Variation Brownian Kernel Ladders

• VBKL recursively composes lower-level atoms with unit-ball Brownian RKHS profiles, then forms the depth-L signed-measure variation hull without intermediate convexification.This separates recursive atom geometry, outer linear-combination cost, and finite realization complexity; each dictionary is identified as a union of Brownian pullback RKHS unit balls.Source: §1 · pp. 2, 4
• VBKL differs from recursively convexified variation spaces because nonlinear composition and absolute convexification need not commute, so no general inclusion or equivalence is claimed.Its strict hierarchy and two-stage approximation complement, rather than reproduce, results for Deep Neural Variation Spaces.Source: §1 · p. 3
• The full-space theory establishes compactness, attained minimum-variation representations, variation-controlled 2^-(L-1)-Hölder regularity, and strict depth growth under support-based local non-degeneracy.With full input support, representatives are unique and continuously Hölder embedded.Source: §1 · p. 4
• Every full-space function admits two-stage approximation with M^-1/2 outer discretization error and m^-1/2 profile error, sharp constant sqrt(A/2), and at most 2M active contributions.Outer atom count and profile resolution represent distinct approximation resources, and the sharp interpolation constant is optimal.Source: §1 · p. 4
• Controlled experiments isolate discretization and interpolation mechanisms, compare supervised VBKL realizations with variation-space and kernel baselines, and indicate favorable limited-data accuracy–parameter trade-offs rather than universal dominance.The normalized tent construction exactly attains the worst-case interpolation bound.Source: §1 · p. 4

2 Notation

• Notation sets X ⊆ R^d as a compact input domain with probability measure ν, Ω ⊆ S^{d−1} as admissible directions, and M(Θ) as finite signed measures under total variation.The Brownian kernel is k^(B)(x,x′) = (|x| + |x′| − |x−x′|)/2, with RKHS profiles satisfying g(0) = 0 and g′ ∈ L2(R).Source: §2 · p. 5

2.1 Atomic Brownian Kernel Ladder

• VBKL recursively builds depth-L atoms from one linear projection followed by L−1 normalized Brownian profiles, then applies signed-measure superposition only after completing the path dictionary.Each recursive dictionary is a union of Brownian pullback RKHS unit balls, while variation complexity is the infimal total variation of a representing measure.Source: §2.1 · pp. 5–7

2.2 Finite Lower-Support Architectures

• Finite lower-support architectures replace infinite-dimensional Brownian profiles with grid-based piecewise-linear profiles and bounded mixing/readout parameters, but are not identified with the full infinite-dimensional VBKL variation ball.The construction has PL−1,m,G = md + (L−2)m(G+1) + (L−3)m^2 + m parameters, with normalization constraints ensuring uniformly bounded intermediate supports.Source: §2.2 · pp. 7–10

3.1 Analytical Properties of VBKL spaces

• Full VBKL spaces admit variation-controlled Hölder representatives and pointwise evaluation, with uniqueness and continuous embedding when supp(ν)=X.These results quantify regularity while distinguishing embedding into, rather than equality with, the Hölder space.Source: §3.1 · p. 11
• Strict growth across consecutive depths holds under a local first-layer non-degeneracy condition whose trace lies in supp(ν).The theorem treats this strict-depth conclusion separately from regularity and requires both the local condition and support inclusion.Source: §3.1 · p. 11

3.2 Statistical Properties of Finite-Architecture Brownian Variation Classes

• The finite-architecture Rademacher analysis controls Brownian variation classes through Brownian quadratic chaos and signed threshold entropy of the finite lower-support architecture.The resulting bound separates outer variation radius, recursive Brownian geometry, and the lower-support parameter count, and applies to the finite architecture rather than the full infinite-dimensional variation ball.Source: §3.2 · p. 12, §3.2.1 · p. 13

3.2.1 Statistical Learning Framework

• The finite-architecture framework defines bounded variation-complexity hypothesis classes in which the outer radius controls signed-measure variation while the lower-support architecture controls parameter complexity.The class is built from finite lower-level supports and corresponding outer Brownian dictionary atoms.Source: §3.2.1 · p. 12, §3.2 · p. 12
• Under the theorem’s architecture and sample assumptions, the Rademacher bound yields a high-probability generalization guarantee for bounded Lipschitz losses over the finite Brownian variation class.The guarantee is stated for the finite-architecture class; the same bound extends to the restricted path-only VBKL subclass.Source: §3.2.1 · pp. 12–14

4 Constructive Approximation of VBKL spaces

• The two-stage construction discretizes the outer signed measure and selected outer Brownian profiles while preserving lower-level supports in U_L−1.The resulting approximant has finite outer atomic support and finite outer-profile interpolation; lower-level supports are not discretized.Source: §4 · p. 14, §4.1 · pp. 14, 16, §4.2 · p. 16

4.1 Finite Approximation by Recursive Atomic Representations

• Theorem 4 yields an M^-1/2 + m^-1/2 approximation bound, separating outer-measure discretization from outer-profile interpolation.The M^-1/2 term comes from replacing the outer measure by at most M atoms, while the m^-1/2 term comes from interpolating selected outer Brownian profiles.Source: §4.1 · pp. 15–16
• Evaluating the two-stage approximant requires at most 2M active outer-profile basis contributions, independently of interpolation resolution m.Piecewise-linear hat interpolation has at most two nonzero terms at each evaluation point.Source: §4.1 · p. 15

4.2 Approximation of Brownian Profile Functions

• Uniform-grid interpolation of Brownian RKHS profiles has optimal rate m^-1/2 and sharp constant A/2.The section states that neither the convergence rate m^-1/2 nor the constant A/2 can be improved.Source: §4.2 · p. 16

5.1 Experimental questions and common protocol

• Studies use explicit deterministic teachers for approximation and compare validation-selected VBKL models with DNVS, KRR, and RBF under shared data splits and five random seeds.Supervised results use test mean-squared error, with recursive-path counts, optimization horizon, and, for Energy Efficiency, profile resolution selected by validation.Source: §5.1 · p. 17, §5 · p. 17

5.2 Constructive approximation and interpolation sharpness

• Normalized tent profiles attain the theoretical interpolation bound at every tested resolution, whereas three representative fixed profiles converge substantially faster than the uniform worst-case rate.The tent-profile result demonstrates sharpness of the interpolation estimate and distinguishes worst-case guarantees from fixed-profile asymptotics.Source: §5.2 · pp. 17–18, Figure 1 · p. 18
• With exact profiles, signed-measure discretization follows the M^-1/2 reference; with fixed finite measures, profile refinement is faster than m^-1/2, and balanced M = m = N refinement exhibits N^-1/2 behavior.These experiments separate the roles of outer-measure and outer-profile complexity in the two-stage construction.Source: §5.2 · p. 17, Figure 2 · p. 18

5.3 Statistical learning and parameter efficiency

• The controlled recursive-teacher study finds validation-selected VBKL realizations particularly effective with limited data, while DNVS attains the lowest mean error at the largest sample sizes.Performance differences narrow as sample size grows, consistent with validation adapting explicit finite representation complexity.Source: §5.3 · pp. 18–19, Figure 3 · p. 20
• At n = 100, VBKL has lower mean error than DNVS on Energy Efficiency and wins four of five matched seed-wise comparisons while using approximately 4.6× fewer parameters.At n = 250 and n = 500, DNVS, KRR, and RBF obtain lower mean errors, while VBKL uses 13.7× and 18.3× fewer parameters than DNVS, respectively.Source: §5.3 · p. 19, Figure 3 · p. 20

5.4 Optimization stability and practical realizability

• Finite-difference directional estimates agree increasingly well with automatic differentiation as the interaction scale decreases from 10^-1 to 10^-3, while Monte Carlo averaging reduces estimator variability as sampled directions increase.The tests support stable numerical optimization under the tested conditions but do not establish a global optimization-convergence theorem.Source: §5.4 · p. 19, Figure 4 · p. 21

• VBKL dictionaries are unions of Brownian pullback RKHS unit balls whose variation hulls admit Hölder representatives and form a strict depth hierarchy under local non-degeneracy and trace-support conditions.Under full support, these representatives define a continuous Hölder embedding.Source: §6 · p. 22
• M^-1/2 + m^-1/2 bounds separate outer-measure support discretization from interpolation of selected outer Brownian profiles, with sharp constant √(A/2) and at most 2M active contributions per evaluation.For finite lower-support architectures, Rademacher and generalization bounds separate the outer variation radius from architecture parameter count; lower-level supports remain in U_L−1.Source: §6 · p. 22
• Experiments illustrate the approximation mechanisms and indicate a favorable accuracy–complexity trade-off in limited-data regimes, without establishing universal predictive dominance or global optimization convergence.Future work includes intrinsic full-space statistical bounds, recursive discretization of lower-level supports, and optimization theory for explicit finite architectures.Source: §6 · p. 22

7.1 Proof of Theorem 1(i)

• Theorem 1(i) obtains attainment and regularity by taking near-minimizing signed-measure representations, forming Bochner barycentres, and using Arzelà–Ascoli.Uniform Hölder and pointwise bounds give equicontinuity and boundedness; a uniformly convergent subsequence remains the same L2(ν) element.Source: §7.1 · pp. 22–23

7.2 Proof of Theorem 1(ii)

• Theorem 1(ii) proves uniqueness on supp(ν)=X and a continuous injective map into C0,αL(X).The proof combines the preceding bounds and equality of continuous representatives that agree as L2(ν) elements.Source: §7.2 · p. 23

7.3 Proof of Theorem 1 (iii)

• Theorem 1(iii) establishes strict growth, V(L) ⊊ V(L+1), under local non-degeneracy and trace-support conditions.The contradiction uses a recursively constructed fractional-power witness whose behavior near the trace conflicts with the αL-Hölder regularity of V(L).Source: §7.3 · pp. 24–26

7.4 Proof of Theorem 2

• Theorem 2’s proof derives the stated finite-architecture Rademacher bound by combining variation-hull reduction, envelope estimates, outer-dictionary control, and expectation over random samples.The argument concludes by applying the samplewise estimates to independent draws from ν and taking expected Rademacher complexity.Source: §7.4 · pp. 26–28

7.5 Proof of Theorem 3

• Theorem 3 constructs a finite signed discrete approximation by sampling atoms from the normalized total-variation measure and controlling Hilbert-space variance.Compactness gives a representing Radon measure; signed sampling yields an L2(ν) approximation with support size at most m.Source: §7.5 · pp. 29–31

7.6 Proof of Theorem 4

• Theorem 4’s proof applies Theorem 3 first, then interpolates each selected outer Brownian profile to form the stated two-stage approximant.Each selected atom is represented through a lower-level atom and an RKHS profile, with the nondegenerate case ensuring A0 > 0 before interpolation.Source: §7.6 · p. 32
• Theorem 4’s proof shows that evaluating F_M,m(x) uses at most 2M active outer-profile basis contributions, independently of interpolation resolution m.Each of the M summands contributes at most two active hat functions and corresponding nodal coefficients.Source: §7.6 · pp. 33–34

7.7 Proof of Proposition 1

• Proposition 1 follows by applying Lemma B16 and rewriting (A/2m)^1/2 as (A/2)^1/2 m^-1/2.These steps yield equation (42).Source: §7.7 · p. 34

Appendix A. Experimental Protocols and Extended Results

• The appendix records protocols and numerical details supporting Section 5, while retaining the principal empirical findings and assessment figures in the main body.Source: Appendix A · p. 34
• It reports controlled constructions, validation-selected configurations, exact numerical comparisons, and computational measurements.Source: Appendix A · p. 35

A.1 Constructive-approximation protocol

• The constructive studies use a deterministic teacher F ∈V(L) represented by an explicit finite signed measure on recursive Brownian atoms.Each selected atom has exact recursive support and a deterministic unit-energy Brownian profile.Source: §A.1 · p. 35
• Because the reference representation is known, approximation errors exclude training and statistical-estimation error.Source: §A.1 · p. 35
• The protocol separately varies signed-measure discretization with exact profiles, profile discretization with fixed measures, and balanced refinement with M = m = N.Monte Carlo means and percentile bands describe sampling experiments, while profile-discretization curves are deterministic.Source: §A.1 · p. 35
• Figure 2 isolates the three construction components, while Figure 1 studies profile resolutions m ∈{8, 16, 32, 64, 128, 256, 512}.Figure 1 reports typical fixed-profile behavior and the normalized tent profile attaining the sharp uniform bound.Source: §A.1 · p. 35

A.2 Statistical-learning protocol and extended results

• VBKL architecture selection is performed separately for every training-set size and random seed using a validation set.Source: §A.2 · p. 35
• Each selected configuration is retrained and evaluated on the held-out test set.Source: §A.2 · p. 35
• Results are aggregated over five independent seeds.Source: §A.2 · p. 35
• DNVS, KRR, and RBF follow the same split protocol within each benchmark.Source: §A.2 · p. 35

A.2.1 Recursive-teacher configurations

• The recursive-teacher target uses the proposed model’s hierarchical Brownian mechanism, creating a controlled matched setting.Source: §A.2.1 · p. 35
• Validation jointly selects the number of recursive Brownian paths and the optimization horizon.Source: §A.2.1 · p. 35
• Table 3 reports the selected recursive-teacher configurations in seed order.Source: §A.2.1 · p. 35
• Configuration variation across sample sizes and seeds records the adaptive procedure used to obtain Figure 3a’s learning curve.Source: §A.2.1 · p. 35

A.2.2 Energy Efficiency numerical results

• VBKL validation independently selects path count, Brownian profile resolution, and optimization horizon for every training size and random seed.The architecture and optimization horizon are selected using the validation set.Source: §A.2.2 · p. 35, Table 3 · p. 36
• The Energy Efficiency benchmark applies the same train/validation/test protocol to all compared methods.This common protocol frames the reported cross-method predictive comparison.Source: §A.2.2 · p. 35
• At the two larger training sizes, VBKL has the largest mean test error among the four methods, while at n = 100 its mean is close to the best and below DNVS.Table 4 reports means and standard deviations over five random seeds; predictive performance should be read alongside, not replaced by, parameter-efficiency comparisons.Source: §A.2.2 · p. 35, Table 4 · p. 36
• Table 4 reports Energy Efficiency test MSE as means with standard deviations over five random seeds.These statistics provide the exact numerical basis for Figure 3b.Source: §A.2.2 · p. 35, Table 4 · p. 36

A.3 Optimization protocol

• The consistency experiment compares finite-difference averaged directional estimates with automatic differentiation across interaction scales h ∈{10−1, 10−2, 10−3, 10−4}.Figure 4 shows the usual finite-difference scale trade-off.Source: §A.3 · p. 36
• In the Monte Carlo stability study, increasing sampled directions from 20 to 27 systematically reduces estimator variance.The resulting estimator’s standard deviation is recorded across the sampled-direction settings.Source: §A.3 · p. 36

A.4 Computational characteristics

• Finite VBKL models remain computationally tractable at benchmark scale, although DNVS has lower inference time and feature and kernel baselines have lower training time.The benchmark records training time, complete held-out test inference time, and fitted or trainable parameter count across training sizes and seeds.Source: §A.4 · p. 36
• VBKL’s principal computational distinction is an explicit recursive representation that is substantially more compact than DNVS in the limited-data Energy Efficiency comparison.The parameter-count comparison evaluates representation compactness, with lower values indicating more compact representations.Source: §A.4 · p. 37, Table 8 · p. 38
• Table 5 summarizes mean ± standard deviation training and inference times over all training sizes and seeds, while parameter counts are reported by median when selected architectures vary.The benchmark-specific Tables 6–8 separately report training time, complete held-out test inference time, and parameter counts.Source: §A.4 · p. 36, Table 5 · p. 37, Table 6 · p. 37
• Computational results use five random seeds, with best column entries bolded and second-best entries underlined.Times are reported as means with standard deviations, and parameter counts follow the same format when they vary across selected configurations.Source: §A.4 · p. 36

Appendix B. Additional notation

• Appendix B defines norm-ball and sphere notation, including B_d and S^{d−1}, for the proof framework.It introduces B_r(Z) and S_r(Z) in normed vector spaces and specializes them to Euclidean unit balls and spheres.Source: Appendix B · p. 38
• Appendix B also fixes function-space notation for C(X), Hölder spaces, L∞, L2(ν), Bochner integrals, and indicator functions.These spaces are equipped with their supremum, Hölder, bounded-function, and ν-square-integrable norms or inner products as specified.Source: Appendix B · p. 38

Sharpness theorem

• m^-1/2 and sqrt(A/2) are optimal in the uniform Brownian profile interpolation estimate.The result establishes sharpness of both the exponent 1/2 and the constant sqrt(A/2).Source: Appendix C · pp. 38–39

Extremizer construction

• For every m ≥ 1, an extremizing gm ∈ Hk(B) attains the reverse interpolation inequality on the uniform grid [−A, A].The construction uses a weak derivative on each interpolation interval, yielding equal endpoint values and a constant piecewise-linear interpolant.Source: Appendix C · pp. 39–40

Lemma B1

• Lemma B1 proves that finite signed-measure Bochner integrals define elements of L2(ν), making V(L) well-defined with finite nonnegative variation complexity.Strong measurability, the uniform atomic bound, and finite total variation ensure Bochner integrability.Source: Appendix D · p. 41

Lemma B2

• Lemma B2 establishes non-degeneracy: zero variation complexity implies F = 0 in L2(ν).The proof uses the bound ∥F∥L2(ν) ≤ R_L bC_var(L).Source: Appendix D · pp. 41–42

Lemma B3

• Lemma B3 characterizes finite profiles as continuous, gridwise affine, anchored Brownian-RKHS functions that are constant outside [−A_X, A_X], with exact RKHS norms.The finite architecture has parameter count P_{L,m,G} = md + (L −2)m(G + 1) + (L −3)m^2 + m.Source: Appendix D · pp. 42, 47

Lemma B4

• Lemma B4 identifies each recursive dictionary component with a Brownian pullback RKHS represented through the orthogonal complement of the composition operator’s kernel.For u ∈ U_{l−1}, the kernel nullspace is closed, and every pullback-RKHS function has a unique representative in its orthogonal complement.Source: Appendix D · p. 47

Exact dictionary representation

• V(l) is exactly the variation hull generated by the recursively constructed union of Brownian pullback RKHS unit balls.The pullback construction preserves minimum-norm representatives and yields the recursive dictionary identity.Source: This representative satisfies · pp. 48, 50

Variation-hull complexity

• For any symmetric dictionary, the empirical Rademacher complexity of its signed-measure variation hull equals the dictionary complexity scaled by the variation radius.The proof uses Dirac measures for the reverse inequality and total-variation bounds for the upper inequality.Source: This representative satisfies · pp. 50, 53

Finite Brownian specialization

• The finite outer Brownian dictionary is symmetric because each constituent RKHS unit ball is symmetric, enabling the preceding exact reduction.The specialization also bounds evaluations using the Brownian pullback kernel and reproducing-property estimates.Source: This representative satisfies · pp. 51, 53, 54

Variation-hull complexity

• The variation-hull Rademacher complexity is bounded by the variation radius times the dictionary envelope.This gives the stated envelope control for finite samples.Source: This representative satisfies · p. 53

Lemma B6

• Lemma B6 proves that unions of Brownian pullback RKHS balls are symmetric and admits an exact fixed-support RKHS reduction leading to a Brownian quadratic-chaos Rademacher bound.The proof uses Hilbert-space duality, reproducing properties, Jensen’s inequality, and diagonal/off-diagonal decomposition; it also specializes the result to finite lower-support architectures.Source: Lemma B6 (Rademacher bound for unions… · pp. 54, 56, 57, 58

Lemma B7

• Lemma B7 bounds the expected maximum of finitely many Rademacher threshold sums by a universal constant times ln(1 + N).The argument combines exponential moment bounds, a union bound, and tail integration over the finite family of N subsets.Source: Lemma B6 (Rademacher bound for unions… · pp. 58, 60

Lemma B8

• Lemma B8 controls Brownian quadratic chaos through the finite signed threshold trace family induced by the support-function class.It represents the Brownian kernel on [−B, B] using positive and negative threshold integrals, then applies Lemma B7 to obtain the resulting complexity control.Source: Lemma B6 (Rademacher bound for unions… · pp. 61, 63

Lemma B9

• Lemma B9 establishes a VC-dimension bound for signed threshold classes from finite architectures, with a constant C_L depending only on depth L.The proof enlarges the admissible parameter set, adds the threshold as one coordinate, and counts polynomial sign patterns generated by the recursive semialgebraic architecture.Source: Lemma B6 (Rademacher bound for unions… · pp. 63–64

Signed threshold entropy

• The signed threshold entropy satisfies the bound C_L(P_{L−1,m,G}+1) ln(e(P_{L−1,m,G}+1)) ln(en), as stated in (258).Source: Lemma B6 (Rademacher bound for unions… · pp. 69, 71

Brownian chaos bound

• The argument establishes finite Bx, controls signed threshold entropy via Lemma B10, and applies Lemma B8 to the support class A with envelope Bx.The cited bounds introduce constants depending only on L.Source: Lemma B6 (Rademacher bound for unions… · pp. 71–72

Attainment of variation complexity

• If U_L is compact in L2(ν), every F ∈ V(L) attains its variation complexity through a finite signed Radon measure μ_F on U_L.A minimizing sequence of representing measures has uniformly bounded total variation; weak-* sequential compactness and continuity of the representation map produce an attaining measure.Source: Lemma B6 (Rademacher bound for unions… · pp. 73–74

Compactness of restricted Brownian profiles

• The restricted Brownian profile class G_A is compact in C([-A,A]) because its elements are uniformly 1/2-Hölder continuous and uniformly bounded.Arzelà–Ascoli yields uniform subsequential convergence, while weak L2 compactness identifies the limit as another admissible Brownian RKHS profile.Source: Lemma B6 (Rademacher bound for unions… · pp. 75, 76, 78

Compactness of the atomic VBKL dictionary

• For every compact input domain X and compact direction set Ω, each recursive atomic dictionary U_L is compact in C(X) and therefore compact in L2(ν).The proof inducts over recursion depth, combining compact lower-level supports with compact restricted Brownian profile classes; the embedding into L2(ν) is continuous when ν is a probability measure.Source: Lemma B6 (Rademacher bound for unions… · pp. 78–80

Brownian profile interpolation

• Uniform piecewise-linear interpolation of Brownian profiles provides an RKHS-controlled approximation error, and the same bound transfers to compositions g∘a over measurable a:X→[−A,A].Lemma B16 derives the uniform-grid interpolation estimate, while Lemma B18 applies it pointwise and then integrates using ν(X)=1.Source: Lemma B6 (Rademacher bound for unions… · pp. 80–81

Uniform range bound for lower-level atomic supports

• At lower recursive levels, every a ∈ U_{L−1} satisfies |a(x)| ≤ R_X 2^−(L−2), so its range lies in the bounded interval [−A_0,A_0].This depth-dependent range control follows from the unit direction bound at L=2 and the recursive atomic pointwise estimate for L≥3.Source: Lemma B6 (Rademacher bound for unions… · pp. 80–81