Optimizing Basis Function Selection in Constructive Wavelet Neural Networks and Its Applications

Dong Shen; Dunsheng Huang; Lei Lu; Ying Tan

REVIEW 3 major objections 6 minor 41 references

Optimizing Basis Function Selection in Constructive Wavelet Neural Networks and Its Applications

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that a constructive wavelet neural network can estimate an unknown function's dominant frequency from data and then add only high-energy wavelet bases, reaching a preset accuracy with fewer parameters than a generic…

desk verdict A useful two-step constructive WNN idea with real parameter savings on its examples, but the efficiency claim leans on an unproved locality assumption that the experiments never stress. read the letter →

arxiv 2507.09213 v1 pith:K47K5DBP submitted 2025-07-12 cs.LG stat.ML

Dunsheng Huang , Dong Shen , Lei Lu , Ying Tan This is my paper · ORCID

classification cs.LGstat.ML

keywords constructivewaveletneuralnetworksfrequencyestimationbasisselectiontime-frequencylocalizationenergy-basedincreasefunctionapproximationcomputationalcomplexitynonlinearsystemidentification

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 tries to show that a constructive wavelet neural network need not guess where an unknown nonlinear function lives in frequency. Instead, the network first estimates the energy of the function projected onto wavelet subspaces from input-output data, uses a smoothed energy comparison to pick an initial wavelet frequency, and then adds only the wavelet bases whose coefficients are not nearly zero. If this works, it removes a major practical bottleneck: choosing the wrong starting frequency either slows convergence or wastes most of the added basis functions. The authors claim concrete savings in their examples, including 174 versus 420 parameters on a synthetic mapping and 1,510 versus 39,366 parameters on a nine-dimensional milling-machine regression.

What carries the argument

The load-bearing object is the enlarged time-frequency box $B_\varepsilon(T,\Omega_0,\Omega_1)$: an orthonormal wavelet $\psi_{mn}$ with scale $m$ and translation center $n$ is kept only when $2^m$ lies inside the extended frequency band and $|n| \preceq 2^m T + t_\varepsilon$. Corollary 1 says wavelets outside this box have nearly zero coefficients, so the basis-expansion loop can skip them. The energy estimates $\hat E_m$ are computed from the data by training coefficients with one neural-network pass per subspace, then smoothing them with an exponential moving average ($\alpha$ chosen from the target accuracy $\varepsilon$) so the algorithm can stop at the first resolution where the smoothed energy stops rising. The separation factor $\mu$ in Algorithm 2 controls how aggressively the highest-energy bases at the current resolution are used to select the next-resolution bases.

What would settle it

Construct a test function on a bounded interval whose wavelet-subspace energies have two well-separated and comparable peaks, for instance a sum of two sinusoids with frequencies several octaves apart and similar amplitudes; run Algorithm 1 and check whether it stops at the lower peak while Algorithm 2 then needs more bases than a standard wavelet neural network to reach the same accuracy.

Watch

Extended reading notes

Core claim

For functions in $L^2(\mathbb{R}^d)$ whose energy is concentrated in a finite time-frequency box, the paper proves an approximation bound (Theorem 1): truncating the wavelet expansion to an enlarged box $B_\varepsilon$ costs at most the energy of the function outside the box plus $2(2\pi)^{d/2}\varepsilon\|f\|$. Corollary 1 then states that every wavelet coefficient whose time-frequency center lies outside $B_\varepsilon$ is essentially zero at accuracy $\varepsilon$. The paper's constructive wavelet neural network turns this into a practical recipe: Algorithm 1 estimates the subspace energies $\hat E_m$ with one-pass coefficient updates and an exponential moving average to select the initial frequency, and Algorithm 2 repeatedly adds the highest-energy bases from the next resolution level until the loss falls below $\varepsilon$. The central claim is that this frequency-aware construction reaches the target accuracy with far fewer bases than a standard wavelet neural network, because the omitted bases are exactly the ones whose coefficients are near zero.

Load-bearing premise

The load-bearing premise is Assumption 2: the energy of the unknown function projected onto wavelet subspaces rises and then falls with resolution, so the first time the smoothed energy stops increasing the algorithm has already found the dominant frequency; if the energy has several comparable peaks, early stopping can miss the main spectral content.

Editorial extensions

If this is right

For any square-integrable mapping whose energy concentrates in a finite time-frequency box, Theorem 1 justifies dropping every wavelet basis outside an enlarged box without exceeding the target error, apart from energy leaked outside the box plus a small constant times $\|f\|$.
Starting the network at the estimated dominant frequency should avoid both slow convergence from a too-low starting frequency and wasted bases from a too-high one.
Because the wavelet bases are orthogonal, newly added bases do not disturb already-trained coefficients, so the incremental structure can expand without full retraining.
The framework is claimed to work in offline, combined-dataset, online time-varying, and nine-dimensional real-data settings, implying the frequency-estimation step transfers across those regimes.
The parameter counts reported in Examples 1 and 4 give the concrete promise: 174 versus 420 parameters on the synthetic mapping, and 1,510 versus 39,366 on the milling regression.

Reading between the lines

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

If the unimodality assumption fails, a practical extension would be to run Algorithm 1's energy scan from both low and high resolutions and keep the higher-energy peak, which would catch multimodal spectra instead of stopping at the first local peak.
The same energy-first selection principle should transfer to other dictionaries such as Fourier packets, curvelets, or shearlets whenever an analogue of the time-frequency box and a coefficient decay estimate exist.
On the milling dataset, the random 80/20 split probably overstates generalization because tool-wear observations are autocorrelated within each trial; a trial-wise split that trains on some machining configurations and tests on unseen ones would provide a sharper test of the claimed parameter savings.
The complexity argument would be stronger with a demonstration that the estimated initial frequency remains reliable as the input dimension grows beyond nine, since the per-dimension parameter cost is the main scaling bottleneck.

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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

Quick take: the paper has a sensible core—estimate the dominant frequency of the unknown function from data, then grow the wavelet basis by energy ranking—and it shows convincing parameter reductions on its four examples (420 to 174, 39,366 to 1,510). That is a genuine engineering contribution within constructive wavelet networks. The writing is rough, but the ideas are coherent and the empirical work is honest.

What is actually new: the specific combination of Algorithm 1's energy-based initial frequency estimator and Algorithm 2's energy-ranked basis increase is not in the cited literature. The authors also provide a real L2 bound (Theorem 1) connecting approximation error to energy outside a time-frequency box, and Corollary 1 gives a legitimate reason to skip many wavelet bases. For the tested functions, the method does what it claims.

Soft spots, in proportion:

1. The load-bearing assumption in Algorithm 2 is that high-energy wavelet coefficients at scale m+1 sit near the high-energy translation centers at scale m. Corollary 1 does not imply this; it only bounds total energy outside a time-frequency box and says nothing about the spatial arrangement of large coefficients inside it. A function with a low-frequency bump in one region and a high-frequency bump far away would defeat the nearest-neighbor selection rule. The outer loop (increasing mu_up, then m) would eventually add the needed bases, but at much higher parameter cost, undermining the paper's efficiency claim. Remark 8 claims robustness to non-stationary spectra but gives no analysis, and Examples 1 and 4 use functions with concentrated, roughly stationary spectra. Example 3 shifts frequency content in time, not space, and even there the method needed a large delay to recover. This is the biggest gap.

2. Assumption 2 (unimodal subspace energy) supports Algorithm 1's stopping rule. It is justified only by a heuristic bandwidth argument, one plotted example, and a claim that it is "not restrictive"—which is not a proof. If subspace energy is multimodal, the initial frequency can land on a local peak and miss dominant spectral content.

3. The efficiency comparisons are against a generic WNN, not against prior constructive wavelet algorithms like Xu and Yan (2011) or Kugarajah and Zhang (1995), which are the natural baselines. Without those, the incremental improvement is unclear.

4. The manuscript has many editorial inconsistencies: equation (42) is cited before it is defined, equation numbers are off, and reference [37] is dated 2027. These are minor but they make replication harder.

5. Convergence to accuracy ε is demonstrated empirically via training loss; Theorem 1 does not guarantee that the greedy selection reaches ε with the claimed parameter count. That is fine as an experimental claim, but it should be stated as such.

Who this is for: researchers working on constructive wavelet networks or adaptive function approximation. It deserves a serious referee, but a conditional accept at best. I would send it to review and require: release the code, add prior constructive WNN baselines, and address the spatially separated frequency-content case, either by a proof of the locality assumption or by experiments specifically designed to break it.

Referee Report

3 major / 6 minor

Summary. The manuscript proposes a constructive wavelet neural network (CWNN) for approximating unknown nonlinear mappings with a preset accuracy. The method has two components: Algorithm 1 estimates the dominant frequency of the unknown function by projecting data onto wavelet subspaces and estimating subspace energies; Algorithm 2 starts from the estimated initial resolution and adds wavelet bases in the next resolution that are nearest to current high-energy translation centers, stopping when the training loss falls below epsilon. The theoretical part (Theorem 1, Corollary 1) states that, under energy concentration in a time-frequency box, wavelets outside an epsilon-expanded box have negligible coefficients. Four experiments compare the CWNN with standard WNN and fixed GNN baselines on synthetic and real milling data and report a reduced parameter count.

Significance. If the parameter-reduction claim is established, the framework would be a useful practical contribution to constructive wavelet networks. The paper supplies a clean statement of the time-frequency truncation result, and the four examples address offline, multi-dataset, online time-varying, and real-world settings, with the real milling benchmark being a reasonable test. It also provides a code-release link and a direct comparison against standard WNN and several GNN baselines. However, the paper's central efficiency claim is not backed by an end-to-end guarantee: the theoretical truncation result concerns the full set of bases in a known time-frequency box, whereas the algorithm selects bases via local energy heuristics. The current experiments use smooth functions whose significant scale content is co-located, so they do not exercise the main risk identified below.

major comments (3)

[IV-C, Eq. (27)] The basis-expansion rule selects the two nearest W_{m+1} centers around each high-energy W_m center. Corollary 1 does not justify this locality assumption: it only bounds the total energy of coefficients with centers outside B_epsilon and says nothing about where the large coefficients inside B_epsilon lie. For a function with a low-frequency bump at one location and a high-frequency bump at a distant location, Step 2 adds next-scale bases near the low-frequency region and misses the high-energy next-scale bases; the outer loop eventually repairs this only by adding many extra bases, negating the reported parameter reduction. Remark 8 and Example 3 do not address this case: the frequency shift in Example 3 is a change over time of the whole mapping, not spatially separated scale-dependent energy. Please provide a theorem or a dedicated experiment for spatially separated multiscale content before claiming the efficiency result.
[III, Algorithm 1, Assumption 2] Algorithm 1's stopping rule assumes E_m is unimodal in m; if E_m is multimodal, the EMA rule can stop at a local peak and select an initial frequency that misses the dominant spectral content. Assumption 2 is not proved, and the unimodality evidence is a single function in Example 1 (Fig. 5(b)). Since the initial frequency feeds Algorithm 2 and strongly affects the parameter count, the efficiency claim depends on this assumption. Please either prove a sufficient condition, show that the algorithm is robust to multimodal E_m with a dedicated experiment, or characterize the failure mode.
[IV, Theorem 1 and Corollary 1 vs Algorithm 2] The theoretical result bounds the error incurred by omitting all wavelets outside a specified B_epsilon, assuming the concentration box [-T,T] x ([Omega_0,Omega_1] union [-Omega_1,-Omega_0]) is known. Algorithm 2 does not construct B_epsilon and does not use T, Omega_0, Omega_1; it selects bases by thresholding estimated energies and by nearest-neighbor proximity. No argument is given that the bases selected by Algorithm 2 form a superset of the significant bases inside B_epsilon or that the Theorem 1 error bound applies to the algorithm's output. Consequently the statement in the conclusion that the approximation error is bounded by the function's energy and the number of wavelet bases is not connected to the implemented algorithm.

minor comments (6)

[III, Algorithm 1 and IV-C] Equation (42) is used in Algorithm 1 Step 1, Step 2, and in Algorithm 2 before it is defined in Appendix C; the energy-estimate definition should be moved to the main text and the cross-references renumbered.
[II-C and Appendix B] The example in Section II-C refers to 'conditions (4), (35), and (36)' and 'Using (34)', but the conditions are (4)-(6) and the construction is (7); the Appendix numbering (33)-(36) is inconsistent with the main text.
[I and VII] The introduction and the conclusion mention 'five numerical examples' and 'five simulation examples', while Section V presents four examples (Examples 1-4); the counts should be reconciled.
[IV, Remarks 5 and 6] Remarks 5 and 6 give inconsistent guidance for the termination threshold: Remark 5 recommends 0.01 epsilon to 0.001 epsilon, while Remark 6 says zeta should be '10 times smaller' than the desired accuracy; please reconcile the two recommendations.
[IV-C, Step 3 and Algorithm 2] The prose around Step 3 of Algorithm 2 describes selection of N_{2 mu E_m} bases after one network run, but the pseudocode uses an outer loop with mu_up incremented by mu; the description and pseudocode should be aligned.
[III, Eqs. (11)-(12)] The empirical mapping from epsilon to alpha is written as alpha = 2 arctan(-lg epsilon)/pi in Eq. (11), but Eq. (12) and the surrounding text use arctan(-lg epsilon) together with pi/2 factors in a way that is not consistent with Eq. (11); the derivation should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the core bound is an external-frame estimate under explicit assumptions, and the basis-selection heuristic is a correctness concern, not circularity.

full rationale

After tracing the derivation chain, I find no step where an output quantity is identical by construction to an input, or where a fitted parameter is relabeled as a prediction. The theoretical core (Theorem 1 and Corollary 1) is an approximation bound for orthonormal wavelet frames: it assumes L2 membership (Assumption 1), unimodal subspace energy (Assumption 2), and time-frequency concentration (Assumption 3), and proves that coefficients outside a sufficiently enlarged box B_epsilon are small using Lemma 1 from an external frame estimate ([31], itself parameter-free). The reduction from Theorem 1 to Corollary 1 is a direct norm argument, not a restatement of the algorithm's selection rule. Algorithm 1 estimates subspace energies from training data and picks m_init; Algorithm 2 then adds bases nearest to currently high-energy centers via Eq. (27). This nearest-neighbor locality rule is a heuristic justified by Mallat-Hwang regularity ([30]) rather than derived from Corollary 1, and it could fail for spatially separated scale-dependent content; but a false or unproved heuristic is a correctness risk, not circularity. The reported parameter reductions (e.g., 174 vs 420 in Example 1; 1,510 vs 39,366 in Example 4) are comparisons on training/test splits after the same data-driven fitting procedure, so they are empirical outcomes. Self-citations [38], [40], [41] occur in application and context discussions and do not carry the derivation. Limitations are openly stated: Assumption 2 is justified only by Remark 1 and one simulation, and Remark 8 admits that non-stationary frequency handling has no detailed analysis. These weaken generality but do not make the argument circular.

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

The framework introduces no new physical entities. It depends on several user-chosen hyperparameters (κ, ιr, µ, ζ, the EMA rule for α, and the target ε) and on three stated assumptions about the function's integrability, energy unimodality, and time-frequency concentration. The unimodality assumption is the most fragile because Algorithm 1's stopping rule is built directly on it and the paper provides only a heuristic justification.

free parameters (6)

hyperparameter κ = 0.36 (Examples 1 and 3), 2/3 (Example 4)
Controls the number of translation centers selected in subspace W1 for initial frequency estimation. Remark 3 gives a recommended range of (1/3, 2/3) but no principled selection rule.
learning rate ιr = 0.0005 (Example 1), 0.0001 (Example 3), 0.001 (Example 4)
Used in the single-step coefficient updates (15)-(16) and in the neural network training for basis coefficient estimation. The values are chosen by hand for each experiment.
EMA parameter α = Set through the empirical rule α = 2 arctan(-lg ε) / π (equation 11)
The exponential moving average parameter for energy estimation is derived from the accuracy ε through an ad hoc formula; no theoretical justification is provided.
separation factor µ = 1/3 in most experiments; scanned from 1/5 to 1/2 in Example 1
Determines how many high-energy bases are selected from each wavelet subspace. Remark 4 states the choice is case-dependent.
termination threshold ζ = 0.00004 in all experiments
Stops the inner training iterations in Algorithm 2. Remark 5 recommends 0.01ε to 0.001ε but the selection remains empirical.
target accuracy ε = 0.006 (D1, D2), 0.025 (D3), 0.02 (Example 3), 0.015 (Example 4)
User-defined stopping accuracy. It influences α and the number of bases, and its choice is specific to each dataset and noise level.

assumptions (6)

domain assumption The unknown function f is square-integrable over Rd or over a compact domain D (Assumption 1).
Needed for the wavelet expansion (2) to converge in L2 and for the energy calculations. Stated in Section II-B as critical for the framework.
ad hoc to paper The wavelet-subspace energy Em is unimodal with respect to resolution m (Assumption 2).
Used in Algorithm 1 to stop at the first detected energy peak. Remark 1 gives a heuristic bandwidth argument but no proof, and no general verification is provided.
domain assumption The energy of f is concentrated in a finite time-frequency box [−T,T] × ([Ω0,Ω1] ∪ [−Ω1,−Ω0]) (Assumption 3).
Required for Theorem 1 and Corollary 1. Reasonable for band-limited signals but not guaranteed for arbitrary nonlinear maps.
standard math The mother wavelet ψ is generated from a function φ satisfying conditions (4)-(6), yielding an orthonormal wavelet frame.
Taken from [31] and [28]; used to construct the wavelet bases and in the proof of Theorem 1.
domain assumption The wavelet transform regularity property from [30] justifies choosing translation centers in Wm+1 near those in Wm.
Step 2 of Algorithm 1 relies on this regularity-translation relation to reduce the search space; it is not proven for the estimated energy coefficients.
ad hoc to paper The empirical EMA rule (11) maps ε to α and is sufficient to detect the main energy peak.
Introduced in Section III-A without derivation; the behavior of the smoothed energy average is assumed to identify the peak accurately.

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Cite this review

Pith. "Pith review of Optimizing Basis Function Selection in Constructive Wavelet Neural Networks and Its Applications." pith.science (2026). https://pith.science/paper/K47K5DBP

@misc{pith2026250709213,
  author       = {Pith},
  title        = {Pith review of: Optimizing Basis Function Selection in Constructive Wavelet Neural Networks and Its Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K47K5DBP}},
  note         = {Machine review of arXiv:2507.09213}
}

read the original abstract

Wavelet neural network (WNN), which learns an unknown nonlinear mapping from the data, has been widely used in signal processing, and time-series analysis. However, challenges in constructing accurate wavelet bases and high computational costs limit their application. This study introduces a constructive WNN that selects initial bases and trains functions by introducing new bases for predefined accuracy while reducing computational costs. For the first time, we analyze the frequency of unknown nonlinear functions and select appropriate initial wavelets based on their primary frequency components by estimating the energy of the spatial frequency component. This leads to a novel constructive framework consisting of a frequency estimator and a wavelet-basis increase mechanism to prioritize high-energy bases, significantly improving computational efficiency. The theoretical foundation defines the necessary time-frequency range for high-dimensional wavelets at a given accuracy. The framework's versatility is demonstrated through four examples: estimating unknown static mappings from offline data, combining two offline datasets, identifying time-varying mappings from time-series data, and capturing nonlinear dependencies in real time-series data. These examples showcase the framework's broad applicability and practicality. All the code will be released at https://github.com/dshuangdd/CWNN.

Figures

Figures reproduced from arXiv: 2507.09213 by the authors.

**Figure 2.** Flowchart for selecting the initial wavelet frequencies (red dots indicate the wavelet bases chosen in the relative space) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** (a) Positions of wavelets {ψmn(x)} in the time–frequency domain. The horizontal and vertical axes represent the wavelet translation and wavelet frequency, respectively. The black dashed box represents B (T, Ω0, Ω1) and the red dashed box represents the enlarged Bε (T, Ω0, Ω1). (b) represents the domain excluding bases outside Bε (T, Ω0, Ω1). Although the nonlinear function f(·) may lack compact support in both freq… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Flowchart of the wavelet-basis increase algorithm [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: (a) illustrates the three datasets. In all datasets, x1 is uniformly sampled from [0, 1], and x2 = √ x1. Each dataset Dj , j = 1, 2, 3, is split into training (Dj,train) and test (Dj,test) sets, with 80% of the data randomly and uniformly allocated to the training set.…

**Figure 6.** Figure 6: The performance of training procedure using 5 neural networks and dataset [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: The number of parameters used by 5 NNs using three training sets [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 9.** Figure 9: (a) Loss(30). (b) the parameter number used by WNN and CWNN until they reach the predefined accuracy ε = 0.02. 00 0 0 0 0 0 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: the spectra of functions (30) and (31). demonstrating the framework’s adaptability to time-varying mappings. Since the difference between the two functions is small, only a modest adjustment—adding new wavelets at the 493rd iteration—is observed. 0 100 200 300 400 500…

**Figure 11.** Figure 11: shows the loss evolution as the nonlinear mapping changes. The mapping is modified at the 8 th iteration, and new wavelets are added at the 493rd iteration. Despite the change, the loss converges to the target accuracy by the 891st iteration, 0 50 100 150 200 250 10-2…

**Figure 12.** Figure 12: (a) The computed Loss using the training set to learn unknown [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]

Discussion (0). Continue with ORCID to comment.

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