Weaves, Wires, and Morphisms: Formalizing and Implementing the Algebra of Deep Learning

Vincent Abbott; Gioele Zardini

cs.LGApr 8, 2026

Weaves, Wires, and Morphisms: Formalizing and Implementing the Algebra of Deep Learning

Authors: Vincent Abbott, Gioele Zardini

Organizations: Laboratory for Information and Decision Systems, Massachusetts Institute of Technology

Abstract

Despite deep learning models running well-defined mathematical functions, we lack a formal mathematical framework for describing model architectures. Ad-hoc notation, diagrams, and pseudocode poorly handle nonlinear broadcasting and the relationship between individual components and composed models. This paper introduces a categorical framework for deep learning models that formalizes broadcasting through the novel axis-stride and array-broadcasted categories. This allows the mathematical function underlying architectures to be precisely expressed and manipulated in a compositional manner. These mathematical definitions are translated into human manageable diagrams and machine manageable data structures. We provide a mirrored implementation in Python (pyncd) and TypeScript (tsncd) to show the universal aspect of our framework, along with features including algebraic construction, graph conversion, PyTorch compilation and diagram rendering. This lays the foundation for a systematic, formal approach to deep learning model design and analysis.

Figures & tables

Term Type	Associated Data	Constructed Property	Non-constructed Property	Interpretation
Product	$\displaystyle g_{\Pi j}=\Pi_{j\in J}g_{j}$	Wrapping	Method $\displaystyle m_{\Pi j,\ell}:\Pi_{j\in J}G_{j}\rightarrow G_{\ell,i}$ $\displaystyle\pi_{\ell}=m_{c,\ell}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}$	$(\Pi_{j\in J}g_{j})\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}=\Pi_{j\in J}(g_{j}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G})$
Constructed Term	$\displaystyle g_{c,f}\mapsto g_{c,i}$ via $\displaystyle\hat{T}_{c}$	Wrapping	Method $\displaystyle m_{c,\ell}:G_{c,i}\rightarrow G_{\ell,f}$ $\displaystyle\pi_{\ell}=\hat{T}_{c}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}m_{c,\ell}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}$	$g_{c,f}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}=g_{c,f}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}\hat{T}_{c}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}\pi_{c}$
Root Term	$\displaystyle g_{r}\mapsto\mu_{r}\times\Pi_{\ell\in r}g_{\ell,f}$ Where $\displaystyle m_{r}$ is non-core metadata.	Wrapping	Data Extraction $\displaystyle p_{r,\ell}:G_{r}\rightarrow G_{r,\ell}$ $\displaystyle\pi_{\ell}=p_{r,\ell}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}$	$\displaystyle g_{r}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}=\gamma_{r}$ $\displaystyle\forall\ell\in r.g_{r}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}p_{r,\ell}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}V_{G}=\gamma_{r}\mathbin{\raise 2.58334pt\hbox{\vtop{\halign{#\cr\hfil$ \scriptscriptstyle\mathrm{o} $\hfil\cr\hfil$ \scriptscriptstyle\mathrm{9} $\hfil\crcr}}}}\pi_{\ell}$

Table 1: Applicability and interpretation for the main term types.

Panel of Figure 10 — Figure 10: Diagonalization corresponds to the equation $\mathbf{y}[p,:]=\mathbf{x}[p,p,:]$ . This can be expressed using the rearrangement reindexing $p\mapsto(p,p)$ .

Panel of Figure 11 — Figure 11: Repetition corresponds to the equation $\displaystyle\mathbf{y}[p,:]=\mathbf{x}[:]$ . This can be expressed using the rearrangement reindexing $\displaystyle p\mapsto()$ .

Figure 36: Here, we show a GeGLU mixture of experts as used in DeepSeek-V3 ( DeepSeek-AI, 2025 ) . We use a Top-K operation to select the indexes of the relevant arrays. The final output is then computed by summing over the normalized top-k weights, aligning with the chosen indexes.

Figure 37: The expression from Figure 36 can be reexpressed in a mathematically equivalent form which rearranges indexed-inputs to outputs, over which we perform a sparse sum.

Figure 44: At a high-level, the deep seek algorithm follows a standard transformer structure: an embedding maps from a series of tokens indexed across $V\in\mathbb{N}$ options, and is sent through a sequence of attention-feed forward layers. The final result is mapped back through a linear layer, producing a vector $[{\mathbb{R}};{V}]$ indicating the log-weights of output token options. These layers are all residual connections ( He et al., 2015 ) .

Figure 45: Attention blocks correspond to multi-head latent attention which comprises a complex rotary component and a real, latent component. These are used to generate queries and keys, which map through a multi-head $Q-K$ matrix multiplication contracted over the $d$ axes parallelized over $h$ heads. The results of the complex and real components are summed together, passed through a SoftMax and weighted lower triangular normalizer for causality, undergo an $S-V$ matrix muliplication, and are passed through an output linear layer, contracting the heads.

Figure 46: The rotary linear layer generates queries and keys for the complex component. This is supplied by a rotary mask generator, which is not “parallelized” over the $\overline{x}$ axis – the outputs are strictly dependent on which $\overline{x}$ index we are at. This “binds” the $\overline{x}$ axis into the output, meaning the result has positions encoded. This multiplies with a linear layer which generates both a real and complex value for the $m$ inputs. The resulting output is “decomplexed”, and yields an array with integrated positional information.

Figure 47: The multilayer perceptrons used by DeepSeek-V3 is a GeGLU ( Shazeer, 2020 ) . We have two input linear layers, one of which passes through an elementwise operation, the results multiplied together, and followed by an output linear layer. The internal size, in this case $d*f$ , is not dependent on external axes sizes, and therefore can be configured independently.

Figure 48: The Mixture-of-Experts block follows Section 5.5 , with a gate selecting experts added to a multilayer perceptron. This architecture supports multi-GPU computational splits. Here, we show the simplified “sparse axis” presentation which presents a mathematically coherent representation insufficient for direct computation.

Figure 49: The gate for the Mixture-of-Experts involves a gate linear layer which provides weights for each of $n$ layers, a Top-K selection which generates a sparse axis, which is then normalized by a SoftMax-like operation, which, in practice, uses a sigmoid rather than an exponent to weight values.

Figure 50: Finally, the experts component follows Section 5.5 , with each layer having an additional $n$ axis of which $k$ values are non-zero.

⬇ 1 qk_matmul = ops . Einops 2 . template ( 3 ’q h d, x h d -> h q d’ )	⬇ 1 softmax = ops . SoftMax . template () As we did not provide the axes, an unnamed axis is generated. Autoalignment will then rename it.
⬇ 1 sv_matmul = ops . Einops 2 . template ( 3 ’h q x, x h d -> q h d’ )	⬇ 1 mask = ops . 2 WeightedTriangularLower 3 . template ()

Table 26

Figure 51: Applying qk_matmul @ softmax @ mask @ sv_matmul , we perform autoalignment operations at each step. Axes are aligned to be the same. Operations are batch broadcasted when the number of axes mismatch. In the case of sv_matmul , we take the product with an identity rearrangement with the array $\displaystyle[\mathbb{R},xhd]$ . Note that the $\displaystyle h$ , $\displaystyle q$ , $\displaystyle x$ axes of qk_matmul and sv_matmul are separately generated, meaning their equivalence is only realized after composition. The $\displaystyle d_{1}$ and $\displaystyle d_{2}$ axes are never aligned together, meaning that the final configuration of the expression takes $\displaystyle\langle q,h,x,d_{1},d_{2}\rangle$ .

Figure 52: A full ResNet attention block expression constructed through autoalignment has a number of numeric sizes with UIDs. By assigning values to these, we have a configured expression ready for PyTorch compilation.

Figure 53: By converting from a “ProdCategory[L,M]“ to a “Hypergraph[L,M]“, we can perform algebraic manipulation with hypergraph rewrite rules. This allows us to apply general algebraic properties such as associativity, bifunctoriality, and symmetry, which are not easily captured by the composed-product approach.

Figure 54: The modularity of the framework allows for a web of features to be developed and integrated. At any point, new features can be “hooked” into the system, allowing for a powerful and extensible framework for future AI research.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 33

Figure 34

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Lattice theory and algebraic models for deep convolutional learning based on mathematical morphology

Gustavo, Angulo

Jesus

Convolutional Neural Networks

Open paper AskPDF arXiv

We develop a rigorous algebraic framework for deep convolutional architectures, CNNs, ResNets, and encoder--decoder networks such as UNet, grounded in lattice theory and mathematical morphology. The central tool is the Matheron--Maragos--Banon--Barrera (MMBB) universal representation theory for translation-invariant operators, which we apply systematically to every layer of a standard deep network. The principal finding is that the standard CNN pipeline (linear convolution~

+

ReLU~

+

flat max-pooling) is a cross-lattice operator: the convolution is an erosion in the Fourier inf-semilattice while ReLU is a lattice-join closing and max-pooling is a dilation in the pointwise max-plus lattice, and their composition is a morphological opening in neither. A second finding is that the upper adjoint of ReLU in the pointwise lattice is a global (non-local) operator, the identity on globally non-negative functions and

-\infty

otherwise, so no local morphological erosion can form an adjunction pair with ReLU. These two results together provide the precise algebraic reason why depth in standard CNNs introduces genuine representational power: the composed layer is not idempotent. Three layer designs that are genuine idempotent openings are identified and fully characterised: the pure max-plus morphological layer (pointwise lattice), the spectral Wiener layer (Fourier lattice), and the self-dual morphological layer. We establish a complete fixed-point and convergence theory. The framework also unifies max-pooling, strided convolution, and the Laplacian pyramid under the Goutsias--Heijmans adjoint pyramid theory, and gives the Activation--Pooling Dilation (APD) factorisation with its correct adjoint.

Abstract

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