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REVIEW 6 major objections 5 minor 17 references

A Hilbert-space norm on neurons unifies pruning, merging, and block eviction into one data-free decision metric.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 07:39 UTC pith:EL6RNSIF

load-bearing objection A genuinely synthetic theory of data-free compression, but the headline empirical claim is unverifiable in this version. the 6 major comments →

arxiv 2607.21366 v1 pith:EL6RNSIF submitted 2026-07-23 cs.LG cs.AIstat.ML

Hilbert Operator for Progressive Encoding (HOPE): A Mathematical Framework for Deconstructing Learned Representations in Deep Networks

Hossein Mobahi , Peter L. Bartlett This is my paper
classification cs.LG cs.AIstat.ML
keywords neural network compressionHilbert-Schmidt operatorsneuron pruningneuron mergingdata-free compressionbatch normalizationrate-distortiontransfer learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Trained deep networks hide internal knowledge in weights, and compression is the proposed lens for pulling that knowledge out. This paper tries to establish that each neuron can be treated as a continuous operator in a Hilbert space—a function of the data distribution—whose norm is a meaningful measure of the neuron's capacity. From that one norm, pruning, neuron merging, and whole residual-block removal become the same low-rank projection, scored by a single distortion cost and selected greedily by distortion per parameter. The framework is data-free when batch-norm statistics are available, because a maximum-entropy Gaussian surrogate turns the required integrals into closed forms. If the central claim holds, architectural decisions across layers of different types and sizes become directly comparable without data or hyperparameters, and the paper's proof-of-concept experiments report that this beats magnitude-based pruning and improves transfer learning.

Core claim

The paper's central claim is that a neuron's identity can be lifted from its raw parameters to a rank-1 Hilbert-Schmidt operator f_i = g_i ⊗ w_out,i, where g_i(x) = ReLU((w_eff_in,i)^T x + b_i) is the activation landscape after absorbing batch-norm statistics, and w_out,i is the output weight vector. The Hilbert norm of this operator, ∥f_i∥_H = ∥w_out,i∥_2 √K(i,i), is proposed as a scale-invariant, shape-invariant capacity score: the kernel K(i,i) is the expected squared activation under a Gaussian surrogate P_X = N(μ̂_x, Σ̂_x) fitted to batch-norm means and variances, and positive homogeneity of ReLU makes opposing rescalings cancel. With this capacity, pruning is projection onto the zero o

What carries the argument

The load-bearing construction is the neuron-as-operator embedding with the Hilbert-Schmidt inner product ⟨f_i,f_j⟩_H = K(i,j)⟨w_out,i,w_out,j⟩_ℝ^c. The kernel K(i,j) = E_{x∼P_X}[ReLU(y_i)ReLU(y_j)] has closed forms: a self-kernel with bias (Equation 3) and a zero-bias cross-kernel approximation (Equation 5) using a warped correlation ρ̂_ij. The capacity norm factors into a discrete output norm and a continuous activation energy, which is what cancels within-neuron scaling. From this kernel the paper derives the layer-state cost functional J through axioms and a continuous-time relaxation whose straight-line upper bound gives J_prune = N‖f_i‖_H/(E_a−‖f_i‖_H), J_merge = N√(‖f_i−f_p‖²_H + ‖f_j−

Load-bearing premise

Everything rests on the surrogate that every neuron's pre-activation is Gaussian with mean β_i and variance γ_i² computed from batch-norm statistics, and especially on the zero-bias approximation in the cross-kernel; if real pre-activations deviate (post-ReLU non-negativity is acknowledged as a paradox), the closed-form kernel values, capacities, and all compression costs built on them lose their foundation.

What would settle it

Measure the true distribution of pre-activations on a real trained network: for each neuron compute empirical E[ReLU(y)²] from a held-out batch and compare it to the closed-form self-kernel K(i,i). If the relative error is large for a substantial fraction of neurons, or if pruning orders flip when the Gaussian surrogate is replaced by the empirical distribution, the central claim is falsified. A second check: if HOPE's claimed accuracy advantage over magnitude pruning disappears when batch-norm statistics are perturbed (e.g., re-estimated on shifted data), then the data-free premise has not he

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • One decision rule can compare pruning a single neuron, merging a pair, and deleting a whole residual block, removing the need for per-layer heuristics or hyperparameter sweeps.
  • Because the capacity norm is invariant to weight resharding and batch-norm rescaling, neurons in different layers can be ranked on the same scale, so global compression decisions are no longer biased by layer width or tensor shape.
  • The greedy rate-distortion selection (cost per parameter released) with a static parameter footprint avoids the fragmentation trap where dynamically shrinking layer sizes repel later compression.
  • The paper's transfer protocol DEFT uses the same capacity scores to freeze a 'core' and plasticize a 'slack', claiming to balance source retention and target accuracy better than full fine-tuning, head-only fine-tuning, PEFT, and EWC.
  • The framework evaluates each candidate action in O(1) time and performs O(N) local updates, making progressive compression computationally feasible at large scale.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the Gaussian pre-activation assumption holds beyond batch-norm networks, HOPE's capacity score becomes a candidate universal importance measure for any network layer, including Transformers, whenever marginal statistics can be calibrated—a testable extension the paper does not itself establish.
  • The zero-bias approximation in the cross-kernel is a clear stress point: layers with large shifts β will have K(i,j) misestimated, and a direct comparison against the exact bivariate-normal kernel would show whether merge decisions actually change.
  • Since J is analytic, one could differentiate the compression objective with respect to architecture widths and use capacity thresholds to set layer sizes automatically, converting compression into a design principle rather than a post-hoc cleanup.
  • The core/slack partition from DEFT suggests a data-free continual-learning prior: before seeing a new task, the network could protect exactly the operators that progressive encoding has shown resist removal, which may be more reliable than Fisher-information estimates.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

6 major / 5 minor

Summary. The paper presents HOPE, a framework for data-free network compression. Each neuron is embedded as a rank-1 Hilbert–Schmidt operator f_i = g_i ⊗ w_out,i in L2(X,P_X;R^c); the surrogate P_X is a maximum-entropy Gaussian constrained by Batch Normalization statistics. Self- and cross-kernels are obtained in closed form, and pruning, neuron merging, and residual-block eviction are cast as low-rank subspace projections with a scale-invariant distortion cost J. A greedy distortion-rate criterion k* = argmin_k J_k/ΔP_k selects actions in a progressive encoding loop. A second contribution, DEFT, uses pruning costs to split the network into a frozen core and plastic slack for transfer learning. The paper claims that HOPE yields superior compression accuracy and that DEFT outperforms baselines on CIFAR-100→SVHN transfer.

Significance. The Hilbert–Schmidt operator formulation is a genuinely attractive way to unify pruning and merging: the rank-1 tensor structure gives a natural capacity norm, and the axiomatic derivation of J from scale invariance and connectivity preservation is worked out in unusual detail in Appendices C–F. The proposed closed-form kernels (Eqs. 3 and 5) are elegant, and the DEFT bounds in Appendix H are a serious attempt at theoretical guarantees. If the Gaussian-surrogate and zero-bias approximations could be validated empirically, the framework would be a substantial contribution to interpretability-oriented compression. At present, however, the central empirical and technical claims rest on unverified assumptions and a missing experiment.

major comments (6)
  1. [Section 11.1] The only compression experiment is not present. The text says 'As demonstrated in the plot' and then claims 'HOPE yields models with superior accuracy compared to the baselines', but the manuscript contains no plot, table, or numeric result for this experiment. Since this is the central evidence for the paper's headline claim, the figure/data must be supplied before the claim can be assessed.
  2. [Section 4 and Eq. (5)] The entire kernel machinery is built on the maximum-entropy Gaussian surrogate P_X=N(μ̂_x,Σ̂_x) and the derived pre-activation distribution y_i∼N(β_i,γ_i^2) (Appendix E.1). The 'Post-ReLU Support Paradox' is handled by a CLT/Diaconis–Freedman argument for 2D projections; learned weight directions are not random projections, and the fact that the 1D marginals are Gaussian does not determine E[ReLU(y_i)ReLU(y_j)] for non-Gaussian, non-negative inputs. Moreover, the implemented cross-kernel (5) assumes β_i,β_j≈0 even though the self-kernel (3) keeps β_i. Trained BN biases are typically nonzero, so this approximation is systematically biased. No comparison with the exact bivariate kernel (Appendix E.3.2) or with empirical kernel estimates is given. Because J_prune, J_merge, J_evict, and the selection rule (23) all inherit these kernels, the data-free capacity claims are currently unsupported
  3. [Section 6.2, Lemma C.3] The discrete transition cost J_bound is proven as an upper bound only under the condition ρ_ij≥ρ*. The 'Correlation Constraint' practical note argues that the condition is automatically satisfied because the greedy optimizer minimizes J_bound and hence selects highly correlated pairs. This is circular: the optimizer uses J_bound to select actions, while the validity of J_bound as an upper bound is exactly what must be checked for the selected action. No value or estimate of ρ* is given, and no fallback is provided for pairs below the threshold. The merge cost in Eq. (6) is therefore not a demonstrated upper bound on the true projection cost for all pairs considered by the algorithm.
  4. [Appendix F.3 and Eq. (20)] For non-residual architectures, block eviction uses the initial capacity E_init as denominator. The text explicitly concedes that this is 'a heuristic', that it 'violates the Markov property' of the continuous state transition, and that it 'departs from Axiom 2'. Since block eviction is advertised as part of the single unified, hyperparameter-free metric used in Eq. (23), this admission contradicts the paper's central claim. The heuristic must either be derived from the same axioms for non-residual blocks or be clearly labeled as outside the rigorous framework and validated separately.
  5. [Appendix B.3, Eq. (39)] The correction for cross-action parameter overlap assumes a uniform scaling factor α such that ΔP_init≈αΔP_live for all actions, and that this preserves the ordering J_a/ΔP_init_a < J_b/ΔP_init_b. Overlap is a function of local connectivity and layer position (e.g., a 1×1 filter and a 3×3 filter share different numbers of weights with their neighbors), so there is no evident reason for α to be constant across the action space. Without this uniformity, the ordering can change and the 'unbiased' global comparison in Eq. (23) is not justified.
  6. [Section 11.2, Appendix G, Table 2] The source-retention protocol grafts the original classification head back and applies a mask that zeros out connections from upstream slack neurons before measuring source accuracy. This mask is a DEFT-specific mechanism; if it is applied only to DEFT, the comparison is not apples-to-apples, since baseline methods are evaluated with the target-adapted weights still active. The text does not state clearly whether the same protocol is used for all baselines. Please clarify the exact evaluation protocol and, if necessary, report an ablation without the mask.
minor comments (5)
  1. [Abstract and Section 11.2.2] The abstract claims HOPE is 'hyperparameter-free', but DEFT introduces percentile P (Eq. 26) and the compression cost uses the choice c(Φ)=N (Section 6). Clarify the scope of the claim.
  2. [Notation] Eq. (24) uses E_b for remaining capacity, while Eq. (27) uses E_i for elasticity; the same letter E is overloaded. Please rename one.
  3. [Appendix B.1] 'absorbs these edge-effects into a into a single global average' contains a duplicated 'a'.
  4. [Section 11.1] If the missing plot is added, define 'density' precisely and report error bars or multiple runs; the current text gives no uncertainty information.
  5. [Section 9] The transition from binary knapsack to continuous relaxation and then to single-step greedy selection is not fully described; clarify how the one-step selection relates to the continuous relaxation.

Circularity Check

1 steps flagged

HOPE's derivation is mostly self-contained, but the validity of its core cost Jbound is justified by a self-referential correlation assumption.

specific steps
  1. other [Section 6.2, 'Practical Notes — The Correlation Constraint']
    "The assumption E(t)≥E(Φ_b) holds only for highly correlated neurons, but this poses no practical limitation. Because the projection error D(Φ_a,Φ_b) vanishes for collinear candidates, the greedy optimizer naturally minimizes Jbound by actively selecting highly correlated pairs, inherently satisfying this requirement."

    Jbound is derived as a valid upper bound only under Lemma C.3's correlation threshold. To establish that this condition is satisfied, the paper appeals to the greedy optimizer's behavior of minimizing Jbound. But Jbound is precisely the quantity whose validity depends on the correlation condition. This is a self-referential validation: the cost is assumed valid to justify the selection rule, and the selection rule is then used to justify the cost's validity. The same correlation assumption underlies the first-order parent-neuron linearization in Section 7.1, so the loop is load-bearing for the merge cost and parent construction.

full rationale

The central HOPE derivation — Gaussian maximum-entropy surrogate from BN statistics, closed-form self/cross kernels, Hilbert-Schmidt capacities, and pruning/merging/eviction costs — is not equivalent to its inputs. It is an analytical consequence of stated modeling assumptions (Gaussian pre-activations, zero-bias cross-kernel, PH-1 activations), not a fitted parameter renamed as a prediction. The claimed 'superior accuracy' result is currently unverifiable because Section 11.1 references a plot that is absent, but missing evidence is not circularity. The one genuine circular element is the Correlation Constraint: the upper-bound cost Jbound is only guaranteed under a high-correlation condition, and the paper argues this condition is automatically met because the greedy optimizer minimizes Jbound, whose own validity is what is at issue. This is a self-referential validation rather than a reduction of a prediction to a fit, so it does not collapse the framework. Self-citations (e.g., Qiu et al. 2024 for H-Score) are not load-bearing, and the Post-ReLU Support Paradox remains an unvalidated assumption rather than a circular step.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 3 invented entities

The framework rests on BN-derived Gaussian surrogates, a zero-bias cross-kernel approximation, design axioms for J, and two admitted heuristics (c=N and E_init). None of these is externally benchmarked. The invented entities are mathematical constructs rather than new physics.

free parameters (3)
  • c(Φ)=N layer-count normalization = N (number of active neurons)
    Section 6.2 'Choice of c(Φ)=N': chosen by hand to remove width bias; the axioms only constrain c to be scale-invariant and piecewise constant, so N is a modeling choice, not derived.
  • DEFT percentile P = P∈{60,40,30,20} swept
    Section 11.2.2/DEFT threshold J_lock uses a target percentile P; this is a user hyperparameter, contradicting the 'hyperparameter-free' claim for the fine-tuning method.
  • Zero-bias cross-kernel approximation = β_i≈0
    Equation (5) footnote and Appendix E.3.3 set biases to zero to avoid bivariate CDF evaluations; this ad-hoc simplification changes K(i,j) for all merges and is not empirically validated.
axioms (7)
  • domain assumption Pre-activations y_i are Gaussian: y_i ~ N(β_i, γ_i^2)
    Section 4 Step 1 and Appendix E.1; CLT/Diaconis-Freedman is invoked, but non-linear activations and low-dimensional structure are not shown to preserve Gaussianity of the full input vector.
  • ad hoc to paper Gaussian surrogate P_X can cover the full space even though post-ReLU inputs are non-negative
    Section 4 'Post-ReLU Support Paradox': the 2D-slice CLT argument is a plausibility argument, not a proof; kernel values and all J costs depend on it.
  • ad hoc to paper Cross-kernel biases vanish: β_i≈0
    Equation (5): the approximate cross-kernel drops biases to avoid bivariate normal CDFs; no validation is provided.
  • domain assumption Kernel factorization K(x,y)=||y|| k(ρ) under an isotropic reference distribution
    Proposition C.8 assumes an isotropic probability distribution; the BN-constrained surrogate is generally not isotropic after the pseudo-inverse construction in Section 4.
  • standard math The universal approximation theorem makes the neuron set total in L2(X,P_X)
    Appendix A.5 Theorem A.2 applies UAT to ensure span(N) is dense; this requires the surrogate measure to support such approximations, which is not proven for Gaussian over R^n.
  • domain assumption Dantzig item independence and uniform overlap scaling α for static ΔP
    Section 9 and Appendix B.3: cross-action parameter overlap is assumed to act as a uniform scale factor; the ordering-preservation argument relies on this unproved uniformity.
  • ad hoc to paper Non-residual block eviction can use initial capacity E_init as denominator
    Appendix F.3 explicitly admits this violates the Markov property and departs from Axiom 2; it is a heuristic to avoid the extinction divergence.
invented entities (3)
  • Surrogate Gaussian input distribution P_X = N(μ̂_x, Σ̂_x) no independent evidence
    purpose: Enables closed-form Hilbert-space integrals without data by matching BN mean/variance constraints.
    The distribution is uniquely determined by a max-entropy argument, but its fidelity to the true activation distribution is untested; BN statistics are the only anchor.
  • Rank-1 Hilbert-Schmidt neuron f_i = g_i ⊗ w_out,i no independent evidence
    purpose: Abstract neuron as a continuous operator so pruning/merging become low-rank projections.
    A mathematical representation, not an empirical entity; it is a modeling choice.
  • E_identity, the capacity of identity skip operators no independent evidence
    purpose: Provides a non-zero denominator for block eviction in residual networks.
    Computed from downstream BN gamma/beta values; no external validation that RMS energy equals survival capacity.

pith-pipeline@v1.3.0-alltime-deepseek · 59953 in / 15538 out tokens · 158217 ms · 2026-08-01T07:39:04.767114+00:00 · methodology

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read the original abstract

Deep neural networks encode complex representations, but deconstructing this internal knowledge remains a challenge. Given the link between learning and compression, network compression offers a promising lens to analyze this knowledge. However, standard compression heuristics often suffer from scale symmetries and architectural biases. To resolve these, we introduce Hilbert Operator for Progressive Encoding (HOPE), a mathematical framework to gradually deconstruct the representations in trained network weights. HOPE shifts network compression from the discrete domain into a Hilbert space of continuous functions. By modeling individual neurons as rank-1 Hilbert-Schmidt operators, HOPE unifies pruning and neuron merging as low-rank subspace projection. Extending this formulation, HOPE introduces macro block eviction to encompass multi-layer structures like entire residual pathways under the same unified metric. This unified approach enables unbiased architectural decisions across layers with different types and sizes. HOPE is a data-free and hyperparameter-free framework. We present proof-of-concept experiments in model compression and fine-tuning to highlight the practical potential of our theory.

discussion (0)

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Reference graph

Works this paper leans on

17 extracted references

  1. [1]

    ASpacethat contains all possible neural identities

  2. [2]

    closeness

    AMetric(Inner Product) to measure "closeness" and "alignment." 3.Completenessto ensure that our optimizations actually have solutions. A Hilbert spaceHprovides these three pillars. A.2. The Inner Product: The Ruler of Geometry The defining feature of a Hilbert space is theinner product. While a vector space only lets us add and scale elements, an inner pr...

  3. [3]

    We embed this function into a scalar Hilbert spaceHin ≜𝐿 2(X,𝑃X;ℝ)

    The Continuous Input Landscape:The effective input weights (which absorb BN statistics) and the activation function create a continuous scalar landscape𝑔𝑖(𝒙)=Ψ((𝒘 eff in,𝑖)𝑇 𝒙+𝑏 𝑖). We embed this function into a scalar Hilbert spaceHin ≜𝐿 2(X,𝑃X;ℝ)

  4. [4]

    We defined the inner product for any functions𝑓,𝑔 . But in HOPE, we only ever calculate it for single ReLU neurons. Is that enough to define the whole space?

    The Discrete Output:This scalar activation is broadcast to the next layer along a finite- dimensional output weight vector𝒘out,𝑖. We define this output space asHout ≜ℝ 𝑐. By taking the tensor product of these two spaces, we construct the full ambient space mapping: H H in⊗H out. Under this formulation, each individual neuron is modeled as arank-1 Hilbert-...

  5. [5]

    3.Partition Invariance:∀𝑓∈H,∀𝑁∈ℤ ≥1, 𝐸((𝑓))=𝐸 (𝑓/𝑁,..., 𝑓/𝑁 | {z } 𝑁times )

    Symmetry & Separability:∃𝑔 : ℝ≥0→ℝ (continuous and strictly monotonic) and a functionℎsuch that∀Φ∈H 𝑁, 𝐸(Φ)=ℎ Í𝑁 𝑘=1𝑔(∥𝑓𝑘∥H) . 3.Partition Invariance:∀𝑓∈H,∀𝑁∈ℤ ≥1, 𝐸((𝑓))=𝐸 (𝑓/𝑁,..., 𝑓/𝑁 | {z } 𝑁times ) . Then𝐸(Φ)= Í𝑁 𝑘=1∥𝑓𝑘∥H. Proof.For a single-neuron stateΦ=(𝑓), Conditions 1 and 2 imply: 𝐸((𝑓))=ℎ 𝑔(∥𝑓∥H) =∥𝑓∥H =⇒ℎ≡𝑔 −1 on Im(𝑔) Thus, the functional sim...

  6. [6]

    By Lemma C.1, capacity scales linearly𝐸(𝑘Φ)=𝑘𝐸(Φ) , so¤𝐸(𝑘Φ(𝑡))=𝑘 ¤𝐸(𝑡)

    Magnitude Neutrality:For any scalar 𝑘 > 0, Magnitude Neutrality requiresJ(𝑘Φ 𝑎,𝑘Φ 𝑏)= J(Φ 𝑎,Φ𝑏). By Lemma C.1, capacity scales linearly𝐸(𝑘Φ)=𝑘𝐸(Φ) , so¤𝐸(𝑘Φ(𝑡))=𝑘 ¤𝐸(𝑡). Integrating over the scaled path yields: ∫ 1 0 −𝜉(𝑘Φ(𝑡))𝑘 ¤𝐸(𝑡)𝑑𝑡= ∫ 1 0 −𝜉(Φ(𝑡)) ¤𝐸(𝑡)𝑑𝑡 Assumingcontinuousintegrands, sincethisequalityholdsforanyvalidcontinuouspath, theintegrands must...

  7. [7]

    Scaling the state by𝑘yields: 𝑐(𝑘Φ)=𝜉(𝑘Φ)𝐸(𝑘Φ)=(𝑘 −1𝜉(Φ))(𝑘𝐸(Φ))=𝑐(Φ) This shows𝑐(Φ) is scale-invariant

    Scale-Invariant Factor:Define𝑐(Φ)≜𝜉(Φ)𝐸(Φ). Scaling the state by𝑘yields: 𝑐(𝑘Φ)=𝜉(𝑘Φ)𝐸(𝑘Φ)=(𝑘 −1𝜉(Φ))(𝑘𝐸(Φ))=𝑐(Φ) This shows𝑐(Φ) is scale-invariant. Substituting𝜉(Φ)=𝑐(Φ)/𝐸(Φ) back into the differential form gives the integral: J(Φ 𝑎,Φ𝑏)= ∫ 1 0 −𝑐(Φ(𝑡)) ¤𝐸(𝑡) 𝐸(Φ(𝑡)) 𝑑𝑡 43 Hilbert Operator for Progressive Encoding (HOPE)

  8. [8]

    Connectivity Preservation:This axiom mandates an infinite cost barrier against layer extinction: lim𝐸𝑏→0+J(Φ 𝑎,Φ𝑏)=∞ , where 𝐸𝑏 =𝐸(Φ 𝑏) and 𝐸𝑎 =𝐸(Φ 𝑎). Applying the change of variables 𝑑𝐸= ¤𝐸(𝑡)𝑑𝑡 and reversing the limits (which absorbs the negative sign since𝐸𝑏 < 𝐸𝑎 due to¤𝐸(𝑡)< 0) gives: J(Φ 𝑎,Φ𝑏)= ∫ 𝐸𝑏 𝐸𝑎 −𝑐(Φ) 𝑑𝐸 𝐸 = ∫ 𝐸𝑎 𝐸𝑏 𝑐(Φ) 𝑑𝐸 𝐸 If𝑐(Φ)is bounded...

  9. [9]

    Thus, 𝑘(1)=𝑘 ′(1)

    Substituting these yields𝑘( 1)= 1 2(𝐶 2 ++𝐶 2 −) and 𝑘′(1)= 1 2(𝐶 2 ++𝐶 2 −). Thus, 𝑘(1)=𝑘 ′(1). □ Proposition C.10(Kernel Positivity at Boundary).For any non-trivial PH-1 functionΨ(𝑥). 0, the induced angular kernel satisfies𝑘(1)>0. Proof. As derived in Proposition C.9,𝑘( 1)= 1 2(𝐶 2 ++𝐶 2 −). Since Ψ is non-trivial, at least one of𝐶+ or 𝐶− is non-zero. C...

    2009

  10. [10]

    2.Cauchy-Schwarz Compliance:The magnitude is bounded:|𝐾(𝑖, 𝑗)|≤ √︁ 𝐾(𝑖,𝑖)𝐾(𝑗, 𝑗)

    Diagonal Consistency:If ˆ𝜌𝑖𝑗 = 1and the marginals are identical (𝛽𝑖 =𝛽 𝑗,|𝛾𝑖|=|𝛾 𝑗|), the interaction recovers the self-kernel:𝐾(𝑖, 𝑗)=𝐾(𝑖,𝑖). 2.Cauchy-Schwarz Compliance:The magnitude is bounded:|𝐾(𝑖, 𝑗)|≤ √︁ 𝐾(𝑖,𝑖)𝐾(𝑗, 𝑗). 3.Weight-Space Correlation Dependency:The interaction is monotonic with respect toˆ𝜌𝑖𝑗. E.3.2. Exact Bivariate Cross-Kernel for Bias...

    2017

  11. [11]

    Block 3:Conv2D (512 filters, 3x3)→ BN→ ReLU→ Conv2D (512 filters, 3x3)→ BN→ ReLU →MaxPool (2x2)

  12. [12]

    Note that BN layers are positionedbetweenthe affine transformations (Conv2D/Dense) and the ReLU non-linearities

    Block 4:Conv2D (512 filters, 3x3)→ BN→ ReLU→ Conv2D (512 filters, 3x3)→ BN→ ReLU →MaxPool (2x2) 5.Transition:GlobalAveragePooling2D 6.Bottleneck:Dense (512 neurons)→BN→ReLU 7.Classification Head:Dense (10 or 20 neurons) for multi-class logit output. Note that BN layers are positionedbetweenthe affine transformations (Conv2D/Dense) and the ReLU non-lineari...

  13. [13]

    Thus,∥𝒘core,𝑗∥2≤∥𝒘 out,𝑗∥2

    Bounded Initialization Shock (Theorem H.1).Severing the slack-to-core connections at𝑡= 0 introduces a static error.As visualized in Figure 5(a), the severed core-directed weights𝒘core,𝑗 of a slack neuron𝑗 form a sub-vector of its total outgoing weights𝒘out,𝑗. Thus,∥𝒘core,𝑗∥2≤∥𝒘 out,𝑗∥2. Because we only sever connections from slack neurons (where∥𝑓 𝑗∥H ≤𝜏 ...

  14. [14]

    Dynamic Decoupling (Theorem H.2).During training𝑡 >0, slack neurons drift to learn the target task. Because the structural mask severs cross-connections at initialization𝒘(0) core,𝑗 = 0and the zero elasticity prevents gradient updates𝜕𝒘core,𝑗/𝜕𝑡= 0, these weights remain zero.As illustrated in Figure 5(b), a changing signal multiplied by zero is zero; thus...

  15. [15]

    Deep networks often fragment a feature across𝑀 correlated neurons

    Freeing Space Safely (Proposition H.4).Freezing the core might leave insufficient parameter space for the target task. Deep networks often fragment a feature across𝑀 correlated neurons. Statically freezing them incorrectly locks redundant volume. Instead, DEFT compresses these𝑀 62 Hilbert Operator for Progressive Encoding (HOPE) neurons into a single rank...

  16. [16]

    However, because Theorem H.2 guarantees zero interference during training, the network’s total error does not compound exponentially

    Unified Cumulative Bound (Corollary H.5).Cutting connections and merging neurons across layers introduces multiple distortions. However, because Theorem H.2 guarantees zero interference during training, the network’s total error does not compound exponentially. By the triangle inequality inH(𝑙), the global degradation in the function space is static and b...

  17. [17]

    Therefore, the functional distortion is strictly bounded by: Δ𝒔init core H(𝑙) ≤ ∑︁ 𝑗∈N(𝑙) slack 𝜏(𝑙) =𝜏(𝑙)|N(𝑙) slack|(112) □ H.2.1

    Substituting these properties gives: Δ𝒔init core H(𝑙) ≤ ∑︁ 𝑗∈N(𝑙) slack 𝒘out,𝑗 2 √︁ 𝐾(𝑗, 𝑗)= ∑︁ 𝑗∈N(𝑙) slack ∥𝑓 𝑗∥H(𝑙) (111) By Definition H.1, the capacity of every slack neuron satisfies∥𝑓 𝑗∥H(𝑙)≤𝜏(𝑙). Therefore, the functional distortion is strictly bounded by: Δ𝒔init core H(𝑙) ≤ ∑︁ 𝑗∈N(𝑙) slack 𝜏(𝑙) =𝜏(𝑙)|N(𝑙) slack|(112) □ H.2.1. Dynamic Decoupling T...