Common Covariance Geometry for Brownian Kernel Ladders

Overview

A fixed RKHS produces one empirical Gram matrix and one ellipsoid of attainable predictions on a sample. A representation-adaptive kernel class is different: the kernel may be selected together with the predictor, so the empirical class is a union of RKHS ellipsoids. This project asks whether that complete union can be represented by one positive-semidefinite covariance without erasing the geometry needed for statistical analysis, simultaneous approximation, and numerical certification.

Brownian Kernel Ladders provide a structured instance of this unrestricted adaptive-union problem. Unlike the controlled-family analysis in the foundational BKL work, the ladder representation is allowed to vary inside the Gaussian supremum. The result is a finite-sample theory centered on one minimum-trace common covariance.

The central covariance object

For a sample \(x=(x_1,\ldots,x_n)\), let \(\mathcal K_L(x)\subseteq\mathbb S_+^n\) be the closed family of empirical depth-\(L\) BKL Gram matrices. The central value is

\[ \tau_L(x) = \min_{N\succeq 0} \left\{ \operatorname{tr}N: N\succeq K \ \text{for every }K\in\mathcal K_L(x) \right\}. \]

An optimizer \(N_\star\) defines one ellipsoid containing the unit trace ellipsoid of every admissible ladder. With \(S_L(x)=\sum_i \rho_L(x_i)^2\), the normalized profile \(\Lambda_L(x)=\sqrt{\tau_L(x)/S_L(x)}\) separates the scale of the universal diagonal envelope from the extra cost of common domination. The trace objective is adapted to Gaussian average radius, but the full matrix carries more information than its trace.

Statistical scale

Its trace controls the Gaussian support of the complete adaptive ball: \(\widehat{\mathcal G}_x(\mathcal B_L(r)) \le r\sqrt{\tau_L(x)}/n\).

Common feature space

Its leading eigenspaces give one sample-dependent subspace that approximates every function in the unrestricted adaptive ball, not only one kernel or one target.

Certification

Its active constraints support finite contact certificates, semidefinite lower bounds, verified separation upper bounds, and conservative graph-based certificates.

Exact finite-sample identities

Empirical covariance body. The closed trace body of the BKL unit ball is exactly the closure of the union of the ellipsoids \(E(K)=K^{1/2}B_2^n\), \(K\in\mathcal K_L(x)\).
Operator factorization. If \(T_{L,x}\) is the adaptive evaluation operator, then \[ \pi_2(T_{L,x})^2=\tau_L(x). \] Thus the common-covariance problem is exactly an absolutely two-summing norm at finite resolution, not merely a generic upper bound.
Robust multiplier geometry. The same value is the exact supremum of the squared support function over all centered multiplier laws whose covariance is dominated by the identity.
Gaussian defect. Replacing the worst covariance-dominated multiplier by a standard Gaussian creates a measurable defect. Bounded defect means that the covariance certificate is tight up to constants; it does not by itself imply a fast learning rate.

Brownian reductions: Dirac, thresholds, and graphs

The BKL-specific step removes the final mixing measure. If \(\mathcal S_{L-1}(x)\) is the closed family of previous-layer unit traces and \(B(a)\) is the Brownian pullback matrix generated by \(a\), then

\[ h_{L,x}(z)^2 = \sup_{a\in\mathcal S_{L-1}(x)} z^\top B(a)z, \] \[ N\succeq K\quad\forall K\in\mathcal K_L(x) \quad\Longleftrightarrow\quad N\succeq B(a)\quad\forall a\in\mathcal S_{L-1}(x). \]

The signed Brownian layer-cake formula then writes each \(B(a)\) as an integral of positive and negative threshold outer products:

\[ B(a) = \int_0^\infty \left[ v_t^+(a)v_t^+(a)^\top + v_t^-(a)v_t^-(a)^\top \right]\,\mathrm dt. \]

This converts unrestricted kernel adaptation into the geometry of signed threshold set systems. Recursive decomposition dictionaries yield upper profiles, stable and integrated thresholds give reverse information, and graph coarea supplies deterministic Loewner-order majorants through effective resistance.

A phase diagram for unrestricted adaptation

Fixed and finite families. One fixed ladder has no common-domination inflation. Finite controlled families and samples with only \(k\) distinct locations admit explicit covariance envelopes.
Hierarchical and ordered traces. A laminar threshold hierarchy with bounded decomposition width gives a near-\(n^{-1/2}\) full-class rate up to a logarithmic factor. Bounded empirical oscillation in one ordering gives an \(O(\log n/\sqrt n)\) rate under the stated hypotheses.
Orthogonal and well-conditioned designs. Orthogonal samples exhibit the worst-case depth law \(\Lambda_L\asymp n^{1/2-2^{-L}}\) and complexity \(n^{-2^{-L}}\). Well-conditioned and high-dimensional spherical samples have a bounded Gaussian defect, while retaining this slow sign-rich rate.
Random support and perturbation transfer. Balanced repeated or categorical supports preserve the covariance and Gaussian-richness geometry. Matched perturbation theorems quantify when those conclusions survive noisy inputs.

Tight certificate is not the same as fast rate. The theory separates the tightness of the common covariance from the actual size of the unrestricted adaptive class. This distinction is visible on both orthogonal and spherical designs.

One subspace for the complete adaptive ball

The common covariance provides an exact simultaneous-approximation interpretation. For an \(m\)-dimensional subspace \(V\subseteq \mathbb R^n\), the empirical Kolmogorov width satisfies

\[ d_{m,L}(x;r)^2 = \frac{r^2}{n} \inf_{\dim V\le m} \sup_{K\in\mathcal K_L(x)} \lambda_{\max}\!\left(P_{V^\perp}KP_{V^\perp}\right). \]

If \(V_m^\star\) is spanned by the first \(m\) eigenvectors of an optimal common covariance \(N_\star\), then

\[ d_{m,L}(x;r) \le \frac{r}{\sqrt n} \sqrt{\lambda_{m+1}(N_\star)} \le r\sqrt{\frac{\tau_L(x)}{n(m+1)}}. \]

The quantifier order is the key point: one subspace is chosen and then works for every function in the full adaptive ball. Under a balanced threshold hierarchy, the subspace can be chosen explicitly from coarse discrete Haar modes, with no logarithmic tree-height penalty in the approximation bound.

Certifiable computation

Finite contact

Although the covariance program has infinitely many constraints, an exact optimizer is determined by at most \(n(n+1)/2+1\) Brownian contact matrices from the closed last-layer Dirac trace family. This finite certificate is exact but existential.

Active semidefinite programs

A finite active family gives a certified lower bound. A rigorously verified global separation value converts a floored active solution into an upper bound. Local witness optimization may improve the lower certificate, but it is not itself evidence of global domination.

Resistance design

For pairwise-distinct sample locations, a graph-coarea majorant leads to a fully convex effective-resistance design problem. It supplies a conservative global upper certificate, explicit optimality conditions, a computable a posteriori gap, and a sparse Frank–Wolfe implementation. On a fixed tree, the optimal weights and certificate are available in closed form.

Finite covariance-path illustration

The manuscript includes a finite depth-two Brownian path between the identity and a training-only Fisher diagonal on frozen ImageNet ResNet-18 representations. The path is selected by repeated cross-validation using only the labelled training sample. Its role is to illustrate the distinction between covariance certification and predictive model selection.

Covariance certificates: 45/45 succeed Every selected fit passes the finite five-kernel path domination check after the explicitly reported numerical repair. Forty-four of the 45 selected covariance strengths are strictly positive.

Predictive gate: not passed On the locked suite, the macro-average difference from the strongest tuned RBF/Laplace control is \(+0.163\) percentage points, but the predeclared requirement was at least 8/15 paired wins or ties and the observed count is 7/15.

The finite path certificate concerns only five path kernels on an empirical anchor. It is not a certificate for the unrestricted recursive BKL family, and the successful covariance checks are not a claim of universal predictive superiority.

Position in the Brownian ladder program

Relation to BKL

The foundational BKL project develops the recursive integral-RKHS hierarchy and separates the full adaptive envelope from realized or finite controlled families. The present project studies the complementary case in which the canonical ladder may vary inside the empirical Gaussian supremum.

Relation to VBKL

VBKL places signed-measure superposition only at the outermost layer and produces a path-atomic variation hull. The common-covariance results here concern the canonical integral-RKHS BKL family and are not asserted to transfer automatically to the full VBKL space.

Exact, conditional, and computational conclusions

Exact: the empirical union-of-ellipsoids representation, the minimum-trace common covariance, the two-summing identity, the robust multiplier formulation, the empirical width formula, and finite contact.
Conditional: Gaussian reverses and sharper rates under explicit threshold, hierarchy, ordering, conditioning, occupancy, or perturbation assumptions.
Computational: active SDP lower certificates, verified-separation upper certificates, resistance-design upper bounds, and the finite covariance-path study.

Open directions include population and out-of-sample common covariances, intrinsic oscillation bounds beyond depth two, less restrictive large-depth perturbation theory, localized profiles, globally verified recursive separation, and reusable common covariances across tasks.

Manuscript and citation

Common Covariance Geometry and Certification for Brownian Kernel Ladders
Mahdi Mohammadigohari.
Figshare preprint, 2026.
doi:10.6084/m9.figshare.33386035

@misc{mohammadigohari2026commoncovariance, title = {Common Covariance Geometry and Certification for Brownian Kernel Ladders}, author = {Mohammadigohari, Mahdi}, year = {2026}, howpublished = {Figshare preprint}, doi = {10.6084/m9.figshare.33386035}, url = {https://doi.org/10.6084/m9.figshare.33386035} }