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REVIEW 3 major objections 7 minor 72 references

Fisher Information, Training and Bias in Fourier Regression Models

T0 review · 3 major / 7 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper shows that whether a higher effective dimension improves training depends on whether the model's function space contains the target: for biased models lower effective dimension trains better, while for unbiased models higher effec

desk verdict The analytical FIM–effective-dimension results are solid and worth keeping; the headline ED-bias empirical claim is a genuine but narrow counterexample, and the biased-case construction partly builds in the answer. read the letter →

arxiv 2510.06945 v2 pith:M2MNKBFK submitted 2025-10-08 cs.LG cond-mat.dis-nnphysics.data-anquant-ph

Lorenzo Pastori , Veronika Eyring , Mierk Schwabe This is my paper · ORCID
classification cs.LGcond-mat.dis-nnphysics.data-anquant-ph
keywords FisherinformationmatrixeffectivedimensionFourierregressionquantumneuralnetworksmodelbiasstructureconstantstensortrainability
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 a model's effective dimension, computed from the Fisher information matrix, does not uniformly predict training performance. For Fourier regression models that are biased toward the target function—meaning the target lies inside the space of functions the model can represent—a lower effective dimension leads to faster, better training. For unbiased models, whose function space does not contain the target, a higher effective dimension is more likely to help. The authors support this by deriving an analytic expression for the Fisher information matrix in terms of the model's structure constants and show that effective dimension is controlled by the decay of the singular values of these constants.

What carries the argument

The central object is the structure-constant matrix Γ, which expresses the model output as a bilinear expansion in input basis functions e_μ(x) and parameter basis functions ι_ν(θ). Its singular value decomposition Γ = U S V^T yields the correlation spectrum—the singular values s_ρ—which the paper shows controls the normalized Fisher information matrix and hence the effective dimension. The argument's load-bearing construction is the biased/unbiased dichotomy: biased models share the singular vectors U(d), V(d) used to generate the target function, and cutoff models impose exponential decay on the tail singular values to lower effective dimension.

What would settle it

Construct a biased model where the target function spans the full D-dimensional input basis (not just an R-dimensional subspace) and give both high- and low-effective-dimension versions access to that full space; if the low-effective-dimension model still trains better, the paper's mechanism of constrained exploration is not the whole story. Alternatively, in an unbiased setting with a target that is random but has a strong Fourier component along the low-effective-dimension model's subspace, check whether low effective dimension occasionally wins.

Watch

Extended reading notes

Core claim

Working in a parameterized Fourier regression framework that includes quantum neural networks, the paper shows that effective dimension is not an unconditional proxy for trainability. It proves that the Fisher information matrix spectrum, and therefore effective dimension, is governed by the singular-value decay (the 'correlation spectrum') of the structure-constant matrix that connects input basis functions to parameter basis functions. Using this, the authors construct pairs of models with high and low effective dimension that either contain the data-generating function in their hypothesis space ('biased') or do not ('unbiased'). Across numerical experiments with gradient descent, lower-ef

Load-bearing premise

The conclusion that low effective dimension helps biased models rests on defining 'bias' as exact containment of the target in the model's space, and on generating the target from the same low-rank singular subspace that the low-effective-dimension model retains; if bias is defined differently (approximate alignment, full-rank target), the direction of the effective-dimension effect is not established.

Editorial extensions

If this is right

  • High effective dimension is not universally a sign of better trainability; its effect depends on model-task alignment.
  • For quantum neural networks, if the ansatz is known to be biased toward the target, reducing effective dimension via spectral decay could accelerate training.
  • Effective dimension and bias can be controlled independently in Fourier models, enabling principled comparisons of capacity metrics.
  • The tensorized representation extends these experiments to larger feature/parameter counts, pointing toward scalable diagnosis of QNN trainability.

Reading between the lines

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

  • The paper's construction of 'bias' places the target function inside the low-effective-dimension model's accessible subspace by design; the reported advantage of low effective dimension in the biased case may partly reflect this experimental alignment rather than a general geometric law.
  • A more general notion of bias—such as spectral bias or approximate alignment—could change the direction of the effect; the paper's all-or-nothing definition does not cover partially misaligned targets that still have strong overlap with the model's subspace.
  • For practitioners, this suggests that capacity metrics like effective dimension should always be interpreted relative to the target; a truly task-independent evaluation metric may indeed be impossible, as the paper hints.
  • A testable extension: vary the overlap between the model's singular subspace and the target's Fourier support beyond the all-or-nothing biased/unbiased cases, and measure where the effective-dimension effect reverses.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper studies the relationship between Fisher-information-based effective dimension (ED) and training performance for Fourier regression models, motivated by quantum neural networks. The authors derive an analytic expression for the FIM of models that are linear in a product basis, show that the FIM spectrum is controlled by the decay of the singular-value spectrum of the structure constants (the purity tr(S^4)), and use this to construct full and cutoff models with high and low ED. They define a model as biased towards a target y when y is exactly representable at some parameter value θ*, and unbiased otherwise. Numerical experiments compare gradient-descent training of full vs cutoff models; they report that for biased targets the low-ED cutoff model attains lower minimum MSE, whereas for unbiased targets the high-ED full model is better. A tensor-network representation is introduced to scale the experiments to more features and parameters.

Significance. The analytical part is a useful contribution: Eq. (14) together with Properties 1 and 2 provide a direct, and apparently novel, connection between the structure constants of Fourier regression models and the FIM/ED, and the tensor-network construction may be independently useful for the analysis of QNNs. The empirical finding, if it generalizes, would be a counterexample to the blanket statement that higher ED always improves training. However, the generality of the claim is currently tied to a narrow operationalization of bias. Because the biased target is placed in the exact R-dimensional subspace preserved by the low-ED cutoff models, the experiments establish the claim only for the exact-representability notion of bias with matched basis structure, not for broader notions such as spectral bias or partial alignment. Within that scope the paper is sound and well supported by detailed appendices.

major comments (3)
  1. [Section 2.3, Eqs. (18)–(20)] The biased-case conclusion is substantially constructed. Eq. (18) generates y from U(d), V(d), and s_ρ for ρ=1..R, with V(d) chosen so that Σ_ν V(d)_{ν,ρ} ι_ν(θ*) = 0 for ρ>R. The cutoff model Γ(c) in Eq. (20) uses the same U(d), V(d) and retains the first R singular values unchanged, while the tail is exponentially suppressed. Hence the target lies exactly in the function space spanned by the R retained components of the low-ED model, and the extra tail directions of the full model are silent at θ* by construction. The comparison is therefore between a model whose inductive bias exactly matches the target and a model with additional irrelevant directions; it does not test more general forms of bias, e.g., approximate alignment, spectral bias, or a target spanning the full D-dimensional space. Please either add experiments with alternative bias constructions (such as targets with compone
  2. [Section 4, Conclusion and outlook] The abstract and Section 1.1 state the result as a general dichotomy for 'biased' vs 'unbiased' models. In Section 4 the authors acknowledge that the analysis compares models with the same underlying structure and say they 'expect' the results not to depend on this choice. This is an explicit untested extrapolation. Given the construction in Eqs. (18)–(20), the burden is on the authors to show that the sign of ΔMSE_min is not an artifact of the shared U(d), V(d) and the placement of the target. Either new numerical evidence or a sharpened statement of scope is needed to support the abstract's general claim.
  3. [Section 3.1, Eq. (24) and Figs. 4–5] The empirical sign of Δ^{f-c} MSE_min is the basis of the central conclusion, but no error bars or confidence intervals are reported. In Fig. 4 each point is from a single random model realization (averaged over 30 training runs), so the scatter across model realizations is not visible; in Fig. 5, averaging over 30 realizations is done but the spread is not shown. Please report the standard deviation or standard error over both model realizations and training seeds, and state how many points are shown. This is necessary to assess whether the positive/negative separation is statistically robust.
minor comments (7)
  1. [Eq. (21)] The definition of δ_data is unclear: specify what S is (the singular-value matrix?) and which norm is used. The notation 'S(V(d)⊤ − V(d)⊤_ϵ)ι(θ*)' should be expanded.
  2. [Section 3.2] Typo: 'in the same manner as those in in the previous section' should read 'in the same manner as those in the previous section'.
  3. [Author affiliations] The string 'f¨ ur' appears to be an encoding artifact; it should be 'für'.
  4. [Figure 6 caption] Only 7 training instances are averaged per point; please specify whether multiple model realizations were used and report the corresponding variance.
  5. [Eq. (22)] The underbrace 'ortho. rotation + isometry' is ambiguous. Clarify the dimensions of T and how the isometry condition is imposed in the tensorized construction.
  6. [Section 2.1.2] The statement 'This results in no loss of generality' for assuming the same local basis sets for all features and parameters needs a brief justification; in general QNNs the frequency sets may differ per feature/parameter.
  7. [Reproducibility] No code is released. Releasing the training and tensor-network code would improve reproducibility, though it is not required for correctness.

Circularity Check

1 steps flagged · score 6.0 of 10

Biased-case result is partly built into the experimental design: targets are generated from the same low-rank subspace that low-ED cutoff models retain (Eqs. 18-20), so the ED advantage in the biased regime is largely a consequence of construction rather than a general law.

  1. self definitional [Section 2.3, Eqs. (18)-(20); used in Section 3.1, Fig. 4]
    "we consider a data-generating function y(x) of the form [Eq. (18): y(x)=Σ_{μ,ν} e_μ(x) ι_ν(θ*) Σ_{ρ=1}^R s_ρ U^{(d)}_{μ,ρ} [V^{(d)⊤}]_{ρ,ν}] ... with V^{(d)} constructed in order to satisfy the property Σ_ν V^{(d)}_{ν,ρ} ι_ν(θ*)=0 for ρ=R+1,...,K. ... one can define several such biased models with different properties of s_ρ, hence different effective dimensions, by choosing different values for s_{ρ>R}, which have no influence on the model prediction for θ=θ*."

    The biased target y(x) is generated from the first R singular directions U^{(d)},V^{(d)} and the orthogonality condition makes all ρ>R directions silent at θ*. Consequently both the full model Γ^{(f)} (Eq. 19) and the cutoff model Γ^{(c)} (Eq. 20) satisfy f_{θ*}(x)=y(x) by construction, and the only difference between them is the decay of singular values in directions that do not affect the target. The low-ED cutoff model's function space therefore contains y exactly, while the high-ED model only adds dimensions irrelevant at θ*. The positive ΔMSE for biased models in Fig. 4(a) is thus substantially enforced by placing the target inside the low-ED subspace, not by a general relation between ED and task alignment. The paper itself limits the comparison to models sharing the same basis struc

full rationale

The analytical part (Section 2.2) is not circular: the FIM expression (Eq. 14), rank bound (Eq. 16), and the tr(S^4) control of ED are derived from the model definition and standard FIM definitions, with external random-matrix theory support. The tensor-network representation is an independent technical contribution. The circularity is concentrated in the empirical ED-bias comparison. The definition of bias is a natural one (exact representability), but the operational construction of biased experiments places y inside the exact R-dimensional subspace retained by the low-ED cutoff model, and makes the extra high-ED directions silent at θ* via the V^{(d)} orthogonality property. The central result for the biased regime is therefore a consequence of the experimental design: it demonstrates that a low-ED model wins when the target is placed in its retained subspace. The unbiased regime, where random high-ED models have more candidate directions to approximate a random target, is a coverage artifact of the same construction. The authors explicitly acknowledge the restriction to same-structure models in Section 4 ('our analysis is based on comparing ED, bias and training performance of models with the same underlying structure') and only state an expectation that results transfer to different model classes. For these reasons the headline asymmetry is partially circular, though the analytic FIM-ED derivation and the existence of a valid counterexample to 'high ED is always better' keep it from being fully circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central quantitative claim rests on a small set of hand-chosen knobs: the low-rank subspace dimension R, the cutoff decay xi, the special parameter point theta*, and the perturbation epsilon. The 'bias' definition itself is an ad hoc modeling choice that makes the biased-case result a near-consequence of construction.

free parameters (5)
  • R (rank of data-generating subspace) = R=6 (Fig. 4), R=2 (Fig. 6), R=7 (Fig. 7), R=4 (Fig. S8/S10)
    Controls the dimension of the subspace in which y(x) lives and the number of singular components retained undecayed in cutoff models; the biased-case result depends on y being in this R-dimensional subspace.
  • xi (decay rate of cutoff spectrum) = positive values used; exact values not stated in main text
    Sets the decay exp(-(rho-R)/xi) of singular values rho>R and thereby tunes the ED difference between full and cutoff models.
  • theta* (target parameter configuration) = randomly chosen parameter configuration
    The point at which biased models exactly reproduce y; the kernel condition V(d)^T iota(theta*)=0 for tail components is enforced via Gram-Schmidt.
  • epsilon (perturbation strength for partially biased models) = used to generate delta_data; exact values in figures
    Creates approximate bias; the deviation metric delta_data in Eq. (21) depends on it.
  • singular values s_rho for full models = s_rho = 1/sqrt(D) uniform
    Uniform spectrum chosen to give high ED; an ad hoc normalization choice that affects the ED comparison.
assumptions (6)
  • domain assumption QNN outputs of data re-uploading circuits can be expanded in finite Fourier series in inputs and parameters with frequencies determined by gate eigenvalues (Eq. 6, Appendix A).
    Bridges QNNs to Fourier models; relies on known results [28-31] and standard diagonalization of rotation gates.
  • domain assumption The input and parameter basis functions form orthonormal bases on [-pi,pi]^N and Theta with uniform measure (Eqs. 2, 4).
    Needed for the FIM expression and SVD; restricts to bounded/uniform input and parameter distributions.
  • standard math In Property 2, V is treated as Haar-random orthogonal; FIM expectations and variances over V and theta are used to infer ED control by tr(S4).
    Random matrix theory (Collins-Sniady / Weingarten) supplies Eqs. 84-85; for fixed non-random V the variance statement is not proven.
  • domain assumption The FIM for MSE regression is defined via a Gaussian noise model with sigma=1 (Supplementary S3.1).
    Justifies Eq. (11); the constant factor does not affect the normalized ED.
  • ad hoc to paper A model is called biased towards y iff there exists theta* with f_theta*(x)=y(x), and unbiased otherwise (Section 2.3).
    This exact-containment definition, together with the construction of y in the R-dimensional singular subspace, drives the empirical conclusions.
  • standard math Rank(F(theta)) <= D for M>D because F is a sum of at most D projections (Property 1, Appendix B and S4.2).
    Used to upper-bound the effective dimension by D.

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Pith. "Pith review of Fisher Information, Training and Bias in Fourier Regression Models." pith.science (2026). https://pith.science/paper/M2MNKBFK

@misc{pith2026251006945,
  author       = {Pith},
  title        = {Pith review of: Fisher Information, Training and Bias in Fourier Regression Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M2MNKBFK}},
  note         = {Machine review of arXiv:2510.06945}
}
read the original abstract

Motivated by the growing interest in quantum machine learning, in particular quantum neural networks (QNNs), we study how recently introduced evaluation metrics based on the Fisher information matrix (FIM) are effective for predicting their training and prediction performance. We exploit the equivalence between a broad class of QNNs and Fourier models, and study the interplay between the \emph{effective dimension} and the \emph{bias} of a model towards a given task, investigating how these affect the model's training and performance. We show that for a model that is completely agnostic, or unbiased, towards the function to be learned, a higher effective dimension likely results in a better trainability and performance. On the other hand, for models that are biased towards the function to be learned a lower effective dimension is likely beneficial during training. To obtain these results, we derive an analytical expression of the FIM for Fourier models and identify the features controlling a model's effective dimension. This allows us to construct models with tunable effective dimension and bias, and to compare their training. We furthermore introduce a tensor network representation of the considered Fourier models, which could be a tool of independent interest for the analysis of QNN models. Overall, these findings provide an explicit example of the interplay between geometrical properties, model-task alignment and training, which are relevant for the broader machine learning community.

Figures

Figures reproduced from arXiv: 2510.06945 by the authors.

Figure 1
Figure 1. Illustration of main results. (a) Schematic behavior of the difference in the training loss (here the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. (a) Scaling of normalized ED with the purity tr( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. (a) Schematic illustration of the construction of biased and unbiased models. The data-generating [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: (a) ∆f−cMSEmin for different values of ˆd full eff − ˆd cut eff , for biased (blue points) and unbiased (yellow points) models. Each point corresponds to ∆f−cMSEmin averaged over 30 training instances starting from randomly chosen parameters, for a single random model …
Figure 5
Figure 5. Figure 5: (a) ∆f−cMSEmin for different values of ˆd full eff − ˆd cut eff , for different values of δdata (color scale). Each point corresponds to ∆f−cMSEmin averaged over 30 training instances for 30 random model realization. (b) Same as panel (a) but resolved as a function of …
Figure 6
Figure 6. Figure 6: (a) ∆f−cMSEmin for different values of ˆd full eff − ˆd cut eff , for biased (blue points) and unbiased (yellow points) models. Each point corresponds to ∆f−cMSEmin averaged over 7 training instances starting from randomly chosen parameters, for a single random model r…
Figure 7
Figure 7. Figure 7: (a) ∆f−cMSEmin for different values of ˆd full eff − ˆd cut eff , for biased (blue points) and unbiased (yellow points) models. Each point corresponds to ∆f−cMSEmin averaged over 30 training instances starting from randomly chosen parameters, for a single random model …
Figure 8
Figure 8. Figure 8: (a) ∆f−cMSEmin for different values of ˆd full eff − ˆd cut eff , for different values of δdata (color scale). Each point corresponds to ∆f−cMSEmin averaged over 30 training instances for 30 random model realization. (b) Same as panel (a) but resolved as a function of …
Figure 9
Figure 9. Figure 9: (a) and (b) ∆f−cMSEmin vs. M for biased and unbiased models, respectively. Here N = 1, D = 25, Ω = ˜ {1} ( ˜d = 3), R = 3 and χ = 50. (c) and (d) ∆f−cMSEmin vs. D for biased and unbiased models, respectively. Here N = 1, M = 50, Ω = ˜ {1} ( ˜d = 3), R = 2 and χ = 120. …

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