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REVIEW 3 major objections 6 minor 34 references

Catalyst: a Novel Regularizer for Structured Pruning with Auxiliary Extension of Parameter Space

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

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Pith's one-line read A catalyst regularizer is claimed to make structured pruning lossless, magnitude-unbiased, and robust.

desk verdict A genuinely new regularizer with correct algebraic lemmas, but the 'provably zero bias' claim is asserted, not proved, and the actual training dynamics are not analyzed. read the letter →

arxiv 2507.14170 v1 pith:CNU4VUF3 submitted 2025-07-10 cs.LG cs.AI

Jaeheun Jung , Donghun Lee This is my paper · ORCID
classification cs.LGcs.AI
keywords structuredpruningfilterregularizationcatalystvariableslosslessmagnitudebiasbifurcationextendedparameterspace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that structured pruning can be made lossless and fair by minimizing a carefully chosen regularizer in an extended parameter space. It identifies the exact algebraic condition for pruning without performance loss — a filter being exactly zero, encoded as $DW=0$ with an auxiliary diagonal matrix $D$ — and constructs Catalyst, a regularizer $\|DW\|_{2,1}$ over those auxiliary variables. The paper argues, with a bifurcation theorem, that this regularizer drives the ratio $D_{ii}/\|F_i\|_2$ to exponentially separate pruned from preserved filters, so pruning decisions are robust and carry provably zero bias toward small filter magnitudes. If these claims are right, they remove the root cause of magnitude bias in L1/Group-Lasso pruning and give stable, near-lossless compression at state-of-the-art accuracy.

What carries the argument

The load-bearing object is the product $\|DW\|_{2,1} = \sum_i \|D_{ii}F_i\|_2$ over a diagonal matrix of auxiliary 'catalyst' variables $D$ inserted into a learnable activation $\psi_{D,\bar D}(x)=Dx-\bar D x+\sigma(x)$. The diagonal $D$ encodes the pruning-invariant set: $DW=0$ with $D\neq 0$ forces each zeroed filter to be exactly zero, so minimizing this term is a tractable proxy for reaching $X_{\mathrm{tgt}}$. The initialization $D_{ii}=\|F_i\|_2$ sets the ratio $c_i=D_{ii}/\|F_i\|_2$ to 1, and the bifurcation dynamics of Theorem 3.4 — driven by the recurrence $c_{t+1}=f(c_t,\lambda_t)c_t$ with $f$ defined by gradient descent with weight decay — push $c_i$ exponentially toward $0$ or very large values. The pruning rule $P=\{i: D_{ii}>\|F_i\|_2\}$ then selects filters whose ratio crossed above 1, and the extend–train–contract scheme removes $D$ and $\bar D$ in two stages.

What would settle it

Take a layer, multiply each filter's initial norm by a known constant, run Catalyst from the same $c=1$ initialization, and compare which filters are pruned: if pruning probability varies with the scaling, the zero-bias claim is false. In parallel, measure the output difference between the pruned and unpruned network at the stopping epsilon; any nonzero activation error shows that pruning at approximate $DW\approx 0$ is not truly lossless.

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Extended reading notes

Core claim

The paper's central claim is that structured pruning can be made lossless and magnitude-fair by regularizing in an extended parameter space. It first defines the pruning-invariant set $X_{\mathrm{tgt}}$ as the union of subspaces in which at least one filter $F_i$ is zero, and proves that this set is exactly the projection of $\{(W,D): DW=0,\ D\neq 0\}$ for a diagonal auxiliary matrix $D$. Minimizing $\|DW\|_{2,1}$ therefore drives $W$ toward the pruning-invariant set, and the paper shows that the regularizer's global minima are nontrivial and all its critical points are global minima. Initializing $D_{ii}=\|F_i\|_2$ places every filter on the decision boundary $c_i=1$, and Theorem 3.4 shows that under the pure regularizer the ratio $c_i=D_{ii}/\|F_i\|_2$ exponentially bifurcates away from 1, so a filter is pruned when $c_i>1$ and preserved when $c_i<1$. Because the escape from $c_i=1$ is attributed to the performance loss $L$, the paper concludes that all channels get an equal chance to be pruned, with wide-margin separation between pruned and preserved filters, and reports empirically that pruning at $\|DW\|_{2,1}<\epsilon$ is lossless and beats state-of-the-art baselines.

Load-bearing premise

The whole fairness argument rests on the unproven assumption that the performance-loss gradient pushes each filter across the decision boundary $c=1$ without systematic preference for filter magnitude, and that stopping at a tiny nonzero $\|DW\|_{2,1}$ behaves like the exact lossless condition $DW=0$.

Editorial extensions

If this is right

  • If the zero-bias claim holds, pruning decisions track a filter's contribution to the loss instead of its magnitude, so small-norm but functionally important filters are no longer preferentially removed.
  • The exponential bifurcation of $c_i$ gives a wide margin between preserved and pruned filters, so small perturbations in filter weights will not flip a pruning decision.
  • Because $DW=0$ is both the lossless-pruning condition and the contraction constraint, the prune operation itself causes negligible accuracy drop, and the training loss curve stays smooth across the pruning steps.
  • The regularizer applies to different structured targets — batch-normalization scaling factors or grouped channels — and across architectures, as confirmed on ResNet-56, VGG-19, and ResNet-50.
  • At comparable speedups the method matches or exceeds state-of-the-art filter-pruning accuracy, for instance gaining +0.47 on ResNet-56/CIFAR-10 at 2.06x speedup and +0.36 on ResNet-50/ImageNet at 1.49x.

Reading between the lines

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

  • A testable consequence the paper leaves implicit: the set of pruned filters should be nearly invariant under a global rescaling of a layer's filter norms, since the decision variable is the ratio $c_i$ rather than the norm itself.
  • If the escape from $c_i=1$ is not in fact magnitude-unbiased under the full objective $L+\gamma_t\|DW\|_{2,1}$, the method would degrade toward the very bias it claims to remove; measuring the correlation between final pruning status and initial filter norm across many seeds would settle this.
  • The same algebraic encoding $DW=0$ could be applied to other structured units — heads in attention layers, channels in grouped convolutions, or blocks of layers — by choosing a coarser diagonal structure for $D$, which suggests a generalization to transformer pruning.
  • Because the hyperparameter $c$ at initialization determines the sign of the initial ratio offset from 1, it effectively sets a continuous sparsity target before training; a controlled sweep of $c$ would test whether pruning ratio and accuracy trade off predictably.
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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 / 6 minor

Summary. The paper introduces Catalyst, a regularizer for structured pruning defined in an extended parameter space via auxiliary diagonal matrices D and D-bar. The main theoretical claim is that minimizing ||DW||_{2,1} with the initialization Dinit = diag(||F_i||_2) drives the ratio c_i = D_ii/||F_i||_2 to bifurcate exponentially away from 1, so that filters with c_i > 1 are pruned and filters with c_i < 1 are preserved, yielding a wide decision margin. The paper further claims that this process gives each filter a provably equal chance of being pruned, independent of filter magnitude, and that pruning under the condition DW=0 is lossless. An algorithm, Catalyst Pruning, is proposed based on a bypass-style extension and contraction pipeline, and experiments are reported on ResNet-56/CIFAR-10, VGG-19/CIFAR-100, and ResNet-50/ImageNet, with comparisons to several structured pruning baselines.

Significance. If the advertised guarantees were valid, the work would be significant: a regularizer whose pruning decisions are provably decoupled from filter magnitude, and which provably produces a wide bifurcation, would address well-known limitations of L1/group-Lasso-based pruning. The algebraic characterization of the pruning-invariant set (Theorem 3.3) and the explicit bypass-style construction are interesting and appear self-contained. The empirical study covers multiple standard benchmarks and reports competitive accuracies, and the loss curves shown are smooth. However, the central advertised properties—provably zero magnitude bias and provably lossless pruning under the actual algorithm—are not established by the theorems given. Theorem 3.4 analyzes only the pure regularizer, and Theorem A.5 requires exact equality conditions that the algorithm does not enforce. The significance of the paper therefore hinges on unsupported claims, which substantially reduces its contribution as a theoretical pruning method.

major comments (3)
  1. [Section 3.4 and Section 4.1; Eq. (12) and Algorithm 1 line 4] The central advertised guarantee of 'provably equal chance' / 'provably zero bias' is not supported by any theorem. Theorem 3.4 analyzes gradient descent on the pure regularizer ||dM||_2 (Appendix E, Eqs. E1–E6), and its part (1) states that when c0=1, ct=1 for all t; hence the Catalyst dynamics alone never selects a filter. The paper's text asserts that 'due to performance loss L' the pair (D_ii, F_i) escapes this decision boundary almost surely and that this makes all channels equally likely to be pruned, but the actual training objective L + gamma_t ||DW||_{2,1} is never analyzed. In particular, no probability space, symmetry condition, or bound on the gradient of L is provided to justify that the side of the boundary reached by a filter is independent of its magnitude. Since the initialization places every filter exactly on the boundary, the pruning decision is entirely delegated to this unmodeled escape mechanism, so the fairness claim is an assertion rather than a proven property.
  2. [Section 3.3 / Theorem A.5 vs. Algorithm 1, lines 6 and 13] There is a mismatch between the lossless-pruning theorem and the algorithm's actual pruning step. Theorem A.5 guarantees that prune(P) is function-preserving only when DW=0 exactly and P={i: D_ii != 0} = {i: W_i=0}. Algorithm 1 stops when ||DW||_{2,1} < epsilon and sets P={i: D_ii > ||F_i||_2}. At that point, the rows F_i for i in P are not required to be zero, so the equality condition of Theorem A.5 is not satisfied. Table 2 reports small measured accuracy changes after pruning, which is good empirical evidence, but the paper does not provide any bound on the error introduced by pruning at epsilon>0. Consequently the claim of 'lossless pruning' as a theoretical property is not established for the implemented algorithm.
  3. [Section 4.4 and Appendix I] The empirical support for 'fair pruning chance' is qualitative and does not test the claimed property. Appendix I shows histograms of initial filter magnitudes for pruned and preserved filters under L1, Group Lasso, and Catalyst regularization. The figures suggest that Catalyst's pruned filters span a wider range of initial magnitudes, but no statistical test is performed (e.g., a two-sample test of the initial-magnitude distributions, or a correlation coefficient between initial magnitude and pruning outcome). The paper claims a provable property, which experiments alone cannot establish; without a quantitative comparison, the evidence remains suggestive rather than conclusive.
minor comments (6)
  1. [Section 3.4] The text says 'Dinit_ii = c||F_i||_2 with the hyperparameter c=1 where N is the number of parameters in Fi'; the role of N is unclear, and Dinit_ii is a scalar norm, not scaled by N. Please clarify or remove the reference to N.
  2. [Notation, Section 3.3 and Algorithm 1] The two auxiliary matrices D and D-bar are both rendered as 'D' in parts of the text and in Algorithm 1 (e.g., line 2 initializes theta, D, D-bar, and later 'D=0' appears). Please distinguish the two clearly throughout, as the current notation is confusing amid the pruning steps.
  3. [Theorem 3.4] The proof of Theorem 3.4 relies on the assumption that the signs of the entries of M and d remain unchanged along the trajectory (Appendix E, first paragraph), but this condition is not stated in the theorem. Please include it explicitly in the statement.
  4. [Figure 2] The caption says 'Simulation on ||DW||_{2,1} minimization' but the subfigures show (d,F) dynamics, ct, and ||Wt||; please clarify what the red arrows in subfigure (a) represent and ensure axis labels are readable.
  5. [Section 4.4] The sentence 'The ratio of c between pruned filters and preserved filters are extremely large, around 10^8' should be rephrased, since the ratio is derived from log-scale plots and the actual ratios may be much larger or the statement refers to the log-ratio; please state the quantity precisely.
  6. [Appendix F] The hyperparameter table appears to have garbled entries in the 'LR decay epoch' row, with repeated or misplaced values across columns. Please reformat the table for clarity.

Circularity Check

1 steps flagged · score 4.0 of 10

The algebraic core is self-contained, but the headline 'provably zero bias' fairness claim reduces to the c=1 initialization plus an unproved assumption about the loss-gradient escape.

  1. other [Section 3.4; Algorithm 1 lines 2 and 7]
    "During training, due to performance loss L (red arrows in Fig. 2a), the (Dii, Fi) escapes this decision boundary almost surely ... Since the pruning decision of ith filter is made by the position of (Dii, Fi) induced by performance loss L, all channels get provably equal chance to be pruned."

    The zero-bias claim is presented as a theorem-backed prediction, but the only proved dynamics, Theorem 3.4(1), has c0=1 as a fixed point, so the Catalyst regularizer alone never moves any filter off the boundary. The escape is attributed to the unmodeled gradient of L, and the paper asserts, without any equation or test, that this escape is independent of filter magnitude. That assertion is exactly the fairness property being claimed ('filters with smaller initial magnitudes are not any more preferred'); it is an assumption about ∇L, not a derived consequence. The observed c-bifurcation then merely illustrates the decision rule P={i|Dii>||F_i||} that defines the pruned and preserved classes.

full rationale

Theorem 3.3 (pruning-invariant set as projection of DW=0), Theorem A.5 (function-preserving prune under DW=0), and the convergence/bifurcation analysis of the pure regularizer in Theorem 3.4 are derived from the paper's own definitions and are not circular. The self-citation of the authors' Bypass pipeline [21] is not load-bearing because the lossless contraction used here is re-proved in Theorem A.5. The circular reduction is confined to the advertised fairness guarantee: every filter is initialized at c0=1 by construction, and the proof that all channels have equal pruning chance is replaced by the statement that the pruning decision is 'induced by performance loss L,' with no analysis of whether ∇L's boundary crossing is magnitude-independent. Since Theorem 3.4's c0=1 branch is a fixed point, the fairness prediction does not follow from the stated theory and is effectively an assumption about the unmodeled loss gradient rather than a theorem. The empirical sections show qualitative histograms but do not quantify this escape bias. Score 4 reflects that the pruning mechanism and algebra have independent content, while the paper's strongest theoretical claim is not derived.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central claims rest on unproven assumptions about the loss gradient and on several hand-chosen hyperparameters; the pure-algebra theorems are sound, but their transfer to the algorithm is assumed.

free parameters (5)
  • c (Catalyst scale) = 1
    Multiplier for D_init = c*diag(||F_i||_2); c=1 places all filters on the decision boundary. The paper says practitioners can set c>1 or c<1 to control pruning ratio, but the fairness theory only holds at c=1; c is a tuning knob.
  • gamma_t regularization schedule = 0.007(1+0.25t) (R56); 3e-4(1+0.25t) (R50); [2e-3,9e-3,12e-3](1+0.25t) (VGG19)
    Weights of the Catalyst regularizer, chosen per dataset/model; controls sparsification pressure.
  • epsilon stopping threshold = 1e-6 to 3e-6 depending on dataset and phase
    Threshold on ||DW||_{2,1} for early stopping; chosen by hand and directly affects when pruning happens.
  • kappa bifurcation threshold = 1 (ImageNet), infinity (CIFAR)
    Early-stopping criterion on |log(c_t)|; set differently per dataset with no stated principled rule.
  • weight decay pair (alpha_theta, alpha_D) = (5e-4,5e-5) R56; (5e-4,0) VGG19; (1e-4,0) R50
    Different decay on weights vs catalysts; the theory in Theorem 3.4 assumes a single alpha for both d and M, so the algorithm operates outside the theorem's assumptions.
assumptions (4)
  • domain assumption Sign of d_t and entries of M_t remain unchanged during the relevant training interval
    Used in the proof of Theorem 3.4 (Appendix E, Eq. E1) to derive the c_{t+1} recurrence; not guaranteed in the actual training with loss L.
  • ad hoc to paper The loss gradient causes (D_ii, F_i) to escape the decision boundary unbiasedly
    Stated informally in Section 3.4 ('escapes this decision boundary almost surely'); no proof that the escape direction is independent of filter magnitude, which the fairness claim requires.
  • domain assumption Weight decay term alpha is identical for d and M in Theorem 3.4
    Theorem 3.4 assumes a single alpha; the experiments use alpha_theta != alpha_D (Appendix F), so the theorem does not directly describe the implemented dynamics.
  • domain assumption Comparison baselines in Table 1 are directly comparable despite different pretrained baselines
    Table 1 pools methods with different baseline accuracies (e.g., 93.80 vs 93.53 for ResNet56) and excludes unspecified baselines that use extra training skills, which can change the ranking.
invented entities (1)
  • Auxiliary diagonal catalyst matrices D and D-bar (rendered identically as D in the text)
    purpose: Extends the parameter space so that the pruning-invariant set can be expressed as DW=0, enabling the regularizer and the lossless contraction.
    These are internal training variables, initialized from filter norms and removed after pruning; no falsifiable prediction outside the paper. The duplicate notation for two different matrices hurts clarity but is not itself a flaw.

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Cite this review

Pith. "Pith review of Catalyst: a Novel Regularizer for Structured Pruning with Auxiliary Extension of Parameter Space." pith.science (2026). https://pith.science/paper/CNU4VUF3

@misc{pith2026250714170,
  author       = {Pith},
  title        = {Pith review of: Catalyst: a Novel Regularizer for Structured Pruning with Auxiliary Extension of Parameter Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNU4VUF3}},
  note         = {Machine review of arXiv:2507.14170}
}
read the original abstract

Structured pruning aims to reduce the size and computational cost of deep neural networks by removing entire filters or channels. The traditional regularizers such as L1 or Group Lasso and its variants lead to magnitude-biased pruning decisions, such that the filters with small magnitudes are likely to be pruned. Also, they often entail pruning results with almost zero margin around pruning decision boundary, such that tiny perturbation in a filter magnitude can flip the pruning decision. In this paper, we identify the precise algebraic condition under which pruning operations preserve model performance, and use the condition to construct a novel regularizer defined in an extended parameter space via auxiliary catalyst variables. The proposed Catalyst regularization ensures fair pruning chance for each filters with theoretically provable zero bias to their magnitude and robust pruning behavior achieved by wide-margin bifurcation of magnitudes between the preserved and the pruned filters. The theoretical properties naturally lead to real-world effectiveness, as shown by empirical validations of Catalyst Pruning algorithm. Pruning results on various datasets and models are superior to state-of-the-art filter pruning methods, and at the same time confirm the predicted robust and fair pruning characteristics of Catalyst pruning.

Figures

Figures reproduced from arXiv: 2507.14170 by the authors.

Figure 1
Figure 1. Intuitive plot of geometric mismatch in structured pruning. If [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Simulation on ∥DW∥2,1 minimization, Using the catalyst D and ∥DW∥2,1 requires a proper training algorithm that includes model extension and contraction methods. In Section 3.3 we present Catalyst Pruning algorithm. Interestingly, the constraint for lossless pruning DW = 0 is also a constraint for the contraction to original parameter space. 3.3 bypassing algorithm for Catalyst pruning The algebraic constraint for th… view at source ↗
Figure 3
Figure 3. The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of Resnet50-1.8G model trained on Imagenet. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last layer. The benchmark models using extra skills (and compared to models trained with vanilla settings) were excluded from [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (25 more)
Figure 4
Figure 4. Figure 4: Accuracy graph of Resnet56+CIFAR10 training, compared to Lasso and Group Lasso regularization. To check the proposed pruning operations were lossless, we report the average of test accuracy drop and test loss difference for each pruning operations in opt1 and opt2 in …
Figure 5
Figure 5. Figure 5: Learning curves of CIFAR10 experiment with Resnet56 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of Resnet56 model trained on CIFAR10. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last layer. . H.2 VGG19+C…
Figure 7
Figure 7. Figure 7: The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of VGG19 model (3x speedup) trained on CIFAR100. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last layer. . …
Figure 8
Figure 8. Figure 8: The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of VGG19 model (8.96x speedup) trained on CIFAR100. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last layer.…
Figure 9
Figure 9. Figure 9: The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of VGG19 model (11.84x speedup) trained on CIFAR100. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last layer…
Figure 10
Figure 10. Figure 10: The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of Resnet50 model (1.49x speedup) trained on Imagenet. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last la…
Figure 11
Figure 11. Figure 11: The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of Resnet50 model (1.96x speedup) trained on Imagenet. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last la…
Figure 12
Figure 12. Figure 12: The histograms of ratio c = Dii ∥Fi∥ , filter vector Fi of weight tensor W, and Dii’s for each layers of Resnet50 model (2.33x speedup) trained on Imagenet. The z-axis represents the frequency. The layer indices of results in opt2 are shifted to start from the last la…
Figure 13
Figure 13. Figure 13: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 1st [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 2nd [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]
Figure 15
Figure 15. Figure 15: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 3rd [PITH_FULL_IMAGE:figures/full_fig_p023_15.png]
Figure 16
Figure 16. Figure 16: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 4th [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 5th [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]
Figure 18
Figure 18. Figure 18: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 6th [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 7th [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]
Figure 20
Figure 20. Figure 20: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 8th [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]
Figure 21
Figure 21. Figure 21: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on 9th [PITH_FULL_IMAGE:figures/full_fig_p026_21.png]
Figure 22
Figure 22. Figure 22: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on [PITH_FULL_IMAGE:figures/full_fig_p027_22.png]
Figure 23
Figure 23. Figure 23: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on [PITH_FULL_IMAGE:figures/full_fig_p027_23.png]
Figure 24
Figure 24. Figure 24: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on [PITH_FULL_IMAGE:figures/full_fig_p028_24.png]
Figure 25
Figure 25. Figure 25: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on [PITH_FULL_IMAGE:figures/full_fig_p028_25.png]
Figure 26
Figure 26. Figure 26: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on [PITH_FULL_IMAGE:figures/full_fig_p029_26.png]
Figure 27
Figure 27. Figure 27: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on [PITH_FULL_IMAGE:figures/full_fig_p029_27.png]
Figure 28
Figure 28. Figure 28: Initial magnitude and regularized magnitude of L1, Group Lass and our regularizer on [PITH_FULL_IMAGE:figures/full_fig_p030_28.png]

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Reviewed August 6, 2026 · model on record in the stance chip above.