Kernel Slotwise Composition

• Kernel slotwise composition is a structural paradigm that composes kernel objects at specific slots using closure rules (e.g., non-negative sums and pointwise products) to maintain kernel validity.
• In Gaussian process models, differentiable composition via linear and product layers enables efficient, expressive kernel learning and pattern discovery across various applications.
• In RKHS, operator-valued, and Markov kernel settings, slotwise composition facilitates pullback operations, coordinated gating for feature selection, and stochastic diagram semantics.

Searching arXiv for the cited papers to ground the article. Kernel slotwise composition denotes a family of constructions in which kernel objects are composed by acting on designated “slots” while preserving the underlying positivity or probabilistic semantics. Across current arXiv usage, the phrase appears in several technically distinct settings: as differentiable composition of kernel-valued network units in Gaussian processes, as pullback and weighting of reproducing kernels under composition operators, as coordinate-wise gating before a kernel predictor, as overlap-subset decomposition in compositionally structured kernel models, as iteration of operator-valued kernels under completely positive maps, and as the measure-theoretic operation that connects one output slot of a Markov kernel to one input slot of another and marginalizes the internal wire (Sun et al., 2018, Kumari et al., 18 Sep 2025, Ruan et al., 17 Sep 2025, Lippl et al., 2024, Tian, 17 Nov 2025, Papamarkou, 30 Apr 2026).

1. Common formal pattern

A unifying feature of these constructions is that the “slot” is a local position at which composition is applied without abandoning kernel validity. In the Gaussian-process setting of the Neural Kernel Network (NKN), each unit in each layer is a slot that itself holds a valid kernel, and layers compose slots by non-negative weighted sums and pointwise products. In RKHS composition theory, the slots are the two arguments of a reproducing kernel, and composition acts by pullback,

$K_\phi(s,t)=K(\phi(s),\phi(t)),$

or by weighted pullback,

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$

In compositional feature-learning models, the slots are input coordinates, with a kernel predictor acting on $\beta\circ x$. In compositional-generalization theory, the slots are orthogonal components $z_c$ of a representation and the relevant structure is which subsets of slots match across two inputs. In operator-valued kernel theory, the slots are the left and right operator arguments acted on by Kraus operators. In Markov polycategories, the slots are explicitly ordered input and output positions of many-input, many-output kernels (Sun et al., 2018, Kumari et al., 18 Sep 2025, Ruan et al., 17 Sep 2025, Lippl et al., 2024, Tian, 17 Nov 2025, Papamarkou, 30 Apr 2026).

What these settings share is not a single syntax but a preservation principle. For positive definite kernels, the relevant closure rules include non-negative linear combinations, pointwise products, pullbacks, and rank-one weightings. For operator-valued kernels, complete positivity and subunitality or unitality control validity, domination, and asymptotics. For Markov kernels, the corresponding preservation mechanism is measure-theoretic composition with marginalization over the internal variable. This suggests that “kernel slotwise composition” is best treated as a structural paradigm rather than a single formalism.

2. Differentiable kernel slots in Gaussian processes

In Gaussian-process regression, the covariance function $k(x,x')$ determines smoothness, periodicity, linear trends, and related inductive biases. The NKN replaces discrete kernel-grammar search by a differentiable architecture in which every hidden unit is itself a kernel. The input layer consists of primitive kernel families such as RBF, PER, LIN, and RQ, instantiated as trainable primitive slots

$h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$

A Linear layer produces new kernel slots by

$$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$

with $\mathbf W_l=\log(1+\exp(\mathbf A_l))$ to enforce non-negativity. A Product layer produces

$$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$

The paper interprets the former as “OR-like” and the latter as “AND-like” slotwise composition. Optional kernel-preserving activations include polynomials with non-negative coefficients and $f(z)=e^z$ (Sun et al., 2018).

The architecture is grounded in the closure lemma that for kernels $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$0, both $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$1 with $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$2 and the pointwise product $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$3 are kernels. Because all primitive parameters, linear-layer parameters, and product operations are differentiable, the final kernel $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$4 is trainable end-to-end by backpropagation through the GP marginal likelihood. This removes the non-differentiable, combinatorial search over subexpressions used by the Automatic Statistician, where kernel grammar search can take hours for short time series (Sun et al., 2018).

The expressiveness analysis is explicit. The paper defines a positive-weighted polynomial (PWP) of base kernels and proves that an NKN with width $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$5 can represent any PWP of $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$6 primitive kernels, while an NKN with width $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$7 and $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$8 Linear–Product modules can represent any PWP with degree at most $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$9. Using Bochner’s theorem and Gaussian-mixture approximation of spectral densities, it further proves stationary universality: for any $\beta\circ x$0-dimensional complex-valued stationary kernel $\beta\circ x$1 and any $\beta\circ x$2, there exist primitive kernels

$\beta\circ x$3

and an NKN of width at most $\beta\circ x$4 whose output $\beta\circ x$5 satisfies

$\beta\circ x$6

Complex-valued internal slots are important here: the paper shows examples in which a real representation fails as a PWP of cosine primitives but a complex representation is exactly a PWP of a single primitive $\beta\circ x$7 (Sun et al., 2018).

Empirically, the slotwise perspective is used to explain pattern discovery in time series, 2D synthetic surfaces, texture extrapolation, UCI regression, and Bayesian optimization. On Airline, the reported training-time comparison is about $\beta\circ x$8 faster, with 201s versus 6147s. A typical small architecture is “6 primitives: 2 RQ, 2 RBF, 2 LIN” followed by “Linear8 – Product4 – Linear4 – Product2 – Linear1”, with about $\beta\circ x$9 parameters in dimension $z_c$0, compared with SM-4’s $z_c$1. The paper also notes limitations: interpretability is weaker than explicit symbolic kernels, optimization can face local minima or initialization sensitivity, and the strongest universality results are for stationary kernels (Sun et al., 2018).

3. Pullback kernels and composition operators on RKHSs

A second usage of kernel slotwise composition arises in RKHS theory, where composition acts directly on the arguments of the reproducing kernel. If $z_c$2 is a kernel on $z_c$3 and $z_c$4, then the pullback

$z_c$5

is again a kernel on $z_c$6. With an additional weight $z_c$7, the transformed kernel is

$z_c$8

The paper identifies this as the exact kernel counterpart of the weighted composition operator

$z_c$9

and proves that $k(x,x')$0, with the norm in $k(x,x')$1 given by the minimum norm among preimages in $k(x,x')$2 (Kumari et al., 18 Sep 2025).

The boundedness criterion is a kernel domination statement. For kernels $k(x,x')$3 on $k(x,x')$4, map $k(x,x')$5, and weight $k(x,x')$6, the operator

$k(x,x')$7

is bounded with norm at most $k(x,x')$8 if and only if

$k(x,x')$9

is a kernel on $h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$0. Setting $h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$1 yields the criterion for the ordinary composition operator $h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$2. At the level of kernel sections, the adjoint acts by

$h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$3

provided $h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$4. This is the paper’s core “kernel slotwise” identity: the adjoint composes the base point in the reproducing kernel (Kumari et al., 18 Sep 2025).

The theory is developed concretely for Hardy and Bergman spaces on the unit disc, polydisc, and ball, with explicit kernels

$h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$5

and analogous weighted Bergman formulas. A central application is an alternative proof that for every holomorphic $h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$6, the composition operator on $h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$7 is bounded. The proof uses the positivity of

$h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$8

together with multiplier arguments, to show that

$h^{(0)}_j(x,x'):=k_j(x,x';\phi_j).$9

is a kernel, which yields

$$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$0

The paper emphasizes that this proof uses kernel positivity rather than Littlewood subordination, Carleson measures, or more traditional analytic techniques (Kumari et al., 18 Sep 2025).

A more general template appears in Theorem 4.2: if

$$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$1

is a kernel and $$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$2 is a multiplier of $$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$3, then $$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$4 is bounded. The same framework yields kernel-based sufficient conditions on $$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$5 and $$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$6 and recovers classical boundedness for automorphism-induced composition operators. Lower bounds on norms follow from $$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$7 and the adjoint identity, for example

$$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$8

on $$h_i^{(l)}(x,x')=\sum_j W_{l,ij}h_j^{(l-1)}(x,x'), \qquad W_{l,ij}\ge 0,$$9 (Kumari et al., 18 Sep 2025).

4. Coordinate-wise slotwise gating for feature learning

A third formalization treats slotwise composition as an input-side gating layer before an RKHS predictor. The model studies

$\mathbf W_l=\log(1+\exp(\mathbf A_l))$0

where $\mathbf W_l=\log(1+\exp(\mathbf A_l))$1 is the Hadamard product. The inner map

$\mathbf W_l=\log(1+\exp(\mathbf A_l))$2

is a coordinate-wise weighting layer, and the outer map is a translation-invariant kernel ridge regressor. If $\mathbf W_l=\log(1+\exp(\mathbf A_l))$3, the $\mathbf W_l=\log(1+\exp(\mathbf A_l))$4-th coordinate is removed from the effective input. The paper presents this as a minimal compositional architecture and a testbed for feature learning in kernel methods (Ruan et al., 17 Sep 2025).

Relevance is formalized through the core sufficient feature set

$\mathbf W_l=\log(1+\exp(\mathbf A_l))$5

assuming it exists and that $\mathbf W_l=\log(1+\exp(\mathbf A_l))$6 and $\mathbf W_l=\log(1+\exp(\mathbf A_l))$7 are independent. Theorem 3.4 shows that for every $\mathbf W_l=\log(1+\exp(\mathbf A_l))$8,

$\mathbf W_l=\log(1+\exp(\mathbf A_l))$9

with strict inequality under nontrivial predictive power and nondegenerate irrelevant coordinates. Under an additional nontrivial-regression assumption and nondegenerate covariance of $$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$0, Theorem 3.8 states that any global minimizer $$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$1 satisfies

$$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$2

For sufficiently small $$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$3, Theorem 3.7 gives the reverse inclusion,

$$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$4

so that

$$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$5

In the population problem, global minimizers therefore achieve exact feature selection (Ruan et al., 17 Sep 2025).

The paper also analyzes stationary points through the directional derivative

$$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$6

where $$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$7. If the irrelevant coordinates $$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$8 are jointly Gaussian with nondegenerate covariance, Theorem 4.1 shows that the direction $$h_i^{(l)}(x,x')=\prod_{j\in\mathcal S_{l,i}}h_j^{(l-1)}(x,x').$$9 satisfies

$f(z)=e^z$0

with strict inequality when the predictor is nontrivial and $f(z)=e^z$1. Corollary 4.2 then states that every directional stationary point is either noise-free,

$f(z)=e^z$2

or trivial,

$f(z)=e^z$3

Thus every nontrivial stationary point eliminates noise coordinates (Ruan et al., 17 Sep 2025).

Kernel choice changes what this slotwise gating layer can detect. For $f(z)=e^z$4-type kernels such as the Laplace kernel,

$f(z)=e^z$5

Theorem 5.3 shows that at $f(z)=e^z$6,

$f(z)=e^z$7

So these kernels detect any main effect through first-order variation. For radial kernels, including the Gaussian

$f(z)=e^z$8

Lemma 5.6 gives

$f(z)=e^z$9

and the leading second-order term depends only on $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$00. The paper’s central contrast is therefore that $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$01-type kernels recover features contributing to nonlinear effects at stationary points, whereas Gaussian kernels recover only linear ones (Ruan et al., 17 Sep 2025).

5. Compositionally structured kernels and the limits of compositional generalization

A fourth usage concerns inputs decomposed into orthogonal component slots,

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$02

with $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$03 for $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$04. For compositionally structured representations of this form, the kernel depends only on which slots match between two inputs. Writing

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$05

the kernel has the form

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$06

The paper shows that wide random neural feature maps preserve this structure, so the induced kernel can be written as a function of the overlap set rather than the specific values (Lippl et al., 2024).

The main theorem is conjunction-wise additivity. For any kernel model with a compositionally structured representation, trained on a set $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$07, the prediction on any $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$08 can be written as

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$09

where $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$10 collects the slot subsets realized as overlaps with training points. Each $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$11 depends only on the values in the subset $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$12. This gives a precise functional description of what slotwise kernel composition can express: additive sums over conjunction functions indexed by overlapping slot subsets (Lippl et al., 2024).

To quantify which conjunctions matter, the paper defines overlap salience through a Möbius-type decomposition of $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$13. In symmetric settings it depends only on $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$14, written $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$15. For deep random (leaky) ReLU networks, the paper proves that as depth $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$16,

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$17

Depth therefore drives the representation toward full conjunctions. This is favorable for memorization of training combinations but unfavorable for generalization to unseen combinations, because unseen full conjunctions do not appear in the training overlap structure (Lippl et al., 2024).

The theory distinguishes tasks that are learnable in principle from tasks that are not. Symbolic addition and several context-dependence tasks are conjunction-wise additive and can therefore be represented by such kernels. By contrast, transitive equivalence cannot, because on unseen pairs the full-conjunction term is unavailable and the remaining additive form

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$18

cannot in general encode arbitrary equivalence relations. The paper also identifies two failure modes even for learnable tasks. The first is memorization leak: in symbolic addition with two slots and training anchor set size $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$19, the test prediction is

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$20

so nonzero salience of the full conjunction forces systematic underestimation. The second is shortcut bias, illustrated in CD-3, where context-only and full-conjunction features can dominate the norm-minimizing solution and block the intended context–feature conjunctions (Lippl et al., 2024).

The paper reports that convolutional networks, residual networks, and Vision Transformers trained on symbolic addition and context-dependence tasks are extremely well fitted by a conjunction-wise additive model, with $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$21 and often $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$22. It also reports the same compressed-value phenomenon predicted by memorization leak and the same shortcut patterns predicted by salience analysis. This suggests that, on these tasks, practical deep networks often operate close to the kernel-slotwise regime unless feature learning pushes them into a richer, non-additive regime (Lippl et al., 2024).

6. Operator-valued kernels and stochastic diagram semantics

In operator-valued kernel theory, slotwise composition is realized by completely positive maps. Starting from a positive definite operator-valued kernel

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$23

the paper forms the scalar lift

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$24

and its RKHS $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$25. With CP maps

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$26

the iterated kernel for a word $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$27 is

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$28

All iterates are realized inside a single Hilbert space

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$29

using creation operators

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$30

The resulting representation is

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$31

with $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$32. In this sense, composition of kernels is realized as composition of bounded operators on a fixed feature space. Under unitality, $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$33 decreases strongly to a projection $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$34, yielding a limit kernel

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$35

The paper also proves a Stein-type decomposition

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$36

and under subunitality a Radon–Nikodym representation

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$37

which implies $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$38. For random compositions, Kingman’s theorem gives an almost-sure growth law for $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$39 and hence exponential bounds for $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$40 (Tian, 17 Nov 2025).

A measure-theoretic notion of kernel slotwise composition appears in colored Markov polycategories. Here a morphism

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$41

is a many-input, many-output Markov kernel, and slotwise composition connects the $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$42-th output slot of one kernel to the $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$43-th input slot of another when the objects agree. If

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$44

with $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$45, the composite

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$46

is defined by integrating over the internal wire:

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$47

Operationally, one samples $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$48 from $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$49, feeds the internal value $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$50 into the chosen input slot of $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$51, samples $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$52 from $$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$53, retains the dangling outputs, and marginalizes the internal variable. The paper proves unitality, associativity, and interchange by giving trace semantics to finite acyclic diagrams and showing independence of topological order (Papamarkou, 30 Apr 2026).

The colored extension relaxes equality of wired objects to compatibility of colors. If an interface witness

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$54

is available, with interface kernel

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$55

the colored composite is

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$56

Typed stochastic diagrams are then interpreted by expanding interface wires into explicit unary interface vertices. For finite acyclic parameterized diagrams, the paper proves a reverse-mode differentiation theorem: if each parameterized vertex admits an admissible local gradient operator, then the derivative of the expected scalar objective is the tuple of expected local reverse contributions

$$\rho(x,y)=\psi(x)\overline{\psi(y)}K(\phi(x),\phi(y)).$$57

Stochastic vertices can use score-function local rules and deterministic vertices can use pathwise local rules. In this setting, kernel slotwise composition is literally the wiring rule of a typed stochastic calculus (Papamarkou, 30 Apr 2026).

These operator-valued and stochastic formulations are mathematically distinct from the GP and RKHS constructions. The first acts on operator values through CP maps; the second acts on ordered input/output positions of Markov kernels by wiring and marginalization. The common feature is again structural locality: a composite is built by operating on specified slots and then recovering a valid kernel object through positivity or probabilistic integration.