Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent

Fengmiao Bian; Jian-Feng Cai; Xiaoqun Zhang; Yuanwei Zhang

REVIEW 2 major objections 6 minor 43 references

Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that a data-driven preconditioner makes Riemannian gradient descent provably linear for low TT-rank tensor completion, with a contraction constant independent of the target tensor's condition number, and that the method…

desk verdict A simple, useful preconditioner for TT completion, but the main theorem's sample bound is stated in terms of undefined iteration-dependent quantities and the paper skips a comparison with the prior preconditioned-TT method. read the letter →

arxiv 2501.13385 v1 pith:WWXL6OV2 submitted 2025-01-23 cs.LG cs.NAmath.NAmath.OC

Fengmiao Bian , Jian-Feng Cai , Xiaoqun Zhang , Yuanwei Zhang This is my paper · ORCID

classification cs.LGcs.NAmath.NAmath.OC MSC 15A2315A8368Q2590C2690C53

keywords low-ranktensorcompletiontraindecompositionpreconditionedRiemanniangradientdescentlinearconvergencequantumstatetomographyhyperspectralimagedata-drivenmetricTT-rank

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 aims to show that a simple data-driven preconditioner turns Riemannian gradient descent into a provably fast algorithm for completing tensors known to have low tensor-train (TT) rank. The proposed algorithm, PRGD, weighs each iterate's gradient by the row norms of its mode-wise unfoldings, defines a new metric on the tangent space, and then takes the usual retraction by TT-SVD. The central theoretical claim is linear convergence: under a good initialization and enough random samples, the squared error contracts by at most $0.3574$ per step, with a rate that does not depend on the condition number of the target tensor. If true, this matters because tensor completion is used in quantum state tomography and image processing, and speedups of this size make the method practical at larger scales.

What carries the argument

The load-bearing object is the data-driven weighted inner product $\langle X, Y\rangle_{W_l} = \langle W_l X, Y\rangle$, with $W_l$ acting by scaling each mode-$i$ slice of a tensor by $G_{l,i}^{1/(2m)}$, where $G_{l,i} = \varepsilon_l I + \operatorname{diag}(M_i(G_l) M_i(G_l)^\top)$ and $G_l = P_\Omega(T_l - T^*)$. This makes the Riemannian gradient equal to $P_{T_l}(W_l^{-1} G_l)$, and the paper proves the projection factors as $W_l^{-1/2} P_{\hat T_l} W_l^{1/2}$; hence the preconditioned update is almost RGD run on the weighted tensor $\hat T_l = W_l^{1/2} T_l$. The diagonal scaling is cheap to apply ($O(m|\Omega|)$ for the sparse gradient) and does not change TT-rank, so the TT-SVD retraction keeps the same cost as in RGD. This mechanism is what removes the condition-number dependence from the contraction rate.

What would settle it

Run PRGD on a synthetic TT-rank tensor with a large condition number, using the theorem's sampling and initialization, and record the ratios $\|T_l - T^*\|_F^2 / \|T_{l-1} - T^*\|_F^2$; a single ratio above $0.3574$ contradicts the claimed contraction. Also monitor $\max_{i,k} \|M_i(G_l)(k,:)\|_2$: if this quantity grows with $l$, the fixed sample-size bound cannot be maintained and the proof's premise fails.

Watch

Extended reading notes

Core claim

The paper's discovery is that a preconditioned metric built from the current gradient makes the Riemannian gradient descent iterate on the fixed TT-rank manifold behave like ordinary RGD on a weighted tensor, and that this weighted view yields a clean contraction bound. The proof shows that with initial error bounded by $\sigma/(C_m \kappa_0 \bar{r}^{1/2})$, incoherence bounded by $2\kappa_0^2 \nu$, and sample size $n$ at least a polylogarithmic factor times the stated rank-dependent expression, Algorithm 2 produces iterates satisfying $\|T_l - T^*\|_F^2 \le 0.3574\|T_{l-1} - T^*\|_F^2$ for every $l$. The constant $0.3574$ is independent of the condition number $\kappa(T^*)$, which is the main improvement over standard RGD. The numerical sections report that PRGD with an optimal constant step size is faster than RGD by two to three orders of magnitude on synthetic tensors, and that it reaches higher PSNR on hyperspectral images and far fewer iterations in quantum state tomography.

Load-bearing premise

The theorem's sample-size condition is written in terms of $\rho_l$ and $\varepsilon_l$, quantities that depend on the current gradient and are only fixed later in the proof ($\varepsilon_l = \|G_l\|_\vee^2$), so the argument assumes these data-dependent quantities stay uniformly controlled at every iteration; if the gradient's largest row norm grows, the required sample size grows and the fixed linear rate no longer follows.

Editorial extensions

If this is right

PRGD converges linearly to the target tensor with a per-step squared-error contraction of at most $0.3574$, under the initialization and sample-size conditions of Theorem 4.1.
The contraction rate is independent of the condition number of the target tensor, so ill-conditioned TT tensors do not slow the asymptotic convergence.
The preconditioner adds only $O(m|\Omega|)$ work per iteration for the sparse gradient scaling and does not change the TT-rank, so the per-iteration cost stays comparable to RGD.
On synthetic data PRGD with optimal constant step size reduces computation time by two orders of magnitude (and up to three at larger sizes) compared with standard RGD.
In hyperspectral image completion and quantum state tomography, PRGD reaches the same recovery quality with substantially fewer iterations and less wall-clock time.

Reading between the lines

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

A natural extension, not pursued in the paper, is to make the sample-size condition iteration-independent by proving uniform bounds on $\|G_l\|_\vee$; without such a bound, the theorem's $n$ is a per-iteration requirement rather than a single upfront sample budget.
The observation that a constant step size beats the adaptive steepest step suggests the preconditioner changes the local geometry in a way that makes exact line search over-optimistic; the same phenomenon should be testable in the matrix PRGD of the cited work.
The same diagonal row-norm preconditioning could be applied to other tensor formats, such as Tucker or CP, where the manifold geometry and retraction differ but the row-norm scaling is format-agnostic.
Experimental validation of the $0.3574$ constant is missing; reporting per-iteration error ratios on ill-conditioned tensors would test the rate directly rather than only final recovery quality.

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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

Colleague, here's my take on 2501.13385.

The paper has a genuine idea: a data-driven diagonal preconditioner for TT-format tensor completion, built from row norms of the mode-unfolded gradient, and a proof that the resulting PRGD inherits RGD's linear convergence with a rate independent of the condition number. The preconditioner is simple, the empirical gains on synthetic problems are large, and the proof is extensive, if heavily indebted to the machinery of [5] and [11]. This is a real contribution, and the core mechanism—weighting each mode-unfolded row by its norm—is natural and worth testing.

The soft spots are in proportion. The central theorem, Theorem 4.1, states a sample-size condition in terms of rho_l and epsilon_l without defining them; only in Appendix B.2 do the authors set epsilon_l = ||G_l||^2_v, which makes rho_l = sqrt(2) and the bound uniform. As stated, Algorithm 2 leaves epsilon as a free parameter, so a user running the algorithm with fixed epsilon has no guarantee that the required n is bounded across iterations. This is a load-bearing gap in the theorem's statement, though it looks repairable by stating the choice of epsilon_l and replacing rho_l with its uniform bound. The paper also does not compare with [22], which already treats TT completion under a preconditioned metric; the authors cite it but never delineate the difference. No code or error bars are provided, and the QST section shows the CPU-time advantage is much smaller than the abstract implies—indeed the adaptive-step version becomes the slowest. The conclusion's "two orders of magnitude" claim is based on the best synthetic cases, not the general behavior.

The proof itself is long and mostly follows [5]; the new analysis is the handling of the data-dependent norm in I2 and I3. I did not find a serious error there, but the theorem statement must be fixed before the result is checkable.

Who is this for? People working on tensor completion or Riemannian optimization on TT manifolds will find the preconditioner useful and the theory close to what they need. It deserves a serious referee: the idea is valuable, the gap is technical, and the missing comparison to [22] is addressable. I would send it to review with a request for revision, not desk-reject it.

Referee Report

2 major / 6 minor

Summary. The paper proposes a preconditioned Riemannian gradient descent algorithm (PRGD) for low-rank tensor completion in the tensor-train (TT) format. The preconditioner is derived from a data-dependent weighted inner product built from the row norms of the current gradient, and the algorithm uses TT-SVD as a retraction together with a trimming step. The main theoretical claim is Theorem 4.1: under a sample-size condition and a sufficiently good initialization, PRGD converges linearly to the target tensor with a contraction factor 0.3574 that is independent of the condition number of the target. Numerical experiments on synthetic tensors, hyperspectral images, and quantum state tomography report substantial reductions in iterations and CPU time relative to the RGD algorithm of [5].

Significance. If the main theorem is correct, the result is significant: it gives a provable, condition-number-independent linear convergence rate for TT-format tensor completion with a computationally cheap preconditioner, and the experiments indicate large practical speedups. The paper builds on the established machinery of [5], provides an explicit algorithm with a detailed proof structure, and evaluates the method on realistic applications including quantum state tomography. However, the central theorem, as stated, has a load-bearing well-posedness gap: its sample-complexity condition depends on quantities that are not defined in the theorem and are data-dependent, so the guarantee cannot be checked as written. This is repairable, but it must be fixed before the main claim can be accepted.

major comments (2)

[Theorem 4.1, Lemmas 4.2–4.4, Appendix B.2] Theorem 4.1 states the sample-size condition in terms of ρ_l and ε_l, but neither is defined in Section 4. Lemmas 4.3–4.4 use ρ_l, and the proof of Lemma 4.3 in Appendix A shows that ρ_l = μ_l/ν_l = (( ε_l + ‖G_l‖²_∨ ) / ε_l)^{1/2}. Appendix B.2 later chooses ε_l = ‖G_l‖²_∨, which gives ρ_l = √2, but this choice is not part of the theorem statement or of Algorithm 2. Since n is fixed in advance while ρ_l and ε_l are iteration- and gradient-dependent, the displayed sample-size condition cannot be checked as written, and the estimates (B.16) and (B.22) require a uniform upper bound on ‖G_l‖_∨ that is never established. The theorem must either state the choice ε_l = ‖G_l‖²_∨, define ρ_l explicitly, and supply such a uniform bound, or state the sample complexity with ρ_l replaced by its worst-case bound.
[Algorithm 2, Eq. (3.2), Eq. (4.3), Appendix B.2] Algorithm 2 specifies the metric in Eq. (3.2) with a fixed parameter ε, whereas the proof of Theorem 4.1 (Appendix B.2, before (B.16)) sets ε_l = ‖G_l‖²_∨ and the theorem’s step size is α_l = 1.001 ε_l^{1/2} p^{-1}. These are different specifications: with a fixed positive ε the sample-complexity bound still involves ρ_l = (1 + ‖G_l‖²_∨/ε)^{1/2}, so a uniform gradient-norm bound is needed, while with ε_l = ‖G_l‖²_∨ the metric degenerates when G_l = 0 and the division by ν_l = ε_l^{1/2} in (4.3) is not justified without a limiting or safeguarding argument. The algorithm, the theorem, and the proof must be aligned, and the precise dependence of ε_l on G_l must be stated.

minor comments (6)

[Section 2.1] The definition of the tensor condition number is garbled: it reads κ(T) := σ(T)/σ(T) := max / min, which should presumably be σ_max(T)/σ_min(T).
[Theorem 4.1] The theorem uses r = (r_1, ..., r_m) and the word “spkiness”, but the TT-rank should be (r_1, ..., r_{m-1}) and the intended word is “spikiness”. Also, the quantities Spiki(T*), Incoh(T0), σ, and κ_0 are used without being defined in the main text; σ and κ_0 only appear later in Appendix A.
[Algorithm 2] The symbol W_l is overloaded: it denotes the metric operator in Section 3.2.1 and also the tensor W_l = T_l - α_l fP_{T_l} W_l^{-1} G_l inside Algorithm 2. This makes the pseudocode confusing and should be fixed.
[Lemma 4.2 proof] In the proof of Lemma 4.2, the inequality ε_l < |G_{i,l}(j,k)| < ε_l + ‖G_l‖²_∨ refers to diagonal entries of a diagonal matrix and appears to have a square/exponent typo; it should be stated as a bound on the diagonal entries of G_{l,i}.
[Section 5] The numerical comparisons are only against the RGD algorithm from [5]; since the paper cites the quotient-geometry preconditioned method [22] as closely related, a comparison with that method or another recent TT-completion algorithm would strengthen the empirical claims.
[Title] The title contains a typo: “Precondtioned” should be “Preconditioned”.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the preconditioner is built from observed residuals, the rate proof uses independently published lemmas, and the undefined ρ_l/ε_l condition is a rigor gap rather than a self-referential reduction.

full rationale

The derivation chain is not circular. The preconditioner W_l in Eqs. (3.2)–(3.3) is constructed from G_l = P_Ω(T_l − T*); its restriction to Ω is exactly the observed data, so no target quantity is fitted to force the theorem true. Theorem 4.1's contraction bound is obtained by bounding I1, I2, and I3 in Eq. (4.2) using the events E_1–E_5 and Lemmas A.1–A.6, which are quoted from the independently published paper [5]; although a coauthor of the present paper is an author of [5], those lemmas are parameter-free facts with stated assumptions that do not include the target convergence and are not assumptions invented for this paper. The preconditioned scheme is motivated by [11], but the new-left-orthogonal decomposition, tangent-space parameterization, and projection formulas (Lemmas 3.3–3.5) are proved in this manuscript rather than imported by citation. There is no fitted-input-called-prediction: the step size α_l = 1.001 ε_l^{1/2} p^{-1} and the sample-size condition are stated in the theorem before the proof, and the proof later fixes ε_l = ||G_l||^2_∨ in Appendix B.2. The main concern is a well-posedness gap, not circularity: ρ_l and ε_l are not defined in Theorem 4.1, and the sample-size condition is therefore iteration-dependent as written, with no quantified sup_l ρ_l in the theorem statement. Since ε_l = ||G_l||^2_∨ would make ρ_l = sqrt(1 + ||G_l||^2_∨/ε_l) = sqrt(2), the condition could be made uniform, but the theorem does not state that choice. No uniqueness theorem from the authors' prior work is invoked to forbid alternatives, and no known empirical pattern is merely renamed in new coordinates. Hence the central claim has independent content and the observed issues are rigor/presentation concerns rather than circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard Riemannian geometry for fixed TT-rank tensors, sampling concentration events inherited from [5], and incoherence/spikiness bounds on the target and iterates. The algorithm introduces a data-dependent metric and trimming step, but no new physical or mathematical entities are postulated. The main unstated burden is the uniform control of rho_l and epsilon_l in the sample complexity, which is neither defined nor proved.

free parameters (3)

epsilon_l (metric regularization) = set to ||G_l||^2_vee in Appendix B; otherwise left unspecified in Theorem 4.1
Appears in G_{l,i} = epsilon_l I + diag(M_i(G_l)M_i(G_l)^T). The convergence proof requires choosing it as a function of the gradient norm, but the theorem states the sample bound in terms of epsilon_l without defining it.
step size alpha_l = 1.001 epsilon_l^{1/2} p^{-1} in theory; 'optimal constant step size' tuned empirically in experiments
The practical speedups reported in Section 5 rely on an empirically chosen constant step size, while the theorem uses a specific data-dependent value. This means the practical gains depend on a tuned parameter.
trimming threshold xi_l = 10 ||W_l||_F / (9 sqrt(d_*) nu)
Introduced in Algorithm 2 for the proof of incoherence preservation. The paper states in Section 3.2.2 that experiments are almost the same without trimming, so it is a proof artifact rather than a data-fitted parameter.

assumptions (5)

standard math The set of fixed TT-rank tensors is an embedded Riemannian manifold.
Used throughout Section 3 to define tangent spaces, projections, and retractions; imported from Holtz-Rohwedder-Schneider [17].
domain assumption The target tensor has bounded spikiness Spiki(T*) <= nu and each iterate maintains incoherence Incoh(T_l) <= sqrt(mu).
Required by Lemmas 4.3 and 4.4 and by the sample complexity. The trimming step is asserted to maintain incoherence, but the incoherence bound mu is not derived from the initialization.
domain assumption The sampling set Omega is drawn uniformly with replacement and the events E1-E5 from [5] hold with high probability.
All concentration inequalities in Appendix A and B assume these events, which are inherited from the RGD analysis in [5] rather than proved in this paper.
standard math TT-SVD is a quasi-optimal retraction satisfying Lemma A.2.
The first inequality in the proof of Theorem 4.1 uses Lemma A.2 (Lemma 26 of [5]) to bound the retraction error by ||W_l - T*||_F^2 plus a cubic term.
standard math The preconditioning matrices G_{l,i} are positive definite, so W_l^{1/2} and the weighted tangent space projection are well-defined.
Positive definiteness holds because epsilon_l > 0 is added to the diagonal, which is stated in Equation (3.2).

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Pith. "Pith review of Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent." pith.science (2026). https://pith.science/paper/WWXL6OV2

@misc{pith2026250113385,
  author       = {Pith},
  title        = {Pith review of: Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWXL6OV2}},
  note         = {Machine review of arXiv:2501.13385}
}

read the original abstract

Low-rank tensor completion aims to recover a tensor from partially observed entries, and it is widely applicable in fields such as quantum computing and image processing. Due to the significant advantages of the tensor train (TT) format in handling structured high-order tensors, this paper investigates the low-rank tensor completion problem based on the TT-format. We proposed a preconditioned Riemannian gradient descent algorithm (PRGD) to solve low TT-rank tensor completion and establish its linear convergence. Experimental results on both simulated and real datasets demonstrate the effectiveness of the PRGD algorithm. On the simulated dataset, the PRGD algorithm reduced the computation time by two orders of magnitude compared to existing classical algorithms. In practical applications such as hyperspectral image completion and quantum state tomography, the PRGD algorithm significantly reduced the number of iterations, thereby substantially reducing the computational time.

Figures

Figures reproduced from arXiv: 2501.13385 by the authors.

**Figure 1.** Illustration of scaling operation on Gl ’s elements. Utilizing TT-SVD as the retraction operator, the pseudocode of PRGD is proposed in Algorithm 2. Algorithm 2: Preconditioned Riemannian Gradient Descent Initialization T0 ∈ Mtt r , spikiness parameter ν and maximum iteration numbers lmax. for l = 0, 1, . . . , lmax do Gl = PΩ(Tl − T ∗ ). Compute stepsize αl . Wl = Tl − αlPfTlW −1 l Gl . Set Wl = Trimξl (Wl) with ξl… view at source ↗

**Figure 2.** Results of CPU-time (in seconds) and the number of iterations for TT-rank tensor completion [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Results of CPU-time (in seconds) and the number of iterations for TT-rank tensor completion [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Results of CPU-time (in seconds) and the number of iterations for TT-rank tensor completion [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Results of relative error and the number of iterations for TT-rank tensor completion on different [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: RGB representations of hyperspectral images. [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

Discussion (0). Continue with ORCID to comment.

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