none of the paper's equations show an internal contradiction.

Who should read this: anyone working in deterministic optimal feedback synthesis, tensor methods for PDEs, or high-dimensional dynamic programming. It deserves a serious referee. My recommendation is to send it to peer review and push for a modest revision: add a discussion of when TT ranks can be expected to stay low for nonlinear systems, and benchmark at least one high-dimensional case against a policy computed by another method or against the HJB residual itself.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a deterministic tensor-train (TT) method for stationary Hamilton-Jacobi-Bellman (HJB) equations arising in infinite-horizon optimal feedback control. The method combines a Legendre Galerkin discretization, TT representations of the value function and system functions via the TT-Cross algorithm, continuous policy iteration, and a shifted AMEn linear solver. Theorem 3.1 provides a TT-rank bound for quadratic (linear-quadratic) value functions by connecting the Riccati equation to low-rank approximations of a Lyapunov inverse. Numerical experiments cover feedback stabilization of the Allen-Cahn equation in one and two spatial dimensions (up to d=121) and of a reduced Fokker-Planck model (d=10), including one constrained-control example. The paper's central claims are that the method scales at most polynomially in the state dimension (largest observed order 4) and that accurate optimal feedback maps can be synthesized for nonlinear dynamics with over 100 state variables.

Significance. If the numerical claims hold, the paper is a useful step toward deterministic high-dimensional HJB solution; the combination of policy iteration with TT algebra is natural and the implementation is nontrivial. The manuscript's strengths are the explicit LQ rank theorem in Theorem 3.1, the reproducibility statement with a public code repository, and the systematic numerical convergence studies in the polynomial degree, the TT threshold, the spatial dimension, and the number of finite-difference points. Its main deficit is that the nonlinear low-rank and accuracy claims are not established beyond the tested dissipative examples and are in part explicitly acknowledged as open in the Conclusion. The significance is therefore moderate: the method is promising, but the advertised scope should be qualified to match the evidence.

major comments (3)

[Sec. 3.1, Theorem 3.1 and Sec. 1.3] The polynomial-in-dimension claim in Sec. 1.3 is not established by Theorem 3.1 for the PDE examples. The bound r_k ≲ (M+r_b)(log(1/ε)+C)^(7/2) assumes that the off-diagonal blocks of A D^{-1} have rank at most M uniformly in k and d. For the Chebyshev pseudospectral Allen-Cahn discretization (4.3), A is not banded, and Fig. 4.2 reports the maximal TT rank growing linearly in d; hence the hypothesis rank AD(k+1:d,1:k) ≤ M is not satisfied in the regime advertised. The linear-in-d rank observation is an empirical finding for the tested parabolic problems, not a consequence of Theorem 3.1, and the Conclusion ("identification of a class ... open") says exactly this. I request that the claims of a "wide class" and "polynomial scaling" be reworded to the class of problems for which the off-diagonal rank assumption or an analogous nonlinear rank bound holds, or that the theorem be extended to the semidiscrete elliptic operators used in Section 4.
[Sec. 4.1, Figs. 4.3 and 4.8] The statement "accurate synthesis of optimal feedback maps ... over 100 dimensions" is not benchmarked against an independent HJB solution for the 121-dimensional case. The comparisons in Figs. 4.3 and 4.8 are against LQR and uncontrolled trajectories; they demonstrate that the computed law stabilizes the sampled initial state and gives lower cost than LQR, but they do not quantify the error in the value function or the feedback law. For the one-dimensional problem a dimension-refinement cost error is shown (J_d − J_64 ∼ d^{−2.3}), but no analogous indicator is provided for the two-dimensional d=121 run. I suggest adding a residual-based error indicator, a refinement study in (n,d), or a test with a manufactured or known solution to support the accuracy claim at d=121.
[Sec. 3.3, Algorithm 3.2] Algorithm 3.2 relies on heuristic choices of the shift μ and its reduction factor q, and the manuscript provides no convergence analysis for the shifted AMEn iteration on the nonsymmetric, degenerate matrix (3.3). The argument that the transition matrix μ(A+μI)^{-1} has spectral radius less than 1 assumes Re λ(A) ≥ 0, while the text then acknowledges that the eigenvalues are only in the right half-plane "for a suitable choice" of domain and polynomial order and that larger shifts or domain sizes can make the stiffness matrix indefinite (Sec. 4.2). Since the policy iteration and the linear solver are not separated in the reported CPU times, the reliability of the method in the advertised high-dimensional regime depends on this heuristic. Please state precise conditions under which Algorithm 3.2 converges, or reformulate the algorithm's guarantees as numerical observation.

minor comments (5)

[Eq. (2.7)] The notation P^{-1}(µ) is ambiguous; from the surrounding text it is the inverse function of P, not the reciprocal, and this should be stated explicitly.
[Algorithm 3.2, line 3] The phrase "optionally µ := µq" is unclear about when the shift is reduced relative to the policy iteration loop; a precise schedule (e.g., every outer iteration after the first) would help reproducibility.
[Remark 3.3 and Fig. 4.2] The statement that TT ranks "grow very mildly" for one-dimensional PDEs is hard to reconcile with Fig. 4.2, which shows a roughly linear growth in d; "mildly" should be quantified (e.g., linear with a small slope) to avoid confusion.
[Sec. 2] The control set U is introduced as a compact subset of R and then the unconstrained case U ≡ R is used; the transition between the two settings should be explicit.
[Fig. 4.2 and Sec. 4.1] The text reports "time ∼ d^4", whereas the complexity estimate O(d n^2 r^4) with r ∼ d would give O(d^5); the discrepancy is attributed to non-uniform ranks and is a useful observation, but it should be stated as an empirical fit rather than a complexity bound.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation chain: the LQ rank bound is proved from independent tensor results and the nonlinear rank growth is reported as numerical observation, not as a fitted prediction.

full rationale

The main derivation chain is self-contained. The TT-rank bound in Theorem 3.1 is proved for quadratic value functions using the Riccati/Lyapunov structure and external tensor results, namely [20, Thm. 4.2], [28, Thm. 9], and [28, Lemma 16]; the value function is not defined in terms of the TT rank, and the rank bound is not fitted to match a desired output. The policy iteration in Algorithm 2.1 solves the linearized HJB equation by Galerkin residual projection, so the control update is computed from the current value iterate, and the closed-loop cost comparisons with LQR are downstream evaluations of the resulting feedback map, not restatements of the input data. For the nonlinear Allen-Cahn and Fokker-Planck examples, the polynomial scaling claim rests on numerically observed TT rank growth in Figs. 4.2, 4.7, and 4.11; the authors explicitly state in the Conclusion that identification of the low-rank class is open, and Remark 3.3 limits Theorem 3.1 to linear systems, so the empirical rank behavior is not disguised as a proven result. The paper contains self-citations, for example [20] and [39], but they support standard tensor rank facts and a previously published polynomial approximation baseline; they do not import the TT-rank ansatz or the target accuracy claims. Algorithmic parameters such as domain size, polynomial degree, shift mu, and tolerance delta are tuning choices and are studied by sensitivity experiments, not fitted constants used to manufacture agreement. A potential steep growth of TT ranks for other nonlinearities would threaten the robustness of the 100-dimensional feasibility claim, but that is a correctness risk, not a circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The method's performance rests on standard PDE and tensor-analysis assumptions plus an empirically observed low-rank property for nonlinear value functions. The only genuinely unproven load-bearing premise is the small TT rank of nonlinear value functions, which the authors themselves identify as an open question. Algorithmic parameters such as polynomial degree, threshold, shift, and domain size are tuning choices that affect but do not define the scientific claim.

free parameters (4)

Legendre polynomial degree n = n=5 (Allen-Cahn and Fokker-Planck); n=3-7 swept in Fig. 4.4
Chosen by hand; larger n increases accuracy and TT ranks, and the observed algebraic error decay justifies n=5 for the reported runs.
TT approximation threshold delta = 10^-3 for Allen-Cahn, 10^-4 for Fokker-Planck
Controls all TT truncations and AMEn stopping; the paper shows cost errors eventually track delta.
Initial shift mu and reduction factor q in shifted AMEn = mu=50 (Allen-Cahn), mu=5 (Fokker-Planck), q=0.98
Heuristic stabilization parameters; no theory fixes their values, and convergence depends on choosing them suitably.
State domain half-width a = a=3 for Allen-Cahn, a=20 for the right-sided Fokker-Planck example
Chosen large enough to contain trajectories; the paper notes larger a can make the stiffness matrix indefinite and cause policy iteration to fail.

assumptions (6)

standard math Equation (2.4) is the unique viscosity solution characterization of the value function.
Used to justify solving the HJB PDE rather than the original control problem.
domain assumption Continuous policy iteration converges for admissible initial feedback and bounded domains.
Algorithm 2.1 requires J(u0,x)<infinity; convergence is cited from [8,9,54,3], not proved here.
domain assumption The value function is sufficiently smooth for Legendre spectral accuracy.
Relies on C-infinity or C-1 regularity results [15,44]; if the value function is nonsmooth, the fast convergence in n fails.
domain assumption The value function has small TT ranks for the nonlinear test problems.
Empirically observed in Figs. 4.2, 4.7, and 4.11; proven only for quadratic value functions in Theorem 3.1. The paper flags this as the open core question.
domain assumption Shifted AMEn solves the linearized Galerkin systems to the stated accuracy.
The paper argues spectral radius less than 1 if the real parts of eigenvalues are nonnegative, but for general nonlinear problems the definiteness condition is heuristic and the shift schedule q=0.98 is empirical.
domain assumption Bilinear balanced truncation yields a faithful reduced model for the Fokker-Planck example.
The paper explicitly notes that [10] gives no a priori error bound for bilinear model reduction; accuracy is checked only via cost differences.

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Pith. "Pith review of Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations." pith.science (2026). https://pith.science/paper/UA2AGO6L

@misc{pith2026190801533,
  author       = {Pith},
  title        = {Pith review of: Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UA2AGO6L}},
  note         = {Machine review of arXiv:1908.01533}
}

read the original abstract

A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton-like iterative method for the solution of the resulting nonlinear system. The tensor approximation leads to a polynomial scaling with respect to the dimension, partially circumventing the curse of dimensionality. A convergence analysis for the linear-quadratic case is presented. For nonlinear dynamics, the effectiveness of the high-dimensional control synthesis method is assessed in the optimal feedback stabilization of the Allen-Cahn and Fokker-Planck equations with a hundred of variables.

Figures

Figures reproduced from arXiv: 1908.01533 by the authors.

**Figure 4.** (left), resulting from a larger condition number of the linearised system. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 4.** [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]

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