Aspects of density approximation by tensor trains

Ji\v{r}\'i Ajgl; Ond\v{r}ej Straka

REVIEW 3 major objections 4 minor 26 references

Aspects of density approximation by tensor trains

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper shows that tensor-train approximations of probability densities built by SVD or cross decomposition carry negative values at many grid points, that correlation inflates ranks in a predictable pattern, and that decorrelation and…

desk verdict A careful, useful cautionary paper on tensor-train density approximation, with a genuine insight about decorrelation raising ranks and a solid analytic core for quadratic functions, but the sweeping 'negative values are practically unavoidable' claim outruns the evidence. read the letter →

arxiv 2505.22218 v2 pith:VLEW4BYM submitted 2025-05-28 eess.SP

Jiří Ajgl , Ondřej Straka This is my paper

classification eess.SP MSC 15A6965F30

keywords tensortrainspoint-massfiltersnegativevalueslow-rankapproximationsingularvaluedecompositioncrossdensitygridinterpolation

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

This paper argues that tensor-train approximations of probability densities, built by the standard cheap and adaptive SVD or cross-decomposition algorithms, unavoidably produce negative values at a substantial fraction of grid points, in some experiments tens of percent. It further establishes that correlation between state variables inflates tensor-train ranks over the interval between the correlated coordinates, with overlapping correlated pairs nearly multiplying ranks, and that decorrelation (whitening) does not reliably reduce ranks. It also shows that interpolating density values from one grid to another inserts noise that caps the achievable approximation precision. A sympathetic reader should care because these findings discipline what point-mass filtering with tensor trains can promise, and point to the specific missing tool: a cheap, adaptive, nonnegative low-rank rounding procedure.

What carries the argument

The central object is the tensor-train decomposition, which stores a $d$-dimensional array as a chain of small matrix products, one matrix-valued 'core' per coordinate, with ranks $r_j$ controlling storage and cost. The paper contrasts two ways of building the cores: SVD truncation, which is optimal in Frobenius norm but must touch every array entry, and cross/pseudoskeleton decomposition, which evaluates only selected rows and columns. The sign mechanism that carries the negative-value claim is that a truncated SVD update of a nonnegative matrix is built from orthogonal singular vectors, and all but the first must change sign; in the cross decomposition, the inverse of a symmetric two-by-two submatrix with equal diagonal entries introduces differences of rows or columns, again forcing sign changes. For the rank analysis, the paper uses an exact functional tensor train of the quadratic form $x^TQx$, factorised into coordinate-dependent matrices, whose nonzero off-diagonal blocks directly determine which ranks grow.

What would settle it

Repeat the Section IV experiment on a multimodal or heavy-tailed density using the same whitening and interpolation procedures; if for some such density the tensor-train ranks in the decorrelated coordinates are lower than in the original coordinates, or if a different interpolation scheme makes the singular values decay as fast as exact sampling, then the paper's negative conclusions about decorrelation and interpolation would not generalise.

Watch

Extended reading notes

Core claim

On its own terms, the paper makes three claims. First, for a nonnegative density sampled on a grid, the SVD-based and cross-based low-rank tensor-train approximations are contaminated by negative values at many grid points; the mechanism is that a rank-1 approximation of a nonnegative matrix is nonnegative, so every correction supplied by later singular vectors or by the inverse submatrix in cross decomposition must carry sign changes and hence creates chequered negative regions, whose area shrinks only slowly with rank. Second, in exact functional tensor trains of a quadratic form $x^TQx$, each nonzero off-diagonal entry $Q_{k,l}$ raises the ranks $r_k,\ldots,r_{l-1}$, and overlapping correlated pairs interact almost multiplicatively; the same rank pattern appears for discretised Gaussian densities with correlation coefficients of 0.5. Third, transforming to decorrelated coordinates can make the singular-value spectrum decay more slowly rather than faster, and interpolating between grids makes every approximation beyond the first few ranks track interpolation noise rather than the density.

Load-bearing premise

The claim that decorrelation does not help and interpolation caps precision rests on a single radar-like density example, one grid spacing, and one interpolation scheme (linear interpolation with nearest-neighbour extrapolation) assumed to represent general point-mass filter densities.

Editorial extensions

If this is right

A point-mass filter that builds tensor-train cores by SVD or cross decomposition will, at any practical rank, report negative probability mass at a nontrivial fraction of grid points, so downstream uses of the density need sign handling.
Ordering the state variables so that strongly correlated pairs are adjacent keeps the extra ranks confined to a short interval of the tensor-train chain; non-adjacent correlated pairs raise all ranks in between.
Whitening the state before tensor-train approximation is not a safe general preprocessing step; in the paper's radar example it slows singular-value decay, so the rank savings from independence can be wiped out.
Any grid change in a point-mass filter should prefer exact resampling over interpolation; linear interpolation lets high-order tensor-train terms model interpolation noise, capping precision.

Reading between the lines

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

The paper does not state this, but the negative-value result gives a concrete acceptance test for any future tensor-train filter: report the fraction of negative grid points per update, and abort or postprocess when it exceeds a threshold.
A natural next step, beyond the paper, is the variable-ordering problem: finding the state permutation that minimises total tensor-train rank adds an optimisation layer to filter design, suggested by the near-multiplicative rank growth for overlapping correlated pairs.
Testable extension: repeat the interpolation experiment with shape-preserving or monotone interpolation on the same density; the paper's conclusion predicts slow singular-value decay persists, but a monotone scheme might confine the damage to fewer modes.
If a cheap adaptive nonnegative tensor-train algorithm were found, the paper's 'practically unavoidable' conclusion would be overturned; the paper implicitly frames this as the open problem by noting that existing nonnegative methods are expensive and need preselected ranks.

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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

Short version: this is a careful, mostly empirical paper about tensor-train approximations of probability densities, and the two things to remember are (1) the sign-switching explanation of negative values is plausible but not proven, and (2) the demonstration that decorrelation can increase TT ranks is genuinely useful and counterintuitive. The paper deserves a serious referee, but the universal "practically unavoidable" claim needs softening or more evidence.

What's actually new: Section III-B gives exact functional TT cores for quadratic functions and a clean table of how ranks grow with off-diagonal couplings; that's a solid, reproducible analytic contribution. The rank tables for sampled Gaussians in Table III are consistent with the functional analysis. The observation in Section IV that decorrelation via a square-root factor can make the singular spectrum decay slower (so higher ranks) is a real caution for anyone who assumes whitening always helps. The interpolation experiment, showing that linear interpolation into a new grid quickly hits a noise floor, is also a practical warning that point-mass filter designers should heed.

Where the soft spots are: the sign-switching argument in Section III-A is not a proof. A signed rank-one update can add negative entries while the sum with the previous approximation stays nonnegative; the counterexample M = u1 v1^T + ε u2 v2^T with well-chosen signs shows the mechanism alone doesn't force negativity. So the "practically unavoidable" wording in the summary overreaches. The numerical evidence covers one radar-like density, one run of greedy2_cross (which the authors admit is often worse), and Gaussian tensors at one correlation level and precision. That's enough to suggest a tendency, not an inevitability. The interpolation section uses one density and one interpolation scheme, so the conclusion that "any interpolation significantly limits achievable precision" is too broad. No code or data is shipped, so the numbers can't be independently checked.

Bottom line: this is a useful cautionary paper for the grid-based filtering/TT audience. It doesn't resolve a long-open question, but it steers algorithm choices and flags pitfalls. It should go to peer review; the referee should ask for code, more varied examples, and a rephrasing of the universal claims. I'd bring it to our reading group.

Referee Report

3 major / 4 minor

Summary. The paper investigates three issues that arise when tensor-train (TT) decompositions are used to represent probability densities on Cartesian grids in point-mass filters: the appearance of negative values in low-rank SVD and cross decompositions; the relation between correlation of Gaussian components and TT ranks; and the effect of grid interpolation and decorrelation on TT approximability. It presents a functional TT factorization of quadratic forms with explicit cores (Eqs. (7)–(10), Tables I–II), numerical TT rank counts and negative-value percentages for four-dimensional Gaussian densities (Table III), and a two-dimensional radar-density study of SVD and cross approximations and of decorrelation by different matrix square roots (Figs. 1–4). The paper concludes that negative values are practically unavoidable with cheap and adaptive TT techniques, that correlation raises intermediate ranks, and that interpolation significantly limits achievable precision.

Significance. The analytical construction of functional TT cores for quadratic functions is a concrete, parameter-free contribution that could be reused in filtering implementations, and Tables I–II provide a useful catalog of how sparse off-diagonal quadratic terms inflate ranks. The numerical experiments are clearly presented and illustrate the phenomena. If the broad conclusions were supported, the paper would provide a useful caution for TT-based point-mass filtering and motivate research on nonnegative TT rounding. However, the evidence is largely anecdotal: the central claims that negative values are 'practically unavoidable' and that interpolation significantly limits precision rest on one or two examples and one interpolation scheme, and the sign-switching explanation in Section III-A is not logically sufficient. The contribution is therefore more exploratory than definitive.

major comments (3)

[Section III-A, Eq. (6)] The claim that signed rank-one SVD updates 'inevitably contain areas of negative values, which introduces such areas to the density approximations' does not follow from the sign-switching observation. A signed update sigma_i u_i v_i^T can have negative entries that are entirely masked by the positive rank-one baseline; for example, a nonnegative matrix M = u_1 v_1^T + epsilon u_2 v_2^T with positive u_1, v_1 and signed orthonormal u_2, v_2 is its own rank-2 SVD and has no negative entries for sufficiently small epsilon. The sign-switching argument therefore shows only that updates are signed, not that truncated SVD approximations of nonnegative tensors must contain negative values. Since the Section V summary relies on this explanation for 'practically unavoidable', the authors should either provide a dominance or bound argument specific to SVD truncation or substantially broaden the empirical evidence.
[Section IV and Section V] The statement in Section V that 'any interpolation significantly limits the achievable precision' is overgeneralized. The supporting evidence consists of a single radar-like density, one grid spacing, and one interpolation rule (linear interpolation with nearest-neighbor extrapolation), and the paper itself notes that the results depend heavily on the choice of R. To support the universal conclusion, the authors should test several densities, grid resolutions, interpolation methods, and covariance square roots, and report approximation errors as a function of TT rank or precision. Without this, the conclusion should be rephrased as an observation about the tested cases.
[Section III-B, Table III and Section V] The quantitative support for the claims that 'the ratio of negative values can reach tens percent' and that ranks in overlapping parts are 'almost multiplied' consists of one four-dimensional Gaussian density with correlation coefficient 0.5, one grid of 41 points per dimension, one precision parameter of 10^-5, and the two-dimensional radar example. No uncertainty quantification, parameter sweep, or sensitivity analysis is reported. If these are intended as general statements, the experiments should cover a range of correlations, grid sizes, precisions, and density shapes; otherwise the claims should be explicitly qualified as case-specific.

minor comments (4)

[Section III-A] The sentence 'It is therefore considerably slower' is ambiguous; please specify whether the cross iteration is slower to converge, slower per rank increase, or computationally slower, and support the statement with quantitative evidence.
[Section III-B, Eq. (8)] The case conditions for \bar{g}_j, such as '1<j<d+1/2' and '(d-1)/2<j<d', are misprinted as fractions that overlap for some d; please use explicit floor or ceiling conditions or separate the parity cases as in Eqs. (9) and (10).
[Section III-B, Table III] The phrase 'the ranks in the overlapping parts are almost multiplied' is vague; Table III shows one product of ranks yielding 100 and a rounded rank of 55, so 'almost multiplied' should be quantified or replaced by the observed values.
[Section III-A, Figs. 1 and 2] The statement that 'the size of the areas does not approach zero for low ranks' would benefit from a quantitative measure, such as the fraction of grid points with negative values as a function of rank, rather than visual inspection alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's claims are supported by worked examples and constructive formulas, not by fitted parameters or load-bearing self-citation.

full rationale

This paper performs numerical and analytical explorations rather than deriving predictions from fitted parameters. The central observations—that SVD and cross tensor-train approximations of positive densities can have widespread negative entries, that correlation raises TT ranks between correlated coordinates, and that interpolation degrades the singular-value spectrum—are supported by worked examples (Section III-A, Tables II–III, Figures 1–4) and by constructive rank formulas for quadratic functions. No quantity reported as a 'prediction' is fitted from the data it is compared against. The self-citations ([14], [19], [25]) are contextual or point to the authors' earlier constructions of functional TT decompositions, but the relevant construction is re-derived in Section III-B (Eqs. (7)–(10), Table I), so the citations are not load-bearing for the conclusions. The verbal summary in Section V that negative values are 'practically unavoidable' is a broad induction from the examples and from the contrast with expensive non-negative decompositions; even if the proof sketch in Section III-A is incomplete (sign-switching of later singular vectors does not by itself force the truncated sum to be negative), that is a strength-of-evidence issue, not circularity.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; the central claims are qualitative. The paper relies on standard linear algebra and prior tensor-train constructions, and its numerical conclusions assume the chosen grids, precision, and interpolation scheme are representative.

free parameters (2)

tt_tensor precision parameter = 10^{-5}
Chosen tolerance for the TT-SVD rounding in Section III-B; it affects the reported numerical ranks but not the qualitative pattern.
correlation coefficient = 0.5
Chosen value for all nonzero off-diagonal elements of the covariance matrix in the Gaussian experiments; determines the specific rank counts but not the general conclusion.

assumptions (4)

standard math The leading singular vectors of a non-negative matrix can be chosen non-negative or non-positive, so all later SVD correction terms switch sign.
Used in Section III-A to argue that negative regions in SVD-based approximations are inevitable; a standard Perron-Frobenius-type result, but not explicitly proved in the paper.
domain assumption Equidistant regular grids are representative of point-mass filter density representations.
All numerical examples use equidistant grids, and the discussion of grid placement assumes this is the relevant setting for filters.
standard math The functional tensor-train construction for quadratic functions in Equations (8)-(10) is exact and covers all sparse patterns of off-diagonal entries.
The cores are derived from the block factorization in Equation (7) and are stated without full proof; the rank pattern in Table II depends on this construction.
domain assumption Linear interpolation with nearest-neighbor extrapolation is representative of interpolation between grids in point-mass filters.
Section IV draws conclusions about interpolation limiting precision from this single scheme; no other interpolation method is tested.

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Cite this review

Pith. "Pith review of Aspects of density approximation by tensor trains." pith.science (2026). https://pith.science/paper/VLEW4BYM

@misc{pith2026250522218,
  author       = {Pith},
  title        = {Pith review of: Aspects of density approximation by tensor trains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLEW4BYM}},
  note         = {Machine review of arXiv:2505.22218}
}

read the original abstract

Point-mass filters solve Bayesian recursive relations by approximating probability density functions of a system state over grids of discrete points. The approach suffers from the curse of dimensionality. The exponential increase of the number of the grid points can be mitigated by application of low-rank approximations of multidimensional arrays. Tensor train decompositions represent individual values by the product of matrices. This paper focuses on selected issues that are substantial in state estimation. Namely, the contamination of the density approximations by negative values is discussed first. Functional decompositions of quadratic functions are compared with decompositions of discretised Gaussian densities next. In particular, the connection of correlation with tensor train ranks is explored. Last, the consequences of interpolating the density values from one grid to a new grid are analysed.

Figures

Figures reproduced from arXiv: 2505.22218 by the authors.

**Figure 2.** One run of the cross approximation (4). The meaning corresponds to Fig. 1. The dashed lines in the top-row figures show the grid points with exact [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Interpolation between grids. Top left figure: the function (contour [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Other choices of square roots R of Q. The meaning corresponds to [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

S ¨arkk¨a and L

S. S ¨arkk¨a and L. Svensson, Bayesian Filtering and Smoothing , 2nd ed. Cambride University Press, 2023

work page 2023

[2] [2]

Particle filter theory and practice with positioning ap- plications,

F. Gustafsson, “Particle filter theory and practice with positioning ap- plications,” IEEE Aerospace and Electronic Systems Magazine , vol. 25, no. 7, pp. 53–82, July 2010

work page 2010

[3] [3]

Particle filtering: The first 25 years and beyond,

S. Godsill, “Particle filtering: The first 25 years and beyond,” in 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, UK, May 2019, pp. 7760–7764

work page 2019

[4] [4]

Advanced point-mass method for nonlinear state estimation,

M. ˇSimandl, J. Kr ´alovec, and T. S ¨oderstr¨om, “Advanced point-mass method for nonlinear state estimation,” Automatica, vol. 42, no. 7, pp. 1135–1145, July 2006

work page 2006

[5] [5]

Design of Rao-Blackwellized point-mass filter with application in terrain aided navigation,

J. Dun ´ık, M. Sot ´ak, M. Vesel ´y, O. Straka, and W. Hawkinson, “Design of Rao-Blackwellized point-mass filter with application in terrain aided navigation,” IEEE Transactions on Aerospace and Electronic Systems , vol. 55, no. 1, pp. 251–272, February 2019

work page 2019

[6] [6]

Design of efficient point-mass filter with terrain aided navigation illustration,

J. Matou ˇsek, J. Dun´ık, and M. Brandner, “Design of efficient point-mass filter with terrain aided navigation illustration,” in 26th International Conference on Information Fusion , Charleston, SC, USA, June 2023

work page 2023

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Lagrangian grid-based filters with application to terrain-aided navigation,

J. Matou ˇsek, J. Dun ´ık, and O. Straka, “Lagrangian grid-based filters with application to terrain-aided navigation,” IEEE Signal Processing Magazine, 2025

work page 2025

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Introduction to tensor decompositions and their applications in machine learning,

S. Rabanser, O. Shchur, and S. G ¨unnemann, “Introduction to tensor decompositions and their applications in machine learning,” ArXiv, no. abs/1711.10781, 2017

arXiv 2017